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      Piecewise linear and Boolean models of chemical reaction networks

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          Abstract

          Models of biochemical networks are frequently high-dimensional and complex. Reduction methods that preserve important dynamical properties are therefore essential in their study. Interactions between the nodes in such networks are frequently modeled using a Hill function, \(x^n/(J^n+x^n)\). Reduced ODEs and Boolean networks have been studied extensively when the exponent \(n\) is large. However, the case of small constant \(J\) appears in practice, but is not well understood. In this paper we provide a mathematical analysis of this limit, and show that a reduction to a set of piecewise linear ODEs and Boolean networks can be mathematically justified. The piecewise linear systems have closed form solutions that closely track those of the fully nonlinear model. On the other hand, the simpler, Boolean network can be used to study the qualitative behavior of the original system. We justify the reduction using geometric singular perturbation theory and compact convergence, and illustrate the results in networks modeling a genetic switch and a genetic oscillator.

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          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.
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            Defining network topologies that can achieve biochemical adaptation.

            Many signaling systems show adaptation-the ability to reset themselves after responding to a stimulus. We computationally searched all possible three-node enzyme network topologies to identify those that could perform adaptation. Only two major core topologies emerge as robust solutions: a negative feedback loop with a buffering node and an incoherent feedforward loop with a proportioner node. Minimal circuits containing these topologies are, within proper regions of parameter space, sufficient to achieve adaptation. More complex circuits that robustly perform adaptation all contain at least one of these topologies at their core. This analysis yields a design table highlighting a finite set of adaptive circuits. Despite the diversity of possible biochemical networks, it may be common to find that only a finite set of core topologies can execute a particular function. These design rules provide a framework for functionally classifying complex natural networks and a manual for engineering networks. For a video summary of this article, see the PaperFlick file with the Supplemental Data available online.
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              The Yeast Cell-Cycle Network Is Robustly Designed

              , , (2003)
              The interactions between proteins, DNA, and RNA in living cells constitute molecular networks that govern various cellular functions. To investigate the global dynamical properties and stabilities of such networks, we studied the cell-cycle regulatory network of the budding yeast. With the use of a simple dynamical model, it was demonstrated that the cell-cycle network is extremely stable and robust for its function. The biological stationary state--the G1 state--is a global attractor of the dynamics. The biological pathway--the cell-cycle sequence of protein states--is a globally attracting trajectory of the dynamics. These properties are largely preserved with respect to small perturbations to the network. These results suggest that cellular regulatory networks are robustly designed for their functions.
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                Author and article information

                Journal
                21 August 2013
                Article
                1308.4758
                ebdffba3-6f88-4a38-9016-5e96b65aa07c

                http://arxiv.org/licenses/nonexclusive-distrib/1.0/

                History
                Custom metadata
                This is an extension of previous work arXiv:1201.2072
                q-bio.MN q-bio.QM

                Quantitative & Systems biology,Molecular biology
                Quantitative & Systems biology, Molecular biology

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