1,136 research outputs found

    Petri nets for systems and synthetic biology

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    We give a description of a Petri net-based framework for modelling and analysing biochemical pathways, which uni¯es the qualita- tive, stochastic and continuous paradigms. Each perspective adds its con- tribution to the understanding of the system, thus the three approaches do not compete, but complement each other. We illustrate our approach by applying it to an extended model of the three stage cascade, which forms the core of the ERK signal transduction pathway. Consequently our focus is on transient behaviour analysis. We demonstrate how quali- tative descriptions are abstractions over stochastic or continuous descrip- tions, and show that the stochastic and continuous models approximate each other. Although our framework is based on Petri nets, it can be applied more widely to other formalisms which are used to model and analyse biochemical networks

    Joining and decomposing reaction networks

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    In systems and synthetic biology, much research has focused on the behavior and design of single pathways, while, more recently, experimental efforts have focused on how cross-talk (coupling two or more pathways) or inhibiting molecular function (isolating one part of the pathway) affects systems-level behavior. However, the theory for tackling these larger systems in general has lagged behind. Here, we analyze how joining networks (e.g., cross-talk) or decomposing networks (e.g., inhibition or knock-outs) affects three properties that reaction networks may possess---identifiability (recoverability of parameter values from data), steady-state invariants (relationships among species concentrations at steady state, used in model selection), and multistationarity (capacity for multiple steady states, which correspond to multiple cell decisions). Specifically, we prove results that clarify, for a network obtained by joining two smaller networks, how properties of the smaller networks can be inferred from or can imply similar properties of the original network. Our proofs use techniques from computational algebraic geometry, including elimination theory and differential algebra.Comment: 44 pages; extensive revision in response to referee comment

    Implicit dose-response curves

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    We develop tools from computational algebraic geometry for the study of steady state features of autonomous polynomial dynamical systems via elimination of variables. In particular, we obtain nontrivial bounds for the steady state concentration of a given species in biochemical reaction networks with mass-action kinetics. This species is understood as the output of the network and we thus bound the maximal response of the system. The improved bounds give smaller starting boxes to launch numerical methods. We apply our results to the sequential enzymatic network studied in Markevich et al. (J Cell Biol 164(3):353–359, 2004) to find nontrivial upper bounds for the different substrate concentrations at steady state. Our approach does not require any simulation, analytical expression to describe the output in terms of the input, or the absence of multistationarity. Instead, we show how to extract information from effectively computable implicit dose-response curves, with the use of resultants and discriminants. We moreover illustrate in the application to an enzymatic network, the relation between the exact implicit dose-response curve we obtain symbolically and the standard hysteresis diagram provided by a numerical ode solver. The setting and tools we propose could yield many other results adapted to any autonomous polynomial dynamical system, beyond those where it is possible to get explicit expressions.Fil: Pérez Millán, Mercedes Soledad. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Dickenstein, Alicia Marcela. Consejo Nacional de Investigaciones Científicas y Técnicas. Oficina de Coordinación Administrativa Ciudad Universitaria. Instituto de Investigaciones Matemáticas "Luis A. Santaló". Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Instituto de Investigaciones Matemáticas "Luis A. Santaló"; Argentin

    Bistability in Apoptosis by Receptor Clustering

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    Apoptosis is a highly regulated cell death mechanism involved in many physiological processes. A key component of extrinsically activated apoptosis is the death receptor Fas, which, on binding to its cognate ligand FasL, oligomerize to form the death-inducing signaling complex. Motivated by recent experimental data, we propose a mathematical model of death ligand-receptor dynamics where FasL acts as a clustering agent for Fas, which form locally stable signaling platforms through proximity-induced receptor interactions. Significantly, the model exhibits hysteresis, providing an upstream mechanism for bistability and robustness. At low receptor concentrations, the bistability is contingent on the trimerism of FasL. Moreover, irreversible bistability, representing a committed cell death decision, emerges at high concentrations, which may be achieved through receptor pre-association or localization onto membrane lipid rafts. Thus, our model provides a novel theory for these observed biological phenomena within the unified context of bistability. Importantly, as Fas interactions initiate the extrinsic apoptotic pathway, our model also suggests a mechanism by which cells may function as bistable life/death switches independently of any such dynamics in their downstream components. Our results highlight the role of death receptors in deciding cell fate and add to the signal processing capabilities attributed to receptor clustering.Comment: Accepted by PLoS Comput Bio

