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      Designing Attractive Models via Automated Identification of Chaotic and Oscillatory Dynamical Regimes

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          Abstract

          Chaos and oscillations continue to capture the interest of both the scientific and public domains. Yet despite the importance of these qualitative features, most attempts at constructing mathematical models of such phenomena have taken an indirect, quantitative approach, e.g. by fitting models to a finite number of data-points. Here we develop a qualitative inference framework that allows us to both reverse engineer and design systems exhibiting these and other dynamical behaviours by directly specifying the desired characteristics of the underlying dynamical attractor. This change in perspective from quantitative to qualitative dynamics, provides fundamental and new insights into the properties of dynamical systems.

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          Approximate Bayesian computation scheme for parameter inference and model selection in dynamical systems

          Approximate Bayesian computation methods can be used to evaluate posterior distributions without having to calculate likelihoods. In this paper we discuss and apply an approximate Bayesian computation (ABC) method based on sequential Monte Carlo (SMC) to estimate parameters of dynamical models. We show that ABC SMC gives information about the inferability of parameters and model sensitivity to changes in parameters, and tends to perform better than other ABC approaches. The algorithm is applied to several well known biological systems, for which parameters and their credible intervals are inferred. Moreover, we develop ABC SMC as a tool for model selection; given a range of different mathematical descriptions, ABC SMC is able to choose the best model using the standard Bayesian model selection apparatus.
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            Parameter estimation in biochemical pathways: a comparison of global optimization methods.

            Here we address the problem of parameter estimation (inverse problem) of nonlinear dynamic biochemical pathways. This problem is stated as a nonlinear programming (NLP) problem subject to nonlinear differential-algebraic constraints. These problems are known to be frequently ill-conditioned and multimodal. Thus, traditional (gradient-based) local optimization methods fail to arrive at satisfactory solutions. To surmount this limitation, the use of several state-of-the-art deterministic and stochastic global optimization methods is explored. A case study considering the estimation of 36 parameters of a nonlinear biochemical dynamic model is taken as a benchmark. Only a certain type of stochastic algorithm, evolution strategies (ES), is able to solve this problem successfully. Although these stochastic methods cannot guarantee global optimality with certainty, their robustness, plus the fact that in inverse problems they have a known lower bound for the cost function, make them the best available candidates.
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              Oscillatory expression of the bHLH factor Hes1 regulated by a negative feedback loop.

              Transcription of messenger RNAs (mRNAs) for Notch signaling molecules oscillates with 2-hour cycles, and this oscillation is important for coordinated somite segmentation. However, the molecular mechanism of such oscillation remains to be determined. Here, we show that serum treatment of cultured cells induces cyclic expression of both mRNA and protein of the Notch effector Hes1, a basic helix-loop-helix (bHLH) factor, with 2-hour periodicity. Cycling is cell-autonomous and depends on negative autoregulation of hes1 transcription and ubiquitin-proteasome-mediated degradation of Hes1 protein. Because Hes1 oscillation can be seen in many cell types, this clock may regulate timing in many biological systems.
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                Author and article information

                Journal
                10.1038/ncomms1496
                1108.4746

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