Analyzing the effects of surface distribution of pores in cell electroporation for a cell membrane containing cholesterol

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1 Analyzing the effects of surface distribution of ores in cell electrooration for a cell membrane containing cholesterol Prati Shil 1,3, Salil Bidaye, Pandit B. Vidyasagar 1 1 Biohysics Laboratory, Deartment of Physics, University of Pune, Pune 4117, India Institute of Bioinformatics and Biotechnology, University of Pune, Pune 4117, India 3 Author to whom any corresondence should be addressed. rati@hysics.uniune.ernet.in Abstract This aer resents a model and numerical analysis (simulations) of transmembrane otential induced in biological cell membrane under the influence of externally alied electric field (i.e., electrooration). This model differs from the established models of electrooration in two distinct ways. Firstly, it incororates the resence of cholesterol (~% mole-fraction) in biological membrane. Secondly, it considers the distribution of ores as a function of the variation of φ m from one region of the cell to another. Formulation is based on the role of membrane tension and electrical forces in the formation of ores in a cell membrane, which is considered as an infinitesimally thin insulator. The model has been used to exlore the rocess of creation and evolution of ores and to determine the number and size of ores as a function of alied electric field (magnitude and duration). Results show that the resence of cholesterol enhances oration by changing the membrane tension. Analyses indicate that the number of ores and average ore radii differ significantly from one art of the cell to the other. While some regions of the cell membrane undergo raid and dense oration, others remain unaffected. The method can be a useful tool for a more realistic rediction of ore formation in cells subjected to electrooration. 1. Introduction Electrooration or electroermeabilization is a well known hysical rocess in biological cells involving enhanced ermeability of the cell membrane induced by an externally alied electric field [1-3]. The alication of controlled high voltage ulses to the cell membrane leads to the formation of micro-ores in the cell membrane. This enables exogenous molecules viz drugs and DNA to move into the cells. Thus numerous alications in molecular biology, biotechnology and medicine have emerged [4-1]. Considerable efforts have also been made to understand this rocess and its otimization for various alications. It is established that the electroermeabilization of cells deends on the electric ulse amlitude, ulse duration, the number of ulses used and also on the exerimental conditions such as buffer and temerature. With these arameters chosen, the rocess of ermeabilization is reversible and the cell comes back to the normal hysiological state. The rocess of electrooration is a two hase rocess: first hase is of ermeabilization during the ulse, and second is a longer hase of resealing that begins after the end of the ulse [11]. A biological cell can be assumed to be a non-conducting shere with a equiotential inner side (i.e, the cell membrane is an infinitesimally thin insulator). When a stewise uniform electric field of magnitude E e is alied to a sherical cell of radius R cell in solution, the induced membrane otential is given by φ = 1.5 R E c o s θ (1) m c e l l e where θ is the angle [1,] as shown by Figure1(a). This equation states that at θ= and at θ= π (referred to as oles ), the value of φ m is extremum and the value is deendant on the cell size. 1

2 Also, the value of φ m changes with θ, meaning that different regions of the cell will exerience different φ m. Though reorts of theoretical models and simulations of cell electrooration exist [1-14], major limitations remain as most of the models consider various aroximations such as uniform oration of whole bilayer membrane. The effective models and numerical analysis (simulations) roosed by Neu and Krassowska and Smith et al rovide considerable insight into the understanding of electrooration rocess [15-16]. An earlier reort ublished by Smith et al [16] describes formation and evolution of ores and also rooses a model for DNA utake mediated by electrooration which favoured exerimental observations. However, the major limitations of this model involve: i) the assumtion that the comosition of the membrane is uniform in terms of membrane liids (which does not take into account the resence of cholesterol) and ii) not considering the fact that the different regions of the cell exerience different values of φ m and hence has different ore densities (surface distribution of ores). This work is aimed at overcoming these limitations and roducing simulation for electrooration of mammalian cell membrane with a cholesterol content of % mole-fraction assumed to be distributed uniformly in the membrane.. Calculation method According to the theory of electrooration, [1-3, 16] all the ores formed are initially hydrohobic and are formed at a rate determined by their energy. Most of them are quickly destroyed by liid fluctuations, but if hydrohobic ores of radius r > r* ( where r* is critical radius) are created then they may be converted sontaneously into long-lived hydrohilic ores. Thus, most of the hydrohilic ores are initially created with radius slightly greater than r* and immediately exand to the minimum energy radius r m. In the earlier model [16] for simulation studies, it has been assumed that the ores are created with initial radius r m (r > r m ). It has also been roosed that for any r > r*, the ores will grow. The rate of creation of ores is given by, dn dt φ m = φ e e α 1 N eq N ( φ ) m () where N is the ore density, φ m is the transmembrane otential, φ e is the characteristic voltage of electrooration, α = m - s -1 is the creation rate coefficient, N eq is the equilibrium ore density for a given voltage and is given by : N eq φ q m e ( φ ) = N e (3) m φ N is the equilibrium ore density for φ m = and q = (r m /r*), where r m =.8 nm is the minimum energy radius at φ m = ; and r* =.51 nm is the minimum ossible radius of a hydrohilic ore [16].

