Extension of Nonequilibrium Work Theorems with Applications to Diï¬usion and Permeation in Biological Systems

Nonequilibrium work methods for determining potentials of mean force (PMF) w(z) have recently gained popularity as an alternative to standard equilibrium based methods. Introduced by Kosztin et al., the forward-reverse (FR) method is a bidirectional work method in that it requires the work to be sampled in both forward and reverse directions along the reaction coordinate z. This bidirectional sampling leads to much faster convergence than other nonequilibrium methods such as the Jarzynski equality, and the calculation itself is extremely simple, making the FR method an attractive way of determining the PMF. Presented here is an extension to the FR method that deals with sampling problems along essentially irreversible reaction coordinates. By oscillating a particle as it is steered along a reaction coordinate, both forward and reverse work samples are obtained as the particle progresses. Dubbed the oscillating forward-reverse (OFR) method, this new method overcomes the issue of irreversibility that is present in numerous soft-matter and biological systems, particularly in the stretching or unfolding of proteins. The data analysis of the OFR method is non-trivial however, and to this end a software package named the ‘OFR Analysis Tool’ has been created. This software performs all of the complicated analysis necessary, as well as a complete error analysis that considers correlations in the data, thus streamlining the use of the OFR method for potential end users. Another attractive feature of the FR method is that the dissipative work is collected at the same time as the free energy changes, making it possible to also calculate local diffusion coefficients, D(z), from the same simulation as the PMF through the Stokes-Nernst-Einstein relation Fdrag = −γv, with γ = kB T /D. While working with the OFR method, however, the D(z) results never matched known values or those obtained through other methods, including the mean square displacement (or Einstein) method. After a reformulation of the procedure to obtain D(z), i.e. by including the correct path length and particle speeds, results were obtained that were much closer to the correct values. The 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 being steered through the “stiff-spring” method was incompatible with the macroscopic definition of the drag coefficient, γ. The drag coefficient requires at most a slowly varying velocity so that the assumption of a linearly related dissipative work remains valid at all times. To address this, a new dynamic constraint steering protocol (DCP) was developed to replace the previously used “stiff-spring” method, now referred to as a dynamic restraint protocol (DRP). We present here the results for diffusion in bulk water, and both the PMF and diffusion results from the permeation of a water molecule through a DPPC membrane. We also consider the issue of ergodicity and sampling, and propose that to obtain an accurate w(z) (and D(z)) from even a moderately complex system, the final result should be a weighted average obtained from numerous pulls. An additional utility of the FR and OFR methods is that the permeability across lipid bilayers can be calculated from w(z) and D(z) using the inhomogeneous solubility-diffusion (ISD) model. As tests, the permeability was first calculated for H2O and O2 through DPPC. From the simulations, the permeability coefficients for H2O were found to be 0.129 ± 0.075 cm/s and 0.141 ± 0.043 cm/s, at 323 K and 350 K respectively, while the permeability coefficients for O2 were 114 ± 40 cm/s and 101 ± 27 cm/s, again at 323 K and 350 K respectively. As a final, more challenging system, the permeability of tyramine – a positively charged trace amine at physiological pH – was calculated. The final value of P = 0.89 ± 0.24 Ang/ns is over two orders of magnitude lower than that obtained from experiment (22 ± 4 Ang/ns), although it is clear that the permeability as calculated through the ISD is extremely sensitive to the PMF, as scaling the PMF by ∼ 20% allowed the simulation and experimental values to agree within uncertainty. With accurate predictions for free energies and permeabilities, the OFR method could potentially be used for many valuable endeavors such as rational drug design.

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