Dynamics and dissipation in enzyme catalysis

Abstract

We use quantized molecular dynamics simulations to characterize the role of enzyme vibrations in facilitating dihydrofolate reductase hydride transfer. By sampling the full ensemble of reactive trajectories, we are able to quantify and distinguish between statistical and dynamical correlations in the enzyme motion. We demonstrate the existence of nonequilibrium dynamical coupling between protein residues and the hydride tunneling reaction, and we characterize the spatial and temporal extent of these dynamical effects. Unlike statistical correlations, which give rise to nanometer-scale coupling between distal protein residues and the intrinsic reaction, dynamical correlations vanish at distances beyond 4-6 Å from the transferring hydride. This work finds a minimal role for nonlocal vibrational dynamics in enzyme catalysis, and it supports a model in which nanometer-scale protein fluctuations statistically modulate--or gate--the barrier for the intrinsic reaction.

Fig. 1.

The hydride transfer reaction catalyzed by DHFR. (A) The active site with the hydride (green) shown in the ring-polymer representation of the quantized MD and the donor and acceptor C atoms in purple. (B) The quantized free-energy profile for the reaction. (C) The time-dependent transmission coefficient corresponding to the dividing surface at λ(x) = -4.8 kcal/mol.

Fig. 2.

Statistical and dynamical correlations among enzyme motions during the intrinsic reaction. (A) (Upper triangles) The covariance c_ij among position fluctuations in DHFR, plotted for the reactant, dividing surface, and product regions. Protein residues are indexed according to [Protein Data Bank (PDB) 1RX2]; substrate and cofactor regions are indicated by the hydride acceptor A and donor D atoms, respectively. (Lower triangles) The difference with respect to the plot for the reactant basin. (B–D) The dynamical correlation measure d_ij(t) for (B) the donor and acceptor atom pair, (C) the substrate-based C₇ and acceptor atom pair, and (D) the cofactor-based C_N3 and donor atom pair. Results for additional atom pairs are presented in Fig. S5. (E) (Upper triangle) The integrated dynamical correlation measure d_ij, indexed as in (A). Significant dynamical correlations appear primarily in the substrate and cofactor regions, which are enlarged in the lower triangle.

Fig. 3.

The dynamical correlation measure , plotted for (A) the donor atom, (B) the acceptor atom, and (C) the side-chain O atom in the Y100 residue of the active site. (D) The size and color of atoms in the active-site region are scaled according to the integrated dynamical correlation measure, f_i. (E) (Main panel) The integrated dynamical correlation measure, f_i, as a function of the distance of atom i from the midpoint of the donor and acceptor atoms. (Inset) The statistical correlation measure, , is similarly presented. Atoms corresponding to the protein side chains, the protein backbone, and the substrate/cofactor regions are indicated by color. Values presented in part A are in units of nm/ps, and values in parts D and E are normalized to a maximum of unity. The estimated error in part E is smaller than the dot size.

Fig. 4.

Minimum free-energy pathways (s, white) and the mean pathway of the reactive trajectories (σ, magenta) overlay two-dimensional projections of the free-energy landscape, F(λ,Θ_α). (A) F(λ,Θ₁), where Θ₁ is the distance between the hydride donor and acceptor atoms. (B) F(λ,Θ₂), where Θ₂ is the distance between side-chain atoms I14 C_δ and Y100 O in the active-site residues. The dots in the magenta curves indicate 5 fs increments in time. Nonzero slope in s and σ indicates statistical and dynamical correlations, respectively.