Estimating DNA methylation potential energy landscapes from nanopore sequencing data.

High-throughput third-generation nanopore sequencing devices have enormous potential for simultaneously observing epigenetic modifications in human cells over large regions of the genome. However, signals generated by these devices are subject to considerable noise that can lead to unsatisfactory detection performance and hamper downstream analysis. Here we develop a statistical method, CpelNano, for the quantification and analysis of 5mC methylation landscapes using nanopore data. CpelNano takes into account nanopore noise by means of a hidden Markov model (HMM) in which the true but unknown ("hidden") methylation state is modeled through an Ising probability distribution that is consistent with methylation means and pairwise correlations, whereas nanopore current signals constitute the observed state. It then estimates the associated methylation potential energy function by employing the expectation-maximization (EM) algorithm and performs differential methylation analysis via permutation-based hypothesis testing. Using simulations and analysis of published data obtained from three human cell lines (GM12878, MCF-10A, and MDA-MB-231), we show that CpelNano can faithfully estimate DNA methylation potential energy landscapes, substantially improving current methods and leading to a powerful tool for the modeling and analysis of epigenetic landscapes using nanopore sequencing data.

Converging genetic and epigenetic drivers of paediatric acute lymphoblastic leukaemia identified by an information-theoretic analysis.

In cancer, linking epigenetic alterations to drivers of transformation has been difficult, in part because DNA methylation analyses must capture epigenetic variability, which is central to tumour heterogeneity and tumour plasticity. Here, by conducting a comprehensive analysis, based on information theory, of differences in methylation stochasticity in samples from patients with paediatric acute lymphoblastic leukaemia (ALL), we show that ALL epigenomes are stochastic and marked by increased methylation entropy at specific regulatory regions and genes. By integrating DNA methylation and single-cell gene-expression data, we arrived at a relationship between methylation entropy and gene-expression variability, and found that epigenetic changes in ALL converge on a shared set of genes that overlap with genetic drivers involved in chromosomal translocations across the disease spectrum. Our findings suggest that an epigenetically driven gene-regulation network, with UHRF1 (ubiquitin-like with PHD and RING finger domains 1) as a central node, links genetic drivers and epigenetic mediators in ALL.

A Dysregulated DNA Methylation Landscape Linked to Gene Expression in MLL-Rearranged AML.

Translocations of the KMT2A (MLL) gene define a biologically distinct and clinically aggressive subtype of acute myeloid leukaemia (AML), marked by a characteristic gene expression profile and few cooperating mutations. Although dysregulation of the epigenetic landscape in this leukaemia is particularly interesting given the low mutation frequency, its comprehensive analysis using whole genome bisulphite sequencing (WGBS) has not been previously performed. Here we investigated epigenetic dysregulation in nine MLL-rearranged (MLL-r) AML samples by comparing them to six normal myeloid controls, using a computational method that encapsulates mean DNA methylation measurements along with analyses of methylation stochasticity. We discovered a dramatically altered epigenetic profile in MLL-r AML, associated with genome-wide hypomethylation and a markedly increased DNA methylation entropy reflecting an increasingly disordered epigenome. Methylation discordance mapped to key genes and regulatory elements that included bivalent promoters and active enhancers. Genes associated with significant changes in methylation stochasticity recapitulated known MLL-r AML expression signatures, suggesting a role for the altered epigenetic landscape in the transcriptional programme initiated by MLL translocations. Accordingly, we established statistically significant associations between discordances in methylation stochasticity and gene expression in MLL-r AML, thus providing a link between the altered epigenetic landscape and the phenotype.

Ranking genomic features using an information-theoretic measure of epigenetic discordance.

