Computing free energies of protein conformations from explicit solvent simulations

Abstract

We report a fully general technique addressing a long standing challenge of calculating conformational free energy differences between various states of a polymer chain from simulations using explicit solvent force fields. The main feature of our method is a special mapping variable, a path coordinate, which continuously connects two conformations. The path variable has been designed to preserve locality in the phase space near the path endpoints. We avoid the problem of sampling the unfolded states by creating an artificial confinement “tube” in the phase space that prevents the molecule from unfolding without affecting the calculation of the desired free energy difference. We applied our technique to compute the free energy difference between two native-like conformations of the small protein Trp-cage using the CHARMM force field with explicit solvent. We verified this result by comparing it with an independent, significantly more expensive calculation. Overall, the present study suggests that the new method of computing free energy differences between polymer chain conformations is accurate and highly computationally efficient.

Computing Free Energy of a Large-Scale Allosteric Transition in Adenylate Kinase Using All Atom Explicit Solvent Simulations

“…Regarding the variable ξ (see Figure 3) defined in eq 1 as elevation above this square, it corresponds to a positive circular Gaussian "peak" placed on the basin of conformation B and a negative circular Gaussian "pit" placed on conformational basin A. 30 The ρ in the equation is a parameter which controls the resolution of ξ and has to be decided based on the specific aims of the problem (see Table 1). If its value is too large, the resolution coarsens and might not capture the subtle structural differences in the respective basins leading to more or less random free energy estimates.…”

“…30 The particular form of confinement potential employed in the present work is chosen to have the following functional form (see Figure 4) which also illustrates the idea of phase space "tube":…”

“…30 The latter is a critical requirement missing in many commonly used alternative approaches. Locality means that if for a conformation X the value of the variable ξ(X) is close to the value of the variable in the closed state ξ(closed), then X is structurally similar to the closed state.…”

“…The same statement is also true for the open state. 30 Another important ingredient of the technique is the confinement potential which alleviates the problem of sampling the degrees of freedom transverse to the umbrella sampling variable. Its effect on the calculation amounts to enveloping the dynamic trajectories of the system into a phase space "tube" without affecting the neighborhood of the two allosteric states.…”

Conformational change path between closed and open forms of C2 domain of coagulation factor V on a two-dimensional free-energy surface

“…The advantage of a two-dimensional FES is that one can easily visualize the approximate size of a basin. Once the basins are determined from the two-dimensional FES, the free-energy difference can be obtained by numerical integration for each basin using the partition functions [33],…”