Stochastic models for cell motion and taxis

Certain biological experiments investigating cell motion result in time lapse video microscopy data which may be modeled using stochastic differential equations. These models suggest statistics for quantifying experimental results and testing relevant hypotheses, and carry implications for the qualitative behavior of cells and for underlying biophysical mechanisms. Directional cell motion in response to a stimulus, termed taxis, has previously been modeled at a phenomenological level using the Keller-Segel diffusion equation. The Keller-Segel model cannot distinguish certain modes of taxis, and this motivates the introduction of a richer class of models which is nevertheless still amenable to statistical analysis. A state space model formulation is used to link models proposed for cell velocity to observed data. Sequential Monte Carlo methods enable parameter estimation via maximum likelihood for a range of applicable models. One particular experimental situation, involving the effect of an electric field on cell behavior, is considered in detail. In this case, an Ornstein- Uhlenbeck model for cell velocity is found to compare favorably with a nonlinear diffusion model.     

An investigation of the plausibility of stochastic resonance in tubulin dimers

This paper considers the possibility of stochastic resonance (SR) in tubulin dimers. A formula for the signal-to-noise ratio (SNR) of tubulin as a function of temperature is derived. The effective potential experienced by a delocalized electron in such a dimer is postulated to be a symmetric bimodal well. Inter-well and intra-well motions are described by Kramers rate theory and the Langevin formalism respectively. The frequency-dependent expression for the SNR shows that the response of the electron-tubulin dimer system is enhanced by ambient dipolar oscillations in specific frequency regimes. This is a characteristic of SR. Biophysical implications of this property such as the relevance to 8.085 MHz microtubule resonance and the clocking mechanism are detailed.     

Stochastic model of leukocyte chemosensory movement

Journal of Mathematical Biology - We propose a hypothesis for a unified understanding of the persistent and biased random walk behavior of leukocytes exhibiting random motility and chemotaxis,...     

随机Lotka-Volterra系统的稳定性

利用Lyapunov方法与K．lto公式及鞅的理论,研究了随机Lotka-Volterra系统正平衡点的全局渐近稳定性．得到了随机全局渐近稳定的主要定理,并以确定性系统的全局稳定性作为定理的推论．     

Stochastic models for inferring genetic regulation from microarray gene expression data

Microarray expression profiles are inherently noisy and many different sources of variation exist in microarray experiments. It is still a significant challenge to develop stochastic models to realize noise in microarray expression profiles, which has profound influence on the reverse engineering of genetic regulation. Using the target genes of the tumour suppressor gene p53 as the test problem, we developed stochastic differential equation models and established the relationship between the noise strength of stochastic models and parameters of an error model for describing the distribution of the microarray measurements. Numerical results indicate that the simulated variance from stochastic models with a stochastic degradation process can be represented by a monomial in terms of the hybridization intensity and the order of the monomial depends on the type of stochastic process. The developed stochastic models with multiple stochastic processes generated simulations whose variance is consistent with the prediction of the error model. This work also established a general method to develop stochastic models from experimental information.     

Stochastic resonance as a possible mechanism of amplification of weak electric signals in living cells

The most important but still unresolved problem in bioelectromagnetics is the interaction of weak electromagnetic fields (EMFs) with living cells. Thermal and other types of noise pose restrictions in cell detection of weak signals. As a consequence, some extant experimental results that indicate low-intensity field effects cannot be accounted for, and this renders the results themselves questionable. One way out of this dead end is to search for possible mechanisms of signal amplification. In this paper, we discuss a general mechanism in which a weak signal is amplified by system noise itself. This mechanism was discovered several years ago in physics and is known, in its simplest form, as a stochastic resonance. It was shown that signal amplification may exceed a factor of 1000, which renders existing estimations of EMF thresholds highly speculative. The applicability of the stochastic resonance concept to cells is discussed particularly with respect to the possible role of the cell membrane in the amplification process. © 1994 Wiley-Liss, Inc.     

Exploring the implicit interlayer regulatory mechanism between cells and tissue: Stochastic mathematical analyses of the spontaneous ordering in beating synchronization

The present study focused on beating synchronization, and tried to elucidate the interlayer regulatory mechanisms between the cells and clump in beating synchronization with using the stochastic simulations which realize the beating synchronizations in beating cells with low cell–cell conductance. Firstly, the fluctuation in interbeat intervals (IBIs) of beating cells encouraged the process of beating synchronization, which was identified as the stochastic resonance. Secondly, fluctuation in the synchronized IBIs of a clump decreased as the number of beating cells increased. The decrease in IBI fluctuation due to clump formation implied both a decline of the electrophysiological plasticity of each beating cell and an enhancement of the electrophysiological stability of the clump. These findings were identified as the community effects. Because IBI fluctuation and the community effect facilitated the beating stability of the cell and clump, these factors contributed to the spontaneous ordering in beating synchronization. Thirdly, the cellular layouts in clump affected the synchronized beating rhythms. The synchronized beating rhythm in clump was implicitly regulated by a complicated synergistic effect among IBI fluctuation of each beating cell, the community effect and the cellular layout. This finding was indispensable for leading an elucidation of mechanism of emergence. The stochastic simulations showed the necessity of considering the synergistic effect, to elucidate the interlayer regulatory mechanisms in biological system.     

