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 important for learning the GRN version are then reviewed, specifically Hill services and Schwarzian derivatives. One bankruptcy is dedicated to the research of GRNs lower than adverse suggestions with time delays and a different case of a homogenous GRN is taken into account. Asymptotic balance research of GRNs less 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 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
6, we know that the function r has at most three fixed points. 3, we can conclude that g has exactly three fixed points. u t Final result of this section reduces the process of finding the fixed points of some multidimensional functions defined on the cone RnC to finding the fixed points of a function defined on RC . 8. x/ W RnC ! xi / W RC ! 43) The number of fixed points of the functions h and q have the same cardinality. Proof. x1 ; x2 ; : : : ; xn / be a fixed point of h. x1 /. Hence, x1 is a fixed point of q.
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.
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. 3. 1. e ax 5a2 : 2 /D In real-life problems, we commonly encounter Hill function type nonlinearities. 0; 1/. x/. x/ D 2b 2 < 0: As a corollary of the above, we have the following result. 1. Let a, b > 0, c 0 and m 2 N be constants.