Diffusive reaction dynamics on invariant free energy profiles

Abstract

A fundamental problem in the analysis of protein folding and other complex reactions in which the entropy plays an important role is the determination of the activation free energy from experimental measurements or computer simulations. This article shows how to combine minimum-cut-based free energy profiles (F(C)), obtained from equilibrium molecular dynamics simulations, with conventional histogram-based free energy profiles (F(H)) to extract the coordinate-dependent diffusion coefficient on the F(C) (i.e., the method determines free energies and a diffusive preexponential factor along an appropriate reaction coordinate). The F(C), in contrast to the F(H), is shown to be invariant with respect to arbitrary transformations of the reaction coordinate, which makes possible partition of configuration space into basins in an invariant way. A "natural coordinate," for which F(H) and F(C) differ by a multiplicative constant (constant diffusion coefficient), is introduced. The approach is illustrated by a model one-dimensional system, the alanine dipeptide, and the folding reaction of a double beta-hairpin miniprotein. It is shown how the results can be used to test whether the putative reaction coordinate is a good reaction coordinate.

Fig. 1.

Model PES (= FES) U(x) = −cos(x) (solid line) together with reconstructed F_C and F_H. The distance between F_C and F_H is equal to kT ln($\sqrt{Ddt/\pi }$) = 0.25ln(0.01/2/π) ≈ 1.61 (see below). (A) Bin size of 0.01. (B) Bin size of 0.001 (see text).

Fig. 2.

Transformations of the FEPs. (A) Model PES and reconstructed FEPs along the coordinate y(x) = x + sin(4x)/4. The lines are (top to bottom) −kT ln(D(y)), where D(y) is the diffusion constant as function of y, U(x(y)), F_C, and F_H. (B) F_C (and F_H) from A transformed “back” to natural coordinate z.

Fig. 3.

FEPs for alanine dipeptide. (A) F_C along the φ dihedral angle for various quench time intervals dt (given in MD steps). (B) F_C and F_H along the φ and ψ dihedral angles.

Fig. 4.

F_C/kT = −ln(Z_C) for dt = 1, 2 for Beta3s. (A) rmsd clustering with a 2.5-Å cutoff radius shows dt^−1/2 behavior. (B) Secondary structure clustering shows dt⁻¹ behavior. For brevity we show just the part of the FEP 0 < Z_A/Z < 0.6. The native state occupies the region 0 < Z_A/Z < 0.35, followed by the denatured state, which includes several enthalpic basins.