Predicting stochastic gene expression dynamics in single cells.
ABSTRACT: Fluctuations in protein numbers (noise) due to inherent stochastic effects in single cells can have large effects on the dynamic behavior of gene regulatory networks. Although deterministic models can predict the average network behavior, they fail to incorporate the stochasticity characteristic of gene expression, thereby limiting their relevance when single cell behaviors deviate from the population average. Recently, stochastic models have been used to predict distributions of steady-state protein levels within a population but not to predict the dynamic, presteady-state distributions. In the present work, we experimentally examine a system whose dynamics are heavily influenced by stochastic effects. We measure population distributions of protein numbers as a function of time in the Escherichia coli lactose uptake network (lac operon). We then introduce a dynamic stochastic model and show that prediction of dynamic distributions requires only a few noise parameters in addition to the rates that characterize a deterministic model. Whereas the deterministic model cannot fully capture the observed behavior, our stochastic model correctly predicts the experimental dynamics without any fit parameters. Our results provide a proof of principle for the possibility of faithfully predicting dynamic population distributions from deterministic models supplemented by a stochastic component that captures the major noise sources.
Project description:The existence and implications of alternative stable states in ecological systems have been investigated extensively within deterministic models. However, it is known that natural systems are undeniably subject to random fluctuations, arising from either environmental variability or internal effects. Thus, in this paper, we study the role of noise on the pattern formation of a spatial predator-prey model with Allee effect. The obtained results show that the spatially extended system exhibits rich dynamic behavior. More specifically, the stationary pattern can be induced to be a stable target wave when the noise intensity is small. As the noise intensity is increased, patchy invasion emerges. These results indicate that the dynamic behavior of predator-prey models may be partly due to stochastic factors instead of deterministic factors, which may also help us to understand the effects arising from the undeniable susceptibility to random fluctuations of real ecosystems.
Project description:Transcription is regulated by a multitude of factors that concertedly induce genes to switch between activity states. Eukaryotic transcription involves a multitude of complexes that sequentially assemble on chromatin under the influence of transcription factors and the dynamic state of chromatin. Prokaryotic transcription depends on transcription factors, sigma-factors, and, in some cases, on DNA looping. We present a stochastic model of transcription that considers these complex regulatory mechanisms. We coarse-grain the molecular details in such a way that the model can describe a broad class of gene-regulation mechanisms. We solve this model analytically for various measures of stochastic transcription and compare alternative gene-regulation designs. We find that genes with complex multiprotein regulation can have peaked burst-size distributions in contrast to the geometric distributions found for simple models of transcription regulation. Burst-size distributions are, in addition, shaped by mRNA degradation during transcription bursts. We derive the stochastic properties of genes in the limit of deterministic switch times. These genes typically have reduced transcription noise. Severe timescale separation between gene regulation and transcription initiation enhances noise and leads to bimodal mRNA copy number distributions. In general, complex mechanisms for gene regulation lead to nonexponential waiting-time distributions for gene switching and transcription initiation, which typically reduce noise in mRNA copy numbers and burst size. Finally, we discuss that qualitatively different gene regulation models can often fit the same experimental data on single-cell mRNA abundance even though they have qualitatively different burst-size statistics and regulatory parameters.
Project description:For cellular biochemical reaction systems where the numbers of molecules is small, significant noise is associated with chemical reaction events. This molecular noise can give rise to behavior that is very different from the predictions of deterministic rate equation models. Unfortunately, there are few analytic methods for examining the qualitative behavior of stochastic systems. Here we describe such a method that extends deterministic analysis to include leading-order corrections due to the molecular noise. The method allows the steady-state behavior of the stochastic model to be easily computed, facilitates the mapping of stability phase diagrams that include stochastic effects, and reveals how model parameters affect noise susceptibility in a manner not accessible to numerical simulation. By way of illustration we consider two genetic circuits: a bistable positive-feedback loop and a negative-feedback oscillator. We find in the positive feedback circuit that translational activation leads to a far more stable system than transcriptional control. Conversely, in a negative-feedback loop triggered by a positive-feedback switch, the stochasticity of transcriptional control is harnessed to generate reproducible oscillations.
