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Moment-based metrics for molecules computable from cryogenic electron microscopy images

Biological imaging 2024年 4卷 e3页

作者： Andy Zhang Oscar Mickelin Joe Kileel Eric J Verbeke Nicholas F Marshall Marc Aurèle Gilles Amit Singer Program in Applied and Computational Mathematics Princeton University Princeton NJ USA. Department of Mathematics and Oden Institute University of Texas at Austin Austin TX USA. Department of Mathematics Oregon State University Corvallis OR USA. Department of Mathematics Princeton University Princeton NJ USA.

Single-particle cryogenic electron microscopy (cryo-EM) is an imaging technique capable of recovering the high-resolution three-dimensional (3D) structure of biological macromolecules from many noisy and randomly oriented projection images. One notable approach to 3D reconstruction, known as Kam's method, relies on the moments of the two-dimensional (2D) images. Inspired by Kam's method, we introduce a rotationally invariant metric between two molecular structures, which does not require 3D alignment. Further, we introduce a metric between a stack of projection images and a molecular structure, which is invariant to rotations and reflections and does not require performing 3D reconstruction. Additionally, the latter metric does not assume a uniform distribution of viewing angles. We demonstrate the uses of the new metrics on synthetic and experimental datasets, highlighting their ability to measure structural similarity.

关键词： Kam’s method alignment-free metric method of moments protein structural similarity rotationally invariant distance structural search

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PHYSICS INFORMED MACHINE LEARNING WITH SMOOTHED PARTICLE HYDRODYNAMICS: HIERARCHY OF REDUCED LAGRANGIAN MODELS OF TURBULENCE

arXiv

引用

arXiv 2021年

作者： Woodward, Michael Tian, Yifeng Hyett, Criston Fryer, Chris Stepanov, Mikhail Livescu, Daniel Chertkov, Michael Graduate Interdisciplinary Program in Applied Mathematics UArizona TucsonAZ85721 United States Department of Mathematics UArizona TucsonAZ85721 United States Computer Computational and Statistical Sciences Division LANL Los AlamosNM87544 United States

Building efficient, accurate and generalizable reduced order models of developed turbulence remains a major challenge. This manuscript approaches this problem by developing a hierarchy of parameterized reduced Lagrangian models for turbulent flows, and investigates the effects of enforcing physical structure through Smoothed Particle Hydrodynamics (SPH) versus relying on neural networks (NN)s as universal function approximators. Starting from Neural Network (NN) parameterizations of a Lagrangian acceleration operator, this hierarchy of models gradually incorporates a weakly compressible and parameterized SPH framework, which enforces physical symmetries, such as Galilean, rotational and translational invariances. Within this hierarchy, two new parameterized smoothing kernels are developed in order to increase the flexibility of the learn-able SPH simulators. For each model we experiment with different loss functions which are minimized using gradient based optimization, where efficient computations of gradients are obtained by using Automatic Differentiation (AD) and Sensitivity Analysis (SA). Each model within the hierarchy is trained on two data sets associated with weekly compressible Homogeneous Isotropic Turbulence (HIT): (1) a validation set using weakly compressible SPH;and (2) a high fidelity set from Direct Numerical Simulations (DNS). Numerical evidence shows that encoding more SPH structure improves generalizability to different turbulent Mach numbers and time shifts, and that including the novel parameterized smoothing kernels improves the accuracy of SPH at the resolved scales. Copyright © 2021, The Authors. All rights reserved.

关键词： Parameterization

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Simulated Diffusion Spreadability for Characterizing the Structure and Transport Properties of Two-Phase Materials

SSRN

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SSRN 2023年

作者： Skolnick, Murray Torquato, Salvatore Department of Chemistry Princeton University PrincetonNJ08544 United States Department of Physics Princeton University PrincetonNJ08544 United States Princeton Institute of Materials Princeton University PrincetonNJ08544 United States Program in Applied and Computational Mathematics Princeton University PrincetonNJ08544 United States School of Natural Sciences The Institute for Advanced Study PrincetonNJ08540 United States

