Stochastic Dynamics Research Articles

Modeling Viral Capsid Assembly: A Review of Computational Strategies and Applications

Viral capsid assembly is a complex and critical process, essential for understanding viral behavior, evolution, and the development of antiviral treatments, vaccines, and nanotechnology. Significant progress in studying viral capsid assembly has been achieved through various computational approaches, including molecular dynamics (MD) simulations, stochastic dynamics simulations, coarse-grained (CG) models, electrostatic analyses, lattice models, hybrid techniques, machine learning methods, and kinetic models. Each of these techniques offers unique advantages, and by integrating these diverse computational strategies, researchers can more accurately model the dynamic behaviors and structural features of viral capsids, deepening our understanding of the assembly process. This review provides a comprehensive overview of studies on viral capsid assembly, emphasizing their critical role in advancing our knowledge. It examines the contributions, strengths, and limitations of different computational methods, presents key computational works in the field, and analyzes milestone studies that have shaped current research.

An optimization-based equilibrium measure describing fixed points of non-equilibrium dynamics: application to the edge of chaos

Understanding neural dynamics is a central topic in machine learning, non-linear physics, and neuroscience. However, the dynamics are non-linear, stochastic and particularly non-gradient, i.e., the driving force cannot be written as the gradient of a potential. These features make analytic studies very challenging. The common tool is the path integral approach or dynamical mean-field theory. Still, the drawback is that one has to solve the integro-differential or dynamical mean-field equations, which is computationally expensive and has no closed-form solutions in general. From the associated Fokker–Planck equation, the steady-state solution is generally unknown. Here, we treat searching for the fixed points as an optimization problem, and construct an approximate potential related to the speed of the dynamics, and find that searching for the ground state of this potential is equivalent to running approximate stochastic gradient dynamics or Langevin dynamics. Only in the zero temperature limit, can the distribution of the original fixed points be achieved. The resultant stationary state of the dynamics exactly follows the canonical Boltzmann measure. Within this framework, the quenched disorder intrinsic in the neural networks can be averaged out by applying the replica method, which leads naturally to order parameters for the non-equilibrium steady states. Our theory reproduces the well-known result of edge-of-chaos. Furthermore, the order parameters characterizing the continuous transition are derived, and the order parameters are explained as fluctuations and responses of the steady states. Our method thus opens the door to analytically studying the fixed-point landscape of the deterministic or stochastic high dimensional dynamics.

Bayesian Frequency Estimation Under Local Differential Privacy With an Adaptive Randomized Response Mechanism

Frequency estimation plays a critical role in many applications involving personal and private categorical data. Such data are often collected sequentially over time, making it valuable to estimate their distribution online while preserving privacy. We propose AdOBEst-LDP, a new algorithm for adaptive, online Bayesian estimation of categorical distributions under local differential privacy (LDP). The key idea behind AdOBEst-LDP is to enhance the utility of future privatized categorical data by leveraging inference from previously collected privatized data. To achieve this, AdOBEst-LDP uses a new adaptive LDP mechanism to collect privatized data. This LDP mechanism constrains its output to a subset of categories that ‘predicts’ the next user's data. By adapting the subset selection process to the past privatized data via Bayesian estimation, the algorithm improves the utility of future privatized data. To quantify utility, we explore various well-known information metrics, including (but not limited to) the Fisher information matrix, total variation distance, and information entropy. For Bayesian estimation, we utilize posterior sampling through stochastic gradient Langevin dynamics, a computationally efficient approximate Markov chain Monte Carlo (MCMC) method. We provide a theoretical analysis showing that (i) the posterior distribution of the category probabilities targeted with Bayesian estimation converges to the true probabilities even for approximate posterior sampling, and (ii) AdOBEst-LDP eventually selects the optimal subset for its LDP mechanism with high probability if posterior sampling is performed exactly. We also present numerical results to validate the estimation accuracy of AdOBEst-LDP. Our comparisons show its superior performance against non-adaptive and semi-adaptive competitors across different privacy levels and distributional parameters.

Value-behavior inconsistency is robust to promote cooperative behavior in structured populations.

