Enhanced Sampling in the Well-Tempered Ensemble

We introduce the well-tempered ensemble (WTE) which is the biased ensemble sampled by well-tempered metadynamics when the energy is used as collective variable. WTE can be designed so as to have approximately the same average energy as the canonical ensemble but much larger fluctuations. These two properties lead to an extremely fast exploration of phase space. An even greater efficiency is obtained when WTE is combined with parallel tempering. Unbiased Boltzmann averages are computed on the fly by a recently developed reweighting method [M. Bonomi et al., J. Comput. Chem. 30, 1615 (2009)]. We apply WTE and its parallel tempering variant to the $2d$ Ising model and to a Gō model of HIV protease, demonstrating in these two representative cases that convergence is accelerated by orders of magnitude.

Figure 1

WTE (black) compared to standard ensemble sampling (red) at two temperatures, below ($T_1=1.0$) and above ($T_2=5.0$) the critical temperature $T_c=2.269$. The unit on the $x$ axis is ${10}^3$ MC steps. Each MC move consists of a complete sweep of the $L=20$ site lattice. Gaussians of 0.1 height and 5.0 width were deposited at each step.

Figure 2

Speedup of PT-WTE compared to standard PT as a function of $\gamma$ in the Ising model with $L=10$ (left-hand panel) and $L=20$ (right-hand panel). Twenty-one replicas were distributed in a geometric progression in the interval $0.1\le T\le 10.0$ as in Ref. 31. Exchange moves were attempted after every lattice sweep. Gaussian parameters as in Fig. 1.

Figure 3

Left-hand panel: FES as a function of the magnetization $F(M)$ of the $L=10$ next neighbor ferromagnetic Ising model below and above the critical temperature, compared with an extensive PT calculation. The statistical error for the PT-WTE calculations is smaller than 2%. We have computed a similar curve for $L=20$, but we do not show it here because the PT calculation to compare with could not be converged. It is remarkable that both magnetizations could be explored in spite of a barrier of the order of $110k_{B}T$. Right-hand panels: Specific heat per spin (top), modulus of the magnetization (middle), and magnetic susceptibility (bottom) as a function of temperature ($L=20$). The continuos line in the top panel is the finite size exact solution [30]. In the middle and bottom panels the line is just a guide to the eye. The statistical error found is at worst 1% in all cases.

Figure 4

Left-hand panel: Gō-model FES convergence in PT-WTE run measured as the free-energy difference between folded and unfolded state as a function of time in units of ${10}^5$ MD steps. Sixteen replicas were distributed with a geometric progression in a temperature range between 0.625 and 1.25 in units of $T_f$. Exchanges between configurations were attempted every 200 MD steps. Gaussians of $1.0\mathrm{kJ}/\mathrm{mol}$ height and 5.0 width were deposited every 1000 steps. A $\gamma$ value of 80 was used for all replicas. Right-hand panels: Convergence of PT-WTE without exchanges (top) and of standard PT (bottom).

Figure 5

Gō-model FES of three $\beta$-strand subunits of HIV-1 protease as a function of the native contacts at $T=0.95$. The FES are obtained by reweighting the PT-WTE run (solid lines) and from an umbrella sampling calculation (points).