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 invaluable for learning the GRN version are then reviewed, specifically Hill services and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs less than unfavorable 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, during which stipulations resulting in bi-stability are derived. Graduate and complex undergraduate scholars and researchers on top of things engineering, utilized arithmetic, platforms 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 resources for Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays**

**Example text**

S/ is stable independent of delay. i >0 Proof. 1). 3 A Synthetic Circuit: The Repressilator In this section, we derive a dynamic model of repressilators from mass action law and Michelis–Menten kinetics. The detailed derivation of the chemical reaction equations is beyond the scope of this book. Also, the assumptions made in this section are standard in biochemistry. An interested reader may refer to a popular biochemistry book such as [46]. The repressilator is a synthetic genetic regulatory network first suggested in [47], where the authors used three transcriptional repressors in cascade to build an oscillating network in Escherichia coli.

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. 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.

X0 / and r has another fixed point x1 < x0 . 7. 28). Let x0 be the unique fixed point of the function f . Then, the following properties hold: 1. x0 /j < 1, then r has the unique fixed point x0 . 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.