Bayesian Uncertainty Quantification for Bond Energies and Mobilities Using Path Integral Analysis

Estimating Position-Dependent and Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories: Existing Methods and Future Outlook

Confinement can substantially alter the physicochemical properties of materials by breaking translational isotropy and rendering all physical properties position-dependent. Molecular dynamics (MD) simulations have proven instrumental in characterizing such spatial heterogeneities and probing the impact of confinement on materials' properties. For static properties, this is a straightforward task and can be achieved via simple spatial binning. Such an approach, however, cannot be readily applied to transport coefficients due to lack of natural extensions of autocorrelations used for their calculation in the bulk. The prime example of this challenge is diffusivity, which, in the bulk, can be readily estimated from the particles' mobility statistics, which satisfy the Fokker-Planck equation. Under confinement, however, such statistics will follow the Smoluchowski equation, which lacks a closed-form analytical solution. This brief review explores the rich history of estimating profiles of the diffusivity tensor from MD simulations and discusses various approximate methods and algorithms developed for this purpose. Besides discussing heuristic extensions of bulk methods, we overview more rigorous algorithms, including kernel-based methods, Bayesian approaches, and operator discretization techniques. Additionally, we outline methods based on applying biasing potentials or imposing constraints on tracer particles. Finally, we discuss approaches that estimate diffusivity from mean first passage time or committor probability profiles, a conceptual framework originally developed in the context of collective variable spaces describing rare events in computational chemistry and biology. In summary, this paper offers a concise survey of diverse approaches for estimating diffusivity from MD trajectories, highlighting challenges and opportunities in this area.

Robust Estimation of Position-Dependent Anisotropic Diffusivity Tensors from Molecular Dynamics Trajectories

Confinement breaks translational and rotational symmetry in materials and makes all physical properties functions of position. Such spatial variations are key to modulating material properties at the nanoscale, and characterizing them accurately is therefore an intense area of research in the molecular simulations community. This is relatively easy to accomplish for basic mechanical observables. Determining spatial profiles of transport properties, such as diffusivity, is, however, much more challenging, as it requires calculating position-dependent autocorrelations of mechanical observables. In our previous paper (Domingues, T.S.; Coifman, R.; Haji-Akbari, A. J. Phys. Chem. B 2023, 127, 5273 10.1021/acs.jpcb.3c00670), we analytically derive and numerically validate a set of filtered covariance estimators (FCEs) for quantifying spatial variations of the diffusivity tensor from stochastic trajectories. In this work, we adapt these estimators to extract diffusivity profiles from MD trajectories and validate them by applying them to a Lennard-Jones fluid within a slit pore. We find our MD-adapted estimator to exhibit the same qualitative features as its stochastic counterpart, as it accurately estimates the lateral diffusivity across the pore while systematically underestimating the normal diffusivity close to hard boundaries. We introduce a conceptually simple and numerically efficient correction scheme based on simulated annealing and diffusion maps to resolve the latter artifact and obtain normal diffusivity profiles that are consistent with the self-part of the van Hove correlation functions. Our findings demonstrate the potential of this MD-adapted estimator in accurately characterizing spatial variations of diffusivity in confined materials.

Robust Estimation of Position-Dependent Anisotropic Diffusivity Tensors from Stochastic Trajectories

Materials under confinement can possess properties that deviate considerably from their bulk counterparts. Indeed, confinement makes all physical properties position-dependent and possibly anisotropic, and characterizing such spatial variations and directionality has been an intense area of focus in experimental and computational studies of confined matter. While this task is fairly straightforward for simple mechanical observables, it is far more daunting for transport properties such as diffusivity that can only be estimated from autocorrelations of mechanical observables. For instance, there are well established methods for estimating diffusivity from experimentally observed or computationally generated trajectories in bulk systems. No rigorous generalizations of such methods, however, exist for confined systems. In this work, we present two filtered covariance estimators for computing anisotropic and position-dependent diffusivity tensors and validate them by applying them to stochastic trajectories generated according to known diffusivity profiles. These estimators can accurately capture spatial variations that span over several orders of magnitude and that assume different functional forms. Our kernel-based approach is also very robust to implementation details such as the localization function and time discretization and performs significantly better than estimators that are solely based on local covariance. Moreover, the kernel function does not have to be localized and can instead belong to a dictionary of orthogonal functions. Therefore, the proposed estimator can be readily used to obtain functional estimates of diffusivity rather than a tabulated collection of pointwise estimates. Nonetheless, the susceptibility of the proposed estimators to time discretization is higher at the immediate vicinity of hard boundaries. We demonstrate this heightened susceptibility to be common among all covariance-based estimators.

Xist nucleates local protein gradients to propagate silencing across the X chromosome

The lncRNA Xist forms ∼50 diffraction-limited foci to transcriptionally silence one X chromosome. How this small number of RNA foci and interacting proteins regulate a much larger number of X-linked genes is unknown. We show that Xist foci are locally confined, contain ∼2 RNA molecules, and nucleate supramolecular complexes (SMACs) that include many copies of the critical silencing protein SPEN. Aggregation and exchange of SMAC proteins generate local protein gradients that regulate broad, proximal chromatin regions. Partitioning of numerous SPEN molecules into SMACs is mediated by their intrinsically disordered regions and essential for transcriptional repression. Polycomb deposition via SMACs induces chromatin compaction and the increase in SMACs density around genes, which propagates silencing across the X chromosome. Our findings introduce a mechanism for functional nuclear compartmentalization whereby crowding of transcriptional and architectural regulators enables the silencing of many target genes by few RNA molecules.

