Unified Approach to Enhanced Sampling

The sampling problem lies at the heart of atomistic simulations and over the years many different enhanced sampling methods have been suggested toward its solution. These methods are often grouped into two broad families. On the one hand, are methods such as umbrella sampling and metadynamics that build a bias potential based on few order parameters or collective variables. On the other hand, are tempering methods such as replica exchange that combine different thermodynamic ensembles in one single expanded ensemble. We instead adopt a unifying perspective, focusing on the target probability distribution sampled by the different methods. This allows us to introduce a new class of collective-variables-based bias potentials that can be used to sample any of the expanded ensembles normally sampled via replica exchange. We also provide a practical implementation by properly adapting the iterative scheme of the recently developed on-the-fly probability enhanced sampling method [M. Invernizzi and M. Parrinello, J. Phys. Chem. Lett. 11, 2731 (2020)], which was originally introduced for metadynamicslike sampling. The resulting method is very general and can be used to achieve different types of enhanced sampling. It is also reliable and simple to use, since it presents only few and robust external parameters and has a straightforward reweighting scheme. Furthermore, it can be used with any number of parallel replicas. We show the versatility of our approach with applications to multicanonical and multithermal-multibaric simulations, thermodynamic integration, umbrella sampling, and combinations thereof.

Popular Summary

Atomistic simulations are playing an ever-increasing role in many areas of physics, from studying new materials to illuminating the inner workings of biological systems. However, many interesting physical phenomena, such as crystallization and protein folding, occur on timescales that are out of reach even for the most powerful supercomputers. To mitigate this issue, many advanced simulation techniques have been developed over the years. Here, we propose a novel perspective that unifies many of the previous methods, leading to new and effective simulation protocols that extend the current capabilities of atomistic simulations.

The most popular enhanced sampling methods are often grouped into two families: expanded ensemble methods, which combine multiple thermodynamic conditions, and collective variables methods, which rely on the identification of an order parameter. We show that it is possible to perform expanded ensemble sampling using a collective-variable method, and we develop a general method that allows us to sample various kinds of generalized ensembles.

Our method greatly simplifies—conceptually and practically—the world of enhanced sampling, leading to more robust and reliable computations. It also creates new possibilities, by combining the sampling strategies of two families. In one of the most striking applications, our method can calculate in one single simulation an entire region of the temperature-pressure phase diagram in the presence of a first-order phase transition.

Figure 1

Alanine dipeptide in the multicanonical ensemble ($T_{\min }=300\mathrm{K}$, $T_{\max }=1000\mathrm{K}$). (a) Explored configurations as a function of the dihedral angles. Sampled points are colored according to their reweighting weight at $T=T_0=300\mathrm{K}$, $w_k(\beta )=e^{-(\beta -{\beta }_0)U_k+v_{k-1}^{(k)}}$. Notice how all the points in the transition state, close to $\varphi =0$, have extremely low probability of being sampled in an isothermal simulation at 300 K. (b) Free-energy difference between the two basins $\mathrm{\Delta }{F}_{AB}$ as a function of temperature. (c) Reweighted free-energy surface at two different temperatures.

Figure 2

Multithermal-multibaric simulation of chignolin. (a) Sampled target distribution in the CV space of potential energy and volume. (b) Relative effective sample size at different temperatures and pressures. The bottom corner of high temperatures and low pressures is excluded from the target to avoid vaporization of the system. (c) Time evolution of the two biased CVs and of the $\mathrm{C}\alpha$ RMSD for one of the 40 walkers. A RMSD threshold between folded and unfolded is highlighted with a dashed line.

Figure 3

The fraction folded of chignolin estimated from the multithermal-multibaric simulation. The inset shows the same quantity over a smaller range of temperatures at 1 bar (highlighted with a gray box), in order to compare it with the reference results from Lindorff-Larsen et al. [62].

Figure 4

Calculation of the free energy of liquid TIP4P water using a single-simulation thermodynamic integration. (a) Evolution of the collective variable $\mathrm{\Delta }U$ as a function of simulation time. (b) Integrand for the thermodynamic integration obtained through reweighting.

Figure 5

Double-well model in the multiumbrella ensemble. (a) Potential energy of the double-well 2D model system, and its free energy along the $x$ coordinate. (b) Multiumbrella target distribution, for different values of umbrella width $\sigma$. The black dotted line is the unbiased probability distribution $P_0(x)$.

Figure 6

Configurations of sodium sampled during the multithermal-multibaric-multiumbrella simulation. (a) The points are colored accordingly to the value of the bias $v(U,V,s)$. (b) The points are shown in the energy-volume space and colored accordingly to the value of the crystallinity CV $s$. As reference we also show in gray the region sampled during an unbiased simulation in the bcc phase.

Figure 7

Phase equilibrium of liquid and solid (bcc) sodium using a combination of thermodynamic and order parameter expansions. (a) Free-energy difference between the phases, $\mathrm{\Delta }{F}_{\mathrm{liq}\to \mathrm{bcc}}$, at different thermodynamic conditions. The coexistence line is shown as a gray dashed line. (b) Free-energy surfaces as a function of the crystallinity CV, for three representative thermodynamic conditions. Error bars are smaller than the linewidth.