Two‐dimensional polynomial type canonical relaxation oscillator model for p53 dynamics

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

“…The non-linearities in the models make the analysis complicated and sometimes impossible. Owing to its low dimensionality and simplicity in the non-linear interactions, the two-dimensional polynomial type oscillator model of Demirkran et al [10] is chosen in this study to analyse the exact ranges of the model parameters ensuring the three modes of the p53 network.…”

“…The reason for choosing these two variables is that ATM * , Wip1, and their interaction plays essential roles in the p53 network dynamics. As explained in [10], this two-dimensional model of the p53 network that takes ATM * and Wip1 as state variables comprise two constitutive mechanisms, namely Wip1 and ATM * negative feedback loop and the positive feedback of ATM * into itself causing the bistable behaviour. Since the concentration level of p53 protein is known to be directly proportional to the concentration level of ATM * , p53 could be assumed as a hidden variable by taking ATM * as a state variable…”

“…In [10], the parameter r is taken as either 0 or 1. The combinations (m, r) of parameters m and r are given as (m, r) ∈ {(1.25, 0), (1.25, 1), (0, 1)} for the low-level state, oscillation and high-level state, respectively, in [10]. As stated in [10], these specific parameter values are chosen only to show that the model is capable of explaining three different modes.…”

“…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].…”

“…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]. The second one is a reduced polynomial type gene regulatory network model for p53 as a form of relaxation oscillator exhibiting three different qualitative behaviours to DNA damages in the cell [10].…”