MODELING OF ELECTROPORATION MEDIATED MOLECULAR DELIVERY

Transcription

1 MODELING OF ELECTROPORATION MEDIATED MOLECULAR DELIVERY BY JIANBO LI A dissertation submitted to the Graduate School New Brunswick Rutgers, The State University of New Jersey in partial fulfillment of the requirements for the degree of Doctor of Philosophy Graduate Program in Mechanical & Aerospace Engineering Written under the direction of Professor Hao Lin and approved by New Brunswick, New Jersey October, 2011

2 ABSTRACT OF THE DISSERTATION Modeling of Electroporation Mediated Molecular Delivery by Jianbo Li Dissertation Director: Professor Hao Lin In electroporation, electric fields are used to transiently permeabilize the cellular membrane to facilitate molecular exchange. This technique is extensively employed to deliver biologically active molecules into the cell compartment, to perform a variety of tasks such as electrochemotherapy and directed stem-cell differentiation. Despite its great potential and broad applications, electroporation still suffers from low delivery efficiency and/or significant cell death, in part due to a lack of fundamental understanding and quantitative prediction tools for the complex processes involved. The aim of this dissertation is to overcome these limitations by providing the much need capabilities. We have developed the first spatially- and temporally-resolved numerical model to predict molecular transport via electroporation. The model framework is used to study the delivery of small molecules such as calcium and propidium iodide. The results are compared directly with experimental data in the literature, and reveal that electrophoresis, not diffusion, is the dominant model of delivery. Furthermore, the maximum achievable concentration within the cell is reciprocally correlated with the extracellular electrical conductivity. This behavior is mediated by an electrokinetic mechanism known as fieldamplified sample stacking. Based on the simulation, we have also developed a compact model to predict delivery with a simple formula. This formula can be conveniently used in place of the complex full-model simulation. This dissertation contributes to the ii

3 field by generating predictions that are previously not available, identifying mechanisms for the underlying physical processes, and providing a high-fidelity optimization tool. The results offered are currently utilized to improve and optimize eletroporation as a promising cell-manipulation technique. iii

4 Acronyms Abbreviation Definition ADI ASE BL DNS FASS Alternating-direction implicit (scheme) Asymptotic Smoluchowski equation Boundary layer Direct numerical simulation Field-amplified sample stacking GT99 Reference [26] NP PAD PI SE TMP Nernst-Planck (equation) Pore area density Propidium iodide Smoluchowski equation Transmembrane potential iv

5 List of Symbols Symbol Description A C m D E E 0 F I J N P e R R i R p R tot T U eo U ep V m V rest a c g l h Area element of a control volume Surface capacitance of the membrane Diffusivity Electric field Applied electric field Faraday constant/molecular flux when with subscripts Current through local membrane Molecular flux Local pore number density Péclet number Ideal gas constant Input resistance Pore resistance Total resistance Temperature Electroosmotic velocity Electrophoretic velocity Transmembrane potential Resting potential Cell radius Concentration Surface conductance of the membrane Membrane thickness v

6 i p j k k +, k r 0 r t w z Γ Φ β Current through each individual pore Electrical current density Boltzmann constant Association and dissociation rate constants Pore radius Minimum radius of a hydrophilic pore Time Mechanical mobility Valence number Stacking ratio Electric potential A dimensionless variable characterizing the degree of charge separation (defined in Eq. (2.15)) γ Extra- to intra-cellular conductivity ratio (σ e /σ i ) δ ǫ ζ λ D Pore aspect ratio (r 0 /h) Permittivity Zeta potential Debye length µ Viscosity ξ, η Coordinates of the rotational elliptic system ρ E ρ p σ τ Net charge Pore area density Electrical conductivity Time scale vi

7 Acknowledgements I would like to express my deep and sincere gratitude to my supervisor, Professor Hao Lin. His inspiration, guidance, detailed instruction, and demand for excellence are essential to the completion of this work. I am very grateful to him for giving me the opportunity to work with him. He has not only taught me a number of invaluable lessons on conducting research, but also mentored me on many aspects of my life during my study at Rutgers University. I am grateful to Professor Jerry W. Shan and Professor David I. Shreiber, for giving advices in studying experiments and biology. Their advices have greatly helped me in understanding the complexity of the project and inspired my thinking on different aspects of the problem. I would also like to express my gratitude to Professor Yogesh Jaluria for serving on my committee and providing valuable suggestions on the study in spite of his extremely busy schedule. In addition, I take this opportunity to thank my colleague Mohamed Sadik, Miao Yu, and Jia zhang. Their numerous advises and discussions during the study have greatly strengthened the quality of this work. Finally, I would like to express my gratitude to my wife and my friends, who have been continuously encouraging and supporting me during my graduate study at Rutgers University. vii

8 Table of Contents Abstract Acknowledgements ii vii 1. Introduction The Current-Voltage Relation for Electropores with Conductivity Gradients Introduction Theory Formulation Analytical solution Numerical Simulation Discussions Numerical Simulation of Molecular Uptake via Electroporation Introduction Model Formulation The electrical problem Pore nucleation and evolution Species transport Numerical implementation Results Discussions The effect of the membrane permeabilization on FASS The effect of free diffusion post-pulsation viii

9 Experimental evidences to support FASS Conclusions The Effect of Conductivity on Electroporation Mediated Molecular Delivery Introduction Model formulation Results Simulation of supra-electroporation with varying extracellular conductivity A compact model and comparison with classical electroporation Discussions Conclusions Conclusions Appendix A. The boundary conditions for the governing Eqs. ( ) 69 A.1. The ambient boundary conditions A.2. The boundary conditions on the membrane Appendix B. A brief derivation of Eq. (3.12) Appendix C. The effect of electroosmotic flow on molecular transport 75 Appendix D. The two-dimensional effects in electrophoretic molecular transport Appendix E. A compact model for molecular delivery via electrophoresis 79 References Vita ix

10 1 Chapter 1 Introduction In electroporation, programmed electric pulses are applied to biological cells, so that the lipid membrane becomes permeable. The permeabilization can be either irreversible or reversible. Irreversible permeabilization can permanently damage the cell and lead to cell apoptosis, and therefore can be used in biofouling control, debacterialization, and cancer therapy in a drug-free manner [63, 70, 83, 88]. On the other hand, a reversible permeabilization provides transient access to the cytoplasm while maintaining cell functionality, such that biological agents can be delivered into the cell [27, 60, 102]. The latter is the focus of the current work. Reversible electroporation is mainly used as a means of molecular delivery. Initially developed as a method for gene transfer [67], electroporation now has been used for the transfer of a large variety of other molecules, such as ions, drugs, dyes, tracers, antibodies, proteins, oligonucleotides, RNA, etc. [27, 120]. Compared to conventional methods, electroporation-mediated molecular delivery has many advantages. The technique is non-invasive, non-chemical, fast and easy to perform, and relatively non-toxic. Because the process involves mainly physical interactions between the electric field and cell membrane, electroporation is less dependent on cell type than competitive methods [29, 35, 67, 68]. Electroporation has been shown to be effective both in vitro and in vivo. In clinical studies, electroporation has been harnessed for DNA vaccination, and for the treatment of a variety of cancers including skin, lung, and breast tumors, bone metastases and leukemia, among others [20, 23, 52, 94, 119]. Despite great potential and extensive applications, electroporation still suffers from significant limitations, including low delivery efficiency and/or low cell viability. Although improvements have been sought for individual causes, the approaches were by

11 2 and large empirical and ad hoc, and could not be generalized. As a result, the delivery efficiency is inconsistent, and may vary by several orders of magnitude under different experimental conditions [116]. The main bottle neck in overcoming these limitations is a lack of fundamental understanding and accurate prediction tools of the complex processes involved. The present work aims at developing a quantitative model to provide these much needed capabilities E + + θ Cytoplasm Cell membrane Ambient solution Figure 1.1: A schematic of electroporation-mediated molecular delivery. The process involves two major aspects: 1) permeabilization of the cell membrane; 2) molecular uptake. Electroporation mediated molecular delivery involves two main aspects. The first is the permeabilization of the membrane via an applied electric field, to provide access to the cytoplasm. The second is the transport of biologically active or other molecules into the permeabilized cell, which is a process known as molecular uptake. Research in the past three decades has led to significant advances in the understanding of the first aspect, namely, membrane permeabilization. As revealed by experimental observations, the applied electric pulse introduces an electric potential drop across the cell membrane, which is termed the transmembrane potential(tmp)[34, 36, 46]. When the TMP exceeds a critical threshold, aqueous, conducting pores/defects begin to form on the membrane [10, 100, 104], which significantly increase the membrane conductance

12 3 and allow molecular passage. In the modeling literature, this process is captured by the Smoluchowski equation (SE) [3, 24, 74, 108], which statistically describes the evolution of the pore population as a function of the driving TMP. Based on this equation, various groups developed their respective models to predict membrane permeabilization. In particular, Krassowska and co-authors derived an asymptotic version of the Smoluchowski equation (ASE, [64, 65]), which was subsequently used in a series of whole-cell-level simulations [15, 16, 50]. Weaver and co-authors implemented the ASE in a transport lattice framework [95] to study a variety of problems including conventional and supra-electroporation, electroporation of single and multiple cells, organelles, and tissue, and irreversible electroporation for cancer treatment [22, 30, 31, 32, 91, 93]. Alternatively, Joshi and co-authors solved the SE directly to track the evolution of pore size and population distribution[39, 40, 44, 42]. These works collectively provide a comprehensive understanding of membrane permeabilization, and quantitative prediction tools to correlate with experimental data, and to help improve the technique. An extensive body of work also exists to tackle the second aspect, molecular uptake. Experimental studies has helped identify many of the key elements, but the development of a conclusive understanding is still in progress. In general, the behavior of large molecules (such as DNA) is different from that of small, low-weight molecules (such as calcium ion (Ca 2+ ) and propidium iodide (PI)) [81]. For DNA, the successful delivery requires the continuous presence of an electric field [47, 115], suggesting the importance of electrophoretic forces in the process [9, 61, 96]. However, DNA electrophoresis alone is not sufficient for uptake. Additional mechanisms such as DNA aggregation, DNAmembrane interaction, and intracellular diffusion may also have critical involvements [29, 101]. The transport of small molecules (with molecular weights less than 4 kda, [101]), on the other hand, is presumably less complex. Both electrophoresis and free diffusion are believed to contribute, and the process can be described with the Nernst- Planck (NP) model for electrodiffusion [79]. However, different findings exist on the relative importance of the two mechanisms with respect to each other. Specifically, measurements by Pucihar et al. [79] revealed that free diffusion of PI post-pulsation contributed to most of the fluorescence signal collected, whereas a series of transdermal

13 4 drug delivery experiments suggested that electrophoresis of charged molecules had a major contribution [17, 76, 80, 106, 107]. In this thesis, we focus on studying the transport of small molecules into electroporated cells, aiming at identifying the governing mechanisms. We establish a model framework to predict the delivery of such molecules, including free ions (e.g., Na +, K +, Ca 2+, and Cl ), drug molecules (mostly below a few kda in the transdermal drug delivery studies [77]), and common molecular dyes used for optical measurements (e.g., PI and Fluo-3). Our work results in several important contributions: In contrast to the previous compartment or simplified diffusion models [33, 59, 78, 79, 82, 89, 92], we provide the first spatially- and temporally-resolved model to track the molecular concentrations. Our work creatively combines the ASE for membrane permeabilization with NP equations for ion transport, and generates results and data trends which can be directly compared with experimental data. Such model and results are previously unavailable. We have identified key mechanisms and processes governing delivery. We found that electrophoresis, not diffusion as previously believed, plays a critical role in delivery. Furthermore, field-amplified sample stacking (FASS), an electrokinetic mechanism well-known to the microfluidics research community, determines the achievable concentration within the cell. This contribution is a significant advance toward the development of a full quantitative prediction capability. We have developed a compact model to conveniently predict delivery with a simple formula. This model is particularly useful to the community due to its simplicity and accuracy. This dissertation is organized as follows. In Chapter 2, we first develop a theoretical model to predict the current-voltage relation across a membrane-bound pore induced by electroporation. The model links the development of the TMP and the pore nucleation and evolution processes, and is a necessary component to enable the full-model development in the rest of the thesis. The result was published in Biomicrofluidics [54].

14 5 In Chapter 3, we present details of the model framework to predict molecular delivery. As an example, we use the model to numerically study the transfer of Ca 2+. The results demonstrate good agreement with the fluorescence measurements by Gabriel and Teissié [26]. This work was published in Bioelectrochemistry [55]. In Chapter 4, we use the model developed in Chapter 3 to simulate the entry of PI into murine myeloma cells, using the experimental parameters found in [19, 62]. The main motivation of the work in this chapter is to interpret the inverse correlation between delivery and the extracellular conductivity observed therein. The development of the compact model is also presented in this chapter. This work has been summarized in a manuscript submitted to Biophysical Journal. The above work collectively contributes toward a mechanistic understanding and high-fidelity prediction capability for the eventual improvement and optimization of electroporation. The results are extensively compared with and validated by experimental data. The model is already being employed by our own and other research groups to aid in experimental design. Indeed, I have been continuously engaged in such activities through collaborative efforts. My work during my PhD study at Rutgers University has resulted in six total publications (3 published [54, 55, 84], 2 in review [56, 117], and 1 in preparation [85]). While focusing on my own thesis research, I have actively participated in other projects led by my fellow graduate students and collaborating professors from other institutions. In addition to the work presented in this thesis, I have also worked on a theory for electrodeformation [84], the analysis of electroporation-mediated delivery with AC electric fields [117], and the analysis of PI entry into 3T3 fibroblast cells for the study of conductivity effects [85]. These work are not presented here for brevity, but also are part of my contributions to my scientific research and academic achievement here at Rutgers University.