BACKGROUND: Establishment and maintenance of DNA methylation throughout the genome is an important epigenetic mechanism that regulates gene expression whose disruption has been implicated in human diseases like cancer. It is therefore crucial to know which genes, or other genomic features of interest, exhibit significant discordance in DNA methylation between two phenotypes. We have previously proposed an approach for ranking genes based on methylation discordance within their promoter regions, determined by centering a window of fixed size at their transcription start sites. However, we cannot use this method to identify statistically significant genomic features and handle features of variable length and with missing data. RESULTS: We present a new approach for computing the statistical significance of methylation discordance within genomic features of interest in single and multiple test/reference studies. We base the proposed method on a well-articulated hypothesis testing problem that produces p- and q-values for each genomic feature, which we then use to identify and rank features based on the statistical significance of their epigenetic dysregulation. We employ the information-theoretic concept of mutual information to derive a novel test statistic, which we can evaluate by computing Jensen-Shannon distances between the probability distributions of methylation in a test and a reference sample. We design the proposed methodology to simultaneously handle biological, statistical, and technical variability in the data, as well as variable feature lengths and missing data, thus enabling its wide-spread use on any list of genomic features. This is accomplished by estimating, from reference data, the null distribution of the test statistic as a function of feature length using generalized additive regression models. Differential assessment, using normal/cancer data from healthy fetal tissue and pediatric high-grade glioma patients, illustrates the potential of our approach to greatly facilitate the exploratory phases of clinically and biologically relevant methylation studies. CONCLUSIONS: The proposed approach provides the first computational tool for statistically testing and ranking genomic features of interest based on observed DNA methylation discordance in comparative studies that accounts, in a rigorous manner, for biological, statistical, and technical variability in methylation data, as well as for variability in feature length and for missing data.

The NASA Twins Study: A multidimensional analysis of a year-long human spaceflight.

To understand the health impact of long-duration spaceflight, one identical twin astronaut was monitored before, during, and after a 1-year mission onboard the International Space Station; his twin served as a genetically matched ground control. Longitudinal assessments identified spaceflight-specific changes, including decreased body mass, telomere elongation, genome instability, carotid artery distension and increased intima-media thickness, altered ocular structure, transcriptional and metabolic changes, DNA methylation changes in immune and oxidative stress-related pathways, gastrointestinal microbiota alterations, and some cognitive decline postflight. Although average telomere length, global gene expression, and microbiome changes returned to near preflight levels within 6 months after return to Earth, increased numbers of short telomeres were observed and expression of some genes was still disrupted. These multiomic, molecular, physiological, and behavioral datasets provide a valuable roadmap of the putative health risks for future human spaceflight.

An information-theoretic approach to the modeling and analysis of whole-genome bisulfite sequencing data.

BACKGROUND: DNA methylation is a stable form of epigenetic memory used by cells to control gene expression. Whole genome bisulfite sequencing (WGBS) has emerged as a gold-standard experimental technique for studying DNA methylation by producing high resolution genome-wide methylation profiles. Statistical modeling and analysis is employed to computationally extract and quantify information from these profiles in an effort to identify regions of the genome that demonstrate crucial or aberrant epigenetic behavior. However, the performance of most currently available methods for methylation analysis is hampered by their inability to directly account for statistical dependencies between neighboring methylation sites, thus ignoring significant information available in WGBS reads. RESULTS: We present a powerful information-theoretic approach for genome-wide modeling and analysis of WGBS data based on the 1D Ising model of statistical physics. This approach takes into account correlations in methylation by utilizing a joint probability model that encapsulates all information available in WGBS methylation reads and produces accurate results even when applied on single WGBS samples with low coverage. Using the Shannon entropy, our approach provides a rigorous quantification of methylation stochasticity in individual WGBS samples genome-wide. Furthermore, it utilizes the Jensen-Shannon distance to evaluate differences in methylation distributions between a test and a reference sample. Differential performance assessment using simulated and real human lung normal/cancer data demonstrate a clear superiority of our approach over DSS, a recently proposed method for WGBS data analysis. Critically, these results demonstrate that marginal methods become statistically invalid when correlations are present in the data. CONCLUSIONS: This contribution demonstrates clear benefits and the necessity of modeling joint probability distributions of methylation using the 1D Ising model of statistical physics and of quantifying methylation stochasticity using concepts from information theory. By employing this methodology, substantial improvement of DNA methylation analysis can be achieved by effectively taking into account the massive amount of statistical information available in WGBS data, which is largely ignored by existing methods.

Intracellular production of hydrogels and synthetic RNA granules by multivalent molecular interactions.

Some protein components of intracellular non-membrane-bound entities, such as RNA granules, are known to form hydrogels in vitro. The physico-chemical properties and functional role of these intracellular hydrogels are difficult to study, primarily due to technical challenges in probing these materials in situ. Here, we present iPOLYMER, a strategy for a rapid induction of protein-based hydrogels inside living cells that explores the chemically inducible dimerization paradigm. Biochemical and biophysical characterizations aided by computational modelling show that the polymer network formed in the cytosol resembles a physiological hydrogel-like entity that acts as a size-dependent molecular sieve. We functionalize these polymers with RNA-binding motifs that sequester polyadenine-containing nucleotides to synthetically mimic RNA granules. These results show that iPOLYMER can be used to synthetically reconstitute the nucleation of biologically functional entities, including RNA granules in intact cells.

