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

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.

Microscopic mechanisms that govern the titration response and p K a values of buried residues in staphylococcal nuclease mutants

“…S4, S5, S11) and can be coupled with local structural fluctuations. The diverse timescales of these events highlight the sampling challenge associated with studying the titration of buried residues and calls for continuous development of robust enhanced sampling techniques …”

“…Depending on the identity of the titratable group and its location, the structural perturbations during titration occur either locally on adjacent secondary structure(s) or globally, with timescales ranging from hundreds of nanoseconds to milliseconds . The diverse set of behaviors observed experimentally inspired numerous computational studies and development of novel enhanced sampling techniques . Depending on the degree of sampling, the computations, especially those based on explicit solvent simulations, met with mixed levels of successes; with progress in methodologies such as constant pH simulations coupled with pH replica exchange, encouraging agreement between computed and measured p K a values has been reported …”

Adaptive Landscape Flattening Accelerates Sampling of Alchemical Space in Multisite λ Dynamics

“…While frequently utilized in computer aided drug design, 12–14 such techniques can also be applied to a wide variety of problems including pK a predictions, 15,16 peptide binding, 17,18 and solvation free energies 19 (Figure 1). Within most free energy methods, λ is a reaction coordinate that tunes the potential between two systems of interest.…”

Isotropic periodic sum for multipole interactions and a vector relation for calculation of the Cartesian multipole tensor

“…31,32 IPS is more convenient than PME in decomposing pairwise interactions, which is essential for some new simulation methods, such as the virtual mixture of multiple states (VMMS's) method. 33 It has been demonstrated that the free energies calculated with the IPS method are consistent with that calculated with PME. 33,34 While the IPS potentials of many interaction forms have been derived, multipole interactions have not yet been explored.…”

“…33 It has been demonstrated that the free energies calculated with the IPS method are consistent with that calculated with PME. 33,34 While the IPS potentials of many interaction forms have been derived, multipole interactions have not yet been explored. It is recognized that multipoles have their advantages in describing molecular interactions.…”