Estimating Free Energy Surfaces and their Convergence from multiple, independent static and history-dependent biased molecular-dynamics simulations with Mean Force Integration.

Addressing the sampling problem is central to obtaining quantitative insight from molecular dynamics simulations. Adaptive biased sampling methods, such as metadynamics, tackle this issue by perturbing the Hamiltonian of a system with a history-dependent bias potential, enhancing the exploration of the ensemble of configurations and estimating the corresponding free energy surface (FES). Nevertheless, efficiently assessing and systematically improving their convergence remains an open problem. Here, building on Mean Force Integration (MFI), we develop and test a metric for estimating the convergence of free energy surfaces obtained by combining asynchronous, independent simulations subject to diverse biasing protocols, including static biases, different variants of metadynamics, and various combinations of static and history-dependent biases. The developed metric and the ability to combine independent simulations granted by MFI enable us to devise strategies to systematically improve the quality of FES estimates. We demonstrate our approach by computing FES and assessing the convergence of a range of systems of increasing complexity, including one- and two-dimensional analytical free energy surfaces, alanine dipeptide, a Lennard-Jones supersaturated vapour undergoing liquid droplet nucleation, and the model of a colloidal system crystallizing via a two-step mechanism. The methods presented here can be generally applied to biased simulations and are implemented in pyMFI, a publicly accessible open-source Python library.