The evolution of cooperation is a theme commonly studied in biology, psychology, sociology, and economics. Mechanisms that promote cooperative behavior in structured populations have been intensively studied. However, individuals' values, specifically, their opinions have been rarely taken into account so far. Inspired by cognition dissonance theory, we assume that individuals pay the cost of guiltiness if the behavior is defection but the opinion deviates from defection, and pay the cost of regret if the behavior is cooperation but the opinion deviates from cooperation. For all general stochastic evolutionary dynamics on arbitrary static networks with multiple opinions, we prove in the weak selection limit that: (i) value-behavior inconsistency cost promotes cooperative behavior if and only if the average cost of regret is less than that of guiltiness; (ii) individuals with value-behavior consistency are more abundant than that with value-behavior inconsistency. This is in contrast with other mechanisms that are at work for cooperation for one population structure but not others. Furthermore, it is also validated on an empirical network and for non-weak selection intensity. The value-behavior inconsistency is thus a robust mechanism to promote cooperative behavior in structured populations. Our results shed light on the importance of the co-evolutionary dynamics of opinion and behavior, which opens an avenue for cooperation.

Increasing environmental fluctuations can dampen variability of endogenously cycling populations.

Understanding how populations respond to increasingly variable conditions is a major objective for natural resource managers forecasting extinction risk. The lesson from current modelling is clear: increasing environmental variability increases population abundance variability. We show that this paradigm fails to describe a broad class of empirically observed dynamics, namely endogenously driven population cycles. In contrast to the dominant paradigm, these populations can exhibit reduced long-run population variance under increasing environmental variability. We provide evidence for a mechanistic explanation of this phenomenon that relies on how stochasticity interacts with long transient dynamics present in the deterministic cycling model. This interaction stands in contrast to the often assumed additivity of stochastic and deterministic drivers of population fluctuations. We show evidence for the phenomenon in two cyclical populations: flour beetles and Canadian lynx. We quantify the impact of the phenomenon with new theory that partitions the effects of nonlinear dynamics and stochastic variation on dynamical systems. In both empirical examples, the partitioning shows that the interaction between deterministic and stochastic dynamics reduces the variance in population size. Our results highlight that previous predictions about extinction under environmental variability may prove inadequate to understand the effects of climate change in some populations.

Intelligent Bayesian neural networks for stochastic SVIS epidemic dynamics: Vaccination strategies and prevalence fractions with Wiener process

Intelligent Bayesian neural networks for stochastic SVIS epidemic dynamics: Vaccination strategies and prevalence fractions with Wiener process

Simulation of stochastic discrete dislocation dynamics in ductile Vs brittle materials

Defects are inevitable during the manufacturing processes of materials. Presence of these defects and their dynamics significantly influence the responses of materials. A thorough understanding of dislocation dynamics of different types of materials under various conditions is essential for analysing the performance of the materials. Ductility of a material is directly related with the movement and rearrangement of dislocations under applied load. In this work, we look into the dynamics of dislocations in ductile and brittle materials using simplified two dimensional discrete dislocation dynamics (2D-DDD) simulation. We consider Aluminium (Al) and Tungsten (W) as representative examples of ductile and brittle materials respectively. We study the velocity distribution, strain field, dislocation count, and junction formation during interactions of the dislocations within the domain. Furthermore, we study the probability densities of dislocation motion for both materials. In mesoscale, moving dislocations can be considered as particle diffusion, which are often stochastic and super-diffusive. Classical diffusion models fail to account for these phenomena and the long-range interactions of dislocations. Therefore, we propose the nonlocal transport model for the probability density and obtained the parameters of nonlocal operators using a machine learning framework.

Stabilising CO2 concentration as a channel for global disaster risk mitigation

We investigate the influence of anthropogenic CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {CO}_2$$\\end{document} concentration fluctuations on the likelihood of climate-related disasters. We calibrate annual incidence rates against global disasters and CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {CO}_2$$\\end{document} growth spanning from 1960 to 2022 based on a dynamic panel logit model. We also study the sensitivity of disaster incidence to stochastic carbon dynamics consistent with IPCC-projected climate outcomes for 2100. The key insight is that present and lagged CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {CO}_2$$\\end{document} growth contains valuable information about the likelihood of future disaster events. We further show that lowering carbon stock uncertainty by dampening the persistence or the variability of CO2\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\hbox {CO}_2$$\\end{document} concentration has a first-order impact on mitigating expected disaster risk.