Potential energy landscapes identify the information-theoretic nature of the epigenome.

Epigenetics is the study of biochemical modifications carrying information independent of DNA sequence, which are heritable through cell division. In 1940, Waddington coined the term "epigenetic landscape" as a metaphor for pluripotency and differentiation, but methylation landscapes have not yet been rigorously computed. Using principles from statistical physics and information theory, we derive epigenetic energy landscapes from whole-genome bisulfite sequencing (WGBS) data that enable us to quantify methylation stochasticity genome-wide using Shannon's entropy, associating it with chromatin structure. Moreover, we consider the Jensen-Shannon distance between sample-specific energy landscapes as a measure of epigenetic dissimilarity and demonstrate its effectiveness for discerning epigenetic differences. By viewing methylation maintenance as a communications system, we introduce methylation channels and show that higher-order chromatin organization can be predicted from their informational properties. Our results provide a fundamental understanding of the information-theoretic nature of the epigenome that leads to a powerful approach for studying its role in disease and aging.

Integrative identification of deregulated miRNA/TF-mediated gene regulatory loops and networks in prostate cancer.

MicroRNAs (miRNAs) have attracted a great deal of attention in biology and medicine. It has been hypothesized that miRNAs interact with transcription factors (TFs) in a coordinated fashion to play key roles in regulating signaling and transcriptional pathways and in achieving robust gene regulation. Here, we propose a novel integrative computational method to infer certain types of deregulated miRNA-mediated regulatory circuits at the transcriptional, post-transcriptional and signaling levels. To reliably predict miRNA-target interactions from mRNA/miRNA expression data, our method collectively utilizes sequence-based miRNA-target predictions obtained from several algorithms, known information about mRNA and miRNA targets of TFs available in existing databases, certain molecular structures identified to be statistically over-represented in gene regulatory networks, available molecular subtyping information, and state-of-the-art statistical techniques to appropriately constrain the underlying analysis. In this way, the method exploits almost every aspect of extractable information in the expression data. We apply our procedure on mRNA/miRNA expression data from prostate tumor and normal samples and detect numerous known and novel miRNA-mediated deregulated loops and networks in prostate cancer. We also demonstrate instances of the results in a number of distinct biological settings, which are known to play crucial roles in prostate and other types of cancer. Our findings show that the proposed computational method can be used to effectively achieve notable insights into the poorly understood molecular mechanisms of miRNA-mediated interactions and dissect their functional roles in cancer in an effort to pave the way for miRNA-based therapeutics in clinical settings.

Intrinsic noise induces critical behavior in leaky Markovian networks leading to avalanching.

The role intrinsic statistical fluctuations play in creating avalanches--patterns of complex bursting activity with scale-free properties--is examined in leaky Markovian networks. Using this broad class of models, we develop a probabilistic approach that employs a potential energy landscape perspective coupled with a macroscopic description based on statistical thermodynamics. We identify six important thermodynamic quantities essential for characterizing system behavior as a function of network size: the internal potential energy, entropy, free potential energy, internal pressure, pressure, and bulk modulus. In agreement with classical phase transitions, these quantities evolve smoothly as a function of the network size until a critical value is reached. At that value, a discontinuity in pressure is observed that leads to a spike in the bulk modulus demarcating loss of thermodynamic robustness. We attribute this novel result to a reallocation of the ground states (global minima) of the system's stationary potential energy landscape caused by a noise-induced deformation of its topographic surface. Further analysis demonstrates that appreciable levels of intrinsic noise can cause avalanching, a complex mode of operation that dominates system dynamics at near-critical or subcritical network sizes. Illustrative examples are provided using an epidemiological model of bacterial infection, where avalanching has not been characterized before, and a previously studied model of computational neuroscience, where avalanching was erroneously attributed to specific neural architectures. The general methods developed here can be used to study the emergence of avalanching (and other complex phenomena) in many biological, physical and man-made interaction networks.

Statistically testing the validity of analytical and computational approximations to the chemical master equation.

