Quantitative modeling of stochastic systems in molecular biology by using stochastic Petri nets

Boolean dynamics of genetic regulatory networks inferred from microarray time series data

MOTIVATION

Methods available for the inference of genetic regulatory networks strive to produce a single network, usually by optimizing some quantity to fit the experimental observations. In this article we investigate the possibility that multiple networks can be inferred, all resulting in similar dynamics. This idea is motivated by theoretical work which suggests that biological networks are robust and adaptable to change, and that the overall behavior of a genetic regulatory network might be captured in terms of dynamical basins of attraction.

RESULTS

We have developed and implemented a method for inferring genetic regulatory networks for time series microarray data. Our method first clusters and discretizes the gene expression data using k-means and support vector regression. We then enumerate Boolean activation-inhibition networks to match the discretized data. Finally, the dynamics of the Boolean networks are examined. We have tested our method on two immunology microarray datasets: an IL-2-stimulated T cell response dataset and a LPS-stimulated macrophage response dataset. In both cases, we discovered that many networks matched the data, and that most of these networks had similar dynamics.

SUPPLEMENTARY INFORMATION

Supplementary data are available at Bioinformatics online.

Qualitative networks: a symbolic approach to analyze biological signaling networks

BACKGROUND

A central goal of Systems Biology is to model and analyze biological signaling pathways that interact with one another to form complex networks. Here we introduce Qualitative networks, an extension of Boolean networks. With this framework, we use formal verification methods to check whether a model is consistent with the laboratory experimental observations on which it is based. If the model does not conform to the data, we suggest a revised model and the new hypotheses are tested in-silico.

RESULTS

We consider networks in which elements range over a small finite domain allowing more flexibility than Boolean values, and add target functions that allow to model a rich set of behaviors. We propose a symbolic algorithm for analyzing the steady state of these networks, allowing us to scale up to a system consisting of 144 elements and state spaces of approximately 10(86) states. We illustrate the usefulness of this approach through a model of the interaction between the Notch and the Wnt signaling pathways in mammalian skin, and its extensive analysis.

CONCLUSION

We introduce an approach for constructing computational models of biological systems that extends the framework of Boolean networks and uses formal verification methods for the analysis of the model. This approach can scale to multicellular models of complex pathways, and is therefore a useful tool for the analysis of complex biological systems. The hypotheses formulated during in-silico testing suggest new avenues to explore experimentally. Hence, this approach has the potential to efficiently complement experimental studies in biology.

A stochastic model of Escherichia coli AI-2 quorum signal circuit reveals alternative synthesis pathways

Quorum sensing (QS) is an important determinant of bacterial phenotype. Many cell functions are regulated by intricate and multimodal QS signal transduction processes. The LuxS/AI-2 QS system is highly conserved among Eubacteria and AI-2 is reported as a 'universal' signal molecule. To understand the hierarchical organization of AI-2 circuitry, a comprehensive approach incorporating stochastic simulations was developed. We investigated the synthesis, uptake, and regulation of AI-2, developed testable hypotheses, and made several discoveries: (1) the mRNA transcript and protein levels of AI-2 synthases, Pfs and LuxS, do not contribute to the dramatically increased level of AI-2 found when cells are grown in the presence of glucose; (2) a concomitant increase in metabolic flux through this synthesis pathway in the presence of glucose only partially accounts for this difference. We predict that 'high-flux' alternative pathways or additional biological steps are involved in AI-2 synthesis; and (3) experimental results validate this hypothesis. This work demonstrates the utility of linking cell physiology with systems-based stochastic models that can be assembled de novo with partial knowledge of biochemical pathways.

Qualitatively modelling and analysing genetic regulatory networks: a Petri net approach

MOTIVATION

New developments in post-genomic technology now provide researchers with the data necessary to study regulatory processes in a holistic fashion at multiple levels of biological organization. One of the major challenges for the biologist is to integrate and interpret these vast data resources to gain a greater understanding of the structure and function of the molecular processes that mediate adaptive and cell cycle driven changes in gene expression. In order to achieve this biologists require new tools and techniques to allow pathway related data to be modelled and analysed as network structures, providing valuable insights which can then be validated and investigated in the laboratory.