Stochastic stability analysis method for doubly fed generator sets considering random blade stress

AbstractWith the increasing capacity of doubly fed induction generators (DFIGs), the diameter of the wind turbine is increasing, the blades are getting longer and longer, and in the process of power generation, the tower shadow effect as well as the role of wind shear are more obvious, and the random blade stresses caused by this is also getting bigger. Random blade stresses cause random and cyclic fluctuations in the power generated by the wind turbine, and power fluctuations often cause voltage flicker, which affects the control system and power quality. To address the impact of random blade stresses on the grid‐connected stability of DFIG, the results of the traditional stability analysis methods may be too conservative or lead to too high a dimensionality to be analyzed. To solve the above problems, this paper proposes a grid‐connected stochastic stability analysis method for DFIG sets considering random blade stresses based on the stochastic averaging method under the Hamiltonian system. a stochastic dynamics model of the doubly fed wind farm was established by considering random blade stress. Subsequently, using the proposed generalized Hamiltonian principle, the model and energy functions in the proposed Hamiltonian form H, based on the stochastic averaging method (SAM), were established to obtain the system energy diffusion equation. Probability density function and regional stability probability were obtained from explicit expressions of the mean and regression square root processes. The drift and diffusion coefficients were obtained using the SAM, and the backward Kolmogorov equation was derived from the Ito equation to obtain the conditional reliability function and the probability density of the first crossing time. Finally, the effects of torque fluctuations with different stochastic intensities on the grid‐connected stability of doubly fed wind farms were investigated, and the effectiveness of the proposed generalized Hamiltonian SAM applied to the stochastic stability analysis of DFIG was verified by numerical analysis and Monte Carlo simulation. This provides a theoretical foundation for analyzing the grid‐connected stability of DFIG affected by random blade stresses.

Element-wise and recursive solutions for the power spectral density of biological stochastic dynamical systems at fixed points

Stochasticity plays a central role in nearly every biological process, and the noise power spectral density (PSD) is a critical tool for understanding variability and information processing in living systems. In steady state, many such processes can be described by stochastic linear time-invariant (LTI) systems driven by Gaussian white noise, whose PSD is a complex rational function of the frequency that can be concisely expressed in terms of their Jacobian, dispersion, and diffusion matrices, fully defining the statistical properties of the system's dynamics at steady state. Here, we arrive at compact, element-wise solutions of the rational function coefficients for the auto- and cross-spectrum that enable the explicit analytical computation of the PSD in dimensions n=2,3,4. We further present a recursive Leverrier-Faddeev-type algorithm for the exact computation of the rational function coefficients. Crucially, both solutions are free of matrix inverses. We illustrate our element-wise and recursive solutions by considering the stochastic dynamics of neural systems models, namely, FitzHugh-Nagumo (n=2), Hindmarsh-Rose (n=3), Wilson-Cowan (n=4), and the stabilized supralinear network (n=22), as well as an evolutionary game-theoretic model with mutations (n=5,31). We extend our approach to derive a recursive method for calculating the coefficients in the power-series expansion of the integrated covariance matrix for interacting spiking neurons modeled as Hawkes processes on arbitrary directed graphs. Published by the American Physical Society 2024

A spatially-resolved model of neutron-irradiated tungsten coupling stochastic cluster dynamics and finite deformation plasticity

Structural materials used in nuclear reactors face severe degradation in mechanical properties, such as hardening and embrittlement. At the microscopic scale, this occurs due to creation and accumulation of irradiation-induced defects and their interaction with system dislocations. Although techniques exist which can model evolution of irradiation defects, for instance kinetic transport theory-based models, their interaction with mechanical deformation of the bulk material has not been investigated extensively. In this work, we demonstrate a novel spatially-resolved multiscale coupling between microscopic irradiation defect evolution, modeled using Stochastic Cluster Dynamics (SCD) and macroscopic mechanical deformation modeled using a finite-deformation plasticity model. SCD is used to determine the statistically averaged defect cluster spacing, dependent on operating conditions such as irradiation dose and temperature. This acts as an initial condition that governs the critical resolved shear stress of dislocation glide in the macroscopic plasticity model. This framework is used to predict mechanical behavior in post-mortem test of irradiated Tungsten samples, which has found its importance as structural material used in nuclear reactors. The results obtained using the coupled approach are in good agreement with experimental data of uniaxial tension tests. The model is able to capture the effect of temperature and irradiation dose on the material hardening. Two methods are proposed to estimate hardness – using Tabor's Law relating uniaxial yield stress to hardness and from flat-punch simulations. The results are in reasonable agreement with hardness data from micro-indentation experiments of irradiated Tungsten samples. The model is also able to reveal microstructural details such as spatial variation in defect density and local stress.

