By Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu
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 useful 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 unfavorable 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 on top of things 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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Additional info for Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays
X/ be a type B function. x/ Ä 1. x3 / < 1: Before starting the proof, it is worth noting that for r to have three fixed points, it should be a type B function. Proof. We have already proved that the function r has at most three fixed points. If r has two or less fixed points, then the proposition is proven automatically. xi / Ä 1 for all i D 1; 2; 3. y1 / > 1. x1 ; x2 /. This is contradiction to the fact that x2 is a fixed point of r. y2 / > 1. 1. Let us continue with the second part of the proof.
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. 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 .
X3 / < 1. x1 / 1. x1 ; x2 /, which is contradiction. x/ > 1 in Œx2 of calculus, it follows that ; x2 . x/dx > x1 C x2 x1 ; x1 which again is a contradiction to the fact that x2 is a fixed point of r. x1 / < 1, which completes the proof. x/ has a special form. 3. 29) Moreover, if x0 is a fixed point of r, then one of the following holds: 1. x0 is a fixed point of f . 2. x0 / and r has another fixed point x1 > x0 . 3. x0 / and r has another fixed point x1 < x0 . 7. 28). Let x0 be the unique fixed point of the function f .