Modeling single cell trajectory using forward-backward stochastic differential equations.

Recent advances in single-cell sequencing technology have provided opportunities for mathematical modeling of dynamic developmental processes at the single-cell level, such as inferring developmental trajectories. Optimal transport has emerged as a promising theoretical framework for this task by computing pairings between cells from different time points. However, optimal transport methods have limitations in capturing nonlinear trajectories, as they are static and can only infer linear paths between endpoints. In contrast, stochastic differential equations (SDEs) offer a dynamic and flexible approach that can model non-linear trajectories, including the shape of the path. Nevertheless, existing SDE methods often rely on numerical approximations that can lead to inaccurate inferences, deviating from true trajectories. To address this challenge, we propose a novel approach combining forward-backward stochastic differential equations (FBSDE) with a refined approximation procedure. Our FBSDE model integrates the forward and backward movements of two SDEs in time, aiming to capture the underlying dynamics of single-cell developmental trajectories. Through comprehensive benchmarking on multiple scRNA-seq datasets, we demonstrate the superior performance of FBSDE compared to other methods, highlighting its efficacy in accurately inferring developmental trajectories.

Quantitative analysis and stochastic modeling of osteophyte formation and growth process on human vertebrae based on radiographs: a follow-up study.

Osteophytes are frequently observed in elderly people and most commonly appear at the anterior edge of the cervical and lumbar vertebrae body. The anterior osteophytes keep developing and will lead to neck/back pain over time. In clinical practice, the accurate measurement of the anterior osteophyte length and the understanding of the temporal progression of anterior osteophyte growth are of vital importance to clinicians for effective treatment planning. This study proposes a new measuring method using the osteophyte ratio index to quantify anterior osteophyte length based on lateral radiographs. Moreover, we develop a continuous stochastic degradation model with time-related functions to characterize the anterior osteophyte formation and growth process on cervical and lumbar vertebrae over time. Follow-up data of anterior osteophytes up to 9 years are obtained for measurement and model validation. The agreement test indicates excellent reproducibility for our measuring method. The proposed model accurately fits the osteophyte growth paths. The model predicts the mean time to onset of pain and obtained survival function of the degenerative vertebrae. This research opens the door to future quantification and mathematical modeling of the anterior osteophyte growth on human cervical and lumbar vertebrae. The measured follow-up data is shared for future studies.

Effects of mRNA conformational switching on translational noise in gene circuits.

Intragenic translational heterogeneity describes the variation in translation at the level of transcripts for an individual gene. A factor that contributes to this source of variation is the mRNA structure. Both the composition of the thermodynamic ensemble, i.e., the stationary distribution of mRNA structures, and the switching dynamics between those play a role. The effect of the switching dynamics on intragenic translational heterogeneity remains poorly understood. We present a stochastic translation model that accounts for mRNA structure switching and is derived from a Markov model via approximate stochastic filtering. We assess the approximation on various timescales and provide a method to quantify how mRNA structure dynamics contributes to translational heterogeneity. With our approach, we allow quantitative information on mRNA switching from biophysical experiments or coarse-grain molecular dynamics simulations of mRNA structures to be included in gene regulatory chemical reaction network models without an increase in the number of species. Thereby, our model bridges a gap between mRNA structure kinetics and gene expression models, which we hope will further improve our understanding of gene regulatory networks and facilitate genetic circuit design.

Effective vaccination strategies for human papillomavirus (HPV) infection and cervical cancer based on the mathematical model with a stochastic process.

Human papillomavirus (HPV) infection poses a significant risk to women's health by causing cervical cancer. In addition to HPV, cervical cancer incidence rates can be influenced by various factors, including human immunodeficiency virus and herpes, as well as screening policy. In this study, a mathematical model with stochastic processes was developed to analyze HPV transmission between genders and its subsequent impact on cervical cancer incidence. The model simulations suggest that both-gender vaccination is far more effective than female-only vaccination in preventing an increase in cervical cancer incidence. With increasing stochasticity, the difference between the number of patients in the vaccinated group and the number in the nonvaccinated group diminishes. To distinguish the patient population distribution of the vaccinated from the nonvaccinated, we calculated effect size (Cohen's distance) in addition to Student's t-test. The model analysis suggests a threshold vaccination rate for both genders for a clear reduction of cancer incidence when significant stochastic factors are present.

