A model of regulatory dynamics with threshold-type state-dependent delay

Qingwen Hu; Qingwen Hu

doi:10.3934/mbe.2018039

Mathematical Biosciences and Engineering

2018, Volume 15, Issue 4: 863-882. doi: 10.3934/mbe.2018039

Previous Article Next Article

A model of regulatory dynamics with threshold-type state-dependent delay

Qingwen Hu ^,

Department of Mathematical Sciences, The University of Texas at Dallas, 800 W. Campbell Road, FO. 35, Richardson, TX, 75080, USA

Received: 10 March 2017 Accepted: 16 December 2017 Published: 01 August 2018
MSC : Primary: 37G05; Secondary: 92D25

We model intracellular regulatory dynamics with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time. A general model which is an extension of the classical differential equation models with constant or zero time delays is developed to study the stability of steady state, the occurrence and stability of periodic oscillations in regulatory dynamics. Using the method of multiple time scales, we compute the normal form of the general model and show that the state-dependent diffusion time may lead to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of Hes1 regulatory dynamics are given to illustrate the general results.
- Diffusion time,
- state-dependent delay,
- Hopf bifurcation,
- multiple time scales,
- normal form
Citation: Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay[J]. Mathematical Biosciences and Engineering, 2018, 15(4): 863-882. doi: 10.3934/mbe.2018039

Related Papers:

Abstract

We model intracellular regulatory dynamics with threshold-type state-dependent delay and investigate the effect of the state-dependent diffusion time. A general model which is an extension of the classical differential equation models with constant or zero time delays is developed to study the stability of steady state, the occurrence and stability of periodic oscillations in regulatory dynamics. Using the method of multiple time scales, we compute the normal form of the general model and show that the state-dependent diffusion time may lead to both supercritical and subcritical Hopf bifurcations. Numerical simulations of the prototype model of Hes1 regulatory dynamics are given to illustrate the general results.

References

[1]	[ Z. Balanov,Q. Hu,W. Krawcewicz, Global hopf bifurcation of differential equations with threshold type state-dependent delay, J. Differential Equations, 257 (2014): 2622-2670.
[2]	[ S. Busenberg,J. M. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression, J. Math. Biol., 22 (1985): 313-333.
[3]	[ Y. Chen,J. Wu, Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. Math. Anal. Appl., 259 (2001): 188-208.
[4]	[ K. Cooke,W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996): 1417-1426.
[5]	[ R. D. Driver, A neutral system with state-dependent delay, J. Differential Equations, 54 (1984): 73-86.
[6]	[ B. C. Goodwin, Oscillatory behavior in enzymatic control process, Adv. Enzyme Regul., 3 (1965): 425-428.
[7]	[ J. Griffith, Mathematics of cellular control processes, ⅰ, negative feedback to one gene, J. Theor. Biol., 20 (1968): 202-208.
[8]	[ -, Mathematics of cellular control processes, ⅱ, positive feedback to one gene, J. Theor. Biol, 20 (1968): 209-216.
[9]	[ H. Hirata,S. Yoshiura,T. Ohtsuka,Y. Bessho,T. Harada,K. Yoshikawa,R. Kageyama, Oscillatory expression of the bhlh factor hes1 regulated by a negative feedback loop, Science, 298 (2002): 840-843.
[10]	[ Q. Hu,W. Krawcewicz,J. Turi, Global stability lobes of a state-dependent model of turning processes, SIAM J. Appl. Math., 72 (2012): 1383-1405.
[11]	[ T. Insperger,G. Stépán,J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dynamics, 47 (2007): 275-283.
[12]	[ M. Jensen,K. Sneppen,G. Tiana, Sustained oscillations and time delays in gene expression of protein hes1, FEBS Lett., null (2003): 176-177.
[13]	[ J. M. Mahaffy,C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 20 (1984): 39-57.
[14]	[ A. H. Nayfeh, Introduction to Perturbation Techniques, John Wiley & Sons, 1981.
[15]	[ R. Nussbaum, Limiting profiles for solutions of differential-delay equations, Dynamical Systems V, Lecture Notes in Mathematics, 1822 (2003): 299-342.
[16]	[ H. L. Smith, Existence and uniqueness of global solutions for a size-structured model of an insect population with variable instar duration, Rocky Mountain J. Math., 24 (1994): 311-334.
[17]	[ H. L. Smith, Oscillations and multiple steady states in a cyclic gene model with repression, J. Math. Biol., 25 (1987): 169-190.
[18]	[ P. Smolen,D. A. Baxter,J. H. Byrne, Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems, Am. J. Physiol., 277 (1999): C777-C790.
[19]	[ P. Smolen,D. A. Baxter,J. H. Byrne, Modeling transcriptional control in gene networks-methods, recent results, and future, Bull. Math. Biol., 62 (2000): 247-292.
[20]	[ M. Sturrock,A. J. Terry,D. P. Xirodimas,A. M. Thompson,M. A. J. Chaplain, Spatio-temporal modeling of the Hes1 and p53 Mdm2 intracellular signaling pathways, J. Theor. Biol., 273 (2011): 15-31.
[21]	[ J. Tyson,H. G. Othmer, The dynamics of feedback control circuits in biochemical pathways, Prog. Theor. Biol., 5 (1978): 2-62.
[22]	[ H.-O. Walther, Stable periodic motion of a system using echo for position control, J. Dynam. Differential Equations, 1 (2003): 143-223.
[23]	[ -, Smoothness properties of semiflows for differential equations with state-dependent delays, J. Math. Sci., 124 (2004): 5193-5207.
[24]	[ A. Wan,X. Zou, Hopf bifurcation analysis for a model of genetic regulatory system with delay, J. Math. Anal. Appl., 356 (2009): 464-476.
[25]	[ B. Wu,C. Eliscovich,Y. J. Yoon,R. H. Singer, Translation dynamics of single mRNAs in live cells and neurons, Science, 352 (2016): 1430-1435.

Reader Comments

Your name:*

Email:*
© 2018 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)