Non-linear dynamics of expression of BMAL1: a mathematical study
Nonlinear dynamics of complex biological systems like expression of protein remained a fascinating field of study. Especially investigation of dynamics of functional proteins like transcription factors offers a great promise to understand many perplexing phenom- ena of cell biology and physiology. This communication aims to capture complex dynamics of a transcription factor called ”Brain and muscle Arnt-like protein-1” (BMAL1)
in the nucleus of a cell. BMAL1 is not only an important component of Circadian cycle of cells but has lately been identified as a key marker in many diseased conditions like cardio-vascular disease, diabetes etc. The system is modelled as a set of non-linear differential equation involving concentrations of the protein and the corresponding m-RNA. In the model Ivlev function has been proposed to describe the translation and allied processing plus transport steps of the protein where the exponential parameter is used as a master control parameter. We have explored the existence and stability of the m-RNA-protein dynamics with the existence of interior equilibria at deferent levels of the density of the two macromolecules. We investigated the stability and bifurcation behavior of the model in presence and absence of time delay. Further the model system
was studied with stochastic perturbation by introducing random variations of the sensitive parameters which is closer to actual cellular system. Lastly, we have estimated the mean square stability of the set of differential equations around the interior equilibrium. The time delay and stochastic perturbation showed crucial impact on the behaviour of expression machinery of the transcription factor.