Green's function and eigenvalue representations of time lag for membrane permeation transport

Abstract

Modeling membrane permeation transport by particles undergoing random walk over a one-dimensional lattice leads to a master equation, which describes the time dependence of particle concentration. With the stipulated conditions appropriate for absorptive permeation, the master equation is then solved in the Laplace domain to find the flux into the receiver, which in turn is used to formulate the time lag, t(L). It turns out t(L) can be represented by trace(A(-1)), where A is the transition matrix associated with the master equation. An analog to this representation for the continuous coordinate space is t(L) = (0)integral(h) G(x,x)dy, where G(x, x) is the Green's function associated with the steady-state diffusion equation with zero value at both boundaries 0 and h, and the source and observation points coinciding at x (0 < x < h). We have also provided a rigorous proof for this new Green's function representation of time lag. A bonus from the latter is that the time lag can be calculated t(L) = Sigma(n) 1/lambda(n) where lambda(n) is the eigenvalue of the transmission matrix. The directional symmetry of the time lag is easy to be interpreted on the basis of Green's function or eigenvalues representations.