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Simulating the electric field mediated motion of ions and molecules in diverse matricies

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Title:
Simulating the electric field mediated motion of ions and molecules in diverse matricies
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Book
Language:
English
Creator:
Hickey, Joseph
Publisher:
University of South Florida
Place of Publication:
Tampa, Fla.
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Subjects

Subjects / Keywords:
Molecular delivery
Electrogenetherapy
Electrochemotherapy
Electroporation
Electrophoresis
Mathematical model
Tissue
Agarose gel
Dissertations, Academic -- Chemical Engineering -- Doctoral -- USF   ( lcsh )
Genre:
government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
non-fiction   ( marcgt )

Notes

Summary:
ABSTRACT: Electroporation is a methodology for the introduction of drugs and genes into cells. This technique works by reducing the exclusionary nature of the cell membrane 125, 129, 186, 189. Electroporation has successfully been used in electrochemotherapy and electrogenetherapy 57, 68, 86, 87, 110, 112, 131. The two major components of electroporation are an induced transmembrane potential and the motion of the deliverable through a compromised cell membrane into the target cell 38, 55, 62, 114, 131. These two components are both dependent on the electrophoretic motion of charged species in an applied electric field 45, 64, 75, 77, 177. Currently, the methods outlined for understanding electroporation have been focused on either a phenomenological perspective, e.g. what works, or modeling the electric fieldstrength in certain regions 12, 56, 87, 129, 146, 204, 205.While this information is necessary for the clinician and the laboratory scientist, it doesn't expand the understanding of how electric field mediated drug and gene delivery works or EFMDGD. To increase the understanding of EFMDGD, new models are required that predict the motion of ions and deliverables through tissues to target areas 75, 77. This document examines the design and creation of an electric field mediated drug and gene delivery model, EFMDGDM. Two example scenarios, ionic motion in tissues and gel electrophoresis, are examined in depth using the EFMDGDM. The model requires tuning for each scenario but only utilizes experimental parameters and one tunable parameter that is computed from regressed experimental data. The EFMDGDM successfully describes the two examples. Future work will incorporate the EFMDGDM as the backbone of an electric field mediated drug and gene delivery modeling package, EFMDGDMP.
Thesis:
Thesis (Ph.D.)--University of South Florida, 2005.
Bibliography:
Includes bibliographical references.
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System requirements: PDF reader.
System Details:
Mode of access: World Wide Web.
Statement of Responsibility:
by Joseph Hickey.
General Note:
Title from PDF of title page.
General Note:
Document formatted into pages; contains 310 pages.

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University of South Florida
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aleph - 001681161
usfldc doi - E14-SFE0001076
usfldc handle - e14.1076
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ABSTRACT: Electroporation is a methodology for the introduction of drugs and genes into cells. This technique works by reducing the exclusionary nature of the cell membrane [125, 129, 186, 189]. Electroporation has successfully been used in electrochemotherapy and electrogenetherapy [57, 68, 86, 87, 110, 112, 131]. The two major components of electroporation are an induced transmembrane potential and the motion of the deliverable through a compromised cell membrane into the target cell [38, 55, 62, 114, 131]. These two components are both dependent on the electrophoretic motion of charged species in an applied electric field [45, 64, 75, 77, 177]. Currently, the methods outlined for understanding electroporation have been focused on either a phenomenological perspective, e.g. what works, or modeling the electric fieldstrength in certain regions [12, 56, 87, 129, 146, 204, 205].While this information is necessary for the clinician and the laboratory scientist, it doesn't expand the understanding of how electric field mediated drug and gene delivery works or EFMDGD. To increase the understanding of EFMDGD, new models are required that predict the motion of ions and deliverables through tissues to target areas [75, 77]. This document examines the design and creation of an electric field mediated drug and gene delivery model, EFMDGDM. Two example scenarios, ionic motion in tissues and gel electrophoresis, are examined in depth using the EFMDGDM. The model requires tuning for each scenario but only utilizes experimental parameters and one tunable parameter that is computed from regressed experimental data. The EFMDGDM successfully describes the two examples. Future work will incorporate the EFMDGDM as the backbone of an electric field mediated drug and gene delivery modeling package, EFMDGDMP.
590
Adviser: Richard Gilbert, Ph.D.
Co-adviser: Mark Jaraszeski, Ph.D.
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Molecular delivery.
Electrogenetherapy.
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Electroporation.
Electrophoresis.
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Agarose gel.
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PAGE 1

Simulating the Electric Field Mediated Motion of Ions and Molecules in Di v erse Matrices by Joseph D. Hick e y a dissertation submitted in partial fulllment of the requirements for the de gree of Doctor of Philosoph y Department of Chemical Engineering Colle ge of Engineering Uni v ersity of South Florida Co-Major Professor: Richard A. Gilbert, Ph.D. Co-Major Professor: Mark J. Jaroszeski, Ph.D. Scott Campbell, Ph.D. Richard Heller Ph.D. Andre w M. Hof f, Ph.D. Date of Appro v al: April 1, 2005 K e yw ords: molecular deli v ery electrogenetherap y electrochemotherap y electroporation, electrophoresis, mathematical model, tissue, ag arose gel c Cop yright 2005, Joseph D. Hick e y

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The Road Not T ak en Tw o roads div erged in a y ello w w o o d, And sorry I could not tra v el b oth And b e one tra v eler, long I sto o d And lo ok ed do wn one as far as I could T o where it b en t in the undergro wth; Then to ok the other, as just as fair, And ha ving p erhaps the b etter claim, Because it w as grassy and w an ted w ear; Though as for that the passing there Had w orn them really ab out the same And b oth that morning equally la y In lea v es no step had tro dden blac k. Oh, I k ept the rst for another da y! Y et kno wing ho w w a y leads up on w a y I doubted if I should ev er come bac k. I shall b e telling this with a sigh Somewhere ages and ages hence: Tw o roads div erged in a y ello w w o o d, and I{ I to ok the one less tra v elled b y And that has made all of the dierence. { Rob ert F rost This document is dedicated to my parents. W ithout their lo ving support, I ne v er w ould ha v e turned this w ay

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Ackno wledgments I am grateful to all the members of the dissertation e xamining committee. The y ha v e all helped me immensely o v er the past fe w years either in a laboratory or course setting. Studying under their e xpert w atch has been the most re w arding and challenging e xperience of my life. I w ould lik e to e xtend much gratitude to the three wise men of the Center for Molecular Deli v ery Richard Heller Richard Gilbert and Mark Jaroszeski for the countless hours of instruction and educationthe y ha v e pro vided o v er the past ten years. I w ould also lik e to thank Richard Gilbert and the Center for Molecular Deli v ery for funding my dissertation. The y ha v e funded the research, the publications and the tra v el to present papers on the research in such w onderful places as Bratisla v a Slo v akia, South Hadle y Massachusetts, Florence Italy and Ne w London Connecticut. The opportunities presented to me by Richard Gilbert ha v e inuenced me greatly

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T able of Contents List of T ables v List of Figures vi List of Abbre viations ix List of Symbols xi Abstract xi v 1 Prelude 1 P art I 5 2 Literature Examination 6 2.1 Historical Re vie w 6 2.2 Current Char ged Species Modeling Practices 10 3 Project Description 20 3.1 Current Status of Electroporation Modeling 20 3.2 Electric Field Mediated Flo w Field Model 21 3.3 Problem Denition 25 4 Material and Methods 26 4.1 Introduction 26 4.2 Materials 26 4.2.1 Research 27 4.2.1.1 Computers 27 4.2.1.2 Operating Systems 27 4.2.1.3 Research Programs 28 4.2.1.4 Gel Electrophoresis 28 4.2.2 Document Preparation 30 i

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5 Modeling Electrophoresis in Fluids 32 5.1 Introduction 32 5.2 Examining Ionic Motion in Solution 32 5.3 Examining Molecular Motion in Solution 34 5.4 Examining a Plasmid' s Motion in Solution 36 5.5 V elocity and Acceleration of Char ged Species in Electric Fields 40 5.6 Summary 43 6 Contrasting In V ivo and In V itr o EFMDGD Flo w Systems 46 6.1 Introduction 46 6.2 Importance of Electrophoresis on Electroporation 46 6.3 Molecular Deli v ery 51 6.4 Summary 51 7 Common Electroporation Electrodes 53 7.1 Introduction 53 7.2 Electroporation Electrode Background 54 7.3 Summary 56 P art II 57 8 Electric Field Mediated Flo w Field Model 58 9 Flo w Field T issue Model 60 9.1 Introduction 60 9.2 Odd Number on a Side Square Array 61 9.3 Randomly Generated Flo w Field V alues 62 9.4 Initial Concentration Prole 64 9.5 T issue Flo w Rules 65 9.5.1 Section 1 : The First Ro w i 11 69 9.5.2 Section 2 : The Rest of the T issue, i > 11 70 9.6 Modeling the Ef fect of an Electric Field on a Flo w P attern 72 10 Flo w Field T issue Model Results and Discussion 76 10.1 Results 76 10.1.1 P arallel Plate Model 77 10.1.2 Needle Electrode Results 79 10.2 Discussion 81 ii

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11 Electric Field Mediated DN A Fragment Deli v ery Model, EFMDFDM 84 11.1 Introduction 84 11.2 Building an EFMDFDM for DN A Deli v ery Prediction 85 11.2.1 Selecting an Array Size 86 11.2.2 Gel Flo w Rules 86 12 Modeling Electrophoresis in Gels Using a F orce Model 89 12.1 Examining DN A Fragment Motion in an Ag arose Gel 89 12.2 DN A Fragment Motion Retardation 90 12.3 Modeling the Retarding F orce of the Gel 94 12.3.1 Exponential Correction DN A Fragment Speed Model 96 13 Simulating DN A Motion in Gels Using the ECDFSM Model 98 13.1 Simulation Frame w ork 98 13.1.1 DN A Fragment Band Selection 98 13.1.2 Simulation Components 99 13.1.2.1 Array Size 99 13.1.2.2 Initial Distrib ution 100 13.1.2.3 Flo w Rules 100 13.1.2.4 Electric Field Distrib ution 102 13.2 Simulation Processing and Results 103 13.3 Simulation Summary 105 14 Conclusion 107 14.1 Introduction 107 14.2 Contrib utions 109 14.3 Future W ork 110 References 113 Bibliograph y 131 Appendicies 132 Appendix A: Potential Dif ference 133 Appendix B: F orce Model of Electroporation 144 Appendix C: Conducti vity 148 Appendix D: Deri ving the T ime Constant: c 152 Appendix E: P acking Analysis for Spheres 154 Appendix F: Computing Acceleration and V elocity of Ions in V iscous Media 155 Appendix G: P acking Analysis for Cylinders 158 iii

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Appendix H: P acking Analysis for Cells in Cuv ette 159 Appendix I: Flo w P attern by Direction 160 Appendix J: P arallel Plate Concentration Contour Plots 161 Appendix K: Needle Array Concentration Contour Plots 165 Appendix L: P arallel Plate Electroporation Applicator Flo w Model Code 169 Appendix M: Needle Array Applicator Electroporation Flo w Model Code 189 Appendix N: Gel-DN A Data 210 Appendix O: T issue V iscosity 213 Appendix P: Gel Retarding F orce and Speed Model Iterations 214 Appendix Q: Gel Electrophoresis F orce Model Code 224 Appendix R: Cross Sectional Area Interaction Model Code 228 Appendix S: Surf ace Area Interaction Model Code 234 Appendix T : Area Correction DN A Fragment Speed Model Code 240 Appendix U: P arabolic Correction DN A Fragment Speed Model Code 246 Appendix V : Exponential Correction DN A Fragment Speed Model Code 251 Appendix W : Radius, Speed, V alenc y and F orce V alues 258 Appendix X: Gel V elocity V alues for F our Models 260 Appendix Y : Gel Electrophoresis Simulation Code 263 Appendix Z: Gel Electrophoresis Simulation Images 286 About the Author End P age i v

PAGE 8

List of T ables T able 4.1: Operating Systems and Softw are Used in the Dissertation 28 T able 4.2: Ag arose Concentrations for the Separation of DN A Fragments 30 T able 5.1: The Drift V elocities of V arious Substances Used in Molecular Deli v ery 39 T able 5.2: Distances T ra v eled for Common Field Conditions 40 T able 5.3: Ionic Mobilities of V arious Substances Used in Molecular Deli v ery 41 T able 5.4 V elocity and Acceleration for K + Ions in a 0.9 Mass % NaCl Solution 45 T able 6.1: V olume and V oid Fraction for Cells of Dif ferent Diameters [88] 48 T able 9.1: T issue Model Lattice 62 T able 12.1: Speeds of DN A Fragments in a 1% Ag arose Gel at 6.56 V cm 93 T able 13.1: Comparison of Simulation Results with the Experimental Data 94 T able A.1: Le gendre Polynomials 134 T able A.2: Boundary Conditions for the VBEM Model 135 T able A.3: Boundary Conditions for the DSDM Model 139 T able B.1: Bilayer Lipid Membrane Permeabilities to Common Biological Substances 145 T able C.1: Concentrations of Critical Ions in Body Fluids [44, 123] 148 T able F .1: V elocity and Acceleration for K + Ions in a 0.9 Mass % NaCl Solution 157 T able N.1: DN A Fragment Experimental and Computed Speeds in an Ag arose Gel 212 T able W .1: V alues Used for the DN A Fragments in a 1% Ag arose Gel at 6.5625 V cm 258 T able W .2: F orce V alues for the DN A Fragments in a 1% Ag arose Gel at 6.56 V cm 259 T able X.1: Speeds of DN A Fragments in a 1% Ag arose Gel at 6.56 V cm 260 v

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List of Figures Figure 1.1: Model Components of the EFMDGDMP 4 Figure 2.1: Potential Distrib utions [25] 9 Figure 2.2: Thin W alled Approximation [133, 172] 13 Figure 2.3: Electric Field Strength V ersus Field Duration [80] 19 Figure 3.1: Applicator Pole Specic Ion Accumulation Ecliptics 23 Figure 5.1: Three Dimensional Structure of Bleomycin [215] 34 Figure 5.2: Plot of Acceleration and Speed in 0.9% NaCl Solution 42 Figure 6.1: Relationship Between Cell Diameter and Electroporation Field Intensity [88] 50 Figure 7.1: Photographs of the P arallel Plate and Six Needle Electrodes 54 Figure 7.2: F our Needle, Needle Array 55 Figure 9.1: Flo w Field V alue and Direction Algorithm 63 Figure 9.2: Random Flo w Direction Axis Labels 64 Figure 9.3: Initial Concentration Prole 65 Figure 9.4: Array Described by Six Re gions 68 Figure 9.5: Electric Field Lines for P arallel Plates and Dipoles in Homogeneous Media [213] 73 Figure 9.6: Modeled Needle Electric Field Shape 74 Figure 10.1: P arallel Plate Induced Motion in a 1500 V cm Field for 450 ms 77 Figure 10.2: P arallel Plate Induced Motion in a 1500 V cm Field for 50 ms 78 Figure 10.3: P arallel Plate Induced Motion in a 1500 V cm Field for 200 ms 79 Figure 10.4: Needle Array Induced Motion in a 1500 V cm Field for 450 ms 80 Figure 10.5: Needle Array Induced Motion in a 1500 V cm Field for 50 ms 81 Figure 10.6: Needle Array Induced Motion in a 1500 V cm Field for 200 ms 81 Figure 12.1: Photograph of a T ypical Electrophoresis Gel 91 Figure 12.2: DN A Fragment Radii as a Function of Base P airs 92 Figure 12.3: F orces Acting on the DN A Fragment 95 vi

PAGE 10

Figure 12.4: Experimental Speed vs ECDFSM Predicted Speed 97 Figure 13.1: DN A Fragment Initial Distrib ution 101 Figure 13.2: DN A Fragment Final Distrib ution 105 Figure 14.1: Model Components of the EFMDGDP 109 Figure A.1: V oltage Breakdo wn Electroporation Model [133] 134 Figure A.2: Dielectric Sphere in Dielectric Media 138 Figure B.1: Sum of the F orces During Electroporation 145 Figure B.2: Simplied Sum of the F orces During Electroporation 146 Figure D.1: Responce Curv e W ith T ime Lag 152 Figure F .1: Plot of Acceleration and Speed in 0.9% NaCl Solution 156 Figure J.1: P arallel Plate Induced Motion, 1500 V cm Field after 0, 50, and 100 ms 161 Figure J.2: P arallel Plate Induced Motion, 1500 V cm Field after 150, 200, and 250 ms 162 Figure J.3: P arallel Plate Induced Motion, 1500 V cm Field after 300, 350, and 400 ms 163 Figure J.4: P arallel Plate Induced Motion, 1500 V cm Field after 450, and 500 ms 164 Figure K.1: Needle Array Induced Motion in a 1500 V cm Field after 0, 50, and 100 ms 165 Figure K.2: Needle Array Induced Motion in a 1500 V cm Field after 150, 200, and 250 ms 166 Figure K.3: Needle Array Induced Motion in a 1500 V cm Field after 300, 350, and 400 ms 167 Figure K.4: Needle Array Induced Motion in a 1500 V cm Field after 450, and 500 ms 168 Figure N.1: Photograph of the Data Electrophoresis Gel 211 Figure P .1: Gel Frictional F orce Model 215 Figure P .2: Experimental Speed vs CSAIM Predicted Speed 217 Figure P .3: Gel Frictional F orce Model 218 Figure P .4: Experimental Speed vs A CDFSM Predicted Speed 220 Figure P .5: Experimental Speed vs PCDFSM Predicted Speed 221 Figure P .6: Experimental Speed vs ECDFSM Predicted Speed 222 Figure X.1: Experimental Speed vs CSAIM Predicted Speed 261 vii

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Figure X.2: Experimental Speed vs A CDFSM Predicted Speed 261 Figure X.3: Experimental Speed vs PCDFSM Predicted Speed 262 Figure X.4: Experimental Speed vs ECDFSM Predicted Speed 262 Figure Z.1: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field Initial State, 3 Minutes and 6 Minutes After Onset of Electric Field 286 Figure Z.2: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 9 Minutes, 12 Minutes and 15 Minutes After Onset of Electric Field 287 Figure Z.3: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 18 Minutes, 21 Minutes and 24 Minutes After Onset of Electric Field 288 Figure Z.4: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 27 Minutes, 30 Minutes and 33 Minutes After Onset of Electric Field 289 Figure Z.5: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 36 Minutes, 39 Minutes and 42 Minutes After Onset of Electric Field 290 Figure Z.6: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 45 Minutes, 48 Minutes and 51 Minutes After Onset of Electric Field 291 viii

PAGE 12

List of Abbre viations A CDFSM Area Correction DN A Fragment Speed Model BLM Bilayer Lipid Membrane CCEM Core Conductor Electroporation Model CMD Center for Molecular Deli v ery CSAIM Cross Sectional Area Interaction Model DI Deionized DN A Deoxyribonucleic acid DSDM Dielectric Sphere in a Dielectric Media ECDFSM Exponential Correction DN A Fragment Speed Model ECEM Electronic Current Electroporation Model EFEM Electric Field Electroporation Model EFMDFDM Electric Field Mediated DN A Fragment Deli v ery Model EFMDGD Electric Field Mediated Drug and Gene Deli v ery EFMDGDM Electric Field Mediated Drug and Gene Deli v ery Model EFMDGDMP Electric Field Mediated Drug and Gene Deli v ery Modeling P ackage EFMDGDP Electric Field Mediated Drug and Gene Deli v ery P ackage EFMFFM Electric Field Mediated Flo w Field Model FBEM F orce Breakdo wn Electroporation Model FFTM Flo w Field T issue Model FSF Free Softw are F oundation GEFM Gel Electrophoresis F orce Model ix

PAGE 13

GNU Gnu' s Not Unix GUI Graphical User Interf ace NaCl Sodium Chloride PCDFSM P arabolic Correction DN A Fragment Speed Model PMRNG P ark and Miller random number generator SAIM Surf ace Area Interaction Model SCEM Single Cell Electroporation Model SD A Self Descripti v e Array RFD Random o w direction RFV Random o w v alue VBEM V oltage Breakdo wn Electroporation Model USF Uni v ersity of South Florida x

PAGE 14

List of Symbols ¢ L Change due to compression, nm ¢ Change, nal initial i Relati v e permiti vity unitless o V acuum permiti vity 8 : 854 £ 10 ¡ 12 F ar ad m V icosity cP i Cell interior conducti vity mS cm m Membrane conducti vity mS cm o Electroporation media conducti vity mS cm E Electric eld, V cm j cs Flux, g cm 2 s j cs ( x; y ; z ) ux of char ged species, g sec ¢ cm 2 R ideal g as constant, 8.314 J mol K s v ector x ^ { + y ^ | + z ^ k Electric potential, v oltage, v m W ork function, electron-v olts, eV 3.14159265358979323846 e e xternal resisti vity m i membrane thickness, m conducti vity S iemen cm sc Induced char ge density C oul ombs m 2 Chemical species, e.g. Na xi

PAGE 15

A area, cm 2 a Acceleration, m s a prolate spheroid minor axis length, nm A g el Gel frictional parameter unit less b prolate spheroid major axis length, nm b o Resolution of the gel in base pairs B g el A CDFSM parameter hr nm 2 cm c Stok e' s La w correction f actor unitless C m Membrane capacitance per unit surf ace area, F cm 2 c cs ( x; y ; z ) Concentration of char ged species, g cm 3 C g el PCDFSM parameter cm 2 hr 2 bp conc [ i; t ] Concentration as a function of array position and time D cs Dif fusi vity of char ged species, m 2 t D g el ECDFSM parameter cm hr D g el ECDFSM parameter cm hr e Char ge of an electron, 1 : 602 £ 10 ¡ 19 coul ombs; C E g el ECDFSM parameter bp E g el ECDFSM parameter bp F F orce, ne wton, N, k g m s 2 f Cell ph ysiology f actor unitless f Coef cient of uid friction, k g s F f Stok e' s La w F orce, Fluid frictional force, N, k g m s 2 F g Resisti v e F orce Due to Gel, N F d f Dri ving force, N F r f Retarding force, N G m Membrane conductance 1 m xii

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I current, A, Amps k b Boltzman' s constant, 1 : 380 £ 10 ¡ 23 J K l Length, cm L o Equilibrium length, nm l cy l length of a c ylinder nm M y Y oung' s modulus, P ascals P R resistance, ohms r cy l radius of a c ylinder nm r por e radius of the gel pore, nm s Speed, m s S A Surf ace area, nm 2 T T emperature, K elvin t c Electric pulse duration, s T h mem membrane thickness, nm u ionic mobility cm 2 s V V V oltage, v olts, v v molecular v olume, pm 3 V T M T ransmembrane v oltage, V olts z V alenc y unitless a Inner membrane edge radius, m b Outer membrane edge radius, m xiii

PAGE 17

Simulating the Electric Field Mediated Motion of Ions and Molecules in Di v erse Matrices Joseph D. Hick e y Abstract Electroporation is a methodology for the introduction of drugs and genes into cells. This technique w orks by reducing the e xclusionary nature of the cell membrane [125, 129, 186, 189]. Electroporation has successfully been used in electrochemotherap y and electrogenetherap y [57, 68, 86, 87, 110, 112, 131]. The tw o major components of electroporation are an induced transmembrane potential and the motion of the deli v erable through a compromised cell membrane into the tar get cell [38, 55, 62, 114, 131]. These tw o components are both dependent on the electrophoretic motion of char ged species in an applied electric eld [45, 64, 75, 77, 177]. Currently the methods outlined for understanding electroporation ha v e been focused on either a phenomenological perspecti v e, e.g. what w orks, or modeling the electric eld strength in certain re gions [12, 56, 87, 129, 146, 204, 205]. While this information is necessary for the clinician and the laboratory scientist, it doesn' t e xpand the understanding of ho w electric eld mediated drug and gene deli v ery w orks or EFMDGD. T o increase the understanding of EFMDGD, ne w models are required that predict the motion of ions and deli v erables through tissues to tar get areas [75, 77]. xi v

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This document e xamines the design and creation of an electric eld mediated drug and gene deli v ery model, EFMDGDM. T w o e xample scenarios, ionic motion in tissues and gel electrophoresis, are e xamined in depth using the EFMDGDM. The model requires tuning for each scenario b ut only utilizes e xperimental parameters and one tunable parameter that is computed from re gressed e xperimental data. The EFMDGDM successfully describes the tw o e xamples. Future w ork will incorporate the EFMDGDM as the backbone of an electric eld mediated drug and gene deli v ery modeling package, EFMDGDMP. This modeling softw are package will be optimized to assist clinicians and scientists in the selection of electric eld signatures for the deli v ery of drugs and genes. By utilizing a softw are package that fully describes the motion of ions and molecules in and around either in vitr o or in vivo cell systems impro v ed deli v ery could be accomplished. xv

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1 Prelude Electroporation is a modern technology used to deli v er nati v e and non-nati v e molecules to in vivo and in vitr o cells by o v ercoming the e xclusionary nature of the cell membrane [76, 77]. This technology w as rst described by Sale and Hamilton in their seminal w ork on irre v ersible poration [66, 128, 171, 172]. Their three paper series e xamined the “Ef fects of Electric Fields on Microor g anisms” primarily the killing of microor g anisms, the lysis of non-w alled cells, and the mechanism of the process. The rst purposeful usage of electroporation w as accomplished by Neumann and Rosenheck. The y demonstrated re v ersible poration via the release of v esicular components and proposed that the method of action w as due to “the density of ions in the ion cloud [on the inside of the cell membrane] is higher than the ion density of the surrounding medium” [132]. Extending the w ork of Neumann and Rosenheck, Zimmerman et al. and Kinosita and Tsong electroporated indi vidual cells, measured ion o w and ascertained the potential difference required for poration [30, 31, 96–98, 162, 221, 222]. Zimmermann et al. and Mir et al. electroporated cells for the uptak e of chemotherapeutic agents for the treatment of cancer [93, 121, 122, 138, 220, 225]. This technique has become a localized treatment of cancer termed electrochemotherap y ECT [59, 60, 70– 72, 122, 196]. Mir et al. and Heller et al. dro v e ECT through phase I and II clinical trials [41, 59, 60, 70, 115, 115]. 1

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W ong and Neumann et al. e xtended the usefulness of electroporation from a scientic curiosity to a laboratory technique via the successful in vitr o transfection of mammalian cells [127, 214, 220]. Electrotransfection' s ef cienc y w as increased, through pulse parameter optimization, allo wing it to produce stable transfectants [27, 47, 52, 91, 100, 155, 156, 195]. T itomiro v et al. and Heller et al. performed and optimized electric eld mediated in vivo gene deli v ery [69, 194]. In V ivo transfection of mammalian cells via DN A electrotransfer is a pro v en and ef fecti v e non-viral technique with results of 10 to 1000 fold o v er direct DN A injection [49, 69, 89, 118, 131, 182, 187, 211]. Electrogenetherap y with inter leukin coding plasmids is a successful treatment for established tumors and tumor gro wth inhibition [73, 109– 111]. Understanding the phenomena of electroporation has been attempted through three basic model types, single cell models, electronic current models and electric eld based models. The single cell electroporation models, SCEMs, ha v e primarily dealt with the required transmembrane breakdo wn v oltage on a circular bilayer lipid membrane [34, 169, 172, 189]. These models break up into tw o camps, the v oltage breakdo wn electroporation models, VBEM, and the force breakdo wn electroporation models, FBEM. The breakdo wn v oltage models propose that once the cell membranes breakdo wn v oltage is e xceeded the membrane porates to reduce the transmembrane potential [37, 38, 131, 133, 190, 201]. The force dependent models propose that the membrane is ruptured by the force applied by the electric eld [4, 21, 22, 84, 125, 186, 220]. The electronic current electroporation models, ECEMs, ha v e primarily dealt with modeling the electronic current past a cell due to an applied electric eld [12, 18, 35, 36, 103, 104, 197]. The electric eld electroporation mod2

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els, EFEMs, predict the electric eld strength at specic points in the tissue as a function of applied electric eld [102, 117, 118, 159, 203, 204]. Each model type has specic strengths and weaknesses when compared to e xperimental data. The SCEM f amily seemingly ignores ho w the transmembrane v oltage dif ference occurs, the ECEM f amily treats the ionic current lik e an electronic current, and the EFEM f amily ignores the current carriers. These three f amilies of models ha v e greatly increased the understanding of electroporation and electric eld mediated deli v ery b ut none pro vide a complete description. The topic of this dissertation is the inception, de v elopment and characterization of an alternati v e method of modeling the process that leads to an electric eld mediated drug or gene deli v ery model. The long term goal for this research is to enunciate clearly the elemental model requirements and subsequent tasks needed to de v elop and assemble an electric eld mediated drug or gene deli v ery modeling package, EFMDGDMP. Figure 1.1, indicates the tw o media re gimes that an electric eld mediated drug and gene deli v ery modeling package w ould ha v e to describe. This softw are tool should allo w the user to model the entrance, mo v ement and subsequent deli v ery of a therapeutic agent within a heterogeneous or homogeneous matrix in an in vivo or in vitr o situation. Although this dissertation research does not culminate with the complete softw are package including an appropriate graphical user interf ace, GUI, it does e xplore, de v elop and demonstrate man y of the model subsystems that constitute an EFMDGDMP. This dissertation establishes an electric eld mediated o w eld model, EFMFFM, as 3

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EFMDGDMP GUI EFMFFM Heterogeneous Matrix Ions, Drugs DN A, Protein Homogeneous Matrix Ions, Drugs DN A, Protein Figure 1.1: Model Components of the EFMDGDMP The electric eld mediated drug and gene deli v ery modeling package, EFMDGDMP, will pro vide a user interf ace allo wing a user to emplo y the electric eld mediated o w eld model to follo w the o w of char ged deli v erables in homogeneous and heterogeneous matricies. the operational system for an EFMDGDMP. This softw are will link the user inputs via the model parameter options to v arious model subsystems. The electric eld mediated o w eld model, EFMFFM, represents a ne w alternati v e method of modeling the processes that lead to electric eld mediated drug and gene deli v ery The electric eld mediated o w eld model considers the motion of ions and char ged molecules in di v erse matrices in an applied electric eld. 4

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P art I 5

PAGE 24

2 Literature Examination 2.1 Historical Re vie w The e xperimental application of electric elds to cells and tissues has a rich history in the scientic canon, be ginning in 1664 with the nerv e stimulation of frog muscles e xperiment of Jan Sw ammerdam. This e xperiment used silv er wires and a copper loop in direct contact with the “motor nerv e of a frog muscle” [114]. This is the rst account of nerv e stimulation via a bimetallic junction. An early statement about the use of electricity w as in 1743 when Johann Gottlob Kr ¨ uger said “ All things must ha v e a usefulness; that is certain. Since electricity must ha v e a usefulness, and we ha v e seen that it cannot be look ed for either in theology or in jurisprudence, there is ob viously nothing left b ut medicine. ” [114, 183] In 1781, an assistant of Luigi Galv ani stimulated a dissected frog le g using an electric machine and a scalpel, as the assistant contacted the table where the electric machine resided and the frog' s le g with the scalpel, the frog' s le g contracted and the electric machine generated a spark [114, 177]. This demonstration is the rst documented e xperiment in neuromuscular stimulation from an e xternal current source [114]. Galv ani continued stimulation e xperiments using atmospheric and bimetallic apparatus. Galv ani set up a cur rent in a frog le g and simultaneously applied “a bimetallic arch of copper and zinc” to the frog' s nerv e and muscle, which resulted in a contraction [114, 177]. Galv ani e xplained this 6

PAGE 25

phenomena as the bimetallic arch dischar ging “animal electricity” [114, 177]. This is the historical reference most often cited to demonstrate e xternal electric nerv e stimulation. Alessandro V olta continued the w ork of Galv ani and recognized that the source of the char ge w as not the frog b ut the tw o metals [114, 177]. V olta substituted inor g anic material for the frog muscle and produced a similar ef fect [177]. His ne xt mo v e w as to create a continuous current source via dissimilar metals separated by a cloth soak ed in a salt solution [177], this is kno w as a V oltaic pile or battery Medical application of electric elds came into e xperimental usage during the 1870' s. The rst successful medical application of electric elds w as “cardiorespriatory resuscitation” [114]. This w as accomplished by T Green in 1872 using 200 of V olta' s piles. The piles generated approximately 300 V olts and were applied to the patient between the neck and the lo wer ribs [114]. This process successfully resuscitated at least v e patients who had suf fered respiratory arrest due to chloroform anesthesia [114]. In 1874, Dr Robert Bartholomeu be g an using induction coils, in v ented by Michael F araday in 1831, for neurostimulation [114, 177]. Dr Bartholomeu e xposed a patients cerebral corte x and stimulated it with F aradic currents [114]. The result of this e xposure w as the motion of the patients limbs on the opposite side of their body and the turning of the head [114]. Cardiac debrillation w as rst reported on in 1899 by Jean Louis Pre v ost and Fr ed eric Battelli. Their report stated that lo w v oltage electric shocks induced v entricular brillation while high v oltage electric shocks restored normal heart rh ythm [114]. As the technology used in measuring the b ulk electrical properties of cells and tissues emanated in the late 19 th and early 20 th century the associated science gre w dramatically 7

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The adv ent of the electron tube in 1906 by Lee de F orest allo wed for the amplication of electric signals [114]. Researchers in the elds of bioelectronics and bioph ysics used this technology to sho wn that the electrical properties of tissues v ary as a function of eld frequenc y [50, 209]. DuBois, as reported by F oster and Schw ann, found that the skin of animals “beha v ed lik e a capacitor if subjected to DC currents” [50]. F or DC elds, cells and tissues act as capacitors char ging up while in A C elds of 1 kHz or greater the cells and tissues act as conductors with a much lo wer resistance [50]. During WWI increased understanding of the ef fects of electric elds on cells and tissues w as accomplished through the actions of cell biologists and ph ysiologists, rather than ph ysicians, ph ysicists and electrical engineers as in the pre vious three centuries [50]. After WWII the studies continued and the sodium-potassium pump protein components of the cell membrane were elucidated through v oltage clamp studies and mathematical modeling of the ef ux of K + and the inux of Na + [95, 123]. Maintenance of the concentration gradient is sustained by the A TP controlled sodium potassium pump [44, 95]. This pump w as the rst documented e xample of acti v e transport, it w as disco v ered in the mid 50' s [11] and w as later found to be a transmembrane protein. Other transmembrane proteins act as sodium co-transporters, transporting membrane impermeable x enomolecules through the cell membrane po wered by the e xtracellular/intracellular Na + ions imbalance [11]. This interaction between the membrane channels and the sodium-potassium pump act to maintain both the concentration and v oltage gradients in cells. The dif ferent resting potentials 8

PAGE 27

in a cell are graphically displayed in gure 2.1 1 The dotted line at the bottom of gure 2.1 is the b ulk intracellular potential and is designated by o The b ulk e xtracelluar potential, 1 as referenced to the b ulk intracellular potential is sho w on the left hand side of the gure. 2 at the membrane-e xtracellular interf ace is the potential dif ference between the b ulk e xtracellular potential and the e xternal membrane potential. The membrane component of the model acts as a leak y conductor with the v oltage rise acting lik e simple dif fusion, the potential rise between the e xtracellular membrane f ace and the intracellular membrane f ace is designated by 3 4 is the dif ference between the b ulk intracellular potential and the internal membrane potential. The right hand side of the gure is the intracellular section. It represents the change from the membrane to the intracellular b ulk. Both the intracellular and the e xtracellular graphs are similar to dif fusion graphs from the b ulk to a surf ace [15]. 6 ? 6 ? + ¡ 1 3 4 2 o g Extracellular Membrane Intracellular Figure 2.1: Potential Distrib utions [25] 1 Note: In gure 2.1 a posti v e potential is up and a ne g ati v e potential is do wn. This follo ws with the literature con v ention. 9

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2.2 Current Char ged Species Modeling Practices Electrophoresis and dielectrophoresis are methods of mo ving nati v ely or induced char ged species through liquids [6, 151, 152, 184, 193]. These techniques were adv anced by Arne T iselius and Herbert Pohl respecti v ely [28, 151, 193]. Electrophoresis is the “motion of [char ged] suspended particles produced by the action of electrostatic elds” [151]. Dielectrophoresis is the motion of char ge induced suspended particles in the presence of an electric eld [151]. Electrophoretic motion is a function of the sign of the electric eld and dielectrophoretic motion is a function of the square of the eld' s magnitude [6, 150, 153]. Unchar ged particles can also be indirectly mo v ed by electrophoresis and dielectrophoresis. The unchar ged particles respond to the tidal motion produced by the motion of the nati v ely or induced char ged particles in the suspensor and the motion of the suspensor itself in an applied electric eld [150, 153]. Electroporation is an increased permeability of a cell' s membrane to typically impermeable molecules [68, 69, 110, 121, 190, 220, 223]. Electroporation is belie v ed to be the result of a rapid intense char ge dif ference spanning a cell' s bilayer lipid membrane, BLM [132, 169, 172, 189]. This rapid intense char ge dif ference o v ercomes the ability of the cell to maintain near membrane homeostasis via typical cell channeling systems and the Na + -K + pump [37, 38, 131, 190]. Figure 2.1 represents the resting state of a cell. This resting state v aries for dif ferent cells b ut typically cells ha v e a ne g ati v e tranmembrane potential dif ference. This potential dif ference is sho w in gure 2.1 as 1 The dif ference in the electric eld between the intracellular re gion and the e xtracellular re gion can be deduced 10

