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

ISBN-10: 3319156055

ISBN-13: 9783319156057

ISBN-10: 3319156063

ISBN-13: 9783319156064

This short examines a deterministic, ODE-based version for gene regulatory networks (GRN) that includes nonlinearities and time-delayed suggestions. An introductory bankruptcy presents a few insights into molecular biology and GRNs. The mathematical instruments important for learning the GRN version are then reviewed, particularly Hill features and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs less than destructive suggestions with time delays and a distinct case of a homogenous GRN is taken into account. Asymptotic balance research of GRNs lower than optimistic suggestions is then thought of in a separate bankruptcy, during which stipulations resulting in bi-stability are derived. Graduate and complicated undergraduate scholars and researchers up to the mark 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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**Example text**

X/ > 0 for all x 2 RC . Suppose that r is bounded and at least three times continuously differentiable. Let xf denote the greatest positive fixed point of g. 25) Proof. xf / > 1. xf ; xf C /. xf ; xf C /. xf C / > xf C . Since the function r is bounded and continuous, there should exist a fixed point x0 36 3 Functions with Negative Schwarzian Derivatives of r such that x0 > xf C . This is contradiction to the fact that xf represents the greatest fixed point of r. xf / Ä 1. u t The second result shows that r has at most two fixed points if r is of type A.

0/ > 0, then h is of type B. 21) then h is of type A. 2 Fixed Points In this section, we analyze fixed points of functions with NSD. In this manuscript, we assume that the nonlinearity functions are monotonic, bounded functions, which take positive values. x/ < 0; 8x > 0: 32 3 Functions with Negative Schwarzian Derivatives We allow the derivative of r to be zero at x D 0 as this does not violate the monotonicity assumption of r. Moreover, we will assume that r has negative Schwarzian derivative.

In fact, the local stability analysis of an equilibrium point will provide us useful insights in the general behavior of the system. In particular, in Chapter 5, it is shown that if the unique equilibrium point of the GRN is locally unstable, then the system exhibits periodic solutions for large enough delay. s/. 2. 13) where for all i D 1; : : : n. s/ is stable independent of delay. i >0 Proof. 1). 3 A Synthetic Circuit: The Repressilator In this section, we derive a dynamic model of repressilators from mass action law and Michelis–Menten kinetics.

### Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays by Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu

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