15 6 Chapter 2 The Current-Voltage Relation for Electropores with Conductivity Gradients 2.1 Introduction In this chapter, we present a model and formula for the current-voltage relation across a membrane-bound pore. This model is a necessary component for the full, whole-cell level model presented in the rest of the thesis. Research in the past two decades has made significant progress in model understanding of the complex processes involved in electroporation. In the current literature, the permeabilization process is understood as the nucleation and evolution of aqueous, conducting pores (a.k.a. the electropores) on the membrane, and is described by the Smoluchowski equation (SE) [3, 24, 74]. The application of an electric field leads to the build-up of a transmembrane potential (TMP). When the TMP reaches a critical threshold, stable hydrophilic pores begin to form on the membrane. Ionic conduction through these pores subsequently limits the growth of the TMP due to the reduced effective membrane resistance [50]. The evolution of the TMP in turn affects further evolution of the pores. A key element coupling these two processes is the current-voltage relation across the membrane-bound electropores. This relation, however, is difficult to measure directly due to the typical nano-meter pore-sizes, although measurements of the integrated currents over the entire cell/membrane can be achieved [12, 36, 38]. Various modeling approaches on the current-voltage relation can be found in the literature. In some of their earlier works, Joshi and co-authors used a simple, linear correlation between membrane resistance and the fractional area occupied by the pores [41, 44]. Krassowska and co-authors calculated the pore resistance based on the electrical conductivity of the pore-filling electrolyte solution; they have also included the

16 7 effects of the input (or spreading ) resistance which is due to the contributions from the half-spaces surrounding the pore [50, 92]. A more sophisticated formula was first proposed by Chernomordik and co-authors which was based on a one-dimensional, NP model for the electrolyte solutions within the pore [12, 28]. This model was further more rigorously developed by Barnett [2], and has been widely used by various groups pursuing electroporation modeling research [15, 30, 39, 42, 43, 45, 91, 93, 108]. A salient feature of this model is the inclusion of the effects due to membrane polarization (the Born energy) and steric hindrance. The most complete formula following this approach is that by Vasilkoski et al. [108], in which the authors also take into account the spreading resistance in a manner similar to Krassowska and co-authors [50, 92]. It is important to note that nearly all these models require the specification of the electrical conductivity for the electrolyte solution in and/or around the pore as an input parameter. For some applications where the intra- and extra-cellular conductivities are equal and known, this task is straightforward, as the conductivity field around the pore can be set to be uniform. For most cases, however, the conductivities do differ. An example is sea urchin electroporation where the conductivity of the buffer is an order of magnitude higher than that of the cytoplasm [36, 50]. In this situation, strong conductivity and electrolyte concentration gradients exist across the membrane-bound pore, and the effective pore conductivity (or resistance) needs to be calculated as a function of the specific environmental parameters. The current work attempts to achieve this goal. We solve the NP equations for a binary electrolyte solution in a three-dimensional geometry describing a cylindrical, membrane-bound electropore and its surrounding spaces (see Fig. 2.1a below). Our approach follows the direct solution method of Neu et al. [66]. However, instead of assuming a uniform conductivity field, we allow different electrolyte concentrations, and hence electrical conductivities, in the intra- and extra-cellular spaces. As a consequence, the conductivity gradient may induce charge separation, and both electrically neutral and non-neutral regimes exist. In 2.2, the governing equations are first solved analytically in the electrically neutral limit. The

17 8 resulting formula, an effective pore resistance as a function of the ambient conductivities, the pore radius, and the membrane thickness, is a main contribution of this chapter. In 2.3, the analytical solution is compared with results from direct numerical simulations of the original NP equations. We find that the agreement between the two solutions is excellent when charge separation is negligible, and reasonable even when electroneutrality is violated, in particular in terms of the effective pore resistance. The results suggest the use of the analytical solution as a convenient and practical formula to calculate the current-voltage relation across an electropore for system-level electroporation modeling. In 2.4, we briefly discuss the present results in relation to the class of methods following Glaser et al. [28] and Barnett [2]. A main difference is that the electric potential in the present work is solved consistently from the ionic current conservation equations, and does not require an empirical specification. We also discuss the limitations of the present model, and possible approaches for extension. In Appendix A, we document the detailed arguments leading to the boundary conditions. These boundary conditions implicitly include the effect of membrane polarization within the continuum framework. 2.2 Theory In Fig. 2.1a, an isolated pore is idealized as a cylindrical, electrolyte-filled channel embedded in an impermeable membrane. The radius of the pore is r 0, the membrane thickness is h, and the geometry is axisymmetric with respect to the z-axis. σ and Φ denote the ambient electrical conductivity and electric potential, and the superscripts i and e denote intra- and extra-cellular, respectively. Specific experimental conditions often give rise to differences in σ i and σ, e resulting in non-uniform conductivity distributions. The difference in the ambient potentials defines the TMP, V m Φ e Φ i. (2.1)

18 9 An effective pore resistance is defined as R tot = R p +R i = V m /i p, (2.2) where i p is the total ionic current through the pore, R p denotes the resistance due to the electrolyte solution within the pore, and R i is the spreading or input resistance due to contributions from the half-spaces surrounding the pore [69, 92, 108]. The effective resistance characterizes the current-voltage relation across the pore-embedded membrane, which is the subject of the current study. σ e, Φe membrane h z (a) extracellular space r 0 r σ e, Φ e η = 1 (b) extracellular space intracellular space ξ = 0 σ e m, Φe m σ i, Φ i Figure 2.1: Schematic of the problem. (a) The geometry of an isolated pore. (b) The rotational elliptic coordinates for the extracellular space: ξ (solid) and η (dashed) contours. Here, σ e m and Φ e m are the conductivity and potential at the extracellular-pore interface, respectively Formulation We use a binary electrolyte model to describe the fluid around the electropore. The steady-state NP equation for ionic transport reads: J ± = 0, J ± w ± Fz ± c ± Φ+D ± c ±. (2.3) Here the subscripts + and denote the cation and anion, respectively. J denotes the molecular flux which includes contributions from both electrophoresis and diffusion. w is the ion mechanical mobility, F is the Faraday constant, z is the ion valence number, c is the molar concentration, D is the molecular diffusivity, and Φ is the electric

19 10 potential. Eq. (2.3) can be re-written in terms of the electrical conductivity and charge density, which are defined as: σ F 2 (w + z 2 +c + +w z 2 c ), ρ E F(z + c + +z c ). (2.4) For simplicity, we shall assume that the ions are monovalent with symmetric properties, i.e., z + = z = 1, w + = w = w. Using Einstein s relation, D = wrt, where R is the molar gas constant, and T is the temperature, we also have D + = D = D. Straightforward manipulation reveals [w 2 F 2 ρ E Φ+D σ] = 0, (2.5) [σ Φ+D ρ E ] = 0. (2.6) These equations are completed with Gauss s law relating the charge density to the electric potential: ǫ Φ = ρ E, (2.7) with ǫ denoting the electrical permittivity of the aqueous solution. For boundary conditions we assume σ σ e,i, ρ E 0, Φ Φ e,i, farfromthepore, σ/ n = ρ E / n = Φ/ n = 0, onthemembrane. (2.8) That is, the ambient solution is assumed to be electrically neutral, and the membrane is impenetrable for both the electric field and ionic transport. Note that the ambient conditions are specified in a manner similar to that of a boundary layer problem in fluid mechanics [51]. The Neumann conditions on the membrane are derived from a leaky-dielectric model [86], and include implicitly the effects of charge accumulation and membrane polarization. Further details are presented in Appendix A. Eqs. ( ) define our governing equation system.

20 11 Further simplifications can be made if we consider a scaling argument. We nondimensionalize the equations with the following scales: [x,y,z] = L, [σ] = σ o, [Φ] = E o L, [ρ E ] = ǫe o /L. (2.9) Here L is a length scale, σ o is a characteristic electrical conductivity, and E o is a characteristic electric field strength. The dimensionless equations read: [( τr τ D )ρ E ] Φ + σ = 0, (2.10) τ 2 ep where, [ ( ) ] σ τr Φ+ ρ E = 0, (2.11) τ D τ r ǫ/σ o, τ ep L/wFE o, τ D L 2 /D, (2.12) are respectively the time scales for charge relaxation, ion electrophoresis, and molecular diffusion. Note that u ep = wfe o is the characteristic electrophoretic velocity. The dimensionless variables are denoted by the superscript. For a physiological solution with an electrical conductivity in the S/m range, τ r is on the order of a few nanoseconds or less. On the other hand, using L = 20 nm and D = m 2 /s (the value for chloride in aqueous solutions), τ D is estimated to be 200 ns, much greater than τ r. Eq. (2.6) can be therefore simplified to the Ohmic equation (we hereby will always work with the dimensional equations, although the simplifications are argued in the dimensionless form): σ Φ = 0. (2.13) Furthermore, using w = mol s/kg (the mobility for a chloride ion), and an electric field strength of E o = 10 5 V/m, the electrophoretic time scale, τ ep, is estimated to be a few micro-seconds. Under this circumstance, the condition τ r τ D τ 2 ep is

21 12 satisfied, and Eq. (2.5) can be reduced to a pure diffusive equation for the conductivity: D σ = 0. (2.14) Eqs. (2.13, 2.14) constitute the reduced governing equations for the potential and the conductivity, and are decoupled from Gauss s law (2.7). Note that the reduction from Eqs. (2.5, 2.6) to (2.13, 2.14) is often (equivalently) achieved by assuming the electroneutrality condition, c + c. This assumption is valid when the criterion β (c + c )/(c + +c ) = (ρ E /F)/(σ/wF 2 ) τ r /τ ep 1 (2.15) is satisfied. Here β is a dimensionless variable characterizing the degree of charge separation. In fact, Eqs. (2.13, 2.14) are in general valid for an arbitrary binary electrolyte (i.e., not restricted to the symmetric binary electrolyte assumed here), with D replaced by an effective diffusivity, D eff D + D (z + z )/(D + z + D z ). We refer interested readers to further details in [11, 53, 57]. We remark that in the case of σ = constant (corresponding to c + = c =constant), Eq. (2.5) becomes trivial, and the charge density, ρ E, is automatically zero. For this case Eq. (2.6) is simplified to be the Laplace equation for Φ. This equation, together with boundary conditions similar to Eq. (2.8), has been solved to obtain the electric potential by Neu et al. [66] in a study of the electrical energy required for pore formation. In what follows, we will first derive an approximate analytical solution using the reduced equation system (2.13, 2.14). The solution is rigorously valid for electrically neutral regime, where the reduced governing equations are valid. For parameters relevant to electroporation applications, the effective electric field is often much greater than the estimate we used above, and non-negligible charge separation is often present. For these cases, we solve the original Eqs. ( ) directly using numerical methods. The details of the numerical solution, as well as comparisons with the analytical solution are presented in 2.3.

