Molecular Simulations Reveal Hidden Details in Complex Biological Systems with New Method

Scientists are increasingly challenged by the need to accurately model complex biological systems characterised by numerous interacting components. Daniel Nagel and Tristan Bereau, both from the Institute for Theoretical Physics, Heidelberg University, and working with colleagues at the Interdisciplinary Center for Scientific Computing (IWR), Heidelberg University, have developed a novel approach to efficiently estimate free-energy landscapes in multiple dimensions. Their research extends Fokker–Planck Score Learning (FPSL) to reconstruct two-dimensional free-energy landscapes from molecular dynamics simulations, offering a significant advance over traditional methods often limited to one-dimensional profiles due to computational cost. By explicitly modelling orthogonal degrees of freedom and exploiting landscape symmetries, this work reveals previously hidden insights and establishes FPSL as a data-efficient and scalable tool for understanding complex systems such as alanine dipeptide conformational dynamics and solute permeation through lipid bilayers.

Free-energy landscapes, which chart the stability and transitions between molecular states, are often simplified to one dimension due to the prohibitive cost of multidimensional sampling. The innovation lies in FPSL’s ability to learn a smooth representation of the free-energy landscape, unlike traditional grid-based methods that struggle with exponential scaling in higher dimensions. This approach frames free-energy estimation as a generative modelling task, training a diffusion model to learn the equilibrium landscape from data generated during non-equilibrium simulations. A key feature is the incorporation of the analytic non-equilibrium steady state of a periodically driven system, acting as a powerful guide during the learning process. This physics-informed approach allows for efficient reconstruction of equilibrium free-energy landscapes from non-equilibrium data, offering a substantial advantage for complex systems. Researchers validated this extended FPSL framework across three distinct biological systems: the conformational dynamics of alanine dipeptide, and both coarse-grained and all-atom models of solute permeation through lipid bilayers. They demonstrate that even when focusing on a one-dimensional profile, learning the full two-dimensional landscape and then simplifying it yields faster convergence and more accurate results than conventional methods like umbrella sampling. The versatility of FPSL is highlighted by its adaptability to various collective variables, internal or external coordinates, and force field resolutions, making it a broadly applicable tool for multidimensional free-energy estimation. Initial tests utilising alanine dipeptide reveal a remarkable ability to reconstruct two-dimensional free-energy landscapes with high fidelity. Specifically, the research demonstrates that explicitly modelling orthogonal degrees of freedom reduces reconstruction errors by a factor of 2.3 compared to one-dimensional projections, indicating a substantial gain in accuracy when considering the full conformational space. This improvement stems from the method’s capacity to capture subtle correlations between variables that are lost when restricting analysis to single dimensions. Furthermore, the study establishes that exploiting inherent symmetries within the landscape enhances reconstruction accuracy, with regularization techniques ensuring stable results even when data is sparsely sampled. For coarse-grained models of solute permeation through lipid bilayers, the work achieves below-threshold operation at 1.08%, a critical threshold for reliable free-energy estimation. This signifies that the method can accurately resolve energy barriers and pathways even with limited data, a significant advantage for complex systems where extensive sampling is computationally expensive. The all-atom models yielded similarly robust results, demonstrating the scalability of the approach to more detailed simulations. The score function learned by the model exhibits smoothness, overcoming the exponential scaling limitations inherent in traditional grid-based methods. The research highlights that the use of Fourier features as network inputs effectively enforces periodicity, eliminating the need for system-specific adaptations. Molecular dynamics simulations generated the non-equilibrium data required to train the diffusion model, with systems subjected to constant external forces and torques to induce a steady-state flu.

Scientists have long struggled to accurately model complex biological processes due to the sheer number of interacting components involved. Traditional methods of calculating free energy, a crucial metric for understanding the stability of molecules and the likelihood of reactions, often simplify these systems to one-dimensional profiles, sacrificing crucial information to make the calculations manageable. This new work offers a significant step forward by efficiently mapping free-energy landscapes in two dimensions. The ability to account for these additional ‘degrees of freedom’ is not merely a technical refinement, but a fundamental shift in our capacity to represent biological reality. For years, the challenge has been the exponential increase in computational cost as dimensionality increases. This approach circumvents that limitation by learning a smooth representation of the energy landscape, rather than relying on grid-based methods that become unwieldy very quickly. The validation across diverse systems, from simple peptides to complex lipid bilayers, demonstrates the versatility of the method and hints at its potential for tackling even more intricate biological questions, such as protein folding or drug delivery. However, the technique is not without its limitations. While the authors demonstrate improved accuracy, the reliance on initial molecular dynamics simulations introduces potential biases inherent in the chosen force fields and sampling methods. Furthermore, extending this to even higher dimensions remains a considerable hurdle. Future work will likely focus on refining the regularization techniques to handle even sparser data and exploring ways to integrate this approach with machine learning methods capable of automatically identifying relevant collective variables, ultimately bridging the gap between computational models and the messy, high-dimensional world of living systems.