Time-dependent diffusion processes between phases are ubiquitous in physical, chemical, and biological materials. Examples of such materials include composite materials, porous materials, cellular solids, polymer blends, colloids, gels, and biological materials. The recently developeddiffusion spreadability, S(t), provides a direct link between time-dependent interphase diffusive transport and the microstructure of two-phase materials across length scales [Torquato, S., Phys. Rev. E., 104 054102 (2021)];thus making S(t) a powerful dynamic means for classifying all statistically homogeneous microstructures, spanning from anti-hyperuniform to hyperuniform. It was shown that the small-, intermediate-, and long-time behaviors of S(t) are directly determined by the small-, intermediate-, and large-scale structural features of the material. Moreover, the spreadability can be applied as a physical-property based tool for microstructural characterization in the absence of or as supplement to scattering information. In this work, we develop a computationally efficient algorithm for ascertaining s(t) directly from digitized representations of material microstructures via random-walk techniques. Our algorithm computes the time-dependent local walker concentration field c(x, t), a quantity not previously examined in the context of the spreadability, enabling us to compute the entropy production rate s(t) of the associated diffusion process which is a quantity related to the rate of dissipation. We also derive exact analytical expressions for s(t), and find that hyperuniform materials have smaller dissipation than any nonhyperuniform materials. Lastly, we use our algorithm to compute, for the first time, the more general case of the spreadability in which the phase diffusion coefficients are distinct and provide a method for extracting the effective diffusion coefficient of the two-phase material from such data. We apply our algorithm to a variety of two- and three-dimensional s

关键词： Microstructure

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Numerical implicitization

引用

Journal of Software for Algebra and Geometry 2019年第1期9卷 55-63页

作者： Chen, Justin Kileel, Joe School of Mathematics Georgia Institute of Technology Atlanta GA United States Program in Applied and Computational Mathematics Princeton University Princeton NJ United States

We present the NumericalImplicitization.m2 package for Macaulay2, which allows for user-friendly computation of the invariants of the image of a polynomial map, such as dimension, degree, and Hilbert function values. This package relies on methods of numerical algebraic geometry, including homotopy continuation and monodromy. © 2019, Mathematical Sciences Publishers. All rights reserved.

关键词： homotopy continuation implicitization interpolation monodromy numerical algebraic geometry

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On the particle approximation of lagged Feynman–Kac formulae

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Stochastic Processes and their Applications 2025年 188卷

作者： Awadelkarim, Elsiddig Caffarel, Michel Del Moral, Pierre Jasra, Ajay Applied Mathematics and Computational Science Program Computer Electrical and Mathematical Sciences and Engineering Division King Abdullah University of Science and Technology Thuwal23955-6900 Saudi Arabia University of Toulouse CNRS-Lab. de Chimie et Physique Quantiques Toulouse31062 France Centre de Recherche Inria Bordeaux Sud-Ouest Talence33405 France School of Data Science The Chinese University of Hong Kong Shenzhen Shenzhen China

In this paper we examine the numerical approximation of the limiting invariant measure associated with Feynman–Kac formulae. These are expressed in a discrete time formulation and are associated with a Markov chain and a potential function. The typical application considered here is the computation of eigenvalues associated with non-negative operators as found, for example, in physics or particle simulation of rare-events. We focus on a novel lagged approximation of this invariant measure, based upon the introduction of a ratio of time-averaged Feynman–Kac marginals associated with a positive operator iterated l∈N times;a lagged Feynman–Kac formula. This estimator and its approximation using Diffusion Monte Carlo (DMC) are commonly used in the physics literature. In short, DMC is an iterative algorithm involving N∈N particles or walkers simulated in parallel, that undergo sampling and resampling operations. In this work, it is shown that for the DMC approximation of the lagged Feynman–Kac formula, one has an almost sure characterization of the L1-error as the time parameter (iteration) goes to infinity and this is at most of O(exp{−κl}/N), for κ>0. In addition a non-asymptotic in time, and time uniform L1−bound is proved which is O(l/N). We also prove a novel central limit theorem to give a characterization of the exact asymptotic in time variance. This analysis demonstrates that the strategy used in physics, namely, to run DMC with N and l small and, for long time enough, is mathematically justified. Our results also suggest how one should choose N and l in practice. We emphasize that these results are not restricted to physical applications;they have broad relevance to the general problem of particle simulation of the Feynman–Kac formula, which is utilized in a great variety of scientific and engineering fields. © 2025 Elsevier B.V.