The master equation is used extensively to model chemical reaction systems with stochastic dynamics. However, and despite its phenomenological simplicity, it is not in general possible to compute the solution of this equation. Drawing exact samples from the master equation is possible, but can be computationally demanding, especially when estimating high-order statistical summaries or joint probability distributions. As a consequence, one often relies on analytical approximations to the solution of the master equation or on computational techniques that draw approximative samples from this equation. Unfortunately, it is not in general possible to check whether a particular approximation scheme is valid. The main objective of this paper is to develop an effective methodology to address this problem based on statistical hypothesis testing. By drawing a moderate number of samples from the master equation, the proposed techniques use the well-known Kolmogorov-Smirnov statistic to reject the validity of a given approximation method or accept it with a certain level of confidence. Our approach is general enough to deal with any master equation and can be used to test the validity of any analytical approximation method or any approximative sampling technique of interest. A number of examples, based on the Schlögl model of chemistry and the SIR model of epidemiology, clearly illustrate the effectiveness and potential of the proposed statistical framework.

Numerical integration of the master equation in some models of stochastic epidemiology.

The processes by which disease spreads in a population of individuals are inherently stochastic. The master equation has proven to be a useful tool for modeling such processes. Unfortunately, solving the master equation analytically is possible only in limited cases (e.g., when the model is linear), and thus numerical procedures or approximation methods must be employed. Available approximation methods, such as the system size expansion method of van Kampen, may fail to provide reliable solutions, whereas current numerical approaches can induce appreciable computational cost. In this paper, we propose a new numerical technique for solving the master equation. Our method is based on a more informative stochastic process than the population process commonly used in the literature. By exploiting the structure of the master equation governing this process, we develop a novel technique for calculating the exact solution of the master equation--up to a desired precision--in certain models of stochastic epidemiology. We demonstrate the potential of our method by solving the master equation associated with the stochastic SIR epidemic model. MATLAB software that implements the methods discussed in this paper is freely available as Supporting Information S1.

Thermodynamically consistent model calibration in chemical kinetics.

BACKGROUND: The dynamics of biochemical reaction systems are constrained by the fundamental laws of thermodynamics, which impose well-defined relationships among the reaction rate constants characterizing these systems. Constructing biochemical reaction systems from experimental observations often leads to parameter values that do not satisfy the necessary thermodynamic constraints. This can result in models that are not physically realizable and may lead to inaccurate, or even erroneous, descriptions of cellular function. RESULTS: We introduce a thermodynamically consistent model calibration (TCMC) method that can be effectively used to provide thermodynamically feasible values for the parameters of an open biochemical reaction system. The proposed method formulates the model calibration problem as a constrained optimization problem that takes thermodynamic constraints (and, if desired, additional non-thermodynamic constraints) into account. By calculating thermodynamically feasible values for the kinetic parameters of a well-known model of the EGF/ERK signaling cascade, we demonstrate the qualitative and quantitative significance of imposing thermodynamic constraints on these parameters and the effectiveness of our method for accomplishing this important task. MATLAB software, using the Systems Biology Toolbox 2.1, can be accessed from http://www.cis.jhu.edu/~goutsias/CSS lab/software.html. An SBML file containing the thermodynamically feasible EGF/ERK signaling cascade model can be found in the BioModels database. CONCLUSIONS: TCMC is a simple and flexible method for obtaining physically plausible values for the kinetic parameters of open biochemical reaction systems. It can be effectively used to recalculate a thermodynamically consistent set of parameter values for existing thermodynamically infeasible biochemical reaction models of cellular function as well as to estimate thermodynamically feasible values for the parameters of new models. Furthermore, TCMC can provide dimensionality reduction, better estimation performance, and lower computational complexity, and can help to alleviate the problem of data overfitting.

Reducing experimental variability in variance-based sensitivity analysis of biochemical reaction systems.

Sensitivity analysis is a valuable task for assessing the effects of biological variability on cellular behavior. Available techniques require knowledge of nominal parameter values, which cannot be determined accurately due to experimental uncertainty typical to problems of systems biology. As a consequence, the practical use of existing sensitivity analysis techniques may be seriously hampered by the effects of unpredictable experimental variability. To address this problem, we propose here a probabilistic approach to sensitivity analysis of biochemical reaction systems that explicitly models experimental variability and effectively reduces the impact of this type of uncertainty on the results. The proposed approach employs a recently introduced variance-based method to sensitivity analysis of biochemical reaction systems [Zhang et al., J. Chem. Phys. 134, 094101 (2009)] and leads to a technique that can be effectively used to accommodate appreciable levels of experimental variability. We discuss three numerical techniques for evaluating the sensitivity indices associated with the new method, which include Monte Carlo estimation, derivative approximation, and dimensionality reduction based on orthonormal Hermite approximation. By employing a computational model of the epidermal growth factor receptor signaling pathway, we demonstrate that the proposed technique can greatly reduce the effect of experimental variability on variance-based sensitivity analysis results. We expect that, in cases of appreciable experimental variability, the new method can lead to substantial improvements over existing sensitivity analysis techniques.

