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The geometry of a graph is encoded in dynamical processes on the graph. Many graph neural network (GNN) architectures are inspired by such dynamical systems, typically based on the graph Laplacian. Here, we introduce Dirac--Bianconi GNNs (DBGNNs), which are based on the topological Dirac equation recently proposed by Bianconi. Based on the graph Laplacian, we demonstrate that DBGNNs explore the geometry of the graph in a fundamentally different way than conventional message passing neural networks (MPNNs). While regular MPNNs propagate features diffusively, analogous to the heat equation, DBGNNs allow for coherent long-range propagation. Experimental results showcase the superior performance of DBGNNs over existing conventional MPNNs for long-range predictions of power grid stability and peptide properties. This study highlights the effectiveness of DBGNNs in capturing intricate graph dynamics, providing notable advancements in GNN architectures., Comment: 14 pages, 7 figures

Computationally generating novel synthetically accessible compounds with high affinity and low toxicity is a great challenge in drug design. Machine-learning models beyond conventional pharmacophoric methods have shown promise in generating novel small molecule compounds, but require significant tuning for a specific protein target. Here, we introduce a method called selective iterative latent variable refinement (SILVR) for conditioning an existing diffusion-based equivariant generative model without retraining. The model allows the generation of new molecules that fit into a binding site of a protein based on fragment hits. We use the SARS-CoV-2 Main protease fragments from Diamond X-Chem that form part of the COVID Moonshot project as a reference dataset for conditioning the molecule generation. The SILVR rate controls the extent of conditioning and we show that moderate SILVR rates make it possible to generate new molecules of similar shape to the original fragments, meaning that the new molecules fit the binding site without knowledge of the protein. We can also merge up to 3 fragments into a new molecule without affecting the quality of molecules generated by the underlying generative model. Our method is generalizable to any protein target with known fragments and any diffusion-based model for molecule generation., Comment: paper, 20 paper, 11 figures

The mechanisms by which a protein's 3D structure can be determined based on its amino acid sequence have long been one of the key mysteries of biophysics. Often simplistic models, such as those derived from geometric constraints, capture bulk real-world 3D protein-protein properties well. One approach is using protein contact maps to better understand proteins' properties. Here, we investigate the emergent behaviour of contact maps for different geometrically constrained models and real-world protein systems. We derive an analytical approximation for the distribution of model amino acid distances, $s$, by means of a mean-field approach. This approximation is then validated for simulations using a 2D and 3D random interaction model, as well as from contact maps of real-world protein data. Using data from the RCSB Protein Data Bank (PDB) and AlphaFold~2 database, the analytical approximation is fitted to protein chain lengths of $L\approx100$, $L\approx200$, and $L\approx300$. While a universal scaling behaviour for protein chains of different lengths could not be deduced, we present evidence that the amino acid distance distributions can be attributed to geometric constraints of protein chains in bulk and amino acid sequences only play a secondary role., Comment: 15 pages, 4 figures

Free energy calculations are rapidly becoming indispensable in structure-enabled drug discovery programs. As new methods, force fields, and implementations are developed, assessing their expected accuracy on real-world systems (benchmarking) becomes critical to provide users with an assessment of the accuracy expected when these methods are applied within their domain of applicability, and developers with a way to assess the expected impact of new methodologies. These assessments require construction of a benchmark - a set of well-prepared, high quality systems with corresponding experimental measurements designed to ensure the resulting calculations provide a realistic assessment of expected performance when these methods are deployed within their domains of applicability. To date, the community has not yet adopted a common standardized benchmark, and existing benchmark reports suffer from a myriad of issues, including poor data quality, limited statistical power, and statistically deficient analyses, all of which can conspire to produce benchmarks that are poorly predictive of real-world performance. Here, we address these issues by presenting guidelines for (1) curating experimental data to develop meaningful benchmark sets, (2) preparing benchmark inputs according to best practices to facilitate widespread adoption, and (3) analysis of the resulting predictions to enable statistically meaningful comparisons among methods and force fields.

Alchemical free energy calculations are a useful tool for predicting free energy differences associated with the transfer of molecules from one environment to another. The hallmark of these methods is the use of "bridging" potential energy functions representing \emph{alchemical} intermediate states that cannot exist as real chemical species. The data collected from these bridging alchemical thermodynamic states allows the efficient computation of transfer free energies (or differences in transfer free energies) with orders of magnitude less simulation time than simulating the transfer process directly. While these methods are highly flexible, care must be taken in avoiding common pitfalls to ensure that computed free energy differences can be robust and reproducible for the chosen force field, and that appropriate corrections are included to permit direct comparison with experimental data. In this paper, we review current best practices for several popular application domains of alchemical free energy calculations, including relative and absolute small molecule binding free energy calculations to biomolecular targets., Comment: 48 pages, 14 figures

The intricate three-dimensional geometries of protein tertiary structures underlie protein function and emerge through a folding process from one-dimensional chains of amino acids. The exact spatial sequence and configuration of amino acids, the biochemical environment and the temporal sequence of distinct interactions yield a complex folding process that cannot yet be easily tracked for all proteins. To gain qualitative insights into the fundamental mechanisms behind the folding dynamics and generic features of the folded structure, we propose a simple model of structure formation that takes into account only fundamental geometric constraints and otherwise assumes randomly paired connections. We find that despite its simplicity, the model results in a network ensemble consistent with key overall features of the ensemble of Protein Residue Networks we obtained from more than 1000 biological protein geometries as available through the Protein Data Base. Specifically, the distribution of the number of interaction neighbors a unit (amino acid) has, the scaling of the structure's spatial extent with chain length, the eigenvalue spectrum and the scaling of the smallest relaxation time with chain length are all consistent between model and real proteins. These results indicate that geometric constraints alone may already account for a number of generic features of protein tertiary structures.

