Stochastic simulations of the origins and implications of long-tailed distributions in gene expression

Gene expression noise results in protein number distributions ranging from long-tailed to Gaussian. We show how long-tailed distributions arise from a stochastic model of the constituent chemical reactions and suggest that, in conjunction with cooperative switches, they lead to more sensitive selection of a subpopulation of cells with high protein number than is possible with Gaussian distributions. Single-cell-tracking experiments are presented to validate some of the assumptions of the stochastic simulations. We also examine the effect of DNA looping on the shape of protein distributions. We further show that when switches are incorporated in the regulation of a gene via a feedback loop, the distributions can become bimodal. This might explain the bimodal distribution of certain morphogens during early embryogenesis.

Estimating Frequencies of Minority Nevirapine-Resistant Strains in Chronically HIV-1-Infected Individuals Naive to Nevirapine by Using Stochastic Simulations and a Mathematical Model

Nevirapine forms the mainstay of our efforts to curtail the pediatric AIDS epidemic through prevention of mother-to-child transmission of HIV-1. A key limitation, however, is the rapid selection of HIV-1 strains resistant to nevirapine following the administration of a single dose. This rapid selection of resistance suggests that nevirapine-resistant strains preexist in HIV-1 patients and may adversely affect outcomes of treatment. The frequencies of nevirapine-resistant strains in vivo, however, remain poorly estimated, possibly because they exist as a minority below current assay detection limits. Here, we employ stochastic simulations and a mathematical model to estimate the frequencies of strains carrying different combinations of the common nevirapine resistance mutations K103N, V106A, Y181C, Y188C, and G190A in chronically infected HIV-1 patients naive to nevirapine. We estimate the relative fitness of mutant strains from an independent analysis of previous competitive growth assays. We predict that single mutants are likely to preexist in patients at frequencies (similar to 0.01% to 0.001%) near or below current assay detection limits (>0.01%), emphasizing the need for more-sensitive assays. The existence of double mutants is subject to large stochastic variations. Triple and higher mutants are predicted not to exist. Our estimates are robust to variations in the recombination rate, cellular superinfection frequency, and the effective population size. Thus, with 10(7) to 10(8) infected cells in HIV-1 patients, even when undetected, nevirapine-resistant genomes may exist in substantial numbers and compromise efforts to prevent mother-to-child transmission of HIV-1, accelerate the failure of subsequent antiretroviral treatments, and facilitate the transmission of drug resistance.

Stochastic Simulations Suggest that HIV-1 Survives Close to Its Error Threshold

The use of mutagenic drugs to drive HIV-1 past its error threshold presents a novel intervention strategy, as suggested by the quasispecies theory, that may be less susceptible to failure via viral mutation-induced emergence of drug resistance than current strategies. The error threshold of HIV-1, mu(c), however, is not known. Application of the quasispecies theory to determine mu(c) poses significant challenges: Whereas the quasispecies theory considers the asexual reproduction of an infinitely large population of haploid individuals, HIV-1 is diploid, undergoes recombination, and is estimated to have a small effective population size in vivo. We performed population genetics-based stochastic simulations of the within-host evolution of HIV-1 and estimated the structure of the HIV-1 quasispecies and mu(c). We found that with small mutation rates, the quasispecies was dominated by genomes with few mutations. Upon increasing the mutation rate, a sharp error catastrophe occurred where the quasispecies became delocalized in sequence space. Using parameter values that quantitatively captured data of viral diversification in HIV-1 patients, we estimated mu(c) to be 7 x 10(-5) -1 x 10(-4) substitutions/site/replication, similar to 2-6 fold higher than the natural mutation rate of HIV-1, suggesting that HIV-1 survives close to its error threshold and may be readily susceptible to mutagenic drugs. The latter estimate was weakly dependent on the within-host effective population size of HIV-1. With large population sizes and in the absence of recombination, our simulations converged to the quasispecies theory, bridging the gap between quasispecies theory and population genetics-based approaches to describing HIV-1 evolution. Further, mu(c) increased with the recombination rate, rendering HIV-1 less susceptible to error catastrophe, thus elucidating an added benefit of recombination to HIV-1. Our estimate of mu(c) may serve as a quantitative guideline for the use of mutagenic drugs against HIV-1.

