Stochastic mechanisms in gene expression

Tracking operator state fluctuations in gene expression in single cells

We report the results of operator state fluctuations in gene expression for the entire bacterial growth cycle, using single-cell analysis and synthetic unregulated and negative-feedback transcription regulatory gene circuits. In the unregulated circuit, during the cell cycle, we observe a crossover from log-normal-to-normal distribution of expressed proteins and an unusual linear dependence of their standard deviation on the mean gene expression levels. With negative-feedback circuits we find the existence of bimodality as the cell cycle progresses. We suggest that such long-tail and bimodal distributions may be used as selection mechanisms in developmental switches and for assigning cell identity.

Self-consistent proteomic field theory of stochastic gene switches

We present a self-consistent field approximation approach to the problem of the genetic switch composed of two mutually repressing/activating genes. The protein and DNA state dynamics are treated stochastically and on an equal footing. In this approach the mean influence of the proteomic cloud created by one gene on the action of another is self-consistently computed. Within this approximation a broad range of stochastic genetic switches may be solved exactly in terms of finding the probability distribution and its moments. A much larger class of problems, such as genetic networks and cascades, also remain exactly solvable with this approximation. We discuss, in depth, certain specific types of basic switches used by biological systems and compare their behavior to the expectation for a deterministic switch.

Robustness of cellular functions

Robustness, the ability to maintain performance in the face of perturbations and uncertainty, is a long-recognized key property of living systems. Owing to intimate links to cellular complexity, however, its molecular and cellular basis has only recently begun to be understood. Theoretical approaches to complex engineered systems can provide guidelines for investigating cellular robustness because biology and engineering employ a common set of basic mechanisms in different combinations. Robustness may be a key to understanding cellular complexity, elucidating design principles, and fostering closer interactions between experimentation and theory.

Efficient simulations of gas-grain chemistry in interstellar clouds

Chemical reactions on dust grains are of crucial importance in interstellar chemistry because they produce molecular hydrogen and various organic molecules. Because of the submicron size of the grains and the low flux, the surface populations of reactive species are small and strongly fluctuate. Under these conditions rate equations fail and the master equation is needed for modeling these reactions. However, the number of equations grows exponentially with the number of reactive species, severely limiting its feasibility. Here we present a method which dramatically reduces the number of equations, thus enabling the incorporation of the master equation in models of interstellar chemistry.

Bridging the gap between stochastic and deterministic regimes in the kinetic simulations of the biochemical reaction networks

The biochemical reaction networks include elementary reactions differing by many orders of magnitude in the numbers of molecules involved. The kinetics of reactions involving small numbers of molecules can be studied by exact stochastic simulation. This approach is not practical for the simulation of metabolic processes because of the computational cost of accounting for individual molecular collisions. We present the "maximal time step method," a novel approach combining the Gibson and Bruck algorithm with the Gillespie tau-leap method. This algorithm allows stochastic simulation of systems composed of both intensive metabolic reactions and regulatory processes involving small numbers of molecules. The method is applied to the simulation of glucose, lactose, and glycerol metabolism in Escherichia coli. The gene expression, signal transduction, transport, and enzymatic activities are modeled simultaneously. We show that random fluctuations in gene expression can propagate to the level of metabolic processes. In the cells switching from glucose to a mixture of lactose and glycerol, random delays in transcription initiation determine whether lactose or glycerol operon is induced. In a small fraction of cells severe decrease in metabolic activity may also occur. Both effects are epigenetically inherited by the progeny of the cell in which the random delay in transcription initiation occurred.

A genetic circuit amplifier: design and simulation

A genetic circuit amplifier is designed using an electronic inverting amplifier as a starting point. Two simulation methods are used to analyze circuit performance in terms of the impulse and sinusoidal responses of electrical engineering. The first method is an exact stochastic simulation based on a kinetic model of the circuit. The second method incorporates statistical thermodynamic analysis. The simulations are used to analyze amplifier performance in response to classical systems analysis stimuli: impulses and sine waves. Degradation reactions, analogous to leakage off circuit capacitors, are found to have considerable impact on circuit response. For the nonlinear gain element used in our exemplary circuit, the selection of bias level based on controlling protein degradation rate plays an important role in determining circuit behavior. A parameter without electronic analog, the circuit plasmid copy number, is crucial to circuit operation. These simulations suggest that the copy number must be less than 50 for desired circuit operation.

