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Free energy perturbation, a method for computing the free energy difference between two states, is often combined with non-Boltzmann biased sampling techniques in order to accelerate the convergence of free energy calculations. Here we present a new extension of the Bennett acceptance ratio (BAR) method by combining it with umbrella sampling (US) along a reaction coordinate in configurational space. In this approach, which we call Bennett acceptance ratio with umbrella sampling (BAR-US), the conditional histogram of energy difference (a mapping of the 3N-dimensional configurational space via a reaction coordinate onto 1D energy difference space) is weighted for marginalization with the associated population density along a reaction coordinate computed by US. This procedure produces marginal histograms of energy difference, from forward and backward simulations, with higher overlap in energy difference space, rendering free energy difference estimations using BAR statistically more reliable. In addition to BAR-US, two histogram analysis methods, termed Bennett overlapping histograms with US (BOH-US) and Bennett-Hummer (linear) least square with US (BHLS-US), are employed as consistency and convergence checks for free energy difference estimation by BAR-US. The proposed methods (BAR-US, BOH-US, and BHLS-US) are applied to a 1-dimensional asymmetric model potential, as has been used previously to test free energy calculations from non-equilibrium processes. We then consider the more stringent test of a 1-dimensional strongly (but linearly) shifted harmonic oscillator, which exhibits no overlap between two states when sampled using unbiased Brownian dynamics. We find that the efficiency of the proposed methods is enhanced over the original Bennett's methods (BAR, BOH, and BHLS) through fast uniform sampling of energy difference space via US in configurational space. We apply the proposed methods to the calculation of the electrostatic contribution to the absolute solvation free energy (excess chemical potential) of water. We then address the controversial issue of ion selectivity in the K(+) ion channel, KcsA. We have calculated the relative binding affinity of K(+) over Na(+) within a binding site of the KcsA channel for which different, though adjacent, K(+) and Na(+) configurations exist, ideally suited to these US-enhanced methods. Our studies demonstrate that the significant improvements in free energy calculations obtained using the proposed methods can have serious consequences for elucidating biological mechanisms and for the interpretation of experimental data.

The mechanism by which K(+) channels select for K(+) over Na(+) ions has been debated for the better part of a century. The prevailing view is that K(+) channels contain highly conserved sites that selectively bind K(+) over Na(+) ions through optimal coordination. We demonstrate that a series of alternating sites within the KcsA channel selectivity filter exists, which are thermodynamically selective for either K(+) (cage made from two planes of oxygen atoms) or Na(+) ions (a single plane of four oxygen atoms). By combining Bennett free energy perturbation calculations with umbrella sampling, we show that when K(+) and Na(+) are both permitted to move into their preferred positions, the thermodynamic preference for K(+) over Na(+) is significantly reduced throughout the entire selectivity filter. We offer a rationale for experimental measures of thermodynamic preference for K(+) over Na(+) from Ba(2+) blocking data, by demonstrating that the presence of Ba(2+) ions exaggerates K(+) over Na(+) thermodynamic stability due to the different binding locations of these ions. These studies reveal that K(+) channel selectivity may not be associated with the thermodynamics of ions in crystallographic K(+) binding sites, but requires consideration of the kinetic barriers associated with the different multi-ion permeation mechanisms.

Potassium channels allow K(+) ions to diffuse through their pores while preventing smaller Na(+) ions from permeating. Discrimination between these similar, abundant ions enables these proteins to control electrical and chemical activity in all organisms. Selection occurs at the narrow selectivity filter containing structurally identified K(+) binding sites. Selectivity is thought to arise because smaller ions such as Na(+) do not bind to these K(+) sites in a thermodynamically favorable way. Using the model K(+) channel KcsA, we examined how intracellular Na(+) and Li(+) interact with the pore and the permeant ions using electrophysiology, molecular dynamics simulations and X-ray crystallography. Our results suggest that these small cations have a separate binding site within the K(+) selectivity filter. We propose that selective permeation from the intracellular side primarily results from a large energy barrier blocking filter entry for Na(+) and Li(+) in the presence of K(+), not from a difference of binding affinity between ions.

Potassium channels allow K+ ions to diffuse through their pores while preventing smaller Na+ ions from permeating. Discrimination between these similar, abundant ions enables these proteins to control electrical and chemical activity in all organisms. Selection occurs at the narrow selectivity filter containing structurally identified K+ binding sites. Selectivity is thought to arise because smaller ions such as Na+ do not bind to these K+ sites in a thermodynamically favorable way. Using the model K+ channel KcsA, we examined how intracellular Na+ and Li+ interact with the pore and the permeant ions using electrophysiology, molecular dynamics simulations and X-ray crystallography. Our results suggest that these small cations have a separate binding site within the K+ selectivity filter. We propose that selective permeation from the intracellular side primarily results from a large energy barrier blocking filter entry for Na+ and Li+ in the presence of K+, not from a difference of binding affinity between ions. [ABSTRACT FROM AUTHOR]