22 Analytical solution For the geometry described in Fig. 2.1a, the reduced Eqs. (2.13, 2.14) can be solved with a rotational elliptic coordinate transformation [69] to obtain analytical expressions forσ, ΦandR tot. Fortheextracellularspace, thenewcoordinatesaredefinedinrelation to the cylindrical coordinates as: r = r 0 (1+ξ 2 )(1 η 2 ), z = r 0 ξη + h 2. (2.16) The contour lines for ξ and η are shown in Fig. 2.1b. Note that under such transformation, the ξ = 0 contour demarcates the pore and the extracellular space. The governing Eqs. (2.13, 2.14) become [ (1+ξ 2 ) σ ] + [ (1 η 2 ) σ ] = 0, (2.17) ξ ξ η η [ (1+ξ 2 )σ Φ ] + [ (1 η 2 )σ Φ ] = 0. (2.18) ξ ξ η η We further assume that the variables at the extracellular-pore interface, σ e m and Φ e m, are constant. This assumption is an approximation which we discuss below. This simplification eliminates the variable dependence on η, and the equations can be directly integrated with respect to ξ to yield: σ(ξ) = σm e +2(σ e σm) e tan 1 (ξ), (2.19) π Φ(ξ) = Φe ln(σ(ξ)/σm) Φ e e mln(σ(ξ)/σ ) e ln(σ /σ e m) e. (2.20) The intracellular space can be solved similarly. The solution is obtained by simply replacing the superscripts e with i in the above expressions, and not presented here. Fortheelectrolytessolutioninthepore, theassumptionthatσ andφareconstantat the intra- and extra-cellular interfaces makes the governing equations one-dimensional:

23 14 ( D σ ) = 0, z z ( σ Φ ) = 0, z z σ(z = ±h/2) = σ e,i m, Φ(z = ±h/2) = Φ e,i m. (2.21) The solutions are ( z σ = σm e +(σm e σm) i h 1 ), Φ = Φ i m +(Φ e m Φ i 2 m) ln( σ/σm i ) ln(σm/σ e m) i. (2.22) The interfacial variables, σ e,i m and Φ e,i m, remain unknown. These values are determined by matching the total conductivity flux and the Ohmic current, F r0 0 ( D σ z z=± h 2 ) 2πrdr, i p r0 0 ( σ Φ z z=± h 2 ) 2πr dr, (2.23) from the respective (intracellular, pore, and extracellular) spaces at the upper and lower pore interfaces. The calculation yields σ i m = σ i + σe σ i 2(1+2h/πr 0 ), σe m = σ e σe σ i 2(1+2h/πr 0 ), (2.24) Φ i m = Φ i +V m ln(σ i m/σ i ) ln(σ e /σ i ), Φe m = Φ e V m ln(σ e /σ e m) ln(σ e /σ i ). (2.25) Eqs. (2.19, 2.20, 2.22, 2.24, 2.25) provide the complete solution for σ and Φ. Finally, with these expressions available, the effective resistance can be readily computed from the definition (2.2). The result is R tot = R p +R i = h ln(σm/σ e m) i πr0 2 (σm e σm) i + 1 ln(σ σ e m/σ i σ i m) e 4r 0 σ e σm e = (πr 0 +2h) 2πr0 2σ, (2.26) eff where σ eff σe σ i ln(σ e /σ i ). (2.27) As stated earlier, R p is the resistance from the electrolyte solution within the pore, and

24 15 the spreading resistance, R i, combines contributions from both the intra- and extracellular spaces. This final expression is a function of the membrane thickness, h, the pore radius, r 0, and the ambient conductivities, σ i and σ e. Note that the Eq. (2.26) recovers the formula used in [50, 92], where we have replaced the pore conductivity with the newly derived quantity, σ eff. We remark that although Eqs. (2.19, 2.20, 2.22) are rigorous solutions of Eqs. (2.13, 2.14) in the respective subspaces, the combined solutions need to be understood as an approximate solution for the entire domain. The approximation arises from the fact that we assume σ e,i m and Φ e,i m are constant along the pore interface. In addition, we match the total fluxes, (2.23), not the fluxes at each point. However, as we demonstrate below with direct numerical simulations, these assumptions are well-validated. The analytical and numerical solutions agree well with each other, especially in the electrically neutral regime for which the former is derived. 2.3 Numerical Simulation In this section, we solve the governing Eqs. ( ) numerically. We will denote the results as direct numerical simulation (DNS), in comparison with the analytical solution developed above. Note a rigorous prescription of the boundary condition (2.8) requires an infinite computational domain which is impossible. Instead, we choose a large computational domain (relative to both the pore radius, r 0, and the membrane thickness, h), and prescribe the ambient boundary conditions on the outer boundaries. A schematic of the computational domain is shown in Fig The geometry is axisymmetric with respect to the z-axis such that the simulation is effectively two-dimensional. Typically, L r and L z are set to be 100r 0 or greater to ensure result convergence with respect to domain size. The governing equations are solved with a finite-volume, ADI (Alternating-Direction Implicit) scheme. We use a non-uniform rectangular grid with higher resolution around the pore to optimize computational efficiency. We have tested numerical convergence with respect to resolution by increasing the number of grids. For the test case of σ = constant and r 0 = 10 nm, our result provides a consistent prediction on the electric potential when compared with that from Neu et al. [66].

25 16 Table 2.1: Parameters for numerical simulation Symbol Value Definition F C/mol Faraday constant ǫ C/V m Permittivity for water w mol s/kg Ion electrophoretic mobility D m 2 /s Diffusivity, calculated using Einstein s relation h 5 nm Membrane thickness σ i S/m Intracellular electrical conductivity For convenience, we will present results with respect to the following dimensionless variables: r = r, z = z r 0 h, σ = σ / σi σ e σ i, Φ = Φ Φi h Φ e Φ i, R = R πr0 2σi. (2.28) With these definitions, the pore has an expanse of [0,1] [ 1/2,1/2] in the (r,z) plane, and σ,φ 0,1 in the respective intra- and extra-cellular spaces away from the pore. We adopt constant values for the membrane thickness and the intracellular conductivity, namely, h = 5 nm [28, 50], and σ i = S/m [36, 50]. All relevant parameters are summarized in Table 2.1. The variations in the pore size, r 0, and the extracellular conductivity, σ, e are characterized by those of the dimensionless parameters, δ r 0 /h, γ σ e /σ i. Note that a specific cell type often requires a specific buffer with a defined value of σ. e However, in the generalized study presented here, we vary γ arbitrarily without considering such physiological constraints. We present studies with two typical values of TMP: V m = 0.05 and 1 V, respectively. The former represents a low-tmp regime where the electroneutrality is mostly observed. The latter represents a high-tmp regime, and is a typical critical value required to induce significant membrane electroporation [18, 105, 110]. Fig. 2.3 shows typical results in contours of σ and Φ. For this case, we choose r 0 = 0.8 nm (δ = 0.16), and σ e = 5 S/m (γ = 11). The value for the pore size

26 17 θ σ e, Φ e Insulation Symmetry axis Ambient Pore z Extracellular Space r Intracellular Space σ, i Φ i L z L r Figure 2.2: Schematic of the computational domain. corresponds to that of an equilibrium hydrophilic pore used in standard electroporation theories [64, 65]; the value for the extracellular conductivity is taken from sea urchin egg electroporation experiments [36], which has been studied by various groups [15, 42, 50]. Figs. 2.3a and b show the analytical solution developed in Note that the result does not depend on the magnitude of the TMP. Figs. 2.3c and d show the DNS using Eqs. ( ) and V m = 0.05 V. The result provides excellent agreement with the analytical solution. The maximum errors for the entire computational domain is and for σ and Φ, respectively. Figs. 2.3e and f show the DNS with V m = 1 V. More obvious deviations from the analytical solution are observed. The maximum errors are 0.38 and 0.19 for σ and Φ, respectively. The good agreement between the analytical solution and the DNS for V m = 0.05 is primarily attributed to a small degree of charge separation, shown in Fig. 2.4a. β is negligibly small for most of the computational domain, and reaches a maximum value of 0.14 around the lower pore interface. For this case, electroneutrality is a consistent approximation, and the reduced governing Eqs. (2.13, 2.14) hold. In addition, the assumption that σ m and Φ m (the conductivity and potential at the upper and lower pore interfaces) are constants is also approximately observed, and the flow in the pore is indeed close to one-dimensional. This effect is suggested in Figs. 2.3c and d, and is more directly shown in Fig. 2.6 below. On the other hand, a more significant charge separation is present for the case of V m = 1 V, shown in Fig. 2.4b. This effect is induced

27 18 by a strong electric field present (correlated with the high value of V m ), as we analyze below. By temporarily ignoring the diffusive current, D ρ E, Eq. (2.6) becomes the Ohmic Eq. (2.13), which can be re-written as ρ E = ǫ σ σ Φ. (2.29) Here we have used the Gauss law (2.7). Because β is proportional to the charge density, ρ E, it is subsequently correlated with both the strength of the electric field, Φ, and the conductivity gradient, σ. For large values of β, the governing Eqs. ( ) are more deviant from the reduced Eqs. (2.13, 2.14), and so are the solutions. Fig. 2.5 shows more results along the pore centerline (z-axis) for the case of V m = 0.05 V, and for two pore radii, r 0 = 0.8 nm (δ = 0.16), and r 0 = 50 nm (δ = 10), representing the small and large pore size regimes. We study five conductivity ratios, γ = 0.1, 0.5, 2, 5, and 11, respectively. For each graph, the pore extends from z = 0.5 to 0.5. In the σ and Φ plots, the dashed lines are from the analytical solution for comparison. It is straightforward to verify that the analytical solution for σ depends only on δ, not γ. For δ = 0.16 (Figs. 2.5a and b), the analytical solution agrees well with the DNS. The degree of deviation is clearly correlated with the degree of charge separation, β, plotted in Fig. 2.5c. The worst case occurs for γ = 0.1, due to two combined effects. First, a strong conductivity gradient (correlated with a large conductivity difference such as in this case) introduces an increased amount of net charge as indicated by Eq. (2.29). Eq. (2.29) also correctly predicts that the charge density is positive for γ > 1, and negative for γ < 1. Second, from Eq. (2.15), the magnitude of β is also inversely proportional to c + + c, or, equivalently, σ (see Eq. (2.15)). For γ = 0.1, the extracellular conductivity assumes the smallest value for all cases studied (σ e = S/m), resulting in a relatively large β. This argument also explains why β always reaches the maximum on the side of lower conductivity, i.e., the extracellular side for γ < 1, and the intracellular side for γ > 1.

28 19 z z z σ Analytical r (a) V m =0.05 V r V m =1 V (c) r (e) z z z Φ Analytical r V m =0.05 V (b) r V m =1 V (d) Figure 2.3: Contour plots of σ and Φ from both analytical (a and b) and numerical (c-f) solutions. In each graph the blank space encompassed by the dashed lines is the membrane. For this case δ = 0.16 and γ = 11. r (f)

29 20 z e 06 1e e 06 1e 05 1e β (V m =0.05 V) 2 (a) r 0 z β (V m =1 V) Figure 2.4: Contour plots of β from the DNS for the case of δ = 0.16 and γ = 11. r (b) In contrast, the case of δ = 10 shows uniformly excellent agreement with the analytical solution. This effect is again explained by Eq. (2.29) in terms of the correlation between charge density and electric field strength. The relatively large pore size (r 0 = 50 nm) results in a weaker electric field for the same value of V m. Estimating from Figs. 2.5b and e (and noting z = z/h, where h = 5 nm), the electric field strength for δ = 0.16 is on the order of 10 7 V/m; whereas for δ = 10, 10 5 V/m. The effective electric field is therefore nearly two orders of magnitude lower in this latter case. For all γ-values considered, the maximum absolute value for β does not exceed The maximum deviation does not exceed for both σ and Φ. Fig. 2.6 plots σ and Φ from the DNS at the extracellular-pore interface (z = 1/2) for δ = 0.16 and various values of γ. The analytical solution gives constant values (dashed) by assumption, which are calculated according to Eqs. (2.24, 2.25). In general, the numerical solutions are near-uniform close to the pore centerline (r = 0), and varies more toward the pore edge (r =1). The analytical solution, although missing the detailed variations due to the simplification, provides a reasonable approximation to the DNS (considered more accurate). The only exception is for Φ and for the case of γ = 11. As we explained earlier, this deviation is caused by the deviation from electroneutrality. For this case, and despite the difference in the absolute value when compared with the analytical solution, the solution from DNS for Φ does not

30 21 1 δ= δ= σ 0.4 σ (a) z δ= (d) z δ= Increasing γ Φ Increasing γ Φ (b) z δ= Increasing γ (e) z 5 x 10 4 δ=10 γ=2, 5, 11 0 β (c) z β 5 γ=0.1, 0.5 (f) z Figure 2.5: Centerline plots (at r = 0) for σ, Φ, and β for V m = 0.05 V. Two δ values are presented: 0.16 (a-c) and 10 (d-f). For both cases, γ = 0.1, 0.5, 2, 5, and 11. Solid lines denote the DNS; dashed, analytical; dotted, the position of the pore boundaries.

31 22 σ (a) r Φ Increasing γ (b) r Figure 2.6: Comparison of the numerical (solid) and analytical (dashed) solutions for σ (a) and Φ (b) at the extracellular-pore interface (z = 0.5), for δ = 0.16, and γ = 0.1, 0.5, 2, 5, and 11. For σ, the analytical solution does not depend on γ, and the DNS for all γ values are close to each other. show significant variation along the radial direction. Similar agreements and trends are observed for all cases studied and are not presented for brevity. For the same parametric set studied in Fig. 2.5, Fig. 2.7 presents the results for V m = 1 V, i.e., for the high-tmp regime. For δ = 0.16 (Figs. 2.7a and b) and when compared with Fig. 2.5 (V m = 0.05 V), the DNS shows obvious deviation from the analytical solution for all γ values, due to the high degree of charge separation present (Fig. 2.7c). Note that the distribution of β tends to be uniform within the pore. Interestingly, the DNS for δ = 10 still well agrees with the analytical solution. The reason for the agreement is due to a relatively low effective electric field and degree of charge separation, same as we have previously explained. Last but not least, we compare the total resistance calculated from the DNS and the analytical solution for both of the TMP values, V m = 0.05 and 1 V. We present the results for a wide range of δ and γ values, namely, δ = 0.16, 1, 2, 5, and 10, and γ = 0.1, 0.2, 0.5, 1, 2, 5, and 11. The results are presented in Fig In Figs. 2.8a and b, we show the dimensionless resistance as calculated with the two methods and for the two TMP values. In c and d, we plot the relative percentage errors between the two. The analytical result is calculated from Eq. (2.26), and then non-dimensionalized using Eq. (2.28) to yield:

32 23 1 δ= δ= σ σ Increasing γ 0.2 Increasing γ (a) z δ= (d) z δ= Increasing γ Φ Increasing γ Φ β Increasing γ z δ=0.16 (b) z (c) β x γ=0.1, 0.5 z δ=10 (e) 15 (f) z γ=2, 5, 11 Figure 2.7: Centerline plots (at r = 0) for σ, Φ, and β for V m = 1 V, and for δ = 0.16 (a-c) and 10 (d-f). For all cases γ = 0.1, 0.5, 2, 5, and 11. Solid lines denote the DNS; dashed, analytical; dotted, the position of the pore boundaries.