Fluctuations-induced switch in the gene transcriptional regulatory system

Based on the kinetic model of genetic regulation system proposed by Am. J. Physiol. 274, c531 (1998)], the effects of fluctuations in the degradation reaction rate and the synthesis reaction rate of the transcription factor have been investigated through numerical computation and analysis theory. In the case of uncorrelated noises, it is shown that only the fluctuation of degradation reaction rate can induce a switch process, and the mean first passage time (MFPT) from the high concentration state to the low concentration one is decreased when the noise intensity of degradation reaction rate is increased. In the case of correlations between noises, a switch process can also be induced by the cross-correlation intensity between noises and by the fluctuation of the synthesis reaction rate in the genetic regulatory system. It is found that, under large cross-correlation intensity, a successive switch process (i.e., "on" --> "off" --> "on," which we call the reentrance transition or twice switch ) occurs with an increase of noise intensities, and a critical noise intensity exists at which the MFPT of the switch process is the largest. While the system is initially in the high concentration state with an increase of the cross correlation, the stationary probability distribution (SPD) of the transcription factor activator monomer concentration at the low concentration state is increased, yet the MFPT is increased due to the decreasing of the SPD of the transient states between the two steady stable states.

Mathematical and computational techniques to deduce complex biochemical reaction mechanisms

Time series data can now be routinely collected for biochemical reaction pathways, and recently, several methods have been proposed to infer reaction mechanisms for metabolic pathways and networks. In this paper we provide a survey of mathematical techniques for determining reaction mechanisms for time series data on the concentration or abundance of different reacting components, with little prior information about the pathways involved.

From large networks to small molecules

Large-scale analysis of genetic and physical interaction networks has begun to reveal the global organization of the cell. Cellular phenotypes observed at the macroscopic level depend on the collective characteristics of protein and genetic interaction networks, which exhibit scale-free properties and are highly resistant to perturbation of a single node. The nascent field of chemical genetics promises a host of small-molecule probes to explore these emerging networks. Although the robust nature of cellular networks usually resists the action of single agents, they may be susceptible to rationally designed combinations of small molecules able to collectively shift network behavior.

Insights into the relation between mRNA and protein expression patterns: I. Theoretical considerations

Translation is a central cellular process in every organism and understanding translation from the systems (genome-wide) perspective is very important for medical and biochemical engineering applications. Moreover, recent advances in cell-wide monitoring tools for both mRNA and protein levels have necessitated the development of such a model to identify parameters and conditions that influence the mapping between mRNA and protein expression. Experimental studies show a lack of correspondence between mRNA and protein expression profiles. In this study, we describe a mechanistic genome-wide model for translation that provides mapping between changes in mRNA levels and changes in protein levels. We use our model to study the system in detail and identify the key parameters that affect this mapping. Our results show that the correlation between mRNA and protein levels is a function of both the kinetic parameters and concentration of ribosomes at the reference state. In particular, changes in concentration of free and total ribosomes in response to a perturbation; changes in initiation and elongation kinetics due to competition for aminoacyl tRNAs; changes in termination kinetics; average changes in mRNA levels in response to the perturbation; and changes in protein stability are all important determinants of the mapping between mRNA and protein expression.

Constructive effects of fluctuations in genetic and biochemical regulatory systems

Biochemical and genetic regulatory systems that involve low concentrations of molecules are inherently noisy. This intrinsic stochasticity has received considerable interest recently, leading to new insights about the sources and consequences of noise in complex systems of genetic regulation. However, most prior work was devoted to the reduction of fluctuation and the robustness of cellular function with respect to intrinsic noise. Here, we focus on several scenarios in which the inherent molecular fluctuations are not merely a nuisance, but act constructively and bring about qualitative changes in the dynamics of the system. It will be demonstrated that in many typical situations biochemical and genetic regulatory systems may utilize intrinsic noise to their advantage.

