ABSTRACT Transcriptional regulation is an inherently noisy process. The origins of this stochastic behavior can be traced to the random transitions among the discrete chemical states of operators that control the transcription rate and to finite number fluctuations in the biochemical reactions for the synthesis and degradation of transcripts. We develop stochastic models to which these random reactions are intrinsic and a series of simpler models derived explicitly from the first as approximations in different parameter regimes. This innate stochasticity can have both a quantitative and qualitative impact on the behavior of gene-regulatory networks. We introduce a natural generalization of deterministic bifurcations for classification of stochastic systems and show that simple noisy genetic switches have rich bifurcation structures; among them, bifurcations driven solely by changing the rate of operator fluctuations even as the underlying deterministic system remains unchanged. We find stochastic bistability where the deterministic equations predict monostability and vice-versa. We derive and solve equations for the mean waiting times for spontaneous transitions between quasistable states in these switches.

Protein levels have been shown to vary substantially between individual cells in clonal populations. In prokaryotes, the contribution to such fluctuations from the inherent randomness of gene expression has largely been attributed to having just a few transcriptsofthecorrespondingmRNAs.Bycontrast, eukaryotic studies tend to emphasize chromatin remodeling and burst-like transcription. Here, we study single-cell transcription in Escherichia coli by measuring mRNA levels in individual living cells. The results directly demonstrate transcriptional bursting, similar to that indirectly inferred for eukaryotes.We also measure mRNA partitioning at cell division and correlate mRNA and protein levels in single cells. Partitioning is approximately binomial, and mRNAprotein correlations are weaker earlier in the cell cycle, where cell division has recently randomized the relative concentrations. Our methods further extend protein-based approaches by counting the integer-valued number of transcript with single-molecule resolution. This greatly facilitates kinetic interpretations in terms of the integer-valued random processes that produce the fluctuations.

Biochemical networks in single cells can display large fluctuations in molecule numbers, making mesoscopic approaches necessary for correct system descriptions. We present a general method that allows rapid characterization of the stochastic properties of intracellular networks. The starting point is a macroscopic description that identifies the system’s elementary reactions in terms of rate laws and stoichiometries. From this formulation follows directly the stationary solution of the linear noise approximation (LNA) of the Master equation for all the components in the network. The method complements bifurcation studies of the system’s parameter dependence by providing estimates of sizes, correlations, and time scales of stochastic fluctuations. We describe how the LNA can give precise system descriptions also near macroscopic instabilities by suitable variable changes and elimination of fast variables. [Supplemental material is available online at www.genome.org.] A key element in systems biology is the design of mathematical models that faithfully describe the dynamics of intracellular chemical networks. In general, chemical reactions in single cells occur far from thermodynamic equilibrium (Keizer 1987), and the molecule copy numbers can sometimes be very small (Gup-tasarama 1995). Both these properties make it mandatory to ana-

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Abstract. We propose a compositional approach to the dynamics of gene regulatory networks based on the stochastic π-calculus, and develop a representation of gene network elements which can be used to build complex circuits in a transparent and efficient way. To demonstrate the power of the approach we apply it to several artificial networks, such as the repressilator and combinatorial gene circuits first studied in Combinatorial Synthesis of Genetic Networks [1]. For two examples of the latter systems, we point out how the topology of the circuits and the interplay of the stochastic gate interactions influence the circuit behavior. Our approach may be useful for the testing of biological mechanisms proposed to explain the experimentally observed circuit dynamics. 1

Introduction Metabolism is the collection of chemical interconversions that take place in all cells or organisms and drive life processes. In analogy to the terms "genome", "transcriptome" and "proteome", the comprehensive set of metabolites synthesized by a particular organism or tissue is refered to as its "metabolome". Metabolomic analysis aims at the unbiased identification and quantification of all metabolites in a biological sample [1, 3, 5, 7]. By integrating the data obtained from such large-scale experiments into metabolic correlation networks, its power to significantly extend and enhance existing functional genomics approaches has already been demonstrated [2, 4]. However, the relationship between metabolic correlation networks and the underlying enzymatic system is still largely unclear. Here, we will address the interpretation of these data-generated networks in terms of the underlying biochemical pathways [6]. Methods and Results As a first step, we elucidate possible

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This paper, prepared for a tutorial at the 2005 IEEE Conference on Decision and Control, presents an introduction to molecular systems biology and some associated problems in control theory. It provides an introduction to basic biological concepts, describes several questions in dynamics and control that arise in the field, and argues that new theoretical problems arise naturally in this context. A final section focuses on the combined use of graph-theoretic, qualitative knowledge about monotone building-blocks and steadystate step responses for components.

Ultrasensitive cascades often implement thresholding operations in cell signaling and gene regulatory networks, converting graded input signals into discrete all-or-none outputs. However, the biochemical and genetic reactions involved in such cascades are subject to random fluctuations, leading to noise in output signal levels. Here we prove that cascades operating near saturation have output signal fluctuations that are bounded in magnitude, even as the number of noisy cascade stages becomes large. We show that these fluctuation-bounded cascades can be used to attenuate the noise in an input signal, and we find the optimal cascade length required to achieve the best possible noise reduction. Cascades with ultrasensitive transfer functions naturally operate near saturation, and can be made to simultaneously implement thresholding and noise reduction. They are therefore ideally suited to mediate signal transfer in both natural and artificial biological networks.

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