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21
Computation with finite stochastic chemical reaction networks
 Natural Computing
, 2008
"... Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have ..."
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Cited by 19 (5 self)
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Abstract. A highly desired part of the synthetic biology toolbox is an embedded chemical microcontroller, capable of autonomously following a logic program specified by a set of instructions, and interacting with its cellular environment. Strategies for incorporating logic in aqueous chemistry have focused primarily on implementing components, such as logic gates, that are composed into larger circuits, with each logic gate in the circuit corresponding to one or more molecular species. With this paradigm, designing and producing new molecular species is necessary to perform larger computations. An alternative approach begins by noticing that chemical systems on the small scale are fundamentally discrete and stochastic. In particular, the exact molecular counts of each molecular species present, is an intrinsically available form of information. This might appear to be a very weak form of information, perhaps quite difficult for computations to utilize. Indeed, it has been shown that errorfree Turing universal computation is impossible in this setting. Nevertheless, we show a design of a chemical computer that achieves fast and reliable Turinguniversal computation using molecular counts. Our scheme uses only a small number of different molecular species to do computation of arbitrary complexity. The total probability of error of the computation can be made arbitrarily small (but not zero) by adjusting the initial molecular counts of certain species. While physical implementations would be difficult, these results demonstrate that molecular counts can be a useful form of information for small molecular systems such as those operating within cellular environments. Key words. stochastic chemical kinetics; molecular counts; Turinguniversal computation; probabilistic computation 1. Introduction. Many
Programmability of Chemical Reaction Networks
"... Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard c ..."
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Cited by 8 (2 self)
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Summary. Motivated by the intriguing complexity of biochemical circuitry within individual cells we study Stochastic Chemical Reaction Networks (SCRNs), a formal model that considers a set of chemical reactions acting on a finite number of molecules in a wellstirred solution according to standard chemical kinetics equations. SCRNs have been widely used for describing naturally occurring (bio)chemical systems, and with the advent of synthetic biology they become a promising language for the design of artificial biochemical circuits. Our interest here is the computational power of SCRNs and how they relate to more conventional models of computation. We survey known connections and give new connections between SCRNs and
Robust Stochastic Chemical Reaction Networks and Bounded TauLeaping
, 803
"... The behavior of some stochastic chemical reaction networks is largely unaffected by slight inaccuracies in reaction rates. We formalize the robustness of state probabilities to reaction rate deviations, and describe a formal connection between robustness and efficiency of simulation. Without robustn ..."
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Cited by 3 (2 self)
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The behavior of some stochastic chemical reaction networks is largely unaffected by slight inaccuracies in reaction rates. We formalize the robustness of state probabilities to reaction rate deviations, and describe a formal connection between robustness and efficiency of simulation. Without robustness guarantees, stochastic simulation seems to require computational time proportional to the total number of reaction events. Even if the concentration (molecular count per volume) stays bounded, the number of reaction events can be linear in the duration of simulated time and total molecular count. We show that the behavior of robust systems can be predicted such that the computational work scales linearly with the duration of simulated time and concentration, and only polylogarithmically in the total molecular count. Thus our asymptotic analysis captures the dramatic speedup when molecular counts are large, and shows that for bounded concentrations the computation time is essentially invariant with molecular count. Finally, by noticing that even robust stochastic chemical reaction networks are capable of embedding complex computational problems, we argue that the linear dependence on simulated time and concentration is likely optimal. 1
Noisy Attractors and Ergodic Sets in Models of Genetic Regulatory Networks
"... We investigate the hypothesis that cell types are attractors. This hypothesis was criticized with the fact that real gene networks are noisy systems and thus, do not have attractors (Kadanoff et al, 2002). Given the concept of “ergodic set ” as a set of states from which the system, once entering, d ..."