3 The ores are initially formed at r m and later exand in order to minimize the energy W of the liid bilayer (henceforth referred to as bilayer energy ). If at any oint of time, there be n number of ores with radii r j, with j=1,..n, then, the rate of change of radii is given by drj D W =, j = 1,,..., n dt kbt r j (4) where D = m s -1 is the diffusion coefficient for ore radius. The bilayer energy deends on the reulsion of liid heads, edge energy of ore erimeter, membrane tension and force due to the alied electric field. The bilayer energy W is given by: W = n rj r + ( ) ( ) 1 π r + j πκ ψ c πσ eff A rj F ( rj, φm ) dr j= r h β (5) j 1 In equation (5), the first term reresents the static reulsion of the liid heads, β = J; the second reresents edge energy of ore erimeter; the third term accounts for the effect of ores in the membrane tension, and the last term reresents the contribution of the transmembrane otential. The resence of cholesterol in the membrane has been incororated in the second term in equation (5), which deals with the edge energy of the ore erimeter. This is a major difference between our model and those ublished earlier [16]. The resence of cholesterol is reflected in the change in line tension γ of the ore edge, when comared to the membrane without cholesterol. This is because of the negative sontaneous curvature (c = -.9 nm -1 ) for cholesterol as found by Karatekin et al [17]. Thus, actual line tension of the membrane containing cholesterol is given by, 1 γ = πκ ψ c h (6) where κ = is the bending modulus of the membrane, h = 5 nm (membrane thickness) and ψ is the mole-fraction of cholesterol in the membrane in the rim around the ore-edge. In the resent study, an average mammalian cell membrane containing a mole-fraction of cholesterol of ~% (i.e, ψ=.) has been considered [18]. The variation in the cholesterol content is suosed to be a major factor contributing to differences in the membrane comosition from one cell tye to another. Hence, differences in the membrane roerties may affect the electrooration behaviour of the cells. This may be correlated with the recent finding by Kanduser et al [19], who have demonstrated that different cell tyes having comarable cell geometries but different membrane roerties (evaluated in terms of intrinsic membrane fluorescence) differed in electrooration behaviour when exosed to similar electric ulses. The third term in equation (5) denotes the effective tension of the membrane, σ eff as the function of the combined area of ores A n = j= 1 π r j and is given by 3

4 σ ' σ σ eff ( A ) = σ ' (7) (1 A / A) where σ is the energy er unit area of hydrocarbon water interface and A is the total membrane surface area. The fourth term in equation (5) accounts for contribution for the transmembrane otential, with the force acting on the inner surface of the torridal ores and where F is defined as: Fmax F ( r, φm ) = φm (8) rh 1+ r + r t where F max = N/V is the maximum attainable force when φ m = 1V, r h =.97nm, r t =.31 nm [15]. As ores are formed they exand and this exansion affects the trans-membrane otential φ m. For the surface distribution of ores, that is, to incororate the effect of variation in ore oulation with θ, the cell has been assumed to be divided into segments or discs with θ varying as: θ = to π/8 (segment ); θ = π/8 to π/4 (segment T); θ = π/4 to 3π/8 (segment ); θ = 3π/8 to π/ (segment ) as shown in the figure 1(b). The remaining half of the cell has also been evaluated in the same fashion. For simlicity of numerical imlementation, the mean value of θ for each segment has been considered as the numerical inut (for θ) for that articular segment. This is also a major difference with our model and those ublished earlier [16]. To comute φ m for each segment at each time ste, it is necessary to choose a tye of exerimental rearation to model. The transmembrane otential is given by dφ 1 1 m φ C + m I, dt + φ + = Rs R (9) Rs R s is 1 ohm in series resistance of the exerimental setu (Device for carrying out electrooration, buffer etc,) R = R m /A is the membrane resistance with R m =.53 ohm-m (assumed to be the same for all segments). I is the combined current through all ores at a articular segment and is given by n n n φm φm I = i ( rj, φ m ) = = ( 1) j= 1 j= 1 R + Ri j= 1 h 1 + gπr gr where h = 5 nm is the membrane thickness, g = S/m is the conductivity of the solution. R = h/(π g r j ) is the ore resistance and R i = 1/ gr j is the inut resistance [16]. Total current through the cell has been calculated by adding u of the currents for the various segments. j j 4