Project description:Bistability arises within a wide range of biological systems from the lambda phage switch in bacteria to cellular signal transduction pathways in mammalian cells. Changes in regulatory mechanisms may result in genetic switching in a bistable system. Recently, more and more experimental evidence in the form of bimodal population distributions indicates that noise plays a very important role in the switching of bistable systems. Although deterministic models have been used for studying the existence of bistability properties under various system conditions, these models cannot realize cell-to-cell fluctuations in genetic switching. However, there is a lag in the development of stochastic models for studying the impact of noise in bistable systems because of the lack of detailed knowledge of biochemical reactions, kinetic rates, and molecular numbers. In this work, we develop a previously undescribed general technique for developing quantitative stochastic models for large-scale genetic regulatory networks by introducing Poisson random variables into deterministic models described by ordinary differential equations. Two stochastic models have been proposed for the genetic toggle switch interfaced with either the SOS signaling pathway or a quorum-sensing signaling pathway, and we have successfully realized experimental results showing bimodal population distributions. Because the introduced stochastic models are based on widely used ordinary differential equation models, the success of this work suggests that this approach is a very promising one for studying noise in large-scale genetic regulatory networks.
Project description:In this research we have analyzed functional magnetic resonance imaging (fMRI) signals of different networks in the brain under resting state condition. To such end, the dynamics of signal variation, have been conceived as a stochastic motion, namely it has been modelled through a generalized Langevin stochastic differential equation, which combines a deterministic drift component with a stochastic component where the Gaussian noise source has been replaced with ?-stable noise. The parameters of the deterministic and stochastic parts of the model have been fitted from fluctuating data. Results show that the deterministic part is characterized by a simple, linear decreasing trend, and, most important, the ?-stable noise, at varying characteristic index ?, is the source of a spectrum of activity modes across the networks, from those originated by classic Gaussian noise (? = 2), to longer tailed behaviors generated by the more general Lévy noise (1 ? ? < 2). Lévy motion is a specific instance of scale-free behavior, it is a source of anomalous diffusion and it has been related to many aspects of human cognition, such as information foraging through memory retrieval or visual exploration. Finally, some conclusions have been drawn on the functional significance of the dynamics corresponding to different ? values.
Project description:Reaction rates (fluxes) in a metabolic network can be analyzed using constraint-based modeling which imposes a steady state assumption on the system. In a deterministic formulation of the problem the steady state assumption has to be fulfilled exactly, and the observed fluxes are included in the model without accounting for experimental noise. One can relax the steady state constraint, and also include experimental noise in the model, through a stochastic formulation of the problem. Uniform sampling of fluxes, feasible in both the deterministic and stochastic formulation, can provide us with statistical properties of the metabolic network, such as marginal flux probability distributions. In this study we give an overview of both the deterministic and stochastic formulation of the problem, and of available Monte Carlo sampling methods for sampling the corresponding solution space. We apply the ACHR, OPTGP, CHRR and Gibbs sampling algorithms to ten metabolic networks and evaluate their convergence, consistency and efficiency. The coordinate hit-and-run with rounding (CHRR) is found to perform best among the algorithms suitable for the deterministic formulation. A desirable property of CHRR is its guaranteed distributional convergence. Among the three other algorithms, ACHR has the largest consistency with CHRR for genome scale models. For the stochastic formulation, the Gibbs sampler is the only method appropriate for sampling at genome scale. However, our analysis ranks it as less efficient than the samplers used for the deterministic formulation.
Project description:Oscillatory dynamics are common in biological pathways, emerging from the coupling of positive and negative feedback loops. Due to the small numbers of molecules typically contained in cellular volumes, stochastic effects may play an important role in system behavior. Thus, for moderate noise strengths, stochasticity has been shown to enhance signal-to-noise ratios or even induce oscillations in a class of phenomena referred to as "stochastic resonance" and "coherence resonance," respectively. Furthermore, the biological oscillators are subject to influences from the division cycle of the cell. In this paper we consider a biologically relevant oscillator and investigate the effect of intrinsic noise as well as division cycle which encompasses the processes of growth, DNA duplication, and cell division. We first construct a minimal reaction network which can oscillate in the presence of large or negligible timescale separation. We then derive corresponding deterministic and stochastic models and compare their dynamical behaviors with respect to (i) the extent of the parameter space where each model can exhibit oscillatory behavior and (ii) the oscillation characteristics, namely, the amplitude and the period. We further incorporate division cycle effects on both models and investigate the effect of growth rate on system behavior. Our results show that in the presence but not in the absence of large timescale separation, coherence resonance effects result in extending the oscillatory region and lowering the period for the stochastic model. When the division cycle is taken into account, the oscillatory region of the deterministic model is shown to extend or shrink for moderate or high growth rates, respectively. Further, under the influence of the division cycle, the stochastic model can oscillate for parameter sets for which the deterministic model does not. The division cycle is also found to be able to resonate with the oscillator, thereby enhancing oscillation robustness. The results of this study can give valuable insight into the complex interplay between oscillatory intracellular dynamics and various noise sources, stemming from gene expression, cell growth, and division.
