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.

Gene trajectory inference for single-cell data by optimal transport metrics.

Single-cell RNA sequencing has been widely used to investigate cell state transitions and gene dynamics of biological processes. Current strategies to infer the sequential dynamics of genes in a process typically rely on constructing cell pseudotime through cell trajectory inference. However, the presence of concurrent gene processes in the same group of cells and technical noise can obscure the true progression of the processes studied. To address this challenge, we present GeneTrajectory, an approach that identifies trajectories of genes rather than trajectories of cells. Specifically, optimal transport distances are calculated between gene distributions across the cell-cell graph to extract gene programs and define their gene pseudotemporal order. Here we demonstrate that GeneTrajectory accurately extracts progressive gene dynamics in myeloid lineage maturation. Moreover, we show that GeneTrajectory deconvolves key gene programs underlying mouse skin hair follicle dermal condensate differentiation that could not be resolved by cell trajectory approaches. GeneTrajectory facilitates the discovery of gene programs that control the changes and activities of biological processes.

Rapid fluctuations in functional connectivity of cortical networks encode spontaneous behavior.

Experimental work across species has demonstrated that spontaneously generated behaviors are robustly coupled to variations in neural activity within the cerebral cortex. Functional magnetic resonance imaging data suggest that temporal correlations in cortical networks vary across distinct behavioral states, providing for the dynamic reorganization of patterned activity. However, these data generally lack the temporal resolution to establish links between cortical signals and the continuously varying fluctuations in spontaneous behavior observed in awake animals. Here, we used wide-field mesoscopic calcium imaging to monitor cortical dynamics in awake mice and developed an approach to quantify rapidly time-varying functional connectivity. We show that spontaneous behaviors are represented by fast changes in both the magnitude and correlational structure of cortical network activity. Combining mesoscopic imaging with simultaneous cellular-resolution two-photon microscopy demonstrated that correlations among neighboring neurons and between local and large-scale networks also encode behavior. Finally, the dynamic functional connectivity of mesoscale signals revealed subnetworks not predicted by traditional anatomical atlas-based parcellation of the cortex. These results provide new insights into how behavioral information is represented across the neocortex and demonstrate an analytical framework for investigating time-varying functional connectivity in neural networks.

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.

Doubly Stochastic Normalization of the Gaussian Kernel Is Robust to Heteroskedastic Noise.

A fundamental step in many data-analysis techniques is the construction of an affinity matrix describing similarities between data points. When the data points reside in Euclidean space, a widespread approach is to from an affinity matrix by the Gaussian kernel with pairwise distances, and to follow with a certain normalization (e.g. the row-stochastic normalization or its symmetric variant). We demonstrate that the doubly-stochastic normalization of the Gaussian kernel with zero main diagonal (i.e., no self loops) is robust to heteroskedastic noise. That is, the doubly-stochastic normalization is advantageous in that it automatically accounts for observations with different noise variances. Specifically, we prove that in a suitable high-dimensional setting where heteroskedastic noise does not concentrate too much in any particular direction in space, the resulting (doubly-stochastic) noisy affinity matrix converges to its clean counterpart with rate m -1/2, where m is the ambient dimension. We demonstrate this result numerically, and show that in contrast, the popular row-stochastic and symmetric normalizations behave unfavorably under heteroskedastic noise. Furthermore, we provide examples of simulated and experimental single-cell RNA sequence data with intrinsic heteroskedasticity, where the advantage of the doubly-stochastic normalization for exploratory analysis is evident.

Diffusion Earth Mover's Distance and Distribution Embeddings.