33 24 R,ana = (πδ +2)lnγ. (2.30) 2(γ 1) This expression shows that the dimensionless resistance has a linear dependence on δ. For the DNS, the resistance is first calculated with Eq. (2.2), then non-dimensionalized in the same manner. Note that for this case the total current is calculated as i p = r0 0 ( σ Φ ) z D ρ E 2πrdr. z z=± h 2 That is, the total current now includes both the Ohmic and the diffusive components (cf. Eq. (2.23)). The result demonstrates similar general trends we previously observe in detailed comparisons. The two solutions agree better for smaller TMP, larger δ, and larger γ values. For most cases, the difference is within 5%. It is interesting to note that good agreement is observed for some cases where the analytical and numerical calculation differ in the details, e.g., for V m = 1 V, δ = 0.16, and γ = This observation is not surprising, as the integration process (to obtain the total current) tends to ameliorate and suppress differences. Based on these comparisons, we conclude that the analytical formula (2.26, 2.27) can be used to characterize the current-voltage relation across an electropore. However, caution should be exercised for a few exceptional cases where the difference exceeds 10%. Notably, for δ = 0.16, γ = 0.1, and both TMP values, the discrepancy approaches 25%. These cases represent a regime where charge separation is the most prominent (see Fig. 2.7), and the details of the analytical and numerical calculations differ the most. In such a situation, the former best serves as an estimate for the real solution which is more accurately captured by the latter. 2.4 Discussions In this work we have presented a NP model for the steady-state ion and current transport processes around an electropore. The main purpose is to study the current-voltage relation, or, equivalently, the effective resistance across the pore-embedded membrane. We have developed an analytical solution for the electrically neutral regime. We have

34 (a) V m =0.05 V (b) V m =1 V Rtot Increasing γ Rtot Increasing γ Relative error (%) δ V m =0.05 V (c) γ=0.1 γ=0.2 γ=0.5 γ=1 γ=2 γ=5 γ=11 Relative error (%) δ V m =1 V (d) γ=0.1 γ=0.2 γ=0.5 γ=1 γ=2 γ=5 γ= δ δ Figure 2.8: Comparison of the total resistance, R tot, from the analytical (solid) and numerical (symbols) solutions (a and b). The conductivity ratios are set to be γ = 0.1, 0.2, 0.5, 1, 2, 5, and 11. (c) and (d) show the percentage error between the two calculations.

35 26 also performed DNS of the full NP equations to provide comparison and validation. We found that the analytical solution and the DNS agree well when charge separation is negligible, and deviate more obviously when otherwise. For the effective resistance, the analytical solution provides an accurate prediction for most of the cases studied. The availability of the analytical expression (2.26) is of practical importance for electroporation modeling and applications, and is one of the main contributions of the current work. In a typical case, the electropores form a large and dynamic population with variable sizes. Performing DNS for each individual pore is prohibitively expensive. Using a compact formula such as Eq. (2.26) to compute the total membrane resistance and electric current is the best viable way for whole-cell or planar-membrane electroporation model calculations. The current work differs from the class of approaches following Glaser et al. [28] and Barnett [2]. In these works, the pore conductance is derived from a one-dimensional NP model, and is expressed in terms of an electric potential. This electric potential in general includes a linear, applied component, and a contribution from the membrane polarization effect. The latter is specified empirically, and is different for the respective cat- and an-ions. Such difference in the electric potential felt by the different ions is possibly accounting for the non-continuum effects, e.g., when the ions are passing through the pore one by one (see also a discussion below). The model by Kakorin and Neumann [45] also considers the ion concentration difference across the pore, but follows a similar approach otherwise. The present work solves the NP system directly in a three-dimensional geometry surrounding the pore. The electric potential is obtained from the governing Eqs. ( ). Consistent with the continuum assumption, the electric field is the same for both the cat- and an-ions. The membrane polarization effect is implicitly taken into account by setting appropriate boundary conditions on the membrane (see Appendix A for further details). The present work extends from that of Neu et al. [66] by considering a gradient in the electrical conductivity instead of a uniformly-distributed conductivity field. A salient difference between the physical behavior of these two cases is that charge separation is possible in the former case (Eq. (2.29)), and theoretically zero in the latter.

36 27 The presence of a net charge in the fluid bulk may have an interesting implication on molecular translocation which we speculate below. Inevitably, the present model is a significant idealization of a problem involving great complexities. We have made simplifications such that the model is tractable, but meanwhile have ignored effects that may have important impacts on the results. These effects are the subject of future explorations, and a few of them are discussed below. Non-continuum effects The present work employs a continuum model to capture physical processes at the nano-meter scale. As pointed out by previous authors, the flow variables in this case are understood as ensemble averages [2]. Within this understanding, the continuum model is assumed to be valid. Even the smallest pore size considered (r 0 = 0.8 nm) is a few times greater than the typical ion radii, hence allowing the passing of several ions at the same time. Non-continuum effects are expected to be more prominent for very dilute solutions, where the ions may pass through the pore one by one. For this case, the electric field each ion experiences depends on the charge it carries, and its instantaneous position with respect to the membrane. Such effects can be approximated by including a potential of trapezium shape in a modified continuum model [28, 45], or can be better captured via molecular simulations, to which the continuum models offer a reference of comparison. Multi-ion effects We have assumed that the electrolyte solution composes of two symmetric, mono-valent ions. This approximation is appropriate for salt-based solutions, such as in vesicle electroporation experiments [18]. For biological cells, and in particular in the cytoplasm, the electrolyte solution has a more complex composition involving both small ions [16] and charged macromolecules (proteins, DNA, etc.). The latter often possess relatively large charge numbers and small diffusivities. An extension to a multi-species NP model can take into account possible non-binary behavior. However, the main challenge would be to properly specify the properties of the macromolecules, such as the number of species, the intracellular concentration, the diffusivity and mobility, etc. Electrohydrodynamic flow Fluid motion induced by the electrostatic(maxwell) stress has been ignored in the present model. The consideration of this effect requires the

37 28 coupling of the NP system with the Navier-Stokes equations describing fluid motion, such as in a work by Lin et al. [57]. Without solving these detailed equations here, it is interesting to speculate the impact of such flow should the effect becomes important. The driving force of the flow, the electrical body force, is derived from the divergence of the Maxwell stress, and equals ρ E E (a vector). This forcing term determines the direction of the induced flow. That is, if ρ E > 0, the flow points to the direction of the electric field; if ρ E < 0, the opposite. In the specific geometric configuration considered here, the flow would jet toward the intracellular space if γ > 1, and the opposite if γ < 1. Note that the directions do not change if the direction of the applied electric field reverts. The induced flow always points into the lower conductivity region. This trend has interesting implications for electroporation-mediated molecular delivery: The EHD flow has a favorable effect if transport into the lower conductivity side is desired, and an adverse effect if the opposite. Finally as a concluding remark, direct validation of the model with experimental measurement is possible. Single and stabilized pores have been generated in previous experiments on planar membrane electroporation [113, 49]. However, no conductivity gradient was present in either of the cases. In order to pursue a direct comparison with the present theory, similar experiments need to be performed with controlled and different conductivities across the membrane. On the other hand, indirect evidence does support the present prediction. In the sea-urchin-egg electroporation model study by Krassowska and co-authors, the authors used σ i = S/m, and σ e = 5 S/m [50, 15, 16]. The conductivity for the pore was adopted to be σ = 2 S/m following previous experiments [111]. Using Eq. (2.27) and these intra- and extra-cellular conductivity values, we can calculate the effective conductivity for this case, σ eff = 1.9 S/m. This value is very close to the above one, with which the model provided favorable prediction on the whole-cell level.

38 29 Chapter 3 Numerical Simulation of Molecular Uptake via Electroporation 3.1 Introduction In this chapter, we present the main model-framework for the prediction of molecular delivery into electroporated cells. Electroporation is a complex process involving two major aspects. The first is the permeabilization of the membrane via an applied electric field, to provide access to the cytoplasm [4, 7, 99]. The second is the transport of biologically active or other molecules into the permeabilized cell, which is a process known as molecular uptake. When successful, the delivered agents stay in the cell to perform functional tasks, and the cell remains viable, and eventually recovers from the induced damage. Research in the past three decades has led to significant advances in the understanding of the first aspect, membrane permeabilization, whereas the mechanisms for the second aspect, transport, is still under debate. As discussed in Chapter 1, the current work focuses on building a prediction tool for the delivery of small molecules into electroporated cells. In contrast to the previous compartment or simplified diffusion models in the literature [33, 59, 78, 79, 82, 89, 92], we simulate the spatial and temporal evolution of molecular concentrations using the NP equations, where we also include chemical production terms to capture reactive kinetics. For membrane permeabilization, we adopt the ASE [50, 64, 65], which provides critical information to compute species fluxes across the membrane. As a specific example, we simulate the delivery of calcium ions into a Chinese Hamster Ovary cell. Our results demonstrate good agreement with the

39 30 fluorescence measurement performed by Gabriel and Teissié [26]. In addition, an analysis of the detailed transport dynamics reveals two important observations. First, for the particular case studied, electrophoresis is the dominant mode for molecular delivery. Second, the intracellular concentration of free calcium ions can be higher than the basal concentration in the extracellular solution, which behavior is mediated by an electrokinetic mechanism known as field-amplified sample stacking (FASS) [6, 13]. According to this mechanism, the maximum achievable ion concentration within the cell is reciprocally correlated with the extracellular conductivity. This prediction corroborates well with previous data [19, 62]. The observations suggest that the transport of small ions is much more complex than passive diffusion. 3.2 Model Formulation The electrical problem Φ e, σ e r Φ i, σ i a θ x E 0 Figure 3.1: Schematic of the problem. The x-axis is aligned with the direction of the electric field. r denotes the radial position, and θ is the inclination angle. The problem is axisymmetric with respect to the x-axis. A schematic of the problem is shown in Fig The cell is idealized to be a thin, rigid, and spherical shell of radius a. A spherical coordinate system is adopted. The x-axis is chosen to align with the direction of the applied electric field. σ and Φ denote the electrical conductivity and electric potential, and the subscripts i and e denote intra- and extra-cellular, respectively. Because the charge relaxation time (on the order

40 31 of nano-seconds) is small when compared with the typical time scales we study (microto milli-seconds), we adopt the Ohmic equation for the intra- and extra-cellular electric potentials: j = (σ i,e Φ i,e ) = 0, (3.1) where j is the Ohmic current density vector. This equation is solved for both the intraand extra-cellular spaces, and the solutions are coupled on the membrane through electric current continuity, n (σ i,e Φ i,e ) = C m V m t +g l (V m +V rest )+j p, (3.2) where n is the local unit normal vector on the membrane, C m is the membrane capacitance, g l is the leakage conductivity, and V m (defined as (Φ e Φ i ) r=a ) and V rest are the transmembrane and the rest potential, respectively. The last term in Eq. (3.2), j p, is the total ionic current density through the pores at a specific location on the membrane. Following the treatment by Krassowska and Filev [50], it is calculated using the following formula, j p (t,θ) = K(t,θ) j=1 i p (r j (t,θ),v m )/ A, (3.3) where A is a local area element, i p is the current through an individual pore with radius r j, and K is the total number of pores on the element. Following Eq. (2.26), i p is given by i p = 2πr2 j σ effv m πr j +2h, (3.4) where σ eff = (σ e σ i )/ln(σ e /σ i ) is an effective pore conductivity, and h is the membrane thickness.