Storm: Incorporating transient stochastic dynamics to infer the RNA velocity with metabolic labeling information

The time-resolved scRNA-seq (tscRNA-seq) provides the possibility to infer physically meaningful kinetic parameters, e.g., the transcription, splicing or RNA degradation rate constants with correct magnitudes, and RNA velocities by incorporating temporal information. Previous approaches utilizing the deterministic dynamics and steady-state assumption on gene expression states are insufficient to achieve favorable results for the data involving transient process. We present a dynamical approach, Storm (Stochastic models of RNA metabolic-labeling), to overcome these limitations by solving stochastic differential equations of gene expression dynamics. The derivation reveals that the new mRNA sequencing data obeys different types of cell-specific Poisson distributions when jointly considering both biological and cell-specific technical noise. Storm deals with measured counts data directly and extends the RNA velocity methodology based on metabolic labeling scRNA-seq data to transient stochastic systems. Furthermore, we relax the constant parameter assumption over genes/cells to obtain gene-cell-specific transcription/splicing rates and gene-specific degradation rates, thus revealing time-dependent and cell-state-specific transcriptional regulations. Storm will facilitate the study of the statistical properties of tscRNA-seq data, eventually advancing our understanding of the dynamic transcription regulation during development and disease.

Sequestration of gene products by decoys enhances precision in the timing of intracellular events

Expressed gene products often interact ubiquitously with binding sites at nucleic acids and macromolecular complexes, known as decoys. The binding of transcription factors (TFs) to decoys can be crucial in controlling the stochastic dynamics of gene expression. Here, we explore the impact of decoys on the timing of intracellular events, as captured by the time taken for the levels of a given TF to reach a critical threshold level, known as the first passage time (FPT). Although nonlinearity introduced by binding makes exact mathematical analysis challenging, employing suitable approximations and reformulating FPT in terms of an alternative variable, we analytically assess the impact of decoys. The stability of the decoy-bound TFs against degradation impacts FPT statistics crucially. Decoys reduce noise in FPT, and stable decoy-bound TFs offer greater timing precision with less expression cost than their unstable counterparts. Interestingly, when both bound and free TFs decay at the same rate, decoy binding does not directly alter FPT noise. We verify these results by performing exact stochastic simulations. These results have important implications for the precise temporal scheduling of events involved in the functioning of biomolecular clocks, development processes, cell-cycle control, and cell-size homeostasis.

From interacting agents to Boltzmann-Gibbs distribution of money

Abstract We investigate the unbiased model for money exchanges: agents give at random time a dollar to one another (if they have one). Surprisingly, this dynamics eventually leads to a geometric distribution of wealth (shown empirically by Dragulescu and Yakovenko, and rigorously by several follow-up papers). We prove a uniform-in-time propagation of chaos result as the number of agents goes to infinity, which links the stochastic dynamics to a deterministic infinite system of ordinary differential equations. This deterministic description is then analyzed by taking advantage of several entropy–entropy dissipation inequalities and we provide a quantitative almost-exponential rate of convergence toward the equilibrium (geometric distribution) in relative entropy.

A comparison of stochastic and deterministic dynamics of tuberculosis model

Tuberculosis is a deadly infectious disease leading to a major health concern in Southern India while there are major obstacles to controlling its spread. Here, we present a mathematical model with five compartments. We categorize the infected compartment into two subcategories: latent TB-infected individuals and active TB-infected individuals. By introducing white noise in a deterministic system, we formulate a stochastic system. We investigate the model in deterministic and stochastic framework, followed by data calibration using TB infection data from Kanyakumari District in South India from 2019 to 2023. In the deterministic model, we derive the disease-free and endemic equilibrium points, compute the basic reproduction number, and examine their stability. Moreover, we perform sensitivity analysis to evaluate how variations in model parameters affect TB prevalence. In addition, we study the uniqueness of solutions of stochastic model. After that we derive the conditions for disease extinction and stochastic permanence, and execute extensive simulations to capture the variability and randomness in TB transmission dynamics. This study signifies that stochastic dynamics is richer than deterministic one in the way of mitigation TB transmission.

Dynamical theory for adaptive systems

Abstract The study of adaptive dynamics, involving many degrees of freedom on two separated timescales, one for fast changes of state variables and another for the slow adaptation of parameters controlling the former’s dynamics is crucial for understanding feedback mechanisms underlying evolution and learning. We present a path-integral approach à la Martin–Siggia–Rose-De Dominicis–Janssen to analyse non-equilibrium phase transitions in such dynamical systems. As an illustration, we apply our framework to the adaptation of gene-regulatory networks under a dynamic genotype-phenotype map: phenotypic variations are shaped by the fast stochastic gene-expression dynamics and are coupled to the slowly evolving distribution of genotypes, each encoded by a network structure. We establish that under this map, genotypes corresponding to reciprocal networks of coherent feedback loops are selected within an intermediate range of environmental noise, leading to phenotypic robustness.

