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 presents a few insights into molecular biology and GRNs. The mathematical instruments invaluable for learning the GRN version are then reviewed, specifically Hill features and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs less than destructive suggestions with time delays and a distinct case of a homogenous GRN is taken into account. Asymptotic balance research of GRNs below confident 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, structures 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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Extra info for Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays
1. x/ is well defined. 2) then the following holds: 1. 2. 3. 4. x//. x/. x/ < 0. x/ cannot have positive local minima nor negative local maxima. Proof. 1. x/ C c/. x//. 2. x/: 3. 1 Classification of Functions with Negative Schwarzian Derivatives 27 4. x/ > 0 which is a contradiction. 5) In other words, it was shown that h0 cannot have positive local minima, so f 0 cannot have negative local maxima. u t Let us now calculate Schwarzian derivatives of some functions which are commonly used as nonlinearities in the modeling of physical systems.
15), for the sake of simplicity we assume that those delays are negligible. 0/ denotes the initial state of the gene. , negative feedback. 2. The model contains n proteins Pi ; i D 1; : : : ; n, which Fig. 2 The n-repressilator model. 50 4 Deterministic ODE-Based Model with Time Delay are produced from the respective gene Gi , which in turn is inhibited by PiC1 . The respective chemical kinematics can be described as: 9 > transcription+translation > > > > > > > k3i 1 > = PiC1 C Gi • P G inhibition > > k 3iC1 > > > > > > > k3i ; Pi !
37]. 0/ D 0 is allowed, since it does not violate the monotonicity of gj . 1. 0; 1/: In the next two chapters, we will investigate the GRN under both negative and positive feedback conditions. 3) is infinite dimensional. a/ W i. Œ ; 0; R/ denotes the set of real valued continuous functions defined on the interval Œ ; 0. a/, for a 2 Œ ; 0. 3). 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.
Analysis of Deterministic Cyclic Gene Regulatory Network Models with Delays by Mehmet Eren Ahsen, Hitay Özbay, Silviu-Iulian Niculescu