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 precious for learning the GRN version are then reviewed, particularly Hill capabilities and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs below adverse suggestions with time delays and a different case of a homogenous GRN is taken into account. Asymptotic balance research of GRNs less than 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 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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Additional resources for Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays
Keywords Schwarzian derivatives • Hill functions • Tangent hyperbolic functions • Fixed points This chapter is devoted to the analysis of functions with NSD (negative Schwarzian derivatives) and consists of two sections. In the first section, basic properties of functions with NSD are given and a classification result is proven for such functions. The second section contains an analysis on the fixed points of functions with NSD. 1 Classification of Functions with Negative Schwarzian Derivatives We start this section with the definition of the Schwarzian derivative.
1. 5). Proof. 5). 8, we conclude that h has as many fixed point as g. But each fixed point of h is an equilibrium point of the system, which concludes the proof. x/. 3). s/ D n Y ! 11) ! 13) 48 4 Deterministic ODE-Based Model with Time Delay is stable. Note that k ! 14) frequently for the analysis of the GRN. 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.
6. 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.