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Mapping the conformational dynamics of proteins is crucial for elucidating their functional mechanisms. While Molecular Dynamics (MD) simulation enables detailed time evolution of protein motion, its computational toll hinders its use in practice. To address this challenge, multiple deep learning models for reproducing and accelerating MD have been proposed drawing on transport-based generative methods. However, existing work focuses on generation through transport of samples from prior distributions, that can often be distant from the data manifold. The recently proposed framework of stochastic interpolants, instead, enables transport between arbitrary distribution endpoints. Building upon this work, we introduce EquiJump, a transferable SO(3)-equivariant model that bridges all-atom protein dynamics simulation time steps directly. Our approach unifies diverse sampling methods and is benchmarked against existing models on trajectory data of fast folding proteins. EquiJump achieves state-of-the-art results on dynamics simulation with a transferable model on all of the fast folding proteins.

Enzymes are cellular protein machines using a variety of conformational changes to power fast biochemical catalysis. Our goal is to exploit the single-spin properties of the luminescent NV (nitrogen-vacancy) center in nanodiamonds to reveal the dynamics of an active enzyme complex at physiological conditions with the highest spatio-temporal resolution. Specifically attached to the membrane enzyme FoF1-ATP synthase, the NV sensor will report the adenosine triphosphate (ATP)-driven full rotation of Fo motor subunits in ten consecutive 36{\deg} steps. Conformational dynamics are monitored using either a double electron-electron resonance scheme or NV- magnetometry with optical readout or using NV- relaxometry with a superparamagnetic nanoparticle as the second marker attached to the same enzyme. First, we show how all photophysical parameters like individual size, charge, brightness, spectral range of fluorescence and fluorescence lifetime can be determined for the NV- center in a single nanodiamond held in aqueous solution by a confocal anti-Brownian electrokinetic trap (ABEL trap). Stable photon count rates of individual nanodiamonds and the absence of blinking allow for observation times of single nanodiamonds in solution exceeding hundreds of seconds. For the proposed quantum sensing of nanometer-sized distance changes within an active enzyme, we show that local magnetic field fluctuations can be detected all-optically by analyzing fluorescence lifetime changes of the NV- center in each nanodiamond in solution., Comment: 14 pages, 5 figures

Protein dynamics form a critical bridge between protein structure and function, yet the impact of evolution on ultrafast processes inside proteins remains enigmatic. This study delves deep into nanosecond-scale protein dynamics of a structurally and functionally conserved protein across species separated by almost a billion years, investigating ten homologs in complex with their ligand. By inducing a photo-triggered destabilization of the ligand inside the binding pocket, we resolved distinct kinetic footprints for each homolog via transient infrared spectroscopy . Strikingly, we found a cascade of rearrangements within the protein complex which manifest in three discrete time points of dynamic activity, conserved over hundreds of millions of years within a narrow window. Among these processes, one displays a subtle temporal shift correlating with evolutionary divergence, suggesting reduced selective pressure in the past. Our study not only uncovers the impact of evolution on molecular processes in a specific case, but has also the potential to initiate a novel field of scientific inquiry within molecular paleontology, where species are compared and classified based on the rapid pace of protein dynamic processes; a field which connects the shortest conceivable time scale in living matter (10^-9 s) with the largest ones (10^16 s).

An increasingly common viewpoint is that protein dynamics data sets reside in a non-linear subspace of low conformational energy. Ideal data analysis tools for such data sets should therefore account for such non-linear geometry. The Riemannian geometry setting can be suitable for a variety of reasons. First, it comes with a rich structure to account for a wide range of geometries that can be modelled after an energy landscape. Second, many standard data analysis tools initially developed for data in Euclidean space can also be generalised to data on a Riemannian manifold. In the context of protein dynamics, a conceptual challenge comes from the lack of a suitable smooth manifold and the lack of guidelines for constructing a smooth Riemannian structure based on an energy landscape. In addition, computational feasibility in computing geodesics and related mappings poses a major challenge. This work considers these challenges. The first part of the paper develops a novel local approximation technique for computing geodesics and related mappings on Riemannian manifolds in a computationally feasible manner. The second part constructs a smooth manifold of point clouds modulo rigid body group actions and a Riemannian structure that is based on an energy landscape for protein conformations. The resulting Riemannian geometry is tested on several data analysis tasks relevant for protein dynamics data. It performs exceptionally well on coarse-grained molecular dynamics simulated data. In particular, the geodesics with given start- and end-points approximately recover corresponding molecular dynamics trajectories for proteins that undergo relatively ordered transitions with medium sized deformations. The Riemannian protein geometry also gives physically realistic summary statistics and retrieves the underlying dimension even for large-sized deformations within seconds on a laptop.

