B cells are important components of the adaptive immune system, responsible for antibody production and working as antigen-presenting cells. B cells display protein receptors on their membrane, which bind with foreign antigens and process them before presenting them to T cells. In this work, we present a stochastic process modeling the dynamics of such receptors on the B cell. The model consists of a two-dimensional birth-death process {(X(t),Y(t)),t≥0} having linear transition rates, where X(t) and Y(t) represent the number of free and occupied receptors, respectively. After determining the partial differential equation for the probability generating function of the process, we compute the main moments of the process, including the covariance. The transient and asymptotic behavior of the means of X(t) and Y(t) is also studied. Throughout the paper, we provide insights into the biological significance of each parameter on the system's dynamics. In addition, we conduct a sensitivity analysis to assess how variations in the model parameters affect the first-order moments. Such analysis shows that minimal variations of the parameters representing the binding frequency of antigens and B-cell receptors, when happening in the initial instants of the process, result in noticeable alterations of the number of occupied receptors.

Adaptive physics-informed neural operator and subset simulation for high-dimensional reliability analysis with multiple stochastic processes in stochastic differential equations

Nitrogen loading fluctuations impact microbial community assembly and functional redundancy in anammox reactors.

Moment stability of McKean-Vlasov stochastic recurrent neural networks with mixed delays.

Stochastic process dominated community assembly during high-temperature Daqu storage, leading to reduced microbial co-occurrence network complexity.

Boundary-catalytic branching processes describe a broad class of natural phenomena where the population of diffusing particles grows due to their spontaneous binary branching (e.g., division, fission, or splitting) on a catalytic boundary located in a complex environment. We investigate the possibility of the geometric control of the population growth by compensating for the proliferation of particles due to catalytic branching events by their absorptions in the bulk or on absorbing regions of the boundary. We identify an appropriate Steklov spectral problem to obtain the phase diagram of this out-of-equilibrium stochastic process. The principal eigenvalue determines the critical line that separates an exponential growth of the population from its extinction in a bounded domain. In other words, we establish a powerful tool for calculating the optimal absorption rate that equilibrates the opposite effects of branching and absorption events and, thus, results in steady-state behavior of this diffusion-reaction system. Moreover, we show the existence of a critical catalytic rate above which no compensation is possible, so that the population cannot be controlled and keeps growing exponentially. The proposed framework opens promising perspectives for better understanding, modeling, and control of various boundary-catalytic branching processes, with applications in physics, chemistry, and life sciences.

A control theoretical approach to gene regulation reveals quantitative constraints for dynamic homeostasis in stochastic gene expression.

Chemical Organization Theory (COT) provides a structural and algebraic framework for analyzing reaction-based systems independently of kinetic assumptions. By identifying closed and self-maintaining sets of species, called organizations, the theory characterizes all structurally admissible persistent configurations of a reaction network. This review synthesizes the mathematical foundations of COT, including closure operators, stoichiometric feasibility, and lattice structures, together with their algorithmic and dynamical interpretations. We examine how organizational structure constrains long-term behavior in reaction-based dynamical systems, including ordinary differential equations, stochastic processes, and spatial reaction-diffusion models. Computational methods for enumerating organizations and distributed organizations are reviewed, alongside extensions to discrete and stochastic settings. Applications spanning atmospheric chemistry, virus dynamics, gene regulation, cell cycle models, and data-driven reaction systems illustrate the breadth and versatility of the framework. Relations to chemical reaction network theory, autocatalytic set theory, and constraint-based approaches are clarified to position COT within the broader landscape of mathematical reaction network analysis. We conclude by highlighting open problems related to transient dynamics, structural transitions, computational scalability, and evolutionary processes, and by emphasizing COT as a unifying structural abstraction for persistence and qualitative behavior in complex reaction systems.