An approximation of populations on a habitat with large carrying capacity.

We consider stochastic dynamics of a population which starts from a small colony on a habitat with large but limited carrying capacity. A common heuristics suggests that such population grows initially as a Galton-Watson branching process and then its size follows an almost deterministic path until reaching its maximum, sustainable by the habitat. In this paper we put forward an alternative and, in fact, more accurate approximation which suggests that the population size behaves as a special nonlinear transformation of the Galton-Watson process from the very beginning.

Disease X epidemic control using a stochastic model and a deterministic approximation: Performance comparison with and without parameter uncertainties.

BACKGROUND: The spread of infectious diseases can be modeled using deterministic or stochastic models. A deterministic approximation of a stochastic model can be appropriate under some conditions, but is unable to capture the discrete nature of populations. We look into the choice of a model from the perspective of decision making. METHOD: We consider an emerging disease (Disease X) in a closed population modeled by a stochastic SIR model or its deterministic approximation. The objective of the decision maker is to minimize the cumulative number of symptomatic infected-days over the course of the epidemic by picking a vaccination policy. We consider four decision making scenarios: based on the stochastic model or the deterministic model, and with or without parameter uncertainty. We also consider different sample sizes for uncertain parameter draws and stochastic model runs. We estimate the average performance of decision making in each scenario and for each sample size. RESULTS: The model used for decision making has an influence on the picked policies. The best achievable performance is obtained with the stochastic model, knowing parameter values, and for a large sample size. For small sample sizes, the deterministic model can outperform the stochastic model due to stochastic effects. Resolving uncertainties may bring more benefit than switching to the stochastic model in our example. CONCLUSION: This article illustrates the interplay between the choice of a type of model, parameter uncertainties, and sample sizes. It points to issues to be considered when optimizing a stochastic model.

Exact results for gene-expression models with general waiting-time distributions.

Complex molecular details of transcriptional regulation can be coarse-grained by assuming that reaction waiting times for promoter-state transitions, the mRNA synthesis, and the mRNA degradation follow general distributions. However, how such a generalized two-state model is analytically solved is a long-standing issue. Here we first present analytical formulas of burst-size distributions for this model. Then, we derive an iterative equation for the mRNA moment-generating function, by which mRNA raw and binomial moments of any order can be conveniently calculated. The analytical results obtained in the special cases of phase-type waiting-time distributions not only provide insights into the mechanisms of complex transcriptional regulations but also bring conveniences for experimental data-based statistical inferences.

Exploring Molecular Energy Landscapes by Coupling the DFTB Potential with a Tree-Based Stochastic Algorithm: Investigation of the Conformational Diversity of Phthalates.

Exploring the global energy landscape of relatively large molecules at the quantum level is a challenging problem. In this work, we report the coupling of a nonredundant conformational space exploration method, namely, the robotics-inspired iterative global exploration and local optimization (IGLOO) algorithm, with the quantum-chemical density functional tight binding (DFTB) potential. The application of this fast and efficient computational approach to three close-sized molecules of the phthalate family (DBP, BBP, and DEHP) showed that they present different conformational landscapes. These differences have been rationalized by making use of descriptors based on distances and dihedral angles. Coulomb interactions, steric hindrance, and dispersive interactions have been found to drive the geometric properties. A strong correlation has been evidenced between the two dihedral angles describing the side-chain orientation of the phthalate molecules. Our approach identifies low-energy minima without prior knowledge of the potential energy surface, paving the way for future investigations into transition paths and states.

Longtime evolution and stationary response of a stochastic tumor-immune system with resting T cells.

In this paper, we take the resting T cells into account and interpret the progression and regression of tumors by a predator-prey like tumor-immune system. First, we construct an appropriate Lyapunov function to prove the existence and uniqueness of the global positive solution to the system. Then, by utilizing the stochastic comparison theorem, we prove the moment boundedness of tumor cells and two types of T cells. Furthermore, we analyze the impact of stochastic perturbations on the extinction and persistence of tumor cells and obtain the stationary probability density of the tumor cells in the persistent state. The results indicate that when the noise intensity of tumor perturbation is low, tumor cells remain in a persistent state. As this intensity gradually increases, the population of tumors moves towards a lower level, and the stochastic bifurcation phenomena occurs. When it reaches a certain threshold, instead the number of tumor cells eventually enter into an extinct state, and further increasing of the noise intensity will accelerate this process.