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from gure 2.1 2 There is a lar ge electric eld in the re gion of the intracellular membrane especially near membrane spanning domains. This transmembrane char ge imbalance is the result of an applied electric eld [138, 190, 220]. Modeling the transmembrane v oltage has been a focus of electroporation researchers since the inception of the technique [133]. The rst and most common models deal with a single spherical cell in homogeneous media with applied homogeneous electric elds [34, 133, 169, 172, 189]. The y are aptly named v oltage breakdo wn electroporation models, VBEMs, a subset of the f amily of single cell electroporation models or SCEMs 3 VBEMs relate the transmembrane v oltage, V T M to the applied electric eld, E and the cell radius, b, via adaptations of the Sale and Hamilton equation, see equation 2.1 page 13 [172]. Additions to this model ha v e impro v ed it by relating the of f equatorial dependence of the electric eld to the v ariation of the transmembrane v oltage, see equation 2.2, page 13 [126, 133, 166, 201]. The directional cos term of equation 2.2 e x emplies the dependence of the transmembrane v oltage to the perpendicular electric eld around the cell [32, 99, 177]. The greatest applied electric eld will be equatorial, while the least applied eld will be polar this result has been repeatedly v eried e xperimentally [164, 165, 191, 192]. A ne tuning to equation 2.1 w as the addition of a cell ph ysiology f actor f This f actor' s purpose is tw ofold, rst, to allo w the same equation to describe multiple time frames during the electroporation process and second, to describe the pre-electroporation state of dif ferent 2 3 T h mem where T h mem is membrane thickness. 3 SCEMs are a f amily of electroporation models that deal solely with the electroporation of a single cell in situ 11

PAGE 30

cell types with the same radii b ut dif ferent electric eld response characteristics [128, 166]. The initial v alue of this f actor corresponds to a cell' s resting state conducti vity in the electroporation media, see appendix C [128]. The v alue can also be modeled as a function of pore formation thereby allo wing the conducti vity of the cell membrane to change during the poration process [36–38, 131]. The ne xt step in comple xity for the electroporation transmembrane v oltage equation w as the addition of char ging time, t c and a corresponding time constant, c [125, 166, 189]. The introduction of t c into the VBEM f amily introduced time dependence of electroporation on the applied electric eld duration. This simple addition took the model from being wholly steady state to ha ving dynamic response as a function of pulse length [114]. c reects the char ge b uildup relationship on either side of the cell membrane 4 It is a modied capacitor time constant that relates the membrane conductance, G m and the membrane capacitance per unit surf ace area, C m to the char ging rate of the cell membrane [114, 128, 169, 174, 175, 177]. The introduction of c allo wed the VBEMs, a subset of the SCEM f amily to incorporate a time course analysis of electroporation for dif ferent pulse durations and shapes [116, 128, 166, 189]. F or the VBEM, v oltage breakdo wn electroporation model, f amily of equations, see equations 2.1 through 2.4, a cell is treated as a conducting sphere within a dielectric 5 [32, 172]. As indicated by the number of papers, [116, 128, 131, 133, 146, 166, 189], and the equation' s metamorphosis and continuance, this technique has pro vided great in4 F or more information about the ef fect of c see appendix D 5 Except for c which is a function of the associated conducti vities, for more information see appendices A and D 12

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sight into the electroporation phenomena. V T M = 3 2 b E (2.1) V T M = 3 2 b E cos (2.2) V T M = 3 2 f b E cos (2.3) V T M = 3 2 f b E cos 1 ¡ exp ¡ t c (2.4) The emphasis on V T M the transmembrane v oltage, is primarily due to the f act that membrane breakdo wn has been predicted to occur at V T M v alues in the range of 100 ¡ 500 mV depending on cell radius, viability and suspensory conducti vity [53, 118, 130, 131, 165, 189]. Such lo w transmembrane v oltage v alues ha v e been seen to ef fecti v ely electroporate cells primarily at the on-eld equator of spherical and spheroid cells [53, 169, 192]. Electroporation is h ypothesized to be the most ef fecti v e equitorially because of edge ef fects and transmembrane v oltage amplication as described by the 3 2 cos term of equations 2.1 through 2.4, see gure 2.2 [130, 169]. 6 a bd 1 E Figure 2.2: Thin W alled Approximation [133, 172] 13

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The v oltage breakdo wn electroporation model, VBEM, approach stems from the e xamination of the ef fect of the applied electric eld inducing a transmembrane potential, which when o v er a certain v alue w ould induce increased cellular permability Alternati v ely the FBEMs or force breakdo wn electroporation models, another subf amily of the SCEM, deal with the impact of force due to an applied electric eld on a bimolecular lipid membrane or BLM and are dependent on the transmembrane v oltage [34, 161, 210]. FBEMs either use the results from VBEMs, v oltage breakdo wn electroporation models, or directly model electric elds to predict the forces applied to cell membrane patches [22, 23, 54, 55]. FBEM models are separated into tw o subgroups by their h ypotheses. The rst h ypothesis is “membrane thinning to rupture produces electropores” [30, 220] and the second h ypothesis “electropores are caused through the remo v al of membrane sections or membrane components” [22, 23]. Examples of both FBEMs are pro vided belo w The lipid thin lm literature is studded with research promoting the “thinning to rupture h ypothesis”. This is particularly pre v alent in studies that in v olv e embedded particles [2, 4, 8, 26, 81, 140, 141, 143, 167]. In studies of BLM compression due to electric elds, the FBEM is de v eloped from the breakdo wn v oltage modeled as a relation between a membrane' s Y oung' s modulus and the applied electric eld pressure [34]. The Y oung' s modulus, M y = ¢ F A ¢ L L o of a non-conducti v e insulating layer surrounded by a conducti v e media can be calculated by setting an e xpression for the elastic force, ln l L o £ M y ; equal to the electric compressi v e tension, ¡ "V 2 2 l 2 see equation 2.5 [34, 210]. The BLM is compressed by the pressure produced by the incident electric eld, 14

PAGE 33

gi v en by V L [34, 210]. The v ariables L o and l are the diameters of the uncompressed and compressed membranes, respecti v ely (2.5) ¡ "V 2 2 l 2 = M y ¢ ln l L o The FBEM de v eloped in equation 2.5, assumes that only balanced forces af fect the stability of the resting membrane and does not deal with dynamic systems [34]. If a force is not identically compensated then it will cause the cell to stretch or compress and mak e the membrane thin or b uckle [30, 220]. This alteration of the membrane may produce defects that allo w for increased transport into the cell. The second subgroup of FBEMs h ypothesize that the force of the electric eld on the cell membrane is great enough to forcefully remo v e a sections of the membrane either into or out of the soma [22, 23]. This model incorporates Ne wton' s second la w ~ F = m ¢ ~ a a friction f actor f ( x; T ) and an induced polarization gradient across the membrane, ¡ q d dx [22, 23] when assembled the result is equation 2.6 belo w F or an in depth discussion of the second la w electroporation force model see appendix B. (2.6) ¡ q d dx ¡ f ( x; T ) = m d 2 x dt 2 The v alidity of e xplaining poration due to mechanical forces is reaf rmed by both the lipid thin lm and the sur gical literature [116]. Cell w ounding produced by intense electric elds resembles areas that had “mechanical stress imposed, implicating mechanical stress as the disrupti v e agent” [116]. The opinion that electrical damage is due not only to heating and current b ut also to force, is held for the cell, tissue and or g an le v els. Raf ael Lee of the 15

PAGE 34

Uni v ersity of Chicago Department of Plastic Sur gery has stated that “ph ysicians who ha v e had e xtensi v e clinical e xperience with electrical trauma compare it to a mechanical crush injury” [107]. The ECEM, electronic current electroporation model, approach uses a core conductor electroporation model, CCEM, to describe the induction of electroporation on c ylindrical cells see equation 2.7. The CCEM is the most comple x analytical model currently used to describe electroporation. The core conductor model has traditionally been applied to describing ion and electron motion do wn neurons and coaxial cables. A diagram of the CCEM model is similar to gure 2.2 on page 13 only as a “do wn the barrel” of a c ylinder rather than cross section of a sphere [36]. The CCEM parameters are assembled as equation. 2.7 (2.7) a 2 i + e s @ 2 V m @ x 2 = C m @ V m @ t + I LR d + I ep In this equation, i and e are the internal and e xternal resisti vities respecti v ely C m is the specic membrane capacitance, a the inner radius as in gure 2.2, and s the ratio of intracellular to e xtracellular cross sectional area. The v ariable x represents the distance measured from the center of the ber t is the time, V m is the transmembrane potential, I LR d is the acti v e membrane current and I ep is the electroporation current [36]. The v ariable I LR d is from the Luo-Rudy dynamic membrane kinetics model that is commonly used with the re-polarization of v entricular myoc ytes [36, 217]. The Luo-Rudy model tak es into account electric currents that naturally o w through the cell membrane via the v oltage g ated channels and through the specic pumps [36, 217] and also relates 16

PAGE 35

the change in the transmembrane potential with respect to distance. The Luo-Rudy model is based on data for guinea pig v entricular myoc ytes b ut w as used as a model for all cardiac bers in this study The electroporation current, I ep equation used by DeBruin and Krasso wska is described by I ep = g p N V m The components of this equation are the number of pores, N the transmembrane potential, V m and the conductance of a single pore, g p While, solving for the number of electropores or the conductance of a single pore are both difcult tasks, the y ha v e been deri v ed from both thermodynamic and rst principle directions [128, 130, 140, 141, 143]. The usage of the core conductor model has been e xtended to spheroid or prolate spheroid cells with moderate success [37, 38, 125, 205]. Inte gral to models based on the core conducting model is the h ypothesis that the most common transmembrane potential equation, equation 2.1 [39, 131, 199– 201], only applies until the time that electroporation occurs. Authors that adopt this ar gument therefore contradict the popular thinking outlined in the literature of the pseudo-steady state electroporation h ypothesis and ha v e utilized a time course approach to electroporation [37, 38, 125, 205]. The basis of these time dependent models includes pore density and duration of the applied electric eld [37, 38]. These models represent the rst ef forts to e xamine electroporation kinetics using material balance methods. Equation 2.8 is based on the dif fusion equation and its components are, @ [ ] i @ t which describes the intracellular 6 concentration change of species as a function of time, D r 2 [ ] i 6 The subscript i stands for intracellular not ^ { the normalized x directional v ector 17

PAGE 36

describes the dif fusion of species into the cell, and D denotes the dif fusi vity of species The second section of the model on the right hand side of equation 2.8 is the “drift of ions in the intracellular electric eld” induced by cell polarization [38]. Its components are z the char ge of the ions, F F arraday' s constant, R the uni v ersal g as constant, T temperature in k elvin, i the intracellular potential, and r i the intracellular electric eld, E i 7 (2.8) @ [ ] i @ t = r ¢ D r [ ] i + D z F R T [ ] i r i In summary the objecti v e of this literature search w as to e xamine the emplo yed modeling approaches in the electroporation community and to ascertain their strengths and weaknesses. From that analysis, no v el viable modeling possibilities were ascertained. The ultimate goal for an electroporation model is to e xplain the phenomena and f acilitate protocol de v elopment decisions. Currently there are man y dif ferent protocols for electroporation all being successful to a certain de gree. The dif ferent protocols approach electroporation conditions from tw o dif ferent anks. The tw o dif ferent routes represented are high v oltage short duration, 500 ¡ 1500 V cm for 100 s [68, 89, 90, 176] and lo w v oltage long duration 30 ¡ 200 V cm for 25 ms conditions, see gure 2.3 [90]. T o date, models found in the literature do not account for or e xplain the ef fect of a long duration pulse compared to a short duration pulse. In the transmembrane v oltage models the only time characteristics e xamined are the char ging time constant of the membrane and the duration of the pulse. The only model that actually uses time and depth dependence is the application of the core conducting model by DeBruin and Krasso wska [37] b ut this 7 The subscript i stands for intracellular not indicie. 18

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L ysing of Cells No Ef fectNo Ef fectPulse Length secElectric FieldV cmRelaxation T ime of CellMinimum Membrane Breakdo wn V oltage k V cm ; sec pulse ? V cm ; msec pulse Figure 2.3: Electric Field Strength V ersus Field Duration [80] treats the membrane as an isotropic material with respect to the time dependence of an applied electric eld and does not tak e into account dif ferences in membrane capacitance. The realization that electric eld mediated drug and gene deli v ery is dependent on multiple f actors w as the impetus for the creation of a ne w model idea. A fe w of those f actors are applied electric eld duration, transmembrane v oltage v alue, applied electric eld strength, conducti vity of the media and the membrane re gion, and also a material component for both the motion of the deli v erable and the chage to the cell membrane. Realizing the ef fect of these f actors on deli v ery presents an opportunity to de v elop a ne w electric eld mediated transport phenomenological model. 19

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3 Project Description 3.1 Current Status of Electroporation Modeling Current electroporation modeling research has primarily attack ed the problem of under standing electroporation using single cell electroporation models, SCEMs, from tw o vie ws. The y are the v oltage breakdo wn electroporation models, VBEMs, and the force breakdo wn electroporation models, FBEMs. While both of these methods ha v e e xtended the under standing of the electroporation phenomenon, the y ha v e a v oided the subject of the motion of the ions, the primary char ge carriers [114]. The electroporation literature has e xamined the transmembrane potentials and the transmembrane forces from applied electric elds in great detail b ut these v enues are limited to the e xamination of a single cell or small portions of a cell. F or e xample, the VBEMs calculate transmembrane v oltage v alues as a function of cell radius and see gure 2.2, b ut the limitations of the model require that the cell is spherical, the inside of the cell is a homogeneous lipid with a dielectric constant of approximately 2, and the e xtracellular matrix is a homogeneous saline solution with a constant dielectric constant of approximately 80 [125, 126, 133, 166, 189, 201]. This simplication of reality has produced a model that describes the single spherical cell system and predicts a transmembrane v oltage required for poration b ut requisite renements are dif cult with this method. Alternately the FBEMs 20

PAGE 39

are simple to use, scalable, and cell shape independent b ut e xamine only a patch of the cell membrane [22, 23]. This technique although an interesting theoretical application for por tions of a single cell, cannot be applied to populations of cells and has only been applied to v ery simple lipid bilayer systems. F or an electroporation model to be applicable and useful in a clinical or laboratory sense, it needs to adequately describe the problem of both electroporation and the deli v ery of molecules to the treatment site. 3.2 Electric Field Mediated Flo w Field Model This dissertation describes and implements the mathematical and theoretical base, and descripti v e frame w ork of a ne w model idea, the electric eld mediated o w eld model or EFMFFM, that can simultaneously e xamine the motion of ions, molecules, and/or DN A fragments dri v en by an electric eld in di v erse matrices 1 The ne w idea, rather than utilizing the transmembrane potential of a homogeneous single cell, addresses the problem from an electrophoretic direction. Electrophoresis is the motion of char ged species through a matrix in an electric eld applied by electrodes in contact with the suspension [6, 29]. The motion of ions in applied electric elds causes a circle of ion accumulation at the ecliptic in the eld dir ection 2 see gure 3.1, and thereby increasing the induced polarization and in turn the transmembrane potential, see equation 3.1 [32, 99, 186, 224]. The induced polarization and the transmembrane potential are both functions of the char ge accumulation at the poles 1 The term matrices is used here not in the mathematical sense b ut as a structural frame w ork. The matrices implied here are an y grouping of elements that act to impede o w 2 The ecliptic is the circle that cuts a spherical object in plane in the in-eld direction. The circle of ion accumulation is similar to a circle of illumination created by the homogeneous light emission of the sun on the curv ed surf ace of the earth. 21

PAGE 40

due to the applied electric eld, see gure 3.1. The yello w re gions in gure 3.1 are the areas where the char ge b uildup has tak en place and where electroporation is predicted by equation 3.1. The left side of gure 3.1 is the anode side of the process and the right side is the cathode 3 The re gions of inuence are electrode shape dependent. In this graphic and in equation 3.1, the electrodes are assumed to be nite points. (3.1) r ¢ E = r ¢ ( ¡r ) = o The transmembrane potential, can be calculated from the v olume char ge density 4 If ( x;y ;x ) w as kno w for both sides of the membrane then the transmembrane potential w ould be calculated by inte grating o v er the inner surf ace to compute the inner surf ace v oltage, V is = Z Q is 4 o r 2 dr = Z Z Z is dA 4 o r 2 dr and outer surf ace to compute the outer v oltage, V os = Z Q os 4 o r 2 dr = Z Z Z os dA 4 o r 2 dr 5 Subtracting these tw o v oltages yields the transmembrane potential, The transmembrane potential described here is the same v alue utilized by the VBEMs b ut it is applied in a dif ferent manner for a completely dif fer ent solution, see appendix A. Understanding the motion of the ions in the applied electric eld via electrophoresis also allo ws for a prediction of the applied force. The force due to a static char ge is described 3 This stems from the cosine dependance of the transmembrane v oltage equation, 2.1 4 is the number of char ge carriers per unit v olume. While is typically considered a constant, in the inhomogeneous matrix where electroporation occurs it is e xtremely directional dependent. 5 Z Z dA is used here as the symbol for taking the closed inte gral on the surf ace of a re gion. It can also be represented as I dA The Jacobian is dependent on the coordinate system used, e.g. for a spherical coordinate system the Jacobian is r 2 cos d d and for a rectangular coordinate system the Jacobian is dxdy dy dz or dxdz : [177] 22

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Figure 3.1: Applicator Pole Specic Ion Accumulation Ecliptics by Coulomb' s la w F = 1 4 o q 1 £ q 2 r 2 [177]. The forces applied to the membrane by a distrib ution of ions could be ascertained using Coulomb' s la w and the predicted ionic concentration due to the applied electric eld in a manner similar to those emplo yed in molecular dynamics. The EFMFFM incorporates portions of nodal analysis, time e v olving nite dif ference, molecular and uid dynamics into one descripti v e system. This task is accomplished using a self descripti v e array or SD A, where each node contains the pertinent information about node content, a v ailable o w directions, content o w v elocities and eld strength. The SD A has the capabilities to be e xpanded to a computer dependent size and thereby posses the ability to describe highly comple x systems. The SD A w as used to e xamine ion o w in tissue for tw o electrodes, DN A motion in tissue and DN A motion in electrophoresis gels. Node content be gins with an initial concentration or normalized v alue of the chosen ionic species. This initial distrib ution can be randomly or patterned throughout the array approximating e xperimental conditions, e.g. concentric distrib ution of analyte from 23

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injection. After the electric eld is applied this initial concentration distrib ution will o w between nodes by a time e v olving nite dif ference technique. Biological matrices are modeled as a o weld where the o w of char ged species is limited to allo w able directions and dened o w rates [13, 75, 77, 124]. This permits the creation of matrices with di v erse structures limited only to the resolution of the chosen array size. Flo welds are typically utilized to describe the dynamics of the motion of uids in comple x systems, lik e smok e emer ging from a chimne y or ri vulets creating ca v erns [124]. In this o weld, the force go v erning motion is an applied electric eld gradient. This is dif ferent from most o welds where the motion is go v erned by gra vity temperature or pressure. The force due to the applied electric eld acting on the char ged species is the calculated Lorentz force. This v alue is countered by retarding forces that are dependent on the distinct matrix being e xamined. Field strength w as modeled using a resistor model for the re gion between electrodes and an in v erse distance squared relationship, 1 r 2 model outside of the electrodes. This method allo ws for the electric eld to be modeled as a constant during application. This simplication is justied because the time response of an ion' s motion in an electric eld is on the order of femtoseconds, see table 5.4 [6, 114, 177]. This eld strength modeling method ignores the char ging of the cell membrane because it is concentrated on the motion of the char ged species rather than membrane breakdo wn. 24

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3.3 Problem Denition The challenge associated with the creation of the EFMFFM, electric eld mediated o w eld model, is the de v elopment of the method to be utilized to e xamine electrophoresis in di v erse matrices. Succinctly stated, this task is preformed by solving equation 3.2, a relati v ely simple dif ferential equation for the ux of char ged species, j cs [45, 64] using numerical techniques. Equation 3.2, combines Fickian dif fusion, ¡ D cs r c cs ( x; y ; z ) with Einstein' s absolute mobility equation, M = D AB k b T and the Lorentz force equation, F = q E This dif ferential equation has an analytical solution in liquids and simple systems b ut not in di v erse biological matrices. The EFMFFM produces a time e v olving solution for this equation by e xamining the o w of char ged species in liquids and e xpanding the w ork to tissue and ag arose gels. The model w as v eried by simulation and comparison to e xperimental data for tw o electrode v arieties in a h ypothetical tissue and for DN A motion in ag arose gels, see chapters 8 through 13. The techniques utilized to create the general model and the simulations are also described and characterized, see chapters 5 through 7. (3.2) j cs ( x; y ; z ) = ¡ D cs r c cs ( x; y ; z ) + c cs ( x; y ; z ) D cs q E appl ied k b T 6 6 In the dif fusi vities the carrier suspensory is understood, hence the dif fusi vities are D cs rather than D AB where A is the char ged species and B is the suspensory 25

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4 Material and Methods 4.1 Introduction The operating systems and the softw are used in the research and during the writing of this dissertation w as GNU/Linux created by Red Hat, Fedora, the Free Softw are F oundation, FSF, and the GNU project. The use of this softw are/operating system combination w as chosen because of the stability reliability and portability of the nal results. This follo ws the ideals of C programming where code should be portable with only recompilation required, re g ardless of the users system. 4.2 Materials This section is brok en up into tw o parts, research and document. In the research portion, the computers, operating systems and the utilized programs, as well as the materials used for the gel electrophoresis process are listed. In the document portion, the softw are used to create this dissertation is listed. Although including the document preparation softw are in the materials section of the document is rare, it is included here for completeness. My goal for this research w as to produce a descripti v e model of the motion of char ged species in di v erse matrices in an source open source format from research through documentation 26

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to presentation. This goal w as accomplished and the programs required are reected here in the materials section. 4.2.1 Research The research for this project had four phases. First, literature and e xperimental data w as collected and interpreted. Second, computer based mathematical models were created. Third, the motion of ions and molecules were simulated in silico F ourth, the indi vidual images produced by the simulations were compared to the original data and con v erted into animated gifs. 4.2.1.1 Computers Manuf acturer Model Processor Speed (MHz) Ram (Mb) Dell Dimension v400 Pentium II 400 384 HP ze5170 Pentium IV 2000 512 HJD 1 W orkstation Pentium IV ht 2600 1024 4.2.1.2 Operating Systems Red hat 9.0 and Fedora Core 1 were used in e v ery element of this dissertation. From the initial planning, to the code writing, to processing, post processing, plotting and animation of the data. Choosing open source operating systems w as requisite to producing an open source dissertation. 1 This is Hick e yJosephDesigned, a homeb uilt pc continuing with the open system idea. 27

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4.2.1.3 Research Programs The operating systems, compilers, shells and graphics packages used in the research contained in this dissertation as well as the creation of the dissertation itself are listed in table 4.1 belo w T able 4.1: Operating Systems and Softw are Used in the Dissertation Operating System Red hat 9.0 Fedora Core 1 Compiler GCC-3.2.2-5 GCC-3.3.2-1 Shell B ASH-2.05b-20.1 B ASH-2.05b-34 Scripting Language Perl-5.8.0-88.3 Perl-5.8.3-16 Plotting Program Gnuplot-3.7.3-2 Gnuplot-3.7.3-4 Graphics Program tete x-1.0.7-66 tete x-2.0.2-8 Animation Program ImageMagick-5.4.7-10 ImageMagick-5.5.6-5 4.2.1.4 Gel Electrophoresis Materials Used in Gel Electrophoresis Experiments 1. Electrophoresis Buf fer (T AE or TBE) (a) T AE 50 x Stock Solution pH 8.5 i. 242 g T ris base ii. 57.1 ml glacial acetic acid iii. 37.2 g Na 2 EDT A ¢ 2H 2 O i v H 2 O to 1 liter (b) T AE 1 x W orking Solution i. 40 mM T ris acetate ii. 2 mM EDT A 28

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(c) TBE 50 x Stock Solution pH 8.5 i. 242 g T ris base ii. 55 g boric acid iii. 40ml 0.5 M EDT A, pH 8.0 i v H 2 O to 1 liter (d) TBE 1 x W orking Solution i. 89 mM T ris base ii. 89 mM boric acid iii. 2 mM EDT A 2. Ethidium Bromide Solution (a) 1000 x Stock solution, 0.5 mg ml i. 50 mg ethidium bromide ii. 100 ml H 2 O (b) W orking solution, 0.5 g ml i. Dilute stock 1:1000 for gels or stain solution 3. Ag arose electroporation grade 4. 10 X loading b uf fer (a) 5 mM CaCl 2 (b) 0.4 M mannitol (c) Mak e up solution in solution of PBS 5. DN A molecular weight mark ers 6. Horizontal gel electrophoresis apparatus 7. Gel casting platform 8. Gel combs 9. DC po wer supply Methods Used in Gel Electrophoresis Experiments [7] 1 1. Prepare the gel, using electrophoresis b uf fer and electrophoresis-grade ag arose, see table 4.2, by melting in a micro w a v e o v en or autocla v e, mixing, cooling to 55 C, pouring into a sealed gel casting platform, and inserting the gel comb Ethidium br omide can be added to the g el and electr ophor esis b uf fer at 0.5 g ml 2 1 The methods listed in this section are reproduced v erbatim from Short Protocols in Molecular Biology by Ausubel et al. 1997. 2 CA UTION: Ethidium bromide is a potential carcinogen. W ear glo v es when handling. 29

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2. After the gel has hardened, remo v e the seal from the gel casting platform and withdra w the gel comb Place into an electrophoresis tank containing suf cient electrophoresis b uf fer to co v er the gel mm. 3. Prepare DN A samples with an appropriate amount of 10 x loading b uf fer and load samples into wells with a pipettor Be sure to include appropriate DN A molecular weight mark ers, see gure 12.1 4. Attach leads so that the DN A migrates to the anode or positi v e lead and electrophorese at 1 to 10 V cm of gel. 5. T urn of f the po wer supply when the bromophenol blue dye from the loading b uf fer has migrated a distance judged suf cient for separation of the DN A fragments. 3 6. Photograph a stained gel directly on a UV transilluminator 4 or rst stain with 0.5 g ml ethidium bromide 10 to 30 min, destaining 30 min in w ater if necessary T able 4.2: Ag arose Concentrations for the Separation of DN A Fragments Ag arose (%) Ef fecti v e range of resolution of linear DN A fragments (kbp) 0.5 30 to 1 0.7 12 to 0.8 1.0 10 to 0.5 1.2 7 to 0.4 1.5 3 to 0.2 4.2.2 Document Preparation This dissertation w as typeset, prepared, edited and processed on a Fedora Core 1 machine using L A T E X 2 This document preparation system w as chosen for tw o reasons. The 3 Bromophenol blue comigrates with the 0 : 5 k b fragments. 4 Photograph y of DN A in Ag arose Gels (a) Illuminate the gel with UV light ( > 2500 W cm 2 ) using a UV transilluminator (b) Photograph with a Polaroid MP4 camera using and orange lter (K odak Wratten # 23A) and a clear UV blocking lter (K odak Wratten #2B) with Polaroid type 667 lm (ASA 3000) 30

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rst reason is compatibility with future writings and the control o v er the document preparation process. The second reason is L A T E X 2 as a document preparation system innately possesses a bibliograph y manager B I BT E X, e xtensi v e equation editing capabilities and the ability to produce PDFs, using P D F L A T E X. L A T E X 2 can also be used to create presentations 5 posters, en v elopes and mass mailings as well as all of one' s document needs. Due to its e xpansi v e scaling from add on packages, callable sources, e xhausti v e e xamples, and practicality L A T E X 2 is an all in one system for producing documents from one page letters to tw o hundred page dissertations. 5 Creating presentaitons required a separate add on package called PPo wer4. This package adds pauses to the pdf output of the late x le. PPo wer4 requires the late x ) dvips ) ps2pdf. 31

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5 Modeling Electrophoresis in Fluids 5.1 Introduction There are se v eral computational and logical steps required to plan and ultimately construct an electrophoresis model for uids. This chapter describes the ef forts that focused on force and v elocity calculations. The rst step in modeling the time dependent motion of ionic or char ged molecule concentration in comple x matrices, is to calculate the drift v elocity of dif ferent species in v arious liquids. This is important because mobility is a function of the analyte, the suspensate, and the dri ving force. The analyte' s contrib utions are its char ge and ionic radius, while the suspensate adds drag due to its viscosity The dri ving force is generated by the interaction of the char ged species and the applied electric eld. This chapter e xamines the motion of ions, chemotherapeutic agents, and plasmid DN A in deionized w ater 0.9% saline solution and blood. Drift v elocities, ionic mobilities, and distances tra v eled for common deli v ery conditions are computed for carrier ions and deli v erables. 5.2 Examining Ionic Motion in Solution This section e xplains the motion of a system of potassium ions in deionized, DI, w ater 0.9 % sodium chloride and human blood. This analysis is performed to establish abso32

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lute baseline e xpectations with respect to speed, acceleration, and applied force for small char ged entities. It also accents situations where the h ydrodynamic radius and char ge of the species hea vily inuences the results. The fundamental equations for describing the motion of ions in solution were de v eloped by Lorentz and Stok es. The force on an ion due to the electric eld is described by the Lorentz' s force equation, F = z eE = z e ¢ l while the retarding force on the ion due to the liquid is predicted by Stok e' s la w F f = f s = 6 r s When these tw o forces are in equilibrium, an ion with its gi v en char ge and radius, will be tra v eling at its f astest possible rate for the applied electric eld in that solv ent 1 Solving for the speed of the ion yields s = z eE f = z eE 6 r The viscosities, of the dif ferent solutions are 0.890 cP 1.014 cP and 3.4 cP for DI w ater 0.9 mass % saline and human blood, respecti v ely [19, 108]. An e xample speed calculation for a potassium ion in DI w ater for an electric eld strength of 1500 V cm is presented in equations 5.1 through 5.6. s = z eE f = z eE 6 r = 1 £ 1 : 602 £ 10 ¡ 19 C £ 1500 V cm 6 £ 3 : 14 £ 0 : 890 cP £ 138 pm (5.1) = 1 £ 1 : 602 £ 10 ¡ 19 C £ 1500 J cm C 6 £ 3 : 14 £ 0 : 890 £ 1 £ 10 ¡ 3 k g ms £ 138 pm (5.2) = 1 £ 1 : 602 £ 10 ¡ 19 C £ 1500 k g m 2 s 2 cm C 6 £ 3 : 14 £ 0 : 890 £ 1 £ 10 ¡ 3 k g ms £ 138 pm (5.3) = 1 £ 1 : 602 £ 10 ¡ 19 £ 1500 m 3 s cm 100 cm m 6 £ 3 : 14 £ 0 : 890 £ 1 £ 10 ¡ 3 £ 138 pm 1 £ 10 ¡ 12 m pm (5.4) = 1 £ 1 : 602 £ 10 ¡ 19 £ 1500 £ 100 6 £ 3 : 14 £ 0 : 890 £ 1 £ 10 ¡ 3 £ 138 £ 1 £ 10 ¡ 12 m s (5.5) s = 0 : 0104 m s = 1 : 04 cm s (5.6) 1 When the dri ving force and the retarding force are in equilibrium an object is said to be mo ving at its terminal v elocity the maximum possible v elocity for a gi v en size object, in a gi v e media, and for a gi v en attracti v e or repulsi v e force 33

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F or a 1500 V cm electric eld, the speed v alues of a potassium ion in 0.9 mass % sodium chloride solution and blood are 0.912 cm s and 0.272 cm s respecti v ely F or an applied electric eld of 150 V cm the speed v alues are 0.104 cm s 0.0912 cm s and 0.0272 cm s for DI w ater 0.9 mass % sodium chloride and blood respecti v ely In liquids, a ten fold increase in electric eld produces a ten fold increase in v elocity This is a direct result of the linearity of Stok e' s la w Figure 5.1: Three Dimensional Structure of Bleomycin [215] 5.3 Examining Molecular Motion in Solution Molecular systems add a le v el of comple xity to the modeling because of their lar ger more comple x three dimensional shape. Lar ge comple x molecules may ha v e multiple three dimensional conformations for the same primary structure. Therefore, a molecular motion 34

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model must be tuned to a specic molecule. Bleomycin w as chosen as the molecule to be e xamined because it is commonly used for both in vivo and clinical electrochemotherap y treatments [41, 60, 70, 86, 87]. Bleomycin demonstrates the additional elements and the increased le v el of comple xity in a molecular mobility calculation. The rst step to calculating speeds of three dimensional objects in uids is the calculation of their h ydrodynamic shape. F or this analysis the A form of Bleomycin w as chosen, it has a molecular weight of 1415.5 g mol This decision required the Le wis structure diagram [215] of the molecular structure of the A form of Bleomycin, see gure 5.1. From the Le wis structure, bond angles and lengths were collected from chemistry refer ences [6, 94, 184] and a lookup table [108]. From the literature data an approximate length and width for the molecule w as computed, 2460 pm and 1935 pm respecti v ely the third dimension is based on isomeraization and w as approximated as equal to the width. The prolate spheroid shaped molecule w as then approximated by an equal v olume sphere with a radius of 2200 pm. An equal v olume radius w as chosen because the char ge w as approximated as being at the center of the molecule [6]. The ne xt step w as to calculate an approximate v alence number for this form of bleomycin. Bleomycin is a highly polar molecule b ut the only sections that will af fect the ionic char acteristics of the molecule are at the edges [6]. The central sections are shielded by other atoms. As a molecule becomes lar ger its ionic characteristics are mask ed by its solv ation radius [6, 85]. A v alue of se v en electron char ges w as calculated for Bleomycin' s o v erall ionic char ge. This v alue w as calculated by a v eraging the v alenc y of the molecule' s constituents in solution, e.g ¡ OH ) ¡ O ¡ + ¡ H + and ¡ NH 2 + ¡ H + ) ¡ NH +3 Bleomycin' s h ydro35

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dynamic radius, a v eraged v alenc y the applied electric eld strength and the viscosity of a 0.9 mass % sodium chloride solution were used to compute the approximate speed limit. s = z eE f = z eE 2 r = 7 £ 1 : 602 £ 10 ¡ 19 C £ 1500 V cm 6 £ 3 : 14 £ 1 : 014 cP £ 2200 pm (5.7) = 7 £ 1 : 602 £ 10 ¡ 19 C £ 1500 J cm C 6 £ 3 : 14 £ 0 : 890 £ 1 £ 10 ¡ 3 k g ms £ 2200 pm (5.8) = 7 £ 1 : 602 £ 10 ¡ 19 C £ 1500 k g m 2 s 2 cm C 6 £ 3 : 14 £ 1 : 014 £ 1 £ 10 ¡ 3 k g ms £ 2200 pm (5.9) = 7 £ 1 : 602 £ 10 ¡ 19 £ 1500 m 3 s cm 100 cm m 6 £ 3 : 14 £ 1 : 014 £ 1 £ 10 ¡ 3 £ 2200 pm 1 £ 10 ¡ 12 m pm (5.10) = 7 £ 1 : 602 £ 10 ¡ 19 £ 1500 £ 100 6 £ 3 : 14 £ 1 : 014 £ 1 £ 10 ¡ 3 £ 2200 £ 1 £ 10 ¡ 12 m s (5.11) s = 0 : 00357 m s = 0 : 357 cm s (5.12)The speed of bleomycin in 0.9 mass % sodium chloride is 0.357 cm s for a 1500 V cm eld while the v elocity in blood and DI w ater for the same conditions w ould be 0.107 cm s and 0.407 cm s respecti v ely F or the 150 V cm cases the v alues are 0.0357 cm s 0.0107 cm s and 0.0407 cm s respecti v ely 5.4 Examining a Plasmid' s Motion in Solution Modeling the motion of a plasmid in the dif ferent solutions and applied v oltages is a more comple x task than predicting the theoretical speed for bleomycin. This is due mainly to the size of the molecule and the intricacies of the tertiary structure. The plasmid can be assumed to be supercoiled and 1000 base pairs long. P art of the dif culty in modeling the motion of a plasmid in solution is due to an applied electric eld. T o use Stok e' s la w for 36

