Complex membrane transport systems. A non-Markovian approach

This paper suggests a method of how to deal with complex membrane transport systems such as ion channels or ion pumps formed by proteins. The complexity of these systems results from the fact that proteins may undergo an internal dynamics of conformational changes and may thereby affect the transmembrane transport. Usually, complex transport systems are mapped into multi-state graphs and couched in terms of Markovian master equations. It is shown in this paper how the dimensionality of such multi-state systems can be reduced. The resulting description may be expressed in the form of a generalized master equation with a memory function as integral kernel. The memory function reflects the protein's own dynamics and its overall effect on the transport. This formalism, non-Markovian in nature, is applied to describe the time-dependent action of ion pumps. A general model is constructed on the basis of the rate theory which contains all the essential parts of ion pumps such as a catalytic unit and a channel-like conduit for ion translocation and which is still analytically tractable. The short-time behaviour of the pumping process turns out to be of particular interest, since it reveals the dynamics of the catalytic unit itself. A strong correlation of the particle's motion over times less than a certain correlation time has been found. This result is compared with experimental findings on the proton pump of Halobacterium halobium. It is concluded that such a perfect short-time memory could be a generic property of active transport systems.