41 Pore nucleation and evolution The ASE model to describe the evolution of the pore statistics follows closely that by Krassowska and Filev [50], ( dn dt = αe(v m/v ep ) 2 1 N N 0 e q(v m/v ep ) 2 ), (3.5) dr j dt = U(r j,v m,τ), j = 1,2,,k. (3.6) Here N(t,θ) is the local pore number density, α, N 0, q, and V ep are constants, U is the advection velocity, and τ is an effective membrane tension. According to this model, pores nucleate at an initial radius, r = 0.51 nm, and at a rate described by Eq. (3.5). They then evolve in size according to Eq. (3.6), to minimize the total energy of the lipid membrane. Resealing effects are also captured by the ASE. Further details of the model, as well as relevant parameters, are found in [50], and are not presented here for brevity. Note here we have adopted one particular model among many permeabilization models present in the modern literature [44, 50, 87, 95]. However, the detailed permeabilization process has no significant effect on molecular delivery as we demonstrate in below Species transport We adopt a generalized NP system to simulate species transport. In the following, we consider three specific species, Ca 2+, Fluo-3, and the compound CaFluo. These species are involved in typical Ca 2+ entry experiments [8, 26, 103, 112]. We assume that the cell is preloaded with Fluo-3 at a molar concentration of [Fluo] i,o, where the subscript i again denotes intracellular, and o denotes an initial value. The initial intra- and extracellular Ca 2+ concentrations are [Ca 2+ ] i,o and [Ca 2+ ] e,o, respectively (Table 3.1). When the free calcium ions enter the cell through the permeabilized membrane, they bind to Fluo-3 to form the compound CaFluo, which emits fluorescent signal upon excitation.

42 33 The reaction is described by Ca 2+ +Fluo k + CaFluo, k where k + and k are the association and dissociation rate constants, respectively, and we use Fluo to denote Fluo-3. The NP equations are modified with production terms to capture this process: [Ca 2+ ] t = (w Ca 2+Fz Ca 2+[Ca 2+ ] Φ ) (3.7) + (D Ca 2+ [Ca 2+ ] ) k + [Fluo][Ca 2+ ]+k [CaFluo], [Fluo] t = (w Fluo Fz Fluo [Fluo] Φ) (3.8) + (D Fluo [Fluo]) k + [Fluo][Ca 2+ ]+k [CaFluo], [CaFluo] t = (w CaFluo Fz CaFluo [CaFluo] Φ) (3.9) + (D CaFluo [CaFluo])+k + [Fluo][Ca 2+ ] k [CaFluo]. Here [X] denotes the molar concentration of the corresponding species X (Ca 2+, Fluo, or CaFluo), F is the Faraday constant, z X is the valence number, D X is the diffusion coefficient, and w X is the mechanical mobility (calculated from D X using Einstein s relation, D = wrt, where R is the universal gas constant, and T is temperature). Eqs. ( ) are solved for both the intra- and extra-cellular spaces, and are coupled on the membrane by continuity of molar flux density for every species: F i,e = F m, (3.10) where, F i,e n (w X Fz X [X] Φ+D X [X]) i,e, (3.11) F m ρ p D X (Pe X lnγ) h (γ 1) lnγ ([X] e [X] i exp(pe X )). (3.12) (γ exp(pe X ))

43 34 Here F i,e are the flux densities from the intra- and extra-cellular spaces, respectively, F m is the flux density across the membrane, Pe X w X Fz X V m /D X is an effective Péclet number for each species, and γ = σ e /σ i the extra-to-intra-cellular conductivity ratio. Eq. (3.12) is derived assuming that the sum of the electrophoretic and diffusive fluxes is constant along the axis within the pore, and a detailed derivation is found in Appendix B. Note that the Péclet number characterizes the relative importance of electrophoretic to diffusive transport. Equivalently, it can be rewritten as Pe X = Fz X V m /RT using Einstein s relation, which characterizes the relative importance of electrical to thermal energy. Eqs. ( ) are coupled to ( ) through two variables, Φ and ρ p. The latter is calculated for every area element after the pore statistics is obtained, ρ p (t,θ) = K(t,θ) j=1 πr 2 j/ A. Here we name this quantity the pore area density (PAD). Evidently, it is the local fractional opening area occupied by the pores, and is a measure of membrane permeabilization. We remark that by adopting the NP framework, i.e., Eqs. ( ), we have ignored the effects of electro-osmotic flow on species transport. A discussion is found in Appendix C, where its importance and impact are estimated Numerical implementation Eqs. ( ) are solved numerically using a finite volume, alternative direction implicit (ADI) scheme. The problem is effectively two-dimensional as we assume axisymmetry about the x-axis. For initial conditions, we assume N(0,θ) = 0, Φ i (0,r,θ) = V rest, Φ e (0,r,θ) = 0. The initial concentrations for Ca 2+ and intracellular Fluo-3 ([Ca 2+ ] i,o, [Ca 2+ ] e,o, and [Fluo] i,o ) are given in Table 3.1. The initial concentrations for extracellular Fluo-3, and CaFluo everywhere are zero. The simulation domain is a large sphere 20a in radius.

44 35 Table 3.1: List of model parameters. Symbol Definition Value a Cell radius 8 µm [26] h Membrane thickness 5 nm σ i Intracellular conductivity 0.5 S/m [37] σ e Extracellular conductivity 0.15 S/m [26] F Faraday constant C/mol R Universal gas constant J/K mol T Room temperature K k + Association rate constant 80 (µms) 1 [90] k Dissociation rate constant 90 s 1 [90] D Ca 2+,i Diffusion coefficient of Ca 2+ in the cytoplasm 250 µm 2 /s [90] D Ca 2+,e Diffusion coefficient of Ca 2+ in the extracellular solution 790 µm 2 /s [14] D Fluo,i Diffusion coefficient of Fluo-3 in the cytoplasm 20 µm 2 /s [90] D Fluo,e Diffusion coefficient of Fluo-3 in the extracellular solution 90 µm 2 /s [90] D CaFluo,i Diffusion coefficient of CaFluo in the cytoplasm 20 µm 2 /s [90] D CaFluo,e Diffusion coefficient of CaFluo in the extracellular solution 90 µm 2 /s [90] z Ca 2+ Valence number of Ca z Fluo Valence number of Fluo-3-5 [90] z CaFluo Valence number of CaFluo -3 [Ca 2+ ] i,o Initial Ca 2+ concentration in the cytoplasm 220 nm [26] [Ca 2+ ] e,o Initial Ca 2+ concentration in the extracellular solution 1 mm [26] [Fluo] i,o Initial Fluo-3 concentration in the cytoplasm 2.2 µm [26] E 0 Applied electric field strength 1.0 kv/cm[26] On the outer boundary, we prescribe Φ e (t,r = 20a,θ) = E 0 rcosθ, where E 0 is the strength of the applied field. This prescription well-approximates the ambient condition of a uniform electric field. To implement Eq. (3.10), the flux density across the membrane, F m, is first computed given [X] e, [X] i, ρ p, V m, etc. The resulting value is then used to prescribe the boundary flux densities, F i and F e, for the intra- and extra-cellular spaces, respectively. A non-uniform spherical grid with higher resolution around the membrane is adopted to optimize computational efficiency. The numerical convergence is tested with respect to resolution by increasing the number of grids. All general parameters pertinent to the permeabilization model ( ) are taken from [50], and are not repeated here for brevity. Parameters specific to this study (e.g., σ i,e,

45 36 a, and E 0 ), and the rate constants are summarized in Table 3.1. They are specified to best approximate the experimental conditions in [26]. This problem is the primary focus of the study presented below. 3.3 Results In this section, we first present the simulated results of a cell permeabilized with a single, 6-ms-long pulse. The cell radius is a = 8 µm, and the field strength is E 0 = 1 kv/cm. The species transport is examined in detail, and is compared with the fluorescence measurements by Gabriel and Teissié [26] (denoted as GT99 in the following). The effect of FASS, and the specific roles of electrophoresis and diffusion in delivery, are then studied and discussed. Fig. 3.2 summarizes the results on the electric potential and membrane permeabilization. Fig. 3.2a shows the electric potential contour along the cell center-plane at t = 20 µs. Outside the cell, the contours become uniformly distributed a couple of diameters away, indicating convergence of the electric field to the ambient, constant condition. The contours are less populated within the cell, suggesting a lower electric field, which we will further examine below. The presence of a TMP is evident as the contour lines are discontinuous at the membrane. In Fig. 3.2b, the TMP is shown as a function of θ, the location along the membrane. At t = 0.5 µs, the TMP exhibits a co-sinusoidal shape, consistent with a non-permeabilized membrane. Further increase in the TMP results in permeabilization around the polar caps (t = 20 µs and 6 ms), and a depression in the profile due to an increased membrane conductivity. Fig. 3.2c shows the PAD, also as a function of θ. Only the areas around the cathode- and anodefacing poles (θ = 0, π) are permeabilized. (Note here the anode is the positive, and the cathode is the negative electrode, as our device consumes power.) The slight asymmetry in the profile is due to the existence of a rest potential, V rest. Fig. 3.2d shows the development of the PAD for θ = 0, π. At t = 6 ms, the PAD at the polar caps are approximately equal in magnitude ( 0.008). After the electric field ceases, the PAD drops dramatically to For both V m and ρ p, a rapid rise occurs within the first 20 µs, followed by a slow and gradual adjustment for the rest of the pulse

46 37 duration. Other basic trends agree with the predictions by Krassowska and Filev [50], although different cell type, size, and pulsing scheme are used here. In this work, we focus on the study of species transport, and further details on the pore dynamics and permeabilization characteristics are not presented for brevity. y (µm) (a) Vm (V) µs 20 µs 6.0 ms (b) x (µm) 1 -π -π/2 0 π/2 π θ (rad) 10 3 ρp µs 20 µs 6.0 ms (c) ρp t=20 µs t=6.0 ms (d) 0 -π/2 0 π/2 π 3π/2 θ (rad) t (ms) θ = 0 θ = π Figure 3.2: Simulated cell permeabilization. (a) The electric potential contour at t = 20µs. The direction of the applied field is from left to right. (b) TMP distribution along the membrane at various times. Here θ is the polar angle, and θ = 0, π correspond to the cathode- and anode-facing pole, respectively (see Fig. 3.1). (c) PAD distribution along the membrane at various times. (d) The time course of PAD at θ = 0, π. Fig. 3.3 shows the evolution of molar concentration for the various species involved. The concentrations are shown in color contours, where red denotes high, and blue denotes low values. The specific values are plotted in Fig For the first three rows ([Ca 2+ ], [Fluo], and [CaFluo]), the contour is taken at the center-plane of the cell. The times correspond to those in GT99. The results for [Ca 2+ ] shows clearly that the ions enter from the anode-facing (positive) side during the first 6 ms, driven

47 38 by electrophoretic forces. After the pulse ceases, they slowly diffuse away within the cell, as indicated by the decrease in intensity from 10 to 16.7 ms. The peak concentration is around 10.5 mm, much higher than the basal extracellular concentration, [Ca 2+ ] e,o = 1 mm. The cause for this phenomenon is explained later. Because [Ca 2+ ] is much higher than the initial Fluo-3 concentration, [Fluo] i,o = 2.2 µm, the product concentration, [CaFluo], is limited by the latter. The contour plot for [Fluo] shows a near-uniform value (see also Fig. 3.4b), with a receding front due to consumption. The CaFluo concentration is at a similar level (see also Fig. 3.4c), but with an advancing front complementary to that of [Fluo]. Because the reaction is relatively fast (with a characteristic time of µs), the spreading of the CaFluo profile is mainly driven by that of Ca 2+, i.e., by electrophoresis for t < 6 ms, and by diffusion afterwards. A comparison of [CaFluo] evolution with the fluorescence measurement by GT99 (Fig. 3.3, bottom row) shows both agreement and discrepancy. The spreading dynamics in general agree with each other, although the simulation slightly over-predicts the rate of front propagation. The discrepancy lies in that the experimental result shows a gradual increase in the fluorescence intensity, whereas the simulation predicts a near-uniform CaFluo concentration. We argue that the non-uniformity observed in the experiments is mainly from an optical effect. Note that we plot [CaFluo] at the (infinitesimally thin) center-plane from the simulation, whereas the fluorescence measurement results from light emission integrated over a finite optical depth. To reconcile the difference, we define a convolved concentration, [CaFluo] conv σz /2 σ z /2 [CaFluo]e z2 /2σ 2 z dz (3.13) where σ z = 18 µm is the focal depth from an earlier work by the same authors, and the z-direction is perpendicular to the focal plane [25]. The result (Fig. 3.3, the fourth row) shows a desired characteristic comparable to that in the experiment. An explanation for this behavior is found below. In Fig. 3.4, the concentration profiles are plotted along the cell centerline for various times. The cell symmetrically spans from x = 8 to 8 µm. As stated above, the Ca 2+

48 39 Figure 3.3: Evolution of species concentration at the cell center-plane (top 3 rows), at times corresponding to those in GT99 (bottom row, adapted with permission). The fourth row shows a convolved concentration, [CaFluo] conv, computed with Eq. (3.13). For the simulated results, red denotes high, and blue denotes low values. White circles denote the membrane. concentration reaches 10.5 mm soon after entering the cell (at t = 4 µs, not shown), and propagates rightward electrophoretically. After the pulse ceases (t > 6 ms), the peak decays, and the profile spreads due to free diffusion. It is interesting to note the depletion outside the cathode-facing cap. In Fig. 3.4b and c, the peak values for both [Fluo] and [CaFluo] are around 2.2 µm, which are determined by the preloaded Fluo-3 concentration. Note that in Fig. 3.4c, a slight leakage is found left to the cell. This effect is induced again by electrophoresis, because CaFluo has a valence number of -3 (Table 3.1). In Fig. 3.4d, the convolved CaFluo concentration is shown. Peaks in the value appear in the profiles, which gradually increase over time. This effect is explained