RESULTS

We propose a new technique for constructing and analysing qualitative models of genetic regulatory networks based on the Petri net formalism. We take as our starting point the Boolean network approach of treating genes as binary switches and develop a new Petri net model which uses logic minimization to automate the construction of compact qualitative models. Our approach addresses the shortcomings of Boolean networks by providing access to the wide range of existing Petri net analysis techniques and by using non-determinism to cope with incomplete and inconsistent data. The ideas we present are illustrated by a case study in which the genetic regulatory network controlling sporulation in the bacterium Bacillus subtilis is modelled and analysed.

AVAILABILITY

The Petri net model construction tool and the data files for the B. subtilis sporulation case study are available at http://bioinf.ncl.ac.uk/gnapn.

The cognitive phenotype of Down syndrome: insights from intracellular network analysis

Down syndrome (DS) is caused by trisomy of chromosome 21. All individuals with DS exhibit some level of cognitive dysfunction. It is generally accepted that these abnormalities are a result of the upregulation of genes encoded by chromosome 21. Many chromosome 21 proteins are known or predicted to function in critical neurological processes, but typically they function as modulators of these processes, not as key regulators. Thus, upregulation in DS is expected to cause only modest perturbations of normal processes. Systematic approaches such as intracellular network construction and analysis have not been generally applied in DS research. Networks can be assembled from high-throughput experiments or by text-mining of experimental literature. We survey some new developments in constructing such networks, focusing on newly developed network analysis methodologies. We propose how these methods could be integrated with creation and manipulation of mouse models of DS to advance our understanding of the perturbed cell signaling pathways in DS. This understanding could lead to potential therapeutics.

Modelling, property verification and behavioural equivalence of lactose operon regulation

Understanding biochemical pathways is one of the biggest challenges in the field of molecular biology nowadays. Computer science can contribute in this area by providing formalisms and tools to simulate and analyse pathways. One formalism that is suited for modelling concurrent systems is Milner's Calculus of Communicating Systems (CCS). This paper shows the viability of using CCS to model and reason about biochemical networks. As a case study, we describe the regulation of lactose operon. After describing this operon formally using CCS, we validate our model by automatically checking some known properties for lactose regulation. Moreover, since biological systems tend to be very complex, we propose to use multiple descriptions of the same system at different levels of abstraction. The compatibility of these multiple views can be assured via mathematical proofs of observational equivalence.

Computational modelling of the receptor-tyrosine-kinase-activated MAPK pathway

The MAPK (mitogen-activated protein kinase) pathway is one of the most important and intensively studied signalling pathways. It is at the heart of a molecular-signalling network that governs the growth, proliferation, differentiation and survival of many, if not all, cell types. It is de-regulated in various diseases, ranging from cancer to immunological, inflammatory and degenerative syndromes, and thus represents an important drug target. Over recent years, the computational or mathematical modelling of biological systems has become increasingly valuable, and there is now a wide variety of mathematical models of the MAPK pathway which have led to some novel insights and predictions as to how this system functions. In the present review we give an overview of the processes involved in modelling a biological system using the popular approach of ordinary differential equations. Focusing on the MAPK pathway, we introduce the features and functions of the pathway itself before comparing the available models and describing what new biological insights they have led to.

Reconstructing the regulatory network controlling commitment and sporulation in Physarum polycephalum based on hierarchical Petri Net modelling and simulation

We reconstruct the regulatory network controlling commitment and sporulation of Physarum polycephalum from experimental results using a hierarchical Petri Net-based modelling and simulation framework. The stochastic Petri Net consistently describes the structure and simulates the dynamics of the molecular network as analysed by genetic, biochemical and physiological experiments within a single coherent model. The Petri Net then is extended to simulate time-resolved somatic complementation experiments performed by mixing the cytoplasms of mutants altered in the sporulation response, to systematically explore the network structure and to probe its dynamics. This reverse engineering approach presumably can be employed to explore other molecular or genetic signalling systems where the activity of genes or their products can be experimentally controlled in a time-resolved manner.

