The internal dynamics of biomolecules, and hence their function, is governed by the structure of their free-energy landscape. Early flash-photolysis experiments on myoglobin suggested that the free-energy landscapes of proteins are hierarchically structured, with a characteristic distribution of free-energy barriers which gives rise to anomalous diffusion. Analytical results have been derived for one-dimensional or high-dimensional hierarchical free-energy landscapes. Recent improvements in methods and computer performance enable generating sufficiently long molecular dynamics (MD) trajectories to extract dynamics information covering many orders of magnitude, such that the broad distributions of energy barriers of proteins become accessible to quantitative studies of intermediate dimensions. In this work, we present a nonequilibrium method to estimate barrier height distributions from microsecond-long MD simulations. It infers barrier height distributions from anomalous diffusion exponents derived from principal component analysis and by comparison to simple hierarchical lattice models. These models are $d$-dimensional lattices of states separated by free-energy barriers, the heights of which are distributed as $p\left(\mathrm{\Delta }G\right)=1/\gamma exp(-\mathrm{\Delta }G/\gamma )$. The parameter $\gamma$ quantifies the “ruggedness” of the free-energy landscape in such models. We show that both parameters, i.e., ruggedness and effective dimensionality $d$, can be inferred from anomalous diffusion exponents. Assuming a similar dependency of anomalous diffusion exponents on $\gamma$ and $d$ for proteins, we estimate the ruggedness of the free-energy landscapes of 500 small, single-domain globular proteins between 15 and 20 kT per dimension with an estimated accuracy of 4.2 kT and dimensionality between 40 and 60 with an estimated accuracy of 10 dimensions. Remarkably, neither effective dimensionality nor the ruggedness correlates with protein size and both ruggedness and effective dimensionality are much smaller than the scatter of protein sizes. From this finding, we conclude that these two properties of the free-energy landscape of a protein are rather adapted to the particular function of each single protein and that, quite generally, are functionally relevant for globular proteins.

• Received 18 August 2021
• Accepted 28 February 2022

DOI:https://doi.org/10.1103/PhysRevE.105.044404

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article's title, journal citation, and DOI. Open access publication funded by the Max Planck Society.

Published by the American Physical Society