Thermodynamically consistent Bayesian analysis of closed biochemical reaction systems.

BACKGROUND: Estimating the rate constants of a biochemical reaction system with known stoichiometry from noisy time series measurements of molecular concentrations is an important step for building predictive models of cellular function. Inference techniques currently available in the literature may produce rate constant values that defy necessary constraints imposed by the fundamental laws of thermodynamics. As a result, these techniques may lead to biochemical reaction systems whose concentration dynamics could not possibly occur in nature. Therefore, development of a thermodynamically consistent approach for estimating the rate constants of a biochemical reaction system is highly desirable. RESULTS: We introduce a Bayesian analysis approach for computing thermodynamically consistent estimates of the rate constants of a closed biochemical reaction system with known stoichiometry given experimental data. Our method employs an appropriately designed prior probability density function that effectively integrates fundamental biophysical and thermodynamic knowledge into the inference problem. Moreover, it takes into account experimental strategies for collecting informative observations of molecular concentrations through perturbations. The proposed method employs a maximization-expectation-maximization algorithm that provides thermodynamically feasible estimates of the rate constant values and computes appropriate measures of estimation accuracy. We demonstrate various aspects of the proposed method on synthetic data obtained by simulating a subset of a well-known model of the EGF/ERK signaling pathway, and examine its robustness under conditions that violate key assumptions. Software, coded in MATLAB®, which implements all Bayesian analysis techniques discussed in this paper, is available free of charge at http://www.cis.jhu.edu/~goutsias/CSS%20lab/software.html. CONCLUSIONS: Our approach provides an attractive statistical methodology for estimating thermodynamically feasible values for the rate constants of a biochemical reaction system from noisy time series observations of molecular concentrations obtained through perturbations. The proposed technique is theoretically sound and computationally feasible, but restricted to quantitative data obtained from closed biochemical reaction systems. This necessitates development of similar techniques for estimating the rate constants of open biochemical reaction systems, which are more realistic models of cellular function.

A comparison of approximation techniques for variance-based sensitivity analysis of biochemical reaction systems.

BACKGROUND: Sensitivity analysis is an indispensable tool for the analysis of complex systems. In a recent paper, we have introduced a thermodynamically consistent variance-based sensitivity analysis approach for studying the robustness and fragility properties of biochemical reaction systems under uncertainty in the standard chemical potentials of the activated complexes of the reactions and the standard chemical potentials of the molecular species. In that approach, key sensitivity indices were estimated by Monte Carlo sampling, which is computationally very demanding and impractical for large biochemical reaction systems. Computationally efficient algorithms are needed to make variance-based sensitivity analysis applicable to realistic cellular networks, modeled by biochemical reaction systems that consist of a large number of reactions and molecular species. RESULTS: We present four techniques, derivative approximation (DA), polynomial approximation (PA), Gauss-Hermite integration (GHI), and orthonormal Hermite approximation (OHA), for analytically approximating the variance-based sensitivity indices associated with a biochemical reaction system. By using a well-known model of the mitogen-activated protein kinase signaling cascade as a case study, we numerically compare the approximation quality of these techniques against traditional Monte Carlo sampling. Our results indicate that, although DA is computationally the most attractive technique, special care should be exercised when using it for sensitivity analysis, since it may only be accurate at low levels of uncertainty. On the other hand, PA, GHI, and OHA are computationally more demanding than DA but can work well at high levels of uncertainty. GHI results in a slightly better accuracy than PA, but it is more difficult to implement. OHA produces the most accurate approximation results and can be implemented in a straightforward manner. It turns out that the computational cost of the four approximation techniques considered in this paper is orders of magnitude smaller than traditional Monte Carlo estimation. Software, coded in MATLAB, which implements all sensitivity analysis techniques discussed in this paper, is available free of charge. CONCLUSIONS: Estimating variance-based sensitivity indices of a large biochemical reaction system is a computationally challenging task that can only be addressed via approximations. Among the methods presented in this paper, a technique based on orthonormal Hermite polynomials seems to be an acceptable candidate for the job, producing very good approximation results for a wide range of uncertainty levels in a fraction of the time required by traditional Monte Carlo sampling.