Learning Petri net models of non-linear gene interactions

Understanding how an individual's genetic make-up influences their risk of disease is a problem of paramount importance. Although machine-learning techniques are able to uncover the relationships between genotype and disease, the problem of automatically building the best biochemical model or "explanation" of the relationship has received less attention. In this paper, I describe a method based on random hill climbing that automatically builds Petri net models of non-linear (or multi-factorial) disease-causing gene-gene interactions. Petri nets are a suitable formalism for this problem, because they are used to model concurrent, dynamic processes analogous to biochemical reaction networks. I show that this method is routinely able to identify perfect Petri net models for three disease-causing gene-gene interactions recently reported in the literature.

A quantitative model of the switch cycle of an archaeal flagellar motor and its sensory control

By reverse-engineering we have detected eight kinetic phases of the symmetric switch cycle of the Halobacterium salinarum flagellar motor assembly and identified those steps in the switch cycle that are controlled by sensory rhodopsins during phototaxis. Upon switching the rotational sense, the flagellar motor assembly passes through a stop state from which all subunits synchronously resume rotation in the reverse direction. The assembly then synchronously proceeds through three subsequent functional states of the switch: Refractory, Competent, and Active, from which the rotational sense is switched again. Sensory control of the symmetric switch cycle occurs at two steps in each rotational sense by inversely regulating the probabilities for a change from the Refractory to the Competent and from Competent to the Active rotational mode. We provide a mathematical model for flagellar motor switching and its sensory control, which is able to explain all tested experimental results on spontaneous and light-controlled motor switching, and give a mechanistic explanation based on synchronous conformational transitions of the subunits of the switch complex after reversible dissociation and binding of a response regulator (CheYP). We conclude that the kinetic mechanism of flagellar motor switching and its sensory control is fundamentally different in the archaeon H. salinarum and the bacterium Escherichia coli.

Dizzy: stochastic simulation of large-scale genetic regulatory networks

We describe Dizzy, a software tool for stochastically and deterministically modeling the spatially homogeneous kinetics of integrated large-scale genetic, metabolic, and signaling networks. Notable features include a modular simulation framework, reusable modeling elements, complex kinetic rate laws, multi-step reaction processes, steady-state noise estimation, and spatial compartmentalization.

The Evolutionary Genetics of Canalization

Evolutionary genetics has recently made enormous progress in understanding how genetic variation maps into phenotypic variation. However why some traits are phenotypically invariant despite apparent genetic and environmental changes has remained a major puzzle. In the 1940s, Conrad Hal Waddington coined the concept and term "canalization" to describe the robustness of phenotypes to perturbation; a similar concept was proposed by Waddington's contemporary Ivan Ivanovich Schmalhausen. This paper reviews what has been learned about canalization since Waddington. Canalization implies that a genotype's phenotype remains relatively invariant when individuals of a particular genotype are exposed to different environments (environmental canalization) or when individuals of the same single- or multilocus genotype differ in their genetic background (genetic canalization). Consequently, genetic canalization can be viewed as a particular kind of epistasis, and environmental canalization and phenotypic plasticity are two aspects of the same phenomenon. Canalization results in the accumulation of phenotypically cryptic genetic variation, which can be released after a "decanalizing" event. Thus, canalized genotypes maintain a cryptic potential for expressing particular phenotypes, which are only uncovered under particular decanalizing environmental or genetic conditions. Selection may then act on this newly released genetic variation. The accumulation of cryptic genetic variation by canalization may therefore increase evolvability at the population level by leading to phenotypic diversification under decanalizing conditions. On the other hand, under canalizing conditions, a major part of the segregating genetic variation may remain phenotypically cryptic; canalization may therefore, at least temporarily, constrain phenotypic evolution. Mechanistically, canalization can be understood in terms of transmission patterns, such as epistasis, pleiotropy, and genotype by environment interactions, and in terms of genetic redundancy, modularity, and emergent properties of gene networks and biochemical pathways. While different forms of selection can favor canalization, the requirements for its evolution are typically rather restrictive. Although there are several methods to detect canalization, there are still serious problems with unambiguously demonstrating canalization, particularly its adaptive value.

