Theoretical Study of Molecular Transport Through a Permeabilized Cell Membrane in a Microchannel

Abstract

A two-dimensional model is developed to study the molecular transport into an immersed cell in a microchannel and to investigate the effects of finite boundary (a cell is suspended in a microchannel), amplitude of electric pulse, and geometrical parameter (microchannel height and size of electrodes) on cell uptake. Embedded electrodes on the walls of the microchannel generate the required electric pulse to permeabilize the cell membrane, pass the ions through the membrane, and transport them into the cell. The shape of electric pulses is square with the time span of 6 ms; their intensities are in the range of 2.2, 2.4, 2.6, 3 V. Numerical simulations have been performed to comprehensively investigate the molecular uptake into the cell. The obtained results of the current study demonstrate that calcium ions enter the cell from the anodic side (the side near positive electrode); after a while, the cell faces depletion of the calcium ions on a positive electrode-facing side within the microchannel; the duration of depletion depends on the amplitude of electric pulse and geometry that lasts from microseconds to milliseconds. By keeping geometrical parameters and time span constant, increment of a pulse intensity enhances molecular uptake and rate of propagation inside the cell. If a ratio of electrode size to cell diameter is larger than 1, the transported amount of Ca ²⁺ into the cell, as well as the rate of propagation, will be significantly increased. By increasing the height of the microchannel, the rate of uptake is decreased. In an infinite domain, the peak concentration becomes constant after reaching the maximum value; this value depends on the intra–extracellular conductivity and diffusion coefficient of interior and exterior domains of the cell. In comparison, the maximum concentration is changed by geometrical parameters in the microchannel. After reaching the maximum value, the peak concentration reduces due to the depletion of Ca ²⁺ ions within the microchannel. Electrophoretic velocity has a significant effect on the cell uptake.

Appendix: The Effect of Electroosmotic Flow on Molecular Uptake

Here we present a brief concept of neglect of electrophoresis for mass transport. Firstly, we are simulated Li and Lin’s model by our method and validate with their procedure;. From the literature (Li and Lin 2011), the electroosmotic flow is induced by the charged surface and its velocity can be found by

$$U_{\text{eo}} = \frac{{ - \varepsilon \varsigma E_{\text{t}} }}{\mu },$$

where ε, μ, E _t, and ζ are the permittivity, viscosity, tangential field, and zeta potential, respectively.

Also, they calculate the electrophoretic velocity in one-dimensional argument to simplify and their calculations were limited in axial vector where E _x. Therefore, the axial electrophoretic velocity is given by

$$U_{\text{ep}} = wFzE_{x}.$$

The importance of influence of flows can be compared by estimating the ratio of the two velocities:

$$\frac{{U_{\text{eo}} }}{{U_{\text{ep}} }} = \frac{ - \varepsilon \varsigma }{\mu wFz}.$$

Regards to Li and Lin 2011 and Zhang et al. 2008 (Zhang et al. 2008), the zeta potential of mammalian cell is in the value range of 20–30 mV. By using these article values, ɛ = 6.9 × 10⁻¹⁰ F/m, μ = 10⁻³ kg/m s, $\omega_{{Ca^{2 + } }} = 3.2 \times 10^{ - 13} {\text{mol}} \,{\text{s/kg}},$ and Z = +2, the resulting ratio is 0.2–0.3. Consequently, the electroosmotic transport does not have significant impact on mass transfer in comparison with electrophoretic transport.