Sustained oscillations via coherence resonance in SIR

Sustained oscillations in a stochastic SIR model are studied using a new multiple scale analysis. It captures the interaction of the deterministic and stochastic elements together with the separation of time scales inherent in the appearance of these dynamics. The nearly regular fluctuations in the infected and susceptible populations are described via an explicit construction of a stochastic amplitude equation. The agreement between the power spectral densities of the full model and the approximation verifies that coherence resonance is driving the behavior. The validity criteria for this asymptotic approximation give explicit expressions for the parameter ranges in which one expects to observe this phenomenon.     

Synchronization and stochastic resonance of the small-world neural network based on the CPG

According to biological knowledge, the central nervous system controls the central pattern generator (CPG) to drive the locomotion. The brain is a complex system consisting of different functions and different interconnections. The topological properties of the brain display features of small-world network. The synchronization and stochastic resonance have important roles in neural information transmission and processing. In order to study the synchronization and stochastic resonance of the brain based on the CPG, we establish the model which shows the relationship between the small-world neural network (SWNN) and the CPG. We analyze the synchronization of the SWNN when the amplitude and frequency of the CPG are changed and the effects on the CPG when the SWNN’s parameters are changed. And we also study the stochastic resonance on the SWNN. The main findings include: (1) When the CPG is added into the SWNN, there exists parameters space of the CPG and the SWNN, which can make the synchronization of the SWNN optimum. (2) There exists an optimal noise level at which the resonance factor Q gets its peak value. And the correlation between the pacemaker frequency and the dynamical response of the network is resonantly dependent on the noise intensity. The results could have important implications for biological processes which are about interaction between the neural network and the CPG.     

Stochastic modeling of nonlinear epidemiology

The objectives of this paper to analyse, model and simulate the spread of an infectious disease by resorting to modern stochastic algorithms. The approach renders it possible to circumvent the simplifying assumption of linearity imposed in the majority of the past works on stochastic analysis of epidemic processes. Infectious diseases are often transmitted through contacts of those infected with those susceptible; hence the processes are inherently nonlinear. According to the classical model of Kermack and McKendrick, or the SIR model, three classes of populations are involved in two types of processes: conversion of susceptibles (S) to infectives (I) and conversion of infectives to removed (R). The master equations of the SIR process have been formulated through the probabilistic population balance around a particular state by considering the mutually exclusive events. The efficacy of the present methodology is mainly attributable to its ability to derive the governing equations for the means, variances and covariance of the random variables by the method of system-size expansion of the nonlinear master equations. Solving these equations simultaneously along with rates associated influenza epidemic data yields information concerning not only the means of the three populations but also the minimal uncertainties of these populations inherent in the epidemic. The stochastic pathways of the three different classes of populations during an epidemic, i.e. their means and the fluctuations around these means, have also been numerically simulated independently by the algorithm derived from the master equations, as well as by an event-driven Monte Carlo algorithm. The master equation and Monte Carlo algorithms have given rise to the identical results.     

Stochastic instability and Liapunov stability

Summary The relationship between the deterministic stability of nonlinear ecological models and the properties of the stochastic model obtained by adding weak random perturbations is studied. It is shown that the expected escape time for the stochastic model from a bounded region with nonsingular boundary is determined by a Liapunov function for the nonlinear deterministic model. This connection between stochastic and deterministic models brings together various notions of persistence and vulnerability of ecosystems as defined for deterministically perturbed or randomly perturbed models.     

Stochastic dynamical aspects of neuronal activity

A stochastic model of neuronal activity is proposed. Some stochastic differential equations based on jump processes are used to investigate the behavior of the membrane potential at a time scale small with respect to the neuronal states time evolution. A model for learning, implying short memory effects, is described.     

