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 offers a few insights into molecular biology and GRNs. The mathematical instruments helpful for learning the GRN version are then reviewed, particularly Hill capabilities and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs below detrimental suggestions with time delays and a distinct 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, during which stipulations resulting in bi-stability are derived. Graduate and complex undergraduate scholars and researchers up to speed 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**

**Example text**

0/ which implies that r is bounded. 2. Let x0 be the unique fixed point of r. 0/ > 0, we have x0 > 0. 23) Since r is bounded, the function f is also bounded. 1, it follows that the function f also has negative Schwarzian derivative. Moreover, by the chain rule, we have f 0 D r 0 r 0 > 0. 1, we know that the function f is either of type A or B. x0 //2 D 1: We know 0 < x0 . 0/ > 0 ; 0 leading to a contradiction. x0 / D 1: We know that f can have either a unique fixed point or three fixed points. 0/ > 0; 0 leading to a contradiction.

E. 2) 53 54 5 Gene Regulatory Networks Under Negative Feedback where gD. 1 g1 / ı . 1 1 g2 / ı 2 ı. 3) n We start by first showing that g has a unique fixed point, which implies that the system has a unique equilibrium point. 1. 1) under negative feedback. Then, the system has a unique equilibrium point in RnC . Proof. 3 it is easy to see that g has a unique fixed point. 1) has as many equilibrium points as the fixed points of g. Since g has a unique fixed point, the system has a unique equilibrium point.

Xf / Ä 1. u t The second result shows that r has at most two fixed points if r is of type A. 5. x/ W RC ! X Â RC be a type A function. Suppose that r is bounded and continuously differentiable. Then, r has at most two fixed points. e. the origin is a fixed point. 26) Proof. 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.

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

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