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describing the motion of the char ged molecule both the h ydrodynamic radius and molecular char ge are required. In the e xample discussed belo w both the h ydrodynamic radius and the molecular char ge v alues were calculated from kno wn v alues using e xtensions of common methods, see appendix E [108, 181, 184]. The rst step in computing the h ydrodynamic radius is to calculate an approximate v olume for a representati v e purine-p yrimidine base pair One molecule of Deoxyadeonsine is approximately 634 pm by 280 pm and one molecule of Deoxyth ymine is approximately 280 pm by 430 pm [184]. The tw o of them joined together ha v e a footprint that is approximately 560 pm by 634 pm. The second step to computing the h ydrodynamic radius is to approximate the representati v e purine-p yrimidine pair with a 600 pm radius sphere. The third h ydrodynamic radius computation step is to calculate an approximate v olume for the entire plasmid. Assuming that the plasmid will be supercoiled and utilize the minimum ef fecti v e space possibles, the DN A molecule' s v olume can be elucidated with the approximated radius. No w after some simple packing analysis, see appendix E, the plasmid system w as approximated by a sphere with a radius of 6000 pm. This v olume approximation is in agreement with literature v alues and the bleomycin approximation presented abo v e [185]. A radial doubling produces a v olumetric f actor of eight change, and a radial increase of 2.7 causes a 20.3 fold v olume increase. Therefore, the h ydrodynamic sphere of the plasmid is approximately twenty times lar ger than for bleomycin. Computing the v alenc y of the DN A plasmid w as accomplished with the aid of a literature v alue that w as measured and re gressed by e xamining the motion of an attached linear strand of DN A [181]. The con v ersion f actor tak en from literature v alues, w as used to cal37

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culate the char ge of the DN A fragment from the number of base pairs 0 : 06 e ¡ basepair [181] 2 Therefore the v alenc y on a 1000 base pair plasmid is 60 for a char ge of 60 e ¡ A sample calculation is sho w belo w in equations 5.13 through 5.18. s = z eE f = z eE 2 r = 60 £ 1 : 602 £ 10 ¡ 19 C £ 1500 V cm 6 £ 3 : 14 £ 1 : 014 cP £ 6000 pm (5.13) = 60 £ 1 : 602 £ 10 ¡ 19 C £ 1500 J cm C 6 £ 3 : 14 £ 0 : 890 £ 1 £ 10 ¡ 3 k g ms £ 6000 pm (5.14) = 60 £ 1 : 602 £ 10 ¡ 19 C £ 1500 k g m 2 s 2 cm C 6 £ 3 : 14 £ 1 : 014 £ 1 £ 10 ¡ 3 k g ms £ 6000 pm (5.15) = 60 £ 1 : 602 £ 10 ¡ 19 £ 1500 m 3 s cm 100 cm m 6 £ 3 : 14 £ 1 : 014 £ 1 £ 10 ¡ 3 £ 6000 pm 1 £ 10 ¡ 12 m pm (5.16) = 60 £ 1 : 602 £ 10 ¡ 19 £ 1500 £ 100 6 £ 3 : 14 £ 1 : 014 £ 1 £ 10 ¡ 3 £ 6000 £ 1 £ 10 ¡ 12 m s (5.17) s = 0 : 0126 m s = 1 : 26 cm s (5.18) The speed of the plasmid, as de v eloped in equations 5.13 through 5.18, in 0.9 mass % sodium chloride is 1.26 cm s for a 1500 V cm eld while the v elocity in blood and DI w ater for the same conditions are 0.376 cm s and 1.44 cm s F or the 150 V cm cases the v alues are 0.126 cm s 0.0376 cm s and 0.144 cm s respecti v ely T able 5.1 summarizes the drift v elocity calculation results for ions, bleomycin and plasmids in DI w ater 0.9% NaCl and blood. Ionic mobility u is the modeling parameter typically used when comparing the motion of ions or molecules in uids, it is the v elocity of that ion or molecule in a gi v en uid times the char ged species' radius di vided by the applied electric eld. The assumptions in calculating ionic mobility are steady state dri v en and include constant viscosity of the 2 The constant e ¡ in the units of this con v ersion f actor is the char ge of an electron, the v alue used in this document w as 1 : 602 £ 10 ¡ 19 C oul ombs 38

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T able 5.1: The Drift V elocities of V arious Substances Used in Molecular Deli v ery Substance and V oltage DI W ater Saline Blood K + @ 1500 V cm 1.04 cm s 0.912 cm s 0.272 cm s K + @ 150 V cm 0.104 cm s 0.0912 cm s 0.0272 cm s Bleomycin @ 1500 V cm 0.406 cm s 0.357 cm s 0.107 cm s Bleomycin @ 150 V cm 0.0406 cm s 0.0357 cm s 0.0107 cm s 1000 bp Plasmid @ 1500 V cm 1.44 cm s 1.26 cm s 0.376 cm s 1000 bp Plasmid @ 150 V cm 0.144 cm s 0.126 cm s 0.0376 cm s solution and constant v elocity of the ion or molecule. The ionic mobilities were calculated from the data listed in table 5.1 and are presented in table 5.3. The calculations of this section included blood as one of the suspensors to help e xplain through comparison what happens in an in vivo situation. This computation assumes that the interstitial uid has the same or similar viscosity as blood. After the applied electric eld is e xtinguished the cells will slo wly come to a stop and then be af fected solely by dif fusion, Bro wnian motion, and clearance. A recent paper in the journal Gene Therap y [216] e xamined the e xact case of the electrophoretic motion of DN A in tissue. The authors concluded that electrophoretic motion is 5 to 6 orders of magnitude f aster than straight dif fusion for ions and char ged molecules. 39

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T able 5.2: Distances T ra v eled for Common Field Conditions Substance, V oltage and T ime DI W ater Saline Blood K + @ 1500 V cm for 100 s 1.04 m 0.912 m 0.272 m K + @ 150 V cm for 150 s 0.156 m 0.137 m 0.0408 m Bleomycin @ 1500 V cm for 100 s 0.406 m 0.357 m 0.119 m Bleomycin @ 150 V cm for 150 s 0.0609 m 0.0536 m 0.0176 m 1000 bp Plasmid @ 1500 V cm for 100 s 1.44 m 1.26 m 0.376 m 1000 bp Plasmid @ 150 V cm for 150 s 0.216 m 0.189 m 0.0564 m This means that the greatest distance tra v eled for molecules lik e DN A and Bleomycin is during an electrophoretic pulse. 5.5 V elocity and Acceleration of Char ged Species in Electric Fields Kno wing the nal v elocity of char ged species in a liquid is important since it can be used to calculate the absolute distance tra v eled. Alternati v ely the instantaneous v elocity and acceleration as a function of time are v ery useful when e xamining the motion of char ged species for short time interv als. The computation of the acceleration and v elocity as a function of time w as accomplished utilizing the rst principle methods outlined in equations 5.19 and 5.23. 40

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T able 5.3: Ionic Mobilities of V arious Substances Used in Molecular Deli v ery Substance and V oltage DI W ater Saline Blood K + 6.93 £ 10 ¡ 4 cm 2 s V 6.08 £ 10 ¡ 4 cm 2 s V 1.81 £ 10 ¡ 4 cm 2 s V Bleomycin 2.76 £ 10 ¡ 4 cm 2 s V 2.38 £ 10 ¡ 4 cm 2 s V 0.713 £ 10 ¡ 4 cm 2 s V 1000 bp Plasmid 9.60 £ 10 ¡ 4 cm 2 s V 8.40 £ 10 ¡ 4 cm 2 s V 2.51 £ 10 ¡ 4 cm 2 s V F = F e ¡ F f = ma (5.19) F = z e E ¡ 6 r s = ma (5.20) s = s o + a t (5.21) a = z e E 6 r t + m (5.22) s = s o + z e E 6 r t + m t (5.23) Solving for the acceleration and speed as a function of time produces a system of tw o independent equations, equations 5.22 and 5.23, with three unkno wns, speed, acceleration and time. This system of equations w as solv ed by setting the initial speed, s o to zero and selecting a range of times from 1 £ 10 ¡ 16 to 1 £ 10 ¡ 13 seconds, see appendix F for the calculation technique emplo yed. The results of these calculations are listed in table 5.4. The interaction between an ion' s acceleration and speed can be seen in gure 5.2. The maximum speed that the K + ion w ould achie v e for a 1500 V cm 100 s pulse is 1 : 038 £ 10 ¡ 2 m s 41

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and it w ould reach that speed in 1 : 23 £ 10 ¡ 13 seconds or so, see table 5.4 3 on page 45 and gure 5.2 belo w 0 5e+10 1e+11 1.5e+11 2e+11 2.5e+11 3e+11 3.5e+11 4e+11 1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 0 0.002 0.004 0.006 0.008 0.01 0.012 Acceleration (m/s2) Speed (m/s)Time (s) Acceleration Speed Figure 5.2: Plot of Acceleration and Speed in 0.9% NaCl Solution From equations 5.22 and 5.23, it can be seen that the greatest acceleration is between time equal to zero and time equal to zero +¢ t o which ph ysically mak es sense because that is also when the greatest change in speed also tak es place. T able 5.4 lists the ion' s v elocity prole from an initial speed of 0 m s and slo wly increasing to a maximum v alue 3 In table 5.4 a lar ge number of digits are e xpressed after 4 : 96 £ 10 ¡ 14 seconds, this is to demonstrate the ef fect of the acceleration term on the v elocity term. The ph ysical signicance is limited to 3 signicant gures. 42

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of 1 : 038 £ 10 ¡ 2 m s see table 5.4. The ef fect of the mass of the ion or molecule on the acceleration or v elocity is minimal e xcept when m 6 r t Potassium ions in a 0.9% NaCl solution reach their drift v elocity in 10 ns. This type of analysis could be performed for other deli v erables and the pulse strength and duration could be tuned per molecule. 5.6 Summary Modeling electrophoresis in uids can be accomplished using Stok e' s la w and Lorentz' s force equation. This process requires the h ydrodynamic radius and v alenc y of the char ged species and the applied electric eld strength. The h ydrodynamic radii and v alenc y of ions are listed in v arious books b ut molecules are a more dif cult subject. The three dimensional structure of molecules requires that their structure be computed from the constituent elements and that a representati v e h ydrodynamic radius be approximated. The v alenc y of molecules can be approximated by a v eraging the v alenc y of the end groups. F or the demonstration e xamples presented here, the h ydrodynamic radius and v alenc y for Potassium w as tak en from literature v alues while the h ydrodynamic radius for bleomycin and a one kilobasepair plasmid were calculated. From these v alues the v elocities for common deli v erables in typical carrier uids were computed for high eld and lo w eld electroporation/electrophoresis signature components, see table 5.1. From the drift v elocities, the distances tra v elled for common deli v ery conditions were calculated, see table 5.2. From the distances tra v elled per unit time, ionic mobilities were calculated, see table 5.3. Ionic mobilities were reported because the y are time and v elocity independant for ions tra v eling in steady state, and could be used for comparision ag ainst literature v alues. Since 43

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ionic mobilities are only useful after the ions ha v e reached steady state, a time series analysis on potassium ions w as performed to understand the motion of ions 4 after the onset of the electric eld. The results for this time series analysis are presented in table 5.4. These calcuations and these calculated v alues are useful to understand the motion of div erse molecules in v arious carrier uids. The w ork and results from this chapter ha v e been presented and published [75, 77]. 4 As well as create a method for other char ged species, utilizing the method of appendix F and v alues for r E z m and t 44

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T able 5.4: V elocity and Acceleration for K + Ions in a 0.9 Mass % NaCl Solution T ime (sec) Acceleration ¡ m s 2 ¢ V elocity ¡ m s ¢ 0 3.700 £ 10 11 0 1.0 £ 10 ¡ 16 3.687 £ 10 11 3.687 £ 10 ¡ 5 3.0 £ 10 ¡ 16 3.648 £ 10 11 1.463 £ 10 ¡ 4 6.0 £ 10 ¡ 16 3.571 £ 10 11 3.605 £ 10 ¡ 4 1.05 £ 10 ¡ 15 3.273 £ 10 11 1.836 £ 10 ¡ 3 3.6 £ 10 ¡ 15 2.454 £ 10 11 3.495 £ 10 ¡ 3 6.6 £ 10 ¡ 15 1.431 £ 10 11 6.364 £ 10 ¡ 3 1.05 £ 10 ¡ 14 6.154 £ 10 10 8.653 £ 10 ¡ 3 3.0 £ 10 ¡ 14 2.795 £ 10 8 1.0372 £ 10 ¡ 2 4.96 £ 10 ¡ 14 5.337 £ 10 5 1.037973 ¤ £ 10 ¡ 2 7.03 £ 10 ¡ 14 5.152 £ 10 2 1.037975 £ 10 ¡ 2 8.2 £ 10 ¡ 14 9.541 £ 10 0 1.037975 £ 10 ¡ 2 9.03 £ 10 ¡ 14 5.557 £ 10 ¡ 1 1.037975 £ 10 ¡ 2 1.04 £ 10 ¡ 13 5.978 £ 10 ¡ 3 1.037975 £ 10 ¡ 2 1.23 £ 10 ¡ 13 9.0531 £ 10 ¡ 6 1.037975 £ 10 ¡ 2 ¤ The number of signicant gures increases to sho w the reduction in ef fect of acceleration on speed, these v alues are not stated as signicant and are only included for pedagogical purposes. 45

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6 Contrasting In V ivo and In V itr o EFMDGD Flo w Systems 6.1 Introduction Chapter 5 described the motion of ions, molecules and plasmids in electric elds in liquids. While the calculations and kno wledge are useful and interesting, the motion described is not entirely representati v e of electric eld mediated drug and gene deli v ery EFMDGD, conditions [68, 89]. EFMDGD is performed in both in vivo and in vitr o e xperimental v enues and while in vivo and in vitr o e xperimental procedures for EFMDGD are dif ferent the methods of action are primarily the same, with electrophoresis being the dri ving force for deli v ery Electrophoresis is the motion of char ged species in an applied electric eld [193]. This chapter will re vie w the dif ferent v enues for electrophoresis in EFMDGD and e xamine the ef fect of v oid fraction on EFMDGD. 6.2 Importance of Electrophoresis on Electroporation Understanding the equations that go v ern the motion of ions and molecules in suspension yields insight into ho w to control their motion with electric elds. Examination of the in vitr o case produces the greatest result for the least intellectual ener gy e xpenditure since it is a simpler case. In this type of e xperiment, cells are placed into a cuv ette with an analyte that is selecti v ely e xcluded by the cell membrane. The cell-analyte solution is then e xposed 46

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to a single, or multiple, poration pulse(s) and the measurement of analyte uptak e be gins. In vitr o e xperiments are used to measure the relati v e ef fecti v eness of a treatment before mo ving to a more e xpensi v e model, e.g. to test the ef cac y of a drug-electric eld combination, to test anticancer drugs ag ainst dif ferent cell lines [88] or to measure the rate of uptak e of small uorescent or radioacti v e tracer molecules [74]. These studies are typically done with the analyte in e xcess, thereby maintaining rst order kinetics 1 The positi v es of in vitr o e xperiments lie in the f act that the y can be done easily ine xpensi v ely and rapidly The ne g ati v es include the f act that the resultant v alues are not al w ays indicati v e of in vivo systems. T w o possible reasons wh y in vitr o systems are not e xact models of in vivo systems for electroporation e xperiments are packing f actor and tissue structure. P acking f actor is dened as the ratio of the sum of cell v olumes to the system v olume [5]. In cuv ette e xperiments, with 5 £ 10 6 cel l s mL the packing f actor of the cells is small and highly cell diameter dependent. F or mammalian cells with an a v erage cell diameter of 50 m [3], the total v olume tak en up by cells is 327 L, see table 6.1. As the cell diameter decreases the v oid fraction increases rapidly see table 6.1. F or diameters less than 32 m the cells comprise only 10% of the system v olume. An increased v oid fraction reduces cell-cell 1 This is actually a pseudo-rst order system because holding the concentration of analyte in great e xcess remo v es it from the rate equation by holding it constant.If [ anal y te ] is constant, it can be di vided out of both sides of the second order equation, see equation 6.1 and the result is equation 6.2, a rst order equation with respect to concentration of cells, [ cel l s ] (6.1) Second order system r ate = k [ cel l s ][ anal y te ] (6.2) First order system r ate = k [ cel l s ] 47

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T able 6.1: V olume and V oid Fraction for Cells of Dif ferent Diameters [88] Diameter V olume of 5E6 cells (mL) % V oid Fraction of 1 mL Suspension 15 0.0088 99.12 20 0.0209 97.91 25 0.0409 95.91 30 0.0706 92.94 35 0.1123 88.77 40 0.1676 83.24 45 0.2385 76.15 50 0.3273 67.27 interaction which in turn af fects in solution cell rotation and ion o w past the cell. Both of these f actors reduce residence time for the ions contacting the cell membrane. A 10% packing f actor is v ery small when compared with an in vivo system. On the or g anism le v el, the human body is 70% intracellular space and 30% e xtracellular space [123]. Therefore, a much lar ger fraction of the body is cells rather than free space for the ions to roam [123]. While this dif ference does not af fect the applied electric eld, it af fects the induced transmembrane potential, see appendix A, the current pathw ays and the motion of the molecule being deli v ered. In cuv ette e xperiments the ions are free to mo v e as the y please and produce a pseudo-homogeneous eld. Ho we v er in tissue the current must o w 48

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in either the intracelluar or e xtracellular uid via the intracellular or e xtracellular pathw ays respecti v ely If cell diameter w as the only f actor af fecting electroporation then selecting the requisite electric eld strength w ould be simple. Figure 6.1 sho ws that f actors other than cell diameter play a lar ge role in permeabilizing cell membranes. Another possible f actor is tissue structure. A closer e xamination of a fe w of the cell types in gure 6.1 yields insight into the tissue structure parameter Le wis Lung 2 cells are loosely adherent to asks, yet typically suspend in clumps while Ishika w a and B16 cells are strongly adherent to ask and typically suspend as single cells. A single cell radii is small b ut a group of cells may act lik e a lar ge single cell, thereby reducing required electric eld. Cell membrane components may also af fect poration. Zahrof f et al ha v e associated electroporation with cell membrane collagen content [216]. Clumping of cells or cell membrane components may help to e xplain the random scatter shape of the eld intensity plot sho wn in gure 6.1 and the tortuosity of the pathw ays in undissociated cell clumps will increase the ef fect of an applied electric eld on the participating cells 2 The tortuosity and porosity of tissue may also e xplain wh y rotating the electric eld between pulses produces increased poration for in vivo electroporation treatments [58, 59, 69]. In in vivo e xperiments, the electric eld is in one direction for a gi v en period of time. Rotating the electric eld mobilizes the ions at a right angle to the pre vious path of tra v el. Changing the eld direction causes the ions to ha v e a more uniform interaction around 2 Depending on the size of the undissociated cell clump, the applied electric eld ef fect could range from full single f aced eld, for the cells on the outside of the clump in line of sight with an electrode, to no eld felt, for cells on the inside of the clump. 49

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700 800 900 1000 1100 1200 1300 10 11 12 13 14 15 16 17 18 19 20 Field Intensity (V/cm)Cell Diameter ( m m) Capan-2 LN1 T47D Lewis Lung 2 N1-S1 B16 Ishikawa Figure 6.1: Relationship Between Cell Diameter and Electroporation Field Intensity [88] each cell. The rotating eld is similar to turning the cell in a homogeneous electric eld. Rotating the eld also allo ws the greatest number of cells to e xperience the greatest local ef fect while promoting tw o dimensional tra v el of deli v erables through the tissue. Three dimensional tra v el of the ions is caused by local interaction with other ions and the tortuosity of the tissue itself. Ions in an electric eld continue to tra v el in a straight line, b uilding up speed until the y achie v e terminal v elocity in that media or until the y interact with the oppositely char ged electrode, ion or a cell [135, 177, 213]. Once the eld is switched of f the ions will continue 50

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to tra v el in the same direction until their momentum runs out via the frictional drag of the media or intersections with cells. The ions will then return to their lo west ener gy state if not acted on by an outside force. 6.3 Molecular Deli v ery Control of electrophoretic molecular deli v ery is done via control of the applied electric eld. While homogeneity of the electric eld produces positi v e results for electroporation, electrophoresis in tissue prots more from an inhomogeneous eld. Ionic molecules may be “dri v en” an ywhere in the tissue between the tw o electrodes by altering the direction of the electric eld. Alternati v ely a neutral dipolar compound could be dri v en via an oscillating electric eld. The electric eld w ould apply a torque to the dipole and gi v en the correct combination, typically a cosine function, the dipolar compound w ould seemingly cartwheel along in the direction of tra v el. A joint electrophoresis-electroporation protocol that prots from the syner gy of the tw o techniques w ould not only impro v e deli v ery to indi vidual cells b ut also distrib ution of the analyte in the tissue. 6.4 Summary While drift v elocities and ionic mobilities are important in liquid systems, their relev ance diminishes when applied to electric eld mediated drug and gene deli v ery EFMDGD, conditions and models. This chapter discussed the ef fects of cell radii, cell membrane composition and cell adherence as possible f actors inuencing the minimum electroporation 51

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signature 3 parameters. The v oid fraction of the cells in tissue or an electroporation cuv ette w as e xamined and proposed as one of the possible reasons for the dif ference between ef fecti v e in vivo and in vitr o electroporation signatures. The last topic presented in this chapter focused on the control of the molecular deli v ery by an applied electric eld and has intimated the dependence of EFMDGD on the applicator F or deli v ery of lar ge molecules and plasmids the v oltage dif ference imposed across the applicator plays a signicant role [58]. This chapter briey illustrates wh y the deli v ery of drugs and genes to cells by electric elds is a comple x problem dependent on the applied electric eld component of the electroporation signature, applicator design, tissue pathw ays, and cell membrane components. Chapter 7 focuses on tw o common electroporation applicators, an emphasis is placed on the impact of applicator geometry and conguration on the induced electric eld. 3 Electroporation signatures are dened as the pulse parameters utilized to induce electroporation. The y include b ut are not limited to pulse duration, width, shape, number of pulses and time between pulses. 52

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7 Common Electroporation Electrodes 7.1 Introduction The dissertation premise that a “ne w model” is needed that e xamines the electric eld induced motion of ions, molecules, and DN A fragments w as introduced in chapter 3. This electric eld mediated o w eld model, EFMFFM, be gins with an inducti v e understanding of the Lorentz force dri ving ions, Stok e' s la w acting to retard the o w of particles though a viscous uid, and Maxwell' s equation that correlates the di v er gence of the electric ux density v ector D 1 with the v olume char ge density r ¢ D = This axiom of Maxwell connects the electric potential scalar with the electric eld v ector E via the gradient operator E = ¡r The di v er gence of the electric eld, r ¢ E = ¡r 2 = o usually kno w as Poisson' s equation pro vides information about the v olume char ge density of the re gion af fected by the applied electric eld 2 Chapters 5 and 6 introduced the concept that the components of an electroporationelectrophoretic signature, the applied electric eld strength, duration, and shape, af fect the v elocity and direction of the deli v erables in liquids, cuv ettes, and tissues. This chapter 1 The electric ux density v ector has units of C m 2 A constituti v e relation is used to con v ert between D and E D = E The units for E are V m and the units for are C 2 N m 2 The units of D = E are C m 2 = C 2 N m 2 V m since a C = J V and a J = N m the units w ork out. 2 A special case of Poisson' s equation is Laplace' s equation r 2 = 0 this equation describes re gions where there is no char ge distrib ution 53

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introduces and describes the shape and use of tw o of the most commonly used electrodes in clinical and laboratory treatments, the parallel plate and the six needle array [58, 65, 69, 72, 198]. This chapter also e xpands on the idea that applicator geometry and conguration inuence the shape of the eld produced by the applied potential. 7.2 Electroporation Electrode Background The primary electrodes used for this research were parallel plate caliper electrodes and needle electrodes. The parallel plate and 4 needle, needle electrodes were chosen because the y are commonly used for in vivo electroporation e xperiments and clinical treatments [56, 73, 86, 87, 173]. The plates of the parallel plate caliper electrode, see gure 7.1-a [58], are tw o stainless steel electrodes af x ed to the arms of a plastic V ernier caliper The tw o (a) (b) Figure 7.1: Photographs of the P arallel Plate and Six Needle Electrodes 54

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electrodes can then be set to a specic separation for easy calculation of the v oltage required for a chosen applied electric eld strength, V cm The parallel plate electrodes were some of the rst tissue electrodes used for animal w ork because of their non-in v asi v e nature and the homogeneous eld that the y produce. The 4 needle, needle array conguration w as chosen for its simplicity and ease of e xpansion to an alternati v e design, e.g. 6 needle array The electroporation research group at USF uses a 6 needle applicator 3 see gure 7.1-b, b ut customarily only acti v ate four needle subgroups, in a square conguration, at a time. This is wh y a 4 needle electrode model is representati v e [58, 59, 144]. T w o graphical representations of this applicator are sho w in gure 7.2. Figure 7.2-a is a “do wn the barrel look” at the applicator' s needle electrodes. It displays the center to center distance between the dif ferent needles. While gure 7.2-b depicts the needle array rotated 40 from the horizontal and 80 from the v ertical. 1 cm 6 1 cm ? (a) 6 ? 1 cm (b) Figure 7.2: F our Needle, Needle Array 3 In addition to other applicators. 55

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7.3 Summary The electric eld is a function of the geometry and conguration of the applicator T w o common electroporation applicators are the parallel plate and multi-needle electrodes. These electrode applicator congurations are currently used in both laboratory and clinical settings [59– 61, 70– 72, 119, 196]. The induced electric eld from these electrodes has been modeled e xtensi v ely in electroporation literature [37, 38, 51, 101, 157, 203, 204, 208]. While the resultant electric eld from these electroporation/electrophoresis applicators has been modeled e xtensi v ely besides calculating induced transmembrane v oltages v ery little has been done with the results of these calculations. Kno wn electric eld patterns were utilized in modeling the electric eld for the distrib ution of ions and plasmid DN A in the ne xt fe w chapters. 56

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P art II 57

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8 Electric Field Mediated Flo w Field Model P art I of the document spans chapters 2 to 7 and has dealt with the electric eld aspect of the electric eld mediated o w eld model, EFMFFM. It has pro vided a re vie w of the electroporation literature and ascertained an area or deeper e xamination, specically the motion of ions, drugs, and DN A in tissues. It also described the motion of ions, drugs, and DN A fragments in viscous liquid systems. The e xamination of the motion of deli v erables in electric elds w as e xamined in detail at dif ferent v oltages and suspensories. Ionic mobilities, drift v elocities, and distances tra v eled for common eld conditions were calculated for the suspensates in the diluents. Ne xt, a time series analysis of the acceleration and speed of a K + ion in a 0.9% NaCl solution w as performed for tw o reasons, rst to ascertain the eld duration required for steady state speed of ions in an applied electric eld and second to understand the process and to create and implement adaptable computer code for use in other systems, e.g. plasmids in b uf fer drugs in tissues. In chapter 6 the dif ferences between in vitr o and in vivo systems were e xplored. The importance of electrophoresis on electroporation w as introduced. The ef fect of cell density w as e xamined and the common packing f actors for dif ferent cells in in vitr o electroporation cuv ette e xperiments were described. The dif ference between a common cuv ette e xperiment' s packing f actor 10%, and a human w as compared. The nal result w as that in 58

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tissue ions ha v e much less room to tra v el than in cuv ettes. This may partially account for the reduced required in vivo electroporation v oltages when compared to in vitr o v alues. Chapter 7 described tw o common electroporation electrodes and the electric eld shapes created by an applied potential. It described ho w the electric eld shape and density is a direct cause of applicator shape. A brief description of ho w the modeling of the electric elds from the dif ferent electrodes w as also included. In part II of this document the o w eld model, FFM, is introduced and its inte gral and requisite elements, and their e xtension into an EFMFFM are described. Chapter 9 describes the creation of a simple, scalable tissue model for e xamining the motion of a single char ged species. The o w eld tissue model, FFTM, is of immediate interest because of its scalability and e xtensibility to a wide v ariety of electric eld mediated deli v ery problems. The FFTM consists of 5 interacting elements that combine to describe tw o dimensional o w of ions through a tissue. Chapter 10 contains the main results and conclusions for the FFTM. Appendices I to O contain all of the simulation results in graphical form, supplemental information on the model, re gressed parameters, and the accompan ying computer code. Chapters 11 to 13 describe the background, creation, and analysis of a no v el application of a o w eld model applied to gel electrophoresis. Appendices P to Z contain the simulation results in graphical form, supplemental information on the model, re gressed parameters, and the computer code for the gel electrophoresis o w eld model. 59

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9 Flo w Field T issue Model 9.1 Introduction Modeling the motion of ions in a tar get tissue is more dif cult than modeling the motion of ions in solution. Motion in all directions is allo w able for ions in a liquid through either Fickian dif fusion, Bro wnian motion, or the inuence of an applied electric eld, con v ersely in a tissue ions are more constrained to mo v e in the e xtracellular space around cells, refered to as e xtracellular channels, or in the intracellular channels of cells. Modeling the motion of ions in the e xtracellular and intracellular channels is one of the unique features of the o w eld tissue model, FFTM, a primary component of the electric eld mediated o w eld model, EFMFFM, and this process is described in this chapter The o w eld tissue model, FFTM, component of the EFMFFM is composed of v e interacting se gments. These v e elements include an odd number on a side, square element self descripti v e array SD A, a set of randomly generated o w eld v alues, an initial concentration prole of analyte, a set of o w rules, and a set of force v alues associated with the applied electric eld. The SD A is self descripti v e because it con v eniently packages the other elements of the FFTM into one array while pro viding a surf ace for the simulation to time e v olv e on. The FFTM can be e xtended to study a v ariety of ions and molecules in 60

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di v erse matrices. The SD A is described briey belo w while the other four elements of the FFTM are then e xamined in detail. 9.2 Odd Number on a Side Square Array The array for the FFTM is the most important element of the model. It is based on an inhomogeneous, one square centimeter slice of tissue 1 This tissue block w as assumed to be of a homogeneous tissue type that had a random o w pattern between the dif ferent cells [64]. By imposing random o w obstructions between the dif ferent array nodes the analyte is randomly distrib uted throughout the tissue yet in the direction of the applied electric eld. The e xample array geometry sho w in table 9.1 is an ele v en node by ele v en node square, where each node is a model focal point. This format w as selected for tw o reasons, rst for a sparse array the creation of the o w rules w as simplied and could be coded easier and second an odd number per side array f acilitates the selection of a true center see table 9.1. A true center is requisite to the creation of balanced rules and to pro vide an area for the initial concentration to be centered around. It is also useful when modeling intra-tumor injection electroporation e xperiments in tissue since the injection sight is commonly chosen as the re gistration mark for the placement of the electroporation electric eld applicators, be the y needle or parallel plate e xperiments. In table 9.1 the center point is box ed and bold for visual ease. The array center point can also be used for orienting the initial concentration prole. 1 This geometry and area is typical of tissue treatment areas for testing electrode designs [58]. 61

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T able 9.1: T issue Model Lattice 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 110 111 112 113 114 115 116 117 118 119 120 121 9.3 Randomly Generated Flo w Field V alues The random o w v alues alluded to in section 9.2 represent the possible normalized o w in the forw ard, re v erse, right, or left directions 2 and the order of o w choice from each node 3 The random o w v alues, RFVs, and directions, RFDs, are the tw o components that describe the o w eld and are generated using tw o P ark and Miller random number generators 4 PMRNGs, that ranged from 0 to 1 and right or left respecti v ely 5 [158]. A v alue of 0 related to completely obstructed o w while a v alue of 1 corresponded to completely 2 The possible o w directions for a Cartesian o w eld are forw ard, re v erse, right, or left. The sum of the o w in these four directions adds up to one. 3 The order of o w choice can be an y of the directions are chosen rst, follo wed by an y other direction e xcept the rst one chosen, there are only tw o choices for the third o w direction and the fourth o w direction is x ed after the random selection of the other three. 4 P ark and Miller random number generators were chosen for this application because the y are rob ust and initial seed independent. PMRNG pass all of the tests for random number generators and ha v e accumulated a lar ge amount of successful use [158]. 5 Flo w in direction of the electric eld w as selected rst and designated as the forw ard direction, o w ag ainst the electric eld w as not allo wed, the last tw o allo w able directions were right and left. 62

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unobstructed o w Obstructions act to impede o w and cause the streams of ions to di v er ge partially or wholly in another direction. In this manner a tissue block is treated as re gions with a v eraged o w parameters rather than indi vidual cells. The re gions could be shrunk do wn to an y realistic size desired, e.g. indi vidual cells or portions of cells b ut for e xploring transport through tissue that resolution is not required. F V 1 = Flo w V alue 1 R V = Random V alue S F D = Second Flo w Direction F V 2 = Flo w V alue 2 T F D = Third Flo w Direction F V 3 = Flo w V alue 3 First Flo w Direction Along E-Field Flo w V alue 1 1 > R V > 0 F V 1 = R V SFD LR Second Flo w Direction S F D = Lef t Second Flo w Direction S F D = R ig ht Flo w V alue 2 F V 2 = R V ¢ (1 ¡ F V 1) Flo w V alue 2 Third Flo w Direction T F D = R ig ht Third Flo w Direction T F D = Lef t Flo w V alue 3 F V 3 = 1 ¡ F V 1 ¡ F V 2 Flo w V alue 3 Figure 9.1: Flo w Field V alue and Direction Algorithm The FFTM w orks by an iterati v e repeated selection of a random o w direction, RFD, follo wed by generating a random o w v alue, RFV The algorithm for generating the RFVs and RFDs used in the FFTM is presented in gure 9.1. The rst selection of a RFD for a node w as al w ays along the in-eld direction, or direction A, see gure 9.2. This w as designated the forw ard direction, and a v alue, F V 1 from 0 to 1 w as randomly generated and multiplied by the concentration of the node. The second RFD, S F D w as randomly chosen, see the conditional diamond in gure 9.1, the choices were right or left, and are sho wn in gure 9.2 as C or E respecti v ely The second RFV w as then randomly generated and multiplied by the quantity one minus rst o w v alue, F V 2 = R V ¢ (1 ¡ F V 1) for 63

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a possible v alue between 0 and (1 ¡ F V 1) The v alue for the third o w direction is the unused direction 6 either right or left, S F D and the the third o w v alue is the residual, F V 3 = 1 ¡ F V 1 ¡ F V 2 C E BD A F E Field Figure 9.2: Random Flo w Direction Axis Labels F or describing the motion of ions out of each node, the o w w as dened to be 100% of the total v alue of o w into that node 7 Therefore, for each computational iteration the contents of each node of the array w as emptied into the three surrounding nodes and then w as lled to some de gree from the three nodes in the F C and E directions. When the electric eld mediated o w eld model, EFMFFM, is used to model 2-D o w eld situations, the other three o w directions, re v erse, F up, B, and do wn, D. 9.4 Initial Concentration Prole An e xample initial concentration prole for a 2-D o w eld is illustrated in gure 9.3. The analyte is constrained as a 3 node on a side, square of equal concentration v alue and is placed in the middle of the model tissue. This concentration distrib ution is the best ap6 The road less tra v eled [106] 7 The o w into a node equals the o w e xiting a node equals the v olume of a node. This means that each node empties and lls each iteration. 64

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proximation of a typical single injection deli v erable prole possible within a sparse array A deli v erable prole can easily be changed into an y shape desired limited only by the neness of the mesh. A circular initial concentration distrib ution with tapering concentration v alues from the center is closer to the usual clinical or in vivo system where the deli v erable w ould be injected into the tissue.Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure 9.3: Initial Concentration Prole 9.5 T issue Flo w Rules The tissue o w rules adopted for the mo v ement of analyte from node to node simplify the solution of equation 3.2, from page 25, by mathematically describing the v elocity and direction of the ionic o w through the nodes that mak e up the h ypothetical tissue. F ol65

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lo wing the time e v olving motion of the analyte is accomplished through nodal analysis techniques, commonly used in the numerical analysis of heat and mass transport, and the Gauss-Seidel iterati v e method [32, 147]. Nodal analysis is performed by calculating steadystate pictures of the system under in v estig ation [147]. This type of numerical solution is well tuned for solving electrophoresis type situations where deli v erables tra v el slo wly predictably and their pathw ays are primarily unidirectional with secondary tra v el stemming from o w impediments. (3.2) j cs ( x; y ; z ) = ¡ D cs r c cs ( x; y ; z ) + c cs ( x; y ; z ) D cs q E applied k b T Equation 3.2 summarizes the ux, j cs calculation of char ged species mo v ement through a node as arranged in table 9.1. The rst half of equation 3.2 is an e xtension of Fick' s rst la w of dif fusion [15, 78]. Fick' s rst la w j D cs ( x; y ; z ) = ¡ D cs r c cs ( x; y ; z ) characterizes the random w alk of an indi vidual particle in a suspensory in the concentration gradient of the char ged species, cs 8 If the analyte is homogeneously distrib uted in the suspensate then the random w alk of the indi vidual particles maintains the systemic homogeneity of the distrib ution, local homogeneity may temporarily be o v ercome. The second half of equation 3.2 is deri v ed from the denition of mobility 9 see equation 9.1, which correlates the a v erage v elocity 10 v associated with a particle with the applied force times the particle' s absolute mobility [48, 105]. Char ged particles in an electric 8 Gradient in this instance is being dened as, “The rate at which a ph ysical quantity such as temperature or pressure, increases or decreases relati v e to change in a gi v en v ariable, especially distance” [137] 9 Einstein link ed ionic mobility with Bro wnian motion in 1905 [48] 10 The bar o v er the v means use the a v erage instantaneous v alue. 66