Noise, delays, robustness, canalization and all that

A biological system such as a developing embryo can withstand many perturbations. What is the basis of this robustness both against noise and mutation? Recent advances in modeling may throw new light on this old problem. First, recent theoretical and experimental work clearly demonstrates the importance of noise and time delays for the proper functioning of genetic networks: noise and delays are simply part of the normal operating constraints. By contrast, sweeping statements have been made recently about a so-called 'robustness' of biological processes, based on work that neglects noise and delays completely. I submit that studying the stability of complex biological systems with such omissions is an unnecessary, inadequate and potentially disastrous simplification. I review the existing alternatives and propose using them to construct a modeling framework that overcomes all serious limitations.

Multivariate analysis of noise in genetic regulatory networks

Stochasticity is an intrinsic property of genetic regulatory networks due to the low copy numbers of the major molecular species, such as, DNA, mRNA, and regulatory proteins. Therefore, investigation of the mechanisms that reduce the stochastic noise is essential in understanding the reproducible behaviors of real organisms and is also a key to design synthetic genetic regulatory networks that can reliably work. We use an analytical and systematic method, the linear noise approximation of the chemical master equation along with the decoupling of a stoichiometric matrix. In the analysis of fluctuations of multiple molecular species, the covariance is an important measure of noise. However, usually the representation of a covariance matrix in the natural coordinate system, i.e. the copy numbers of the molecular species, is intractably complicated because reactions change copy numbers of more than one molecular species simultaneously. Decoupling of a stoichiometric matrix, which is a transformation of variables, significantly simplifies the representation of a covariance matrix and elucidates the mechanisms behind the observed fluctuations in the copy numbers. We apply our method to three types of fundamental genetic regulatory networks, that is, a single-gene autoregulatory network, a two-gene autoregulatory network, and a mutually repressive network. We have found that there are multiple noise components differently originating. Each noise component produces fluctuation in the characteristic direction. The resulting fluctuations in the copy numbers of the molecular species are the sum of these fluctuations. In the examples, the limitation of the negative feedback in noise reduction and the trade-off of fluctuations in multiple molecular species are clearly explained. The analytical representations show the full parameter dependence. Additionally, the validity of our method is tested by stochastic simulations.

Pathway databases

Network representations of biological pathways offer a functional view of molecular biology that is different from and complementary to sequence, expression, and structure databases. There is currently available a wide range of digital collections of pathway data, differing in organisms included, functional area covered (e.g., metabolism vs. signaling), detail of modeling, and support for dynamic pathway construction. While it is currently impossible for these databases to communicate with each other, there are several efforts at standardizing a data exchange language for pathway data. Databases that represent pathway data at the level of individual interactions make it possible to combine data from different predefined pathways and to query by network connectivity. Computable representations of pathways provide a basis for various analyses, including detection of broad network patterns, comparison with mRNA or protein abundance, and simulation.

Stochastic computing with biomolecular automata

Stochastic computing has a broad range of applications, yet electronic computers realize its basic step, stochastic choice between alternative computation paths, in a cumbersome way. Biomolecular computers use a different computational paradigm and hence afford novel designs. We constructed a stochastic molecular automaton in which stochastic choice is realized by means of competition between alternative biochemical pathways, and choice probabilities are programmed by the relative molar concentrations of the software molecules coding for the alternatives. Programmable and autonomous stochastic molecular automata have been shown to perform direct analysis of disease-related molecular indicators in vitro and may have the potential to provide in situ medical diagnosis and cure.

Stochastic approaches for modelling in vivo reactions

In recent years, stochastic modelling has emerged as a physically more realistic alternative for modelling in vivo reactions. There are numerous stochastic approaches available in the literature; most of these assume that observed random fluctuations are a consequence of the small number of reacting molecules. We review some important developments of the stochastic approach and consider its suitability for modelling intracellular reactions. We then describe recent efforts to include the fluctuation effects caused by the structural organisation of the cytoplasm and the limited diffusion of molecules due to macromolecular crowding.