49 40 by the following considerations. 1) The focal depth, σ z = 18 µm, is greater than the cell diameter, 2a = 16 µm. Light emission from the entire cell is thus collected. 2) However, the cell is thicker at the center, and thinner on the edge. Therefore, a near-uniform [CaFluo] would give rise to a non-uniform [CaFluo] conv, which we assume to correlate with the fluorescence intensity. Fig. 3.4d provides a favorable comparison withfig.2bingt99, whichisnotreproducedhere. Thisresultsuggeststhatcareneeds to be taken when attempting to calculate molar concentration values from fluorescence measurements in such situations. [Ca 2+ ] (mm) [CaFluo] (µm) Cell x (µm) Cell (a) 0.0 ms 3.3 ms 6.7 ms 10.0 ms 13.3 ms 16.7 ms x (µm) (c) 0.0 ms 3.3 ms 6.7 ms 10.0 ms 13.3 ms 16.7 ms [Fluo] (µm) [CaFluo]conv (nmol/m 2 ) Cell x (µm) Cell (b) 0.0 ms 3.3 ms 6.7 ms 10.0 ms 13.3 ms 16.7 ms x (µm) (d) 0.0 ms 3.3 ms 6.7 ms 10.0 ms 13.3 ms 16.7 ms Figure 3.4: Species concentrations along the cell centerline. Fig. 3.4d provides a favorable agreement when compared with Fig. 2B in GT99. The high intracellular Ca 2+ concentration observed in Figs. 3.3 and 3.4 can be explained by a known mechanism termed field-amplified sample stacking (FASS) [6, 13]. The concentration elevation is caused by a non-uniform electrophoretic velocity across the membrane. Here we present a simplified, one-dimensional argument. We restrict

50 41 our considerations to the cell centerline, along which all vectors are uni-directional due to symmetry. The axial electrophoretic velocity, U ep, is shown in Fig. 3.5a, and is calculated with the formula U ep = wfze x, where E x is the axial electric field. The variation in U ep has contributions from two sources. The first one is in E x, which is shown in Fig. 3.5b. At the cell interfaces (x = ±8 µm), the Ohmic law (3.1) requires that σ e E x,e = σ i E x,i, such that E x,e = E x,i (σ i /σ e ). The extracellular field is thus higher due to a lower buffer conductivity (Table 3.1). Further variations of the field inside and outside of the cell are due to geometric effects. The second is from the fact that calcium ions diffuse approximately three times more slowly within the cell (Table 3.1) [14, 90]. Using Einstein s relation, w = D/RT, the mechanical mobility within the cell is also smaller by the same proportion. The combined contributions give rise to an order of magnitude decrease in U ep across the membrane. Qualitatively, when the ions enter from the anode-facing side electrophoretically, they experience a sudden slow-down, and hence stack in a manner similar to that of a traffic jam. Quantitatively, the concentration enhancement can be estimated via a simple calculation. By flux continuity, F i = F e (Eq. (3.10)), and temporarily ignoring diffusion, we have (U ep c) i = (U ep c) e, at the membrane, (3.14) where c is a generic species concentration. If we further assume that c e remains to be the ambient, initial value (denoted by c e,o ), and that c i at the membrane is the maximum achievable concentration (denoted by c i,max ), we then have Γ = c i,max c e,o = ( )( σi De σ e D i ), (3.15) where D i and D e are respectively the intra- and extra-cellular diffusion coefficient, and Γ is the stacking ratio. Eq. (3.15) is derived following a standard FASS theory [6, 13]. Although significant simplifications are involved, it does provide a reasonable and convenient estimate. For example, for the present case, Eq. (3.15) yields Γ = 10.5, in close agreement with the value from the simulation. Moreover, it clearly indicates

51 42 that c i,max 1/σ e. (3.16) Further discussions and comparisons with experiments are found in the next section. Uep (mm/s) Cell (a) 1 µs 20 µs 6.0 ms Ex (kv/cm) µs 20 µs 6.0 ms Cell (b) x (µm) x (µm) Figure 3.5: a) The axial electrophoretic velocity. b) The axial electric field. Both quantities are plotted along the cell centerline. Interestingly, Eq. (3.15) also indicates that Γ < 1, or rarefaction, can occur. Indeed, when we set (σ i /σ e ) = 0.2, but keeping all other parameters identical, we observe a concentration decrease within the cell(fig. 3.6). For this case, Eq.(3.15) gives Γ = 0.63, and the simulation gives Γ = Note that in Fig. 3.6a, although in general it appears that U ep,e > U ep,i, what dictates rarefaction is the velocity difference right across the membrane (at x = ±8 µm), where U ep,e < U ep,i. Uep (mm/s) Cell 6.0 ms x (µm) (a) 1 µs 20 µs [Ca 2+ ] (mm) Cell 0.0 ms 3.3 ms 6.7 ms 10.0 ms 13.3 ms 16.7 ms x (µm) (b) Figure 3.6: Concentration rarefaction, for σ i /σ e = 0.2. a) The axial electrophoretic velocity. b) Ca 2+ concentration. Both quantities are plotted along the cell centerline.

52 43 Last but not least, the respective effects of electrophoresis and free diffusion on delivery are investigated. In Fig. 3.7a, the solid curve shows the total moles of Ca 2+ (denoted by [Ca 2+ ] tot ) within the cell as a function of time. The molar number increases to 1.9 fmol ( mol) during the 6-ms pulse, and changes only slowly afterwards due to diffusive leakage in/out of the cell. In contrast, if we artificially remove electrophoresis from Eqs. ( ), such that only diffusion is driving molecular transport (Fig. 3.7a, dashed), the resulting value is much lower at 0.1 fmol. This calculation demonstrates that for this particular case, electrophoresis, and not diffusion, is the dominant mode of transport. A further study on the contribution of diffusion is found in 3.4.2, where the effect of post-pulsation PAD is examined, and the results are discussed in relation to the experimental observation by Pucihar et al. [79]. Finally, molecular delivery can be further enhanced by using an extended pulse. The dot-dashed curve in Fig. 3.7a shows the evolution of [Ca 2+ ] tot under a continuous, 150-ms pulse (E 0 = 1 kv/cm). The final value reaches 22.4 fmol, which is another order of magnitude higher than that with a 6-ms pulse. For this case, the bulk of Ca 2+ continues to spread within the cell until reaching the opposite side (Fig. 3.7b), filling the entire cell at a concentration of approximately 10.5 mm. At this point, a steady-state is reached, as the influx and efflux from respectively the anode- and cathode-facing caps equate. Note that such a long pulse and high Ca 2+ concentration will likely irreversibly damage the cell. However, here we are simply theoretically delineating the upper limit of electrophoretically-mediated molecular delivery using Ca 2+ as a generic example. 3.4 Discussions The effect of the membrane permeabilization on FASS In the previous section, we presented results on simulated ionic transport in, across, and around an electroporated cell. Although our emphasis has been on the species concentrations, the results do depend on the permeabilization calculation via the ASE model. In what follows, such dependence is discussed. When deriving the relation (3.15), we presented an idealized argument by ignoring

53 44 [Ca 2+ ] tot (fmol) ms pulse t (ms) 6-ms pulse diffusion only (a) [Ca 2+ ] (mm) Cell x (µm) (b) increasing time Figure 3.7: a) Total Ca 2+ within the cell ([Ca 2+ ] tot ) with different mechanisms. b) Ca 2+ concentration along the cell centerline with a 150-ms-long pulse at 1 kv/cm. The times correspond to t = 0, 1.1 ms, 3.3 ms, 10 ms, 30 ms, 50 ms, 70 ms, 90 ms, and 120 ms. diffusion and assuming c e = c e,o, and we have not considered the presence of the membrane. The latter, however, can have complex effects on the detailed transport dynamics. Fig. 3.8a shows an up-close view of Ca 2+ concentration around the anodefacing membrane (x = 8 µm) at the end of a 6-ms pulse, and again we focus our study along the cell centerline. A clear depletion is shown outside the cell. This is because if the initial extracellular concentration, [Ca 2+ ] e,o, is used to calculate F m and F e, the condition F m > F e will result which violates the flux continuity (3.10). Therefore, the extracellular concentration near the membrane has to adjust (decrease) to lower F m. Meanwhile, although the electrophoretic component in F e decreases along with [Ca 2+ ] e, the diffusive component increases as the concentration gradient becomes favorable, resulting in a final balance between F m and F e. Conversely, if using [Ca 2+ ] e,o in evaluating F m and F e results in the condition F m < F e, then ion accumulation outside the membrane occurs, as shown in Fig. 3.8b. For these cases, we artificially reduce ρ p by 10 (denoted by 0.1ρ p ) and 100 (denoted by 0.01ρ p ) times, respectively, to demonstrate this effect. That is, we multiply the ρ p variable calculated from the ASE model by 0.1 and 0.01, respectively, while keeping all other parameters un-changed. In such situations, the decrease in ρ p requires an increase in [Ca 2+ ] e (accumulation) to balance the fluxes. These two effects in part compensate

54 45 each other in determining the final F m. The extracellular diffusive fluxes for both cases are opposite to the electrophoretic fluxes. The Γ values from the simulation are 10.7 for the case of 0.1ρ p, and 14.2 for the case of 0.01ρ p, respectively. Most interestingly, these values are higher when compared to the original case shown in Fig. 3.8a, for which Γ = 10.5 from the simulation. (For all cases, the theory predicts Γ = 10.5 from Eq. (3.15).) This behavior is counter-intuitive, as the membrane is actually less permeabilized. An explanation is found in Appendix D, where we show that the twodimensional effect which we have not considered so far also has important contributions. The above results indicate that molecular transport into an electroporated cell involves complex dynamic processes. On the other hand, they also demonstrate that FASS is a robust mechanism which does not strongly depend on the specific state of the membrane. That is, we have varied the degree of membrane permeabilization by two-orders of magnitude, and observed similar stacking behavior in the intracellular concentration, although some quantitative variations exist in the final Γ values. In a sense, the depletion/accumulation effects are means to modulate the membrane fluxes toward a stable equilibrium value. In our simulations, the predicted maximum ρ p value corresponds to a membrane area loss of around 0.8% during the presence of the electric field. This value is high when compared with those from the literature, e.g., % as estimated by Kinosita et al. [46]. The cases studied in Fig. 3.8b bring the ρ p values to exactly this range. We therefore argue that FASS should be observed across a membrane with a realistic degree of permeabilization. On the other hand, the discrepancy in ρ p between the simulation and the experiment is possibly due to the difference in the parametric configuration in the current study and that used by Kinosita et al. [46]. Using σ e = 5 S/m, σ i = S/m, a = 50 µm, and E 0 = 100 V/cm [46, 50], the ASE model yields a ρ p value of 0.2%, which is in much closer agreement with the experimental data. The details are not shown for brevity.

55 46 12 (a) ρ p (b) [Ca 2+ ] (mm) depletion Cell [Ca 2+ ] (mm) ρ p Cell x (µm) x (µm) Figure 3.8: a) Concentration depletion outside the cell at the end of a 6-ms pulse. The data is taken from the simulation shown in Figs b) Concentration accumulation outside the cell due to decreased ρ p -values (denoted by 0.1ρ p and 0.01ρ p, respectively). For all cases, the dashed line denotes a jump in value across the membrane The effect of free diffusion post-pulsation In Fig. 3.7, we have shown that diffusion plays a secondary role in transport, and the delivered molecules through this mechanism is an order of magnitude less than those by electrophoresis during a 6-ms pulse. More importantly, once the pulse ceases, the total molar content within the cell remains approximately constant, and no significant exchange across the membrane occurs (Fig. 3.7, solid line). This is simply because the PAD decreases immediately (with a characteristic time of 10 µs) by 3 orders of magnitude (from 10 2 to 10 5 ) post-pulsation, which results from the ASE model s capability to predict resealing. However, this result seems to contradict those by Pucihar et al. [79]. In the latter experiment, the intensity of PI within electroporated cells was tracked using a photomultiplier tube both during and after pulsation. The results indicated that PI continued to enter the cell by diffusion after the pulse ceased, and PI signal enhancement was stronger during this second stage when compared with that during the pulsation. We hypothesize that the discrepancy between these studies might be caused by an underestimate of the PAD during resealing in the simulation. To test this hypothesis, weincreaseρ p by10(denotedby10ρ p, dashed, Fig.3.9)and100(denotedby100ρ p, dotdashed, Fig. 3.9) times for t > 6 ms, respectively, while keeping all other parameters