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Cited by 2 (0 self)
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We investigate the hypothesis that cell types are attractors. This hypothesis was criticized with the fact that real gene networks are noisy systems and thus, do not have attractors (Kadanoff et al, 2002). Given the concept of “ergodic set ” as a set of states from which the system, once entering, does not leave when subject to internal noise, first, using the Boolean network model, we show that if all nodes of states on attractors are subject to internal state change with a probability p due to noise, multiple ergodic sets are very unlikely. Thereafter, we show that if a fraction of those nodes are “locked ” (not subject to state fluctuations caused by internal noise), multiple ergodic sets emerge. Finally, we present an example of a gene network, modelled with a realistic model Preprint submitted to Elsevier 13 April 2007of transcription and translation and genegene interaction, driven by a Stochastic Simulation Algorithm with multiple timedelayed reactions, which has internal noise and that we also subject to external perturbations. We show that, in this case, two distinct ergodic sets exist and are stable within a wide range of parameters variations and, to some extent, to external perturbations.
Zero Singularities of Codimension Two and Three in Delay Differential Equations
, 2008
"... We give conditions under which a general class of delay differential equations has a point of BogdanovTakens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to a two or three dimensional systems of ordinary differential equations. W ..."
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Cited by 2 (0 self)
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We give conditions under which a general class of delay differential equations has a point of BogdanovTakens or a triple zero bifurcation. We show how a centre manifold projection of the delay equations reduces the dynamics to a two or three dimensional systems of ordinary differential equations. We put these equations in normal form and determine how the coefficients of the normal forms depend on the original parameters in the model. Finally we apply our results to two neural models and compare the predictions of the theory with numerical bifurcation analysis of the full equations. One model involves a transcritical bifurcation, hence we derive and analyze the appropriate unfoldings for this case. Mathematics Subject Classification: 92B20, 34K20, 34K15 Keywords: delay differential equations, centre manifold, normal form, BogdanovTakens bifurcation, triple zero singularity. 1
Inference in continuoustime changepoint models
"... We consider the problem of Bayesian inference for continuoustime multistable stochastic systems which can change both their diffusion and drift parameters at discrete times. We propose exact inference and sampling methodologies for two specific cases where the discontinuous dynamics is given by a ..."
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Cited by 1 (1 self)
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We consider the problem of Bayesian inference for continuoustime multistable stochastic systems which can change both their diffusion and drift parameters at discrete times. We propose exact inference and sampling methodologies for two specific cases where the discontinuous dynamics is given by a Poisson process and a twostate Markovian switch. We test the methodology on simulated data, and apply it to two real data sets in finance and systems biology. Our experimental results show that the approach leads to valid inferences and nontrivial insights. 1
Induction Level Determines Signature of Gene Expression Noise in Cellular Systems
, 2008
"... Noise in gene expression, either due to inherent stochasticity or to varying inter and intracellular environment, can generate significant celltocell variability of protein levels in clonal populations. To quantify the different sources of gene expression noise several theoretical studies have be ..."
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Noise in gene expression, either due to inherent stochasticity or to varying inter and intracellular environment, can generate significant celltocell variability of protein levels in clonal populations. To quantify the different sources of gene expression noise several theoretical studies have been performed either using a meanfield approximation for the emerging master equation or deriving a timedependent description, when cell division is taken explicitly into account. Here, we give a short overview of the different origins of gene expression noise which were found experimentally and introduce the basic stochastic modelling approaches. Furthermore, we extend and apply the timedependent description of gene expression noise to published, experimental data. The analysis shows that the induction level of the transcription factor can be employed to discriminate the noise profiles and their characteristic signatures. Furthermore, on the basis of experimentally measured cell distributions, our simulations suggest that transcription factor binding and promoter activation can be modelled independently of each other with sufficient accuracy. ∗ Correspondence to JR. Address: Center for Biological Systems Analysis, Habsburgerstr.
Leading Edge Review Nature, Nurture, or Chance: Stochastic Gene Expression and Its Consequences
"... Gene expression is a fundamentally stochastic process, with randomness in transcription and translation leading to celltocell variations in mRNA and protein levels. This variation appears in organisms ranging from microbes to metazoans, and its characteristics depend both on the biophysical parame ..."
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Gene expression is a fundamentally stochastic process, with randomness in transcription and translation leading to celltocell variations in mRNA and protein levels. This variation appears in organisms ranging from microbes to metazoans, and its characteristics depend both on the biophysical parameters governing gene expression and on gene network structure. Stochastic gene expression has important consequences for cellular function, being beneficial in some contexts and harmful in others. These situations include the stress response, metabolism, development, the cell cycle, circadian rhythms, and aging.