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eld are described by Lorentz' s la w 11 see equation 9.2, which correlates the force on a char ged species, F with its char ge times the electric eld that it is e xposed to. Substituting equation 9.2 into equation 9.1 yields an equation relating the v elocity of an ion in an electric eld to that ions mobility see equation 9.3. Substituting Einstein' s denition of mobility M = D AB k b T into equation 9.1 and multiplying by the concentration of the char ged species, C cs under e xamination yields the second term of equation 3.2, which is the ux of that species due to an applied electric eld, see equation 9.4. Summing the tw o ux es yields the ux, j E cs ( x; y ; z ) due to dif fusion and the applied electric eld of a concentration of char ged particle. v = F M (9.1) F = q E (9.2) v = q E M (9.3) j Ecs ( x; y ; z ) = c cs ( x; y ; z ) D cs q E appl ied k b T (9.4) The computation of the o w pattern through the nodes w as accomplished in a tw o fold manner rst the number of re gions w as optimized and then the equations describing those re gions were subsequently optimized. After an iterati v e process the array w as modeled using six re gions, see gure 9.4. The six equations were created for the six re gions by e xamining where the ions w ould o w Each re gion has an equation that describes the to and from motion of deli v erables. 11 Lorentz' s la w is sho wn here ignoring the magnetic eld component, see equation 9.2 67

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5 2 First ro w ? First Column 1 4 ? Last Column 3 6 Section 1: i 11 Section 2: i > 11 Figure 9.4: Array Described by Six Re gions The six equations that describe the o w pattern use the RFDs and RFVs described in section 9.3 and are e xpressed in pseudo-code format for simplicity see equations 9.5 through 9.10. Details for the de v elopment of those equations follo w in sections 9.5.1 and 9.5.2. The nodes that constitute the array were dened to be of equal v olumes with complete mixing, all of the equations are related via concentration as a function of position and time, conc[i,t]. Positions in the array are a function of i because for the small array a unidimensional system with an e xternal accounting method simplied computation time and memory allocation. T ime is specied in nite increments with only the last iteration af fecting the current iteration, per the Gauss-Seidel technique. The formulation of equations 9.5 through 9.10 proceeded in a tw o part f ashion. First, the array w as separated into the i 11 and the i > 11 sections. The i 11 is the rst ro w from gure 9.4 and i > 11 is the rest of the matrix. Separation into these tw o sections 68

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allo wed for the generation of the rst pass lters, ie. i > 11 or i 11 Second, equations were generated that dealt with the specic o w conditions for each of the re gions specied in gure 9.4. The i 11 has the more comple x rule structure and will be described rst. 9.5.1 Section 1 : The First Ro w i 11 The rst ro w is the most dif cult section because on three sides it borders areas outside of the e xperimentally considered space. Flo w into re gions 1, 2, and 3 from abo v e and on the the left and right of re gions 1 and 2 respecti v ely see gure 9.4, w as ignored because those areas are considered on the line of eld in v ersion and o w w ould primarily ha v e been dri v en a w ay The three selectors that indi viduate the re gions of the rst ro w are i = 1 1 < i < 11 and i = 11 The three equations that deal with these re gions are equations 9.5, 9.6 and 9.7, respecti v ely Equations 9.5 through 9.7 describe the concentration of deli v erable in node i at time t by rst referencing the concentration of deli v erable in node i at the pre vious time, t ¡ i Second, the amount of deli v erable that o ws out from node i from the A, C, and E directions, see gure 9.2, are subtracted from node i Third, the amount that o ws into node i from the C and E directions, see gure 9.2, are added. Flo w into node i from the F direction is ignored in section 1 since it is on the other side of the electrode. Flo w into node i in re gion 1, see gure 9.4, and equation 9.5 is only from the E direction of node i which is the C direction of node i + 1 Flo w into node i in re gion 2, see gure 9.4 and equation 9.6, is from the C and E directions of nodes i ¡ 1 and i + 1 respecti v ely Flo w into node i in re gion 3, see gure 9.4 and equation 9.7, is only from the node i ¡ 1 in the 69

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C direction. The rules for computing the time, position and electric eld dependent o w v alues required a tw o prong ltering approach rst by re gion and then by array position. conc [ i; t ] = conc [ i; t ¡ 1] ¡ conc [ i; t ¡ 1] ¤ f [ i ] :A (9.5) ¡ conc [ i; t ¡ 1] ¤ f [ i ] :C ¡ conc [ i; t ¡ 1] ¤ f [ i ] :E + conc [ i + 1 ; t ¡ 1] ¤ f [ i + 1] :C conc [ i; t ] = conc [ i; t ¡ 1] ¡ conc [ i; t ¡ 1] ¤ f [ i ] :A (9.6) ¡ conc [ i; t ¡ 1] ¤ f [ i ] :C ¡ conc [ i; t ¡ 1] ¤ f [ i ] :E + conc [ i ¡ 1 ; t ¡ 1] ¤ f [ i ¡ 1] :C + conc [ i + 1 ; t ¡ 1] ¤ f [ i + 1] :E conc [ i; t ] = conc [ i; t ¡ 1] ¡ conc [ i; t ¡ 1] ¤ f [ i ] :A (9.7) ¡ conc [ i; t ¡ 1] ¤ f [ i ] :C ¡ conc [ i; t ¡ 1] ¤ f [ i ] :E + conc [ i ¡ 1 ; t ¡ 1] ¤ f [ i ¡ 1] :E 9.5.2 Section 2 : The Rest of the T issue, i > 11 The equations describing o w through the rest of the tissue. The rest of the tissue is brok en up into three re gions in a similar manner to that described in section 9.5.1. The three re gions of section 2 sho wn in gure 9.4 break up into columns sharing similar transport scenarios. Re gions 4 and 6 are selected by computing the modulus of i If i mo d 11 = 1 then i is in re gion 4 and if i mo d 11 = 0 then i is in re gion 6. The third selection equation is if i mo d 11 6 = 1 or 0 and i > 11 then i is in re gion 5. Equation 9.8 describes the o w 70

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to and from o w for nodes in re gion 4, see gure 9.4, and has the same characteristics as equation 9.5 e xcept there is o w into node i from node i ¡ 11 the node directly behind it. Re gion 5 is described by equation 9.9. Equation 9.9 describes outo w in the A, C and E directions and ino w from the A, C, and F directions. Flo w in from the F direction is outo w from the A direction of node i ¡ 11 Flo w into and out of re gion 6 is described by equation 9.10. This equation has similar characteristics to equation 9.7, e xcept for the o w in from node i ¡ 11 behind. conc [ i; t ] = conc [ i; t ¡ 1] ¡ conc [ i; t ¡ 1] ¤ f [ i ] :A (9.8) ¡ conc [ i; t ¡ 1] ¤ f [ i ] :C ¡ conc [ i; t ¡ 1] ¤ f [ i ] :E + conc [ i + 1 ; t ¡ 1] ¤ f [ i + 1] :C + conc [ i ¡ 11 ; t ¡ 11] ¤ f [ i ¡ 11] :A conc [ i; t ] = conc [ i; t ¡ 1] ¡ conc [ i; t ¡ 1] ¤ f [ i ] :A (9.9) ¡ conc [ i; t ¡ 1] ¤ f [ i ] :C ¡ conc [ i; t ¡ 1] ¤ f [ i ] :E + conc [ i ¡ 1 ; t ¡ 1] ¤ f [ i ¡ 1] :C + conc [ i + 1 ; t ¡ 1] ¤ f [ i + 1] :E + conc [ i ¡ 11 ; t ¡ 11] ¤ f [ i ¡ 11] :A conc [ i; t ] = conc [ i; t ¡ 1] ¡ conc [ i; t ¡ 1] ¤ f [ i ] :A (9.10) ¡ conc [ i; t ¡ 1] ¤ f [ i ] :C ¡ conc [ i; t ¡ 1] ¤ f [ i ] :E + conc [ i ¡ 1 ; t ¡ 1] ¤ f [ i ¡ 1] :E + conc [ i ¡ 11 ; t ¡ 11] ¤ f [ i ¡ 11] :A 71

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9.6 Modeling the Ef fect of an Electric Field on a Flo w P attern Electric elds inuence the o w of char ged particles [32, 177, 193]. Modeling the effect of the applied electric eld on ions is accomplished using the techniques described in section 5.2. The electric eld shape follo wed from the denition of potential dif ference and from Ohm' s la w see equations 9.11 and 9.12 [178]. This approach assumes that the cur rent pathw ays in tissue are Ohmic conductors [154] and that the tissue can be brok en do wn into a series of resistors. Thereby maintaining the electric eld constant for on-axis current paths [213]. Using this idea, the parallel plate electrodes are treated as char ged parallel plates, see gure 9.5(a), and paired needle electrodes are treated as a pair of dipoles, see gure 9.5(b) [213]. This allo ws the four needle array to be treated as imbricated interacting paired dipoles with the eld from one dipole o v erlapping the other Each anode is indi vidually paired with the opposing cathodes and then the electric eld v alues are summed. (9.11) V b ¡ V a = ¡ Z b a E ¢ d s = E x Z l 0 dx = E x l (9.12) V = I l A = I R F or this dissertation, the parallel plate electrode' s homogeneous eld w as incorporated into the tissue model as a ubiquitous o w eld between the tw o electrodes. This w as accomplished by multiplying each o w v alue by a set v alue of 1, see appendix L for the code that implements this. This allo wed the o w eld electrophoresis model to be created 72

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+ + + + + + + + + + + + + + + + 1 (a) (b) Figure 9.5: Electric Field Lines for P arallel Plates and Dipoles in Homogeneous Media [213] from three indi vidual parts. Those parts are the random o w directions, the random o w v alues, and the o w due to the electric eld. The electric eld created by the needle electrode w as a more in v olv ed modeling procedure. This process w as simplied by assuming that the greatest strength electric eld is directly between the cathode and the anode, see gure 9.5(b) [213]. In gure 9.5, the greatest density of eld lines can be seen between the anode and cathode. The gure also depicts the reduction in eld strength at dif ferent distances from the electrode bisection. The tw o pairs of needle electrodes that comprise the needle electroporation applicator see gure 7.1-b, were modeled as tw o interacting dipoles with o v erlapping elds, see gure 9.6. This method w as chosen for tw o reasons. First, because it most closely follo wed the di v erse electric eld shapes reported in the literature for tissue, cells, and non73

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conducting spheres and spheroids [32, 118, 160, 197, 203, 206]. Second, a minimal array density limits the resolution of the possible applied electric eld v alues. The resulting applied electric eld pattern model is sho wn in gure 9.6 and has a hour glass shape.Percent Applied Field Strength 0 2 4 6 8 Row Nodes 2 4 6 8 10 Column Nodes 0 20 40 60 80 100 Figure 9.6: Modeled Needle Electric Field Shape The hour glass shape of gure 9.6 is created through the superposition of the four dif ferent interacting electric elds produced by the four electrodes of the four needle ar ray [32, 177]. The ridge in the center of the graphic tra v ersing point (5,5) 12 see gure 9.6, is due to the interaction of the diagonal electric eld magnitudes. When the electric eld 12 The v alue presented here is in ro w-column format, i.e. (ro w ,column). 74

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v ectors are separated into their respecti v e x and y components, the respecti v e v alues can be summed and in certain re gions, e.g. (5,3), the x-v alues destructi v ely sum to cancel each other while the y-v alues constructi v ely contrib ute to the electric eld v alue. The ef fect of this electric eld contrib ution on the ionic o w w as tw o fold and w as approximated by incorporating a normalized force term into the model equations. This force af fects the speed with which the ions tra v el from one node to the ne xt and as the electric eld strength is reduced so is the v elocity of the ions, see chapter 5. In summary the shape of the electric eld from the needle array adjusted tw o v ariables, electric eld intensity and ionic v elocity The needle array dened re gion felt dif ferent electric eld intensities and the v elocity of the ions through those re gions v aried as dictated by the locally dened electric eld strengths. 75

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10 Flo w Field T issue Model Results and Discussion 10.1 Results The motion of potassium ions in deionized w ater at an applied electric eld of 1500 V cm were the chosen parameters for testing the FFTM. These tw o substances were chosen for tw o reasons. First, the y are simplied representati v es of biological materials. Second, since electroporation is dened as an increase of the permeability of cell membranes due to a transmembrane potential dif ference produced by electric v oltage pulses [131], one half of that potential dif ference w ould be due to cations with potassium being a common cation used in cellular re gulation [184]. V ery little data e xists in the literature that co v ers the o w of ions in tissue under the inuence of an applied electric eld. Due to the sparseness of pree xisting data, the inuence of the electric eld produced by the parallel plate electrode w as compared ag ainst the needle electrode to see the ef fects of the dif ferent electrode shapes on the motion of char ged ions. While ha ving e xperimental data w ould ha v e been the ideal situation, comparing the electric eld induced ion motion of one applicator to another applicator creates a v ery interesting and important study in itself. Comparing the mathematically modeled and computer simulated motion of ions under the inuence of the dif ferent applicators allo ws for a time series analysis of the electroporation phenomenon. 76

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Figures 10.1 and 10.4 belo w sho w the induced ionic motion in a 1500 V cm electric eld for 450 ms as applied across parallel plate and needle electroporation applicators, respecti v ely The x and y ax es correspond to the ro w and column nodes. The z-axis w as nor malized to the initial concentration and the scale ranged from 0 to 2.5. This z-scale range w as selected because it spans the maximum concentration dif ference for both applicator designs.10.1.1 P arallel Plate Model Figure 10.1 displays the ef fect of the parallel plate electrode on the motion of the K + ions. The homogeneity of the applied electric eld is e vident in the undulating motion (a) 0 2 4 6 8 10 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 (b) 0 2 4 6 8 10 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5Normalized Concentration(c) 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 (d) 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure 10.1: P arallel Plate Induced Motion in a 1500 V cm Field for 450 ms of the concentration of ions. Figure 10.1(a) displays the initial concentration prole of the potassium ion after injection into the tar get tissue. The time between sub-gures is 77

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150 ms in the presence of the applied electric eld. Figures 10.1(b), 10.1(c) and 10.1(d) sho w the spread of the analyte as the simulation progresses. Figure 10.1(b) clearly sho ws a smoothing and distrib ution of the initial concentration pattern from gure 10.1(a). At the end of the simulation, see gure 10.1(d), the ions are mo ving uniformly out the back portion of the modeled re gion.Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure 10.2: P arallel Plate Induced Motion in a 1500 V cm Field for 50 ms Figure 10.1 gi v es the impression that the ionic motion produced by parallel plate electrodes is isotropic. This is not the case when the motion of the ions is e xamined at shorter time interv als than the ones e xamined in gure 10.1. Figure 10.2 displays the ionic motion after 50 ms. The concentration of ions is clearly not isotropic, three peaks ha v e formed at nodes (3,5), (6,6), and (7,6), with the peak at node (6,6) be gin almost double the initial concentration. As the simulation progresses, these peaks smooth and 200 ms into the simulation, see gure 10.3, there is one dominant peak at node (7,8). The early on anisotrop y of the ionic motion and subsequent char ge accumulation is possibly what generates the 78

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transmembrane v oltage dif ference 1 that leads to electroporation. The complete series of concentration contour plots for the parallel plate electrodes are in appendix J.Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure 10.3: P arallel Plate Induced Motion in a 1500 V cm Field for 200 ms 10.1.2 Needle Electrode Results Figure 10.4 displays the ef fect of the needle electrode on the motion of the K + ions. The inhomogeneity of the applied electric eld is e vident in the progression from initial condition, gure 10.4(a), to after 450 ms in the applied electric eld, gure 10.4(d). Con v ersely to the ef fect of the parallel plate electrodes, the needle electrodes dri v e ionic o w to create tw o dense re gions of concentration change within the applied electric eld. As the simulation progresses, the anisotrop y of the electric eld is clearly present, see gure 10.4(b) compared to the parallel plate electrode, see gure 10.1(b). The needle electrode seems to ha v e a more intense ef fect on the ions in a smaller re gion than the parallel plate electrode, compare gures 10.1 and 10.4. 1 aka transmembrane potential 79

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(a) 0 2 4 6 8 10 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5 (b) 0 2 4 6 8 10 0 2 4 6 8 10 0 0.5 1 1.5 2 2.5Normalized Concentration(c) 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 (d) 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure 10.4: Needle Array Induced Motion in a 1500 V cm Field for 450 ms The duration of a needle array generated electric eld has a profound ef fect on char ged species distrib ution. On lar ge time scales, the needle array creates a drastic change in the distrib ution of ions, see gure 10.4. By contrast, on shorter time scales the electric eld has a more gentle ef fect on ionic distrib ution, see gure 10.5. The contour plot 2 belo w the zero concentration plane in gure 10.5 sho ws an intact initial distrib ution with the change occurring primarily at the center The contour plot belo w the zero concentration plane in gure 10.6 sho ws the time e v olution of the distrib ution and a general motion to the f ar left corner of the graphic. The complete series of concentration contour plots for the needle array electrode are in presented appendix K, 2 A contour plot is a 2-D mapping of a 3-D graphic, using lines to e xpress a gradient change. 80

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Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure 10.5: Needle Array Induced Motion in a 1500 V cm Field for 50 msNormalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure 10.6: Needle Array Induced Motion in a 1500 V cm Field for 200 ms 10.2 Discussion Deli v ery of a therapeutic agent to the treatment site is a critical step in an y treatment protocol. Deli v ery can be intra v enous, intratumorally or interstitially [77]. Recent in vitr o research has e xperimentally demonstrated that dif fusi v e transport of lar ge molecules is orders of magnitude slo wer than electric eld mediated transport in tissue [216]. F or plasmid DN A dif fusion coef cients within tissue in the 10 ¡ 12 cm 2 s range suggest that dif81

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fusi v e transport after injection is not a signicant e v ent. In vivo research has indicated that combinations of electric signatures for transport and poration impact gene e xpression [57, 67, 120, 173] In summary the o w eld electrophoresis model has demonstrated the ef fect that different electrode congurations can ha v e on therapeutic deli v ery The importance of this is tw o fold, it can aid in the formulation of electric eld signatures for the deli v ery of char ged species and it can aid in the prototyping of ne w applicators or electrode designs for electric eld mediated deli v ery The simplicity of this model allo ws it to be applied to dif ferent char ged species kno wing only the carrier viscosity and size and char ge of the deli v erable. This allo ws the model to be used to test the motion of ions or char ged molecules in tissue for a gi v en electric eld pattern. Con v ersely a desired electric eld pattern, from a ne w electrode or electric eld signature 3 could be loaded into the model and the ef fecti v eness of that eld pattern could be e xamined in silico F or a nal analysis, this rst principle model w as compared ag ainst DN A electrophoresis literature data for tissue [216]. Zaharof f et al. e xperimentally sho wed that a 5.1 kbp plasmid in a 10 pulse, 50 ms, 465 V cm electric eld w ould tra v el 3.69 m and 1.01 m for B16.F10 and 4T1 tumors respecti v ely [216]. The o w eld tissue model, FFTM, predicts that a 5.1 kpb plasmid in a 500 V cm electric eld for 500 ms w ould tra v el 7.42 m [76] 4 Zaharof f et al stated in their seminal w ork, “Electromobility of Plasmid DN A in T umor T issues During Electric Field-Mediate Gene Deli v ery”, that one possible reason for the 3 More electric eld elements or dif ferent v oltages on the dif ferent elements of an electrode. 4 The viscosity used for this analysis w as 0.9 poise, see appendix O 82

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dif ference in the distance tra v eled by the DN A w as due to tissue collagen [216]. Collagen is belie v ed to increase the structural stability of certain tissue types by reducing the v oid fraction of the e xtracellular space [44, 123]. One possible reason wh y the o w eld tissue model predicted slightly f aster linear v elocities than the e xperimental v alues is path radii and tortuosity between cells. As the path between cells becomes more torturous the plasmid DN A has to w ork its w ay through and its progression slo ws. This is analogous to the phenomena used in the laboratory technique gel electrophoresis [7]. The separation of the single DN A band into multiple bands is a function of the smaller DN A fragments tra v eling f aster through the homogeneous gel [7]. Thus, the inuence of the tortuosity dif ference between the B16.F10 and 4T1 tumors is mirrored by the dif ferent ag arose concentrations 5 in gel electrophoresis separation of DN A fragments b ut not ackno wledged in the FFTM. This code could be written into the FFTM b ut that model w as specically designed to e xamine the motion of small char ged particles tra v eling through heterogeneous spaces. 5 Dif ferent ag arose concentrations are used to increase the tortuosity and decrease the porosity of an ag arose gel for DN A fragment separation. The dif ferent concentrations are used to increase gel resolution for dif ferent band sizes. As the ag arose concentration increases, the gel has a higher resolution for smaller fragment sizes. Therefore a higher ag arose gel concentration will retard the o w of lar ger fragments and allo w for greater band separation of the smaller fragment sizes. 83

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11 Electric Field Mediated DN A Fragment Deli v ery Model, EFMDFDM 11.1 Introduction The w ork of Zaharof f et al. on the motion of DN A in tissue suggests the limitations of the FFTM, o w eld tissue model, described in chapter 9 [216]. The FFTM is limited to one char ged species at a time and is dependent on e xternal v alues for the viscosity of the carrier uid. These characteristics reduce the FFTM' s functionality for tw o reasons. First, modeling the electric eld mediated o w eld deli v ery of a single char ge/radius species is only useful for prototyping deli v ery conditions and electrodes. Second, the e xternal carrier viscosity v alues may not aptly describe the motion of lar ge molecules, lik e lar ge plasmids and proteins, through the conned interstitial space. Inte gral to the creation of a v ersatile softw are based electric eld mediated drug and gene deli v ery modeling package, EFMDGDP is the demonstration that the v e main elements of the FFTM, the SD A, o w eld v alues, o w rules, initial concentration prole, and the set of force v alues, see chapter 9, can be utilized to produce a simultaneous and reliable multiparticle/multichar ge/multiradius/multi v elocity characterization model. Once created, this tool could be used to more accurately pro vide a descriptions of ion, drug, plasmid and protein motion in electric eld mediated deli v ery by simultaneously simulating their motion through comple x biological matrices. In pursuit of this objecti v e a 1% ag arose gel w as 84

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chosen as the representai v e matrix. Gels ha v e similar properties to tissue [149, 212], and although gels and tissues do not ha v e the e xact same characteristics, the ber -matrix netw ork of the tw o systems are similar enough that for o w modeling gels produce an acceptable 1-D model of tissue [134, 136, 149, 212]. DN A w as chosen as the simulation deli v erable for tw o reasons. First, DN A is the deli v erable in electrogenetherap y [67, 110, 112, 218]. Second, electrophoretic separation of DN A is a common biology lab procedure [113, 193]. Creating a model that describes the motion of DN A in an electrophoresis gel is the rst step to w ards the better understanding of site specic electric eld mediated deli v ery of drugs and genes into tissue. This chapter describes the o w eld tissue model frame w ork issues for the creation of an electric eld mediated DN A fragment deli v ery model, EFMDFDM, the element of the EFMDGDP that is responsible for the v elocity calculations and simulates the motion of multiple DN A fragments in a homogeneous material under the inuence of an electric eld. This chapter describes the specic modications in the SD A, self descripti v e array and the rational behind the model o w rules for electrophoresis gel applications. 11.2 Building an EFMDFDM for DN A Deli v ery Prediction The electric eld mediated DN A fragment deli v er model, EFMDFDM, w as created utilizing the same techniques emplo yed in the de v elopment of the o w eld tissue model, FFTM, presented in chapter 9. The elements utilized in the EFMDFDM are a SD A, an initial concentration prole of DN A 1 appropriate o w rules and the force from the applied 1 The initial concentration prole describes the dif ferent sizes of DN A fragments and the respecti v e concentration of those fragments. 85

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electric eld. While these structural elements of the EFMDFDM are the same as those used in the FFTM, subtle dif ferences in the array shape and size, o w rules, and the initial concentration prole of the deli v erable change the simulation dramatically 11.2.1 Selecting an Array Size The coarseness of the SD A in the FFTM, while useful for prototyping electrode designs or electric pulse signatures, w as prohibiti v e when e xamining di v erse populations of char ged species. The selected array size for this demonstration of the EFMDFDM w as a 1331 by 50 element array This size array w as selected for both visual and o w resolution reasons. The visual reason w as the nal graphics needed to ha v e a similar aspect ratio and visual representation to a picture of a gel, length much greater than width, while not requiring an immensely lar ge array size. The resolution required a ner mesh because each species tra v els at a v elocity dependent on its v alenc y and h ydrodynamic radius [6, 184]. A second resolution issue dealing with the time e v olving separation between the deli v erables is dependent on the mesh density [6, 184]. W ithout a lar ge enough mesh the dif ferent speeds of the deli v erables w ould not be resolv able. The selected array size attempts to mimic the aspect ratio of a DN A lane in an ag arose gel while attaining the required resolution for visual separation and maintaining a minimal array size. 11.2.2 Gel Flo w Rules The rst generation electric eld mediated DN A fragment deli v ery model, EFMDFDM for DN A deli v ery prediction and motion simulation w as de v eloped for the characterization 86

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of DN A mo v ement in a gel matrix. It utilized the rst principle char ge/v elocity force equations introduced and described in chapter 5 [75– 77]. P acking f actors were calculated and speed predictions for DN A in gels were computed and compared ag ainst e xperimental data, see columns 1 and 3 2 of table 12.1 on page 93. Since the computed data did not correlate well with the e xperimental results a force, unaccounted for by the model, w as h ypothesized to be limiting the motion of the DN A fragments through the ag arose gel and af fecting the o w rules emplo yed at this time. This de viation of prediction from e xperiment required a ree xamination of the mathematical model emplo yed for the compuation of v elocity of DN A fragments in an ag arose gel. Reptation is an e xcellent starting point for e xamining the motion of DN A fragments through an ag arose gel [9, 40, 42, 43]. In this approach an ag arose gel is described utilizing a tube model because the actual topological interaction is e xtremely dif cult [40]. The tube geometry assumes that the topological constraints of the ag arose gel and the 1-D applied electric eld produce a unidirectional tube netw ork [40]. In the reptation approach DN A fragments are simulated by di viding the linear fragment into N units called reptons [20]. The space between the reptons is between 400 to 800 A, the approximate persistence length of a DN A fragment, a fe w hundred base pairs or so [134]. An e xact reptation model w as discarded because of the intense computational time requirements requisite for each, instead a o w netw ork based on a tube model with the DN A simulated as a deformable “blob” 3 w as utilized to b uild an appropriate set of gel o w rules on [9, 134]. 2 Special thanks to Dr Loree Heller from the CMD at USF for pro viding the DN A v elocity data 3 T reating the DN A as a group of link ed blobs is the basis of the reptation model. The blobs congeal at lo w v elocities, eld strengths, or lar ge diameter tubes and are then described by equation 11.1. Therefore, 87

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The modied o w rules de v eloped for the EFMDFDM are similar to the tissue o w rules described in section 9.5. The main dif ferences are the FFTM w as focused on a single deli v erable and the random o w through the tissue. In the EFMDFDM there are multiple deli v erables, yet the y are constrained to o w in one dimension as dictated by the tube model geometry The Gauss-Seidel method w as used in conjunction with nodal analysis to approximate the iterati v e motion of the DN A fragments through the ag arose gel [32, 147]. F or this demonstration a v e le v el array w as created to follo w the mo v ement of v e representati v e DN A fragment sizes. The representati v e fragments were 200, 600, 1000, 2500 and 5000 base pairs. These 5 bands were selected because the y span the range of fragment sizes yet the y don' t clutter the nal graphic. the EFMDFDM is utilizing portions of the reptation model b ut in a limiting case. (11.1) v E N F Equation 11.1 links the applied electric eld times the number of reptons to the force applied to the DN A fragment' s a v erage v elocity for further analysis of the theory behind equation 11.1 `see section 9.5 88

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12 Modeling Electrophoresis in Gels Using a F orce Model 12.1 Examining DN A Fragment Motion in an Ag arose Gel Understanding the motion and the go v erning equations of a molecule' s path in a tissue is an ultimate requirement for the repeatable, site specic deli v ery of molecules in a tissue [75, 76, 148]. Producing an accurate, descripti v e, tw o dimensional lar ge molecule through tissue model from rst principles is a v ery comple x and dif cult task due to a lack of literature data to support model precepts. As in the v elocity and acceleration de v elopment discussed in section 5.5, DN A position predictions in gel electrophoresis requires good estimates of e xperimental parameters. This chapter describes the process undertak en to use dissertation de v eloped model elements to predict the motion of kno wn lar ge molecular weight molecules 1 in a 1% ag arous gel for a gi v en applied electric eld duration and strength with a prioi kno wledge of the viscosity of the running b uf fer Since representati v e v alues of the electric eld strength, duration, and distance tra v eled for common DN A fragment sizes were not a v ailable in the literature the y were e xperimentally determined for this demonstration ef fort. A gel w as ran using DN A standards at a specic v oltage, 6.56 V cm 2 for a specic time, 1 hr 3 An image of this gel is sho wn in 1 More specically DN A fragments. 2 A v alue of 105 V olts w as measured across the 16 cm gel apparatus 3 Special thanks to Dr Loree Heller of the USF Center for Molecular Deli v ery CMD 89

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gure 12.1. Lane 3 of the gel w as loaded with HyperLadder I, made by Bioline USA Inc., Canton Ma., catalog number BIO-33025 [14]. The band sizes in HyperLadder I are 10k, 8k, 6k, 5k, 4k, 3k, 2500, 2000, 1500, 1000, 800, 600, 400, and 200 basepairs [14]. Lane 5 of the gel w as loaded with a 100 bp ladder made by Bayou Biolabs, Harahan, La., catalog number L-101 [10]. The 100 bp ladder spans from 100 bp to 4000 bp in increments of 100 bp, e.g. 100 bp, 200 bp, 300 bp, ..., 3900 bp, and 4000 bp [10]. The gel matrix w as a 1% ag arose gel and the carrier solution w as 1x T AE b uf fer [7] and the applied electric eld w as 6.56 V cm The DN A fragment displacements sho wn in lane 3 of gure 12.1 were selected as the data frame for the model because of the wider separation between bands both in the number of base pairs and the separation between bands o v er a lar ger re gion, see gure 12.1. Clear delineation between bands in the range from 200 bp to 5000 bp w as observ ed. Kno wing the electric eld strength, duration and the number of base pairs of each of the se gments allo wed for this rst generation DN A fragment motion model with e xperimental renement. 12.2 DN A Fragment Motion Retardation Section 5.4 described a method for e xamining the motion of a plasmid in solution under the ef fect of an applied electric eld. This technique utilized the radius of the plasmid along with other e xperimental parameters. A simple w ay of approximating the radius of an amalg am is through an equal v olume sphere. Appendix G describes a technique for calculating the v olume of a DN A molecule from the number of base pairs. This method assumes that the DN A purine-p yrimidine pairs join to form a c ylindrical shape that minimizes v oid 90

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Cathode Side 10000 bp 2000 bp 800 bp 200 bp Anode Side Lane 3 5 Figure 12.1: Photograph of a T ypical Electrophoresis Gel space. By minimizing v oid space, the c ylinders are deformed into amorphous bodies. The c ylinder' s v olume w as then multiplied by the number of base pairs of each DN A fragment. The resultant v olume w as then related to an equal v olume sphere, the assumed resting state condition of the DN A fragments used in this research [1, 136, 180]. The radius of the equal v olume spheres w as calculated for each of the base pairs and used throughout the rest of calculations. 91

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6 8 10 12 14 16 18 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 DNA Sphere Radii (nm)Molecular Weight (bp) Figure 12.2: DN A Fragment Radii as a Function of Base P airs This gel w as used to prototype the data rules and speeds. Figure N.1, in appendix N on page 211, is an image of the gel used for optimizing the model. As in chapter 5, the computation for a particle' s speed from char ge, radius and carrier solution viscosity w as deri v ed. This method w as used to calculate the drift v elocity of the DN A fragments in the gi v en conditions. The calculated v alues were then compared with e xperimental v alues and the dif ferences were noted, see table 12.1. The e xperimental v elocity of the 100 bp fragment w as f aster than the v alue predicted by the rst principle model. T o account for this discrepanc y a shape correction f actor w as utilized 4 This shape correction f actor c w as re gressed and deconstructed using c = a b 5 where a is the length of the minor ax es and b is the major axis of a prolate spheroid [6] 6 The f actor v alues were 4 The ph ysical basis behind the shape correction f actor is due to the de viation from sphericity as a function of v elocity this type of shape correction f actor is typically kno w as eccentricity 5 The v ariable c is used here rather than the typical e that is used for eccentricity because e has already been used for the char ge of an electron and since the speed of light doesn' t sho w up in this dissertation, c w as unclaimed. 6 When describing eccentricity usually the three axis lengths of the spheroid are dened. The common 92

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T able 12.1: Speeds of DN A Fragments in a 1% Ag arose Gel at 6.56 V cm DN A Fragment Experimental Calculated Corrected Dif ference Size (bp) Speed ¡ cm hr ¢ Speed ¡ cm hr ¢ Speed ¡ cm hr ¢ (Cor -Exp) 100 6.70 2.80 6.70 0.00 200 6.34 4.45 10.64 4.30 300 6.09 5.83 13.95 7.86 400 5.78 7.06 16.90 11.12 600 5.28 9.26 22.14 16.86 800 4.80 11.21 26.82 22.02 1000 4.50 13.01 31.12 26.62 1500 3.83 17.05 40.78 36.95 2000 3.40 20.66 49.40 46.00 2500 3.11 23.97 57.33 54.22 3000 2.89 27.07 64.74 61.85 4000 2.60 32.79 78.42 75.82 5000 2.39 38.05 91.00 88.61 ascertained by comparing the v elocity of the 100 bp line of lane 5 in gure 12.1 to the calculated v elocity of a 100 bp DN A fragment. A dif ference between the shape f actor corrected and the e xperimental v alues w as e xpected b ut a di v er gence between the e xperimental and calculated v alues w as not. Experimental speed v alues decrease with an increase in number of base pairs while the calculated speeds increase with an increase in number of base pairs. Upon e xamination of the model description is e = a b or e = a c or e = b c where a and b are the tw o minor axis lengths and c is the major axis length. F or this analysis the tw o minor axis were set equal to each other a = b and designated as a and the major axis designation w as changed from c to b 93

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structure, this di v er gence w as theorized to be due to the model' s inability to describe the gel' s retarding ef fect on the motion of the DN A fragment. Column 5 of table 12.1 sho ws the di v er gence between the shape f actor corrected model and the e xperimental v alues. This discrepanc y between predicted and e xperimental v alues indicated that a modication of the equations that describe` the motion of ions, molecules, and plasmids in electric elds in liquids is required, see chapter 5. This force balance, as described in chapter 5, had tw o terms, one that represented the dri ving force of the electric eld, Coulombic, and the other the uid frictional retarding force of the suspensory Stok e' s la w the v ector sum of these tw o forces w as set equal to zero, F E ¡ F f = 0 F or the data in table 12.1, the sum of these tw o forces w as not zero b ut w as proportional to the dif ference in the speed between the shape f actor corrected model and the e xperimental data. F or this representati v e system the Coulombic, F E and Stok e' s la w F f forces were calculated, see gure 12.3, and then subtracted, see equation 12.1. The result w as the e xperimentally de v eloped gel DN A retarding force, F g see equation 12.1. Coulom bic F orces ¡ Stok e 0 s La w F orces = Gel DNA Retarding F orce F E ¡ F f = F g (12.1)12.3 Modeling the Retarding F orce of the Gel The retarding force of the gel w as modeled using tw o techniques. Both techniques utilized the interaction of the DN A fragment with the ag arose gel, b ut the model de v elopment principles were decidedly dif ferent. The rst technique directly modeled the retarding force 94

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0 50 100 150 200 250 300 350 4 6 8 10 12 14 16 18 Forces ( 10-16 N)DNA Frgment Radius (nm) Electric Field Force Gel Frictional Force Fluid Frictional Force Figure 12.3: F orces Acting on the DN A Fragment interaction of the DN A fragment with the ag arose gel, see appendix P, pages 214 through 217. The second technique modeled the speed of the DN A fragment directly and then back related the results to the retarding force on the DN A fragment as e x erted by the ag arose gel, see appendix P, pages 218 through 222. Although an in depth description of each of these modeling ef forts to describe the speed of the DN A fragments through the ag arose gel is presented in appendix P only the utilized model with its appropriate parameters is described here in section 12.3.1. Ho we v er the octa v e source code for each of the models is gi v en in appendices R through V for reference and comparison. 95

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12.3.1 Exponential Correction DN A Fragment Speed Model This document subsection re vie ws the characteristics of the e xponential correction DN A fragment speed model, ECDFSM, that w as ultimately used to obtain gel retardation force v alues, F g The single tunable parameter ECDFSM has an e xponential correction f actor that compensates for the dif ferences between the predicted and the e xperimental speed v alues. The ECDFSM uses the shape distortion corrected model initial speed v alue and adjusts the speeds using the number of base pairs as an independent v ariable. Equation 12.2 w as used to calculate the predicted speeds, s The equation also describes the damping of the DN A fragment' s speed, the rst term in equation 12.2, by the ag arose gel. While, the second term represents the damping contrib ution of the gel to the DN A fragment' s speed. V alues for D g el represent the maximum decrease in fragment speed for the smallest fragment caused by the gel. The parameter E g el represents the attenuation f actor on this maximum speed decrease as reected by the number of base pairs in the DN A fragment. The v alue for E g el w as 750 bp and the v alue for D g el w as 4.91915 cm hr the speed of a 750 bp DN A fragment in this gel at this electric eld strength. The f act that D g el and E g el are both dependent on the number of base pairs means that there is only 1 tunable parameter in the ECDFSM. Examination of gure 12.4 re v eals that at 750 bp the graph lea v es a linear re gion and transitions to a re gion of curv ature. From 100 to 750 bp the graph of the e xperimental data is essentially a straight line b ut from 750 to 3000 bp the graph has a distinct curv ature. At 3000 bp the graph becomes linear ag ain. Therefore there 96