5 (a) A φm C θ B E e Electrode (+) Electrode (-) (b) π/ 3π/8 π/4 θ π/8 T Figure 1. (a) Sherical cell in an alied external electric field E e. (b) The cell is divided into segments as: θ = to π/8 (segment ); θ = π/8 to π/4 (segment T); θ = π/4 to 3π/8 (segment ); θ = 3π/8 to π/ (segment ). 3. Comutational model A rogram has been develoed in C ++ language for the dynamic simulation of cell electrooration. Simulations were carried out to obtain the numerical solutions for the equations. For simulations, our model reresents the ores in oulations as: i) small ores, whose radii congregate near the minimum energy radius (r m ), are accounted for by ore density N(t), and ii) large ores with radii greater than r m are accounted by number of ores n (r j, j =1,, 3, n). It has been assumed that most of the hydrohilic ores are created at the minimum energy radius (r m =.8 nm) [16]. Then, the transmembrane otential and all other electrical arameters like current are calculated searately for each of the four regions as shown by figure 1(b). Equation (9) has been used for calculating φ m, which is assumed to be constant throughout each region. The only difference is in the values of φ used for the different regions. φ is the maximum otential that can be develoed across the membrane and is calculated using equation (1). Thus, each half of the cell is treated as four indeendent regions as far as the electrical roerties of the cell are concerned. But the mechanical roerties like membrane surface tension, etc were considered the same for all the segments of the cell. Thus, the hysical integrity of the cell is not lost. The entire time duration of the simulation is divided into small time stes, each of duration t =. ns = 1-4 µs. Simulations were carried out for a total time duration of 4 us which is divided into i) time for alication of the electric ulse T ON = µs and ii) ost ulse 5

6 duration, T OFF = µs during which the membrane tries to rearrange. It has also been assumed that at time t =, assume φ m =, N = N and n =. N is the equilibrium ore density at φ m = and deends on the rate of thermal fluctuations of liid molecules in the membrane. A single time ste in the simulation for each region consists of the following calculations: 1. Sontaneous creation of ores according to equation () simultaneously in all the segments of the cell.. If the number of ores N(t) x A (membrane area for given region) >1, then n is increased by an integer number. For examle, if N(t) A=.3, then increase n by and decrease N(t) by /A. 3. Udate radii (r j ) of n ores according to equation (4). If any of these ores have new radii equal to minimum energy radius r m then value of n is decreased by integer number of ores and make corresonding increase in the value of N(t). 4. Membrane tension σ eff is calculated only once for each time ste, according to the equation (6). This value is used while udating the ore radii according to equation (4). 5. The transmembrane otential, φ m has to be calculated according to equation (9). Current through each individual ore is comuted and ores from both oulations are considered during this calculation using density of ores N(t), and the radii of ores r j, (j =1,, 3, n). Then total current I is calculated using equation (1). 6. For each segment, the mean value of θ was considered as the numerical inut for calculation of φ m. For examle, for segment 1 (θ = to π/8), θ = π /16 was considered at the inut. 7. The numerical integration of the equations (), (4) and (9) has been erformed using the Runge-Kutta 4 th order method. 8. To incororate the resence of cholesterol in the membrane, the numerical imlementation of equation (4) has been erformed using the value of bilayer energy W as defined by the equation (5). 9. Values of R avg (arithmetic mean of all ore radii in a segment), R max (max ore radius in a segment), φ m and ore-count are recorded in the data files. Simulations were erformed for two different values of alied electric field: i) Ee =.833 kv/cm (for comaring results with revious model roosed by[16]) and ii) E e = 1. kv/cm (for understanding the henomenon of electrooration at higher alied electric field magnitudes). The rogram for simulations has been used on a Pentium 4 PC with Windows XP, 1.8 GHz., 56 MB RAM. The rogram generates data files (φ m vs t, R avg vs t, R max vs t, Pore count vs t) for all segments. 4. Results and discussion In this study, our model incororates the effects of surface distribution of ores for a sherical cell of radius (R cell = 1 µm) whose membrane consists of cholesterol (% molefraction). Results for simulations erformed with two different values of alied electric field, i) E e =.833 kv/cm and ii) E e = 1. kv/cm, are discussed below. 4.1 Electrooration with E e =.833 kv/cm: The simulation of cell electrooration, in which electric ulses of µs duration have been alied, is shown by Figure. The total time interval is divided into i) time for electric ulse alication (duration µs) and ii) a ost-ulse duration of µs. As shown by the figure (a), the transmembrane otential, φ m, for various segments of the cell increases to a maximum value and then falls to V at the end of the ulse. For segment, the value of φ m attains a value of 6

