The Simultaneous Determination of Diffusion Coefficients and PMFs Through the OFR Method

Abstract

Progress in nonequilibrium, bidirectional work theorems have lead to the development of an important theory, known as the forward-reverse (FR) method, that allows the PMF to be extracted from nonequilibrium dynamics. In addition, the FR method allows the simultaneous determination of the reaction coordinate dependent diffusion coefficient, D(z). We recently extended the utility of the FR method through the use of an oscillating steering protocol that we named the oscillating forward-reverse (OFR) method. While working with OFR, the D(z) results did not match known values or those obtained through other methods. After reformulating the procedure to obtain D(z), we were able to obtain results close to the correct values. These results however showed very little variation over the length of the reaction coordinate, even when D(z) was known to vary drastically. It seemed that the highly variable and noncontinuous velocity function of the particle - a consequence of being steered using the “stiff-spring” method - was incompatible with the macroscopic definition of the drag coefficient through which D(z) is calculated. To address this, a new dynamic constraint steering protocol (DCP) was developed to replace the previously used “stiff-spring” method. We present here the results for D(z) in bulk water, and both the PMF and D(z) results from the permeation of a water molecule through a DPPC membrane. We also consider the issue of sufficient sampling, and propose that to obtain an accurate PMF (and D(z)) from even a moderately complex system, the final result should be a weighted average of numerous pulls. This is actually an advantage of nonequilibrium over equilibrium methods, the latter having a limited ability to sample much phase space beyond their initial conditions - especially with the time-scales currently available in computer simulations.