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seem to be three re gions to the ef fect of the gel retarding the speed of the DN A fragment depending on the number of base pairs. (12.2) S = F E 6 r por e ¡ D g el e ¡ E g el bp 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Speed (cm/hr)DNA Fragment Size (bp) Experimental Model Predicted Figure 12.4: Experimental Speed vs ECDFSM Predicted Speed Figure 12.4 displays the predicted and the e xperimental speeds for the DN A fragments as a function of base pairs. The t of this model is visually quite good, as can be seen in Figure 12.4. Although, there is some v ariation at the edges of resolution of the gel in the w orking re gion of the 1% ag arose gel 7 the model performs well. The R 2 v alue for this model is 0.99409. The code for the re gression is listed in appendix V. 7 T able 4.2 on page 30 states that a 1% ag arose gel is used for DN A fragments in the range of 500 to 10,000 bps 97

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13 Simulating DN A Motion in Gels Using the ECDFSM Model 13.1 Simulation Frame w ork The e xponential correction DN A fragment speed model, ECDFSM model, described in section 12.3.1 w as used in conjunction with the nodal analysis techniques, chapter 9, to simulate the motion of DN A fragments in a 1% ag arose gel in a 6.56 V cm electric eld. T o impliment the simulation, some initial element v alues needed to be determined. Those elements were DN A fragment band sizes, array size, initial distrib ution of the DN A fragments, descripti v e and applicable o w rules through the gel and electric eld distrib ution. The folo wing sections of this chapter describe the process of creating and running the simulation and then processing the nal data. 13.1.1 DN A Fragment Band Selection The simulation is initiated after the selection of a representati v e sample of DN A fragments to run in the theoretical gel. F or this demonstration the sample sizes chosen were 200, 600, 1000, 2500, 5000 bps. These samples were chosen for tw o reasons, rst this range of DN A fragments spans the resolution of a 1% ag arose gel and second, the y sho w distinct dif ferences or similarities when compared to the DN A separation distances of the e xperimental gels. If the bp v alues selected were closer together in number of fragements 98

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then the resolution of the simulation w ould ha v e reduced and the bands w ould ha v e “ran” together lik e bands do in e xperimental gel, see gure 12.1. 13.1.2 Simulation Components Simulating the motion of DN A fragments in an ag arose gel utilized four elements of the FFTM, uid o w tissue model. Those components were, rst, create an array capable of representing the gel, second, dene an initial distrib ution, third, determine an appropriate set of o w rules, and fourth, select and implement a force model of the electric eld. As e xpected at this point these tasks are similar to the acti vities required to b uild the FFTM model discussed in chapter 9 with minor dif ferences. F or this application o w rules are not as intense because the DN A motion is not obstructed by cells and the applied electric eld is only due to one applicator conguration 1 The primary important distinction between this simulation and the simulation from chapter 9 is the f act that this analysis e xamines the motion of a di v erse array of sized and char ged molecules. 13.1.2.1 Array Size The simulation array is composed of 1331 elements £ 50 elements, for a total of 65,550 elements. This size array w as selected because it pro vides the required resolution for the DN A fragments in an ag arose gel. Since the dimensional resolution of an array is dependent on the number of elements in a gi v en dimension, more elements in one dimension increases the resolution in that dimension. Increased resolution in the dimension of tra v el 1 The wire electrode in the gel apparatus w as modeled as a parallel plated electrode 99

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w as important for this application because each DN A fragment mo v es at its o wn v elocity An increased number of steps allo ws the modeled DN A fragments to mo v e freely at their respecti v e speeds independent of each other 13.1.2.2 Initial Distrib ution The initial DN A fragment distrib ution w as dened in the same manner as described in section 9.4. Each node w as initialized with a v alue for the v e chosen DN A fragment sizes. The initial distrib ution w as based on equal participation for each DN A fragment. Each fragment size w as assumed to be homogeneous and one-fth of the total DN A amount. The summed initial amount of DN A w as normalized for simplied plotting. The layout of DN A in the model matrix w as created to mimic the layout in one lane of an ag arose gel. This meant that all of the DN A w as originally lumped into a well at the cathode side of the ag arose gel, see gure 12.1. The cathode side is located on the 0 node side of the gure and the anode side is at the 1400 node side of the gure. T o model this system and maintain the proper visual aspect the well consisted of a 50 element £ 50 element square. The well in an ag arose gel is not square b ut the nodal resolution mismatch, approximately 133:5, allo wed for a square to visually approximate a high aspect ratio rectangle, see gure 13.1 belo w 13.1.2.3 Flo w Rules The o w rules for this model were chosen to be unidirectional because the electric eld w as assumed to be homogeneous. The primary dif ference in the o w rules for this applica100

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Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 1 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Figure 13.1: DN A Fragment Initial Distrib ution tion compared to the rules de v eloped in section 9.4 lies in the f act that this simulation must deal with the dif ferent speeds due to the dif ferent DN A fragment radii and char ges. The modeled rules were created via a tw o step process. First, the speed v alues were calculated for each DN A fragment size from the ECDFSM, subsection 12.3.1 page 96, and implemented as an inde x, and second, the o w v alues for a specied node were only dependent on the pre vious v alues for that node and the v alues of the node directly behind. These, o w rules were entirely a function of speed. Since speed is dened as distance time this w as accomplished by con v erting the number of nodes into a representati v e distance and then utilizing the predicted speeds to compute the rate at which dif ferent DN A fragments w ould tra v el between the dif ferent nodes. The speed calculated from the ECDFSM w as 101

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con v erted into a rate with units of no des iteration This rate w as then implemented into the c-code as an a function of the ro w inde x. F or this simulation, the f astest components mo v ed at a rate of just greater than three times the slo west components, and the 200 bp DN A fragment tra v eled greater than 3 times as f ar as the 5000 bp DN A fragment. The interaction of speed dif ferences dictates the array' s length. As the array increases in size the speed resolution increases. This allo ws for the DN A fragments to tra v el at their respecti v e speeds. 13.1.2.4 Electric Field Distrib ution The electric eld for this simulation w as modeled as homogeneous. The photograph of the actual gel chromatograph, Figure 12.1, sho ws that the DN A fragments are mo ving in primarily a forw ard direction with v ery slight twist to w ard the center This minor twist to the center w as ignored for the purpose of this simulation. Although it is a common occurrence when running DN A fragments in ag arose gels, it w as primarily due to Joule heating [28, 63, 207]. In e xperimental apparatus, the electric eld that the gel is e xposed to is not perfectly homogeneous and the center of the gel actually recei v e a higher current [16, 17, 145, 207]. A homogeneous model w as utilized for this simulation because the dif ference is minimal and Joule heating can be minimized if the gel is run at a lo wer v oltage for a longer period of time. The simulation used an equal electric eld being spread across the entire gel for the duration of the applied electric eld for the analysis. 102

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13.2 Simulation Processing and Results From a technical perspecti v e, the simulation w orks in a ternary format. First, the c-code, see appendix Y, w as e x ecuted to collect the simulation data. Second, the simulation data w as processed into tw o graphics formats using Perl 2 and Bash 3 scripts. The output graphic formats were encapsulated postscript, EPS 4 and portable netw ork graphics, PNG 5 in both monochrome and color formats using Gnuplot 6 Third, the color PNGs were con v erted into an animated GIF using Con v ert 7 a utility of the ImageMagick 8 tool and library collection. The result is an animated GIF that steps through the frames of the simulation. The indi vidual graphics that comprise this demonstration simulation are sho wn in appendix Z. The nal results for the simulation are sho wn in gure 13.2 belo w T o illustrate the modeling tool' s ef fecti v eness, the graphic combines the nal slide of the simulation juxtaposed with the lane from the ag arose gel. The simulation results visually match the e xperimental results. The rst four DN A fragments clearly match within the error of the data collection, see table 13.1 nd gure 13.2, while the 5000 bp fragment is seemingly lost in a sea of DN A at the lo wer resolution of the ag arose gel. This graphic reinforces the impact of using only v e DN A fragments in the simulation. 2 F or information about Perl, see chapter 4 3 F or information about Bash, see chapter 4 4 EPS is a high resolution image format for print 5 PNG is the open source answer to Compuserv e' s GIF 6 F or information about Gnuplot, see chapter 4 7 F or information about Con v ert, see chapter 4 8 F or information about ImageMagick, see chapter 4 103

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T able 13.1: Comparison of Simulation Results with the Experimental Data Fragment size 200 600 1000 2500 5000 Experimental Distance 6.34 5.28 4.50 3.11 2.39 T ra v eled (cm) Simulation Distance 6.340 5.280 4.499 3.044 2.283 T ra v eled (cm) % error 0.00 0.06 1.13 2.12 4.45 T able 13.2 sho ws the maximum error as 2.73%. While this is a considerable error it is consistant with both the e xperimental error of the electrophoresis gel system and the resolution of a small array P art of this error could be easily reduced by using a lar ger array for the gel. As stated in section 13.1.2.1 the demonstration array w as only 1331 nodes by 50 nodes. Increasing the number of nodes w ould increase the possible resolution of the system b ut w ould require either more ram, more processor time or a f aster processor The original code w as written as a marraige of e xperimental data, speed of e x ecution, ability to perform on di v erse hardw are and simulation. The interaction of these four elements w as inte gral to the design and e x ecution of this simulation softw are. 9 9 Primary to the creation of this code w as the nal product which w ould be run on all forms of computers from pentium III' s to 64 bit pentium IV' s. While 64 bit pentium IV' s will possibly be supercomputers in their o wn right, most labs w on' t ha v e adv anced machines for a fe w years yet and this code will run on a pentium II 400, the prototype machine for this code. 104

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Figure 13.2: DN A Fragment Final Distrib ution 13.3 Simulation Summary The nal result for this DN A fragment modeling ef fort w as a graphic simulation that described the real time progress of DN A fragments through an ag arose gel at 6.56 V cm for 1 hour The technical de v elopment of this simulation of DN A fragment motion is an ag arose gel required man y of the FFTM components. Those components included the usage of a modied force model, the selection of a representati v e assortment of DN A fragment sizes, a lattice array of a capable size, a representati v e initial distrib ution, o w rules, and an electric eld model. The c-code written to handle these components w as e x ecuted and follo wed 105

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by scripted data processing. The data processing w as accomplished using Gnuplot via both Bash and Perl scripts to mak e identical graphs in the multiple formats used for print graphics and animation. Colored PNGs were con v erted into an animated gif using Con v ert. 106

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14 Conclusion 14.1 Introduction The research associated with this dissertation resulted in the creation of an alternati v e method of modeling the processes that lead to electric eld mediated drug and gene deli v ery The long term goal for this research ef fort is to create, de v elop, characterize, and ultimately implement the v arious modeling elements required for a practical electric eld mediated drug and gene deli v ery modeling package, EFMDGDMP. The EFMDGDMP is a tool kit that describes the motion of particles of di v erse sizes and char ges in di v erse matrices. This tool kit contains a primary model, the electric eld mediated o w eld model, EFMFFM, which describes the motion of char ged particles using Lorentz' s force equation, Stok es' La w and e xperimental renement. Primary model usage options depend on whether the matrix being modeled is homogeneous, e.g. a uid or a gel, or heterogeneous, e.g. stratied liquid or gel or a tissue 1 See gure 14.1. The dif ferences in the modeling options demonstrated in gure 14.1 reect ho w the matrix is handled. This dissertation v eried the feasibility of the EFMDGDMP by demonstrating model simulations for both matrix situations, each with a dif ferent particle size re gime. 1 The tissue may be homogeneous b ut from the perspecti v e of o w a tissue is herterogenous. It has dif ferences in o w depending on position. 107

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Homogeneous matrices were modeled using a 1-D homogeneous o w model with tra v el restricted to the in eld direction of the applied electric eld. The applied electric eld w as a function of the applicator geometry and each node held a eld force v alue which directed o w The homogeneous model treated the pathw ays identically with no constrictions to o w and is described in chapter 5 Heterogeneous matrices were modeled using a randomly generated 2-D o w eld tissue model, which is describe in chapter 9. This treated the pathw ays through tissue as random constrictions with a possible normalized o w range from 0 to 1. The electric eld ef fect on the o w of char ged species w as incorporated into the model to allo w for the prediction of the ef fect of dif ferent electric eld applicator geometries and pulse signatures. The v alue of the EFMDGDMP is more clearly demonstrated when the selection e xibility of entity motion to be in v estig ated is considered. Figure 14.1 indicates the optional entity selection menu path. These entity options include small molecules and ions, or lar ge molecules, lik e DN A or proteins. The reason for these EFMDGDMP is the interaction of the molecules and the v arious paths that the y might tra v el through. F or e xample, in liquids the motion of molecules and ions is only inhibited by the viscosity of the suspensory deformation only occurs for lar ge molecules tra v eling at v ery high v elocities. In gels and tissues small molecules and ions will also tra v el freely because their ef fecti v e radii is much smaller than the pore that the y are tra v eling through. By contrast, lar ge molecules will deform in gels and tissues as the ef fecti v e radius of the molecule approaches that of the pore the molecule will deform. This deformation is an interaction between the retarding force of the matrix and the structural stability of the molecule. 108

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EFMDGDT EFMFFM Heterogeneous Matrix Ions, Drugs DN A, Protein Homogeneous Matrix Ions, Drugs DN A, Protein Figure 14.1: Model Components of the EFMDGDP The electric eld mediated drug and gene deli v ery tool, EFMDGDP, utilizes the electric eld mediated o w eld model, EFMFFM, to simulate the o w of char ged deli v erables through homogeneous and heterogeneous matricies. 14.2 Contrib utions The contrib ution of this w ork to the eld of electroporation research is three fold. The rst, from a human perspecti v e, a document w as created that coalesced much of the infor mation referred and alluded to in the electroporation modeling and associated ph ysics and chemistry literature. Although this contrib ution may not be percei v ed as glamorous, it will be e xtremely useful to future researchers de v eloping and rening mathematical models for electroporation. The second contrib ution is the creation of a ne w mathematical model that follo ws the motion of char ged deli v erables through random paths created in heterogeneous matrices. This technique had not been applied in electroporation research pre viously and it allo ws researchers to address dif cult questions that arise when e xamining electroporation results; e.g. “When will the applied electric eld produce non-isotropic char ged species o w?”, “Where will the char ged species congre g ate to produce an electroporation ef fect?”, “Where 109

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will deli v ery occur?”, “What electroporation signature is required to ensure deli v ery of this molecule?”. Although, these questions ha v e not been completely e xplored by the demonstrations here utilizing the EFMDGDP, the models associated with this package ha v e been created and characterized. A third signicant contrib ution to the discipline is the de v elopment of a process to e xamine the motion of multiple deli v erables with dif ferent char ges and sizes in a homogeneous matrix simultaneously This process w as successfully demonstrated with the e xamination of the motion of dif ferent DN A fragments through a 1% ag arose gel. While this demonstration w as not an e xact rst principle model, because it utilizes re gressed e xperimental parameters, it successfully describes the motion of DN A in ag arose gels. 14.3 Future W ork One of the future applications of the electric eld mediated drug and gene deli v ery tool will be to e xamine the ef fects of kilo v olt microsecond pulses v ersus me g a v olt nanosecond pulses. The EFMDGDP will need to be adapted to a lar ge array co v ering a small scale thereby producing the proper time scale for deli v ery The w ork done in chapter 5 will need to be ree xamined and adjusted for short time scale b ut the theory describes the motion of ions in that time scale for kilo v olt pulses, see table 5.4. A second application of this w ork w ould be to e xpand the electric eld mediated DN A fragment deli v ery model to identify unkno wns. Gi v en the distances tra v eled for the DN A fragment standards and a desired DN A fragment size in base pairs the model could sho w the theoretical placement of that fragment. Con v ersely gi v en the distances tra v eled for the 110

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DN A fragment standards and the distance tra v eled for dif ferent bands of DN A the model could predict the number of base pairs and produce a 3-D graphic. The third application of this w ork w ould be to e xamine the motion of DN A and proteins through gels and tissues. This could be used to e xamine matrix-deli v erable interaction. A model could be created using the methods outlined in chapters 8 and 9. The purpose of this model w ould be to characterize the transport phenomena of lar ge molecules in tissues for deli v ery applications. T o complete this application, o w rules for the h ypothetical tissue and v elocity proles for dif ferent molecules through dif ferent tissues are required. The EFMDFDM could also be used as a stand alone tool for the analysis and character ization of electrophoresis gels. The required elements for producing images of theoretical DN A bands are innate in the code of the EFMDGDM. Re w orking the code is required to acquire inputs from the user via either a command line interf ace or a GUI. The necessary inputs from a user w ould be the distance tra v eled of a representati v e set of standards, the % ag arose of the gel, the applied electric eld strength, the duration of the applied electric eld, the number of base pairs, and the column postion of the fragments whose predicted motion is desired. The purpose of this “stand-alone” program w ould be to create underlays or o v erlays that w ould aid in representing the data. The current method for presenting DN A gel data is a picture of the gel and the standards. The standards are spread o v er a range and typically either run together or are spread too f ar Either w ay the y represent a v ery imprecise method of presenting the data. If a prediction underlay or o v erlay could be included with the gel 111

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picture documenting where the selected standards sho w up and where the data bands e xisted, the gel picture w ould then be more precise. A second purpose of the “stand-alone” softw are w ould be to decon v olv e “messy” or “smeared” bands. When tw o bands do not fully separate the y form a single band of e xtended length in the tra v el direction. This e xtended length band can be decon v olv ed into its subsequent bands and the number of base pairs of each of the fragments could be ascertained through an alteration of the e xisting code. Also, the percentage of the each data band that made up the “smeared” band could also be decon v olv ed from the gel data. In this instance resolution of the model w ould be greater than for the gel. 112

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References [1] Beha vior of comple x knots in single DN A molecules. Physical Re vie w Letter s 91(26):265506/1–4, 2003. [2] I. G. Abidor V B. Arak elyan, L. V Chernomordik, Y A. Chizmadzhe v V F P astushenk o, and M. R. T arase vich. Electrical breakdo wn of bilayer lipid membranes I. the main e xperimental f acts and their qualitati v e discussion. Bioelectr oc hemistry and Bioener g etics 6:37–52, 1979. [3] I. Edw ard Alcamo. Fundamentals of Micr obiolo gy Benjamin Cummings, Ne w Y ork, NY 1997. [4] V B. Arak elyan, Y A. Chizmadzhe v and V F P astushenk o. Electrical breakdo wn of bilayer lipid membranes V. consideration of the kinetic stage in the case of the membrane containing an arbitrary number of defects. Bioelectr oc hemistry and Bioener g etics 6:81–87, 1979. [5] Donald R. Ask eland. The Science and Engineering of Materials International Thomson Publishing, 1994. [6] Peter Atkins. Physical Chemistry W H. Freeman and Compan y Ne w Y ork, 1994. [7] Frederick M. Ausubel, Roger Brent, Robert E. Kingston, Da vid D. Moore, J. G. Seidman, John A. Smith, and K e vin Struhl. Short Pr otocols in Molecular Biolo gy Harv ard Medical Books, Ne w Y ork, NY second edition, 1997. [8] A. V Babak o v L. N. Ermishkin, and E. A. Liberman. Inuence of electric eld on the capacity of phospholipid membranes. Natur e 210:148–160, 1966. [9] G. T Bark ema, J.F Mark o, and B. W idom. Electrohporesis of char ged polymers: Simulation and scaling in a lattice model of reptation. Physical Re vie w E 49(6):5303–5309, 1994. [10] Bayou Biolabs, Harahan, La. 100 bp Ladder. www.bayoubiolabs.com [11] W ayne M. Beck er and Da vid W Deamer The W orld of the Cell Benjamin/Cummings Publishing Compan y Redw ood City California, 1991. [12] Anna O. Bilska, Katherine A. DeBruin, and W anda Krasso wska. Theoretical modeling of the ef fects of shock duration, frequenc y and strength on the de gree of electroporation. Bioelectr oc hemistry 51:133–143, 2000. 113

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[65] Laurent Grossin, Nad ege Gaborit, Lluis Mir P atrick Netter and Pierre Gillet. Gene therap y in cartilage using electroporation. J oint Bone Spine 70:480–482, 2003. [66] W A. Hamilton and A. J. H. Sale. Ef fects of high electric elds on micro-or g anisms II. lysis of erythroc ytes and protoplasts. Bioc himica Et Biophysica Acta 148:789– 800, 1967. [67] Richard Heller Deli v ery of plasmid DN A using in vivo electroporation. Pr eclinica 1:198–208, 2003. [68] Richard Heller Richard Gilbert, and Mark J. Jaroszeski. Clinical applications of electrochemotherap y Advanced Drug Delivery Re vie ws 35:119–129, 1999. [69] Richard Heller Mark Jaroszeski, Andre w Atkin, Darius Moradpour Richard Gilbert, Jack W ands, and Claude Nicolau. In V ivo gene electroinjection and e xpression in rat li v er FEBS Letter s 389:225–228, 1996. [70] Richard Heller Mark Jaroszeski, L. Frank Glass, Jane L. Messina, Da vid P Rapaport, Ronald C. DeConti, Neil A. Fensk e, Richard Gilbert, Lluis M. Mir and D. S. Reintgen. Phase I/II trial for the treatment of cutaneous and subcutaneous tumors using electrochemotherap y Cancer 77(5):964–971, 1996. [71] Richard Heller Mark Jaroszeski, Jane Leo-Messina, Ron Perrot, Nanc y V an V oorhis, Doug Reintgen, and Richard Gilbert. T reatment of B16 mouse melanoma with the combination of electropermeabilization and chemotherap y Bioelectr oc hemistry and Bioener g etics 36:83–87, 1995. [72] Richard Heller Mark Jaroszeski, Ronald Perrott, Jane L. Messina, and Richard Gilbert. Ef fecti v e treatment of b16 melanoma by direct deli v ery of bleomycin using electrochemotherap y Melanoma Resear c h 7(1):10–18, 1997. [73] Richard Heller Jan Schultz, M. Lee Lucas, Mark J. Jaroszeski, Loree C. Heller Richard A. Gilbert, Karin Moelling, and Claude Nicolau. Intradermal delviery of interleukin-12 plasmid dna by in vivo electroporation. DN A and Cell Biolo gy 20(1):21–26, 2001. [74] Joseph D. Hick e y Creating an instrument system to electroporate and monitor the uptak e of uoresecent molecules into cells. Master' s thesis, Uni v ersity of South Florida, Colle ge of Engineering, Department of Chemical Engineering, T ampa, Florida, 2000. [75] Joseph D. Hick e y Modelling the Motion of Ions and Molecules in Electroporation and Electrophoresis Field Conditions. Uni v ersity of South Florida, Colle ge of Arts and Sciences, Department of Ph ysics, T ampa, Florida, 2003. [76] Joseph D. Hick e y and Richard Gilbert. Modeling the electromobility of ions in a tar get tissue. DN A and Cell Biolo gy 22(12):823–828, 2003. 118

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[77] Joseph D. Hick e y and Richard Gilbert. Fluid o w electrophoresis model. Bioelectr oc hemistry 63(2):365–367, 2004. [78] Anthon y L. Hines and Robert N. Maddox. Mass T r ansfer : Fundamentals and Applications Prentice-Hall, Engle w ood Clif fs, Ne w Jerse y 1985. [79] G. A. Hofmann, S. B. De v S. Dimmer and G. S. Nanda. Electroporation therap y: A ne w approach for the treatment of head and neck cancer IEEE T r ansactions on Biomedical Engineering 46(6):752–759, 1999. [80] Gunter A. Hofmann. Instrumentation and electrodes for In V ivo electroporation. [88], pages 37–61. [81] C. Huang, L. Wheeldon, and T E. Thompson. The properties of lipid bilayer membranes seperating tw o aqueous phases : F ormation of a membrane of simple composition. J ournal of Molecular Biolo gy 8:148–160, 1964. [82] IEEE. Pr oceedings of the 1995 IEEE Engineering in Medicine and Biolo gy 17 th Annual Confer ence and 21 st Canadian Medical and Biolo gical Engineering Confer ence Montr eal, Quebec, Canada 1995. [83] IEEE. Pr oceedings of the 18 th Annual International Confer ence of the IEEE Engineering in Medicine and Biolo gy Society Amster dam, Holland, 1996 1996. [84] Herv e Isambert. Understanding the electroporaion of cells and articial bilayer membranes. Physical Re vie w Letter s 80(15):3404–3407, 1998. [85] Jacob N. Israelachvili. Intermolecular and Surface F or ces Academic Press, London, UK, 1997. [86] M. J. Jaroszeski, D. Coppola, G. Nesmith, C. Pottinger M. Hyacinthe, K. Benson, R. Gilbert, and R. Heller Ef fects of electrochemotherap y with bleomycin on normal li v er tissue in a rat model. Eur opean J ournal of Cancer 37:414–421, 2001. [87] M. J. Jaroszeski, D. Coppola, C. Pottinger K. Benson, R. A. Gilbert, and R. Heller T reatment of hepatocellular carcinoma in a rat model using electrochemotherap y Eur opean J ournal of Cancer 37:422–430, 2001. [88] M. J. Jaroszeski, V Dang, C. Pottinger J. Hick e y R. Gilbert, and R. Heller T oxicity of anticancer agents mediated by electroporation in vitr o Anti-Cancer Drugs 11:201–208, 2000. [89] Mark J. Jaroszeski, Richard Gilbert, Claude Nicolau, and Richard Heller In vi v o gene deli v ery by electroporation. Advanced Drug Delivery Re vie ws 35:131–137, 1999. [90] Mark J. Jaroszeski, Richard Gilbert, Claude Nicolau, and Richard Heller Deli v ery of genes in vi v o. [88], pages 173–186. 119

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Appendicies 132

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Appendix A: Potential Dif ference 1 ; 2 A.1 Problem Description Potential dif ference problems are typically go v erned by Laplace' s equation, see equation A.1. Problems v ary as a function of the initial and boundary conditions. (A.1) r 2 = 0 From Laplace' s equation we can deduce that is a constant or a rst order function. F or conductors in electrostatic elds or for electrodes under going o wing current in a conductor = 0. But in a dielectric is a function of position relati v e to the electrode and R @ @ t dS is related to the total char ge on a conductor or the total current between tw o electrodes. Since in electroporation the electric eld conditions are nite in both strength and distrib ution, 0 at innity Another limitation for electroporation conditions are the multiple medias, e.g. the electrodes, the e xtracellular media, the intracellular media and cell membrane. This structure is 2 conductors and 3 dielectrics with v aried dielectric constants from 81 for w ater to 2 for cholesterol [85, 202, 208]. F or simple systems lik e spheres or parallel plates in a homogeneous media, the v alue of is kno wn or can be simply deri v ed b ut for more comple x systems is a series of harmonic functions. The simplest harmonic function is 1 r ¡ r o with r o = 0 Other e xamples of simple harmonic functions are r cos and cos r 2 The three e xamples presented here are cases of the spherical harmonic functions kno wn as Le gendre polynomials. The most commonly used electroporation models used are the VBEMs and the y are directly deri v ed from the spherical harmonics. Figure A.1 is the basis for the VBEMs. It sho ws the initial and boundary conditions that are applied to the model. 133

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Appendix A. (Continued) 6 a bd 1 E Figure A.1: V oltage Breakdo wn Electroporation Model [133] Solving this model is accomplished using the spherical harmonics as described before. Spherical harmonics are typically created from the Le gendre polynomials, P l ( u ) see table A.1 and the associated coef cients, a l [163, 170, 179]. The Le gendre polynomials can be calculated from the Rodrigue formula P l ( u ) = 1 2 l ¢ l d l du l ( u 2 ¡ 1) l [163]. The coef cients of the spherical harmonics used to solv e Laplace' s equation are of the form, a l = A l ¢ r l or A l r l +1 Since needs to be nite at 1 a l = A l r l +1 for l 1 is calculated by taking an innite sum of the associated coef cients times the Le gendre polynomials, see equations A.2 A.5. T able A.1: Le gendre Polynomials Short Hand Notation Expanded F orm P 0 ( u ) 1 P 1 ( u ) u P 2 ( u ) 1 2 (3 u 2 ¡ 1) P 3 ( u ) 1 2 (5 u 3 ¡ 3 u ) 134

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Appendix A. (Continued) A.2 Conducting Sphere in a V acuum Deri ving the VBEM is done by rst selecting initial and boundary conditions. The initial condition is that the cell membrane is intact and that Laplace' s equation is satised, see boundary condition BC.1.1. If the cell is treated lik e a sphere the surf ace closest to the electrodes will feel the greatest amount of v oltage, see gure BC.1.A.1. Therefore the on-equitoral boundary condition must ha v e an cos term, see boundary conditions BC.1.3 and BC.1.2. The tw o boundary conditions BC.1.3 and BC.1.2 deal with the v alue of at r = 0 and r = 1 At both of those re gions the v alue of must be nite. The third boundry applied to the system is that the v alue of is equal to 0 at the surf ace of the cell. T able A.2: Boundary Conditions for the VBEM Model BC.1.1 r 2 = 0 In a v acuum BC.1.2 + E o r cos = nite at r = 0 BC.1.3 + E o r cos = nite at r = 1 BC.1.4 = 0 for r < a The v oltage inside a conductor is 0. The general equation that fullls the requirements of the model is equation A.2. This equation is the combination of an e xact part from boundary conditions 2and 3 and an innite sum. The coef cents of the innite sum need to be solv ed using the boundary conditions listed in table A.2. (A.2) + E o r cos = 1 X 0 a l ¢ P l (cos ) 135

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Appendix A. (Continued) Expanding the summation for the Le gendre polynomial terms. + E o r cos = a 0 ¢ P 0 (cos ) + a 1 ¢ P 1 (cos ) + a 2 ¢ P 2 (cos ) + ¢ ¢ ¢ (A.3) + E o r cos = A 0 r 1 ¢ P 0 (cos ) + A 1 r 2 ¢ P 1 (cos ) + A 2 r 3 ¢ P 2 (cos ) + ¢ ¢ ¢ (A.4) + E o r cos = A 0 r 1 ¢ P 0 (cos ) + A 1 r 2 ¢ P 1 (cos ) + A 2 r 3 ¢ P 2 (cos ) + ¢ ¢ ¢ (A.5) Substituting the Le gendre Polynomial v alues into equation A.5 yields equation A.6 + E o r cos = A 0 r 1 ¢ 1 + A 1 r 2 ¢ cos + A 2 r 3 ¢ 3 2 (cos 2 ¡ 1) + ¢ ¢ ¢ (A.6) Simplifying equation A.6 and setting A 0 = 0 because it does not ha v e c ylindrical symmetry produces equation A.7. = ¡ E o r cos + A 1 r 2 ¢ cos + A 2 r 3 ¢ 3 2 (cos 2 ¡ 1) + ¢ ¢ ¢ (A.7) Further simplication of equation A.7 and the application of boundary condition 4 yields equation A.8. 0 = A 1 a 2 ¡ E o a ¢ cos + A 2 a 3 ¢ 3 2 (cos 2 ¡ 1) + ¢ ¢ ¢ (A.8) Setting A 1 = E o ¢ a 3 and setting A 2 ¢ ¢ ¢ A 1 = 0 yields 0 = E o ¢ a 3 a 2 ¡ E o a ¢ cos (A.9) Substituting E o ¢ a 3 in for A 1 produces = a 3 r 2 ¡ r ¢ E o ¢ cos (A.10) Therefore, the nal result is = a 3 r 2 ¡ r ¢ E o ¢ cos = a 3 ¡ r 3 r 2 ¢ E o ¢ cos (A.11) Once has been calculated, the induced surf ace char ge density of the sphere can easily be calculated. This v alue relates directly to the transmembrane v oltage. Calculating the induced surf ace char ge, sc via the substitution of equation A.11 into sc o = ¡ ¡ @ @ r ¢ r = a [33] 136

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Appendix A. (Continued) sc o = ¡ @ @ r r = a = ¡ @ @ r a 3 r 2 ¡ r ¢ E o ¢ cos r = a (A.12) = ¡ @ @ r a 3 r 2 ¡ ¡ @ @ r r r = a ¢ E o ¢ cos (A.13) = ¡ a 3 @ @ r r ¡ 2 + 1 r = a ¢ E o ¢ cos (A.14) = £ ¡ a 3 ¡ ¡ 2 ¢ r ¡ 3 ¢ + 1 ¤ r = a ¢ E o ¢ cos (A.15) = ¡ a 3 ¡ 2 r 3 + 1 r = a ¢ E o ¢ cos (A.16) = 2 ¢ a 3 r 3 + 1 r = a ¢ E o ¢ cos (A.17) = 2 ¢ a 3 a 3 + 1 ¢ E o ¢ cos (A.18) sc o = 3 ¢ E o ¢ cos (A.19) sc = 3 ¢ o ¢ E o ¢ cos (A.20) Therefore the induced surf ace char ge density described in equation A.20 is only dependant on the applied elecctric eld strength and the incident angle. This is not a true to life scenario for electroporation b ut it is descripti v e of a V an de Graf f generator which is described by = a 3 ¡ r 3 r 2 ¢ E o ¢ cos + Q 4 r A.3 Dielectric Sphere in Dielectric Media Creating a mathematical model closer to actual systems is accomplished using a dielectric sphere in dielectric media, DSDM, is accomplished by dening the initial and boundary conditions. The DSDM model treats the system as a solid sphere of dielectric constant 1 imbedded in a inifnite b ulk media with a dielectric constant of 2 see gure A.2. This 137

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Appendix A. (Continued) model emplo ys spherical symetry which is not sho wn in the 2-D representation of gure A.2. The radius of the dielectric sphere is dened as a see gure A.2. R I a 1 2 Figure A.2: Dielectric Sphere in Dielectric Media The initial and boundary conditions are listed in table A.3. The boundary conditions are Laplace' s equation applied to the tw o re gions, inside and outside the dielectric sphere, BC.2.1. The potential outside the dielectric sphere, 1 will be nite at ininity BC.2.2. The potential inside the dielectric sphere, 2 will be nite, BC.2.3. The tw o potentials will be equal at the surf ace of the dielectric sphere in e v ery direction, BC.2.4. The last boundary condition is deals with the electric displacement, D n and is dened to be continuous at the interf ace between the tw o re gions, BC.2.5. Creating the mathematical model for this system is similar to the method of section A.2. The same techniques are emplo yed, e.g. using the spherical harmonics with the Le gendre polynomials and then applying the boundary conditions of table A.3 one at a time to eliminate or e v alutate constants. The process starts with the general solution of potential difference problems with spherical symetry as sho wn in equation A.2 abo v e. The dif ference 138

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Appendix A. (Continued) T able A.3: Boundary Conditions for the DSDM Model BC.2.1 r 2 1 = 0 r 2 2 = 0 BC.2.2 1 + E 1 r cos = nite at r = 1 BC.2.3 2 = nite at r a BC.2.4 1 = 2 at r = a for all BC.2.5 1 @ 1 @ r = 2 @ 2 @ r at r = a for all is that it is solv ed simultaneously for the tw o dif ferent re gions, inside the dielectric sphere and the b ulk, outside of the dielectic sphere. 1 + E o r cos = 1 X 0 a l ¢ P l (cos ) Solution for b ulk (A.21) 2 = 1 X 0 b l ¢ P l ( cos ) Solution inside sphere (A.22) The solution inside the sphere lacks the electric eld term because the ef fect of the applied electric eld is damped by the b ulk dielectric and the eld inside the sphere is uniform. The ne xt step is to solv e for the constants in the e xpanded forms of the general solutions. The same techniques used abo v e are repeated for the b ulk solution. Inside the sphere is a dif ferent story Since the sphere contains the point r = 0 it requires the other form of the Le gendre polynomials r n and not r ¡ n lik e before. 139

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Appendix A. (Continued) 1 + E o r cos = A 1 r 2 cos (A.23) 2 = 1 X 0 r l ¢ B l ¢ P l ( cos ) (A.24) 2 = r 0 ¢ B 0 ¢ P 0 (cos ) + r 1 ¢ B 1 ¢ P 1 (cos ) (A.25) + r 2 ¢ B 2 ¢ P 2 (cos ) + r 3 ¢ B 3 ¢ P 3 (cos ) ¢ ¢ ¢ Solving for the v alues of the associated constants, B l s, is accomplished by applying the boundary conditions. Primarily BC.2.1, r 2 2 = 0 which requires that B l for l 2 be set = 0 The constant B 0 is set equal to zero because the associated terms are of constant form, r 0 = 1 and p 0 (cos ) = 1 Therefore, the solution for inside the dielectric sphere is 2 = B 1 ¢ r 1 ¢ cos this satises boundary conditions BC.2.1 and BC.2.3. Boundary condition 4 is satised by setting = 0 r = a and 1 = 2 see equation A.28. 1 = 2 (A.26) A 1 r 2 cos ¡ E o r cos = B 1 ¢ r ¢ cos (A.27) A 1 a 2 ¡ E o a = B 1 ¢ a (A.28) The ne xt step is to solv e for the constants A 1 and B 1 using boundary conditions BC.2.4 and BC.2.5. First, the boundary condition BC.2.5 is applied to the respecti v e sides of equation A.27. This ef fti v ely remo v es the radial dependance of the equation, while simultanously introducing the dielectric constants, see equation A.30. Remo ving the common 140