Stochastic variation in the concentration of a repressor activates GAL genetic switch: implications in evolution of regulatory network

In Saccharomyces cerevisiae, a recessive mutation in the signal transducer encoded by GAL3 leads to a significant lag in the induction of GAL genes, referred to as long term adaptation phenotype (LTA). Further, gal3 mutation in combination with other genetic defects leads to the non-inducibility of GAL genes. It was shown that the expression of GAL1 encoded galactokinase, a redundant GAL3 like signal transducer, eventually substitutes for the lack of GAL3 signal transduction function. However, how GAL1 gets induced in the absence of GAL3 is not clear. We hypothesize that GAL1 induction in gal3 cells exposed to galactose is due to a stochastic decrease in the repressor, Gal80p concentration, leading to heterogeneity in the population. This observation explains not only LTA observed in gal3 cells but also explains the non-inducibility of gal3 mutants in combination with other genetic defects. By recruiting a dedicated signal transducer, GAL3, S. cerevisiae GAL switch has evolved to overcome the fortuitous induction, which occurs due to low signal to noise ratio in certain mutants of Escherichia coli and Kluveromyces lactis.

Using Petri Net tools to study properties and dynamics of biological systems

Petri Nets (PNs) and their extensions are promising methods for modeling and simulating biological systems. We surveyed PN formalisms and tools and compared them based on their mathematical capabilities as well as by their appropriateness to represent typical biological processes. We measured the ability of these tools to model specific features of biological systems and answer a set of biological questions that we defined. We found that different tools are required to provide all capabilities that we assessed. We created software to translate a generic PN model into most of the formalisms and tools discussed. We have also made available three models and suggest that a library of such models would catalyze progress in qualitative modeling via PNs. Development and wide adoption of common formats would enable researchers to share models and use different tools to analyze them without the need to convert to proprietary formats.

Connecting the dots between genes, biochemistry, and disease susceptibility: systems biology modeling in human genetics

Understanding how DNA sequence variations impact human health through a hierarchy of biochemical and physiological systems is expected to improve the diagnosis, prevention, and treatment of common, complex human diseases. We have previously developed a hierarchical dynamic systems approach based on Petri nets for generating biochemical network models that are consistent with genetic models of disease susceptibility. This modeling approach uses an evolutionary computation approach called grammatical evolution as a search strategy for optimal Petri net models. We have previously demonstrated that this approach routinely identifies biochemical network models that are consistent with a variety of genetic models in which disease susceptibility is determined by nonlinear interactions between two or more DNA sequence variations. We review here this approach and then discuss how it can be used to model biochemical and metabolic data in the context of genetic studies of human disease susceptibility.

The selective values of alleles in a molecular network model are context dependent

Classical quantitative genetics has applied linear modeling to the problem of mapping genotypic to phenotypic variation. Much of this theory was developed prior to the availability of molecular biology. The current understanding of the mechanisms of gene expression indicates the importance of nonlinear effects resulting from gene interactions. We provide a bridge between genetics and gene network theories by relating key concepts from quantitative genetics to the parameters, variables, and performance functions of genetic networks. We illustrate this methodology by simulating the genetic switch controlling galactose metabolism in yeast and its response to selection for a population of individuals. Results indicate that genes have heterogeneous contributions to phenotypes and that additive and nonadditive effects are context dependent. Early cycles of selection suggest strong additive effects attributed to some genes. Later cycles suggest the presence of strong context-dependent nonadditive effects that are conditional on the outcomes of earlier selection cycles. A single favorable allele cannot be consistently identified for most loci. These results highlight the complications that can arise with the presence of nonlinear effects associated with genes acting in networks when selection is conducted on a population of individuals segregating for the genes contributing to the network.

