Electrokinetic transport through the nanopores in cell membrane during electroporation

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Abstract

In electroporation, applied electric field creates hydrophilic nanopores in a cell membrane that can serve as a pathway for inserting biological samples to the cell. It is highly desirable to understand the ionic transfer and fluid flow through the nanopores in order to control and improve the cell transfection. Because of submicron dimensions, conventional theories of electrokinetics may lose their applicability in such nanopores. In the current study, the Poisson–Nernst–Planck equations along with modified Navier–Stokes equations and the continuity equation are solved in order to find electric potential, fluid flow, and ionic concentration through the nanopores. The results show that the electric potential, velocity field, and ionic concentration vary with the size of the nanopores and are different through the nanopores located at the front and backside of the cell membrane. However, on a given side of the cell membrane, angular position of nanopores has fewer influences on liquid flow and ionic transfer. By increasing the radius of the nanopores, the averaged velocity and ionic concentration through the nanopores are increased. It is also shown that, in the presence of electric pulse, electrokinetic effects (electroosmosis and electrophoresis) have significant influences on ionic mass transfer through the nanopores, while the effect of diffusion on ionic mass flux is negligible in comparison with electrokinetics. Increasing the radius of the nanopores intensifies the effect of convection (electroosmosis) in comparison with electrophoresis on ionic flux.

Highlights

► Mass transfer and fluid flow in nanopores during cell electroporation were studied. ► Poisson–Nernst–Planck and modified Navier–Stokes equations were simulated directly. ► Electrokinetics has great influences on ionic mass transfer through nanopores. ► Bigger nanopores have greater convective ionic flux. ► Ionic flux and fluid flow are different on two sides of cell membrane.

Introduction

Electroporation (or electropermeabilization) is an increase in cell membrane’s permeability by applying a sufficiently strong electric field [1]. One of the most important applications of cell electroporation is cell transfection: nanopores created in the cell membrane are used as a pathway to insert biological molecules into the cell. Although cell electroporation was reported in early 1970s [2], the first successful reversible electroporation and DNA electrotransfer can trace its roots back to 30 years ago, in 1982 [3]. Nowadays, microscale cell electroporation has been demonstrated to have the best cell viability and transfection efficiency among all recognized gene transfection methods [4]. More and more experimental studies on microfluidic cell electroporation have been reported in recent years [4], [5], [6], [7], [8], [9], [10].

Presence of applied electric field near the cell membrane causes the extra transmembrane potential (Um) across the cellular membrane. If there is not any hydrophilic nanopore on the cell membrane, the Um is linearly proportional to the cell radius and external electric field. For spherical cells surrounded by sufficiently high conductive media, the steady state Um can be evaluated as Um = 1.5Eea cos (θ), where Ee is the external electric field, a is the cell radius, and θ is the polar angle measured with respect to the direction of the external field Ee, see Fig. 1. This equation is usually stated as Schwan equation [11]. If the medium is not highly conductive, the constant of Schwan equation should be less than 1.5. Excessive theoretical studies have been done to investigate the transient response of the Schwan equation [12], also the effects of different parameters such as alternating electric fields, conductivity of the media, and shape of the cell on the induced transmembrane potential [13], [14], [15], [16], [17]. The sinusoidal dependency of Um to the angular position on the cell membrane is proved experimentally [18], [19], [20]. Um causes random hydrophobic pores in the cell membrane. These pores grow under the stress from Um and become hydrophilic at the threshold value of Um = 0.5  1 V [5]. Upon creating the first hydrophilic nanopore, Um remains constant in the vicinity of the created pores, and further increase in the external electric field will no longer have any effects on Um. In this stage, further increase in the electric field has two effects: first, increasing the electroporated area (the area with hydrophilic nanopores) on the cell membrane and second, increasing the radius of the created nanopores [21]. Under the controlled condition (restricted pulse duration and intensity), these nanopores are reversible and can act as a pathway for either inserting hydrophilic molecules such as membrane-impermeant molecules [22], gene [23], and DNA plasmid [24] to the cell or releasing internal contents of the cell [25], [26]. By removing the electric pulse, the hydrophilic nanopores are present on the cell membrane from seconds to minutes [6].

