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 helpful for learning the GRN version are then reviewed, specifically 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 below confident 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, platforms 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
X/. x/ D 2b 2 < 0: As a corollary of the above, we have the following result. 1. Let a, b > 0, c 0 and m 2 N be constants. 6). Then, one of the following holds: 1. x/ D 0. 2. x/ < 0. 3. bx/, then S(h(x))< 0. 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.
X/ representing its first, second, and third derivatives. 1). E. 1. x/ is well defined. 2) then the following holds: 1. 2. 3. 4. x//. x/. x/ < 0. x/ cannot have positive local minima nor negative local maxima. Proof. 1. x/ C c/. x//. 2. x/: 3. 1 Classification of Functions with Negative Schwarzian Derivatives 27 4. x/ > 0 which is a contradiction. 5) In other words, it was shown that h0 cannot have positive local minima, so f 0 cannot have negative local maxima. u t Let us now calculate Schwarzian derivatives of some functions which are commonly used as nonlinearities in the modeling of physical systems.
Moreover, suppose that h is bounded. Then h0 cannot be a strictly increasing function, otherwise h cannot be bounded. a; 1/. 1 0 0 2 4 6 8 0 1 2 3 4 5 6 x x Fig. 1 Typical h0 vs x graphs for type A and B functions. 1. For a bounded function h with a negative Schwarzian derivative, we will say h is of type A if h0 is a strictly decreasing function, and of type B otherwise. 1. 2. It is easy to determine whether a function h is of type A or B. 0/ D 0, then it is clear that the function h is of type B.