A Derivative Matching Approach to Moment Closure for the Stochastic Logistic Model

Continuous-time birth-death Markov processes serve as useful models in population biology. When the birth-death rates are nonlinear, the time evolution of the first n order moments of the population is not closed, in the sense that it depends on moments of order higher than n. For analysis purposes, the time evolution of the first n order moments is often made to be closed by approximating these higher order moments as a nonlinear function of moments up to order n, which we refer to as the moment closure function. In this paper, a systematic procedure for constructing moment closure functions of arbitrary order is presented for the stochastic logistic model. We obtain the moment closure function by first assuming a certain separable form for it, and then matching time derivatives of the exact (not closed) moment equations with that of the approximate (closed) equations for some initial time and set of initial conditions. The separable structure ensures that the steady-state solutions for the approximate equations are unique, real and positive, while the derivative matching guarantees a good approximation, at least locally in time. Explicit formulas to construct these moment closure functions for arbitrary order of truncation n are provided with higher values of n leading to better approximations of the actual moment dynamics. A host of other moment closure functions previously proposed in the literature are also investigated. Among these we show that only the ones that achieve derivative matching provide a close approximation to the exact solution. Moreover, we improve the accuracy of several previously proposed moment closure functions by forcing derivative matching.     

Modeling of Nonlinear Chemical Reaction Systems and Two-Parameter Stochastic Resonance

The idea of stochastic resonance (SR) is extended to two-parameter dynamical systems based on the Oregonator model of the Belousov-Zhabotinsky (BZ) reaction. The first case presents the photosensitivity of the reaction, and light flux and a flow rate are the two control parameters. The second case presents the effect of temperature on the oscillatory behaviors, and temperature and a flow rate are the control parameters. Stochastic resonance is demonstrated in the first case in which a signal and noise are applied to the different inputs, respectively. The scenario and novel aspects of SR in two-parameter systems are discussed, and the possibility of the analogous SR in biological systems is also pointed out.     

Stochastic dynamics of metastasis formation

Tumor metastasis accounts for the majority of deaths in cancer patients. The metastatic behavior of cancer cells is promoted by mutations in many genes, including activation of oncogenes such as RAS and MYC. Here, we develop a mathematical framework to analyse the dynamics of mutations enabling cells to metastasize. We consider situations in which one mutation is necessary to confer metastatic ability to the cell. We study different population sizes of the main tumor and different somatic fitness values of metastatic cells. We compare mutations that are positively selected in the main tumor with those that are neutral or negatively selected, but faster at forming metastases. We study whether metastatic potential is the property of all (or the majority of) cells in the main tumor or only the property of a small subset. Our theory shows how to calculate the expected number of metastases that are formed by a tumor.     

Stochastic modeling of controlled-drug release

A drug release process by the oral route is random in nature and thus is subject to constant fluctuations. Moreover, individuals have varied tolerances to such fluctuations. The objective of this work is to characterize these fluctuations by a stochastic formalism. The system under consideration, i.e., the gastrointestinal tract consists of four consecutive compartments, i.e., stomach, duodenum, jejunum, and ileum. The master equation of the system as well as the governing equations for the means, variances, and covariances of the random variables, each representing the number of microspheres in an individual compartment, have been derived through the probabilistic population balance. These equations have been numerically solved to predict the total release fraction of drug and its internal fluctuations, and the dynamic statistics (means, variances, and covariances) of the amount of drug in each compartment at any time after administration. The dissolution-intensity functions in the model have been recovered from the available in vitro dissolution data from controlled-release pellets of isosorbide-5-nitrate (IS-5-N) by assuming that the rate of release is of the first order. The residence times and transition-intensity functions of drug in the individual compartments have been estimated from the available data generated by the gamma scintigraphies of IS-5-N pellets labeled by ¹¹¹In. Based on these parameters, the total numbers of dissolved drug microspheres and their fluctuations at any instance have been calculated. The model is in accord with the existing in vivo dissolution data of the same drug independently obtained through plasma analysis. More important, the model predicts that fluctuations in terms of the standard deviations of the numbers of particles in the duodenum, jejunum, and ileum can be of the same orders of magnitude as the corresponding mean numbers when 100 microspheres are simultaneously administered orally; in practice, such fluctuations characterized by these deviations could result in an undesirable release profile. Discussion is given of the potential direct clinical application of the results obtained as well as the plausible indirect application of these results and the model derived to the analyses of chemical and biochemical reactors.     

Stochastic payoff evaluation increases the temperature of selection

We study stochastic evolutionary game dynamics in populations of finite size. Moreover, each individual has a randomly distributed number of interactions with other individuals. Therefore, the payoff of two individuals using the same strategy can be different. The resulting "payoff stochasticity" reduces the intensity of selection and therefore increases the temperature of selection. A simple mean-field approximation is derived that captures the average effect of the payoff stochasticity. Correction terms to the mean-field theory are computed and discussed.