Deterministic and stochastic approaches to a minimal model for the transition from autophagy to apoptosis.

Autophagy and apoptosis are crucial cellular mechanisms. The cytoprotective function of autophagy is mediated by the negative regulation of apoptosis, which in turn inhibits autophagy. Although research into the molecular connection between autophagy and apoptosis is booming, the intricate regulatory mechanisms of this process are still not completely understood. Therefore, the objective of this study was to develop a minimal model to explore the transition from autophagy to apoptosis. This biological system was analyzed by comprehensively integrating both the deterministic and the stochastic dynamics of the cells. The system exhibited bistability, and the statistical properties of cells undergoing autophagy and apoptosis were analyzed at two different stress levels with varying noise strengths. Moreover, we investigated how noise affected the double negative feedback loops between autophagy and apoptosis and further triggered transitions at two different stress levels and initial conditions. Finally, the effect of noise on transition was comprehensively studied under continuous stress variations and the two different initial conditions, showing that stronger noise results in more randomness during the switching process. Our work may provide novel insights for further experiments and modeling.

How do stochastic processes and genetic threshold effects explain incomplete penetrance and inform causal disease mechanisms?

Incomplete penetrance is the rule rather than the exception in Mendelian disease. In syndromic monogenic disorders, phenotypic variability can be viewed as the combination of incomplete penetrance for each of multiple independent clinical features. Within genetically identical individuals, such as isogenic model organisms, stochastic variation at molecular and cellular levels is the primary cause of incomplete penetrance according to a genetic threshold model. By defining specific probability distributions of causal biological readouts and genetic liability values, stochasticity and incomplete penetrance provide information about threshold values in biological systems. Ascertainment of threshold values has been achieved by simultaneous scoring of relatively simple phenotypes and quantitation of molecular readouts at the level of single cells. However, this is much more challenging for complex morphological phenotypes using experimental and reductionist approaches alone, where cause and effect are separated temporally and across multiple biological modes and scales. Here I consider how causal inference, which integrates observational data with high confidence causal models, might be used to quantify the relative contribution of different sources of stochastic variation to phenotypic diversity. Collectively, these approaches could inform disease mechanisms, improve predictions of clinical outcomes and prioritize gene therapy targets across modes and scales of gene function. This article is part of a discussion meeting issue 'Causes and consequences of stochastic processes in development and disease'.

Detecting changes in the transmission rate of a stochastic epidemic model.

Throughout the course of an epidemic, the rate at which disease spreads varies with behavioral changes, the emergence of new disease variants, and the introduction of mitigation policies. Estimating such changes in transmission rates can help us better model and predict the dynamics of an epidemic, and provide insight into the efficacy of control and intervention strategies. We present a method for likelihood-based estimation of parameters in the stochastic susceptible-infected-removed model under a time-inhomogeneous transmission rate comprised of piecewise constant components. In doing so, our method simultaneously learns change points in the transmission rate via a Markov chain Monte Carlo algorithm. The method targets the exact model posterior in a difficult missing data setting given only partially observed case counts over time. We validate performance on simulated data before applying our approach to data from an Ebola outbreak in Western Africa and COVID-19 outbreak on a university campus.

Chronic arsenic exposure induces malignant transformation of human HaCaT cells through both deterministic and stochastic changes in transcriptome expression.

Biological processes are inherently stochastic, i.e., are partially driven by hard to predict random probabilistic processes. Carcinogenesis is driven both by stochastic and deterministic (predictable non-random) changes. However, very few studies systematically examine the contribution of stochastic events leading to cancer development. In differential gene expression studies, the established data analysis paradigms incentivize expression changes that are uniformly different across the experimental versus control groups, introducing preferential inclusion of deterministic changes at the expense of stochastic processes that might also play a crucial role in the process of carcinogenesis. In this study, we applied simple computational techniques to quantify: (i) The impact of chronic arsenic (iAs) exposure as well as passaging time on stochastic gene expression and (ii) Which genes were expressed deterministically and which were expressed stochastically at each of the three stages of cancer development. Using biological coefficient of variation as an empirical measure of stochasticity we demonstrate that chronic iAs exposure consistently suppressed passaging related stochastic gene expression at multiple time points tested, selecting for a homogenous cell population that undergo transformation. Employing multiple balanced removal of outlier data, we show that chronic iAs exposure induced deterministic and stochastic changes in the expression of unique set of genes, that populate largely unique biological pathways. Together, our data unequivocally demonstrate that both deterministic and stochastic changes in transcriptome-wide expression are critical in driving biological processes, pathways and networks towards clonal selection, carcinogenesis, and tumor heterogeneity.

