Coexistence of Deterministic and Stochastic Bistability in a 1-D Birth-Death Process with Hill Type Nonlinear Birth Rates

Bifurcation analysis of bistable and oscillatory dynamics in biological networks using the root‐locus method

“…These values are not enough to see the whole picture about the ranges of the parameters ensuring three distinct behaviours of the p53 network. In the sequel of this paper, by applying a thorough analysis similar in toggle switch case, the exact parameter ranges for m and r providing three qualitative behaviours of the p53 gene regulatory model in (14) and (15) are explored. Choosing m and r as bifurcation parameters in this study relies on the fact that these two parameters are the determinant factors of the p53 network; r is the driving input and m adjusts the strength of the Wip1-ATM * negative feedback loop controlling the behaviour of the network.…”

“…For three different combinations of parameters m and r given in [10], the dynamical behaviours of the model in (14) and (15), which correspond to three distinct modes, are examined by drawing the nullclines as depicted in Figs. 14-16.…”

“…For the considered values of the parameters, the concentration levels of ATM * and Wip1 proteins are computed as in Figs. 14b, 15b, and 16b by solving equations in (14) and (15). The dynamical behaviours of the model change depending on the qualitative and quantitative properties of the equilibrium points, namely their locations and stabilities, which are indeed the intersections of the nullclines.…”

“…These dynamics are modelled and analysed experimentally and computationally to get insights into the functioning of complex biological mechanisms in gene regulatory networks and signalling pathways [3,[5][6][7][8]. The mathematical models of biological systems, which might be derived based on chemical interactions or directly from experimental data by employing signal processing/machine learning methods, are vital not only for understanding the underlying mechanism of complex dynamics but also for predicting the behaviour of these systems via computational studies [8][9][10][11][12][13][14][15].…”

“…A study on two basic gene regulatory network models both of which are given as ODEs with rational non‐linearities only is presented: one of them is Gardner's genetic toggle switch model [7] and the other one is the polynomial type relaxation oscillator model for p53 gene network [10]. The dynamical behaviours of these models are analysed by using the root‐locus based method of [12, 15]. The first model considered in the study is the most studied synthetic gene network, the so‐called genetic toggle switch, which is built by a pair of mutually inhibitory genes demonstrating bistability due to the positive feedback [7].…”