7 ~1. V and then stabilizes at 1.14 V. For the segment T and, φ m attains constant values of 1.4 V and.69 V resectively through-out the ulse duration. The least value of φ m is obtained for segment (~.4 V). Figure (b) shows the variation of average ore radius R avg with time during the electrooration. For segment, R avg increases and stabilizes at nm 64 nm. At the end of the ulse the value of R avg decreases and reaches ~15 nm (at t= 4 µs). For segment T, the R avg increases to a maximum value of ~3 nm and falls sharly at the end of the ulse, indicating the resealing of ores. Fig (c) shows that for the segment, R max reaches a maximum value of ~8 nm and then gradually decreases to ~77 nm towards the end of the ulse duration. For segment T, R max attained a value of ~57 nm initially and then, stabilizes at ~55 nm. The difference in values of R max and R avg is less, indicating that majority of the ores in each segment attain similar radii close to the R avg for that segment. Φ m (volts) R max (nm) (a) T (c) T Pore Count R avg (nm) T (b) T (d) Figure. The figures (a), (b) and (c) reresent the variation of transmembrane otential (φ m ), average ore radius (R avg ) and maximum ore radius (R max ) with time resectively. Each figure shows 4 curves corresonding to 4 regions of cell laced in an electric field of.8333kv/cm: (θ = to θ = π/8), T(θ = π/8 to θ = π/4), (θ = π/4 to θ = 3π/8) and (θ = 3π/8 to θ = π/). Figure (d) indicates the ore distribution for various segments. 7

8 The grah also shows that there is no significant ore formation in the segments and (R avg = nm) (Fig (d)). The rogram also redicts that the number of ores formed in segments and T are 99 and 6 resectively, whereas there is no ore formation in the other segments. Our model considers the effective electric fields in different segments of the cell surface. The total number of ores formed in a cell as er this model is far less as comared to that redicted by a model considering the cell as a uniform membrane atch (model of Smith et al). Our model redicts that most of the ores are formed in the segment, leading to a slight dro in the value of φ m during the alication of the ulse for that segment. Still a high value of φ m is maintained during the entire ulse-duration leading to an increase in the ore size. Hence, the average radius for ores in this region reaches a maximum of ~64 nm, which is significantly greater (~1.9 times) than that redicted (~34 nm) by [16]. It is evident that the number of ores redicted by this model is lesser as comared to that redicted by [16]. 4. Electrooration with E e = 1. kv/cm: Simulations of cell electrooration have also been carried out for an alied field of 1 kv/cm and results are dislayed by Figure 3. Figure 3(a) shows that for segment, φ m increases abrutly to ~1.45 V with the alication of the electric ulse and then stabilizes at ~.84 V. On withdrawal of the ulse, φ m decreases to V. This initial sike is similar to that reorted by Smith et al [16]. But for segment T, φ m increases to ~1.4 V and stabilizes at ~1.15 V till the end of the ulse and then falls to V. For segments and, φ m attains constant values of.83 V and.9 V resectively. Figure 3(b) shows the variation of R avg with time for the various segments of the cell. For segment, the radius of the ore increases abrutly and stabilizes at ~8 nm for the duration of the ulse. On withdrawal of the ulse, R avg falls to ~.8 nm, indicating closing of the ores. For segment T, the R avg is maximum (~ 3 nm). The value of R avg decreases on withdrawal of ulse. It is evident from the figure 3 that larger ores are generated in the segment T whereas smaller ores are formed in the segment. Figures 3 (b & c) show that for each segment values of R max and R avg are very close. This signifies that in each segment almost all ores attain the same radius, ~ R avg for that segment. Also, Figure 3(d) indicates that the ore oulation is more in segment (34985) as comared to that in segment T (11,19). The grah also indicates that there is no significant ore formation in segments and. 8

