System Theory

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

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 offers a few insights into molecular biology and GRNs. The mathematical instruments worthy for learning the GRN version are then reviewed, particularly Hill services and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs less than detrimental suggestions with time delays and a distinct 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 complex undergraduate scholars and researchers up to speed engineering, utilized arithmetic, structures biology and artificial biology will locate this short to be a transparent and concise creation 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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0/ which implies that r is bounded. 2. Let x0 be the unique fixed point of r. 0/ > 0, we have x0 > 0. 23) Since r is bounded, the function f is also bounded. 1, it follows that the function f also has negative Schwarzian derivative. Moreover, by the chain rule, we have f 0 D r 0 r 0 > 0. 1, we know that the function f is either of type A or B. x0 //2 D 1: We know 0 < x0 . 0/ > 0 ; 0 leading to a contradiction. x0 / D 1: We know that f can have either a unique fixed point or three fixed points. 0/ > 0; 0 leading to a contradiction.

S/ is stable for all 0 Ä < m and unstable for all m , where m is the delay margin and it is computed as m ! c arctan. g > 0 is the smallest ! satisfying  n X ! arctan a i iD1 à D : 3. s/ is unstable independent of delay. 4. s/ is stable independent of delay. 5. s/ is unstable independent of delay. Proof. 17). c C an / ˇ iD1 Qn iD1 .! 2 C ai2 / is an increasing function of ! 1. 20) has a root on the imaginary axis [44]. 1. Moreover, if K l < k < Ku ; then the delay free system is stable. s/ has a root on the imaginary axis [44].

X/ D 1C2x 2 and h D f ı f . Compute the fixed points of h. 0 What are the values of h at these fixed points? Problem 3. 5. Chapter 4 Deterministic ODE-Based Model with Time Delay Abstract This chapter is devoted to the derivation of the ODE-based model with time delay that is to be analyzed in the forthcoming chapters. In particular, an equivalent simplified mathematical model of the GRN model is proposed through some interpretations of the interconnection scheme. Next, the stability conditions for the linearized system around an equilibrium point are discussed.

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