Oscillatory Behaviors in Genetic Regulatory Networks Mediated by MicroRNA With Time Delays and Reaction-Diffusion Terms

Abstract

In this paper, we investigate the oscillatory expression in Escherichia coli mediated by microRNA with time delays and reaction-diffusion terms. First of all, the integrated effects of delays and diffusions are first introduced into the genetic regulatory networks involving microRNAs, and a general model of genetic regulatory networks is then formulated. Second, two functional issues on gene regulatory networks, i.e. stability and oscillation of such model, are addressed in detail, and an explicit algorithm determining the properties of periodic oscillation is also presented. We demonstrate that the oscillatory expression of Escherichia coli is not only crucially dependent on the transcriptional and translational delays, but also heavily influenced by the diffusion coefficients. The conclusion is practically verified by a lot of biological experiments and observations. We also find that if the diffusion coefficients of miRNA, mRNA, and protein are suitably small, it can predict that inhomogeneous periodic oscillations can occur unless there only exhibits spatially homogeneous periodic oscillations. The obtained results indicate that the effects of transcriptional and translational delays are essential factors for designing or controlling genetic regulatory networks, in the meantime the functions of reaction-diffusion must beconsidered. Finally, numerical examples are presented to illustrate and visualize theoretical results.

Oscillation induced by Hopf bifurcation in the p53–Mdm2 feedback module

“…On the one hand, from a dynamical perspective, the time delay can be used as a bifurcation parameter to study the induced Hopf bifurcation. On the other hand, the time delay is inherent and important for biological systems, especially in gene regulatory network [22, 27, 32]. In this subsection, we will investigate how time delay affects the oscillations in Model 1 through numerical simulations.…”

Bifurcation and oscillatory dynamics of delayed CDK1‐APC feedback loop

“…Due to the W 20 (ξ) and W 11 (ξ) in ν 21 , we shall calculate them further. According to (13) and (26)…”

“…C * ) − 2A * e −iτ 0 ω 0 × f ′′ 2 (C * ))v 1 W 11 1 ( − 1) + ((1 − A * ) e −iτ 0 ω 0 × f ′′′ 2 (C * ) − 2 f ′′ 2 (C * ) − e −2iτ 0 ω 0 f ′′ 2 (C * ))v 1 2 v ¯1 −( f ′ 2 (C * ) − e iτ 0 ω 0 f ′′ 2 (C * ) + A * e iτ 0 ω 0 f ′′ 2 (C * )) × v 1 W 20 1 ( − 1) − 2 e −iτ 0 ω 0 f ′ 2 (C * )v 1 W 11 2 (0) −e iτ 0 ω 0 f 2′ (C * )v ¯1W 20 2 (0)}α 2 ,n 32 = − 2(2 f ′ 1 (C * )α 3 − f ′′ 1 (C * )α 3 + C * f ′′ 1 (C * )α 3 + f ′ 3 (A * )v 1 β 1 )W 11 1 (0) − (2 f ′ 1 (C * )α 3 − f ′′ 1 (C * )α 3 + C * f ′′ 1 (C * )α 3 + f ′ 3 (A * )v ¯1β 1 ) × W 20 1 (0) − ( f ′′ 3 (A * )v 1 + 2 f ′′ 3 (A * )v ¯1 +C * f ′′′ 3 (A * )v 1 v ¯1)v 1 β 1 − (3 f ′′ 1 (C * ) − f ′′′ 1 (C * ) + C * f ′′′ 1 (C * ))α f ′ 3 (A * ) + C * f ′′ 3 (A * )v 1 )β 1 W 11 2 (0) −( f ′ 3 (A * ) + C * f ′′ 3 (A * )v ¯1)β 1 W 20 2 (0) .Due to the W 20 (ξ) and W 11 (ξ) in ν 21 , we shall calculate them further. According to(13) and(26), one can getW ˙= A(0)U t + R(0)U t − (iω 0 τ 0 z(t) +ρ ¯ * (0) f 0 (z, z ¯))ρ(ξ) − [ − (iω 0 τ 0 z ¯(t) +ρ * (0) f ¯0(z, z ¯))ρ ¯(ξ)] = AW − 2ℜ ρ ¯ * (0) f 0 ρ(ξ) , ξ ∈ [ − 1, 0)AW − 2ℜ ρ ¯ * (0) f 0 ρ(ξ) + f 0 , ξ = 0…”

Stability and bifurcation analysis of a gene expression model with small RNAs and mixed delays

“…In system theory, bifurcation theory can be used to discuss the generation and disappearance of bifurcation phenomena in nonlinear systems [3][4][5][6][7][8]. For gene networks, the bifurcation theory is a useful tool for studying the dynamic performance in regulation process [9][10][11][12][13][14][15].…”

“…On the basis of a comprehensive interpretation of cell metabolism, it has played a great role in exploring the mechanism of life activities, the cause and treatment of the disease. A mathematical model of gene expression mediated by sRNAs is put forward in [13][14][15]: ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ẋ (t) = -cx(t)dy(t)x(t) + g(z(tτ 1 )), y(t) = edy(t)x(t)fy(t), z(t) = -bz(t) + ax(tτ 2 ), (1) where x(t), y(t), and z(t) represent the densities of mRNA, sRNA, and protein, respectively. a represents the synthesis rate of protein.…”