Discrete event, multi-level simulation of metabolite channeling

Typically differential equations are employed to simulate cellular dynamics. To develop a valid continuous model based on differential equations requires accurate parameter estimations; an accuracy which is often difficult to achieve, due to the lack of data. In addition, processes in metabolic pathways, e.g. metabolite channeling, seem to be of a rather qualitative and discrete nature. With respect to the available data and to the perception of the underlying system, a discrete rather than a continuous approach to modeling and simulation seems more adequate. A discrete approach does not necessarily imply a more abstract view on the system. If we move from macro to micro and multi-level modeling, aspects of subsystems and their interactions, which have been only implicitly represented, become an explicit part of the model. To start exploring discrete event phenomena within metabolite channeling we choose the tryptophan synthase. Based on a continuous macro model, a discrete event, multi-level model is developed which allows us to analyze the interrelation between structural and functional characteristics of the enzymes.

Intrinsic and external noise in an auto-regulatory genetic network

A single gene auto-regulatory network is analysed. The main goal is to investigate the effects of the negative and positive feedbacks on the intrinsic and external noises. The central finding of this paper is that: for the intrinsic noise, both the negative and positive feedback regulations increase the fluctuation strength of mRNA levels (where the fluctuation strength is measured by the Fano factor for both the fluctuations of mRNAs and proteins), and the negative feedback decreases, but the positive feedback increases, the fluctuation strength of proteins; for the external noise, the negative feedback not only increase the fluctuation strength of mRNA levels but also the fluctuation strength of proteins, and though the effect of the positive feedback on the fluctuation strength of mRNA levels depends on the size of positive feedback parameter k, the positive feedback must decrease the fluctuation strength of proteins.

Setting the pace: mechanisms tying Caulobacter cell-cycle progression to macroscopic cellular events

The bacterium Caulobacter crescentus divides asymmetrically, producing daughter cells with differing polar structures, different cell fates and asymmetric regulation of the initiation of chromosome replication. Complex intracellular signaling is required to keep the organelle developmental processes at the cell poles synchronized with other cell cycle events. Two recently characterized switch mechanisms controlling cell cycle progress are triggered by relatively large-scale developmental events in the cell: the progress of the DNA replication fork and the physical compartmentalization of the cell that occurs well before division. These mechanisms invoke rapid, precisely timed and even spatially differentiated regulatory responses at important points in the cell cycle.

Noise-reduction through interaction in gene expression and biochemical reaction processes

We demonstrate that interaction in gene expression and biochemical reaction processes has a significant influence on reducing fluctuations. Especially, we have found that the interaction between synthesized proteins and background molecules can reduce the fluctuation level in gene expression, which is a counter example to the intuition that background factors disturb information processing in genetic networks by increasing the noise level. This fact also indicates that the macromolecular crowding observed in actual cells can contribute to reduce the noise level. In addition, the noise-reduction phenomenon is not limited to the interaction between the proteins and background molecules, but can be applied to other reactions such as a dimerization process and the coupling of reactions with large fluctuations by intrinsic noise. Finally, on the basis of these results, we propose a new and plausible method for reducing the fluctuations generated in synthesized genetic networks, and also discuss the applicability of this method to the stabilization of system dynamics by using a toggle switch model.

The Karyote physico-chemical genomic, proteomic, metabolic cell modeling system

Modeling approaches to the dynamics of a living cell are presented that are strongly based on its underlying physical and chemical processes and its hierarchical spatio-temporal organization. Through the inclusion of a broad spectrum of processes and a rigorous analysis of the multiple scale nature of cellular dynamics, we are attempting to advance cell modeling and its applications. The presentation focuses on our cell modeling system, which integrates data archiving and quantitative physico-chemical modeling and information theory to provide a seamless approach to the modeling/data analysis endeavor. Thereby the rapidly growing mess of genomic, proteomic, metabolic, and cell physiological data can be automatically used to develop and calibrate a predictive cell model. The discussion focuses on the Karyote cell modeling system and an introduction to the CellX and VirusX models. The Karyote software system integrates three elements: (1) a model-building and data archiving module that allows one to define a cell type to be modeled through its reaction network, structure, and transport processes as well as to choose the surrounding medium and other parameters of the phenomenon to be modeled; (2) a genomic, proteomic, metabolic cell simulator that solves the equations of metabolic reaction, transcription/translation polymerization and the exchange of molecules between parts of the cell and with the surrounding medium; and (3) an information theory module (ITM) that automates model calibration and development, and integrates a variety of data types with the cell dynamic computations. In Karyote, reactions may be fast (equilibrated) or slow (finite rate), and the special effects of enzymes and other minority species yielding steady-state cycles of arbitrary complexities are accounted for. These features of the dynamics are handled via rigorous multiple scale analysis. A user interface allows for an automated generation and solution of the equations of multiple timescale, compartmented dynamics. Karyote is based on a fixed intracellular structure. However, cell response to changes in the host medium, damage, development or transformation to abnormality can involve dramatic changes in intracellular structure. As this changes the nature of the cellular dynamics, a new model, CellX, is being developed based on the spatial distribution of concentration and other variables. This allows CellX to capture the self-organizing character of cellular behavior. The self-assembly of organelles, viruses, and other subcellular bodies is being addressed in a second new model, VirusX, that integrates molecular mechanics and continuum theory. VirusX is designed to study the influence of a host medium on viral self-assembly, structural stability, infection of a single cell, and transmission of disease.