Dynamic modeling of biochemical reactions with application to signal transduction: principles and tools using Mathematica

Modeling of biochemical phenomena is based on formal reaction kinetics. This requires the translation of the original reaction systems into sets of differential equations expressing the effects of the various reaction steps. The temporal behavior of the system is obtained by solving the differential equations. We present the main concepts on which the formal approach of these two problems is based and we show how the amount of work needed to treat them can be significantly reduced by using a mathematical program package (Mathematica). Symbolic and numerical calculations can be performed with the programs presented and graphic presentations of the behavior of the system be obtained. The basic ideas are illustrated with three examples taken from the area of signal transduction and ion signaling.

A Formal Language for Computational Systems Biology

The post-genomic era has opened new insights into the complex biochemical reaction systems present in the cell and has generated huge amount of information. The biological systems are highly complex and can overwhelm the numerically computable models. Therefore, models employing symbolical techniques might provide a faster insight. This paper presents some preliminary results and recent trends in the above direction. Specifically, it presents an overview of the main features of some formalisms and techniques from the field of specification languages for concurrency and mobility, which have been proposed to model and simulate the dynamics of the interaction of complex biological systems. The ultimate goal of these symbolic approaches is the modeling, analysis, simulation, and hopefully prediction of the behavior of biological systems (vs. biological components).

The ubiquitous nature of epistasis in determining susceptibility to common human diseases

There is increasing awareness that epistasis or gene-gene interaction plays a role in susceptibility to common human diseases. In this paper, we formulate a working hypothesis that epistasis is a ubiquitous component of the genetic architecture of common human diseases and that complex interactions are more important than the independent main effects of any one susceptibility gene. This working hypothesis is based on several bodies of evidence. First, the idea that epistasis is important is not new. In fact, the recognition that deviations from Mendelian ratios are due to interactions between genes has been around for nearly 100 years. Second, the ubiquity of biomolecular interactions in gene regulation and biochemical and metabolic systems suggest that relationship between DNA sequence variations and clinical endpoints is likely to involve gene-gene interactions. Third, positive results from studies of single polymorphisms typically do not replicate across independent samples. This is true for both linkage and association studies. Fourth, gene-gene interactions are commonly found when properly investigated. We review each of these points and then review an analytical strategy called multifactor dimensionality reduction for detecting epistasis. We end with ideas of how hypotheses about biological epistasis can be generated from statistical evidence using biochemical systems models. If this working hypothesis is true, it suggests that we need a research strategy for identifying common disease susceptibility genes that embraces, rather than ignores, the complexity of the genotype to phenotype relationship.

Using the Unified Modelling Language (UML) to guide the systemic description of biological processes and systems

One of the main issues in Systems Biology is to deal with semantic data integration. Previously, we examined the requirements for a reference conceptual model to guide semantic integration based on the systemic principles. In the present paper, we examine the usefulness of the Unified Modelling Language (UML) to describe and specify biological systems and processes. This makes unambiguous representations of biological systems, which would be suitable for translation into mathematical and computational formalisms, enabling analysis, simulation and prediction of these systems behaviours.

The Selective Values of Alleles in a Molecular Network Model Are Context Dependent

Classical quantitative genetics has applied linear modeling to the problem of mapping genotypic to phenotypic variation. Much of this theory was developed prior to the availability of molecular biology. The current understanding of the mechanisms of gene expression indicates the importance of nonlinear effects resulting from gene interactions. We provide a bridge between genetics and gene network theories by relating key concepts from quantitative genetics to the parameters, variables, and performance functions of genetic networks. We illustrate this methodology by simulating the genetic switch controlling galactose metabolism in yeast and its response to selection for a population of individuals. Results indicate that genes have heterogeneous contributions to phenotypes and that additive and nonadditive effects are context dependent. Early cycles of selection suggest strong additive effects attributed to some genes. Later cycles suggest the presence of strong context-dependent nonadditive effects that are conditional on the outcomes of earlier selection cycles. A single favorable allele cannot be consistently identified for most loci. These results highlight the complications that can arise with the presence of nonlinear effects associated with genes acting in networks when selection is conducted on a population of individuals segregating for the genes contributing to the network.

