Research Article: A Virtual Mixture Approach to the Study of Multistate Equilibrium: Application to Constant pH Simulation in Explicit Water

Date Published: October 27, 2015

Publisher: Public Library of Science

Author(s): Xiongwu Wu, Bernard R. Brooks, Qiang Cui

Abstract: Chemical and thermodynamic equilibrium of multiple states is a fundamental phenomenon in biology systems and has been the focus of many experimental and computational studies. This work presents a simulation method to directly study the equilibrium of multiple states. This method constructs a virtual mixture of multiple states (VMMS) to sample the conformational space of all chemical states simultaneously. The VMMS system consists of multiple subsystems, one for each state. The subsystem contains a solute and a solvent environment. The solute molecules in all subsystems share the same conformation but have their own solvent environments. Transition between states is implicated by the change of their molar fractions. Simulation of a VMMS system allows efficient calculation of relative free energies of all states, which in turn determine their equilibrium molar fractions. For systems with a large number of state transition sites, an implicit site approximation is introduced to minimize the cost of simulation. A direct application of the VMMS method is for constant pH simulation to study protonation equilibrium. Applying the VMMS method to a heptapeptide of 3 ionizable residues, we calculated the pKas of those residues both with all explicit states and with implicit sites and obtained consistent results. For mouse epidermal growth factor of 9 ionizable groups, our VMMS simulations with implicit sites produced pKas of all 9 ionizable groups and the results agree qualitatively with NMR measurement. This example demonstrates the VMMS method can be applied to systems of a large number of ionizable groups and the computational cost scales linearly with the number of ionizable groups. For one of the most challenging systems in constant pH calculation, SNase Δ+PHS/V66K, our VMMS simulation shows that it is the state-dependent water penetration that causes the large deviation in lysine66’s pKa.

Partial Text: Chemical and thermodynamic equilibrium of multiple states is a fundamental phenomenon in biological systems. Typical examples of multiple state equilibria are protonation equilibrium, ligand binding equilibrium, and phosphorylation equilibrium. For example, through the equilibrium of different protonation states, protein can fold or unfold in different pH environments. By changing phosphorylation states, protein can activate or deactivate certain functions. Ligand binding equilibrium controls the inhibition of enzyme activities.

The VMMS simulation method uses multiple techniques to achieve efficient simulation. Through several examples, we validate the techniques and demonstrate the application of the VMMS method in the study of protonation equilibrium.

This work presents the VMMS simulation method for directly simulating the equilibrium of multiple chemical states. Through including explicit solvent environments of all states, this method avoids the time-consuming solvent reorientation between states, allowing free energy differences between different states to be efficiently estimated during simulation. Using model systems, we examined the conformational distribution of the VMMS ensemble and validated the reweighting formula from the VMMS distributions to pure state distributions.