We propose a new fast method of measuring distances between large numbers of related high dimensional datasets called the Diffusion Earth Mover's Distance (EMD). We model the datasets as distributions supported on common data graph that is derived from the affinity matrix computed on the combined data. In such cases where the graph is a discretization of an underlying Riemannian closed manifold, we prove that Diffusion EMD is topologically equivalent to the standard EMD with a geodesic ground distance. Diffusion EMD can be computed in $\tilde{O}(n)$ time and is more accurate than similarly fast algorithms such as tree-based EMDs. We also show Diffusion EMD is fully differentiable, making it amenable to future uses in gradient-descent frameworks such as deep neural networks. Finally, we demonstrate an application of Diffusion EMD to single cell data collected from 210 COVID-19 patient samples at Yale New Haven Hospital. Here, Diffusion EMD can derive distances between patients on the manifold of cells at least two orders of magnitude faster than equally accurate methods. This distance matrix between patients can be embedded into a higher level patient manifold which uncovers structure and heterogeneity in patients. More generally, Diffusion EMD is applicable to all datasets that are massively collected in parallel in many medical and biological systems.

Local conformal autoencoder for standardized data coordinates.

We propose a local conformal autoencoder (LOCA) for standardized data coordinates. LOCA is a deep learning-based method for obtaining standardized data coordinates from scientific measurements. Data observations are modeled as samples from an unknown, nonlinear deformation of an underlying Riemannian manifold, which is parametrized by a few normalized, latent variables. We assume a repeated measurement sampling strategy, common in scientific measurements, and present a method for learning an embedding in [Formula: see text] that is isometric to the latent variables of the manifold. The coordinates recovered by our method are invariant to diffeomorphisms of the manifold, making it possible to match between different instrumental observations of the same phenomenon. Our embedding is obtained using LOCA, which is an algorithm that learns to rectify deformations by using a local z-scoring procedure, while preserving relevant geometric information. We demonstrate the isometric embedding properties of LOCA in various model settings and observe that it exhibits promising interpolation and extrapolation capabilities, superior to the current state of the art. Finally, we demonstrate LOCA's efficacy in single-site Wi-Fi localization data and for the reconstruction of three-dimensional curved surfaces from two-dimensional projections.

Two-sample statistics based on anisotropic kernels.

The paper introduces a new kernel-based Maximum Mean Discrepancy (MMD) statistic for measuring the distance between two distributions given finitely many multivariate samples. When the distributions are locally low-dimensional, the proposed test can be made more powerful to distinguish certain alternatives by incorporating local covariance matrices and constructing an anisotropic kernel. The kernel matrix is asymmetric; it computes the affinity between [Formula: see text] data points and a set of [Formula: see text] reference points, where [Formula: see text] can be drastically smaller than [Formula: see text]. While the proposed statistic can be viewed as a special class of Reproducing Kernel Hilbert Space MMD, the consistency of the test is proved, under mild assumptions of the kernel, as long as [Formula: see text], and a finite-sample lower bound of the testing power is obtained. Applications to flow cytometry and diffusion MRI datasets are demonstrated, which motivate the proposed approach to compare distributions.

Author Correction: Visualizing structure and transitions in high-dimensional biological data.

An amendment to this paper has been published and can be accessed via a link at the top of the paper.

Visualizing structure and transitions in high-dimensional biological data.

The high-dimensional data created by high-throughput technologies require visualization tools that reveal data structure and patterns in an intuitive form. We present PHATE, a visualization method that captures both local and global nonlinear structure using an information-geometric distance between data points. We compare PHATE to other tools on a variety of artificial and biological datasets, and find that it consistently preserves a range of patterns in data, including continual progressions, branches and clusters, better than other tools. We define a manifold preservation metric, which we call denoised embedding manifold preservation (DEMaP), and show that PHATE produces lower-dimensional embeddings that are quantitatively better denoised as compared to existing visualization methods. An analysis of a newly generated single-cell RNA sequencing dataset on human germ-layer differentiation demonstrates how PHATE reveals unique biological insight into the main developmental branches, including identification of three previously undescribed subpopulations. We also show that PHATE is applicable to a wide variety of data types, including mass cytometry, single-cell RNA sequencing, Hi-C and gut microbiome data.

A REMARK ON THE ARCSINE DISTRIBUTION AND THE HILBERT TRANSFORM.

