Deformation of an elastic membrane interacting electrostatically with a rigid curved domain: implications to biosystems

Abstract

Deformation of thin elastic sheet due to electrostatic forces play important role in engineering and biological systems. In this work, we analyze the deformation of thin elastic sheet, while interacting with a rigid curved domain in the presence of dielectric fluid. Mechanical deformation of the sheet is coupled with the electrostatic interaction in its equilibrium configuration. We consider small deformation of the sheet, which obeys Hooke’s law. The electrostatic forces acting between the sheet and curved domain are calculated by using Debye–Hückel equation. It is observed that the sheet deformation is proportional to the electrostatic forces acting on it. Increase in inverse Debye length (which signifies the strength of electrostatic field) of the dielectric fluid decreases the sheet deformation. With the help of present model, binding mechanism between peripheral BAR proteins and cell membrane is studied by treating cell membrane as an elastic membrane and BAR protein as a rigid curved domain. For small curvature of BAR protein, the present model captures very well the scaffolding mechanism of protein binding. This model can also be used to analyze problems in electrophotography, powder technology, semiconductor and pharmaceutical industries.

Appendices

Appendix A: Variation of attractive and repulsive forces

The attractive and repulsive forces in z-direction are calculated numerically using Eqs. (33) and (29), respectively. The absolute value of attractive force is maximum near to the pinned support ($\bar{X} = \pm b/2$) of the membrane as shown in Fig. 10. The values of the repulsive forces are close to zero along the width of the membrane.

Fig. 10

Variation of attractive ($\bar{f}_{az}$) and repulsive ($\bar{f}_{rz}$) force acting in z-direction of membrane at $\bar{Y} = 0.5$ for $\bar{\sigma }_c = 1$ with $\bar{b} = 0.3$. Other parameter values are $\bar{H} = 0.01$, $\theta = \pi$, $\bar{k} = 1$, $\bar{R} = 0.15$, $\bar{t} = 0.04$, and $\bar{\mu } = 644$

Appendix B: Variation of traction associated with the membrane tensions

The tractions associated with the membrane tensions $\bar{p}_x$ and $\bar{p}_y$ are calculated using Equilibrium Eqs. (38) and (39), respectively. The total electrostatic traction $\bar{f}_{ex}$ and $\bar{f}_{ey}$ are obtained from Eqs. (35) and (36), respectively, after calculating integrals $\bar{f}_{ax}$, $\bar{f}_{ay}$, $\bar{f}_{rx}$, and $\bar{f}_{ry}$ numerically from Eqs. (31)–(32) and (27)–(28). It is observed that the traction $\bar{p}_x$ does not have significant changes as we approach along the length of the membrane as shown in Fig. 11a.

Fig. 11

Variation of traction associated with the membrane tension in a x and b y directions for different values of $\bar{Y}$. The tractions are depicted in $x-z$ plane. The parameter values are $\bar{H} = 0.01$, $\theta = \pi$, $\bar{k} = 1$, $\bar{R} = 0.15$, $\bar{t} = 0.04$, and $\bar{\sigma }_c = 1$

If we do not consider the edge effect of the membrane near $\bar{Y} = 0$ and 1, then the traction $\bar{p}_y$ will have almost constant value along the width of the membrane as shown in Fig. 11b. It is more prominent near to the middle of the length of the membrane (i.e., at $\bar{Y} = 0.5$). The value of $\bar{p}_y$ approaches zero near to the middle. It will be equal to zero at $\bar{Y} = 0.5$ because of symmetry (not shown in figure). As we approach toward the edges of the membrane, the value of $\bar{p}_y$ increases to maintain the deformation of membrane only in z-direction. Here, negative value of the traction component signifies the opposite nature of it.