Experimental and Theoretical Brownian Dynamics Analysis of Ion Transport During Cellular Electroporation of E. coli Bacteria

Abstract

Escherichia coli bacterium is a rod-shaped organism composed of a complex double membrane structure. Knowledge of electric field driven ion transport through both membranes and the evolution of their induced permeabilization has important applications in biomedical engineering, delivery of genes and antibacterial agents. However, few studies have been conducted on Gram-negative bacteria in this regard considering the contribution of all ion types. To address this gap in knowledge, we have developed a deterministic and stochastic Brownian dynamics model to simulate in 3D space the motion of ions through pores formed in the plasma membranes of E. coli cells during electroporation. The diffusion coefficient, mobility, and translation time of Ca²⁺, Mg²⁺, Na⁺, K⁺, and Cl⁻ ions within the pore region are estimated from the numerical model. Calculations of pore’s conductance have been validated with experiments conducted at Gustave Roussy. From the simulations, it was found that the main driving force of ionic uptake during the pulse is the one due to the externally applied electric field. The results from this work provide a better understanding of ion transport during electroporation, aiding in the design of electrical pulses for maximizing ion throughput, primarily for application in cancer treatment.

Appendices

Appendix

Derivations of Induced Image Charges at the Pores

Green’s function of the interior of a cylinder of radius ρ₀ and length 2z₀ [63, 64] is given by:

$$G\left(\rho ,\mathrm{\varnothing },z,{\rho }^{{{\prime}}},{\mathrm{\varnothing }}^{{{\prime}}},z{^{\prime}}\right)=\frac{1}{2\pi {\varepsilon }_{0}{\rho }_{0}}\sum_{m=0}^{\infty }\sum_{n=1}^{\infty }\left\{\frac{{\epsilon }_{m}\mathrm{cos}\left[m\left(\mathrm{\varnothing }-\mathrm{\varnothing^ {\prime}}\right)\right]{J}_{m}\left({j}_{m,n}\rho \right)}{}\right.\frac{{J}_{m}\left({j}_{m,n}\rho{^{\prime}}\right)}{{J}_{m,n}{J}_{m+1}^{2}\left({j}_{m,n}\right)}\left.\frac{\mathrm{sinh}\left[{j}_{m,n}\left(\alpha +z<\right)\right]\mathrm{sinh}\left[{j}_{m,n}\left(\alpha -z>\right)\right]}{\mathrm{sinh}\left({2j}_{m,n}\alpha \right)}\right\}$$

(42)

where ${\epsilon }_{m}=2-{\delta }_{m,0}$. ${\delta }_{\mathrm{0,0}}=1$ , and ${\delta }_{m,0}=0$ otherwise. j_m,n is the nth zero of the Bessel function J_m(x); $\alpha ={z}_{0}/{\rho }_{0}$ is the aspect ratio of the cylinder. The ρ and z cylindrical coordinates are scaled by ρ₀ and their origin is the center of the cylinder. The coordinate z_< is the lesser of z and z′, and z_> is the greater.

The z component on a charge q can be found by evaluating Green’s function G [Eq. (42)] at $\rho ^{\prime}= \rho ,\varnothing ^{\prime}=\varnothing ,z^{\prime}=z$ and taking the gradient following Smythe [59]:

$${F}_{z}=-\frac{{q}^{2}\nabla G\left(\rho ,\varnothing ,z,{\rho }^{\prime},{\varnothing }^{\prime},{z}^{\prime}\right)}{2}=\frac{{q}^{2}}{2\pi {\varepsilon }_{0}{\rho }_{0}}\sum_{m=0}^{\infty }\sum_{n=1}^{\infty }\frac{{\epsilon }_{m}{{J}_{m}}^{2}\left({j}_{m,n}{\rho }^{\prime}\right)}{{J}_{m+1}^{2}\left({j}_{m,n}\right)}\frac{\mathrm{sinh}\left[{2j}_{m,n}{z}^{\prime}\right]}{\mathrm{sinh}\left[{2j}_{m,n}{\alpha }^{\prime}\right]}.$$

(43)

Since this approach yields a divergent series when calculating the radial force F_ρ, a different technique by **ang et al. (**a93) is followed using an alternate form for the Green’s function:

$$G\left(\rho ,\varnothing ,z,{\rho }^{\prime},{\varnothing }^{\prime},z^{\prime}\right)=\frac{1}{2\pi {\varepsilon }_{0}{z}_{0}}\sum_{m=0}^{\infty }\sum_{n=1}^{\infty }\left.\left\{{\epsilon }_{m}\mathrm{cos}\left[m\left(\varnothing -\varnothing ^{\prime}\right)\right]\mathrm{sin}\left[\frac{n\pi \left(z+1\right)}{2}\right]\mathrm{sin}\left[\frac{n\pi \left(z^{\prime}+1\right)}{2}\right]{T}_{n}\left(\rho ,\rho ^{\prime}\right)\right.\right\},$$