Microbial community assembly (MCA) is the key to biological wastewater treatment by dynamically shaping the functional populations responsible for pollutant removal through deterministic selection and stochastic ecological processes, but identifying temporal drivers for the MCA remains a challenge. This study reported a stochastic physics-informed deep learning (SPI-DL) framework that embedded generalized Lotka-Volterra (gLV) models, stochastic differential equations (SDEs), and SDE integrators to identify deterministic and stochastic dynamics for MCA based on limited-view and sparsely sampled data. The effectiveness was verified by MCA for nitrifying groups of ammonia-oxidizing bacteria (AOB) and nitrite-oxidizing bacteria (NOB) as a representative case. The log-likelihood decoupling (LLD) method coupled with SHAP analysis was proposed to unravel the time-resolved relative contribution of the deterministic and stochastic factors. As indicated by driver decomposition based on LLD/SHAP analysis, stochastic variability was mainly linked to flow rate and hydraulic retention time, whereas deterministic succession was associated with selective covariates (e.g., DO for NOB; NH4-N/TN for AOB). The SPI-DL+LLD framework demonstrates satisfactory representability, predictability, and generalizability in explaining and identifying temporal drivers for MCA in WWTPs, which has important implications for precise process control and optimization of smart wastewater treatment systems.

Financial time series forecasting poses significant challenges due to the diverse risk profiles and dynamic behaviors of assets such as the S&P 500, NASDAQ, and Bitcoin, especially across different market periods. This study introduces a novel framework, waveletstochastic-chaos informed machine learning, that integrates wavelet transforms, stochastic processes, and chaos theory to improve machine learning prediction accuracy over a decade (2015–2025). The analysis is divided into four distinct periods: All Time, PreCOVID, COVID, and Post-COVID. The aim is to capture the multi-scale patterns, volatility, and complexity inherent in financial data, which will be assessed across various market conditions. The framework outperforms the baseline maximum likelihood models in most scenarios, achieving significant root mean squared errors for the scaled price predictions of S&P 500 (e.g., from 0.0348 to 0.0122 in All Time), NASDAQ (e.g., from 0.0284 to 0.0180 in All Time) and Bitcoin (e.g., from 0.0838 to 0.0288 in All Time) based on 1000 experimental trials. It excels in volatile periods like COVID and for high-risk Bitcoin, though it slightly underperforms in the stable Post-COVID recovery for S&P 500. Wavelet features are found to be critical for accuracy. Additionally, stochastic and chaos-based elements enhance performance in volatile and complex contexts, respectively, as confirmed by ablation studies. This study provides empirical evidence of predictive utility for financial time-series forecasting in assets with different dynamics and market regimes. The results indicate that multi-scale, stochastic, and complexity-based feature representations can improve forecasting performance within the examined datasets, suggesting that the framework may apply to other non-stationary time-series settings, although such extensions remain for future investigation.

Abstract Stochastically gated interfaces play an important role in a variety of cellular transport processes, including diffusion through membrane ion channels and intercellular gap junctions. Most studies of stochastically-gated interfaces are based on macroscopic models that track the particle concentration averaged with respect to different realisations of the gate dynamics. In this paper we develop a novel probabilistic model of single-particle Brownian motion (BM) through a stochastically gated interface. We proceed by constructing a renewal equation for one-dimensional BM with an interface at the origin, which effectively sews together a sequence of BMs on the half-line with a totally absorbing boundary at $x=0$. Each time the particle is absorbed, the stochastic process is immediately restarted according to the following rule: if the gate is closed then BM restarts on the same side of the interface, whereas if the gate is open then BM restarts on either side of the interface with equal probability. In order to ensure that diffusion restarts in a state that avoids immediate re-absorption. we assume that whenever the particle reaches the interface it is instantaneously shifted a distance $\epsilon$ from the origin. We explicitly solve the renewal equation for $\epsilon>0$ and show how the solution of a corresponding forward Kolmogorov equation is recovered in the limit $\epsilon\rightarrow 0$. However, the renewal equation provides a more general mathematical framework for modelling a stochastically gated interface by explicitly separating the first passage time problem of detecting the gated interface (absorption) and the subsequent rule for restarting BM. We illustrate this by calculating the non-equilibrium stationary state across an interface in the presence of stochastic resetting. We conclude by discussing some of the mathematical challenges in extending the theory to higher-dimensional interfaces.

Vertical distribution pattern, preferred life strategies and environmental response of prokaryotic microbiome in the eastern tropical indian ocean.

