Abstract
MiRNA regulation is involved in many important biological processes such as cell proliferation, apoptosis and metabolism. Computational predictions of miRNA targets suggest that up 30 % of human proteins coding genes may be regulated by miRNAs. This makes miRNAs one of the most abundant classes of regulatory genes in humans. In the present paper we develop a time delay model of a feedback system regulated via miRNA. The model resulted in three DDEs with three discrete time delays. Since this system is a classical case study, covering several essential features of miRNA and genetic regulatory mechanisms, general conclusions about design principles and role of time delays in the stability of gene circuits can be suggested. The basic view that time delays are a key factor in the dynamic behaviour of the system was confirmed by the analytical calculations and numerical simulations.
Highlights
Cells are the structural and functional units of all living organisms as protein synthesis is an essential function of a cell
We investigate the bifurcation structure- the Andronov-Hopf bifurcation- for system (2), using time delays τ1, τ2 or τ3 as bifurcation parameters
We proposed the analytical tools and used them for a qualitative analysis of the system, obtaining predictions about dynamics of the system, i.e. the stability and existence of periodic solutions via Andronov-Hopf bifurcation in time delay model (2)
Summary
Cells are the structural and functional units of all living organisms as protein synthesis is an essential function of a cell. In the recent years ordinary differential equations (ODEs) with retarded argument(s) have been widely used in modelling and analyzing regulatory systems involve many genes, factors and complex interactions (Gene Regulatory Networks) In these models, the gene and mRNA concentrations are timedependent variables, interactions are represented by functional and differential between variables, and retarded arguments are usually the time of transcription and translation, or time delay feedback loop. An interesting and important problem appears due to the nonlinearity of fi, since an analytical solution of the system is usually not possible In this sense, the recent molecular biological discoveries (like the miRNAs and their complex regulatory effects) clearly show the need to develop mathematical models that take into consideration the post-transcriptional regulation. The system can be represented by the following mathematical model in time delayed differential equations dy dt k2
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