It is known that if (p n ) n ∈ℕ is a sequence of orthogonal polynomials in L 2([-1,1],w(x)dx), then the roots are distributed according to an arcsine distribution π -1(1 - x 2)-1 dx for a wide variety of weights w(x). We connect this to a result of the Hilbert transform due to Tricomi: if f(x)(1 - x 2)1/4 ∈ L 2(-1,1) and its Hilbert transform Hf vanishes on (-1,1), then the function f is a multiple of the arcsine distribution f ( x ) = c 1 - x 2 χ ( - 1 , 1 ) where c ∈ ℝ . We also prove a localized Parseval-type identity that seems to be new: if f(x)(1-x 2)1/4 ∈ L2(-1, 1) and f ( x ) 1 - x 2 has mean value 0 on (-1, 1), then ∫ - 1 1 ( H f ) ( x ) 2 1 - x 2 d x = ∫ - 1 1 f ( x ) 2 1 - x 2 d x . .

Data-Driven Tree Transforms and Metrics.

We consider the analysis of high dimensional data given in the form of a matrix with columns consisting of observations and rows consisting of features. Often the data is such that the observations do not reside on a regular grid, and the given order of the features is arbitrary and does not convey a notion of locality. Therefore, traditional transforms and metrics cannot be used for data organization and analysis. In this paper, our goal is to organize the data by defining an appropriate representation and metric such that they respect the smoothness and structure underlying the data. We also aim to generalize the joint clustering of observations and features in the case the data does not fall into clear disjoint groups. For this purpose, we propose multiscale data-driven transforms and metrics based on trees. Their construction is implemented in an iterative refinement procedure that exploits the co-dependencies between features and observations. Beyond the organization of a single dataset, our approach enables us to transfer the organization learned from one dataset to another and to integrate several datasets together. We present an application to breast cancer gene expression analysis: learning metrics on the genes to cluster the tumor samples into cancer sub-types and validating the joint organization of both the genes and the samples. We demonstrate that using our approach to combine information from multiple gene expression cohorts, acquired by different profiling technologies, improves the clustering of tumor samples.

Reconstruction of normal forms by learning informed observation geometries from data.

The discovery of physical laws consistent with empirical observations is at the heart of (applied) science and engineering. These laws typically take the form of nonlinear differential equations depending on parameters; dynamical systems theory provides, through the appropriate normal forms, an "intrinsic" prototypical characterization of the types of dynamical regimes accessible to a given model. Using an implementation of data-informed geometry learning, we directly reconstruct the relevant "normal forms": a quantitative mapping from empirical observations to prototypical realizations of the underlying dynamics. Interestingly, the state variables and the parameters of these realizations are inferred from the empirical observations; without prior knowledge or understanding, they parametrize the dynamics intrinsically without explicit reference to fundamental physical quantities.

Heterogeneity in Early Responses in ALLHAT (Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial).

Randomized trials of hypertension have seldom examined heterogeneity in response to treatments over time and the implications for cardiovascular outcomes. Understanding this heterogeneity, however, is a necessary step toward personalizing antihypertensive therapy. We applied trajectory-based modeling to data on 39 763 study participants of the ALLHAT (Antihypertensive and Lipid-Lowering Treatment to Prevent Heart Attack Trial) to identify distinct patterns of systolic blood pressure (SBP) response to randomized medications during the first 6 months of the trial. Two trajectory patterns were identified: immediate responders (85.5%), on average, had a decreasing SBP, whereas nonimmediate responders (14.5%), on average, had an initially increasing SBP followed by a decrease. Compared with those randomized to chlorthalidone, participants randomized to amlodipine (odds ratio, 1.20; 95% confidence interval [CI], 1.10-1.31), lisinopril (odds ratio, 1.88; 95% CI, 1.73-2.03), and doxazosin (odds ratio, 1.65; 95% CI, 1.52-1.78) had higher adjusted odds ratios associated with being a nonimmediate responder (versus immediate responder). After multivariable adjustment, nonimmediate responders had a higher hazard ratio of stroke (hazard ratio, 1.49; 95% CI, 1.21-1.84), combined cardiovascular disease (hazard ratio, 1.21; 95% CI, 1.11-1.31), and heart failure (hazard ratio, 1.48; 95% CI, 1.24-1.78) during follow-up between 6 months and 2 years. The SBP response trajectories provided superior discrimination for predicting downstream adverse cardiovascular events than classification based on difference in SBP between the first 2 measurements, SBP at 6 months, and average SBP during the first 6 months. Our findings demonstrate heterogeneity in response to antihypertensive therapies and show that chlorthalidone is associated with more favorable initial response than the other medications.

