Statistically unbiased free energy estimates from biased simulations

ABSTRACT

Estimating the free energy in molecular simulation requires, implicitly or explicitly, counting how many times the system is observed in a finite region. If the simulation is biased by an external potential, the weight of the configurations within the region can vary significantly, and this can make the estimate numerically unstable. We introduce an approach to estimate the free energy as a simultaneous function of several collective variables starting from data generated in a statically biased simulation. The approach exploits the property of a free energy estimator recently introduced by us, which provides by construction of the estimate in a region of infinitely small size. We show that this property allows removing the effect of the external bias in a simple and rigorous manner. The approach is validated on model systems for which the free energy is known analytically and on a small peptide for which the ground truth free energy is estimated in an independent unbiased run. In both cases the free energy obtained with our approach is an unbiased estimator of the ground-truth free energy, with an error whose magnitude is also predicted by the model.

GRAPHICAL ABSTRACT

Keywords:

• Free energy landscapes
• reweighting
• umbrella sampling
• unbiased estimators

Acknowledgements

We thank Alex Rodriguez and Aldo Glielmo for several important suggestions and fruitful discussions.

Disclosure statement

No potential conflict of interest was reported by the authors.

Notes

1 Notice the PAk approach is valid also in the case of no explicit dimensional reduction $\mathit{\sigma }(\mathbf{x}):=\mathbf{x}$, i.e. when $\mathrm{\Sigma }\equiv {\mathbb{R}}^N$. In this case the free energy $F_i$ is equivalent to the potential energy of the configuration ${\mathbf{x}}_i$ entering the canonical Boltzmann factor. Even in this case, however, it is still appropriate to refer to this quantity as ”free energy”, since $\rho (\mathit{\sigma })$ is estimated in a space of dimension $d\le D$.