Clusters of regulatory signals for RNA polymerase II transcription associated with Alu family repeats and CpG islands in human promoters

Primate genomes contain a very large number of short interspersed GC-rich repeats of the Alu family, which are abundant in introns and intergenic spacers but also present in 5' flanking regions of genes enriched in binding motifs (BMs) for transcription factors and frequently containing CpG islands. Here we studied whether CpG islands located in promoters of human genes overlap with Alu repeats and with clusters of BMs for the zinc-finger transcription factors Sp1, estrogen receptor alpha, and YY1. The presence of estrogen-response elements in Alu was shown earlier and here we confirm the presence in the consensus Alu sequence of the binding sites for Sp1 and YY1. Analyzing >5000 promoters from the two databases we found that Alu sequences are underrepresented in promoters compared to introns and that approximately 4% of CpG islands located within the -1000 to +200 segments of human promoters overlap with Alu repeats. Although this fraction was found to be lower for proximal segments of promoters (-500 to +100), our results indicate that a significant number (>1000) of all human genes may be controlled by Alu-associated CpG islands. Analysis of clustering of potential BMs for the indicated transcription factors within some promoters also suggests that the Alu family contributed to the evolution of transcription cis-regulatory modules in the human genome. It is important that among Alu sequences overlapping with CpG islands in promoters a large fraction of members of the old Alu subfamilies is found, suggesting extensive retroposon-assisted regulatory genome evolution during the divergence of the primates.

Living Bacterial Cell Array for Genotoxin Monitoring

A biosensor composed of a high-density living bacterial cell array was fabricated by inserting bacteria into a microwell array formed on one end of an imaging fiber bundle. The size of each microwell allows only one cell to occupy each well. In this biosensor, E. coli cells carrying a recA::gfp fusion were used as sensing components for genotoxin detection. Each fiber in the array has its own light pathway, enabling thousands of individual cell responses to be monitored simultaneously with both spatial and temporal resolution. The biosensor was capable of performing cell-based functional sensing of a genotoxin with high sensitivity and short incubation times (1 ng/mL mitomycin C after 90 min). Dose-response curves for several genotoxins were obtained. The biosensors demonstrated an active sensing lifetime of more than 6 h and a shelf lifetime of two weeks.

Stochastic processes influence stationary-phase decisions in Bacillus subtilis

It has recently been proposed that phenotypic variation in clonal populations of bacterial species results from intracellular "noise," i.e., random fluctuations in levels of cellular molecules, which would be predicted to be insensitive to selective pressure. To test this notion, we propagated five populations of Bacillus subtilis for 5,000 generations with selection for one phenotype: the decision to sporulate. In support of the noise hypothesis, we report that none of the populations responded to selection by improving their efficiency of sporulation, indicating that intracellular noise is independent of heritable genotype.

Nonlinear kinetics and new approaches to complex reaction mechanisms

This paper reviews recent developments in the field of nonlinear chemical kinetics. Five topics are dealt with: (a) new approaches to complex reaction mechanisms, stoichiometric network analysis, classification of chemical oscillators and formulation of their mechanisms by deduction from experiments, and correlation metric construction of reaction pathways from measurements; (b) thermodynamic and stochastic theory of nonequilibrium processes, the eikonal approximation, the evaluation of stochastic potentials, experimental tests of the thermodynamic and stochastic theory of relative stability, and fluctuation-dissipation relations in nonequilibrium chemical systems; (c) chemical kinetics and cellular automata and lattice gas automata; (d) theoretical approaches and experimental studies of stochastic resonance in chemical kinetics; and (e) rate processes in disordered systems, stochastic Liouville equations, stretched exponential relaxation in disordered systems, and universality classes for rate processes in systems with static or dynamic disorder.