Describing the performance of U.S. hospitals by applying big data analytics.

Public reporting of measures of hospital performance is an important component of quality improvement efforts in many countries. However, it can be challenging to provide an overall characterization of hospital performance because there are many measures of quality. In the United States, the Centers for Medicare and Medicaid Services reports over 100 measures that describe various domains of hospital quality, such as outcomes, the patient experience and whether established processes of care are followed. Although individual quality measures provide important insight, it is challenging to understand hospital performance as characterized by multiple quality measures. Accordingly, we developed a novel approach for characterizing hospital performance that highlights the similarities and differences between hospitals and identifies common patterns of hospital performance. Specifically, we built a semi-supervised machine learning algorithm and applied it to the publicly-available quality measures for 1,614 U.S. hospitals to graphically and quantitatively characterize hospital performance. In the resulting visualization, the varying density of hospitals demonstrates that there are key clusters of hospitals that share specific performance profiles, while there are other performance profiles that are rare. Several popular hospital rating systems aggregate some of the quality measures included in our study to produce a composite score; however, hospitals that were top-ranked by such systems were scattered across our visualization, indicating that these top-ranked hospitals actually excel in many different ways. Our application of a novel graph analytics method to data describing U.S. hospitals revealed nuanced differences in performance that are obscured in existing hospital rating systems.

Intrinsic map dynamics exploration for uncharted effective free-energy landscapes.

We describe and implement a computer-assisted approach for accelerating the exploration of uncharted effective free-energy surfaces (FESs). More generally, the aim is the extraction of coarse-grained, macroscopic information from stochastic or atomistic simulations, such as molecular dynamics (MD). The approach functionally links the MD simulator with nonlinear manifold learning techniques. The added value comes from biasing the simulator toward unexplored phase-space regions by exploiting the smoothness of the gradually revealed intrinsic low-dimensional geometry of the FES.

Quantitative arbor analytics: unsupervised harmonic co-clustering of populations of brain cell arbors based on L-measure.

This paper presents a robust unsupervised harmonic co-clustering method for profiling arbor morphologies for ensembles of reconstructed brain cells (e.g., neurons, microglia) based on quantitative measurements of the cellular arbors. Specifically, this method can identify groups and sub-groups of cells with similar arbor morphologies, and simultaneously identify the hierarchical grouping patterns among the quantitative arbor measurements. The robustness of the proposed algorithm derives from use of the diffusion distance measure for comparing multivariate data points, harmonic analysis theory, and a Haar-like wavelet basis for multivariate data smoothing. This algorithm is designed to be practically usable, and is embedded into the actively linked three-dimensional (3-D) visualization and analytics system in the free and open source FARSIGHT image analysis toolkit for interactive exploratory population-scale neuroanatomic studies. Studies on synthetic datasets demonstrate its superiority in clustering data matrices compared to recent hierarchical clustering algorithms. Studies on heterogeneous ensembles of real neuronal 3-D reconstructions drawn from the NeuroMorpho database show that the proposed method identifies meaningful grouping patterns among neurons based on arbor morphology, and revealing the underlying morphological differences.

Diffusion methods for aligning medical datasets: location prediction in CT scan images.

The purpose of this study is to introduce diffusion methods as a tool to label CT scan images according to their position in the human body. A comparative study of different methods based on a k-NN search is carried out and we propose a new, simple and efficient way of applying diffusion techniques that is able to give better location forecasts than methods that can be considered the current state-of-the-art.