Probabilistic sensitivity analysis of biochemical reaction systems.

Sensitivity analysis is an indispensable tool for studying the robustness and fragility properties of biochemical reaction systems as well as for designing optimal approaches for selective perturbation and intervention. Deterministic sensitivity analysis techniques, using derivatives of the system response, have been extensively used in the literature. However, these techniques suffer from several drawbacks, which must be carefully considered before using them in problems of systems biology. We develop here a probabilistic approach to sensitivity analysis of biochemical reaction systems. The proposed technique employs a biophysically derived model for parameter fluctuations and, by using a recently suggested variance-based approach to sensitivity analysis [Saltelli et al., Chem. Rev. (Washington, D.C.) 105, 2811 (2005)], it leads to a powerful sensitivity analysis methodology for biochemical reaction systems. The approach presented in this paper addresses many problems associated with derivative-based sensitivity analysis techniques. Most importantly, it produces thermodynamically consistent sensitivity analysis results, can easily accommodate appreciable parameter variations, and allows for systematic investigation of high-order interaction effects. By employing a computational model of the mitogen-activated protein kinase signaling cascade, we demonstrate that our approach is well suited for sensitivity analysis of biochemical reaction systems and can produce a wealth of information about the sensitivity properties of such systems. The price to be paid, however, is a substantial increase in computational complexity over derivative-based techniques, which must be effectively addressed in order to make the proposed approach to sensitivity analysis more practical.

Classical versus stochastic kinetics modeling of biochemical reaction systems.

We study fundamental relationships between classical and stochastic chemical kinetics for general biochemical systems with elementary reactions. Analytical and numerical investigations show that intrinsic fluctuations may qualitatively and quantitatively affect both transient and stationary system behavior. Thus, we provide a theoretical understanding of the role that intrinsic fluctuations may play in inducing biochemical function. The mean concentration dynamics are governed by differential equations that are similar to the ones of classical chemical kinetics, expressed in terms of the stoichiometry matrix and time-dependent fluxes. However, each flux is decomposed into a macroscopic term, which accounts for the effect of mean reactant concentrations on the rate of product synthesis, and a mesoscopic term, which accounts for the effect of statistical correlations among interacting reactions. We demonstrate that the ability of a model to account for phenomena induced by intrinsic fluctuations may be seriously compromised if we do not include the mesoscopic fluxes. Unfortunately, computation of fluxes and mean concentration dynamics requires intensive Monte Carlo simulation. To circumvent the computational expense, we employ a moment closure scheme, which leads to differential equations that can be solved by standard numerical techniques to obtain more accurate approximations of fluxes and mean concentration dynamics than the ones obtained with the classical approach.

A hidden Markov model for transcriptional regulation in single cells.

We discuss several issues pertaining to the use of stochastic biochemical systems for modeling transcriptional regulation in single cells. By appropriately choosing the system state, we can model transcriptional regulation by a hidden Markov model (HMM). This opens the possibility of using well-known techniques for the statistical analysis and stochastic control of HMMs to mathematically and computationally study transcriptional regulation in single cells. Unfortunately, in all but a few simple cases, analytical characterization of the statistical behavior of the proposed HMM is not possible. Moreover, analysis by Monte Carlo simulation is computationally cumbersome. We discuss several techniques for approximating the HMM by one that is more tractable. We employ simulations, based on a biologically relevant transcriptional regulatory system, to show the relative merits and limitations of various approximation techniques and provide general guidelines for their use.

Stochastic transcriptional regulatory systems with time delays: a mean-field approximation.

Modeling transcriptional regulation with time delays is an important problem of computational cell biology. In this paper, we propose a computational tool for studying transcriptional regulation in single cells based on a mean-field approximation method. The main idea is to replace the occurrence probabilities of the underlying transcriptional events by their mean values and use appropriately chosen additive noise terms to model statistical variations not accounted by this approximation. The proposed methodology allows us to characterize the transient and steady-state behavior of transcriptional regulation. Moreover, it provides a rather simple and computationally attractive tool for rapid statistical characterization of the dynamic behavior of a nonlinear transcriptional regulatory system with time delays.