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 offers a few insights into molecular biology and GRNs. The mathematical instruments worthwhile for learning the GRN version are then reviewed, particularly Hill services and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs below unfavourable suggestions with time delays and a distinct case of a homogenous GRN is taken into account. Asymptotic balance research of GRNs lower than 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, 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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This short examines a deterministic, ODE-based version for gene regulatory networks (GRN) that includes 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, specifically Hill capabilities and Schwarzian derivatives.
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Additional info for Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays
8). The questions of whether we have a unique or multiple equilibrium points, and whether these points are stable or not, are the subject of the remaining chapters. 2 Analysis of the Linearized Model Let xeq D Œx1 ; : : : ; xn T be an equilibrium point of the system. 8, we obtain the following result. 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.
2. x0 / < 1. 3. x0 / > 1, then r has exactly three fixed points. Proof. 2 Fixed Points 39 So, f is a bounded function, which implies that the function r is bounded. x0 / ¤ 1. x0 / > 1: Having a negative Schwarzian derivative, r is either of type A or type B. Therefore, if we prove second and third part of the Proposition, then the first part will follow. 5 by noting that 0 cannot be a fixed point of r. For the third part, we assume that r is of type B. 32) Note that the zero crossings of h and the fixed points of r are the same.
Assume that r is of type A. Then r 0 is strictly decreasing in RC . Let x0 > 0 be any fixed point of r. x0 / 1. x/ > 1; 8x 2 Œ0; x0 ; leads to a contradiction. x0 / < 1. 0/ D 0. 0; 1/. 0; 1/. 0; /. x/dx > 0 C ; r. 0; 1/. x/ has another fixed point greater than 0. x/. x/ < 1 for x > x1 . x/ < 1 for x > x1 . Therefore, r can only have a unique fixed point greater than 0. t u Our final result deals with the case when r is of type B and has exactly three fixed points. This result is especially useful for studying the bistable behavior discussed in Chapter 6.
Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays by Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu