Biomedical subjects

Mark F Schumaker

Publications and source records attributed to Mark F Schumaker.

4 recordsLinked to original sources

PubMed2006

Framework models of ion permeation through membrane channels and the generalized King-Altman method.

A modern approach to studying the detailed dynamics of biomolecules is to simulate them on computers. Framework models have been developed to incorporate information from these simulations in order to calculate properties of the biomolecules on much longer time scales than can be achieved by the simulations. They also provide a simple way to think about the simulated dynamics. This article develops a method for the solution of framework models, which generalizes the King-Altman method of enzyme kinetics. The generalized method is used to construct solutions of two framework models which have been introduced previously, the single-particle and Grotthuss (proton conduction) models. The solution of the Grotthuss model is greatly simplified in comparison with direct integration. In addition, a new framework model is introduced, generalizing the shaking stack model of ion conduction through the potassium channel.

Algorithms↗

PubMed2004

A framework model based on the Smoluchowski equation in two reaction coordinates.

The general form of the Smoluchowski equation in two reaction coordinates is obtained as the diffusion limit of a random walk on an infinite square grid using transition probabilities that satisfy detailed balance at thermodynamic equilibrium. The diffusion limit is then used to construct a generalization of the single-particle model to two reaction coordinates. The state space includes a square on which diffusion takes place and an isolated empty state. Boundary conditions on opposite sides of the square correspond to transitions between the empty state and the square. The two-dimensional (2D) model can be reduced to a 1D single-particle model by adiabatic elimination. A finite element solution of the 2D boundary value problem is described. The method used to construct the 2D model can be adapted to state spaces that have been constructed by other authors to model K+ conduction through gramicidin, proton conduction through dioxolane-linked gramicidin, and chloride conduction through the bacterial H(+)-Cl- antiporter.

Journal Article↗

PubMed2003

Numerical framework models of single proton conduction through gramicidin.

A framework model of single-proton conduction through gramicidin was previously designed to incorporate potentials of mean force and diffusion coefficients computed by the molecular dynamics simulations of Pom s and Roux (1). The resulting diffusion model was solved analytically using the lumped state approximation (LSA), allowing a detailed comparison to be made with conductance data from gramicidin A and two Trp--> Phe analogs (2). The comparison included a sensitivity analysis which required over 1 million current evaluations. A numerical method for constructing framework models is now introduced which involves finding the steady states of random walks using a trapezoid rule closely related to the rule for numerical integration. The method is described and then applied directly to the LSA. Convergence of the results to the analytical solution is seen as the number of random walk sites increase. The numerical method is then used to construct a more elaborate framework model which avoids the LSA. This is also in very good agreement with the analytical solution under the experimental conditions, confirming the accuracy of the LSA. The numerical method remains fast enough to allow an extensive comparison with conductance data.

Computer Simulation↗

PubMed2002

The role of Trp side chains in tuning single proton conduction through gramicidin channels.

We present an extensive set of measurements of proton conduction through gramicidin A (gA), B (gB), and M (gM) homodimer channels which have 4, 3, or 0 Trp residues at each end of the channel, respectively. In gA we find a shoulder separating two domains of conductance increasing with concentration, confirming the results of Eisenman, G., B. Enos, J. Hagglund, and J. Sandblom. 1980. Ann. NY. Acad. Sci. 339:8-20. In gB, the shoulder is shifted by approximately 1/2 pH unit to higher H(+) concentrations and is very sharply defined. No shoulder appears in the gM data, but an associated transition from sublinear to superlinear I-V values occurs at a 100-fold higher [H(+)] in gM than in gA. The data in the low concentration domain are analyzed using a configuration space model of single-proton conduction, assuming that the difference in the proton potential of mean force (PMF) between gA and its analogs is constant, similar to the results of Anderson, D., R. B. Shirts, T. A. Cross, and D. D. Busath. 2001. Biophys. J. 81:1255-1264. Our results suggest that the average amplitudes of the calculated proton PMFs are nearly correct, but that the water reorientation barrier calculated for gA by molecular dynamics using the PM6 water model (Pomès, R., and B. Roux. 1997. Biophys. J. 72:246a) must be reduced in amplitude by 1.5 kcal/mol or more, and is not rate-limiting for gA.

Biophysical Phenomena↗