(225a) Time-Independent Free Energy Surfaces from Time-Dependent Biased Simulations Via Thermodynamic Integration

Free energy surfaces, i.e. projections of the free energy of a system in a low-dimensional set of collective variables, are often employed to convey quantitative information on the thermodynamics of molecular processes¹.

History-dependent biased sampling methods, such as metadynamics^2,3, provide a convenient way of adaptively exploring the configuration space of a molecular system. However, in order to efficiently leverage the sampling obtained from such protocols, time-independent estimates of free energy surfaces are often desirable^4,5.

Here, inspired by thermodynamic integration⁶, we propose a general method for the calculation of time-independent free energy surfaces from history-dependent biased simulations. In contrast with other methods, our proposed approach is not limited to specific sampling protocols, for instance we demonstrate its applicability on different variants of metadynamics including standard, Well Tempered, and Adaptive Gaussians.

In all cases the method displays faster convergence compared to conventional estimates, in which free energy surfaces are computed as a function of the total bias in the long-time limit. Moreover, our approach circumvents the need for computing the time-dependent constant c(t) as done with the weighted histogram analysis method (WHAM)⁷ or through the reweighing procedure proposed by Tiwary and Parrinello⁵.

Remarkably, the proposed method extends naturally to aggregate information obtained from independent simulations carried out with different biasing protocols in a single estimate of the free energy surface, allowing us to efficiently compute free energy surfaces from swarms of independent metadynamics runs.We note that, in order to achieve convergence with our method, while sampling barrier crossing events is necessary, it is not compulsory to observe re-crossing events in a single continuous trajectory. This feature allows to efficiently estimate and refine free energy surfaces with short, independent calculations.

In this work we validate our approach against one and two-dimensional model potentials, and by comparing the conformational free energy surface of ibuprofen in the solid state⁸ computed from a single, long well-tempered metadynamics simulation and from a swarm of short, independent metadynamics runs.

Finally, we discuss the application of our proposed method to efficiently estimate the reversible work necessary to form a kink site on a series of ibuprofen crystal surfaces in different solvents.

References

1. Chipot, C., & Pohorille, A. (2007). Free energy calculations. Springer-Verlag Berlin Heidelberg.
2. Laio, A. and M. Parrinello. Proceedings of the National Academy of Sciences 99.20 (2002): 12562-12566.
3. Barducci, A., G. Bussi, and M. Parrinello. Physical review letters 100.2 (2008): 020603.
4. Bonomi, M., A. Barducci, and M. Parrinello. Journal of computational chemistry 30.11 (2009): 1615-1621.
5. Tiwary, P., and M. Parrinello. The Journal of Physical Chemistry B 119.3 (2014): 736-742.
6. Kästner, Johannes, and Walter Thiel. The Journal of chemical physics 123.14 (2005): 144104.
7. Kumar, S., Rosenberg, J. M., Bouzida, D., Swendsen, R. H., & Kollman, P. A. (1992). Journal of computational chemistry, 13(8), 1011-1021.
8. Marinova, V., G. Wood, I. Marziano, M. Salvalaglio, Journal of chemical theory and computation 14.12 (2018): 6484-6494.