Petri net modeling of high-order genetic systems using grammatical evolution

Understanding how DNA sequence variations impact human health through a hierarchy of biochemical and physiological systems is expected to improve the diagnosis, prevention, and treatment of common, complex human diseases. We have previously developed a hierarchical dynamic systems approach based on Petri nets for generating biochemical network models that are consistent with genetic models of disease susceptibility. This modeling approach uses an evolutionary computation approach called grammatical evolution as a search strategy for optimal Petri net models. We have previously demonstrated that this approach routinely identifies biochemical network models that are consistent with a variety of genetic models in which disease susceptibility is determined by nonlinear interactions between two DNA sequence variations. In the present study, we evaluate whether the Petri net approach is capable of identifying biochemical networks that are consistent with disease susceptibility due to higher order nonlinear interactions between three DNA sequence variations. The results indicate that our model-building approach is capable of routinely identifying good, but not perfect, Petri net models. Ideas for improving the algorithm for this high-dimensional problem are presented.

Modeling transmission of directly transmitted infectious diseases using colored stochastic Petri nets

In order to improve our understanding of directly transmitted pathogens within host populations, epidemic models should take into account individual heterogeneities as well as stochastic fluctuations in individual parameters. The associated cost results in an increasing level of complexity of the mathematical models which generally lack consistent formalisms. In this paper, we demonstrate that complex epidemic models could be expressed as colored stochastic Petri nets (CSPN). CSPN is a mathematical tool developed in computer science. The concept is based on the Markov Chain theory and on a standard well codified graphical formalism. This approach presents an alternative to other computer simulation methods since it offers both a theoretical formalism and a graphical representation that facilitate the implementation, the understanding and thus the replication or modification of the model. We explain how common concepts of epidemic models--such as the incidence function--can be easily translated into an individual based point of view in the CSPN formalism. We then illustrate this approach by using the well documented susceptible-infected model with recruitment and death.

Light- and carbon-signaling pathways. Modeling circuits of interactions

Here, we report the systematic exploration and modeling of interactions between light and sugar signaling. The data set analyzed explores the interactions of sugar (sucrose) with distinct light qualities (white, blue, red, and far-red) used at different fluence rates (low or high) in etiolated seedlings and mature green plants. Boolean logic was used to model the effect of these carbon/light interactions on three target genes involved in nitrogen assimilation: asparagine synthetase (ASN1 and ASN2) and glutamine synthetase (GLN2). This analysis enabled us to assess the effects of carbon on light-induced genes (GLN2/ASN2) versus light-repressed genes (ASN1) in this pathway. New interactions between carbon and blue-light signaling were discovered, and further connections between red/far-red light and carbon were modeled. Overall, light was able to override carbon as a major regulator of ASN1 and GLN2 in etiolated seedlings. By contrast, carbon overrides light as the major regulator of GLN2 and ASN2 in light-grown plants. Specific examples include the following: Carbon attenuated the blue-light induction of GLN2 in etiolated seedlings and also attenuated the white-, blue-, and red-light induction of GLN2 and ASN2 in light-grown plants. By contrast, carbon potentiated far-red-light induction of GLN2 and ASN2 in light-grown plants. Depending on the fluence rate of far-red light, carbon either attenuated or potentiated light repression of ASN1 in light-grown plants. These studies indicate the interaction of carbon with blue, red, and far-red-light signaling and set the stage for further investigation into modeling this complex web of interacting pathways using systems biology approaches.

