Rate theory models for ion transport through rigid pores. III. Continuum vs discrete models in single file diffusion

In studying the single file model in its discrete as well as in its continuum form the relationship between the phenomenological continuum theory of diffusion and the rate theory approach is analyzed. The single file model in its original form is discrete and represents the most general rate theory model for ion transport through rigid pores in biological membranes. In neglecting the interionic interactions which the single file model takes into account, the Nernst-Planck equation of macroscopic free diffusion can be derived from single file by means of the procedure n leads to infinity (where n is the number of binding sites within a pore) and the classical diffusion theory can thereby be integrated into the more general concept of single filing transport. Moreover, the single file model has been transformed in the limit n leads to infinity into the corresponding continuum form involving interionic interactions. The essential differences between the two derived continuum forms are: In the macroscopic diffusion model, the interionic interactions are regarded in the form of a "mean field". Thus we only get one equation of motion (Nernst-Planck equation) for the ionic concentration c(x, t) within the membrane. In the continuum version of the single file model, however, we obtain a hierarchy of Fokker-Planck equations for the probability density functions Pm(x1, . . . , xm, t) (where m is the number of ions within a pore). The interactions of the single file system are incorporated in detail into the Fokker-Planck equation as well as into the corresponding boundary conditions. As a consequence, the boundary conditions are highly complex in comparison with periodic conditions or Dirichlet conditions often used for the Nernst-Planck equation in electrophysiology. Two types of boundary conditions have been found which are principally different: The first one is to regulate the entry and exit of the ions at the pore mouth by a negative feedback mechanism, the second one describes the collisions of the ions within multiply occupied pores. In this context the question is discussed of whether the continuum version of single file has advantages over the discrete one.