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Appendix A. (Continued) denominator of cos from equation A.30 produces equation A.31 1 @ @ r A 1 r 2 cos ¡ E o r cos r = a = 2 @ @ r ( B 1 ¢ r ¢ cos ) r = a (A.29) 1 ¡ 2 A 1 a 3 cos ¡ E o cos = 2 ( B 1 cos ) (A.30) 1 ¡ 2 A 1 a 3 ¡ E o = 2 B 1 (A.31)Dropping the subscripts on the coef fcients and rearranging equations A.31 and A.28 results is a 2 equation, 2 unkno wn system that can be solv ed through simple substitution. A a 2 ¡ E o a = B ¢ a (A.32) 1 ¡ 2 A a 3 ¡ E o = 2 B (A.33) No w to solv e for A. 1 2 ¡ 2 A a 3 ¡ E o = A a 3 ¡ E o (A.34) 1 2 ¡ 2 A a 3 ¡ 1 2 E o = A a 3 ¡ E o (A.35) 1 2 ¡ 2 A a 3 ¡ 2 A 2 a 3 = 1 2 E o ¡ E o (A.36) 2 1 + 2 2 a 3 A = 2 ¡ 1 2 E o (A.37) A = 2 ¡ 1 2 1 + 2 a 3 E o (A.38)No w to solv e for B. This is accomplished by rst solving equation A.32 for A, see equa141

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Appendix A. (Continued) tion A.39, and then substituting the result into equation A.33, see equation A.40. from equation A.32 A = ( B + E o ) ¢ a 3 (A.39) 1 ( ¡ 2 ( B + E o ) ¡ E o ) = 2 B (A.40) ¡ 2 ( B + E o ) ¡ E o = 2 1 B (A.41) ¡ 3 E o = 2 + 2 1 1 B (A.42) B = ¡ 3 1 2 + 2 1 E o (A.43)The last step is to substitute equations A.42 and A.43 into equation A.27. The resulting equation, A.44 is the specic solution for the dielectric sphere in dielectric media system at the interf ace between the tw o discrete elements. 2 ¡ 1 2 1 + 2 a 3 E o r 2 cos ¡ E o r cos = ¡ 3 1 2 + 2 1 E o ¢ r ¢ cos (A.44) Simplifying the abo v e equation yields 2 ¡ 1 2 1 + 2 a 3 r 2 E o cos ¡ r E o cos = ¡ 3 1 2 + 2 1 E o ¢ r ¢ cos (A.45) Since this is an area of interest for the electroporation modeler a deeper e xamintation is in order V alues for the dielectric constant of saline and cholesterol, 79.8 and 5.41, were found in the literature and were used to simplify the system of equations at the sphere-b ulk interf ace [85, 168, 202]. The nal result of equation A.47 is the result listed in most of the electroporation literature and w as rst described by T urnb ull in 1973 for the transmembrane 142

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Appendix A. (Continued) potential of a cell in a dielectric media. 1 = 5 : 41 ¡ 79 : 8 2 ¢ 79 : 8 + 5 : 41 a 3 r 2 E o cos ¡ r E o cos (A.46) 2 = ¡ 3 ¢ 79 : 8 5 : 41 + 2 ¢ 79 : 8 E o ¢ r ¢ cos ¡ 3 2 E o ¢ r ¢ cos (A.47) 143

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Appendix B: F orce Model of Electroporation The acceleration of an object is directly proportional to the net force acting on it and is in v ersely proportional to its mass. -Sir Issac Ne wton Sir Issac Ne wtion is stating that a = F m or more precisely a = P F m Not only does this simple equation go v ern billiard balls and car wrecks, it also leads directly to electroporation because as long as the sum of the forces acting on a re gion of cell membrane are equal to zero, there is no acceleration and the membrane may not rupture. The rst step to understanding the forces applied by an electric eld on a cell is to learn the permeability of a bilayer lipid membrane, BLM, to common biological substances. These v alues range from 1 £ 10 ¡ 12 cm s for small ions to 5 £ 10 ¡ 2 cm s for w ater [184], see table B.1. Since an a v erage biological membrane is 40 A [85, 184], or 4 £ 10 ¡ 7 cm in thickness most of these substances can rapidly tra v el e xcept the ions. The limited dif fusi v e properties of the ions is the reasons that cells ha v e created transmembrane proteins to usher in ions. The ef fect of the electric eld on these ions may be what causes electroporation and blebbing of the cell membrane. The force model that this appendix refers to is essentially a force balance that e v entually ends in imbalance if the eld duration or strength e xceeds the membrane' s capability for elastic deformation. Examining this force balance from the anode side of the membrane has the form of F T = F ef ¡ ¡ F ef + ¡ F f ¡ F mr schematically this is represented in gure B.1. The four forces as described here are the attracti v e force of the anode on anions, F ef ¡ the force of cations pushing to w ards the cathode, to satisfy the char ge imbalance in the cell, 144

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Appendix B. (Continued) T able B.1: Bilayer Lipid Membrane Permeabilities to Common Biological Substances Biological Substance Na + K + Cl ¡ Glucose & T ryptophan Permeability cm s 1 £ 10 ¡ 12 6 £ 10 ¡ 11 2 £ 10 ¡ 10 4 £ 10 ¡ 7 Biological Substance Urea Glycerol Indole H 2 O Permeability cm s 5 £ 10 ¡ 5 5 £ 10 ¡ 5 8 £ 10 ¡ 3 5 £ 10 ¡ 2 F ef + the frictional force k eeping the membrane patch from slipping its bonds, F f and the restoring force of the membrane to e xternal forces, F mr The restoring force is analogous to Hook e' s la w F = ¡ k £ x This model is an adaptation of e xisting FBEMs [22, 23, 54, 55]. F ef ¡ F ef + F f F mr Figure B.1: Sum of the F orces During Electroporation Deri ving the model in question from rst principles is done through Ne wton' s second la w the dri ving force F d f and the retarding forces, F r f The dri ving force in this instance is Coulomb' s la w This process is described in equations B.1 through B.3 belo w There are a fe w tricks used belo w E = ¢ V ¢ x ; a = d 2 x dt 2 and separation of v ariables w as used to 145

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Appendix B. (Continued) simplify f ( x; t ) to f ( x; T ) = f 1 ( T ) f 2 ( x ) F T = X F = ma (B.1) F T = F d f ¡ F r f = ¡ q E ¡ f ( x; T ) = ma (B.2) F T = ¡ q dV dx ¡ f 1 ( T ) f 2 ( x ) = m d 2 x dt 2 (B.3) 0 L q dV dx w exp ¡ ¢ E d R T ¢ x Figure B.2: Simplied Sum of the F orces During Electroporation Setting up the inte grals to solv e equation B.3 yields equation B.4. Solving these integrals required tw o small tricks. The rst trick is the identication of almost the result of a chain rule operation in the right hand side of equation B.4, d 2 xdx The actual chain rule result is d ( dx 2 ) = 2 dxd 2 x since there is not a free 2 in equation B.4 both a 2 and a 1 2 w as introduced. The second trick is a change of limits, the dv 2 term in equation B.6 w as changed to a v and the limit w as changed to v 2 d 146

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Appendix B. (Continued) ¡ q V e Z V i dV ¡ f 1 ( T ) L Z 0 f 2 ( x ) dx = m v d Z 0 d 2 x dt 2 dx (B.4) ¡ q V e Z V i dV ¡ f 1 ( T ) L Z 0 f 2 ( x ) dx = 1 2 m v d Z 0 d dx 2 dt 2 = 1 2 m v d Z 0 d dx dt 2 (B.5) ¡ q V e Z V i dV ¡ f 1 ( T ) L Z 0 f 2 ( x ) dx = 1 2 m v d Z 0 d ¡ v 2 ¢ (B.6) q ( V i ¡ V e ) ¡ f 1 ( T ) [ f 2 ( L ) ¡ f 2 (0)] = 1 2 mv 2 d (B.7) The nal solution w as simplied with tw o substitutions for the separation of v ariables components. The tw o substitions are for the temperature dependant motion of the membrane patch and the ener gy required to remo v e the membrane patch. The Arrhenius equation w as substituted in for the temperature rate of motion of the phospholipid-protein patch, f 1 ( T ) = exp ¡ ¢ E d R T ¢ This is consistant with the material science method of ionic and crystal motion [5]. The terminology for the ener gy requried to remo v e the membrane patch w as absconded from the materials science term for the ener gy required to remo v e an electron from a material. This is kno w as the w ork function and is designated by The substitution here w as ¢ f 2 = w [5, 188]. The result after these tw o substitutions and a rearrangement is equation B.8. This equation can be used to calculate the transmembrane v oltage required to porate cells. (B.8) ¢ V = 1 ¡ q w exp ¢ E d R T + 1 2 mv 2 d This equation agrees with e xperimental data, r = 0.914 [22]. 147

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Appendix C: Conducti vity [128] C.1 Introduction Conducti vity is the scientic measure of conductance. It typically refers to the o w of electric char ge, ionic char ge or heat from a dense re gion to a sparce re gion. This appendix deals with the ionic conducti vity of a three part system. The three components of the syster are intracellular membrane, and e xtracellular of a cell. The dif ferent re gions ha v e their o wn specic conducti vity The inside of a cell is an aqueous cationic en vironment with high concentrations of potassium and lo w concentrations of sodium, chloride, magnesium and calcium [123], see table C.1. The uid bilipid membrane is a leak y conductor with acti v e and passi v e tranport for dif ferent ions. By maintaining this concentration gradient a cell creates an inate transmembrane electrical potential, kno w as the resting potential [44]. T able C.1: Concentrations of Critical Ions in Body Fluids [44, 123] Re gion Na + K + Cl ¡ Nondif fusable Char ged ions sum Extracellular 145 4 120 None +29 Fluid Membrane None None None None None Intracellular 12 155 4 155 -8 Fluid The concentration units for the ions presented here are in mil imol es l This electrical and chemical gradient is maintained through both acti v e and passi v e transport mechanisms [123]. P assi v e transport is accomplished via ion channels that w ork148

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Appendix C. (Continued) ing with a concentraion gradient and may contain a selecti vity re gion for a specic species. Acti v e transport is mediated by carriers, which act to carry ions or molecules ag ainst a concentration gradient. Carriers utilize tw o main methods of transport for mo ving ions and molecules, primary transport or gradient f acilitated transport. The most common e xample of primary transport is the Na + -K + A TP ase 1 This acti v e transport system maintains the electrical and chemical gradient by using A TP to po wer a protein that mo v es three Na + ions out of the cell, while mo ving tw o K + ions into the cell along with a dephosphorylation of and A TP molecule. An increase in the concentration of ions in a re gion decreases the resistance to current o w and increases the conducti vity of a re gion [44, 45]. T able C.1 sho ws the dif ference in the concentration of ions in the dif ferent re gions associated with a cell. This in turn means that the dif ferent re gions will respond dif ferently to applied electric elds. C.2 Examining the Conducti vity in a Three P art System 2 (C.1) f ( ) = 2 o 2 m + i + ( m ¡ i ) a ¡ d a 3 ¡ 3 m a ¡ d a # (2 m + i ) (2 o + m ) + 2 a ¡ d a 3 ( i ¡ m ) (2 m ¡ o ) The conducti vity f actor f described in chapter 2 is dependant on the conducti vity of the cell membrane, m the e xtracellular media, o and the intracellular media, i Equa1 Other e xamples of A TP ases are the H + -K + A TP ase responsible for H + secretion in the g astic mucosa and the Ca ++ A TP ase that pumps Ca ++ into the sarcoplasmic reticulum [123] 2 This section is based loosely on the w ork of Eberhard Neumann in Electroporation and Electrofusion in Cell Biology [126] 149

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Appendix C. (Continued) 6 a bd 1 E Figure 2.2: Thin W alled Approximation tion C.1 describes the conducti vity term for the mix ed three part system. The deri v ation of this equation is be yond the scope of the document and is only presented here for completeness [128]. Equation C.1 incorporates the three s as well as the associated radii, see gure 2.2 from the introduction section. F or most mammalian cells a >> d and this allo ws for the substitution a ¡ d a 3 1 ¡ 3 d a and that substitution simplies C.1 to equation C.2. (C.2) f ( ) = o i 2 d a (2 o + i ) m + 2 d a ( o ¡ m ) ( i ¡ o ) If the intracellular ionic concentration drops or the e xtracellular ionic concentration increases this conducti vity f actor f ( ) increases. This f actor is typically ignored when dealing with electroporation because the describing equations were created for phospholipid membranes rather than cells [24, 25, 140– 143, 190]. The leakiness or transmission of ions through the membrane is typically ignored for electroporation b ut it should not be. If 150

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Appendix C. (Continued) m is lar ge that w ould ha v e a signicant contrib ution of the equations. This is especially true if small re gions of a cell were modeled. 151

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Appendix D: Deri ving the T ime Constant: c Equation 2.4 from chapter 2 alluded to a time constant associated with membrane char ging prior to electroporation. This time constant stems from relaxation theory where applied or remo v ed elds, while ha ving instantaneous ef fects on the re gions encompased by the elds, the elements af fected by those elds do not respond instantaneously [174]. This time lag results in a distortion of the responce curv e when compared to the input, see gure D.1. (2.4) V T M = 3 2 f b E cos 1 ¡ exp ¡ t c 6Applied V oltageT ime 6T ransmembrane PotentialT ime a b Figure D.1: Responce Curv e W ith T ime Lag This time constant for the polarization of the transmembrane eld is a possible reason for the ef fecti v eness of dif ferent v oltage/duration electroporation signatures. Current literature has the electroporation signature from hundreds of v olts for millisecond pulses to kilo v olts at microsecond pulses to me g a v olts at nanosecond pulses [79, 80, 92, 145]. The 152

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Appendix D. (Continued) time constant of the transmembrane potential describes the responce of the ions to the applied electric eld. One description of the time constant of a cell membrane w as postulated by Scha wn in 1957. This equation describes the transmembrane time constant as a function of the specic conducti vities of the inside, i membrane, m and outside, o re gions, and the cell membrane' s capacitance, C m and conductance, G m see equation D.1. (D.1) c = a ¢ C m i + 2 o 2 i o + a ¢ G m ( i + 2 o ) 153

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Appendix E: P acking Analysis for Spheres The packing analysis presented here represents a sphere created by the interaction of 1000 equal sized spheres. It assumes that the spheres are deformable and produce zero v oid space. Since the initial sphere radii are an approximation this approximation may or may not af fect the analysis. The initial radii for the indi vidual spheres is 600 pm. v olume of a individual sphere = 4 3 r 3 (E.1) = 4 3 (600 pm ) 3 (E.2) = 4 3 £ 2 : 16 £ 10 8 pm 3 (E.3) equal v olume of 1000 spheres = 1000 £ 4 3 £ 2 : 16 £ 10 8 pm 3 (E.4) = 4 3 £ 2 : 16 £ 10 11 pm 3 (E.5) radius of a sphere of kno w v olume = 3 r 3 4 v (E.6) radius of an equal v olume sphere = 3 r 3 4 4 3 £ 2 : 16 £ 10 11 pm 3 (E.7) radius of an equal v olume sphere = 3 p 2 : 16 £ 10 11 pm 3 = 6000 pm (E.8) 154

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Appendix F: Computing Acceleration and V elocity of Ions in V iscous Media This appendix is an e xtension of chapter 5, solving for the speed and acceleration of potassium ions in a 0.9% NaCl solution in a 1500 V cm electric eld. s = s o + a t (F .1) F = F e ¡ F f = ma (F .2) F = z e E ¡ 6 r s = ma (F .3) s = ma ¡ z e E ¡ 6 r (F .4) Substituting equation F .4 into equation F .1 for s yields equation F .5. ma ¡ z e E ¡ 6 r = s o + a t (F .5) ma ¡ z e E = ¡ 6 r s o ¡ 6 r a t (F .6) ma + 6 r a t = z e E ¡ 6 r s o (F .7) a ¢ ( m + 6 r t ) = z e E ¡ 6 r s o (F .8) a = z e E + 6 r s o m + 6 r t (F .9) Rearranging equation F .9 and substituting it into equation F .1 a = z e E 6 r t + m (F .10) s = s o + z e E 6 r t + m t (F .11) (F .12) The tw o equations are solv ed by assuming that s o is zero and a table of times. The rst acceleration v alue is computed using equation F .10 with s o equal to zero and time is t i ¡ t o 155

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Appendix F (Continued) where t o = 0 The result of that is plugged into F .1 with s o equal to zero and time is t i ¡ t o where t o = 0 This process is repeated by setting s o = s incrementing t i and subtracting the current v alue of t i from the pre vious v alue of t i e.g. in pseudocode t i ( i ) ¡ t i ( i ¡ 1) Using this v alue for s o This process w as repeated for time v alues ranging from 1 £ 10 ¡ 16 to 1 £ 10 ¡ 10 see gure 1. 0 5e+10 1e+11 1.5e+11 2e+11 2.5e+11 3e+11 3.5e+11 4e+11 1e-16 1e-15 1e-14 1e-13 1e-12 1e-11 1e-10 0 0.002 0.004 0.006 0.008 0.01 0.012 Acceleration (m/s2) Speed (m/s)Time (s) Acceleration Speed Figure F .1: Plot of Acceleration and Speed in 0.9% NaCl Solution 156

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Appendix F (Continued) T able F .1: V elocity and Acceleration for K + Ions in a 0.9 Mass % NaCl Solution T ime (sec) Acceleration ¡ m s 2 ¢ V elocity ¡ m s 2 ¢ 0 3.700 £ 10 11 0 1.0 £ 10 ¡ 16 3.687 £ 10 11 3.687 £ 10 ¡ 5 3.0 £ 10 ¡ 16 3.648 £ 10 11 1.463 £ 10 ¡ 4 6.0 £ 10 ¡ 16 3.571 £ 10 11 3.605 £ 10 ¡ 4 1.05 £ 10 ¡ 15 3.273 £ 10 11 1.836 £ 10 ¡ 3 3.6 £ 10 ¡ 15 2.454 £ 10 11 3.495 £ 10 ¡ 3 6.6 £ 10 ¡ 15 1.431 £ 10 11 6.364 £ 10 ¡ 3 1.05 £ 10 ¡ 14 6.154 £ 10 10 8.653 £ 10 ¡ 3 3.0 £ 10 ¡ 14 2.795 £ 10 8 1.0372 £ 10 ¡ 2 4.96 £ 10 ¡ 14 5.337 £ 10 5 1.037973 £ 10 ¡ 2 7.03 £ 10 ¡ 14 5.152 £ 10 2 1.037975 £ 10 ¡ 2 8.2 £ 10 ¡ 14 9.541 £ 10 0 1.037975 £ 10 ¡ 2 9.03 £ 10 ¡ 14 5.557 £ 10 ¡ 1 1.037975 £ 10 ¡ 2 1.04 £ 10 ¡ 13 5.978 £ 10 ¡ 3 1.037975 £ 10 ¡ 2 1.23 £ 10 ¡ 13 9.0531 £ 10 ¡ 6 1.037975 £ 10 ¡ 2 157

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Appendix G: P acking Analysis for Cylinders The packing analysis presented here represents a sphere created by the interaction of 1000 equal sized c ylinders. It assumes that the c ylinders are deformable and produce zero v oid space. Since the initial c ylinder length and radii are e xperimental and accepted literature v alues, this is a better approximation than appendix E. The initial radii and length for the indi vidual c ylinders is 2.1 nm and 0.34 nm respecti v ely [113, 185]. v olume of a individual cylinder = £ r 2 cy l £ l cy l (G.1) = £ (2 : 1 nm ) 2 £ 0 : 34 nm (G.2) = £ 4 : 41 nm 2 £ 0 : 34 nm (G.3) = £ 1 : 50 nm 3 (G.4) equal v olume of 1000 cylinders = 1000 £ £ 1 : 5 nm 3 (G.5) = £ 1500 nm 3 (G.6) radius of a sphere of kno wn v olume = 3 r 3 4 v (G.7) radius of an equal v olume sphere = 3 r 3 4 £ 1500 nm 3 (G.8) radius of an equal v olume sphere = 3 p 1125 nm 3 = 10 : 4 nm (G.9) 158

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Appendix H: P acking Analysis for Cells in Cuv ette The packing analysis presented here represents 5 £ 10 6 cells in one mL of media. It is used to calculated the v oid fraction for electroporation cuv ette e xperiments. The cells ha v e a diameter of 50 m. V cel l = 4 3 r 3 = 4 3 (25 m ) 3 = 4 3 £ 15625 m 3 = 6 : 545 £ 10 4 m 3 (H.1) V T otal = # of cel l s £ v ol ume of a C el l (H.2) = 5 £ 10 6 £ 6 : 545 £ 10 4 m 3 = 3 : 273 £ 10 11 m 3 (H.3) = 0 : 3273 mL (H.4) The packing fraction is only 32.7%. Therefore, the cells only tak e up 1 3 of the solution v olume for cells with a diameter of 50 m. 159

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Appendix I: Flo w P attern by Direction F orw ard Right Left 0.120818 0.510931 0.368252 0.267348 0.235076 0.497576 0.635589 0.122484 0.241927 0.091262 0.150115 0.758624 0.652168 0.344283 0.003549 0.533845 0.313671 0.152485 0.762832 0.216919 0.02025 0.000932 0.32818 0.670888 0.13544 0.569263 0.295297 0.565513 0.180995 0.253491 0.655744 0.031831 0.312425 0.030192 0.430353 0.539455 0.120139 0.155263 0.724598 0.824013 0.031342 0.144646 0.159548 0.44086 0.399592 0.137434 0.124618 0.737948 0.840371 0.018462 0.141167 0.829565 0.085598 0.084837 0.992458 0.005727 0.001815 0.106606 0.236403 0.65699 0.65731 0.138888 0.203802 0.676188 0.098387 0.225424 0.345125 0.313913 0.340962 F orw ard Right Left 0.24235 0.135212 0.622438 0.420033 0.292815 0.287152 0.457975 0.441016 0.101009 0.074456 0.568375 0.357169 0.853056 0.099949 0.046995 0.132008 0.565399 0.302594 0.855436 0.046053 0.098511 0.118903 0.531226 0.349871 0.814216 0.086536 0.099248 0.502735 0.263967 0.233298 0.206651 0.149451 0.643897 0.110444 0.683865 0.205691 0.260276 0.338376 0.401348 0.112048 0.169936 0.718015 0.528267 0.274382 0.197351 0.748096 0.062105 0.189799 0.629983 0.045464 0.324553 0.062928 0.596564 0.340508 0.762834 0.01343 0.223736 0.270766 0.558205 0.171029 0.216025 0.212937 0.571038 0.021564 0.416175 0.562261 0.804714 0.033695 0.161591 160

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Appendix J: P arallel Plate Concentration Contour PlotsNormalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure J.1: P arallel Plate Induced Motion, 1500 V cm Field after 0, 50, and 100 ms 161

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Appendix J. (Continued)Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure J.2: P arallel Plate Induced Motion, 1500 V cm Field after 150, 200, and 250 ms 162

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Appendix J. (Continued)Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure J.3: P arallel Plate Induced Motion, 1500 V cm Field after 300, 350, and 400 ms 163

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Appendix J. (Continued)Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure J.4: P arallel Plate Induced Motion, 1500 V cm Field after 450, and 500 ms 164

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Appendix K: Needle Array Concentration Contour PlotsNormalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure K.1: Needle Array Induced Motion in a 1500 V cm Field after 0, 50, and 100 ms 165

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Appendix K. (Continued)Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure K.2: Needle Array Induced Motion in a 1500 V cm Field after 150, 200, and 250 ms 166

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Appendix K. (Continued)Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure K.3: Needle Array Induced Motion in a 1500 V cm Field after 300, 350, and 400 ms 167

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Appendix K. (Continued)Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5Normalized Concentration 0 2 4 6 8 10 Row Nodes 0 2 4 6 8 10 Column Nodes 0 0.5 1 1.5 2 2.5 Figure K.4: Needle Array Induced Motion in a 1500 V cm Field after 450, and 500 ms 168

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Appendix L: P arallel Plate Electroporation Applicator Flo w Model Code This appendix contains all of the code required to repeat the paralell plate electroporation applicator o w model project. Just type it in and compile it using mak e. It is brok en up into the separate sections, the y are each a dif ferent c code or header le. The follo wing sections are Mak ele, see page 170, it contains a list of all the ccode and header les required to run the code. It mak e compilation simple using gnu mak e program. Ne xt is rndsqr .c on page 171, this is the main program. It initializes the arrays, prototypes the v ariables, calls all of the subprograms and handles all of the returns and reads from and writes to les. F ollo wing that are the P ark and Miller random number generator les. There is a header le and a code le, ran0.h and ran0.c see pages 183 and 184 respecti v ely This code randomly select a o w v alue between 0 and 1. The ne xt set of included les are rand2.h and rand2.c, see pages 186 and 187 respecti v ely This code is used to select the direction of tra v el. There is a safety check in rndqr .c to guarantee that the direction is only chosen once. 169

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Appendix L. (Continued) L.1 Mak ele # makefile for the random number generator ran: ran0.o rndsqr.o rand5.o rand2.o g++ -o ran ran0.o rndsqr.o rand5.o rand2.o rndsqr.o: rndsqr.c ran0.h rand5.o rand2.o g++ -c rndsqr.c ran0.o: ran0.c ran0.h g++ -c ran0.c rand5.o: rand5.c rand5.h g++ -c rand5.c rand2.o: rand2.c rand2.h g++ -c rand2.c 170

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Appendix L. (Continued) L.2 P arallel Plate Electroporation Model Ccode, rndsqr .c /* rndsqr.c update 20030315 */ #include #include #include #include "ran0.h" #include "rand5.h" #include "rand2.h" struct flow {float A; float C; float E;}; /* this program calls the random number generator ran0.c and */ /* uses that program to calculate random numbers to fill a matrix */ /* the matrix is going to be used as a tissue model. */ /* the first random number is dropped because it is not all that random */ float conc[121]; main(){ int i, k, l; float m, n; float sconc[121], conc_ph = 0.0; float j; long int t, u; 171

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Appendix L. (Continued) float result; FILE* datafile; FILE* datafile2; FILE* datafile3; FILE* datafile4; FILE* datafile5; FILE* datafile6; FILE* datafile7; FILE* datafile8; /* priming the random number generator seeds */ u = sqrt(time(NULL))+cbrt(time(NULL))+time(NULL); t = sqrt(time(NULL))+cbrt(time(NULL))+time(NULL); /* result = ran0(&t);*/ m = ran0(&u); n = rand2(&t); /* This is to randomly generate a number between 1 and 2 */ /* and to decide if that number has alread been used */ datafile = fopen("test.data","a"); datafile2 = fopen("f_A.data","a"); datafile3 = fopen("f_C.data","a"); 172

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Appendix L. (Continued) datafile4 = fopen("f_E.data","a"); datafile5 = fopen("f_out_right.data","a"); datafile6 = fopen("f_out_left.data","a"); datafile7 = fopen("init_conc.data","w"); datafile8 = fopen("first_change.data","w"); /* setting up counters */ k=0;/* I am using an 11 by 11 square */ /* This allows for easily finding the center and having a midpoint to start from */ /* the value of of x in f[x] has to be 1 greater than i */ struct flow f[122]; for(i=1; i<=121; i++) { /* o[1]=0; o[2]=0;o[3]=0;*/n = 0; f[i].A = ran0(&u); while (n != 1 && n != 2) { n= (int) rand2(&t); 173

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Appendix L. (Continued) } /* these two conditional statements describe the pick process */ /* if the first # chosen is a 1 then the first statement executes */ /* if the second # chosen is a 1 then the second statement executes */ if(n==1) { f[i].C=(1-f[i].A) ran0(&u); f[i].E=(1-f[i].A-f[i].C); } if( n==2) { f[i].E=(1-f[i].A) ran0(&u); f[i].C=(1-f[i].A-f[i].E); } /* This next line puts all of the data into one file */ fprintf(datafile,"%i %f\t%f\t%f\t%f\n",i,f[i].A, f[i].C, f[i].E, f[i].A+f[i].C+f[i].E);/* This next line puts the f.A data into one file, one 174

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Appendix L. (Continued) data point per line */ /* It also puts a second carraige return every 11 data points */ fprintf(datafile2,"%f\n",f[i].A);if ( i % 11 == 0) fprintf(datafile2,"\n"); /* This next line puts the f.C data into one file, one data point per line */ /* It also puts a second carraige return every 11 data points */ fprintf(datafile3,"%f\n", f[i].C); if ( i % 11 == 0) fprintf(datafile3,"\n"); /* This next line puts the f.E data into one file, one data point per line */ /* It also puts a second carraige return every 11 data points */ fprintf(datafile4,"%f\n", f[i].E); if ( i % 11 == 0) fprintf(datafile4,"\n"); } for ( i=1; i <=121; i++) { 175

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Appendix L. (Continued) /* this set of code keeps the sides from cycling back on themselves */ if (i%11 !=0) fprintf(datafile5," %0.3f \n",f[i].C-f[i+1].E); if(i%11 == 0) fprintf(datafile5,"%0.3f \n",f[i].C); if(i %11 ==0 ) fprintf(datafile5,"\n"); if ((i)%11!=1) fprintf(datafile6," %0.3f \n",f[i].E-f[i-1].C); /* printf(" %0.3f \n",f[i].E-f[i-1].C); */ if((i)%11==1) fprintf(datafile6,"%0.3f \n",f[i].E); /* printf(" %0.3f \n",f[i].E);*/ if(i %11 ==0 ) fprintf(datafile6,"\n"); } /* This subroutine creates the initial conc. profile and writes it to 2 files */ /* The first file is the initial conc. profile file and the 2nd file is the moving conc. profile */ 176

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Appendix L. (Continued) for(i=1; i<=121; i++) { if (i==49 || i==50 || i==51 || i==60 || i==61 || i==62 || i==71 || i==72 || i==73) { j=1;sconc[i]=j; } else { j=0;sconc[i]=j; } fprintf(datafile7,"%f\n",sconc[i]);fprintf(datafile8,"%f\n",sconc[i]);/* printf("%1.0f ",sconc[i]);*/ if (i%11==0) { fprintf(datafile7,"\n");fprintf(datafile8,"\n");/* printf("\n");*/ } if(i==121) fprintf(datafile8,"\n"); 177

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Appendix L. (Continued) conc_ph = conc_ph + sconc[i]; } /* printf("%f \n",conc_ph);*/ /* conc[i] has to be a function of position i and time t so it should be conc[i,t] the more general but harder to write to a file and call from gnuplot or conc[i].t where t = 0 -> 5 for this first example */ printf("\n");/* for(i=1; i<=121; i++) { printf("%0.3f ",sconc[i]); if (i%11==0) printf("\n");}*/ conc_ph=0;for (k=1; k<=10; k++) { for(i=1; i<=121; i++) { if(i<=11) { if (i==1) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E sconc[i]*f[i].A 178

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Appendix L. (Continued) sconc[i]*f[i].C sconc[i]*f[i].E; } if (i>1 && i<11) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E + sconc[i-1]*f[i-1].C sconc[i]*f[i].A sconc[i]*f[i].C sconc[i]*f[i].E; } if (i==11) { conc[i] = sconc[i] + sconc[i-1]*f[i-1].C sconc[i]*f[i].A sconc[i]*f[i].C sconc[i]*f[i].E; } } if(i>11) { if(i%11!=0 && i%11!=1) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E + sconc[i-1]*f[i-1].C + sconc[i-11]*f[i-11].A sconc[i]*f[i].A sconc[i]*f[i].C sconc[i]*f[i].E; } if( i%11==1) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E + sconc[i-11]*f[i-11].A sconc[i]*f[i].A sconc[i]*f[i].C sconc[i]*f[i].E; 179

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Appendix L. (Continued) } if(i%11==0) { conc[i] = sconc[i] + sconc[i-1]*f[i-1].C + sconc[i-11]*f[i-11].A sconc[i]*f[i].A sconc[i]*f[i].C sconc[i]*f[i].E; } } fprintf(datafile8,"%f \n",conc[i]); /* printf("%f ",conc[i]); */ if(i%11==0) { /* printf("\n");*/ fprintf(datafile8,"\n"); } /* This next line writes the conc[i] values to a holder variable that can be accessed */ /* in the next loop to be used as last time values */ /* sconc[i] = conc[i];*/ /* this line has to be after all of the tests but before the close of the i indicie */ conc_ph = conc_ph + sconc[i]; if(i==121) { 180

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Appendix L. (Continued) fprintf(datafile8,"\n");/* printf("\n"); */ } } for(i=1;i<=121;i++) { /* This next line writes the conc[i] values to a holder variable that can be accessed */ /* in the next loop to be used as last time values*/ sconc[i] = conc[i]; } printf("%f \n",conc_ph); conc_ph =0; /* printf("\n");*/ } /* this block of code is a test to see what the system is outputing printf("%1.0f ",conc[i,k]); if(i%11==0) printf("\n"); if(i==121) { 181

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Appendix L. (Continued) printf("\n"); } /* this block of code is a test to see what the system is outputing */ fclose(datafile);fclose(datafile2);fclose(datafile3);fclose(datafile4);fclose(datafile5);fclose(datafile6);fclose(datafile7);fclose(datafile8); } 182

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Appendix L. (Continued) L.3 Flo w V alue Random Number Generator Header File, ran0.h /* this is the header file for ran0.c a random number generator */ float ran0(long *idum); 183

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Appendix L. (Continued) L.4 Flo w V alue Random Number Generator Ccode File, ran0.c #include /* this program generates random numbers using the method presented */ /* in the book "Numerical Recipies in C" */ /* it is listed on page 278 by Park and Miller */ /* this generates a random number between 0.0 and 1.0 */ #include "ran0.h" #define IA 16807 #define IM 2147483647 #define AM (1.0/IM) #define IQ 127773 #define IR 2836 #define MASK 123459876 float ran0 (long *idum) { long k; float ans; /* printf("%12i\t", *idum); */ /* the ˆ= is and exclusive or */ *idum ˆ= MASK; /* The ˆ= is an XOR */ k = (*idum)/IQ; 184

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Appendix L. (Continued) *idum = IA (*idum k IQ) IR k; if (*idum < 0) *idum += IM; ans = AM (*idum); *idum ˆ= MASK; return ans; } 185

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Appendix L. (Continued) L.5 Directional Random Number Generator rand2.h /* this is the header file for ran0.c a random number generator */ float ran0(long *idum); 186

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Appendix L. (Continued) L.6 Directional Random Number Generator Ccode File, rand2.c #include /* this program generates random numbers using the method presented */ /* in the book "Numerical Recipies in C" */ /* it is listed on page 278 by Park and Miller */ /* this generates a random number between 0.0 and 1.0 */ #include "ran0.h" #define IA 16807 #define IM 2147483647 #define AM (1.0/IM) #define IQ 127773 #define IR 2836 #define MASK 123459876 float ran0 (long *idum) { long k; float ans; /* printf("%12i\t", *idum); */ /* the ˆ= is and exclusive or */ *idum ˆ= MASK; /* The ˆ= is an XOR */ k = (*idum)/IQ; 187

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Appendix L. (Continued) *idum = IA (*idum k IQ) IR k; if (*idum < 0) *idum += IM; ans = AM (*idum); *idum ˆ= MASK; return ans; } 188

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Appendix M: Needle Array Applicator Electroporaiton Flo w Model Code This appendix contains all of the code required to repeat the needle array applicator electroporation o w model project. Just type it in and compile it using mak e. It is brok en up into the separate sections, the y are each a dif ferent c code or header le. The follo wing sections are Mak ele, see page 190, it contains a list of all the ccode and header les required to run the code. It mak e compilation simple using gnu mak e program. Ne xt is needle.c on page 191, this is the main program. It initializes the arrays, prototypes the v ariables, calls all of the subprograms and handles all of the returns and reads from and writes to les. F ollo wing that are the P ark and Miller random number generator les. There is a header le and a code le, ran0.h and ran0.c see pages 204 and 205 respecti v ely This code randomly select a o w v alue between 0 and 1. The ne xt set of included les are rand2.h and rand2.c, see pages 207 and 208 respecti v ely This code is used to select the direction of tra v el. There is a safety check in needle.c to guarantee that the direction is only chosen once. 189

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Appendix M. (Continued) M.1 Mak ele # makefile for the random number generator ned: ran0.o needle.o rand5.o rand2.o g++ -o ned ran0.o needle.o rand5.o rand2.o needle.o: needle.c ran0.h rand5.h rand2.h g++ -c needle.c ran0.o: ran0.c ran0.h g++ -c ran0.c rand5.o: rand5.c rand5.h g++ -c rand5.c rand2.o: rand2.c rand2.h g++ -c rand2.c 190