56 47 unchanged. This manipulation brings the membrane area loss from around 10 5 to 0.01%-0.1%,whichfallswithintheestimatedrangebyKinositaet al. [46]. Theresulted evolution of [Ca 2+ ] tot is shown in Fig For reference, the original case without a manipulation in ρ p is also shown (denoted by ρ p, solid). We observe that for the case of10ρ p, thetotalmolarcontentdecreasesslowly, indicatingaslightleakageofca 2+ from the cell to the extracellular space. For the case of 100ρ p, a decrease is followed by an increase at a later time. The initial decrease is mainly mediated by a leakage outward on the anode-facing cap, because [Ca 2+ ] i > [Ca 2+ ] e, whereas the eventual increase is mediated by a leakage inward on the opposite side, where [Ca 2+ ] e > [Ca 2+ ] i (see Fig. 4a). However, neither case corroborates well with the qualitative trend observed by Pucihar et al. [79]. We speculate that the difference might be caused again by the different configurations used, i.e., by the differences in pulse length, dye molecule (PI vs. Ca 2+ ), among others. A simulation with a configuration better approximating the experimental conditions in this case may help reconcile the discrepancy, and is the scope of Chapter 4. 2 [Ca 2+ ] tot (fmol) ρ p 10ρ p 100ρ p t (ms) Figure 3.9: The effect of post-pulsation PAD on total molar content within the cell. For the cases of 10ρ p and 100ρ p, the PAD is increased by 10 and 100 times, respectively Experimental evidences to support FASS In the previous section, we have demonstrated that our model prediction satisfyingly reproduced the trends observed in the fluorescence measurements in GT99. However,

57 48 the data collected in the latter is insufficient to support FASS, because the preloaded Fluo-3 concentration was at a low level of 2.2 µm, whereas our predicted Ca 2+ concentration is on the order of 10 mm. Such an experiment therefore cannot detect any enhancement beyond the limiting dye concentration. However, other experimental evidences do exist in the literature to strongly support our prediction. In Eq. (3.16), the theory predicts that the maximum intracellular concentration for a generic ionic species, c i,max, is reciprocally proportional to the extracellular conductivity, σ e. This correlation has been observed by Djuzenova et al. [19]. In this experiment, PI was delivered into murine myeloma cells, with the buffer conductivity ranging from 1 to 5 ms/cm. The PI molar content within the cells, c PI, was estimated from fluorescence signals, and its correlation with σ e was well-fitted by a reciprocal curve, c PI 1/σ e. Although c i,max (maximum concentration) and c PI (total molar number) are not exactly the same variable, the experimental data does corroborate with our theory. In a latter work by the same group [62], an inverse correlation between PI uptake and medium conductivity was again observed for nano-second pulsations, although the fitted curve was not exactly reciprocal. Interestingly, the FASS mechanism also explains why efflux was not observed in GT99. In this work, the same authors also investigated the situation where the initial intracellular Ca 2+ concentration ([Ca 2+ ] i,o = 220 nm) was higher than that in the buffer ([Ca 2+ ] e,o =100 nm). For this case, the buffer was loaded with Fluo-3 at 10 µm to detect any elevation in the extracellular Ca 2+ concentration, whereas the cytoplasm hadnofluo-3initially. Intuitively, Ca 2+ shouldbepushedelectrophoreticallyoutofthe cathode-facing cap, such that efflux is observed. However, the experiment showed only enhanced fluorescence outside the cell on the anode- (positive-electrode-) facing side. This result nonetheless can be explained with the current model. The enhancement outside the anode-facing cap could be interpreted as accumulation similar to those shown in Fig. 3.8b. The extracellular region next to the cathode-facing cap, on the other hand, was possibly depleted by means of FASS. In fact, a simulation predicts that Ca 2+ concentration in this region should be approximately 21 nm, much lower than the background value of 100 nm (not shown). Certainly, this region would be

58 49 refilled by diffusion once the pulse ceases. 3.5 Conclusions In this chapter, we have presented a numerical simulation to explore the spatio-temporal dynamics of Ca 2+ -delivery via electroporation. The main conclusions are: Electrophoresis can play an important role in molecular delivery via electroporation; it provides a faster and more efficient means of transport when compared with free diffusion. FASS, which mechanism arises from a gradient in the electrophoretic velocity, determines the achievable molecular concentration within the cell. The latter is reciprocally correlated with the extracellular electrical conductivity. In addition, we have also discovered that a uniform dye concentration may give rise to a non-uniform fluorescent signal, due to the three-dimensional cell geometry combined with a finite focal depth comparable to the cell size. This result suggests that careful de-convolution and calibration need to be performed in order to extract molar concentration values from fluorescence data. Finally, the current model framework is suitable to study the transport of small molecules including free ions, drug molecules, and molecular dyes. The interaction of large molecules (such as DNA) with the membrane is much more complex, and a more sophisticated model is needed to describe these processes.

59 50 Chapter 4 The Effect of Conductivity on Electroporation Mediated Molecular Delivery 4.1 Introduction In the previous chapter, we have established a basic framework to predict molecular entry. We have discovered the importance of electrophoresis, as well as the effects of field-amplified sample stacking on intracellular concentration. In this chapter, we further extend the study, and use the model to systematically explore the effects of extracellular conductivity. The work is motivated by experimental observations from previous research, namely, by Zimmermann and co-authors in two companion papers [19, 62]. In these experiments, the delivery of PI into murine myeloma cells was investigated under both classical and supra-electroporation conditions, and an inverse correlation between the percentage of PI uptake (or total amount of delivery) and the extracellular conductivity was discovered. Although several possible mechanisms have been proposed by the authors [19, 62, 98], a causal relation between them and the data trends has yet to be established. WewillusethemodeldevelopedinChapter3tosimulatetheentryofPIintomurine myeloma cells, using experimental parameters found in [19, 62]. In 4.2, the formulation and simulation method are briefly outlined. In 4.3.1, we use simulation to investigate membrane permeabilization and PI delivery with nano-second pulses, and compare the results with those from [62]. Our results indicate that only a weak correlation exists between the degree of permeabilization and the extracellular conductivity. In 4.3.2, a compact formula is developed to approximate results predicted by the full model, and is subsequently used to analyze the data trends under classical electroporation conditions from [19]. For both cases, the simulation and the compact formula produce

60 51 an expected inverse correlation between delivery and conductivity, which is compared in detail with the experimental data. The results reveal good agreement, confirming that electrophoresis may be the root cause of the observed trends. On the other hand, quantitative discrepancies still exist between the prediction and the data, which points toward mechanisms not included in the present model. In 4.4 several possible effects including those of charging times, electrodeformation forces, and diffusion are addressed. We draw our conclusions in Model formulation The overall model formulation follows Chapter 3. The cell is idealized to be a thin, rigid, and spherical shell of radius a, and a spherical coordinate system is adopted (see Fig. 3.1). The x-axis is chosen to align with the direction of the applied electric field. We solve Eqs. ( ) for the electric field and pore nucleation and evolution. The species transport on the other hand are different due to the different ionic species considered. In the following, we consider three specific species, free PI ions (denoted by PI 2+ ), DNA and RNA binding sites (denoted by B), and bound PI (denoted by PI-B). We assume that the binding sites are uniformly distributed inside the cell with a molar concentration of [B] i,o, where the subscript i again denotes intracellular, and o denotes an initial value. The extracellular PI 2+ concentration is [PI 2+ ] e,o (Table 4.1). When free PI ions enter the cell through the permeabilized membrane, they bind to DNA and RNA molecules at the binding sites. Upon excitation the complex emits fluorescent signal which is observed by microscopy. The reaction is described by PI 2+ +B k + PI-B, k where k + and k are the association and dissociation rate constants, respectively. The

61 52 Table 4.1: List of model parameters. Symbol Definition Value a Cell radius 7 µm [19, 62] h Membrane thickness 5 nm σ e Extracellular conductivity S/m [19, 62] σ i Intracellular conductivity 0.4 S/m [19, 62] F Faraday constant C/mol R Universal gas constant J/K mol T Room temperature K k + Association rate constant 1.54 (µms) 1 [97, 114] k Dissociation rate constant 5.17 s 1 [97, 114] D e Diffusivity of PI 2+ in the extracellular solution 437 µm 2 /s a D i Diffusivity of PI 2+ in the cytoplasm 146 µm 2 /s b z Valence number for PI [PI 2+ ] e,o Initial PI 2+ concentration in the extracellular solution 37.4 µm [19, 62] [B] i,o Initial binding site concentration in the cytoplasm 6.93 mm [19, 97] a Direct measurement, courtesy of M. M. Sadik. b This value is taken to be one-third of D e. NP equations are modified with production terms to capture this process: [ PI 2+] t [B] t [PI-B] t = (wfz [ PI 2+] Φ ) (4.1) + D [ PI 2+] k + [B] [ PI 2+] +k [PI-B], = k + [B] [ PI 2+] +k [PI-B], (4.2) = k + [B] [ PI 2+] k [PI-B]. (4.3) Here [PI 2+ ], [B], and [PI-B] denote the molar concentrations, F is the Faraday constant, z, D, and w are the valence number, diffusion coefficient, and the mechanical mobility of PI 2+, respectively. Because in general DNA and RNA have very low mobility within the cell, they are assumed to be fixed. The continuity of molar flux density Eqs. ( ) are still adopted as boundary conditions except that X now represents PI. All general parameters pertinent to the permeabilization model ( ) are taken from [50], and are not repeated here for brevity. Parameters specific to this study (e.g., σ i,e, a, and E 0 ), and the rate constants are summarized in Table 4.1. They are specified to best approximate the experimental conditions in [19, 62].

62 Results We first present simulated results on cell permeabilization and PI delivery under supraelectroporation conditions, and compare with experimental data from [62] ( 4.3.1). In 4.3.2, we develop a compact formula to interpret the predicted and observed trends, and use it to further correlate with experimental data under classical electroporation conditions from [19] Simulation of supra-electroporation with varying extracellular conductivity Fig. 4.1 shows the permeabilization of a cell membrane under an 160 kv/cm electric pulse 95 ns in length. In Fig. 4.1a, the evolution of the TMP as a function of time at the anode facing pole (θ = π) is shown, for the various extracellular conductivities (σ e = S/m). All cases exhibit an initial increase in the magnitude of the potential, followed by a subsequent decrease due to the permeabilization of the cell membrane. The permeabilization increases the effective conductivity of the membrane caps, such that further growth in the TMP is limited. After the pulse ceases at 95 ns, the TMP drops to near-zero. Fig. 4.1b shows the distribution of the TMP along the membrane at t = 95ns. In Fig. 4.1c, we plot the evolution of the PAD (ρ p ) as a function of time, also at the anode-facing pole (θ = π). Consistent with the behavior of V m, the increase in ρ p goes through a rapid stage followed by a slower one. The pores begin to shrink immediately after the pulse ceases, and the majority of them vanish between 180 to 200 ns. In Fig. 4.1d, ρ p is plotted as a function of θ at the end of the 95-ns-long pulse. The salient feature of Fig. 4.1 is that both the TMP and PAD are insensitive to the extracellular conductivity. (The intracellular conductivity is considered to be a constant at σ i = 0.4S/m.) This insensitivity can be explained by the interdependence of the TMP and the PAD in the model framework. According to Eqs. (3.5) and (3.6), the PAD exponentially depends on the square of the TMP. On the other hand, the TMP has an approximate linear dependence on the PAD through Eq. (3.2). As a result,

63 54 a small increase in the TMP leads to a great increase of in the PAD, which in turn limits further growth of the former parameter. This dynamic process leads to a final equilibration for both ρ p and V m, and the equilibrium values depend only weakly on σ e. The result suggests that within the validity of the current model framework, the strong dependence of molecular delivery on the extracellular conductivity is not explained by the variations in permeabilization. We remark that when compared with results for classical electroporation (E 0 1 kv/cm) [55], the permeabilized areas at the anode- and cathode-facing caps are larger. In addition, the maximum PAD is 1 2 orders greater than that in the classical cases. However, these results are consistent with model predictions by other authors for supraelectroporation [93, 108]. Next we present results for PI delivery into the permeabilized cell, also for E 0 = 160kV/cm and a pulse length of t pulse = 95ns. For the exemplary results shown in Fig. 4.2, σ e = 0.1S/m. The concentrations for the dye in free ([PI 2+ ]) and bound ([PI- B]) forms are plotted along the cell centerline (the x-axis). Due to the small time scale involved, both exhibit very narrow peaks inside the cell immediately next to the anodefacing cap (x = a = 7µm, Figs. 4.2a, b). An enlarged view in Figs. 4.2c and d shows more details of the development. In Fig. 4.2c, the free-pi concentration reaches a peak value around 0.43 mm by the end of the pulse, mediated mainly by electrophoretic transport. This peak value is much higher than the basal extracellular concentration, [PI 2+ ] e,o = 37.4µM. The cause for this concentration enhancement is electrokientic in nature as we have previously demonstrated, and is termed field-amplified sample stacking (FASS) [55]. After the pulse ceases, the concentration slowly diffuses away, during which we also observe leakage through the permeabilized membrane. On the contrary, the concentration for bound PI continues to increase even after the pulse ends, due to continuous binding of the free ions with the available sites. (Note the binding time scale is 0.1ms, much longer than the time scale presented.) In Fig. 4.3, the time course of total PI delivery is examined for the various conductivities studied in [62]. The total delivered molecular content within the cell, PI tot, is calculated by summing that of both the free and bound PI. The total delivery first

64 Figure 4.1: Membrane permeabilization under an 160 kv/cm pulse 95 ns in length, for various extracellular conductivities ( S/m). (a) The development of the TMP, V m, at the anode-facing pole (θ = π). (b) The distribution of the TMP as a function of the position along the membrane (θ) at t = 95ns. (c) The development of the PAD, ρ p, at the anode-facing pole (θ = π). (d) The distribution of the PAD as a function of θ at t = 95ns. 55

65 56 Figure 4.2: Concentration evolution for free and bound PI along the cell centerline (x-axis) for E 0 = 160kV/cm, t pulse = 95ns, and σ e = 0.1S/m. The cell extends from x = 7 to 7 µm. (c) and (d) zoom in around the anode facing pole (x = 7µm) to demonstrate detailed profile evolution. Figure 4.3: The time course of simulated total PI content (PI tot ). E 0 = 160 kv/cm and t pulse = 95ns. The numbers 1-5 denote σ e in the unit of ms/cm.