Stochastic simulations of the Madden-Julian oscillation activity

The Madden-Julian oscillation (MJO) is the most prominent form of tropical intraseasonal variability. This study investigated the following questions. Do inter-annual-to-decadal variations in tropical sea surface temperature (SST) lead to substantial changes in MJO activity? Was there a change in the MJO in the 1970s? Can this change be associated to SST anomalies? What was the level of MJO activity in the pre-reanalysis era? These questions were investigated with a stochastic model of the MJO. Reanalysis data (1948-2008) were used to develop a nine-state first order Markov model capable to simulate the non-stationarity of the MJO. The model is driven by observed SST anomalies and a large ensemble of simulations was performed to infer the activity of the MJO in the instrumental period (1880-2008). The model is capable to reproduce the activity of the MJO during the reanalysis period. The simulations indicate that the MJO exhibited a regime of near normal activity in 1948-1972 (3.4 events year(-1)) and two regimes of high activity in 1973-1989 (3.9 events) and 1990-2008 (4.6 events). Stochastic simulations indicate decadal shifts with near normal levels in 1880-1895 (3.4 events), low activity in 1896 1917 (2.6 events) and a return to near normal levels during 1918-1947 (3.3 events). The results also point out to significant decadal changes in probabilities of very active years (5 or more MJO events): 0.214 (1880-1895), 0.076 (1896-1917), 0.197 (1918-1947) and 0.193 (1948-1972). After a change in behavior in the 1970s, this probability has increased to 0.329 (1973-1989) and 0.510 (1990-2008). The observational and stochastic simulations presented here call attention to the need to further understand the variability of the MJO on a wide range of time scales.

Hybrid stochastic simulations of intracellular reaction-diffusion systems.

With the observation that stochasticity is important in biological systems, chemical kinetics have begun to receive wider interest. While the use of Monte Carlo discrete event simulations most accurately capture the variability of molecular species, they become computationally costly for complex reaction-diffusion systems with large populations of molecules. On the other hand, continuous time models are computationally efficient but they fail to capture any variability in the molecular species. In this study a hybrid stochastic approach is introduced for simulating reaction-diffusion systems. We developed an adaptive partitioning strategy in which processes with high frequency are simulated with deterministic rate-based equations, and those with low frequency using the exact stochastic algorithm of Gillespie. Therefore the stochastic behavior of cellular pathways is preserved while being able to apply it to large populations of molecules. We describe our method and demonstrate its accuracy and efficiency compared with the Gillespie algorithm for two different systems. First, a model of intracellular viral kinetics with two steady states and second, a compartmental model of the postsynaptic spine head for studying the dynamics of Ca+2 and NMDA receptors.

Stochastic simulation of chemical reactions in spatially complex media

Discrete stochastic simulations, via techniques such as the Stochastic Simulation Algorithm (SSA) are a powerful tool for understanding the dynamics of chemical kinetics when there are low numbers of certain molecular species. However, an important constraint is the assumption of well-mixedness and homogeneity. In this paper, we show how to use Monte Carlo simulations to estimate an anomalous diffusion parameter that encapsulates the crowdedness of the spatial environment. We then use this parameter to replace the rate constants of bimolecular reactions by a time-dependent power law to produce an SSA valid in cases where anomalous diffusion occurs or the system is not well-mixed (ASSA). Simulations then show that ASSA can successfully predict the temporal dynamics of chemical kinetics in a spatially constrained environment.

Oscillatory regulation of Hes1: Discrete stochastic delay modelling and simulation

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.