ABSTRACT A collection of time series are “related” if they follow similar stochastic processes and/or they are statistically dependent. This paper proposes a related time series (RTS) forecasting model that exploits these relationships. The model's foundation is a set of univariate Gaussian autoregressions, one for each series, which are then augmented to incorporate stochastic volatility, heavy‐tailed innovations, additive outliers, time‐varying parameters and common factors. The model is estimated and forecasts are computed using Bayesian methods with hierarchical priors that pool information across series. Computationally efficient MCMC methods are proposed. The RTS model is applied to three datasets and yields encouraging pseudo‐out‐of‐sample forecasting results.

Dynamics of number entropy for free fermionic systems in the presence of defects and stochastic processes

Agricultural greenhouses are usually located in the suburbs or remote areas away from towns, and generally speaking, the cost of transmission and power supply is high, and some remote areas do not even have electricity supply. However, traditional greenhouses contain many different electrical equipment and facilities, and a stable power supply is essential for the normal, economical and efficient operation of the greenhouse. Modern agricultural greenhouses also need to be equipped with complete lighting systems, temperature and humidity control systems, ventilation systems, carbon dioxide concentration controlsystems, irrigation sprinkler systems, etc., which are difficult for traditional greenhouses to achievesmoothly. Traditional greenhouses are usually covered with plastic film, which usually needs to be replaced every year, and the discarded plastic film does not meet the requirements of energy conservation and environmental protection. The problem of "thermal insulation" in greenhouses has also been plaguing greenhouse growers. From the perspective of planting cycle, traditional greenhouses are generally onlyplanted twice a year, and the economic benefits are limited.

Extreme acidic environments represent natural laboratories for investigating the mechanisms of microbial community assembly, yet the ecological processes structuring these communities remain incompletely understood. Here, we investigate how spatial partitioning, hydrodynamics, and colonization history shape microbial succession in a unique sulfur-rich, acidic river of volcanic origin in northern Patagonia. We combined 16S rRNA gene profiling and shotgun metagenomics with a multi-scale experimental framework encompassing water column fractionation and colonization assays under native and controlled conditions. Microbial diversity was strongly influenced by spatial fractionation, with free-living communities exhibiting higher richness and temporal variability than particle-associated assemblages. Water flow modulated community structure, increasing evenness in free-living fractions under high-flow conditions, but had limited impact on particle-attached communities. Colonization of sulfur-beads followed a structured successional trajectory, with autotrophic sulfur oxidizers dominating early stages and heterotrophs adapted to biofilm lifestyles increasing over time. Ex situ recolonization assays revealed strong priority effects, with initial colonizers determining successional trajectories. Turnover analyses revealed that the balance among stochastic and deterministic assembly processes shifted across communities with pronounced stochasticity in the water column and flow-dependent effects in free-living communities, while biofilm associated communities on sulfur-beads exhibited stronger contribution of deterministic selection. These ecological patterns were mirrored by functional differentiation, with gene enrichment analyses revealing adaptive signatures of substrate attachment and resource acquisition. By integrating fine-scale environmental variation with colonization dynamics, this study reveals how microscale habitat structure and temporal fluxes jointly modulate microbial community assembly rules, offering a nuanced framework to dissect ecological processes in extreme systems.

BackgroundCrop wild relatives and their microbiomes are essential for sustainable crop production. However, the co-evolution of wild rice species and their microbiomes remains poorly understood. Herein, we investigated microbiome assembly across 17 wild rice and one cultivated rice species under controlled conditions spanning ~15 million years of evolution.ResultsOur data reveal distinct eco-evolutionary patterns for bacteria and fungi. Host divergence time was the predominant driver of root microbiota structure, outweighing polyploidy and life cycle, and exerted a stronger effect on bacteria than fungi. Bacterial community exhibited a significant phylosymbiosis with its host, but fungi did not. Over evolutionary time, bacterial diversity decreased while phylogenetic clustering increased. Deterministic and stochastic processes co-drove bacteria assembly, whereas stochastic processes strongly drove fungi assembly. Potentially functional taxa, including nitrogen-fixing and methane-cycle bacteria, were differentially enriched across evolutionary time and polyploidization events. Notably, co-speciating bacteria better predicted grain weight than fungi, with core species making a major contribution. Using a synthetic community (SynCom) derived from the wild rice core microbiome and four nitrogen-fixing strains enriched in early- and medium-diverging Oryza species, we demonstrated that the SynCom strongly promoted rice growth, with the removal of key members markedly reducing its impact.ConclusionsThese results reveal co-phylogenetic patterns between Oryza and root-associated bacteria, highlighting the closer functional linkage between rice traits and bacteria than fungi, likely due to their co-evolution. Our findings provide new insights into crop–microbiome symbiosis from an eco-evolutionary perspective and underscore the importance of co-speciating microbiomes from wild relatives in supporting crop growth.Video Supplementary InformationThe online version contains supplementary material available at 10.1186/s40168-026-02359-z.