关键词： Iterative methods

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Optimized Large Hyperuniform Binary Colloidal Suspensions in Two Dimensions

引用

Physical Review Letters 2020年第6期125卷 068002-068002页

作者： Zheng Ma Enrique Lomba Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

The creation of disordered hyperuniform materials with extraordinary optical properties (e.g., large complete photonic band gaps) requires a capacity to synthesize large samples that are effectively hyperuniform down to the nanoscale. Motivated by this challenge, we propose a feasible equilibrium fabrication protocol using binary paramagnetic colloidal particles confined in a 2D plane. The strong and long-ranged dipolar interaction induced by a tunable magnetic field is free from screening effects that attenuate long-ranged electrostatic interactions in charged colloidal systems. Specifically, we numerically find a family of optimal size ratios that makes the two-phase system effectively hyperuniform. We show that hyperuniformity is a general consequence of low isothermal compressibilities, which makes our protocol suitable to treat more general systems with other long-ranged interactions, dimensionalities, and/or polydispersity. Our methodology paves the way to synthesize large photonic hyperuniform materials that function in the visible to infrared range and hence may accelerate the discovery of novel photonic materials.

关键词： Applications of soft matter Colloids Disordered systems

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Deep neural network for the dielectric response of insulators

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Physical Review B 2020年第4期102卷 041121(R)-041121(R)页

作者： Linfeng Zhang Mohan Chen Xifan Wu Han Wang Weinan E Roberto Car Department of Chemistry Department of Physics Program in Applied and Computational Mathematics Princeton Institute for the Science and Technology of Materials Princeton University Princeton New Jersey 08544 USA

We introduce a deep neural network to model in a symmetry preserving way the environmental dependence of the centers of the electronic charge. The model learns from ab initio density functional theory, wherein the electronic centers are uniquely assigned by the maximally localized Wannier functions. When combined with the deep potential model of the atomic potential energy surface, the scheme predicts the dielectric response of insulators for trajectories inaccessible to direct ab initio simulation. The scheme is nonperturbative and can capture the response of a mutating chemical environment. We demonstrate the approach by calculating the infrared spectra of liquid water at standard conditions, and of ice under extreme pressure, when it transforms from a molecular to an ionic crystal.

关键词： Electric polarization Electrical properties Electronic structure Ferroelectrics Liquids Piezoelectrics Density functional theory First-principles calculations Machine learning Molecular dynamics Multiscale modeling Wannier function methods

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Generation and structural characterization of Debye random media

引用

Physical Review E 2020年第4期102卷 043310-043310页

作者： Zheng Ma Salvatore Torquato Department of Chemistry Department of Physics Princeton Institute for the Science and Technology of Materials and Program in Applied and Computational Mathematics Princeton University Princeton New Jersey 08544 USA

In their seminal paper on scattering by an inhomogeneous solid, Debye and coworkers proposed a simple exponentially decaying function for the two-point correlation function of an idealized class of two-phase random media. Such Debye random media, which have been shown to be realizable, are singularly distinct from all other models of two-phase media in that they are entirely defined by their one- and two-point correlation functions. To our knowledge, there has been no determination of other microstructural descriptors of Debye random media. In this paper, we generate Debye random media in two dimensions using an accelerated Yeong-Torquato construction algorithm. We then ascertain microstructural descriptors of the constructed media, including their surface correlation functions, pore-size distributions, lineal-path function, and chord-length probability density function. Accurate semianalytic and empirical formulas for these descriptors are devised. We compare our results for Debye random media to those of other popular models (overlapping disks and equilibrium hard disks) and find that the former model possesses a wider spectrum of hole sizes, including a substantial fraction of large holes. Our algorithm can be applied to generate other models defined by their two-point correlation functions, and their other microstructural descriptors can be determined and analyzed by the procedures laid out here.