The biophysics of bacterial infections: Adhesion events in the light of force spectroscopy

Bacterial infections are the most eminent public health challenge of the 21st century. The primary step leading to infection is bacterial adhesion to the surface of host cells or medical devices, which is mediated by a multitude of molecular interactions. At the interface of life sciences and physics, last years advances in atomic force microscopy (AFM)-based force spectroscopy techniques have made possible to measure the forces driving bacteria-cell and bacteria-materials interactions on a single molecule/cell basis (single molecule/cell force spectroscopy). Among the bacteria-(bio)materials surface interactions, the life-threatening infections associated to medical devices involving Staphylococcus aureus and Escherichia coli are the most eminent. On the other hand, Pseudomonas aeruginosa binding to the pulmonary and urinary tract or the Helicobacter pylori binding to the gastric mucosa, are classical examples of bacteria-host cell interactions that end in serious infections. As we approach the end of the antibiotic era, acquisition of a deeper knowledge of the fundamental forces involved in bacteria - host cells/(bio)materials surface adhesion is crucial for the identification of new ligand-binding events and its assessment as novel targets for alternative anti-infective therapies. This article aims to highlight the potential of AFM-based force spectroscopy for new targeted therapies development against bacterial infections in which adhesion plays a pivotal role and does not aim to be an extensive overview on the AFM technical capabilities and theory of single molecule force spectroscopy.

Statistical Tests for Force Inference in Heterogeneous Environments

We devise a method to detect and estimate forces in a heterogeneous environment based on experimentally recorded stochastic trajectories. In particular, we focus on systems modeled by the heterogeneous overdamped Langevin equation. Here, the observed drift includes a "spurious” force term when the diffusivity varies in space. We show how Bayesian inference can be leveraged to reliably infer forces by taking into account such spurious forces of unknown amplitude as well as experimental sources of error. The method is based on marginalizing the force posterior over all possible spurious force contributions. The approach is combined with a Bayes factor statistical test for the presence of forces. The performance of our method is investigated analytically, numerically and tested on experimental data sets. The main results are obtained in a closed form allowing for direct exploration of their properties and fast computation. The method is incorporated into TRamWAy, an open-source software platform for automated analysis of biomolecule trajectories.

Mapping spatio-temporal dynamics of single biomolecules in living cells

We present a Bayesian framework for inferring spatio-temporal maps of diffusivity and potential fields from recorded trajectories of single molecules inside living cells. The framework naturally lets us regularise the high-dimensional inference problem using prior distributions in order to obtain robust results. To overcome the computational complexity of inferring thousands of map parameters from large single particle tracking datasets, we developed a stochastic optimisation method based on local mini-batches and parsimonious gradient calculation. We quantified the gain in convergence speed on numerical simulations, and we demonstrated for the first time temporal regularisation and aligned values of the inferred potential fields across multiple time segments. As a proof-of-concept, we mapped the dynamics of HIV-1 Gag proteins involved in the formation of virus-like particles (VLPs) on the plasma membrane of live T cells at high spatial and temporal resolutions. We focused on transient aggregation events lasting only on tenth of the time required for full VLP formation. The framework and optimisation methods are implemented in the TRamWAy open-source software platform for analysing single biomolecule dynamics.

Single molecule localisation microscopy reveals how HIV-1 Gag proteins sense membrane virus assembly sites in living host CD4 T cells

Monitoring virus assembly at the nanoscale in host cells remains a major challenge. Human immunodeficiency virus type 1 (HIV-1) components are addressed to the plasma membrane where they assemble to form spherical particles of 100 nm in diameter. Interestingly, HIV-1 Gag protein expression alone is sufficient to produce virus-like particles (VLPs) that resemble the immature virus. Here, we monitored VLP formation at the plasma membrane of host CD4+ T cells using a newly developed workflow allowing the analysis of long duration recordings of single-molecule Gag protein localisation and movement. Comparison of Gag assembling platforms in CD4+ T cells expressing wild type or assembly-defective Gag mutant proteins showed that VLP formation lasts roughly 15 minutes with an assembly time of 5 minutes. Trapping energy maps, built from membrane associated Gag protein movements, showed that one third of the assembling energy is due to direct Gag capsid-capsid interaction while the remaining two thirds require the nucleocapsid-RNA interactions. Finally, we show that the viral RNA genome does not increase the attraction of Gag at the membrane towards the assembling site but rather acts as a spatiotemporal coordinator of the membrane assembly process.

Reconstruction of Cell Focal Adhesions using Physical Constraints and Compressive Regularization

We develop a method to reconstruct, from measured displacements of an underlying elastic substrate, the spatially dependent forces that cells or tissues impart on it. Given newly available high-resolution images of substrate displacements, it is desirable to be able to reconstruct small-scale, compactly supported focal adhesions that are often localized and exist only within the footprint of a cell. In addition to the standard quadratic data mismatch terms that define least-squares fitting, we motivate a regularization term in the objective function that penalizes vectorial invariants of the reconstructed surface stress while preserving boundaries. We solve this inverse problem by providing a numerical method for setting up a discretized inverse problem that is solvable by standard convex optimization techniques. By minimizing the objective function subject to a number of important physically motivated constraints, we are able to efficiently reconstruct stress fields with localized structure from simulated and experimental substrate displacements. Our method incorporates the exact solution for a stress tensor accurate to first-order finite differences and motivates the use of distance-based cutoffs for data inclusion and problem sparsification.