Characterizing EEG signal dynamics in healthy, seizure-free, and seizure states using the chaos decision tree algorithm

Abstract Epilepsy is a multifaceted neurological condition marked by repetitive seizures that arise from irregular electrical activity in the brain. To understand this condition, a thorough examination of brain signals captured in different states is needed. In order to examine the dynamic behavior of brain signals in three different conditions: healthy, seizure-free, and seizure periods, this study uses the chaos decision tree algorithm. The findings show notable variations in these situations’ dynamics. Chaos is evident during seizure moments, showing extremely chaotic activity. The signals mostly exhibit stochastic behavior in the healthy condition, which is consistent with typical brain dynamics. It is noteworthy that an intermediate state exhibiting a blend of stochastic and chaotic signal dynamics is exhibited throughout the seizure-free time. Furthermore, the research shows that the frequency of chaotic signals rises with increasing proximity to the epileptogenic zone. These discoveries clarify the complex nature of epilepsy and offer insightful information about the dynamic properties of brain signals in various stages, aiding in improved understanding and potential diagnostic approaches.

Cooperation in the face of crisis: effect of demographic noise in collective-risk social dilemmas.

In deciding whether to contribute to a public good, people often face a social dilemma known as the tragedy of the commons: either bear the cost of promoting the collective welfare, or free-ride on the efforts of others. Here, we study the dynamics of cooperation in the context of the threshold public goods games, in which groups must reach a cumulative target contribution to prevent a potential disaster, such as an environmental crisis or social unrest, that could result in the loss of all private wealth. The catch is that the crisis may never materialize, and the investment in the public good is lost. Overall, higher risk of loss promotes cooperation, while larger group size tends to undermine it. For most parameter settings, free-riders (defectors) cannot be eliminated from the population, leading to a coexistence equilibrium between cooperators and defectors for infinite populations. However, this equilibrium is unstable under the effect of demographic noise (finite population), since the cooperator-only and defector-only states are the only absorbing states of the stochastic dynamics. We use simulations and finite-size scaling to show that cooperators eventually die off and derive scaling laws for the transient lifetimes or half-lives of the coexistence metastable state. We find that for high risk, the half-life of cooperators increases exponentially with population size, while for low risk, it decreases exponentially with population size. At the risk threshold, where the coexistence regime appears in a discontinuous manner, the half-life increases with a power of the population size.

The recurrent temporal restricted Boltzmann machine captures neural assembly dynamics in whole-brain activity

Animal behaviour alternates between stochastic exploration and goal-directed actions, which are generated by the underlying neural dynamics. Previously, we demonstrated that the compositional Restricted Boltzmann Machine (cRBM) can decompose whole-brain activity of larval zebrafish data at the neural level into a small number (∼100-200) of assemblies that can account for the stochasticity of the neural activity (van der Plas et al., eLife, 2023). Here, we advance this representation by extending to a combined stochastic-dynamical representation to account for both aspects using the recurrent temporal RBM (RTRBM) and transfer-learning based on the cRBM estimate. We demonstrate that the functional advantage of the RTRBM is captured in the temporal weights on the hidden units, representing neural assemblies, for both simulated and experimental data. Our results show that the temporal expansion outperforms the stochastic-only cRBM in terms of generalization error and achieves a more accurate representation of the moments in time. Lastly, we demonstrate that we can identify the original time-scale of assembly dynamics by estimating multiple RTRBMs at different temporal resolutions. Together, we propose that RTRBMs are a valuable tool for capturing the combined stochastic and time-predictive dynamics of large-scale data sets.

The recurrent temporal restricted Boltzmann machine captures neural assembly dynamics in whole-brain activity.

Animal behaviour alternates between stochastic exploration and goal-directed actions, which are generated by the underlying neural dynamics. Previously, we demonstrated that the compositional Restricted Boltzmann Machine (cRBM) can decompose whole-brain activity of larval zebrafish data at the neural level into a small number (∼100-200) of assemblies that can account for the stochasticity of the neural activity (van der Plas et al., eLife, 2023). Here, we advance this representation by extending to a combined stochastic-dynamical representation to account for both aspects using the recurrent temporal RBM (RTRBM) and transfer-learning based on the cRBM estimate. We demonstrate that the functional advantage of the RTRBM is captured in the temporal weights on the hidden units, representing neural assemblies, for both simulated and experimental data. Our results show that the temporal expansion outperforms the stochastic-only cRBM in terms of generalization error and achieves a more accurate representation of the moments in time. Lastly, we demonstrate that we can identify the original time-scale of assembly dynamics by estimating multiple RTRBMs at different temporal resolutions. Together, we propose that RTRBMs are a valuable tool for capturing the combined stochastic and time-predictive dynamics of large-scale data sets.