Bio-array images processing and genetic networks modelling

The new tools available for gene expression studies are essentially the bio-array methods using a large variety of physical detectors (isotopes, fluorescent markers, ultrasounds...). Here we present first rapidly an image-processing method independent of the detector type, dealing with the noise and with the peaks overlapping, the peaks revealing the detector activity (isotopic in the presented example), correlated with the gene expression. After this primary step of bio-array image processing, we can extract information about causal influence (activation or inhibition) a gene can exert on other genes, leading to clusters of genes co-expression in which we extract an interaction matrix M and an associated interaction graph G explaining the genetic regulatory dynamics correlated to the studied tissue function. We give two examples of such interaction matrices and graphs (the flowering genetic regulatory network of Arabidopsis thaliana and the lytic/lysogenic operon of the phage Mu) and after some theoretical rigorous results recently obtained concerning the asymptotic states generated by the genetic networks having a given interaction matrix and reciprocally concerning the minimal (in the sense of having a minimal number of non-zero coefficients) matrices having given stationary stable states.

Selective instability: maternal effort and the evolution of gene activation and deactivation rates

We previously used simulations of gene expression to demonstrate that rapid activation and deactivation rates stabilized outcomes in stochastic systems. We hypothesized that transient single allele inactivation of an autosomal gene during gametogenesis or very early embryogenesis could have a selective advantage if it permits the functional sampling of each allele and precludes committing maternal effort to an embryo with a deleterious mutation. To test this hypothesis, we simulated the evolution of gene expression activation and deactivation rates and imposed two different selective pressures on the populations: (a). late selection against individuals that cannot maintain a threshold level of gene product that occurs after the investment of maternal effort (i.e., after birth); or (b). early selection: in addition to late selection, maintenance of the gene product above a threshold level was necessary for early development prior to commitment of maternal effort. We found that the opportunity to save reproductive effort from early selection caused the evolution of higher deactivation rates and lower activation rates than in the late selection condition. Thus, we predict that in the special case where early selection can save maternal investment in non-viable offspring, gene expression activation rates and deactivation rates might be selected to permit sampling of the product from each allele.

Stochastic vs. deterministic modeling of intracellular viral kinetics

Within its host cell, a complex coupling of transcription, translation, genome replication, assembly, and virus release processes determines the growth rate of a virus. Mathematical models that account for these processes can provide insights into the understanding as to how the overall growth cycle depends on its constituent reactions. Deterministic models based on ordinary differential equations can capture essential relationships among virus constituents. However, an infection may be initiated by a single virus particle that delivers its genome, a single molecule of DNA or RNA, to its host cell. Under such conditions, a stochastic model that allows for inherent fluctuations in the levels of viral constituents may yield qualitatively different behavior. To compare modeling approaches, we developed a simple model of the intracellular kinetics of a generic virus, which could be implemented deterministically or stochastically. The model accounted for reactions that synthesized and depleted viral nucleic acids and structural proteins. Linear stability analysis of the deterministic model showed the existence of two nodes, one stable and one unstable. Individual stochastic simulation runs could access and remain at the unstable node. In addition, deterministic and averaged stochastic simulations yielded different transient kinetics and different steady-state levels of viral components, particularly for low multiplicities of infection (MOI), where few virus particles initiate the infection. Furthermore, a bimodal population distribution of viral components was observed for low MOI stochastic simulations. The existence of a low-level infected subpopulation of cells, which could act as a viral reservoir, suggested a potential mechanism of viral persistence.

Modeling and simulation of genetic regulatory systems: a literature review

In order to understand the functioning of organisms on the molecular level, we need to know which genes are expressed, when and where in the organism, and to which extent. The regulation of gene expression is achieved through genetic regulatory systems structured by networks of interactions between DNA, RNA, proteins, and small molecules. As most genetic regulatory networks of interest involve many components connected through interlocking positive and negative feedback loops, an intuitive understanding of their dynamics is hard to obtain. As a consequence, formal methods and computer tools for the modeling and simulation of genetic regulatory networks will be indispensable. This paper reviews formalisms that have been employed in mathematical biology and bioinformatics to describe genetic regulatory systems, in particular directed graphs, Bayesian networks, Boolean networks and their generalizations, ordinary and partial differential equations, qualitative differential equations, stochastic equations, and rule-based formalisms. In addition, the paper discusses how these formalisms have been used in the simulation of the behavior of actual regulatory systems.