Life-on-a-chip

Mechanistic studies of cellular processes are usually carried out with large populations of cells. However, parameters that are measured as averages of large populations can be misleading. For instance, an apparently linear response to a signal could, in fact, reflect an increasing number of cells in the population that have switched from 'off' to 'on', rather than a graded increase in response by all the cells. At present, the study of single cells is challenging, but new technologies mean it might soon be a reality.

Production of cell lines secreting TAT fusion proteins

Transduction of proteins and other macromolecules constitutes a potent technology to analyze cell functions and to achieve therapeutic interventions. In general, fusion proteins with protein transduction domains, such as TAT, are produced in a bacterial expression system. Here we describe the generation of a mammalian expression vector coding for TAT-EGFP fusion protein. Transfection of CHO-K1 cells by this vector and subsequent selection by Zeocin resulted in cell lines that express and secrete EGFP, a variant of the green fluorescent protein GFP. The ultimate cell line was produced by first cloning the stable integrants and subsequent selection of EGFP-expressing cells by flow cytometric sorting. In the resulting cell line approximately 98% of cells express EGFP. Using the same methodology, we generated cell lines that express DsRed fluorescent protein. The advantages of using such a mammalian expression system include the ease of generating TAT fusion proteins and the potential for sustained production of such proteins in vitro and, potentially, in vivo.

Enhancement of the stability of genetic switches by overlapping upstream regulatory domains

We study genetic switches formed from pairs of mutually repressing operons. The switch stability is characterized by a well-defined lifetime, which grows very rapidly, albeit subexponentially, with the number of copies of the most-expressed transcription factor. The switch stability can be drastically enhanced by overlapping the upstream regulatory domains such that competing regulatory molecules mutually exclude each other. Our results suggest that robustness against biochemical noise can provide a selection pressure that drives operons together in the course of evolution.

Network-based analysis of metabolic regulation in the human red blood cell

Reconstruction of cell-scale metabolic networks is now possible. A description of allowable metabolic network functions can be obtained using extreme pathways, which are the convex basis vectors of the solution space containing all steady state flux distributions. However, only a portion of these allowable network functions are physiologically possible due to kinetic and regulatory constraints. Methods are now needed that enable us to take a defined metabolic network and deduce candidate regulatory structures that control the selection of these physiologically relevant states. One such approach is the singular value decomposition (SVD) of extreme pathway matrices (P), which allows for the characterization of steady state solution spaces. Eigenpathways, which are the left singular vectors from the SVD of P, can be described and categorized by their biochemical function. SVD of P for the human red blood cell showed that the first five eigenpathways, out of a total of 23, effectively characterize all the relevant physiological states of red blood cell metabolism calculated with a detailed kinetic model. Thus, with five degrees of freedom the magnitude and nature of the regulatory needs are defined. Additionally, the dominant features of these first five eigenpathways described key metabolic splits that are indeed regulated in the human red blood cell. The extreme pathway matrix is derived directly from network topology and only knowledge of Vmax values is needed to reach these conclusions. Thus, we have implemented a network-based analysis of regulation that complements the study of individual regulatory events. This topological approach may provide candidate regulatory structures for metabolic networks with known stoichiometry but poorly characterized regulation.

Biochemical Network Stochastic Simulator (BioNetS): software for stochastic modeling of biochemical networks

BACKGROUND

Intrinsic fluctuations due to the stochastic nature of biochemical reactions can have large effects on the response of biochemical networks. This is particularly true for pathways that involve transcriptional regulation, where generally there are two copies of each gene and the number of messenger RNA (mRNA) molecules can be small. Therefore, there is a need for computational tools for developing and investigating stochastic models of biochemical networks.

