System Theory

Download Analysis of Deterministic Cyclic Gene Regulatory Network by Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu PDF

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By Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

This short examines a deterministic, ODE-based version for gene regulatory networks (GRN) that comes with nonlinearities and time-delayed suggestions. An introductory bankruptcy presents a few insights into molecular biology and GRNs. The mathematical instruments beneficial for learning the GRN version are then reviewed, specifically Hill services and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs lower than adverse suggestions with time delays and a different case of a homogenous GRN is taken into account. Asymptotic balance research of GRNs below optimistic suggestions is then thought of in a separate bankruptcy, within which stipulations resulting in bi-stability are derived. Graduate and complicated undergraduate scholars and researchers on top of things engineering, utilized arithmetic, structures biology and artificial biology will locate this short to be a transparent and concise advent to the modeling and research of GRNs.

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Extra resources for Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays

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1 Classification of Functions with Negative Schwarzian Derivatives 29 In the sequel, we try to classify functions with negative Schwarzian derivatives. Next result helps this endeavor. 1. 0; 1/ with a < b. 11) Proof. For the first part of Lemma, suppose on the contrary that there exist positive constants a < b such that h0 is constant in Œa; b. c/ D 0; which is a contradiction. Therefore, h0 cannot be constant in any subinterval of RC . d / > 0. x2 / for all x 2 I . x/, 8x 2 I: Hence, by definition, x1 is a positive local minima of the function h0 .

1, it cannot be constant on any interval. B/ < 0 and f is continuous being the summation of two continuous functions. x0 / D x0 . Hence, r has at least one fixed point. To prove the uniqueness of the fixed point, assume x1 < x2 are two different fixed points of the function r. x2 / D x2 . x/ < 0 for all positive x. x/ can have a unique fixed point, which completes the first part of the proof. For the second part, assume that r has four fixed points x1 < x2 < x3 < x4 . y3 / > 1. y2 / D 1. y1 ; y3 /, which is a positive local minima.

3 A Synthetic Circuit: The Repressilator 49 9 > transcription+translation > > > > > > C > k2 > = P C G • P G inhibition > > k2 > > > > > > > k3 ; P ! 15) where G, P , and P G represent the Gene, Protein, and Protein-Gene complex, which provides self-inhibition. 15) is given in [48]. 15), for the sake of simplicity we assume that those delays are negligible. 0/ denotes the initial state of the gene. , negative feedback. 2. The model contains n proteins Pi ; i D 1; : : : ; n, which Fig. 2 The n-repressilator model.

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