(44)

where

$${T}_{n}\left(\rho ,\rho ^{\prime}\right)=\frac{{I}_{m}\left(\frac{n\pi \rho >}{2}\right)}{{I}_{m}\left(\frac{n\pi }{2\alpha }\right)}\left[{I}_{m}\left(\frac{n\pi }{2\alpha }\right){K}_{m}\left(\frac{n\pi \rho <}{2}\right)-{K}_{m}\left(\frac{n\pi }{2\alpha }\right){I}_{m}\left(\frac{n\pi \rho <}{2}\right)\right].$$

(45)

${I}_{m}$ y ${K}_{m}$ are hyperbolic Bessel functions. Scaling is by z₀. The charge distribution σ on the cylinder surface S can be found from Green’s function [Eq. (44)]:

$$\sigma =-{\varepsilon }_{0}q\nabla G\left(\rho ,\varnothing ,z,{\rho }^{\prime},{\varnothing }^{\prime},{z}^{\prime}\right),$$

(46)

where the gradient is taken with respect to $\rho ,\varnothing ,z$. Using the identity

$${I}_{m}\left(u\right){K}_{m}^{\prime}\left(u\right)- {K}_{m}\left(u\right){I}_{m}^{\prime}\left(u\right)=-1/u,$$

(47)

the surface charge on the cylinder is:

$${\sigma }_{\mathrm{c}}=\frac{-q}{2\pi {\varepsilon }_{0}{z}_{0}}\sum_{m=0}^{\infty }\sum_{n=1}^{\infty }\left.\left\{{\epsilon }_{m}\mathrm{cos}\left[m\left(\varnothing -\varnothing ^{\prime}\right)\right]\mathrm{sin}\left[\frac{n\pi \left(z+1\right)}{2}\right]\mathrm{sin}\left[\frac{n\pi \left(z^{\prime}+1\right)}{2}\right]\frac{{I}_{m}\left(\frac{n\pi \rho ^{\prime}}{2}\right)}{{I}_{m}\left(\frac{n\pi }{2\alpha }\right)}\right.\right\}.$$

(48)

The surface charge creates a pressure directed outwards of the conducting membrane pore:

$$p=\frac{{\sigma }^{2}}{2{\varepsilon }_{0}}.$$

(49)

Thus, the force on the ion is given by:

$${F}_{\rho }={\int }_{S}\frac{{\sigma }^{2}\mathrm{d}a}{2{\varepsilon }_{0}}.$$

(50)

The radial force on the ion from the above equation is:

$${F}_{\rho }=-\frac{{\rho }_{0}}{4\pi {\varepsilon }_{0}}{\int }_{0}^{{z}_{0}}\mathrm{d}z{\int }_{0}^{2\pi }\mathrm{d}\varnothing 2\pi {\sigma }^{2}\mathrm{cos}\left(\varnothing -{\varnothing }^{\prime}\right).$$

(51)

Considering the identity:

$${\int }_{0}^{2\pi }\mathrm{cos}\left(x\right)\mathrm{cos}\left(mx\right)\mathrm{cos}\left(nx\right)\mathrm{d}x=\left\{\begin{array}{ll}\pi & {\text{for}}\, m=0\, {\text{and}}\, n=1,\\ \pi & {\text{for}}\, m=1 \,{\text{and}}\, n=0,\\ \frac{\pi }{2} & {\text{for}} m-n=\pm 1\, {\text{and}}\, n,m>0,\\ 0 &{\text{otherwise}},\end{array}\right.$$

(52)

the radial force becomes [70]:

$${F}_{\rho }=\frac{{q}^{2}}{2\pi {\varepsilon }_{0}{{\rho }_{0}z}_{0}}\sum_{m=0}^{\infty }\sum_{n=1}^{\infty }\left.\left\{{\mathrm{sin}}^{2}\left[\frac{n\pi \left(z+1\right)}{2}\right]\mathrm{sin}\left[\frac{n\pi \left(z^{\prime}+1\right)}{2}\right]\frac{{I}_{m}\left(\frac{n\pi \rho ^{\prime}}{2}\right){I}_{m+1}\left(\frac{n\pi \rho ^{\prime}}{2}\right)}{{I}_{m}\left(\frac{n\pi }{2\alpha }\right){I}_{m+1}\left(\frac{n\pi }{2\alpha }\right)}\right.\right\}.$$

(53)