RESULTS

We have developed the software package Biochemical Network Stochastic Simulator (BioNetS) for efficiently and accurately simulating stochastic models of biochemical networks. BioNetS has a graphical user interface that allows models to be entered in a straightforward manner, and allows the user to specify the type of random variable (discrete or continuous) for each chemical species in the network. The discrete variables are simulated using an efficient implementation of the Gillespie algorithm. For the continuous random variables, BioNetS constructs and numerically solves the appropriate chemical Langevin equations. The software package has been developed to scale efficiently with network size, thereby allowing large systems to be studied. BioNetS runs as a BioSpice agent and can be downloaded from http://www.biospice.org. BioNetS also can be run as a stand alone package. All the required files are accessible from http://x.amath.unc.edu/BioNetS.

CONCLUSIONS

We have developed BioNetS to be a reliable tool for studying the stochastic dynamics of large biochemical networks. Important features of BioNetS are its ability to handle hybrid models that consist of both continuous and discrete random variables and its ability to model cell growth and division. We have verified the accuracy and efficiency of the numerical methods by considering several test systems.

Inferring cellular networks using probabilistic graphical models

High-throughput genome-wide molecular assays, which probe cellular networks from different perspectives, have become central to molecular biology. Probabilistic graphical models are useful for extracting meaningful biological insights from the resulting data sets. These models provide a concise representation of complex cellular networks by composing simpler submodels. Procedures based on well-understood principles for inferring such models from data facilitate a model-based methodology for analysis and discovery. This methodology and its capabilities are illustrated by several recent applications to gene expression data.

Bistability and switching in the lysis/lysogeny genetic regulatory network of bacteriophage λ

Bistability and switching are two important aspects of the genetic regulatory network of lambda phage. Positive and negative feedbacks are key regulatory mechanisms in this network. By the introduction of threshold values, the developmental pathway of lambda phage is divided into different stages. If the protein level reaches a threshold value, positive or negative feedback will be effective and regulate the process of development. Using this regulatory mechanism, we present a quantitative model to realize bistability and switching of lambda phage based on experimental data. This model gives descriptions of decisive mechanisms for different pathways in induction. A stochastic model is also introduced for describing statistical properties of switching in induction. A stochastic degradation rate is used to represent intrinsic noise in induction for switching the system from the lysogenic pathway to the lysis pathway. The approach in this paper represents an attempt to describe the regulatory mechanism in genetic regulatory network under the influence of intrinsic noise in the framework of continuous models.

Summing up the noise in gene networks

Random fluctuations in genetic networks are inevitable as chemical reactions are probabilistic and many genes, RNAs and proteins are present in low numbers per cell. Such 'noise' affects all life processes and has recently been measured using green fluorescent protein (GFP). Two studies show that negative feedback suppresses noise, and three others identify the sources of noise in gene expression. Here I critically analyse these studies and present a simple equation that unifies and extends both the mathematical and biological perspectives.

Simulating cellular dynamics through a coupled transcription, translation, metabolic model

In order to predict cell behavior in response to changes in its surroundings or to modifications of its genetic code, the dynamics of a cell are modeled using equations of metabolism, transport, transcription and translation implemented in the Karyote software. Our methodology accounts for the organelles of eukaryotes and the specialized zones in prokaryotes by dividing the volume of the cell into discrete compartments. Each compartment exchanges mass with others either through membrane transport or with a time delay effect associated with molecular migration. Metabolic and macromolecular reactions take place in user-specified compartments. Coupling among processes are accounted for and multiple scale techniques allow for the computation of processes that occur on a wide range of time scales. Our model is implemented to simulate the evolution of concentrations for a user-specifiable set of molecules and reactions that participate in cellular activity. The underlying equations integrate metabolic, transcription and translation reaction networks and provide a framework for simulating whole cells given a user-specified set of reactions. A rate equation formulation is used to simulate transcription from an input DNA sequence while the resulting mRNA is used via ribosome-mediated polymerization kinetics to accomplish translation. Feedback associated with the creation of species necessary for metabolism by the mRNA and protein synthesis modifies the rates of production of factors (e.g. nucleotides and amino acids) that affect the dynamics of transcription and translation. The concentrations of predicted proteins are compared with time series or steady state experiments. The expression and sequence of the predicted proteins are compared with experimental data via the construction of synthetic tryptic digests and associated mass spectra. We present the mathematical model showing the coupling of transcription, translation and metabolism in Karyote and illustrate some of its unique characteristics.