66 Figure 4.4: Simulated total PI delivery and comparison with experimental data. (a) Simulated total PI content within the cell (PI tot, symbols) for t pulse = 95ns, and as a function of the applied field strength, E 0, and the extracellular conductivity, σ e. The numbers 1-5 denote σ e in the unit of ms/cm. (b) Simulated PI tot (symbols) as a function of σ e and t pulse for E 0 = 160kV/cm. For both (a) and (b), the dashed lines are theoretical predictions generated with Eq. (4.4). (c) and (d): experimental data adapted from Fig. 3 in [62]. The dashed lines in (c) are least-square linear fits. 57

67 58 increases linearly with time until the pulse ends at 95ns, implying a dominant mode of electrophoresis. After the pulse ceases, noticeable leakage is observed, especially for the lower conductivity cases where the intracellular specific concentrations are higher due to FASS. PI tot eventually reaches equilibrium values, and after 400 ns no significant changes are observed. In Figs. 4.4a and b, the simulated total PI delivery at t = 800ns is examined and shown as symbols, for the parametric configurations studied in [62]. (The dashed-lines are calculated with a compact model (4.4) which is presented in ) In Fig. 4.4a, the total PI is plotted as a function of the applied field strength, E 0, and for the various extracellular conductivities, σ e. The pulse length is 95ns. In Fig. 4.4b, a single field strength, E 0 = 160kV/cm is chosen, and the effects of pulse length (11-95 ns) is studied. For comparison, the experimental results from [62] are adapted and presented in Figs. 4.4c and d. Very good agreements are observed in the general data trends. First, PI uptake depends linearly on the field strength (Figs. 4.4a and c), and the slope of the linear curves decreases with an increasing conductivity. Second, the total delivery shows an inverse correlation with the extracellular conductivity (Figs. 4.4b and d), and the correlation is stronger with longer pulses. These agreements suggest that transport, and in particular via electrophoresis, plays a key role in mediating the delivery and producing the the results shown. The mechanism is further analyzed with a simplified understanding below A compact model and comparison with classical electroporation The behavior observed in Fig. 4.4 can be understood with a simplified model. In Appendix E and via a control volume analysis, we show that the total molecular delivery, c tot, can be approximated by the following formula, [ ( )] c tot = t pulse c e ωfze o πa 2 3σi. (4.4) 2σ e +σ i Here c e is the extracellular concentration of the target molecule. The simple theory takes into account only the electrophoresis transport, and the term within the square

68 59 brackets represent the total electrophoretic flux into the cell. The calculated results using the same parameters as in the previous section are shown as dashed lines in Figs. 4.4a and b. The simple formula provides a good approximation when compared with results from the full model. The agreement suggests that indeed electrophoresis is the main mode of molecular transport for the cases studied, and that Eq. (4.4) can be used as a compact, convenient formula in place of the more complex and costly full-model simulations. More importantly, Eq. (4.4) correctly captures the linear dependence of c tot on E 0, as well as its inverse dependence on σ e. The former is simply due to the linear dependence of the electrophoretic velocity on the field strength, and the latter is due to the heterogeneous spatial distribution of the electric field, which depends on both the intra- and extra-cellular conductivities. (See Appendix E for the detailed derivation.) The functional dependence reveals that c tot 3σ i 2σ e +σ i, (4.5) where σ i is assumed to depend only on the cell type, and is a fixed constant for all cases studied. The availability of Eq. (4.4) allows us to further analyze the data trends observed in the experiments. Note that a direct comparison with the results from[62] is not possible, because our model predicts the total delivery into a single cell, whereas the experimental data measures the percentage of cells with successful PI uptake. Instead, we attempt to fit the the experimental data with Eq. (4.5). In Fig. 4.5, the black dots are the slopes extracted from the least-square linear fits from Fig. 4.4c. The dashed curve is a least-square fit assuming a functional form of A[3σ i /(2σ e +σ i )], where A = 0.20 is the resulting fit parameter. This fit captures the trend, but tends to produce a somewhat weaker dependence on σ e when compared with the data. However, a correction of the functional form with an added constant produces an excellent agreement, ( ) 3σi f 2 (σ e ) = B +C, (4.6) 2σ e +σ i

69 60 Figure 4.5: Parametric fitting of the experimental data on PI delivery. In (a), the dots are the numerical values of the slopes extracted from the linear fits in Fig. 4.4c. In (b), the experimental data are regenerated from Fig. 3 in [19], where open circles, solid circles, and triangles represent KCl, NaCl, Na 2 SO 4 solutions, respectively. For both cases, the dashed curves are fits with the original functional form (4.5). The dot-dashed are fits from the modified functional form (4.6). where B = 0.44, C = 1.28 are determined to minimize the fitting error. The resulting curve is shown in Fig. 4.5a as a dot-dashed line. Eq. (4.4) also permits us to conveniently analyze the other data, namely, the experimental measurements by Djuzenova et al. [19]. This work proceeds [62], and investigates the effect of extracellular conductivity with classical (E 0 = 3kV/cm and the field exponentially decays with time constant of 40 µs ) instead of supra-electroporation. The experimental configuration is otherwise similar to [62]. The data is adapted and presented in Fig. 4.5b as symbols. Again a least-square fit with (4.5) reveals a weaker dependence on σ e, whereas the modified form (4.6) gives a satisfactory result which well-captures the data trends. Note that interestingly, the fit constant C for the two cases in Fig. 4.5 closely match each other ( 1.28 and 1.46, respectively), pointing toward a consistent functional form. The results presented above lead to two important speculations. First, the formula (4.4) as well as the full model capture an essential part of the physical processes involved. While quantitatively discrepancies are present, qualitatively the trends are well-reproduced. It is particularly interesting to see a simple modification based on the prediction produces satisfactorily the curvature observed in both [19] and [62]. Based

70 61 on this agreement, we believe that electrophoretic ion transport plays a significant role in mediating the delivery of small charged molecules. Second, an unknown mechanism not included in our model also contributes to delivery, which is evidenced by the constant C that is artificially added to Eq. (4.6). We are currently investigating various possibilities for this discrepancy in understanding. Regardless of its nature and origin, our results indicate that the effect is also linearly proportional to the applied field strength. 4.4 Discussions The current work posits that the molecular uptake dependence on extracellular conductivity is mediated by electrophoretic transport. However, the mechanism we propose is not necessarily exclusive. For a complex process as such, multiple mechanisms may simultaneously contribute to the system behavior, and some possibilities are discussed below. The effects on setup parameters When the setup parameters, such as cell radius, external electric field, association and dissociation speeds, and intracellular diffusivity, are not properly specified or directly depends on the extracellular conductivity, the delivery result may be affected. However, according to Eq. (4.4), only the effective charge and mobility of PI 2+, external electric fields, cell radius, and intracellular conductivity may greatly affect the delivery efficiency. In particular, the effective charge and mobility of PI 2+ may only weakly depend on the extracellular conductivity due to the bath solution falls in the dilute region. Therefore, the latter three may play important roles in the strong dependence of deliver efficiency on the external extracellular conductivity. The affect of electric field will be discussed in the next section, and here we focuses the latter two: In this work, we have assumed that the cell radius to be constant through the process. However direct experimental observations suggest that cells expand dramatically during the pulses and then slowly adjust afterwords [1, 71, 109]. Due to the expansion mechanism is still under investigation, the effect of the extracellular conductivity on

71 62 cell size change remains unclear. Preliminary results done by our group indicate that higher extracellular conductivity leads to larger equilibrium cell size, which suggest a stronger inner flow and hence result an opposite trends. Similarly, we have also assumed that the intracellular conductivity to be a constant. Firstly, the influx may greatly change the cellular conductivity and therefore affect the dependence of delivery efficiency on external conductivity. However, this again will leads to a weaker dependence as the leakage eliminates the difference between the intraand extra-cellular conductivity. Secondly, the stacking of ions due to FASS leads to the ionic imbalance between the cell membrane, which in turn affects the electric fields. However, Due to the complexity, this problem is out of the scope of this dissertation and are our group s current focus. Thirdly, the existence of the nucleolus and organ, as well as the heterogeneity of cytoplasm also limit the analysis results in this work. Membrane and electrode charging times The charging time is the most obvious conductivity-dependent parameter in an electroporation system. In Djuzenova et al. [19], the membrane charging process as a possible contributor has been discussed. The relaxation time, τ m, for a near-insulating membrane is given by ( 1 τ charg = ac m + 1 ), σ i 2σ e which shows an inverse dependence on σ e. However, as the authors argued in [19], this dependence would rather produce an opposite trend. According to the formula, a decreasing σ e leads to an increasing charging time. The cell would hence experience less exposure to the field post-permeabilization, and both permeabilization and delivery would more likely decrease in this case. Another charging time is the electrode screening time, τ c = λ DL D, whereλ D isthedebyethicknessfortheelectricaldoublelayer, Listhedistancebetween the electrodes, and D is a characteristic ion diffusivity [5]. This time scale is calculated with a resistor-capacitor model. For electroporation experiments, it characterizes the

72 63 time for significant field reduction to occur due to ion accumulation near the electrodes. Using λ D = 3nm (for typical buffer with an ion concentration around 10mM), L = 1mm, and D = 10 9 m 2 /s, τ c is on the order of 3ms, much longer in general than the pulsing time in both [19] and [62]. This mechanism is therefore also unlikely responsible for the observed conductivity effects. Membrane permeabilization and diffusion In Müller et al. [62] the authors proposed that the conductivity-dependent electrodeformation force determines the degree of electropermeabilization, which in turn causes the observed dependence of delivery on the extracellular conductivity. The normal component of the electrical pressure acting on the membrane, P D, is given by the formula: P D = 9 2 ǫ we0 2 cos 2 θ σ2 i σ2 e (σ i +2σ e ) 2, (4.7) where ǫ w is the permittivity for an aqueous solution. 1 Here we argue that this mechanism is neither likely a viable interpretation. First, although Eq. (4.7) does have a similar dependence on σ e through the denominator, the more conspicuous feature is that P D dependents on E0 2, not E 0 as shown in the experiments. (Note that in Fig. 4.4c the data for the two higher conductivities show some nonlinearity; however those for the two lower conductivities show evidently linear dependence. We believe the latter data are more representative due to their much higher values in the percentage uptake.) Indeed, a companion study by Zimmermann and co-authors and a recent work by us both indicate that deformation depends quadratically on E 0 [84, 98]. Second, our results from Fig. 4.1 show the permeabilization only has a weak correlation with the the varying extracellular conductivity, due to a self-rectifying mechanism we explained earlier. Third, if indeed delivery is dominated by electrophoresis, it is insensitive to the degree of membrane permeabilization. This point has been adequately argued in our previous work [55]. 1 In fact, this formula is incorrect because it does not take into account the potential discontinuity at the membrane (the TMP). The correct formula has a similar dependence on both field and conductivity, and is not presented here for brevity.

73 64 Figure 4.6: Simulated asymmetric PI uptake at 10 s with an exponentially decaying pulse (3 kv/cm, with a decay constant of 40 µs) following experimental conditions in [19]. The contour for the bound PI concentration is shown. The conductivity of the bath solution is 3 ms/cm (left) and 3.7 ms/cm (right), respectively. Other model parameters are from Table 1. The results compare well with Fig. 6 in [19]. However, we do not exclude the possibility that electropermeabilization can indeed depend on the conductivity, through mechanisms not included in the current Krassowska-Neu framework. Furthermore, if diffusion affects transport appreciably, the effects together can result in contributions to the data trend observed. Although our simulation shows that diffusion leads to only a small portion of the total molecules delivered with nano-second pulses, over a long time it may have more significant effects. Fig. 4.6 shows simulated PI entry under conditions from [19], which demonstrates good qualitative agreement with Fig. 6 therein. The molecules accumulated on the anode facing side are mostly from electrophoresis during pulsation, whereas those from the cathode-facing side are from the much slower diffusion over a long time of 10 s. For the two cases studied, diffusion is responsible for 43% and 54% of total delivery, respectively. In this case, because cross-membrane diffusion depends strongly on the degree of permeabilization, a variation in the latter may affect the former. Further experiments as well as model development (to take into account the conductivity-dependent membrane forces) are needed to help quantify the effects of both permeabilization and transport, which is the scope our on-going work.