关键词： Patterns in complex systems

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Hyperuniformity classes of quasiperiodic tilings via diffusion spreadability

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Physical Review E 2024年第6期109卷 064108-064108页

作者： Adam Hitin-Bialus Charles Emmett Maher Paul J. Steinhardt Salvatore Torquato Department of Physics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA. Department of Chemistry <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA. Jefferson Physical Laboratory <a href="https://***/03vek6s52">Harvard University</a> Cambridge Massachusetts 02138 USA and Department of Physics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA. Department of Chemistry <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA Department of Physics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA Princeton Institute of Materials <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA and Program in Applied and Computational Mathematics <a href="https://***/00hx57361">Princeton University</a> Princeton New Jersey 08544 USA.

Hyperuniform point patterns can be classified by the hyperuniformity scaling exponent α>0, that characterizes the power-law scaling behavior of the structure factor S(k) as a function of wave number k≡|k| in the vicinity of the origin, e.g., S(k)∼|k|α in cases where S(k) varies continuously with k as k→0. In this paper, we show that the spreadability is an effective method for determining α for quasiperiodic systems where S(k) is discontinuous and consists of a dense set of Bragg peaks. It has been shown in [Phys. Rev. E 104, 054102 (2021)] that, for media with finite α, the long-time behavior of the excess spreadability S(∞)−S(t) can be fit to a power law of the form t−(d−α)/2, where d is the space dimension, to accurately extract α for the continuous case. We first transform quasiperiodic and limit-periodic point patterns into two-phase media by mapping them onto packings of identical nonoverlapping disks, where space interior to the disks represents one phase and the space in exterior to them represents the second phase. We then compute the spectral density χ̃V(k) of the packings, and finally compute and fit the long-time behavior of their excess spreadabilities. Specifically, we show that the excess spreadability can be used to accurately extract α for the one-dimensional (1D) limit-periodic period-doubling chain (α=1) and the 1D quasicrystalline Fibonacci chain (α=3) to within 0.02% of the analytically known exact results. Moreover, we obtain a value of α=5.97±0.06 for the two-dimensional Penrose tiling and present plausible theoretical arguments strongly suggesting that α is exactly equal to six. We also show that, due to the self-similarity of the structures examined here, one can truncate the small-k region of the scattering information used to compute the spreadability and obtain an accurate value of α, with a small deviation from the untruncated case that decreases as the system size increases. This strongly suggests that one can obtain a good estimate of

关键词： Quasicrystals

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Flagellum Pumping Efficiency in Shear-Thinning Viscoelastic Fluids

arXiv

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arXiv 2023年

作者： Barrett, Aaron Fogelson, Aaron L. Gregory Forest, M. Gruninger, Cole Lim, Sookkyung Griffith, Boyce E. Department of Mathematics University of Utah Salt Lake CityUT United States Departments of Mathematics and Biomedical Engineering University of Utah Salt Lake CityUT United States Department of Mathematics University of North Carolina Chapel HillNC United States Department of Applied Physical Sciences University of North Carolina Chapel HillNC United States Department of Biomedical Engineering University of North Carolina Chapel HillNC United States Carolina Center for Interdisciplinary Applied Mathematics University of North Carolina Chapel HillNC United States Department of Mathematical Sciences University of Cincinnati CincinnatiOH United States Computational Medicine Program University of North Carolina School of Medicine Chapel HillNC United States McAllister Heart Institute University of North Carolina School of Medicine Chapel HillNC United States

Microorganism motility often takes place within complex, viscoelastic fluid environments, e.g., sperm in cervicovaginal mucus and bacteria in biofilms. In such complex fluids, strains and stresses generated by the microorganism are stored and relax across a spectrum of length and time scales and the complex fluid can be driven out of its linear response regime. Phenomena not possible in viscous media thereby arise from feedback between the swimmer and the complex fluid, making swimming efficiency co-dependent on the propulsion mechanism and fluid properties. Here we parameterize a flagellar motor and filament properties together with elastic relaxation and nonlinear shear-thinning properties of the fluid in a computational immersed boundary model. We then explore swimming efficiency, defined as a particular flow rate divided by the torque required to spin the motor, over this parameter space. Our findings indicate that motor efficiency (measured by the volumetric flow rate) can be boosted or degraded by relatively moderate or strong shear-thinning of the viscoelastic environment. Copyright © 2023, The Authors. All rights reserved.

关键词： Shear thinning

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