We propose a discrete transition-based reweighting analysis method (dTRAM) for analyzing configuration-space-discretized simulation trajectories produced at different thermodynamic states (temperatures, Hamiltonians, etc.) dTRAM provides maximum-likelihood estimates of stationary quantities (probabilities, free energies, expectation values) at any thermodynamic state. In contrast to the weighted histogram analysis method (WHAM), dTRAM does not require data to be sampled from global equilibrium, and can thus produce superior estimates for enhanced sampling data such as parallel/simulated tempering, replica exchange, umbrella sampling, or metadynamics. In addition, dTRAM provides optimal estimates of Markov state models (MSMs) from the discretized state-space trajectories at all thermodynamic states. Under suitable conditions, these MSMs can be used to calculate kinetic quantities (e.g. rates, timescales). In the limit of a single thermodynamic state, dTRAM estimates a maximum likelihood reversible MSM, while in the limit of uncorrelated sampling data, dTRAM is identical to WHAM. dTRAM is thus a generalization to both estimators.

Computing the equilibrium properties of complex systems, such as free energy differences, is often hampered by rare events in the dynamics. Enhanced sampling methods may be used in order to speed up sampling by, for example, using high temperatures, as in parallel tempering, or simulating with a biasing potential such as in the case of umbrella sampling. The equilibrium properties of the thermodynamic state of interest (e.g., lowest temperature or unbiased potential) can be computed using reweighting estimators such as the weighted histogram analysis method or the multistate Bennett acceptance ratio (MBAR). weighted histogram analysis method and MBAR produce unbiased estimates, the simulation samples from the global equilibria at their respective thermodynamic state--a requirement that can be prohibitively expensive for some simulations such as a large parallel tempering ensemble of an explicitly solvated biomolecule. Here, we introduce the transition-based reweighting analysis method (TRAM)--a class of estimators that exploit ideas from Markov modeling and only require the simulation data to be in local equilibrium within subsets of the configuration space. We formulate the expanded TRAM (xTRAM) estimator that is shown to be asymptotically unbiased and a generalization of MBAR. Using four exemplary systems of varying complexity, we demonstrate the improved convergence (ranging from a twofold improvement to several orders of magnitude) of xTRAM in comparison to a direct counting estimator and MBAR, with respect to the invested simulation effort. Lastly, we introduce a random-swapping simulation protocol that can be used with xTRAM, gaining orders-of-magnitude advantages over simulation protocols that require the constraint of sampling from a global equilibrium., Comment: 23 pages with appendices, 5 figures

We explore the dynamical large-deviations of a lattice heteropolymer model of a protein by means of path sampling of trajectories. We uncover the existence of non-equilibrium dynamical phase-transitions in ensembles of trajectories between active and inactive dynamical phases, whose nature depends on properties of the interaction potential. When the full heterogeneity of interactions due to the amino-acid sequence is preserved, as in a fully interacting model or in a heterogeneous version of the G\={o} model where only native interactions are considered, the transition is between the equilibrium native state and a highly native but kinetically trapped state. In contrast, for the homogeneous G\={o} model, where there is a single native energy and the sequence plays no role, the dynamical transition is a direct consequence of the static bi-stability between unfolded and native states. In the heterogeneous case the native-active and native-inactive states, despite their static similarity, have widely varying dynamical properties, and the transition between them occurs even in lattice proteins whose sequences are designed to make them optimal folders., Comment: 9 pages, 9 figures, paper + supplementary material

We present a numerical study of the dynamics of the one-dimensional Ising model by applying the large-deviation method to describe ensembles of dynamical trajectories. In this approach trajectories are classified according to a dynamical order parameter and the structure of ensembles of trajectories can be understood from the properties of large-deviation functions, which play the role of dynamical free-energies. We consider both Glauber and Kawasaki dynamics, and also the presence of a magnetic field. For Glauber dynamics in the absence of a field we confirm the analytic predictions of Jack and Sollich about the existence of critical dynamical, or space-time, phase transitions at critical values of the "counting" field $s$. In the presence of a magnetic field the dynamical phase diagram also displays first order transition surfaces. We discuss how these non-equilibrium transitions in the 1$d$ Ising model relate to the equilibrium ones of the 2$d$ Ising model. For Kawasaki dynamics we find a much simple dynamical phase structure, with transitions reminiscent of those seen in kinetically constrained models., Comment: 23 pages, 10 figures