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Appendix M. (Continued) M.2 Needle Applicator Electroporation Model Ccode, needle.c /* needle.c update 20030605 */ #include #include #include #include "ran0.h" #include "rand5.h" #include "rand2.h" struct flow {float A; float C; float E;}; /* this program calls the random number generator ran0.c and */ /* uses that program to calculate random numbers to fill a matrix */ /* the matrix is going to be used as a tissue model. */ /* the first random number is dropped because it is not all that random */ float conc[121]; main(){ int i, k, l,o=121; float m, n, p, nf[o]; float sconc[121], conc_ph = 0.0; float j; long int t, u; 191

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Appendix M. (Continued) float result; FILE* datafile; FILE* datafile2; FILE* datafile3; FILE* datafile4; FILE* datafile5; FILE* datafile6; FILE* datafile7; FILE* datafile8; FILE* needleforce; /* priming the random number generator seeds */ u = sqrt(time(NULL))+cbrt(time(NULL))+time(NULL); t = sqrt(time(NULL))+cbrt(time(NULL))+time(NULL); /* result = ran0(&t);*/ m = ran0(&u); n = rand2(&t); /* This is to randomly generate a number between 1 and 2 */ /* and to decide if that number has alread been used */ datafile = fopen("test.data","a"); datafile2 = fopen("f_A.data","a"); 192

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Appendix M. (Continued) datafile3 = fopen("f_C.data","a"); datafile4 = fopen("f_E.data","a"); datafile5 = fopen("f_out_right.data","a"); datafile6 = fopen("f_out_left.data","a"); datafile7 = fopen("init_conc.data","w"); datafile8 = fopen("first_change.data","w"); needleforce = fopen("needshape.data","r"); /* setting up counters */ k=0;/* I am using an 11 by 11 square */ /* This allows for easily finding the center and having a midpoint to start from */ /* the value of of x in f[x] has to be 1 greater than i */ struct flow f[122]; for(i=1; i<=121; i++) { /* o[1]=0; o[2]=0;o[3]=0;*/n = 0; f[i].A = ran0(&u); while (n != 1 && n != 2) 193

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Appendix M. (Continued) { n= (int) rand2(&t); } /* these two conditional statements describe the pick process */ /* if the first # chosen is a 1 then the first statement executes */ /* if the second # chosen is a 1 then the second statement executes */ if(n==1) { f[i].C=(1-f[i].A) ran0(&u); f[i].E=(1-f[i].A-f[i].C); } if( n==2) { f[i].E=(1-f[i].A) ran0(&u); f[i].C=(1-f[i].A-f[i].E); } /* This next line puts all of the data into one file */ fprintf(datafile,"%i %f\t%f\t%f\t%f\n",i,f[i].A, f[i].C, f[i].E, f[i].A+f[i].C+f[i].E); 194

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Appendix M. (Continued) /* This next line puts the f.A data into one file, one data point per line */ /* It also puts a second carraige return every 11 data points */ fprintf(datafile2,"%f\n",f[i].A);if ( i % 11 == 0) fprintf(datafile2,"\n"); /* This next line puts the f.C data into one file, one data point per line */ /* It also puts a second carraige return every 11 data points */ fprintf(datafile3,"%f\n", f[i].C); if ( i % 11 == 0) fprintf(datafile3,"\n"); /* This next line puts the f.E data into one file, one data point per line */ /* It also puts a second carraige return every 11 data points */ fprintf(datafile4,"%f\n", f[i].E); if ( i % 11 == 0) fprintf(datafile4,"\n"); } for ( i=1; i <=121; i++) 195

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Appendix M. (Continued) { /* this set of code keeps the sides from cycling back on themselves */ if (i%11 !=0) fprintf(datafile5," %0.3f \n",f[i].C-f[i+1].E); if(i%11 == 0) fprintf(datafile5,"%0.3f \n",f[i].C); if(i %11 ==0 ) fprintf(datafile5,"\n"); if ((i)%11!=1) fprintf(datafile6," %0.3f \n",f[i].E-f[i-1].C); /* printf(" %0.3f \n",f[i].E-f[i-1].C); */ if((i)%11==1) fprintf(datafile6,"%0.3f \n",f[i].E); /* printf(" %0.3f \n",f[i].E);*/ if(i %11 ==0 ) fprintf(datafile6,"\n"); } /* This subroutine creates the initial conc. profile and writes it to 2 files */ /* The first file is the initial conc. profile file and the 196

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Appendix M. (Continued) 2nd file is the moving conc. profile */ for(i=1; i<=121; i++) { if (i==49 || i==50 || i==51 || i==60 || i==61 || i==62 || i==71 || i==72 || i==73) { j=1;sconc[i]=j; } else { j=0;sconc[i]=j; } fprintf(datafile7,"%f\n",sconc[i]);fprintf(datafile8,"%f\n",sconc[i]);/* printf("%1.0f ",sconc[i]);*/ if (i%11==0) { fprintf(datafile7,"\n");fprintf(datafile8,"\n");/* printf("\n");*/ } 197

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Appendix M. (Continued) if(i==121) fprintf(datafile8,"\n"); conc_ph = conc_ph + sconc[i]; } /* printf("%f \n",conc_ph);*/ /* conc[i] has to be a function of position i and time t so it should be conc[i,t] the more general but harder to write to a file and call from gnuplot or conc[i].t where t = 0 -> 5 for this first example */ printf("\n");/* for(i=1; i<=121; i++) { printf("%0.3f ",sconc[i]); if (i%11==0) printf("\n");}*/ i =1; while(!feof(needleforce)) { fscanf(needleforce,"%f",&p);nf[i]=p/100;/* printf("%6.1f ",nf[i]); if(i%11 ==0) { 198

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Appendix M. (Continued) printf("\n");}*/ i++; }conc_ph=0;for (k=1; k<=10; k++) { for(i=1; i<=121; i++) { if(i<=11) { if (i==1) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E*nf[i+1] sconc[i]*f[i].A*nf[i] sconc[i]*f[i].C*nf[i] sconc[i]*f[i].E*nf[i]; } if (i>1 && i<11) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E*nf[i+1] + sconc[i-1]*f[i-1].C*nf[i-1] sconc[i]*f[i].A*nf[i] sconc[i]*f[i].C*nf[i] sconc[i]*f[i].E*nf[i]; } if (i==11) 199

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Appendix M. (Continued) { conc[i] = sconc[i] + sconc[i-1]*f[i-1].C*nf[i-1] sconc[i]*f[i].A*nf[i] sconc[i]*f[i].C*nf[i] sconc[i]*f[i].E*nf[i]; } } if(i>11) { if(i%11!=0 && i%11!=1) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E*nf[i+1] + sconc[i-1]*f[i-1].C*nf[i-1] + sconc[i-11]*f[i-11].A*nf[i-1] sconc[i]*f[i].A*nf[i] sconc[i]*f[i].C*nf[i] sconc[i]*f[i].E*nf[i]; } if( i%11==1) { conc[i] = sconc[i] + sconc[i+1]*f[i+1].E*nf[i+1] + sconc[i-11]*f[i-11].A*nf[i-11] sconc[i]*f[i].A*nf[i] sconc[i]*f[i].C*nf[i] sconc[i]*f[i].E*nf[i]; } if(i%11==0) { conc[i] = sconc[i] + sconc[i-1]*f[i-1].C*nf[i-1] 200

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Appendix M. (Continued) + sconc[i-11]*f[i-11].A*nf[i-11] sconc[i]*f[i].A*nf[i] sconc[i]*f[i].C*nf[i] sconc[i]*f[i].E*nf[i]; } } fprintf(datafile8,"%f \n",conc[i]); /* printf("%f ",conc[i]); */ if(i%11==0) { /* printf("\n");*/ fprintf(datafile8,"\n"); } /* This next line writes the conc[i] values to a holder variable that can be accessed in the next loop to be used as last time values */ /* sconc[i] = conc[i];*/ /* this line has to be after all of the tests but before the close of the i indicie */ conc_ph = conc_ph + sconc[i]; if(i==121) { fprintf(datafile8,"\n");/* printf("\n"); */ } } 201

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Appendix M. (Continued) for(i=1;i<=121;i++) { /* This next line writes the conc[i] values to a holder variable that can be accessed in the next loop to be used as last time values*/ sconc[i] = conc[i]; } printf("%f \n",conc_ph); conc_ph =0; /* printf("\n");*/ } /* this block of code is a test to see what the system is outputing printf("%1.0f ",conc[i,k]); if(i%11==0) printf("\n"); if(i==121) { printf("\n"); } /* this block of code is a test to see what 202

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Appendix M. (Continued) the system is outputing */ /* i =1; while(!feof(needleforce)) { fscanf(needleforce,"%f",&p);nf[i]=p; printf("%6.1f ",nf[i]); if(i%11 ==0) { printf("\n");} i++;}*/ fclose(datafile);fclose(datafile2);fclose(datafile3);fclose(datafile4);fclose(datafile5);fclose(datafile6);fclose(datafile7);fclose(datafile8);fclose(needleforce); } 203

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Appendix M. (Continued) M.3 Flo w V alue Random Number Generator Header File, ran0.h /* this is the header file for ran0.c a random number generator */ float ran0(long *idum); 204

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Appendix M. (Continued) M.4 Flo w V alue Random Number Generator Ccode File, ran0.c #include /* this program generates random numbers using the method presented */ /* in the book "Numerical Recipies in C" */ /* it is listed on page 278 by Park and Miller */ /* this generates a random number between 0.0 and 1.0 */ #include "ran0.h" #define IA 16807 #define IM 2147483647 #define AM (1.0/IM) #define IQ 127773 #define IR 2836 #define MASK 123459876 float ran0 (long *idum) { long k; float ans; /* printf("%12i\t", *idum); */ /* the ˆ= is and exclusive or */ *idum ˆ= MASK; /* The ˆ= is an XOR */ k = (*idum)/IQ; 205

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Appendix M. (Continued) *idum = IA (*idum k IQ) IR k; if (*idum < 0) *idum += IM; ans = AM (*idum); *idum ˆ= MASK; return ans; } 206

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Appendix M. (Continued) M.5 Directional Random Number Generator rand2.h /* this is the header file for rand2.c a random number generator */ float rand2(long *idum); 207

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Appendix M. (Continued) M.6 Directional Random Number Generator Ccode File, rand2.c #include /* this program generates random numbers using the method presented */ /* in the book "Numerical Recipies in C" */ /* it is listed on page 278 by Park and Miller */ /* this generates a random number either 1 or 2 selecting flow direction */ #include "rand2.h" #define IA 16807 #define IM 2147483647 #define AM (1.0/IM) #define IQ 127773 #define IR 2836 #define MASK 123459876 float rand2 (long *idum) { long k; float ans; /* the ˆ= is and exclusive or */ 208

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Appendix M. (Continued) *idum ˆ= MASK; /* The ˆ= is an XOR */ k = (*idum)/IQ; *idum = IA (*idum k IQ) IR k; if (*idum < 0) *idum += IM; /* to change this to picking any value add 1 to the value and replace 3.0, 3.0 means 2 choices */ ans = 3.0 AM (*idum); /* printf("%12i\t", ans); */ *idum ˆ= MASK; return ans; } 209

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Appendix N: Gel-DN A Data Modeling the o w of the DN A fragments in the gel matrix w as initially attempted using the rst principle char ge/v elocity model of chapter 5 [75– 77]. From the rst principle char ge/v elocity model and the packing f actor methods, predicted speeds for DN A in gels were calculated, see column 3, table N.1. The ne xt step w as to acquire a reference data set from the literature. After an e xtensi v e literature search a v acuum w as noted. The v alues for both the strength and duration of the applied electric elds were both v acant from all of the DN A gel literature. The e xact or e v en approximate distances tra v eled by the DN A bands w as also absent from the journal articles. Molecular biologists must not nd this information important, b ut for this analysis those three pieces of data were paramount. Since representati v e v alues of the electric eld strength, duration, and distance tra v eled for common DN A fragment sizes were not a v ailable in the literature the y were e xperimentally determined for this demonstration ef fort. A gel w as ran using DN A standards at a specic v oltage, 6.56 V cm 1 for a specic time, 1 hr 2 An image of this gel is sho wn in gure N.1. Lane 3 of the gel w as loaded with HyperLadder I, made by Bioline USA Inc., Canton Ma., catalog number BIO-33025 [14]. The band sizes in HyperLadder I are 10k, 8k, 6k, 5k, 4k, 3k, 2500, 2000, 1500, 1000, 800, 600, 400, and 200 basepairs [14]. Lane 5 of the gel w as loaded with a 100 bp ladder made by Bayou Biolabs, Harahan, La., catalog number L-101 [10]. The 100 bp ladder spans from 100 bp to 4000 bp in increments of 100 bp, e.g. 100 bp, 200 bp, 300 bp, ..., 3900 bp, and 4000 bp [10]. The gel matrix w as a 1% 1 A v alue of 105 V olts w as measured across the 16 cm gel apparatus 2 Special thanks to Dr Loree Heller of the USF Center for Molecular Deli v ery CMD 210

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Appendix N. (Continued) ag arose gel and the carrier solution w as 1x T AE b uf fer [7] and the applied electric eld w as 6.56 V cm Cathode Side 15000 bp 10000 bp 4000 bp 1000 bp 300 bp Anode Side Lane 3 5 Figure N.1: Photograph of the Data Electrophoresis Gel This is a scan out of my lab notebook of the actual gel used to optimize the model. 211

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Appendix N. (Continued) T able N.1: DN A Fragment Experimental and Computed Speeds in an Ag arose Gel DN A Fragment Experimental Calculated Corrected Dif ference Size (bp) Speed ¡ cm hr ¢ Speed ¡ cm hr ¢ Speed ¡ cm hr ¢ (Cor -Exp) 100 6.70 2.80 6.70 0.00 200 6.34 4.45 10.64 4.30 300 6.09 5.83 13.95 7.86 400 5.78 7.06 16.90 11.12 600 5.28 9.26 22.14 16.86 800 4.80 11.21 26.82 22.02 1000 4.50 13.01 31.12 26.62 1500 3.83 17.05 40.78 36.95 2000 3.40 20.66 49.40 46.00 2500 3.11 23.97 57.33 54.22 3000 2.89 27.07 64.74 61.85 4000 2.60 32.79 78.42 75.82 5000 2.39 38.05 91.00 88.61 212

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Appendix O: T issue V iscosity Calculating the a v erage viscosity in a tissue is accomplished by rst solving the v elocity equation for ions in a liquid from chapter 5 and then multiplying by the duration of the electric eld, see equation O.1 belo w In the Zharof f paper a 5.1 kbp plasmid with an applied electric eld of 465 V cm for 10 50 ms pulses. Computing the char ge and approximate radius of the plasmid is performed using the methods of chapter 5 and appendicies G and E. The v alenc y for the 5.1 kbp plasmid is 306 e ¡ and the radius is either 10.3 or 17.9 nm depending on the approximation, computed with equations O.2 and O.3 respecti v ely The viscosities that correspond to these radius v alues are 1.59 and 0.916 poise. (O.1) = z eE f = z eE 6 sr ¢ t eld applied (O.2) radius sphere ( pm ) = 3 p 2 : 16 £ # of base pairs £ 10 8 pm 3 (O.3) radius cyl ( nm ) = 3 p 1 : 125 £ # of base pairs 213

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Appendix P: Gel Retarding F orce and Speed Model Iterations P .1 Modeling the Retarding F orce of the Gel The retarding force of the gel on a DN A fragment w as modeled using the follo wing assumptions, the gel is made up of tubes, the tubes are constant radii, and the DN A is pliable and e xible [42, 43, 134]. T ubes were chosen as the pathw ay for the DN A to tra v el for tw o reasons. First, polymeric knot theory predicts that ag arose gels form a three-dimensional gel with cross-links that interact to approximate a tube [40, 134], and second, treating the DN A pathw ays as tubes allo wed for the utilization of the o w eld model of chapter 8. P .1.1 Cross Sectional Area Interaction Model, CSAIM From these assumptions it w as realized that the dif ference in the cross sectional area of the DN A fragment and the gel tube is the major contrib utor to the retarding force. Therefore the retarding frictional force of the gel on the DN A fragment w as modeled with a F g = £ ( r D N A ¡ r por e ) 2 dependence. The r por e v alue w as ascertained from gel resolution. The ag arose content of gels is chosen as a function of the desired separation gradient. As the DN A fragment size decreases the ag arose content of the gel must increase to maintain band denition. This also minimizes the maximum size of the DN A fragment that can tra v el through the gel. A 1% ag arose gel is commonly used to resolv e DN A do wn to 100 bp [7, 10, 14]. The radius of the 100 bp DN A fragment w as chosen as the gel pore radii, r por e = 4 : 8268 nm. The parameter A g el w as re gressed and its v alue w as calculated to be 5 : 688 £ 10 ¡ 17 N nm 2 The frictional force of the model compared to the e xperimental frictional 214

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Appendix P (Continued) force is sho wn in gure P .1. The R 2 v alue for the model is 0.9989 [139]. The code for the re gression and the statistics in listed in appendix R (P .1) F g = A g el £ £ ( r D N A ¡ r por e ) 2 0 50 100 150 200 250 300 350 4 6 8 10 12 14 16 18 Forces ( 10-16 N)DNA Frgment Radius (nm) Experimental Gel Frictional Force Calculated Gel Frictional Force Figure P .1: Gel Frictional F orce Model P .1.2 Calculating DN A Fragment Speeds Using the CSAIM Theoretically the DN A fragment w ould be at its greatest speed when the sum of the forces in the cumulati v e force equation were in equilibrium. Therefore, equation P .2 w as set equal to zero and solv ed for speed, see equation P .3. The v alues used in equation P .3 to calculate the speed for the dif ferent DN A fragments tra v eling in the gel matrix. The 215

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Appendix P (Continued) calculated v alues are listed in appendix W and the code to compute those v alues is in appendix R. § F or ces = z eE ¡ 6 r sc ¡ A g el ( r D N A ¡ r por e ) 2 (P .2) s = z eE ¡ A g el ( r D N A ¡ r por e ) 2 6 r c (P .3) P .1.3 CSAIM Results and Discussion The e xperimental speeds, the model predicted speeds and the dif ferences are listed in table X.1, appendix X. Figure P .2 displays the e xperimental speeds and the model predicted speeds. The R 2 v alue of the model compared to e xperimental data is 0.66756. As an e xtension of the rst principle model, the cross sectional area interaction model, CSAIM, initially had positi v e results. The agreement between the force due to the gel and the Coulombic force as seen in gure 12.3 seemed promising. The cross sectional area model, described in section P .1.1, represented the e xperimental v alue well. This can be seen in gure P .1 and in the computed R 2 v alue of 0.99870. Figure P .2 displays that the speed computed by the CSAIM does not ef fecti v ely model the speed of the v arious DN A fragments in the DN A gel. The R 2 v alue for the model compared to the e xperimental v alues is 0.66756. This result w as disappointing and surprising because of the superb t of the gel frictional force model, see gure P .1. The cause of the v ariation of the speeds predicted by the CSAIM and the e xperimental speeds is primarily due to the v ariation in between the calculated frictional force and the e xperimental frictional force di vided by 6 r c 216

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Appendix P (Continued) 1 2 3 4 5 6 7 8 9 4 6 8 10 12 14 16 18 Speed (cm/hr)DNA Frgment Radius (nm) Experimental Model Predicted Figure P .2: Experimental Speed vs CSAIM Predicted Speed P .1.3 Surf ace Area Interaction Model, SAIM After the poor result of the CSAIM, it w as h ypothesized that modeling the cross sectional area of the DN A fragment did not suitably and properly describe the DN A fragment, ag arose gel interaction. Using the prolate spheroid eccentricity f actor c the spheroid shape w as subsequently modeled as a c ylinder and the major and minor ax es where decomposed from the dif ferent representati v e v olumes for the indi vidual sized fragments. The interacting surf ace area, I S A of the DN A fragment c ylinder without the end caps w as calculated using equation P .4. The force due to the gel, F g w as calculated with P .5. A single parameter A g el see equation P .5, w as re gressed to t the data. The re gressed v alue of A g el w as 1 : 247 £ 10 ¡ 17 N nm 2 The R 2 v alue for this model w as 0.9640, not as good as the CSAIM b ut 217

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Appendix P (Continued) still an interesting model. After comparing gures P .3 and P .1, the SAIM w as ignored due to poor t with the e xperimental data. I S A = 2 ( r ¡ r por e ) L (P .4) F g = A g el ( r ¡ r por e ) L (P .5) 0 5e-15 1e-14 1.5e-14 2e-14 2.5e-14 3e-14 3.5e-14 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Retarding Force of the Gel (N)DNA Fragment Size (bp) Experimental Model Predicted Figure P .3: Gel Frictional F orce Model P .2 Modeling the Speed of the DN A Fragments in an Gel After the shortcomings of the CSAIM, where modeling the speed of the DN A fragment w as attempted by modeling the error of the rst principle model, the follo wing series of 218

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Appendix P (Continued) models use the rst principle model as a starting point b ut solv es directly for the speed of the DN A fragment. The three follo wing models be gin with the rst principle model solv ed for speed and add a correction f actor as a function of radius or number of base pairs. The three models are the area correction DN A fragment speed model, A CDFSM, the parabolic correction DN A fragment speed model, PCDFSM, and the e xponential correction DN A fragment speed model, ECDFSM. P .2.1 Area Correction DN A Fragment Speed Model The A CDFSM uses a similar technique to the CSAIM e xcept instead of creating a model to solv e for the lost frictional force, the model solv es the error in the speed, see equation P .6 [46, 219]. Mo ving the correction f actor into an equation that directly models the speed, reduces the propag ation of error .The re gressed v alue for B g el w as 10.2427 hr nm 2 cm The computer code written to compute the speed v alues is in appendix T The graph comparing the e xperimental speed to the model predicted speed is gure P .4. The R 2 v alue for this model is 0.90713. This is a mark ed impro v ement o v er the CSAIM model b ut the A CDFSM model does not match the shape of the e xperimental data and di v er ges at radii lar ger than 14 nm. (P .6) S = F E 6 r por e ¡ ¡ r 2 ¡ r 2 por e ¢ B g el 219

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Appendix P (Continued) 1 2 3 4 5 6 7 4 6 8 10 12 14 16 18 Speed (cm/hr)DNA Frgment Radius (pm) Experimental Model Predicted Figure P .4: Experimental Speed vs A CDFSM Predicted Speed P .2.2 P arabolic Correction DN A Fragment Speed Model The PCDFSM is an e xtension of the A CDFSM. It uses a parabola to model the speed er ror in the rst principle model. The other main dif ference is that it uses the number of base pairs as its independent v ariable rather than the DN A fragment radius. This reduces the reliance of the model on approximated v alues and brings the model one step closer to kno wn data. A parabolic equation, see equation P .7, w as used because it closely modeled the er ror between the rst principle model and the e xperimental v alues [46, 219]. Appendix U contains the computer code used to re gress the parameter C g el The graph comparing the model speed to the e xperimental speed is sho wn in gure P .5. The re gressed v alue for C g el w as 0.004513 cm 2 hr 2 bp and the R 2 v alue for the model is 0.96395. This model more closely 220

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Appendix P (Continued) predicted the speed of the DN A fragments and more closely follo ws the shape of the curv e b ut it di v er ges for DN A fragments greater than 3700 basepairs. (P .7) S = F E 6 r por e ¡ q C g el ¤ ( bp ¡ bp o ) 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Speed (cm/hr)DNA Fragment Size (bp) Experimental Model Predicted Figure P .5: Experimental Speed vs PCDFSM Predicted Speed P .2.3 Exponential Correction DN A Fragment Speed Model The ECDFSM utilizes an e xponential correction f actor for modeling the error in the speed between the rst principle model and the e xperimental v alues. It b uilds on the techniques from the pre vious three models. The ECDFSM uses the rst principle model for 221

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Appendix P (Continued) the initial v alue while an e xponential model is used to handle the error in the speed. Lik e the PCDFSM the number of base pairs is used as the independent v ariable. Equation P .8 w as used to calculate the predicted speeds. This equation uses tw o tunable parameters to describe the damping of the DN A fragment' s speed by the ag arose gel. The parameter D g el tunes the rate of change in speed. The parameter E g el acts to change the steepness of the graph. The v alue for D g el w as 4.91915 cm hr and the v alue for E g el w as 750 bp The v alue of D g el is the speed of a 750 bp DN A fragment in this gel at this electric eld strength. The code for the re gression is listed in appendix V. 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Speed (cm/hr)DNA Fragment Size (bp) Experimental Model Predicted Figure P .6: Experimental Speed vs ECDFSM Predicted Speed Figure P .6 displays the predicted and the e xperimental speeds for the DN A fragments as a function of base pairs. The t of this model is quite good as can be seen in Figure P .6. 222

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Appendix P (Continued) There is some v ariation at the edges of resolution of the gel b ut in the w orking re gion of the gel the model performs well. The R 2 v alue for this model is 0.99409. (P .8) S = F E 6 r por e ¡ D g el e ¡ E g el bp 223

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Appendix Q: Gel Electrophoresis F orce Model Code #! /usr/bin/octave -qf ############################################################### frictional_force3.m # Joseph D. Hickey Feb. 23, 2004 # This is an octave program for computing the coulombic and # frictional forces on a DNA fragment in a liquid given the # viscosity, speed and number of basepairs # The results are in Newtons ################################################################format long; time_conversion = 3.6e5; # this is the converstion from m/s to cm/hr eta = 0.890e-12; # viscosity di water # 0.890 cP = 0.890 e -3 kg/(m s) # = 0.890 e -12 kg/(nm s) #e = 1.602e-19; # charge of an electron in Coulombs C E = 656; # electric field 105 V /16 cm # 6.56 V/cm = 6.56 J/(C cm) 224

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Appendix Q. (Continued) # = 6.56 (kg mˆ2)/(sˆ2 C cm) # = 656 (kg m)/(sˆ2 C) # the force of the electric field in Newtons pi = 3.14157; # the value that I like for pi r_c = 2.1; # nm L_c = 0.34; # nm bp = [100, 200, 300, 400, 600, 800, 1000, 1500, 2000, 2500, 3000, 4000, 5000]; s = [6.70, 6.34, 6.09, 5.78, 5.28, 4.80, 4.50, 3.83, 3.40, 3.11, 2.89, 2.60, 2.39]; s = s / time_conversion; # converts speed from cm/hr into m/s r = [5.84, 7.35, 10.60, 12.57, 17.06, 21.50]; # radius in nm z = 0.06 bp; v_c = pi r_cˆ2 L_c .* bp ; # Volume of a cylinder of bp size r = ((3/(4*pi))*v_c).ˆ(1/3) ; # radius of an equal volume sphere ####################### preloading values # ###################### 225

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Appendix Q. (Continued) F_g(1) = 1; a =1.25522; b = 3; form_factor = 0.3333; # was a/b while((F_g(1) > 1e-24) || (F_g(1) < 0)) F_f = 6*pi*eta.*r.*s form_factor; F_e = z*e*E; F_g = F_e F_f; F_g(1)if(F_g(1) < 0) form_factor = form_factor 1e-5 form_factor elseif(F_g(1) > 1e-22) form_factor = form_factor + 1e-5 form_factor; endendwhileform_factor 226

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Appendix Q. (Continued) Forces = [F_f', F_e', F_g'] 227

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Appendix R: Cross Sectional Area Interaction Model Code #! /usr/bin/octave -qf ############################################################# power3.m # Joseph D. Hickey Feb. 24, 2004 # This is an octave program for computing the parameters # for the frictional force due to a gel power equation. It # keys off of the speed of the DNA in the gel # Y = A_gel*(r_dna r_pore)ˆ2 ################################################################################# Constants ## ###############e = 1.602e-19; # charge of an electron in Coulombs C pi = 3.14157; # the value that I like for pi r_c = 2.1; # nm L_c = 0.34; # nm 228

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Appendix R. (Continued) time_conversion = 3.6e5; # this is the converstion # from m/s to cm/hr form_factor = 0.4181; # regressed form factor parameter E = 656; # electric field 105 V /16 cm # 6.56 V/cm = 6.56 J/(C cm) # = 6.56 (kg mˆ2)/(sˆ2 C cm) # = 656 (kg m)/(sˆ2 C) # the force of the electric field in Newtons eta = 0.89e-12; # the viscosity of di water; #0.890e-12; # viscosity of saline; # 0.890 cP = 0.890 e -3 kg/(m s) # = 0.890 e -12 kg/(nm s) bp = [100, 200, 300, 400, 600, 800, 1000, 1500, 2000, 2500, 3000, 4000, 5000]; z = 0.06 bp; # computing valency, unitless s = [6.70, 6.34, 6.09, 5.78, 5.28, 4.80, 4.50, 3.83, 3.40, 3.11, 2.89, 2.60, 2.39]; s = s;# / time_conversion; # converts speed from cm/hr into m/s F_F = [6.3055e-16, 7.5175e-16, 8.2661e-16, 8.6349e-16, 9.0294e-16, 9.0347e-16, 9.1241e-16, 8.8894e-16, 8.6856e-16, 8.5582e-16, 8.4511e-16, 8.3683e-16, 8.2864e-16]; 229

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Appendix R. (Continued) F_E = [6.3055e-16, 12.611e-16, 18.926e-16, 25.222e-16, 37.833e-16, 50.444e-16, 63.055e-16, 94.582e-16, 126.11e-16, 157.64e-16, 189.16e-16, 252.22e-16, 315.27e-16]; F_G = [0, 5.0934e-16, 10.650e-16, 16.587e-16, 28.803e-16, 41.409e-16, 53.931e-16, 85.693e-16, 117.42e-16, 149.08e-16, 180.71e-16, 243.85e-16, 306.99e-16]; v_c = pi r_cˆ2 L_c .* bp ; # Volume of a cylinder of bp size r = ((3/(4*pi))*v_c).ˆ(1/3) ; # radius of an equal volume sphere r_pore = r(1); # the radius of a 100 bp dna sphere A_gel = 5.688e-17;#3.76e-17; # guess force value for the gel N/nm j = 1; s_dx = 1; while((s_dx > 1e-8) && (j < 10000)) f_g = A_gel pi *(r-r_pore).ˆ2; f_E = z.*e*E; num = f_E f_g; f_f = 6*pi*eta.*r.*s.*form_factor; cs = s.*(f_E f_g)./f_f; cs = cs time_conversion; delt = cs s; dx = sqrt(delt.ˆ2); 230

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Appendix R. (Continued) s_dx = sum(dx); s_cs = sum(cs); s_s = sum(s); if(s_cs > s_s) A_gel = A_gel + 0.001*A_gel; endifif(s_cs < s_s) A_gel = A_gel 0.001*A_gel; endifj = j +1; s_dx;endwhilescsdx = dx A_gelavg_speed = s_s/13; 231

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Appendix R. (Continued) se = (s cs).ˆ2; st = (cs avg_speed).ˆ2; sse = sum(se); sst = sum(st); r_sqred = 1 (sse/sst) plot(r,f_g,r,F_G)pause (10) gset terminal postscript enh color 'times-roman' 14 gset output "frictional_force.eps" replotgset terminal x11 plot(r,cs,r,s)pause (10) gset terminal postscript enh color 'times-roman' 14 gset output "speeds.eps" replotf_gF_G 232

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Appendix R. (Continued) # diff_f_g = F_E F_G; #diff_e_g = f_E f_g; # plot(r,diff_f_g,r,diff_e_g,'b*') # plot(r,cs) #plot(r,F_G,r,f_g,'r*');avg_F_G = sum(F_G)/length(F_G); se = (F_G f_g).ˆ2; st = (f_g avg_F_G).ˆ2; sse = sum(se) sst = sum(st) R_sqred = 1 (sse/sst) 233

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Appendix S: Surf ace Area Interaction Model Code #! /usr/bin/octave -qf ######################################################## saim.m # Surface Area Interaction Model Code # Joseph D. Hickey Jul. 6, 2004 # This is an octave program for computing the surface # area interaction between DNA fragments and the # electrophoresis gel tubes. It utilizes the length and # radius of a cylindrical DNA fragment from the number # of base pairs and the length and radius of an individual # base pair. # The results are in nanometers #############################################################bp = [100, 200, 300, 400, 600, 800, 1000, 1500, 2000, 2500, 3000, 4000, 5000]; F_F = [6.3055e-16, 7.5175e-16, 8.2661e-16, 8.6349e-16, 9.0294e-16, 9.0347e-16, 9.1241e-16, 8.8894e-16, 8.6856e-16, 8.5582e-16, 8.4511e-16, 8.3683e-16, 8.2864e-16]; F_E = [6.3055e-16, 12.611e-16, 18.926e-16, 25.222e-16, 37.833e-16, 234

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Appendix S. (Continued) 50.444e-16, 63.055e-16, 94.582e-16, 126.11e-16, 157.64e-16, 189.16e-16, 252.22e-16, 315.27e-16]; F_G = [0, 5.0934e-16, 10.650e-16, 16.587e-16, 28.803e-16, 41.409e-16, 53.931e-16, 85.693e-16, 117.42e-16, 149.08e-16, 180.71e-16, 243.85e-16, 306.99e-16]; pi = 3.14157; # the value that I like for pi r = 2.1; # radius of DNA in nanometers l = 0.34; # length of DNA in nanometers # these values came from strzelecka and rill # J. Am. Chem. Soc. 1987, 109, 4513-4518 # c = a/b = minor/major axis of a prolate spheroid c = 0.41841; # computed in frictional_forces3.m V_sphere = pi rˆ2 l .* bp; # computing the volume of the DNA sphere L = ((1/(pi*cˆ2))*V_sphere).ˆ(1/3) ; R = c*L; SA = 2*pi*R.*L; # surface area ignoring the endcaps, # just the interacting surface area 235

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Appendix S. (Continued) ISA = 2*pi*(R-R(1)).*L; s_dx = 1; # priming the pump A_gel = 1.1255e-17; # the starting guess for the frictional parameter while((s_dx > 1e-15) && (j < 100000)) f_g = A_gel *2*pi*(R-R(1)).*L; delt = F_G f_g; dx = sqrt(delt.ˆ2); s_dx = sum(dx); s_f_g = sum(f_g); s_F_G = sum(F_G); if(s_f_g > s_F_G) A_gel = A_gel 0.001*A_gel; endifif(s_f_g < s_F_G) A_gel = A_gel + 0.0001*A_gel; 236

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Appendix S. (Continued) endifj = j +1; endwhiles_dxs_f_gs_F_GA_gelf_gF_Gavg_F_G = sum(F_G)/length(F_G); se = (F_G f_g).ˆ2; st = (f_g avg_F_G).ˆ2; format long; A_gelsse = sum(se) 237

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Appendix S. (Continued) sst = sum(st) R_sqred = 1 (sse/sst) file = fopen("cylinder-paramters.data","w"); fprintf(file,"# bp \t L\t R\t SA\t \tISA\t\t F_G \t\t f_g_comp\t diff\n"); for i = 1:length(bp) fprintf(file,"%5i\t %2.3f\t %2.3f\t %2.3f\t %2.3e\t %2.3e \t %2.3e\t %2.3e\n",bp(i),L(i),R(i),SA(i),ISA(i), F_G(i),f_g(i),delt(i)); endfprintf(file,"\n\n\n# Rˆ2 value = %2.6f",R_sqred); 238

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Appendix S. (Continued) fprintf(file,"\n\n\n# A_gel = %2.6e",A_gel); fprintf(file,"\n\n # f_g = A_gel *2*pi*(R-R(1)).*L"); fclose(file); 239

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Appendix T : Area Correction DN A Fragment Speed Model Code #! /usr/bin/octave -qf ################################################################### ###### # speeds.m # Joseph D. Hickey Feb. 26, 2004 # This is an octave program for computing the predicted speed of the # DNA fragment in the an agarose gel and the second model # given the viscosity, speed and number of basepairs # The results are in Newtons #################################################################### ###### #format long; time_conversion = 3.6e5; # this is the converstion from m/s to cm/hr eta = 0.890e-12; # viscosity di water # 0.890 cP = 0.890 e -3 kg/(m s) # = 0.890 e -12 kg/(nm s) #e = 1.602e-19; # charge of an electron in Coulombs C E = 656; # electric field 105 V /16 cm # 6.56 V/cm = 6.56 J/(C cm) # = 6.56 (kg mˆ2)/(sˆ2 C cm) 240

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Appendix T (Continued) # = 656 (kg m)/(sˆ2 C) # the force of the electric field in Newtons pi = 3.14157; # the value that I like for pi r_c = 2.1; # nm L_c = 0.34; # nm bp = [100, 200, 300, 400, 600, 800, 1000, 1500, 2000, 2500, 3000, 4000, 5000]; s = [6.70, 6.34, 6.09, 5.78, 5.28, 4.80, 4.50, 3.83, 3.40, 3.11, 2.89, 2.60, 2.39]; s = s;# / time_conversion; # converts speed from cm/hr into m/s z = 0.06 bp; v_c = pi r_cˆ2 L_c .* bp ; # Volume of a cylinder of bp size r = ((3/(4*pi))*v_c).ˆ(1/3) ; # radius of an equal volume sphere r_pore = r(1); form_factor = 0.4181; # guess form factor parameter ####################### preloading values # ######################diff(1)=1#F_e = z*e*E; #F_f = 6*pi*eta.*r.*s*form_factor; format long; 241