Many studies have been done on the creation of the nanopores [27] and the effects on the cell electroporation of different parameters such as applied electric field [28], field shock duration, frequency and strength [29], field strength and rest potential [30], also ionic concentration [31]. However, there are few analytical studies focused on the cell uptake through the created nanopores during the electroporation. Zaharoff et al. used one-dimensional mass transfer equation to determine the cellular uptake of macromolecules [32]. They estimated the dimension of the generated pores due to applied electric field in the cell membrane and consequently treated these pores as a channel to uptake the macromolecules to the cell. They considered only the effect of diffusion in this process. The other study on the modeling of cellular uptake during the electroporation is based on the assumptions that the process of mass transfer happens in every cell in the tissue [33]; the cells are infinitesimally small, and that the drug entering the cell can be modeled as a uniformly distributed reaction rate. Again, only the effect of diffusion on cellular uptake was considered. However, because of the presence of the transmembrane potential, electrokinetic effects may have considerable influence on the ion insertion and flow uptake of the cell. However, the influence of electrokinetics is present until the end of electric pulse. As it was indicated before, by removing the electric pulse, the hydrophilic nanopores are still present on the cell membrane from seconds to several minutes. In this stage, diffusion may play the decisive role on ion insertion. Nevertheless, the effect of electrokinetics is not negligible on cell transfection. Because of the nanometer dimensions of the created pores, conventional electrokinetic theories such as Helmholtz–Smoluchowski model are not applicable. In these small nanochannels, the electric field generated by the surface charge and the ionic distribution no longer obeys the Poisson–Boltzmann model [34], [35]. However, previous experimental studies show that the continuum assumption for liquid flow of aqueous solution is valid in the channels as small as 4 nm [36].

More recently, Li and Lin have conducted a two dimensional numerical study on molecular uptake via electroporation [37]. Although their work is the most comprehensive study on this topic, they have not considered the effects of different parameters such as surface electric charge of cell membrane and intercellular concentrations of Na+, K+, and Cl ions in their simulations. Because of EDL overlapping through the generated nanoscale pores on cell membrane, these parameters have significant influences on electric field and ion transportation through the nanopores [38].

In a recent publication [38], we studied the electric potential, ion distribution, and flow field in nanochannels by solving a set of highly coupled partial differential equations including the Poisson equation, the Nernst–Planck equation, the modified Navier–Stokes equations, and the continuity equation. We have shown that unlike microchannels, the electric potential, ionic concentration, and velocity fields are strongly size-dependent in nanochannels. In the present study, ionic transfer and flow uptake to the cell through the nanopores are investigated during the presence of electric pulse. First, the mathematical model of electrokinetic mass and momentum transfers in the nanopores is presented. Then, the numerical method utilized in the present study is introduced. Finally, the effects of different parameters such as the location of nanopores in the cell membrane and the nanopore size on the ionic mass transfer and the liquid flow in the nanopores are examined. The effects of electroosmosis, electrophoresis, and diffusion on ionic mass transfer through the created nanopores are compared.

Section snippets

Mathematical modeling

The appearance of external electric field in a vicinity of cell surface can create hydrophilic nanopores in the cell membrane. These nanopores can serve as a pathway for ion and fluid transport. The flow and ion transfer in such a nanoscale channel can be analyzed by a combination of equations governing nanopore creation, electrostatics, mass transfer, and momentum transfer. In the present study, we examine the mass and momentum transfer in one nanopore. Fig. 2 depicts the computational domain.

Numerical simulation

All the above-mentioned, highly coupled equations were solved simultaneously with the corresponding boundary conditions, as described in the last section. The present numerical study was conducted by using COMSOL Multiphysics 3.5a; we employed a mesh independent structure to make sure that the results are unique and will not change if any other grid distribution is applied. In order to discretize the solution domain, the structured (mapped) meshes were applied. The meshes must fully cover the

Results and discussion

Table 1 summarizes the quantitative information used in the simulations. We consider a mammalian cell with radius 50 μm and membrane thickness of 5 nm; the liquid is a mixture of NaCl and KCl aqueous solutions. Cell electroporation is usually conducted in vitro. We can modify the extracellular ionic concentrations to appropriate values; however, the assumed ions (Na+, K+, and Cl) have predefined concentrations inside the cell. In the simulations, we consider the typical intercellular

Conclusion

In this paper, ionic mass transfer and fluid flow through the created nanopores in the cell membrane during the electroporation and in the presence of the electric pulse were studied. Previous studies only considered the effect of diffusion on the cell transfection. Because of the transmembrane potential, the electrokinetic effects must consider on the ionic mass transfer through the nanopores. Because of nanoscale dimensions of the created pores, conventional electrokinetic theories such as

Acknowledgments

The authors wish to thank the financial support of the Canada Research Chair program and the Natural Sciences and Engineering Research Council (NSERC) of Canada through a research grant to D. Li.

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