Mathematical Models of Ionic Flow Through Open Protein Channels
The subject of this dissertation is a mathematical and physical study of ion flow through open protein channels of biological membranes. Protein channels are macromolecules embedded in biological membranes, through which almost all transfer of ions into and out of living cells is done. There are hundreds of different protein channels with diverse functions, ranging from the transfer of electrical signals in the nervous system and coordination of muscle contraction, to secretion of urine and transfer of hormones. Over the past twenty years a vast amount of experimental data concerning the permeation characteristics of various channels has been accumulated. However, much theoretical work remains to be done to understand and interpret these data. Therefore, the study of ionic permeation through protein channels is one of the central problems of theoretical biology today. To quantitavely study channels, mathematical and physical models of ionic permeation need to be derived. In this dissertation we consider mathematical models for the determination of the ionic current flowing through a single protein channel, given its structure, the applied electric potential and the ionic bath concentrations. We present three mathematical models of permeation on different scales of resolution. First, we derive coarse grained continuum equations of ionic permeation through a channel from a microscopic model of motion, with long and short range interactions. Then, on a coarser level, we propose a semiMarkovian model, with general probability distributions of residence times, which generalizes Markovian rate theories of channel permeation. Finally, we analyze a Langevin simulation of ionic motion in a region possibly containing a protein channel. The main results of this dissertation are as follows: The first result is a new coupled system of partial differential equations for the ionic densities and electrostatic fields that describe ionic permeation in an electrolyte bath containing a protein channel. These equations are derived from a non equilibrium molecular model by a novel averaging procedure. Starting from a stochastic Langevin description of the coupled motions of all ions in a finite system containing an electrolytic solution, two electrodes, a membrane and a channel, a mathematical averaging procedure is devised to describe the system with averaged electrostatic potentials and averaged charge concentrations. The resulting equations form a coupled system of Poisson and Nernst-Planck equations involving conditional and unconditional charge densities (C-PNP). While derivations of continuum equations from equilibrium molecular models are common in the statistical physics literature, this is the first derivation of Poisson Nernst Planck type equations from a non equilibrium molecular model. The resulting equations differ in many aspects from the standard PNP equations used so far. To start with, the force in the resulting NP equations has two components instead of one. The first component of the force is the gradient of a conditional electric potential, which depends on ionic densities at one location, conditioned on the presence of an ion at another location. In contrast, in previous PNP theories only unconditional densities and an unconditional potential enter the equations. Our conditional densities take into account short range forces and excluded volume effects, which are important for ionic flow through confined regions such as a protein channel. The second component of the force, not present at all in standard treatments, is the self induced force on a single discrete ion due to surface charges induced by that ion at dielectric interfaces. This component is important near dielectric interfaces, for example near all interfaces between the channel, the ionic baths and the bilipid membrane. Another difference from the standard PNP system is that the proposed conditional PNP system is not complete, because Poisson’s equation for the electric potential depends on conditional charge densities, while the NP equations contain unconditional densities. The conditional densities present in our equations are the non equilibrium generalization of the well studied pair-correlation functions of equilibrium statistical mechanics. According to the mathematical derivation, the charge densities in the NP equations are interpreted as time averages over long times, of the motion of a quasi-particle that diffuses with the same diffusion coefficient as that of a real ion, but driven by an averaged force. In this way, our derivation explains how continuum equations with averaged charge densities and mean-fields can be used to describe permeation through a microscopic protein channel. The importance of our derivation lies in the fact that the description of a complex molecular bath and channel system with averaged continuum equations reduces dramatically the number of degrees of freedom, rendering the problem computationally feasible. The second main result in this dissertation is the presentation of semi-Markov chain models, with arbitrary probability distributions of residence times, which generalize Markovian rate models of ionic permeation through a protein channel. A mathematical procedure for