    Graphical Conditions for Rate Independence in Chemical Reaction Networks

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    Chemical Reaction Networks (CRNs) provide a useful abstraction of molecular interaction networks in which molecular structures as well as mass conservation principles are abstracted away to focus on the main dynamical properties of the network structure. In their interpretation by ordinary differential equations, we say that a CRN with distinguished input and output species computes a positive real function f:R+f : R+ \rightarrowR+ R+, if for any initial concentration x of the input species, the concentration of the output molecular species stabilizes at concentration f (x). The Turing-completeness of that notion of chemical analog computation has been established by proving that any computable real function can be computed by a CRN over a finite set of molecular species. Rate-independent CRNs form a restricted class of CRNs of high practical value since they enjoy a form of absolute robustness in the sense that the result is completely independent of the reaction rates and depends solely on the input concentrations. The functions computed by rate-independent CRNs have been characterized mathematically as the set of piecewise linear functions from input species. However, this does not provide a mean to decide whether a given CRN is rate-independent. In this paper, we provide graphical conditions on the Petri Net structure of a CRN which entail the rate-independence property either for all species or for some output species. We show that in the curated part of the Biomodels repository, among the 590 reaction models tested, 2 reaction graphs were found to satisfy our rate-independence conditions for all species, 94 for some output species, among which 29 for some non-trivial output species. Our graphical conditions are based on a non-standard use of the Petri net notions of place-invariants and siphons which are computed by constraint programming techniques for efficiency reasons

    A Unique Transformation from Ordinary Differential Equations to Reaction Networks

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    Many models in Systems Biology are described as a system of Ordinary Differential Equations, which allows for transient, steady-state or bifurcation analysis when kinetic information is available. Complementary structure-related qualitative analysis techniques have become increasingly popular in recent years, like qualitative model checking or pathway analysis (elementary modes, invariants, flux balance analysis, graph-based analyses, chemical organization theory, etc.). They do not rely on kinetic information but require a well-defined structure as stochastic analysis techniques equally do. In this article, we look into the structure inference problem for a model described by a system of Ordinary Differential Equations and provide conditions for the uniqueness of its solution. We describe a method to extract a structured reaction network model, represented as a bipartite multigraph, for example, a continuous Petri net (CPN), from a system of Ordinary Differential Equations (ODEs). A CPN uniquely defines an ODE, and each ODE can be transformed into a CPN. However, it is not obvious under which conditions the transformation of an ODE into a CPN is unique, that is, when a given ODE defines exactly one CPN. We provide biochemically relevant sufficient conditions under which the derived structure is unique and counterexamples showing the necessity of each condition. Our method is implemented and available; we illustrate it on some signal transduction models from the BioModels database. A prototype implementation of the method is made available to modellers at http://contraintes.inria.fr/~soliman/ode2pn.html, and the data mentioned in the “Results” section at http://contraintes.inria.fr/~soliman/ode2pn_data/. Our results yield a new recommendation for the import/export feature of tools supporting the SBML exchange format

    Algebraic Systems Biology: A Case Study for the Wnt Pathway

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    Steady state analysis of dynamical systems for biological networks give rise to algebraic varieties in high-dimensional spaces whose study is of interest in their own right. We demonstrate this for the shuttle model of the Wnt signaling pathway. Here the variety is described by a polynomial system in 19 unknowns and 36 parameters. Current methods from computational algebraic geometry and combinatorics are applied to analyze this model.Comment: 24 pages, 2 figure

    Analytical Mechanics in Stochastic Dynamics: Most Probable Path, Large-Deviation Rate Function and Hamilton-Jacobi Equation

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    Analytical (rational) mechanics is the mathematical structure of Newtonian deterministic dynamics developed by D'Alembert, Langrange, Hamilton, Jacobi, and many other luminaries of applied mathematics. Diffusion as a stochastic process of an overdamped individual particle immersed in a fluid, initiated by Einstein, Smoluchowski, Langevin and Wiener, has no momentum since its path is nowhere differentiable. In this exposition, we illustrate how analytical mechanics arises in stochastic dynamics from a randomly perturbed ordinary differential equation dXt=b(Xt)dt+ϵdWtdX_t=b(X_t)dt+\epsilon dW_t where WtW_t is a Brownian motion. In the limit of vanishingly small ϵ\epsilon, the solution to the stochastic differential equation other than x˙=b(x)\dot{x}=b(x) are all rare events. However, conditioned on an occurence of such an event, the most probable trajectory of the stochastic motion is the solution to Lagrangian mechanics with L=q˙b(q)2/4\mathcal{L}=\|\dot{q}-b(q)\|^2/4 and Hamiltonian equations with H(p,q)=p2+b(q)pH(p,q)=\|p\|^2+b(q)\cdot p. Hamiltonian conservation law implies that the most probable trajectory for a "rare" event has a uniform "excess kinetic energy" along its path. Rare events can also be characterized by the principle of large deviations which expresses the probability density function for XtX_t as f(x,t)=eu(x,t)/ϵf(x,t)=e^{-u(x,t)/\epsilon}, where u(x,t)u(x,t) is called a large-deviation rate function which satisfies the corresponding Hamilton-Jacobi equation. An irreversible diffusion process with ×b0\nabla\times b\neq 0 corresponds to a Newtonian system with a Lorentz force q¨=(×b)×q˙+1/2b2\ddot{q}=(\nabla\times b)\times \dot{q}+1/2\nabla\|b\|^2. The connection between stochastic motion and analytical mechanics can be explored in terms of various techniques of applied mathematics, for example, singular perturbations, viscosity solutions, and integrable systems.Comment: 23 pages, 2 figure
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