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Appendix T (Continued) A_gel = 10.391; s_diff(1)=10;j = 1# form_factorwhile((j < 10000) && (s_diff > 5) || (s_diff < 0)) difference = ((z*e*E)./(6*pi*eta.*r*form_factor))*time_conversion; speed = ((z*e*E)./(6*pi*eta.*r*form_factor))*time_conversion \\ (pi*(r.ˆ2 r(1)ˆ2)/A_gel); diff = speed s; s_diff(j+1) = sum(abs(diff)); values = [j,s_diff(j+1), A_gel]; if(s_diff(j+1) > s_diff(j)) A_gel = A_gel + 1e-5 A_gel; elseif(s_diff(j+1) < s_diff(j)) A_gel = A_gel 1e-5 A_gel; endj = j+1; endwhiledelt = (r-r_pore); m = diff./delt; change = (pi*(r.ˆ2 r(1)ˆ2)/10.391); 242

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Appendix T (Continued) A_gelform_factor;format short; #diff = speed s values = [bp', speed', s', diff']; f_speed = ((z*e*E)./(6*pi*eta.*r*form_factor))*time_conversion; f_diff = f_speed -s; file = fopen("speeds.data","w"); fprintf(file, "# bp\t radius (nm)\t speed\t\t s\t\t delta \t \t \\ f_diff\t\t change\n"); for i = 1:length(bp) fprintf(file,"%5i\t %9f\t%9f\t%9f\t%9f\t%9f\t%9f\n",bp(i),r( i),\\ speed(i),s(i),diff(i),f_diff(i),change(i)); endformat long # calculating Rˆ2 for the speed model avg_s = sum(s)/length(s); se = (s speed).ˆ2; st = (speed avg_s).ˆ2; 243

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Appendix T (Continued) sse = sum(se) sst = sum(st); R_sqred = 1 (sse/sst) # calculating Rˆ2 for the f_g model avg_f_diff = sum(f_diff)/length(f_diff); se = (f_diff change).ˆ2; st = (change avg_f_diff).ˆ2; sse = sum(se) sst = sum(st); R_sqred_2 = 1 (sse/sst) fprintf(file,"\n\n\n\n############################\n");fprintf(file,"#\n");fprintf(file,"# change = (pi(r-r_pore)ˆ2)/A_gel units \\ of A_gel is hr/(cm nm) \n"); fprintf(file,"# A_gel = %6f\n",A_gel); 244

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Appendix T (Continued) fprintf(file,"# form_factor = %6f\n",form_factor); fprintf(file,"# Rˆ2 = %6f\n",R_sqred_2); fprintf(file,"# S = F_E/(6*pi*eta*r*form_factor) \\ pi(rˆ2 r_poreˆ2)/A_gel\n"); fprintf(file,"# Rˆ2 = %6f\n",R_sqred); fprintf(file,"#\n");fprintf(file,"############################\n");fclose(file); 245

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Appendix U: P arabolic Correction DN A Fragment Speed Model Code #! /usr/bin/octave -qf ######################################################### parabola.m # Joseph D. Hickey Mar. 1, 2004 # This is an octave program for computing the parameters # for the frictional force due to a gel power equation. # It keys off of the speed of the DNA in the gel # Y = A sqrt(B*bp C) # A = F_E/(6*pi*eta*r(100bp)), cm/hr # B = cmˆ2/(hrˆ2 bp) # C = B bp(1) ############################################################################## Constants ## ###############e = 1.602e-19; # charge of an electron in Coulombs C pi = 3.14157; # the value that I like for pi r_c = 2.1; # nm L_c = 0.34; # nm 246

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Appendix U. (Continued) time_conversion = 3.6e5; # this is the converstion from m/s to cm/hr form_factor = 0.4181; # regressed form factor parameter E = 656; # electric field 105 V /16 cm # 6.56 V/cm = 6.56 J/(C cm) # = 6.56 (kg mˆ2)/(sˆ2 C cm) # = 656 (kg m)/(sˆ2 C) # the force of the electric field in Newtons eta = 0.89e-12; # the viscosity of di water; #0.890e-12; # viscosity of saline; # 0.890 cP = 0.890 e -3 kg/(m s) # = 0.890 e -12 kg/(nm s) bp = [100, 200, 300, 400, 600, 800, 1000, 1500, 2000, 2500, 3000, \\ 4000, 5000]; z = 0.06 bp; # computing valency, unitless s = [6.70, 6.34, 6.09, 5.78, 5.28, 4.80, 4.50, 3.83, 3.40, 3.11, \\ 2.89, 2.60, 2.39]; #s = s / time_conversion; # converts speed from cm/hr into m/s F_F = [6.3055e-16, 7.5175e-16, 8.2661e-16, 8.6349e-16, 9.0294e-16, \\ 9.0347e-16, 9.1241e-16, 8.8894e-16, 8.6856e-16, 8.5582e-16, \\ 8.4511e-16, 8.3683e-16, 8.2864e-16]; F_E = [6.3055e-16, 12.611e-16, 18.926e-16, 25.222e-16, 37.833e-16, \\ 247

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Appendix U. (Continued) 50.444e-16, 63.055e-16, 94.582e-16, 126.11e-16, 157.64e-16, \\ 189.16e-16, 252.22e-16, 315.27e-16]; F_G = [0, 5.0934e-16, 10.650e-16, 16.587e-16, 28.803e-16, 41.409e-16, \\ 53.931e-16, 85.693e-16, 117.42e-16, 149.08e-16, 180.71e-16, \\ 243.85e-16, 306.99e-16]; v_c = pi r_cˆ2 L_c .* bp ; # Volume of a cylinder of bp size r = ((3/(4*pi))*v_c).ˆ(1/3) ; # radius of an equal volume sphere r_pore = r(1); # the radius of a 100 bp dna sphere avg_s = sum(s)/length(s); B = 0.004515 ;# cmˆ2 / (hrˆ2 bp) cs = (z(1)*e*E)/(6*pi*eta*r(1)*form_factor)*time_conversion \\ sqrt(B*bp); delta = cs s; se = delta.ˆ2; st = (cs avg_s).ˆ2; sse = sum(se); sst = sum(st); R_sqred = 1 (sse/sst); j = 1; s_dx = 1; s_dk = 1; s_r2_p = 1; while((s_dk > 1e-8) && (j < 10000)) 248

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Appendix U. (Continued) cs = (z(1)*e*E)/(6*pi*eta*r(1)*form_factor)*time_conversion \\ sqrt(B*bp B*bp(1)); delta = cs s; se = delta.ˆ2; st = (cs avg_s).ˆ2; sse = sum(se); sst = sum(st); R_sqred = 1 (sse/sst); s_r2 = R_sqred; if(s_r2_p > s_r2) B = B + 0.0001*B ; endifif(s_r2_p < s_r2) B = B 0.0001*B ; endifj = j +1; s_dk = abs(s_r2_p s_r2); s_r2_p = s_r2; endwhile 249

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Appendix U. (Continued) j; BB*bp(1)R_sqredcs;values = [cs', s'] file = fopen("parabola.data","w"); fprintf(file, "# bp\t speed\t\t computed speed\n"); for i = 1:length(bp) fprintf(file,"%5i\t %9f\t%9f\n",bp(i),s(i),cs(i)); endfprintf(file,"\n\n\n\n############################\n");fprintf(file,"#\n");fprintf(file,"# form_factor = %6f\n",form_factor); fprintf(file,"# B = %6f, cmˆ2/(hrˆ2 bp)\n",B); fprintf(file,"# Rˆ2 = %6f\n",R_sqred); fprintf(file,"# S = F_E/(6*pi*eta*r(100bp)*form_factor) \\ sqrt(B*bp)\n"); fprintf(file,"#\n");fprintf(file,"############################\n");fclose(file); 250

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Appendix V : Exponential Correction DN A Fragment Speed Model Code #! /usr/bin/octave -qf ##################################################### exp.m # Joseph D. Hickey Feb. 27, 2004 # This is an octave program for computing the # parameters for the frictional force due to a gel # power equation. It keys off of the speed of the # DNA in the gel # Y = A B*exp(-C/bp) # A = F_E/(6*pi*eta*r(100bp)), cm/hr # B = cm/hr # C = bp #################################################################### ###### ################# Constants ## ###############e = 1.602e-19; # charge of an electron in Coulombs C pi = 3.14157; # the value that I like for pi r_c = 2.1; # nm 251

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Appendix V (Continued) L_c = 0.34; # nm time_conversion = 3.6e5; # this is the converstion from m/s to cm/hr form_factor = 0.4181; # regressed form factor parameter E = 656; # electric field 105 V /16 cm # 6.56 V/cm = 6.56 J/(C cm) # = 6.56 (kg mˆ2)/(sˆ2 C cm) # = 656 (kg m)/(sˆ2 C) # the force of the electric field in Newtons eta = 0.89e-12; # the viscosity of di water; #0.890e-12; # viscosity of saline; # 0.890 cP = 0.890 e -3 kg/(m s) # = 0.890 e -12 kg/(nm s) bp = [100, 200, 300, 400, 600, 800, 1000, 1500, 2000, 2500, 3000, \\ 4000, 5000]; z = 0.06 bp; # computing valency, unitless s = [6.70, 6.34, 6.09, 5.78, 5.28, 4.80, 4.50, 3.83, 3.40, 3.11, \\ 2.89, 2.60, 2.39]; #s = s / time_conversion; # converts speed from cm/hr into m/s F_F = [6.3055e-16, 7.5175e-16, 8.2661e-16, 8.6349e-16, 9.0294e-16, \\ 9.0347e-16, 9.1241e-16, 8.8894e-16, 8.6856e-16, 8.5582e-16, \\ 8.4511e-16, 8.3683e-16, 8.2864e-16]; 252

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Appendix V (Continued) F_E = [6.3055e-16, 12.611e-16, 18.926e-16, 25.222e-16, 37.833e-16, \\ 50.444e-16, 63.055e-16, 94.582e-16, 126.11e-16, 157.64e-16, \\ 189.16e-16, 252.22e-16, 315.27e-16]; F_G = [0, 5.0934e-16, 10.650e-16, 16.587e-16, 28.803e-16, 41.409e-16, \\ 53.931e-16, 85.693e-16, 117.42e-16, 149.08e-16, 180.71e-16, \\ 243.85e-16, 306.99e-16]; v_c = pi r_cˆ2 L_c .* bp ; # Volume of a cylinder of bp size r = ((3/(4*pi))*v_c).ˆ(1/3) ; # radius of an equal volume sphere r_pore = r(1); # the radius of a 100 bp dna sphere avg_s = sum(s)/length(s); A = s(1); B = 4.922; C = 750; cs = A B .*exp(-C./bp) ; se = (s cs).ˆ2; st = (cs avg_s).ˆ2; sse = sum(se) sst = sum(st) R_sqred = 1 (sse/sst) 253

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Appendix V (Continued) format long j = 1; s_dx = 1; s_dk = 1; s_r2_p = 1; while((s_dk > 1e-8) && (j < 100000)) cs = A B .*exp(-C./bp) ; delt = cs s; se = (s cs).ˆ2; st = (cs avg_s).ˆ2; sse = sum(se); sst = sum(st); R_sqred = 1 (sse/sst); s_r2 = R_sqred; if(s_r2_p > s_r2) B = B 0.0001*B; endifif(s_r2_p < s_r2) B = B + 0.0001*B; 254

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Appendix V (Continued) endifj = j +1; s_dk = abs(s_r2_p s_r2); s_r2_p = s_r2; endwhilejs_dkse = (s cs).ˆ2; st = (cs avg_s).ˆ2; sse = sum(se); sst = sum(st); BR_sqred = 1 (sse/sst) #values = [cs',s',delt'] #plot(bp,s,bp,cs)#pause(10) 255

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Appendix V (Continued) file = fopen("exp.data","w"); fprintf(file, "# bp\t speed\t\t computed speed\n"); for i = 1:length(bp) fprintf(file,"%5i\t %9f\t%9f\n",bp(i),s(i),cs(i)); endfprintf(file,"\n\n\n\n############################\n");fprintf(file,"#\n");fprintf(file,"# form_factor = %6f\n",form_factor); fprintf(file,"# B = %6f, cm/hr\n",B); fprintf(file,"# C = %6f, bp\n",C); fprintf(file,"# Rˆ2 = %6f\n",R_sqred); fprintf(file,"# S = F_E/(6*pi*eta*r(100bp)*form_factor) \\ B*exp(-C/bp)\n"); fprintf(file,"#\n");fprintf(file,"############################\n");fclose(file);for i = 1:(length(bp)-1) dx(i) = (bp(i+1) bp(i))/2; x(i) = bp(i) + (bp(i+1) bp(i))/2; 256

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Appendix V (Continued) dy(i) = cs(i+1) cs(i); enddrv = dy./dx; values = [x,drv] plot(x,drv)#pause(10)gset terminal png gset output "dev_exp.png" replotfile = fopen("dev_exp.data","w"); fprintf(file,"#######################################\n");fprintf(file,"bp\t Derivative\n"); for i = 1:length(drv) fprintf(file,"%3i\t%6f\n",x(i),drv(i));endfclose(file); 257

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Appendix W : Radius, Speed, V alenc y and F orce V alues This section holds the e xperimental data v alues, the calculated data v alues, and the computed force v alues that were used to create the dif ferent models in the document. T able W .1: V alues Used for the DN A Fragments in a 1% Ag arose Gel at 6.5625 V cm Size (bp) Radius (nm) Speed cm hr V alenc y 100 4.8268 6.70 6 200 6.0814 6.34 12 300 6.9615 6.09 18 400 7.6621 5.78 24 600 8.7709 5.28 36 800 9.6536 4.80 48 1000 10.3990 4.50 60 1500 11.9039 3.83 90 2000 13.1020 3.40 120 2500 14.1137 3.11 150 3000 14.9980 2.89 180 4000 16.5074 2.60 240 5000 17.7821 2.39 300 258

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Appendix W (Continued) T able W .2: F orce V alues for the DN A Fragments in a 1% Ag arose Gel at 6.56 V cm Size Coulombic Stok e' s La w Gel Gel F orce (bp) F orce (N) F orce (N) F orce (N) Model (N) 100 6.01 £ 10 ¡ 16 6.01 £ 10 ¡ 16 0 0 200 1.20 £ 10 ¡ 15 7.10 £ 10 ¡ 16 4.92 £ 10 ¡ 16 2.69 £ 10 ¡ 16 600 3.60 £ 10 ¡ 15 8.37 £ 10 ¡ 16 2.77 £ 10 ¡ 15 2.67 £ 10 ¡ 15 1000 6.01 £ 10 ¡ 15 8.51 £ 10 ¡ 16 5.16 £ 10 ¡ 15 5.34 £ 10 ¡ 15 2500 1.50 £ 10 ¡ 14 7.14 £ 10 ¡ 16 1.43 £ 10 ¡ 14 1.49 £ 10 ¡ 14 5000 3.00 £ 10 ¡ 14 6.54 £ 10 ¡ 16 2.94 £ 10 ¡ 14 2.98 £ 10 ¡ 16 259

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Appendix X: Gel V elocity V alues for the F our Models This appendix contains the ra w data and the graphs from the four mathematical models e xamined in appendix P. The four models are the cross sectional area interaction model, CSAIM, the area correction DN A fragment speed model, A CDFSM, the parabolic correction DN A fragment speed model, PCDFSM, and the e xponential correction DN A fragment speed model, ECDFSM. The graphs and data are included here for easy reference. T able X.1: Speeds of DN A Fragments in a 1% Ag arose Gel at 6.56 V cm FragmentSize (bp) ExperimentalV alues CSAIM A CDFSM PCDFSM ECDFSM 100 6.70 6.700 6.704 6.705 6.697 200 6.34 8.267 6.446 6.033 6.584 600 5.28 5.857 5.690 5.202 5.291 1000 4.50 3.709 5.100 4.689 4.377 2500 3.11 1.225 3.377 3.413 3.056 5000 2.39 4.346 1.163 2.001 2.466 260

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Appendix X. (Continued) 1 2 3 4 5 6 7 8 9 4 6 8 10 12 14 16 18 Speed (cm/hr)DNA Frgment Radius (nm) Experimental Model Predicted Figure X.1: Experimental Speed vs CSAIM Predicted Speed 1 2 3 4 5 6 7 4 6 8 10 12 14 16 18 Speed (cm/hr)DNA Frgment Radius (pm) Experimental Model Predicted Figure X.2: Experimental Speed vs A CDFSM Predicted Speed 261

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Appendix X. (Continued) 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Speed (cm/hr)DNA Fragment Size (bp) Experimental Model Predicted Figure X.3: Experimental Speed vs PCDFSM Predicted Speed 2 2.5 3 3.5 4 4.5 5 5.5 6 6.5 7 0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 Speed (cm/hr)DNA Fragment Size (bp) Experimental Model Predicted Figure X.4: Experimental Speed vs ECDFSM Predicted Speed 262

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Appendix Y : Gel Electrophoresis Simulation Code This appendix contains all of the code required to repeat the gel electrophoresis simulation project. Just type it in and compile it using mak e. It is brok en up into the separate sections, the y are each a dif ferent c code or header le. The follo wing sections are Mak ele, see page 264, it contains a list of all the ccode and header les required to run the code. It mak e compilation simple using gnu mak e program. Ne xt is gep.c on page 265, this is the main program. It initializes the arrays, prototypes the v ariables, calls all of the subprograms and handles all of the returns and reads from and writes to les. F ollo wing that are the tw o cube related subroutines and their associated header les. The rst header le is cubebyv alue.h, on page 282, the associated ccode is cubebyv alue.c located on page 282. This program essentailly tak es an int and returns the cube of that number as an int. It w as written because math.h doesn' t ha v e a cube function and I w anted a simple pass style cube program. The ne xt header le is cuberoot.h, on page 283, its associated ccode is cuberoot.c and is located on page 284. This program is a little more comple x that cubebyv alue.c. It tak es a v alue as a oat, it could be typecast as a oat, and returns the cube root of that v alue as a oat. Float w as chosen because it is a more ef cient use of resources than double. 263

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Appendix Y (Continued) Y .0 Mak ele # The makefile for the gep electrophoresis program gep: gep.o cubebyvalue.o cuberoot.o gcc -o gep gep.o cubebyvalue.o cuberoot.o -lm gep.o: gep.c cubebyvalue.h cuberoot.h gcc -c gep.c cubebyvalue.o: cubebyvalue.c cubebyvalue.h gcc -c cubebyvalue.c cuberoot.o: cuberoot.c cuberoot.h gcc -c cuberoot.c 264

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Appendix Y (Continued) Y .1 Gel Electrophoresis Model Ccode, gep.c /****************************************************************** **** Name: gep.c Author: Joseph D. Hickey Date: Dec. 16, 2003 Requires: stdio.h, math.h, cubebyvalue.c, cubebyvalue.h, cuberoot.c, cuberoot.h This program takes a (float) variable and returns the cuberoot of that (float) variable as a (float). This program is an extension of gel.c, the main difference is that this program uses a row column format while gel.c used a single column format ******************************************************************** **********************/ #include #include "cubebyvalue.h" #include "cuberoot.h" /* global struct definitions */ struct drawer {int base_pairs; double radius_DNA_sphere; double speed_cmps; double initial_normalized_mass;}; /*struct fragments {double third; double first; double second;}; /* a struct of 2 dna fragments */ double fragments_first[51][1331][50]; 265

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Appendix Y (Continued) double fragments_second[51][1331][50]; double fragments_third[51][1331][50]; double fragments_fourth[51][1331][50]; double fragments_fifth[51][1331][50]; /* double fragments_sixth[51][1331][5];*/ double fragments_sum[51][1331][50]; main(){ int i,j,k,t; /* The three position indicies and time */ /* The number of base pairs */ int base_pairs = 1000; /* Calculating the charge on the DNA sphere */ int z=1; /* a singly charged ion */ double elementary_charge = 1.6022e-19; /* elementary charge in coulombs*/ double charge_per_base_pair = 0.066; /* the fractional charge per base pair of DNA [1] */ double dna_charge; /* The charge of the DNA molecule */ /* Calculating the volume of the representative base pair. */ int radius_ppp = 560; /* radius of a DNA purine pyrimadine pair */ int r_cubed; /* The radius cubed */ double pi = 360/(2 57.29578); double volume_ppp; 266

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Appendix Y (Continued) /* Calculating the apporoximate volume and radius of the DNA sphere from the number of base pairs*/ double volume_DNA_sphere; double three_fourths_volume_DNA_sphere; double radius_DNA_sphere; /* The electric field and associated forces */ int electric_field = 1500; /* V/cm */ double force_coulomibic; double force_fluid_friction; double eta = 1.014; /* The Greek letter eta*/ /* meaning dynamic viscosity in units of centiPoise */ /* speed = (z elementary_charge electric_field)/(6 Pi eta r) units of (kg mˆ2)/(cg pm) */ double speed_mps; /* the speed in meters per second */ double lump_conversion_cg_pm_to_kg_m =1e17; /* lump conversion factor from 1e5 cg/kg 1e12 pm/m */ double speed_cmps; /* the speed in centimeters per second*/ double lump_conversion_cg_pm_to_kg_cm = 1e19; /* conversion factor from 1e5 cg/kg 1e12 pm/m 100 cm/m */ /* concentration volume = mass */ /* initial normalized mass : the sum of the individual masses divided by the total mass */ double initial_normalized_mass = 1; /* Working on the array */ 267

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Appendix Y (Continued) double mass[51][1331]; /* the mass array is 1331 by 51, the bolus is 40 square */ /* defining the struct drawer */ struct drawer d; /*struct fragments past_fragmentsize[51][1331][5]; /* an array of 66551 elements structure of 2 dna fragments */ /*struct fragments current_fragmentsize[51][1331][5]; /* an array of 66551 elements structure of 2 dna fragments */ /* Defining and opening the files */ FILE* datafile1; datafile1 = fopen("array.first.data","a"); /* this is the data from the array */ FILE* datafile2; datafile2 = fopen("array.second.data","a"); /* this is the data from the array */ FILE* datafile3; datafile3 = fopen("array.third.data","a"); /* this is the data from the array */ FILE* datafile4; datafile4 = fopen("array.fourth.data","a"); /* this is the data from the array */ FILE* datafile5; 268

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Appendix Y (Continued) datafile5 = fopen("array.fifth.data","a"); /* this is the data from the array */ FILE* datafilesum; datafilesum = fopen("array.sum.data","a"); /* this is the data from the array */ /* printf("Please enter the number of base pairs of your plasmid. "); scanf("%i",&base_pairs);*/ /* Computing the charge and radius of the DNA sphere */ dna_charge = base_pairs charge_per_base_pair elementary_charge; printf("The charge on your plasmid is %e Coulombs.\n",dna_charge); r_cubed = cubebyvalue(radius_ppp); /* cubing the radius of the purine pyrimadine pair */ volume_ppp = (double) (4*pi)/3 r_cubed; printf("Volume per base pair = %f pmˆ3.\n",volume_ppp); volume_DNA_sphere = volume_ppp base_pairs; printf("Volume DNA sphere = %e pmˆ3.\n",volume_DNA_sphere); three_fourths_volume_DNA_sphere = (double) 3/(4 pi) volume_DNA_sphere; radius_DNA_sphere = cuberoot(three_fourths_volume_DNA_sphere); printf("Radius DNA sphere = %e pm.\n",radius_DNA_sphere); /* Doing the force balance between the electric field and the fluid */ speed_mps = (z elementary_charge electric_field)/(6 pi eta radius_DNA_sphere) lump_conversion_cg_pm_to_kg_m ; printf("Speed = %e m/s\n",speed_mps); speed_cmps = (z elementary_charge electric_field)/(6 pi eta 269

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Appendix Y (Continued) radius_DNA_sphere) lump_conversion_cg_pm_to_kg_cm ; printf("Speed = %e cm/s\n",speed_cmps); d.base_pairs = base_pairs; d.radius_DNA_sphere = radius_DNA_sphere; d.speed_cmps = speed_cmps; d.initial_normalized_mass = initial_normalized_mass; printf("# of base pairs = %d. \n",d.base_pairs); printf("speed in cmps = %e pm.\n",d.speed_cmps); printf("Radius DNA sphere = %e pm.\n",d.radius_DNA_sphere); for (i = 1; i <= 50; i++) /* 1331 50 = 66551 */ { /* The front wall of the dna well */ for (j = 1; j<=35;j++) { fragments_first[i][j][1]=0;fragments_second[i][j][1]=0;fragments_third[i][j][1]=0;fragments_fourth[i][j][1]=0;fragments_fifth[i][j][1]=0;fragments_sum[i][j][1]=0; /*if(i % 50 == 1 || i % 50 == 2 || i % 50 ==3 || i % 50 ==4 || i % 50 == 5) { 270

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Appendix Y (Continued) mass[i][j]=0;fragments_first[i][j][1]=0;fragments_second[i][j][1]=0;fragments_third[i][j][1]=0;fragments_sum[i][j][1]= fragments_first[i][j][1] + fragments_second[i][j][1] // + fragments_third[i][j][1]; } else if(i % 50 == 46 || i % 50 == 47|| i % 50 ==48|| i % 50 ==49 || i % 50 ==0) { mass[i][j]=0;fragments_first[i][j][1]=0.4*mass[i][j];fragments_second[i][j][1]=0.6*mass[i][j];fragments_third[i][j][1]=0.1*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] + fragments_second[i][j][1] // + fragments_third[i][j][1]; } else { mass[i][j]=0;fragments_first[i][j][1]=0.4*mass[i][j];fragments_second[i][j][1]=0.6*mass[i][j]; 271

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Appendix Y (Continued) fragments_third[i][j][1]=0.1*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] + fragments_second[i][j][1] // + fragments_third[i][j][1]; }*/ } /* The dna filled well */ for (j = 36; j<=61;j++) { if(i % 50 == 1 || i % 50 == 2 || i % 50 ==3 || i % 50 ==4 || i % 50 == 5) { mass[i][j]=0;fragments_first[i][j][1]=0.2*mass[i][j];fragments_second[i][j][1]=0.2*mass[i][j];fragments_third[i][j][1]=0.2*mass[i][j];fragments_fourth[i][j][1]=0.2*mass[i][j];fragments_fifth[i][j][1]=0.2*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] + fragments_second[i][j][1] + fragments_third[i][j][1] + fragments_fourth[i][j][1] + fragments_fifth[i][j][1]; } else if(i % 50 == 46 || i % 50 == 47|| i % 50 ==48|| i % 50 ==49 || i % 50 ==0) 272

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Appendix Y (Continued) { mass[i][j]=0;fragments_first[i][j][1]=0.2*mass[i][j];fragments_second[i][j][1]=0.2*mass[i][j];fragments_third[i][j][1]=0.2*mass[i][j];fragments_fourth[i][j][1]=0.2*mass[i][j];fragments_fifth[i][j][1]=0.2*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] + fragments_second[i][j][1] + fragments_third[i][j][1] // + fragments_fourth[i][j][1] + fragments_fifth[i][j][1]; } else { mass[i][j]=1;fragments_first[i][j][1]=0.2*mass[i][j];fragments_second[i][j][1]=0.2*mass[i][j];fragments_third[i][j][1]=0.2*mass[i][j];fragments_fourth[i][j][1]=0.2*mass[i][j];fragments_fifth[i][j][1]=0.2*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] + fragments_second[i][j][1] + fragments_third[i][j][1] // + fragments_fourth[i][j][1] + fragments_fifth[i][j][1]; } } /* The rest of the gel */ 273

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Appendix Y (Continued) for (j = 62; j<=1331;j++) { fragments_first[i][j][1]=0;fragments_second[i][j][1]=0;fragments_third[i][j][1]=0;fragments_fourth[i][j][1]=0;fragments_fifth[i][j][1]=0;fragments_sum[i][j][1]=0;/* if(i % 50 == 1 || i % 50 == 2 || i % 50 ==3 || i % 50 ==4 || i % 50 == 5) { mass[i][j]=0;fragments_first[i][j][1]=0.4*mass[i][j];fragments_second[i][j][1]=0.6*mass[i][j];fragments_third[i][j][1]=0.1*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] // + fragments_second[i][j][1] + fragments_third[i][j][1]; } else if(i % 50 == 46 || i % 50 == 47|| i % 50 ==48|| i % 50 ==49 || i % 50 ==0) { mass[i][j]=0;fragments_first[i][j][1]=0.4*mass[i][j];fragments_second[i][j][1]=0.6*mass[i][j]; 274

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Appendix Y (Continued) fragments_third[i][j][1]=0.1*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] // + fragments_second[i][j][1] + fragments_third[i][j][1]; } else { mass[i][j]=0;fragments_first[i][j][1]=0.4*mass[i][j];fragments_second[i][j][1]=0.6*mass[i][j];fragments_third[i][j][1]=0.1*mass[i][j];fragments_sum[i][j][1]= fragments_first[i][j][1] // + fragments_second[i][j][1] + fragments_third[i][j][1]; }*/ } } /* moving the DNA between the nodes */ t = 1; /* This is presetting the time value */ for(k=2; k<=50; k++) { for(i = 1; i <= 50;i++) { for(j=1;j<=1331;j++) { if(t%1==0 && j>25) 275

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Appendix Y (Continued) { fragments_first[i][j][k] = fragments_first[i][j-25][k-1]; } else { fragments_first[i][j][k] = fragments_first[i][j][k-1]; } if(t%1==0 && j>20) { fragments_second[i][j][k] = fragments_second[i][j-20][k-1]; } else { fragments_second[i][j][k] = fragments_second[i][j][k-1]; } if(t%1==0 && j>18) { fragments_third[i][j][k] = fragments_third[i][j-18][k-1]; } else { fragments_third[i][j][k] = fragments_third[i][j][k-1]; } if(t%1==0 && j>11) { 276

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Appendix Y (Continued) fragments_fourth[i][j][k] = fragments_fourth[i][j-11][k-1]; } else { fragments_fourth[i][j][k] = fragments_fourth[i][j][k-1]; } if(t%1==0 && j>8) { fragments_fifth[i][j][k] = fragments_fifth[i][j-8][k-1]; } else { fragments_fifth[i][j][k] = fragments_fifth[i][j][k-1]; } fragments_sum[i][j][k] = fragments_first[i][j][k]// + fragments_second[i][j][k] + fragments_third[i][j][k]// + fragments_fourth[i][j][k] + fragments_fifth[i][j][k]; } } t++;} printf("Moving the DNA between nodes complete \n"); /* Printing the data to a file */ for(j=1; j <= 1331; j++) { 277

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Appendix Y (Continued) for (i = 1; i <= 50; i++) { for (k = 1; k <= 50; k++) { fprintf(datafile1,"%f\t",fragments_first[i][j][k]);fprintf(datafile2,"%f\t",fragments_second[i][j][k]);fprintf(datafile3,"%f\t",fragments_third[i][j][k]);fprintf(datafile4,"%f\t",fragments_fourth[i][j][k]);fprintf(datafile5,"%f\t",fragments_fifth[i][j][k]);fprintf(datafilesum,"%f\t",fragments_sum[i][j][k]);if(k%50==0) { fprintf(datafile1,"\n");fprintf(datafile2,"\n");fprintf(datafile3,"\n");fprintf(datafile4,"\n");fprintf(datafile5,"\n");fprintf(datafilesum,"\n"); } } if(i%50==0) { fprintf(datafile1,"\n");fprintf(datafile2,"\n");fprintf(datafile3,"\n"); 278

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Appendix Y (Continued) fprintf(datafile4,"\n");fprintf(datafile5,"\n");fprintf(datafilesum,"\n"); } } } fclose(datafile1);fclose(datafile2);fclose(datafile3);fclose(datafile4);fclose(datafile5);fclose(datafilesum); }/*Bibliography: [1] From S. B. Smith and Arnold J. Bendich, Electrophoretic Charge 279

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Appendix Y (Continued) Density and Persistance Length of DNA as Measured by Fluorescence Microscopy, Biopolymers (29) 1167-1173, 1990 */ 280

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Appendix Y (Continued) Y .2 Inte ger Cube Function Header File, cubebyv alue.h /****************************************************************** ** Cube by value header file Joseph D. Hickey Nov. 6th, 2003 ********************************************************************* / int cubebyvalue(int n); 281

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Appendix Y (Continued) Y .3 Inte ger Cube Function Code, cubebyv alue.c /************************************************************* ** Name: cubebyvalue.c Author: Joseph D. Hickey Created: Nov. 6th, 2003 Requires: cubebyvalue.h The program "cubebyvalue" takes an int and returns the cube of that int. ******************************************************************/ #include #include "cubebyvalue.h" int cubebyvalue(int n) { return n n n; } 282

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Appendix Y (Continued) Y .4 Iterati v e Cube Root Solv er Header File, cuberoot.h /************************************************************** Name: cuberoot.h Author: Joseph D. Hickey Date: Nov. 7, 2003 Provides: cuberoot.h ******************************************************************* / float cuberoot(float h); 283

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Appendix Y (Continued) Y .5 Iterati v e Cube Root Solv er Code, cuberoot.c /****************************************************************** Name: cuberoot.c Author: Joseph D. Hickey Date: Nov. 7, 2003 Requires: stdio.h, math.h, cuberoot.h This program takes a (float) variable and returns the cuberoot of that (float) variable as a (float). ******************************************************************* *****/ #include #include #include "cuberoot.h" float cuberoot(float h) { int j,k,i; double a,b=1.0,c,d,e; double ul, ll, mp, cmp; double accept_error = 1e-6; ul = sqrt(h); /* calculating the upper limit value */ ll = sqrt(ul); /* calculating the lower limit value */ mp = (ul + ll) /2; /* calculating the midpoint value */ cmp = mp mp mp; /* calculating the cube of the midpoint value */ 284

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Appendix Y (Continued) j = 0; while ( b > accept_error) /* b is the difference between the guess and the root */ { cmp = mp mp mp; /* cubing the midpoint */ if(cmp > h) /* cube of the midpoint is greater than the passed value */ { j = 1; ul = mp; mp = (ul + ll) / 2; } if(cmp < h) /* cube of the midpoint is less than the passed value */ { j = 2; ll = mp; mp = (ul + ll)/2; } a = (1/accept_error) *(cmp h); b = accept_error sqrt(a*a); /* printf(" the computed values are %9.8f, %9.8f \n",cmp, mp); */ }return mp; } 285

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Appendix Z: Gel Electrophoresis Simulation Images Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 1 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 4 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 7 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Figure Z.1: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field Initial State, 3 Minutes and 6 Minutes After Onset of Electric Field 286

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Appendix Z. (Continued) Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 10 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 13 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 16 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Figure Z.2: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 9 Minutes, 12 Minutes and 15 Minutes After Onset of Electric Field 287

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Appendix Z. (Continued) Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 19 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 22 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 25 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Figure Z.3: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 18 Minutes, 21 Minutes and 24 Minutes After Onset of Electric Field 288

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Appendix Z. (Continued) Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 28 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 31 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 34 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Figure Z.4: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 27 Minutes, 30 Minutes and 33 Minutes After Onset of Electric Field 289

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Appendix Z. (Continued) Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 37 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 40 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 43 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Figure Z.5: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 36 Minutes, 39 Minutes and 42 Minutes After Onset of Electric Field 290

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Appendix Z. (Continued) Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 46 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Normalized IntensityDNA Well Distance Along Gel (45 microns/node) 49 0 5 10 15 20 25 30 35 40 45 50 0 200 400 600 800 1000 1200 1400 0 0.2 0.4 0.6 0.8 1 Figure Z.6: DN A Fragment Motion in a 1% Ag arose Gel in a 6.54 V cm Electric Field 45 Minutes, 48 Minutes and 51 Minutes After Onset of Electric Field 291

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About the Author Joseph Hick e y w as born in T ampa, Florida on February 13 th 1973. He attended H.B. Plant High School in T ampa, Florida from 1988 to 1991. In 1991, H.B. Plant w as noted as an Excellence in Education Select Sc hool and rank ed 17 th in the Nation. In high school, the author focused on Biology Marine Biology and Ph ysics. He attended Eck erd Colle ge in St. Petersb ur g Florida from August 1991 to May 1996, where he earned bachelors of science de grees in Biology and Ph ysics with minors in Math and Chemistry His under graduate research project w as titled “The Increased Ef cac y of Common Electrochemotherap y Agents in Conjunction with Electroporation on Di v erse Cell T ypes”, under the e xpert tutelage of Dr Richard Heller and Dr Mark Jaroszeski. This w ork w as published in 2000 in the journal Anti-Cancer Drugs [88]. He entered into the combined Masters/Bachelors 5 year program in August of 1996. His masters thesis titled “Creating an Instrument System to Electroporate and Monitor the Uptak e of Fluorescent Molecules into Cells” w as accepted on July 26 th 2000. The thesis w as nominated for the outstanding masters thesis a w ard and w on the Outstanding Student Presentation A w ard at the 2000 Suncoast Biomolecular Science Conference. While pursuing his Ph.D. in Chemical Engineering the author earned a Masters of Science in Ph ysics and a graduate certicate in Materials Science and Engineering. The author aspires to continue in biomedical research and to teach at a small pri v ate colle ge that puts a strong emphasis on education rather than publication.