Life Is Uncertain: Inherent Variability Exhibited by Organisms, and at Higher Levels of Biological Organization.

Organisms act stochastically. A not uncommon view in the ecological literature is that this is mainly due to the observer having insufficient information or a stochastic environment-and not partly because organisms themselves respond with inherent unpredictability. In this study, I compile the evidence that contradicts that view. Organisms generate uncertainty internally, which results in irreducible stochastic responses. I consider why: for instance, stochastic responses are associated with greater adaptability to changing environments and resource availability. Over longer timescales, biologically generated uncertainty influences behavior, evolution, and macroecological processes. Indeed, it could be stated that organisms are systems defined by the internal generation, magnification, and record-keeping of uncertainty as inputs to responses. Important practical implications arise if organisms can indeed be defined by an association with specific classes of inherent uncertainty: not least that isolating those signatures then provides a potential means for detecting life, for considering the forms that life could theoretically take, and for exploring the wider limits to how life might become distributed. These are all fundamental goals in astrobiology.

Effects of bursty synthesis in organelle biogenesis.

A fundamental question of cell biology is how cells control the number of organelles. The processes of organelle biogenesis, namely de novo synthesis, fission, fusion, and decay, are inherently stochastic, producing cell-to-cell variability in organelle abundance. In addition, experiments suggest that the synthesis of some organelles can be bursty. We thus ask how bursty synthesis impacts intracellular organelle number distribution. We develop an organelle biogenesis model with bursty de novo synthesis by considering geometrically distributed burst sizes. We analytically solve the model in biologically relevant limits and provide exact expressions for the steady-state organelle number distributions and their means and variances. We also present approximate solutions for the whole model, complementing with exact stochastic simulations. We show that bursts generally increase the noise in organelle numbers, producing distinct signatures in noise profiles depending on different mechanisms of organelle biogenesis. We also find different shapes of organelle number distributions, including bimodal distributions in some parameter regimes. Notably, bursty synthesis broadens the parameter regime of observing bimodality compared to the 'non-bursty' case. Together, our framework utilizes number fluctuations to elucidate the role of bursty synthesis in producing organelle number heterogeneity in cells.

Compartmental exchange regulates steady states and stochastic switching of a phosphorylation network.

The phosphoregulation of proteins with multiple phosphorylation sites is governed by biochemical reaction networks that can exhibit multistable behavior. However, the behavior of such networks is typically studied in a single reaction volume, while cells are spatially organized into compartments that can exchange proteins. In this work, we use stochastic simulations to study the impact of compartmentalization on a two-site phosphorylation network. We characterize steady states and fluctuation-driven transitions between them as a function of the rate of protein exchange between two compartments. Surprisingly, the average time spent in a state before stochastically switching to another depends nonmonotonically on the protein exchange rate, with the most frequent switching occurring at intermediate exchange rates. At sufficiently small exchange rates, the state of the system and mean switching time are controlled largely by fluctuations in the balance of enzymes in each compartment. This leads to negatively correlated states in the compartments. For large exchange rates, the two compartments behave as a single effective compartment. However, when the compartmental volumes are unequal, the behavior differs from a single compartment with the same total volume. These results demonstrate that exchange of proteins between distinct compartments can regulate the emergent behavior of a common signaling motif.

Solving the time-dependent protein distributions for autoregulated bursty gene expression using spectral decomposition.

In this study, we obtain an exact time-dependent solution of the chemical master equation (CME) of an extension of the two-state telegraph model describing bursty or non-bursty protein expression in the presence of positive or negative autoregulation. Using the method of spectral decomposition, we show that the eigenfunctions of the generating function solution of the CME are Heun functions, while the eigenvalues can be determined by solving a continued fraction equation. Our solution generalizes and corrects a previous time-dependent solution for the CME of a gene circuit describing non-bursty protein expression in the presence of negative autoregulation [Ramos et al., Phys. Rev. E 83, 062902 (2011)]. In particular, we clarify that the eigenvalues are generally not real as previously claimed. We also investigate the relationship between different types of dynamic behavior and the type of feedback, the protein burst size, and the gene switching rate.