The circular topology of rhythm in asynchronous random Boolean networks

The analysis of previously evolved rhythmic asynchronous random Boolean networks [Biosystems 59 (2001) 185] reveals common topological characteristics indicating that rhythm originates from a circular functional structure. The rhythm generating core of the network has the form of a closed ring which operates as a synchronisation substrate by supporting a travelling wave of state change; the size of the ring corresponds well with the period of oscillation. The remaining nodes in the network are either stationary or follow the activity of the ring without feeding back into it so as to form a coherent whole. Rings are typically formed early on in the evolutionary search process. Alternatively, long chains of nodes are favoured before they close upon themselves to stabilize. Analysis of asynchronous networks with de-correlated (non-rhythmic, non-stationary) attractors reveals no such common topological characteristics. These results have been obtained using statistical analysis and a specifically developed bottom-up pruning algorithm. This algorithm works from local interactions to global configuration by eliminating redundant links. The suitability of the algorithm has been confirmed by both numerical and single lesion analysis. The ring topology solution for the generation of rhythm implies that it will be harder to evolve rhythmic networks for big sizes and small periods and for bigger number of connections per node. These trends are confirmed empirically. Finally, the identified mechanisms are utilised to handcraft rhythmic networks of different periods showing that a low number of connections suffices for a large variety of rhythms. Random asynchronous update forces the evolved solutions to be highly robust maintaining their performance in the presence of intrinsic noise. The biological implications of such robust designs for molecular clocks are discussed.

Design of genetic networks with specified functions by evolution in silico

Recent studies have provided insights into the modular structure of genetic regulatory networks and emphasized the interest of quantitative functional descriptions. Here, to provide a priori knowledge of the structure of functional modules, we describe an evolutionary procedure in silico that creates small gene networks performing basic tasks. We used it to create networks functioning as bistable switches or oscillators. The obtained circuits provide a variety of functional designs, demonstrate the crucial role of posttranscriptional interactions, and highlight design principles also found in known biological networks. The procedure should prove helpful as a way to understand and create small functional modules with diverse functions as well as to analyze large networks.

Limitations of quantitative gene regulation models: a case study

Understanding the relationship between network structure and behavior is fundamental to the field of computational and systems biology. A particularly important distinction is the extent to which qualitative aspects of network performance are encoded in network topology as opposed to being determined through quantitative details, such as those of kinetics. Here, we develop a general and rigorous mathematical framework for the analysis of genetic networks and apply it to a family of synthetic gene networks. A key feature of our methodology involves determining network behavior that is insensitive to kinetic parameters such as rate constants and nonlinear functional dependencies of rates on molecular concentrations. Results indicate that behavior observed in some networks cannot be reconciled with standard gene expression and regulation models. We explore relaxing model assumptions to explain the observed behavior, allowing for both dynamic and stochastic phenomena, and propose an alternative model. Our alternative model includes the suggestion of a new mechanism by which the counterintuitive behavior could be achieved; central to the model is the assumption that the Clp protein degradation system, which is responsible for the regulatory proteins used in this study, becomes saturated.

A computational model for the identification of biochemical pathways in the krebs cycle

We have applied an algorithmic methodology which provably decomposes any complex network into a complete family of principal subcircuits to study the minimal circuits that describe the Krebs cycle. Every operational behavior that the network is capable of exhibiting can be represented by some combination of these principal subcircuits and this computational decomposition is linearly efficient. We have developed a computational model that can be applied to biochemical reaction systems which accurately renders pathways of such reactions via directed hypergraphs (Petri nets). We have applied the model to the citric acid cycle (Krebs cycle). The Krebs cycle, which oxidizes the acetyl group of acetyl CoA to CO(2) and reduces NAD and FAD to NADH and FADH(2), is a complex interacting set of nine subreaction networks. The Krebs cycle was selected because of its familiarity to the biological community and because it exhibits enough complexity to be interesting in order to introduce this novel analytic approach. This study validates the algorithmic methodology for the identification of significant biochemical signaling subcircuits, based solely upon the mathematical model and not upon prior biological knowledge. The utility of the algebraic-combinatorial model for identifying the complete set of biochemical subcircuits as a data set is demonstrated for this important metabolic process.