Full Length Articles

Volume 1. Structure and Dynamics of Membranes

Chapter 12. Electrostatic properties of membranes: The Poisson-Boltzmann theory

D. Andelman
School of Physics and Astronomy, Tel Aviv University,
Ramat Aviv 69978, Tel Aviv, Israel

1. Introduction

Biological membranes are complex and heterogeneous objects separating living cells from their extra-cellular surroundings. Many of the membrane structural properties depend substantially on electrostatic interactions [1, 2], e.g., rigidity, structural stability, lateral phase transitions (the 'main' transition), and dynamics. Furthermore, electric charges have a very important role in processes involving more than one membrane such as membrane adhesion and cell-cell interaction, as well as the overall interaction of the membrane with other intra- and extra-cellular molecules.

The delicate interplay between charged membranes and their surrounding ionic solution can simply be explained as following. As any charged object immersed in an ionic solution, the membrane attracts a cloud of opposite charges forming a diffusive 'electric double layer' [1-5]. The exact distribution of the charges is given by the competition between the electrostatic interactions and the entropy of the ions in the solution which tends to disperse them. This diffusive electric double layer in turn influences the overall electrostatic interactions of the membrane with its environment as well as the 'internal' membrane properties.

Electrostatic interactions constitute a key component in understanding interactions between charged bodies in ionic solutions. For example, the stability of colloidal particles dispersed in a solvent [1, 2] can be explained by considering the competition between repulsive electrostatic interactions and attractive Van der Waals interactions. Electrostatic interactions are also of importance when considering interactions and adhesion between membranes. Furthermore, strong (unscreened) electrostatic interactions tend to rigidify flexible objects such as membranes and charged polymers (polyelectrolytes). Another characteristic of ions in solutions is that due to entropic effects, temperature is an important parameter controlling equilibrium properties.

The aim of this chapter is to review some of the basic considerations underlying the behavior of charged membranes in aqueous solutions. Due to the tremendous complexity of real biological membranes, we will restrict ourselves to very simple model charged membranes and will rely on the following assumptions and simplifications:

We will mainly be concerned with the interplay between electrostatic interactions and the structure of the membrane as a whole. The membrane will be treated in the continuum limit as an interface with some degree of flexibility and with a given surface charge distribution. Equipotential membranes (of a given surface potential) will also be mentioned in some of the cases. We will not discuss special function regions within real heterogeneous membranes such as ion channels which have a specific biological function.
This review will present in detail only theoretical results. Some experimental results will be mentioned. The reader should look at the chapter by Parsegian and Rand for more details on experiments. The calculations are limited to solutions of the Poisson-Boltzmann equation (mean field theory). The finite size of the ions in the solution and within the membrane is ignored. The electric potential and the charge densities of the various ions are described by continuous variables (using the continuum hypothesis). For theoretical results which go beyond the continuum approach, we provide a few references and a short summary in the next section.
We will not discuss the effect of electrostatics on dynamic properties of membranes and restrict ourselves to the static ones (in thermodynamical equilibrium) as well as fluctuations. These are the properties which are presently better understood. In addition, we will not consider the important interplay between electrostatics and other interactions (Van der Waals, hydration, etc.).

This review is organized in the following way: after some general considerations of charged surfaces in liquids and the derivation of the Poisson-Boltzmann equation (section 2), we present specific solutions of several electrostatic problems starting with a single flat and rigid membrane in section 3, and generalizing it to two flat membranes in section 4. Then, we consider the possibility of having a flexible membrane in various situations: a single membrane (section 6), two membranes, a stack of membranes, etc. (section 7). Special emphasis is given to the coupling between the electrostatic and the elastic properties. Some concluding remarks are presented in section 8.

[Full text] (PDF 507 Kbytes)