Stationary distribution and extinction of a stochastic HIV/AIDS model with nonlinear incidence rate.

This paper studies a stochastic HIV/AIDS model with nonlinear incidence rate. In the model, the infection rate coefficient and the natural death rates are affected by white noise, and infected people are affected by an intervention strategy. We derive the conditions of extinction and permanence for the stochastic HIV/AIDS model, that is, if $ R_0/ < 1, $ HIV/AIDS will die out with probability one and the distribution of the susceptible converges weakly to a boundary distribution; if $ R_0/ > 1 $, HIV/AIDS will be persistent almost surely and there exists a unique stationary distribution. The conclusions are verified by numerical simulation.

Nitrogen deposition mediates more stochastic processes in structuring plant community than soil microbial community in the Eurasian steppe.

Anthropogenic environmental changes may affect community assembly through mediating both deterministic (e.g., competitive exclusion and environmental filtering) and stochastic processes (e.g., birth/death and dispersal/colonization). It is traditionally thought that environmental changes have a larger mediation effect on stochastic processes in structuring soil microbial community than aboveground plant community; however, this hypothesis remains largely untested. Here we report an unexpected pattern that nitrogen (N) deposition has a larger mediation effect on stochastic processes in structuring plant community than soil microbial community (those <2 mm in diameter, including archaea, bacteria, fungi, and protists) in the Eurasian steppe. We performed a ten-year nitrogen deposition experiment in a semiarid grassland ecosystem in Inner Mongolia, manipulating nine rates (0-50 g N m-2 per year) at two frequencies (nitrogen added twice or 12 times per year) under two grassland management strategies (fencing or mowing). We separated the compositional variation of plant and soil microbial communities caused by each treatment into the deterministic and stochastic components with a recently-developed method. As nitrogen addition rate increased, the relative importance of stochastic component of plant community first increased and then decreased, while that of soil microbial community first decreased and then increased. On the whole, the relative importance of stochastic component was significantly larger in plant community (0.552±0.035; mean±standard error) than in microbial community (0.427±0.035). Consistently, the proportion of compositional variation explained by the deterministic soil and community indices was smaller for plant community (0.172-0.186) than microbial community (0.240-0.767). Meanwhile, as nitrogen addition rate increased, the linkage between plant and microbial community composition first became weaker and then became stronger. The larger stochasticity in plant community relative to microbial community assembly suggested that more stochastic strategies (e.g., seeds addition) should be adopted to maintain above- than below-ground biodiversity under the pressure of nitrogen deposition.

Asymptotic properties of the Lotka-Volterra competition and mutualism model under stochastic perturbations.

Stochastically perturbed models, where the white noise type stochastic perturbations are proportional to the current system state, the most realistically describe real-life biosystems. However, such models essentially have no equilibrium states apart from one at the origin. This feature makes analysis of such models extremely difficult. Probably, the best result that can be found for such models is finding of accurate estimations of a region in the model phase space that serves as an attractor for model trajectories. In this paper, we consider a classical stochastically perturbed Lotka-Volterra model of competing or symbiotic populations, where the white noise type perturbations are proportional to the current system state. Using the direct Lyapunov method in a combination with a recently developed technique, we establish global asymptotic properties of this model. In order to do this, we, firstly, construct a Lyapunov function that is applicable to the both competing (and globally stable) and symbiotic deterministic Lotka-Volterra models. Then, applying this Lyapunov function to the stochastically perturbed model, we show that solutions with positive initial conditions converge to a certain compact region in the model phase space and oscillate around this region thereafter. The direct Lyapunov method allows to find estimates for this region. We also show that if the magnitude of the noise exceeds a certain critical level, then some or all species extinct via process of the stochastic stabilization ('stabilization by noise'). The approach applied in this paper allows to obtain necessary conditions for the extinction. Sufficient conditions for the extinction (that for this model occurs via the process that is known as the 'stochastic stabilization', or the 'stabilization by noise') are found applying the Khasminskii-type Lyapunov functions.