Stochastic chemical kinetics and the quasi-steady-state assumption : application to the stochastic simulation algorithm and chemical master equation

Recently the application of the quasi-steady-state approximation (QSSA) to the stochastic simulation algorithm (SSA) was suggested for the purpose of speeding up stochastic simulations of chemical systems that involve both relatively fast and slow chemical reactions [Rao and Arkin, J. Chem. Phys. 118, 4999 (2003)] and further work has led to the nested and slow-scale SSA. Improved numerical efficiency is obtained by respecting the vastly different time scales characterizing the system and then by advancing only the slow reactions exactly, based on a suitable approximation to the fast reactions. We considerably extend these works by applying the QSSA to numerical methods for the direct solution of the chemical master equation (CME) and, in particular, to the finite state projection algorithm [Munsky and Khammash, J. Chem. Phys. 124, 044104 (2006)], in conjunction with Krylov methods. In addition, we point out some important connections to the literature on the (deterministic) total QSSA (tQSSA) and place the stochastic analogue of the QSSA within the more general framework of aggregation of Markov processes. We demonstrate the new methods on four examples: Michaelis–Menten enzyme kinetics, double phosphorylation, the Goldbeter–Koshland switch, and the mitogen activated protein kinase cascade. Overall, we report dramatic improvements by applying the tQSSA to the CME solver.

Parallel implementation of stochastic simulation for large scale cellular processes

Experimental and theoretical studies have shown the importance of stochastic processes in genetic regulatory networks and cellular processes. Cellular networks and genetic circuits often involve small numbers of key proteins such as transcriptional factors and signaling proteins. In recent years stochastic models have been used successfully for studying noise in biological pathways, and stochastic modelling of biological systems has become a very important research field in computational biology. One of the challenge problems in this field is the reduction of the huge computing time in stochastic simulations. Based on the system of the mitogen-activated protein kinase cascade that is activated by epidermal growth factor, this work give a parallel implementation by using OpenMP and parallelism across the simulation. Special attention is paid to the independence of the generated random numbers in parallel computing, that is a key criterion for the success of stochastic simulations. Numerical results indicate that parallel computers can be used as an efficient tool for simulating the dynamics of large-scale genetic regulatory networks and cellular processes

Physicochemical properties and 3D geostatistical simulations of the Herten and the Descalvado aquifer analogs

This data set presents a comprehensive characterisation of the sedimentary structures from important groundwater hosting formations in Germany (Herten aquifer analog) and Brazil (Descalvado aquifer analog). Multiple 2-D outcrop faces are described in terms of hydraulic, thermal and chemical properties and interpolated in 3D using stochastic techniques. For each aquifer analog, multiple 3D realisations of the facies heterogeneity are provided using different stochastic simulations settings. These are unique analogue data sets that can be used by the wider community to implement approaches for characterising aquifer formations.

Binomial leap methods for simulating stochastic chemical kinetics

This paper discusses efficient simulation methods for stochastic chemical kinetics. Based on the tau-leap and midpoint tau-leap methods of Gillespie [D. T. Gillespie, J. Chem. Phys. 115, 1716 (2001)], binomial random variables are used in these leap methods rather than Poisson random variables. The motivation for this approach is to improve the efficiency of the Poisson leap methods by using larger stepsizes. Unlike Poisson random variables whose range of sample values is from zero to infinity, binomial random variables have a finite range of sample values. This probabilistic property has been used to restrict possible reaction numbers and to avoid negative molecular numbers in stochastic simulations when larger stepsize is used. In this approach a binomial random variable is defined for a single reaction channel in order to keep the reaction number of this channel below the numbers of molecules that undergo this reaction channel. A sampling technique is also designed for the total reaction number of a reactant species that undergoes two or more reaction channels. Samples for the total reaction number are not greater than the molecular number of this species. In addition, probability properties of the binomial random variables provide stepsize conditions for restricting reaction numbers in a chosen time interval. These stepsize conditions are important properties of robust leap control strategies. Numerical results indicate that the proposed binomial leap methods can be applied to a wide range of chemical reaction systems with very good accuracy and significant improvement on efficiency over existing approaches. (C) 2004 American Institute of Physics.

Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation

Discrete stochastic simulations are a powerful tool for understanding the dynamics of chemical kinetics when there are small-to-moderate numbers of certain molecular species. In this paper we introduce delays into the stochastic simulation algorithm, thus mimicking delays associated with transcription and translation. We then show that this process may well explain more faithfully than continuous deterministic models the observed sustained oscillations in expression levels of hes1 mRNA and Hes1 protein.

Parallel implementation of stochastic simulation for large scale cellular processes

Experimental and theoretical studies have shown the importance of stochastic processes in genetic regulatory networks and cellular processes. Cellular networks and genetic circuits often involve small numbers of key proteins such as transcriptional factors and signaling proteins. In recent years stochastic models have been used successfully for studying noise in biological pathways, and stochastic modelling of biological systems has become a very important research field in computational biology. One of the challenge problems in this field is the reduction of the huge computing time in stochastic simulations. Based on the system of the mitogen-activated protein kinase cascade that is activated by epidermal growth factor, this work give a parallel implementation by using OpenMP and parallelism across the simulation. Special attention is paid to the independence of the generated random numbers in parallel computing, that is a key criterion for the success of stochastic simulations. Numerical results indicate that parallel computers can be used as an efficient tool for simulating the dynamics of large-scale genetic regulatory networks and cellular processes

Velocity-jump models with crowding effects

Velocity jump processes are discrete random walk models that have many applications including the study of biological and ecological collective motion. In particular, velocity jump models are often used to represent a type of persistent motion, known as a “run and tumble”, which is exhibited by some isolated bacteria cells. All previous velocity jump processes are non-interacting, which means that crowding effects and agent-to-agent interactions are neglected. By neglecting these agent-to-agent interactions, traditional velocity jump models are only applicable to very dilute systems. Our work is motivated by the fact that many applications in cell biology, such as wound healing, cancer invasion and development, often involve tissues that are densely packed with cells where cell-to-cell contact and crowding effects can be important. To describe these kinds of high cell density problems using a velocity jump process we introduce three different classes of crowding interactions into a one-dimensional model. Simulation data and averaging arguments lead to a suite of continuum descriptions of the interacting velocity jump processes. We show that the resulting systems of hyperbolic partial differential equations predict the mean behavior of the stochastic simulations very well.

Post-transcriptional regulation in the nucleus and cytoplasm: study of mean time to threshold (MTT) and narrow escape problem

Messenger RNAs (mRNAs) can be repressed and degraded by small non-coding RNA molecules. In this paper, we formulate a coarsegrained Markov-chain description of the post-transcriptional regulation of mRNAs by either small interfering RNAs (siRNAs) or microRNAs (miRNAs). We calculate the probability of an mRNA escaping from its domain before it is repressed by siRNAs/miRNAs via cal- culation of the mean time to threshold: when the number of bound siRNAs/miRNAs exceeds a certain threshold value, the mRNA is irreversibly repressed. In some cases,the analysis can be reduced to counting certain paths in a reduced Markov model. We obtain explicit expressions when the small RNA bind irreversibly to the mRNA and we also discuss the reversible binding case. We apply our models to the study of RNA interference in the nucleus, examining the probability of mRNAs escaping via small nuclear pores before being degraded by siRNAs. Using the same modelling framework, we further investigate the effect of small, decoy RNAs (decoys) on the process of post-transcriptional regulation, by studying regulation of the tumor suppressor gene, PTEN : decoys are able to block binding sites on PTEN mRNAs, thereby educing the number of sites available to siRNAs/miRNAs and helping to protect it from repression. We calculate the probability of a cytoplasmic PTEN mRNA translocating to the endoplasmic reticulum before being repressed by miRNAs. We support our results with stochastic simulations