Peptide folding kinetics from replica exchange molecular dynamics

We show how accurate kinetic information, such as the rates of protein folding and unfolding, can be extracted from replica-exchange molecular dynamics (REMD) simulations. From the brief and continuous trajectory segments between replica exchanges, we estimate short-time propagators in conformation space and use them to construct a master equation. For a helical peptide in explicit water, we determine the rates of transitions both locally between microscopic conformational states and globally for folding and unfolding. We show that accurate rates in the $\sim 1∕\left(100\mathrm{ns}\right)$ to $\sim 1∕\left(1\mathrm{ns}\right)$ range can be obtained from REMD with exchange times of $5\mathrm{ps}$, in excellent agreement with results from long equilibrium molecular dynamics.

Figure 1

(Color online) REMD simulations. (a) Schematic of transition-path based assignment of conformational states, shown for illustrative purposes in 1D (with the actual assignment done using both $\Phi$ and $\Psi$ [13]). The backbone dihedral angle $\Psi$ of alanine exhibits transitions between helical (blue, $\Psi \approx -50°$) and nonhelical states (green, $\Psi \approx 120°$). Conformations within narrow regions around the two free energy minima (gray) can be assigned as helical or coil with high confidence. For other conformations (black dots), the assigned state changes only if the trajectory crosses between the well-defined regions, but not on equilibrium excursions that revert without actual crossing. (b) State assignment corresponding to (a). (c) Temperatures sampled by a typical Ala5 replica during a $150\mathrm{ns}$ REMD simulation. (d) Conformational states sampled by the Ala5 replica during the same run.

Figure 2

(Color online) REMD equilibrium populations $P_{\mathrm{eq}}$ at 300 and $350\mathrm{K}$. Shading indicates the folded basin. (Inset) Scatter plot of $P_{\mathrm{eq}}$ from standard MD and REMD at $300\mathrm{K}$. Error bars indicate standard deviations of the mean.

Figure 3

(Color online) Relaxation times ${\tau }_2$ (circles, blue) and ${\tau }_3$ (diamonds, green) as a function of temperature. Open squares show ${\tau }_2$ and ${\tau }_3$ from standard MD at 300 and $350\mathrm{K}$ [13]. (Inset) Folding $\left(k_F\right)$ and unfolding $\left(k_U\right)$ rate constants as a function of $1∕T$, together with Arhenius fits.

Figure 4

(Color online) Validation of transition rates estimated from REMD trajectories. (a) Rates $k_{ij}$ from REMD versus those from standard MD [13] at $300\mathrm{K}$ (diamonds, blue) and $350\mathrm{K}$ (circles, red). (b) Dependence of the relaxation times ${\tau }_2$ (top, red), ${\tau }_3$ (middle, green), and ${\tau }_4$ (bottom, blue) on the lag time $\mathrm{\Delta }t$ at $300\mathrm{K}$. REMD results are shown as symbols connected by solid lines. Reference values from standard MD are shown as dashed lines with error bars. Results for $\mathrm{\Delta }t>\delta t_{\mathrm{REMD}}$ were obtained from continuous trajectory segments in which replica exchange attempts had been rejected.