Despite their ubiquity in Nature, spikes or stingers rarely exhibit sharp tips. Instead, a closer inspection of their roughly conical tips reveals a striking similarity in their profiles: They adhere to a power-law, [Formula: see text], where [Formula: see text]. This conformity persists across diverse spatial scales and materials. The mechanistic basis for this universality was recently attributed to evolutionary selection for ease of piercing [H. Quan et al., Proc. Natl. Acad. Sci. U.S.A. 121, e2316320121 (2024)]. However, the transient nature of their morphology, progressively modified by repeated use and inevitable wear, has received little scrutiny. In this work, we combine tabletop experiments with continuum analysis to demonstrate that the universal tip morphology can result from stochastic weathering processes. This finding is particularly significant in light of recent observations of the same tip geometry on dissolving or melting solids and geomorphic structures in addition to biological stingers. Our results suggest that the prevalence of this power-law profile may not be the result of evolutionary selection, but rather an inevitable consequence of exposure to random erosive processes.

Computing power networks play a crucial role in supporting modern digital infrastructure, whose resilience needs to be improved due to the increasingly frequent and severe disruptive events. This paper introduces a phase-based framework for resilience assessment, coupled with an integrated resource allocation model that jointly optimises pre- and post-disaster strategies to systematically quantify and enhance the resilience of computing power networks. The system behaviour is modelled as a semi-Markov chain, capturing the stochastic dynamics of multi-state transitions under disruptions. Overall resilience is evaluated as a synthesis of four principal attributes: resistance, absorption, adaptation, and restoration, each characterised by both inherent and acquired measures. Analytical expressions for these measures are derived using the theory of aggregated stochastic processes, enabling efficient and accurate resilience quantification. Furthermore, a resource allocation model is developed to optimise resilience improvement strategies under resource constraints, incorporating both pre- and post-disaster decisions. Two case studies on homogeneous and heterogeneous computing power networks are conducted to validate the applicability of the proposed evaluation framework and demonstrate the superiority of the integrated resource allocation strategy in enhancing resilience. Finally, some managerial insights are provided to designers and managers responsible for resilient computing power network design and emergency decision-making.

The tumour microenvironment (TME) is a complex dynamic ecosystem where cancer cells, immune responses, and the extracellular matrix (ECM) engage in a delicate battle for dominance. Capturing this interesting interplay within a mathematically tractable yet biologically meaningful framework remains a key challenge. In this work, we construct and analyse a novel three-dimensional ODE model that describes the interaction between cancer cells, immune response, and ECM and reveals an interestingly rich dynamical landscape characterized by multiple bifurcation phenomena, deterministic chaos, and stochastic transitions. Through rigorous bifurcation analysis, we uncover transcritical, saddle-node, cusp, and Bogdanov–Takens bifurcations, each portraying critical transitions between distinct tumour states. The system further shows deterministic chaos, illustrating how intrinsic nonlinear interaction can induce unpredictable tumour behaviour even in the absence of external perturbations. Afterward we investigate how random environmental effects modulate tumour persistence or extinction through stochastic sensitivity by incorporating random fluctuations into biologically relevant parameters. Our findings demonstrate that ECM degradation, coupled with immune suppression, orchestrates a key mechanism driving tumour persistence and escape. Theoretically and biologically, our study provides a unified framework that links deterministic and stochastic processes to better understand and control cancer progression.