Project description:BACKGROUND: Apoptosis is a cell suicide mechanism that enables multicellular organisms to maintain homeostasis and to eliminate individual cells that threaten the organism's survival. Dependent on the type of stimulus, apoptosis can be propagated by extrinsic pathway or intrinsic pathway. The comprehensive understanding of the molecular mechanism of apoptotic signaling allows for development of mathematical models, aiming to elucidate dynamical and systems properties of apoptotic signaling networks. There have been extensive efforts in modeling deterministic apoptosis network accounting for average behavior of a population of cells. Cellular networks, however, are inherently stochastic and significant cell-to-cell variability in apoptosis response has been observed at single cell level. RESULTS: To address the inevitable randomness in the intrinsic apoptosis mechanism, we develop a theoretical and computational modeling framework of intrinsic apoptosis pathway at single-cell level, accounting for both deterministic and stochastic behavior. Our deterministic model, adapted from the well-accepted Fussenegger model, shows that an additional positive feedback between the executioner caspase and the initiator caspase plays a fundamental role in yielding the desired property of bistability. We then examine the impact of intrinsic fluctuations of biochemical reactions, viewed as intrinsic noise, and natural variation of protein concentrations, viewed as extrinsic noise, on behavior of the intrinsic apoptosis network. Histograms of the steady-state output at varying input levels show that the intrinsic noise could elicit a wider region of bistability over that of the deterministic model. However, the system stochasticity due to intrinsic fluctuations, such as the noise of steady-state response and the randomness of response delay, shows that the intrinsic noise in general is insufficient to produce significant cell-to-cell variations at physiologically relevant level of molecular numbers. Furthermore, the extrinsic noise represented by random variations of two key apoptotic proteins, namely Cytochrome C and inhibitor of apoptosis proteins (IAP), is modeled separately or in combination with intrinsic noise. The resultant stochasticity in the timing of intrinsic apoptosis response shows that the fluctuating protein variations can induce cell-to-cell stochastic variability at a quantitative level agreeing with experiments. Finally, simulations illustrate that the mean abundance of fluctuating IAP protein is positively correlated with the degree of cellular stochasticity of the intrinsic apoptosis pathway. CONCLUSIONS: Our theoretical and computational study shows that the pronounced non-genetic heterogeneity in intrinsic apoptosis responses among individual cells plausibly arises from extrinsic rather than intrinsic origin of fluctuations. In addition, it predicts that the IAP protein could serve as a potential therapeutic target for suppression of the cell-to-cell variation in the intrinsic apoptosis responsiveness.
Project description:Dynamic causal modelling (DCM) was introduced to study the effective connectivity among brain regions using neuroimaging data. Until recently, DCM relied on deterministic models of distributed neuronal responses to external perturbation (e.g., sensory stimulation or task demands). However, accounting for stochastic fluctuations in neuronal activity and their interaction with task-specific processes may be of particular importance for studying state-dependent interactions. Furthermore, allowing for random neuronal fluctuations may render DCM more robust to model misspecification and finesse problems with network identification. In this article, we examine stochastic dynamic causal models (sDCM) in relation to their deterministic counterparts (dDCM) and highlight questions that can only be addressed with sDCM. We also compare the network identification performance of deterministic and stochastic DCM, using Monte Carlo simulations and an empirical case study of absence epilepsy. For example, our results demonstrate that stochastic DCM can exploit the modelling of neural noise to discriminate between direct and mediated connections. We conclude with a discussion of the added value and limitations of sDCM, in relation to its deterministic homologue.
Project description:Cells live in uncertain, dynamic environments and have many mechanisms for sensing and responding to changes in their surroundings. However, sudden fluctuations in the environment can be catastrophic to a population if it relies solely on sensory responses, which have a delay associated with them. Cells can reconcile these effects by using a tunable stochastic response, where in the absence of a stressor they create phenotypic diversity within an isogenic population, but use a deterministic response when stressors are sensed. Here, we develop a stochastic model of the multiple antibiotic resistance network of Escherichia coli and show that it can produce tunable stochastic pulses in the activator MarA. In particular, we show that a combination of interlinked positive and negative feedback loops plays an important role in setting the dynamics of the stochastic pulses. Negative feedback produces a pulsatile response that is tunable, while positive feedback serves to amplify the effect. Our simulations show that the uninduced native network is in a parameter regime that is of low cost to the cell (taxing resistance mechanisms are expressed infrequently) and also elevated noise strength (phenotypic variability is high). The stochastic pulsing can be tuned by MarA induction such that variability is decreased once stresses are sensed, avoiding the detrimental effects of noise when an optimal MarA concentration is needed. We further show that variability in the expression of MarA can act as a bet hedging mechanism, allowing for survival in time-varying stress environments, however this effect is tunable to allow for a fully induced, deterministic response in the presence of a stressor.