9 φ m (Volts) T (a) R avg (nm) (b) T R max (nm) T (c) Pore counts (x1 4 ) t (µ s) T (d) Figure 3. The figures (a), (b) and (c) reresent the variation of transmembrane otential (φ m ), average ore radius (R avg ) and maximum ore radius (R max ) with time, resectively. Each figure shows 4 curves corresonding to 4 regions of cell laced in an electric field of 1 kv/cm: (θ = to θ = π/8), T(θ = π/8 to θ = π/4), (θ = π/4 to θ = 3π/8) and (θ = 3π/8 to θ = π/). Figure (d) indicates the segment-wise distribution of ores over the cell surface. This model redicts that for segment, the transmembrane otential φ m abrutly increases to ~1.45 V at the onset of the ulse. This leads to the creation of large number of ores within a very short time (~ µs). This abrut increase in ore oulation increases the ermeability (and hence the conductivity) of the membrane thereby reducing φ m to ~.8 V for the rest of the ulse duration. Such low value of φ m does not favour further exansion of ores. This is evident from a constant value of R avg maintained during the rest of the ulse. However, for segment T, φ m attains an initial value of only ~1.4 V leading to the formation of 11,19 ores, which is much less as comared to that in segment. This causes a higher transmembrane otential (φ m ~1.15 V) to be maintained for the entire ulse duration. Consequently, the ores exand to larger radii, with the R avg being ~3 nm. Such large ores cannot be redicted by models that consider uniform membrane oration [16] for higher values of 9

10 electric fields viz E e ~1 kv/cm or more. The simultaneous existence of large ores (in segment T) and a large number of small ores (in segment ) indicate the ossibility of effective electrooration with the chosen electrooration arameters. 4.3 Future rosect The resent model describes the electrooration of a cell membrane containing cholesterol (~%). The model redicts the occurrence of significant oration in two segments and T of the cell with lower values of θ (i.e, regions with θ = to θ = π/4) and no oration in the other segments. This is in agreement with the exerimental results ublished earlier []. However, the model has not been extended to redict the utake of DNA. This may be taken u as a future roject. Also, the cholesterol content of mammalian cells deends on the tye of the biological cell. General case with (% mole fraction) cholesterol content has been considered for simulation in this model. The deendence of nature of ore formation (oration) on cholesterol content of mammalian cells (all ossible values of ψ) also can be taken u as a future course of study. 5. Conclusion In this article a method for numerical imlementation (simulations) of electrooration of a mammalian cell (with cholesterol containing membrane) has been develoed. Based on the model we have analyzed the variation of transmembrane otential φ m, size of ores R avg, R max and orecount with time for different segments of a biological cell during electrooration. For simulation of electrooration henomenon with a more realistic aroach, two modifications were made to the existing model [16]: a) the incororation of resence of cholesterol, and b) surface distribution of ores (i.e the variation of number of ores with θ). The results are summarized below: i) The consideration of cholesterol in membrane has effectively increased the average ore radius as comared to that redicted for a cholesterol-free membrane [16] for the same magnitude of alied electric field. ii) This model quantitatively redicts the surface distribution of ores i.e., variation in number of ores with θ. The results clearly show greater oration in olar regions (regions with θ = to θ = π/4) as comared to the remaining arts of the cell. Hence, the total number of ores redicted for the whole cell is different as comared to that redicted by other models [16]. iii) The model can redict enhanced oration and effective electrooration for higher magnitudes of alied electric fields (E e ~1 kv/cm onwards). This model can be a useful tool for redicting a more realistic nature of oration in electrooration of mammalian cells at moderate and higher magnitudes of alied electric fields. It can also hel researchers in understanding the deendence of electrooration on cholesterol content of cell membranes. Acknowledgements The authors would like to thank Dr. S. H. Sanghvi and Dr. K.P. Mishra, Ex-Scientists, Radiation Biology and Health Sciences Division, BARC, Mumbai, India for meaningful discussions. This work has been suorted by the School of Basic Medical Sciences at the Deartment of Physics, University of Pune, Pune, India. 1

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