The paradigm that the primary amino acid sequence prescribes structure and thus function has for a long time been central to the understanding of protein science. Though the theory is supported by the behaviour of most structured proteins, it loses much of its applicability when discussing intrinsically disordered proteins (IDPs). These peculiar proteins, whose tertiary structure constantly interconverts between a series of energetically favourable conformations, are the root of many current, pressing scientific mechanisms. Many biological processes that are still yet to be elucidated--the mechanisms of of protein folding, ligand binding, and general protein dynamics--involve IDPs. Because most dynamic protein events are on such short time scales, using experimental methods to observe their action often times doesn't yield useful data. As well, the data resulting from scientific techniques developed for structured, "static proteins" must be presented in conjunction with data from methods tailored specifically to IDPs in order to have significance. A method that models IDPs with shocking accuracy is computer simulation, particularly Molecular Dynamics (MD) simulations. With computational power only recently increasing enough to encompass the timescale needed for protein dynamics, MD simulations are still fairly novel in their implementations. This paper will discuss and consolidate the current methods, problems, and solutions of using MD simulation to model IDPs. Which simulation parameters can be altered to more precisely describe observed biological behaviour? How can one accurately use MD simulation to answer questions that, when using experimental methods, have no answer? How can the data resulting from MD simulation be analyzed and quantified to support the conclusions being drawn?, Comment: Withdrawn because it was submitted without consent of the co-author

Increased ability to predict protein structures is moving research focus towards understanding protein dynamics. A promising approach is to represent protein dynamics through networks and take advantage of well-developed methods from network science. Most studies build protein dynamics networks from correlation measures, an approach that only works under very specific conditions, instead of the more robust inverse approach. Thus, we apply the inverse approach to the dynamics of protein dihedral angles, a system of internal coordinates, to avoid structural alignment. Using the well-characterized adhesion protein, FimH, we show that our method identifies networks that are physically interpretable, robust, and relevant to the allosteric pathway sites. We further use our approach to detect dynamical differences, despite structural similarity, for Siglec-8 in the immune system, and the SARS-CoV-2 spike protein. Our study demonstrates that using the inverse approach to extract a network from protein dynamics yields important biophysical insights., Comment: 5 figures

A phase field model which describes the formation of protein-RNA complexes subject to phase segregation is analyzed. A single protein, two RNA species, and two complexes are considered. Protein and RNA species are governed by coupled reaction-diffusion equations which also depend on the two complexes. The latter ones are driven by two Cahn-Hilliard equations with Flory-Huggins potential and reaction terms depending on the solution variables. The resulting nonlinear coupled system is equipped with no-flux boundary conditions and suitable initial conditions. The former ones entail some mass conservation constraints which are also due to the nature of the reaction terms. The existence of global weak solutions in a bounded (two- or) three-dimensional domain is established. In dimension two, some weighted-in-time regularity properties are shown. In particular, the complexes instantaneously get uniformly separated from the pure phases. Taking advantage of this result, a unique continuation property is proven. Among the many technical difficulties, the most significant one arises from the fact that the two complexes are initially nonexistent, so their initial conditions are zero i.e., they start from a pure phase. Thus we must solve, in particular, a system of two coupled Cahn-Hilliard equations with singular potential and nonlinear sources without the usual assumption on the initial datum, i.e., the initial phase cannot be pure. This novelty requires a new approach to estimate the chemical potential in a suitable $L^p(L^2)$-space with $p\in(1,2)$. This technique can be extended to other models like, for instance, the well-known Cahn-Hilliard-Oono equation., Comment: 56 pages

Prediction of protein-ligand binding affinity is a major goal in drug discovery. Generally, free energy gap is calculated between two states (e.g., ligand binding and unbinding). The energy gap implicitly includes the effects of changes in protein dynamics induced by the binding ligand. However, the relationship between protein dynamics and binding affinity remains unclear. Here, we propose a novel method that represents protein behavioral change upon ligand binding with a simple feature that can be used to predict protein-ligand affinity. From unbiased molecular simulation data, an unsupervised deep learning method measures the differences in protein dynamics at a ligand-binding site depending on the bound ligands. A dimension-reduction method extracts a dynamic feature that is strongly correlated to the binding affinities. Moreover, the residues that play important roles in protein-ligand interactions are specified based on their contribution to the differences. These results indicate the potential for dynamics-based drug discovery.