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August 2018, 15(4): 863-882. doi: 10.3934/mbe.2018039

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

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

* Corresponding author

Received  March 10, 2017 Accepted  December 16, 2017 Published  March 2018

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.

Citation: Qingwen Hu. A model of regulatory dynamics with threshold-type state-dependent delay. Mathematical Biosciences & Engineering, 2018, 15 (4) : 863-882. doi: 10.3934/mbe.2018039
References:
[1]

Z. BalanovQ. Hu and W. Krawcewicz, Global hopf bifurcation of differential equations with threshold type state-dependent delay, J. Differential Equations, 257 (2014), 2622-2670. doi: 10.1016/j.jde.2014.05.053.

[2]

S. Busenberg and J. M. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression, J. Math. Biol., 22 (1985), 313-333. doi: 10.1007/BF00276489.

[3]

Y. Chen and J. Wu, Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. Math. Anal. Appl., 259 (2001), 188-208. doi: 10.1006/jmaa.2000.7410.

[4]

K. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426. doi: 10.1090/S0002-9939-96-03437-5.

[5]

R. D. Driver, A neutral system with state-dependent delay, J. Differential Equations, 54 (1984), 73-86. doi: 10.1016/0022-0396(84)90143-8.

[6]

B. C. Goodwin, Oscillatory behavior in enzymatic control process, Adv. Enzyme Regul., 3 (1965), 425-428. doi: 10.1016/0065-2571(65)90067-1.

[7]

J. Griffith, Mathematics of cellular control processes, ⅰ, negative feedback to one gene, J. Theor. Biol., 20 (1968), 202-208. doi: 10.1016/0022-5193(68)90189-6.

[8]

——, Mathematics of cellular control processes, ⅱ, positive feedback to one gene, J. Theor. Biol, 20 (1968), 209-216. doi: 10.1016/0022-5193(68)90190-2.

[9]

H. HirataS. YoshiuraT. OhtsukaY. BesshoT. HaradaK. Yoshikawa and R. Kageyama, Oscillatory expression of the bhlh factor hes1 regulated by a negative feedback loop, Science, 298 (2002), 840-843. doi: 10.1126/science.1074560.

[10]

Q. HuW. Krawcewicz and J. Turi, Global stability lobes of a state-dependent model of turning processes, SIAM J. Appl. Math., 72 (2012), 1383-1405. doi: 10.1137/110859051.

[11]

T. InspergerG. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dynamics, 47 (2007), 275-283. doi: 10.1007/s11071-006-9068-2.

[12]

M. JensenK. Sneppen and G. Tiana, Sustained oscillations and time delays in gene expression of protein hes1, FEBS Lett., (2003), 176-177. doi: 10.1016/S0014-5793(03)00279-5.

[13]

J. M. Mahaffy and C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 20 (1984), 39-57. doi: 10.1007/BF00275860.

[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. doi: 10.1007/978-3-540-45204-1_5.

[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. doi: 10.1216/rmjm/1181072468.

[17]

H. L. Smith, Oscillations and multiple steady states in a cyclic gene model with repression, J. Math. Biol., 25 (1987), 169-190. doi: 10.1007/BF00276388.

[18]

P. SmolenD. A. Baxter and J. H. Byrne, Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems, Am. J. Physiol., 277 (1999), C777-C790. doi: 10.1152/ajpcell.1999.277.4.C777.

[19]

P. SmolenD. A. Baxter and J. H. Byrne, Modeling transcriptional control in gene networks-methods, recent results, and future, Bull. Math. Biol., 62 (2000), 247-292. doi: 10.1006/bulm.1999.0155.

[20]

M. SturrockA. J. TerryD. P. XirodimasA. M. Thompson and M. A. J. Chaplain, Spatio-temporal modeling of the Hes1 and p53 Mdm2 intracellular signaling pathways, J. Theor. Biol., 273 (2011), 15-31. doi: 10.1016/j.jtbi.2010.12.016.

[21]

J. Tyson and 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. doi: 10.1023/A:1026161513363.

[23]

——, Smoothness properties of semiflows for differential equations with state-dependent delays, J. Math. Sci., 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.

[24]

A. Wan and X. Zou, Hopf bifurcation analysis for a model of genetic regulatory system with delay, J. Math. Anal. Appl., 356 (2009), 464-476. doi: 10.1016/j.jmaa.2009.03.037.

[25]

B. WuC. EliscovichY. J. Yoon and R. H. Singer, Translation dynamics of single mRNAs in live cells and neurons, Science, 352 (2016), 1430-1435. doi: 10.1126/science.aaf1084.

show all references

References:
[1]

Z. BalanovQ. Hu and W. Krawcewicz, Global hopf bifurcation of differential equations with threshold type state-dependent delay, J. Differential Equations, 257 (2014), 2622-2670. doi: 10.1016/j.jde.2014.05.053.

[2]

S. Busenberg and J. M. Mahaffy, Interaction of spatial diffusion and delays in models of genetic control by repression, J. Math. Biol., 22 (1985), 313-333. doi: 10.1007/BF00276489.

[3]

Y. Chen and J. Wu, Slowly oscillating periodic solutions for a delayed frustrated network of two neurons, J. Math. Anal. Appl., 259 (2001), 188-208. doi: 10.1006/jmaa.2000.7410.

[4]

K. Cooke and W. Huang, On the problem of linearization for state-dependent delay differential equations, Proc. Amer. Math. Soc., 124 (1996), 1417-1426. doi: 10.1090/S0002-9939-96-03437-5.

[5]

R. D. Driver, A neutral system with state-dependent delay, J. Differential Equations, 54 (1984), 73-86. doi: 10.1016/0022-0396(84)90143-8.

[6]

B. C. Goodwin, Oscillatory behavior in enzymatic control process, Adv. Enzyme Regul., 3 (1965), 425-428. doi: 10.1016/0065-2571(65)90067-1.

[7]

J. Griffith, Mathematics of cellular control processes, ⅰ, negative feedback to one gene, J. Theor. Biol., 20 (1968), 202-208. doi: 10.1016/0022-5193(68)90189-6.

[8]

——, Mathematics of cellular control processes, ⅱ, positive feedback to one gene, J. Theor. Biol, 20 (1968), 209-216. doi: 10.1016/0022-5193(68)90190-2.

[9]

H. HirataS. YoshiuraT. OhtsukaY. BesshoT. HaradaK. Yoshikawa and R. Kageyama, Oscillatory expression of the bhlh factor hes1 regulated by a negative feedback loop, Science, 298 (2002), 840-843. doi: 10.1126/science.1074560.

[10]

Q. HuW. Krawcewicz and J. Turi, Global stability lobes of a state-dependent model of turning processes, SIAM J. Appl. Math., 72 (2012), 1383-1405. doi: 10.1137/110859051.

[11]

T. InspergerG. Stépán and J. Turi, State-dependent delay in regenerative turning processes, Nonlinear Dynamics, 47 (2007), 275-283. doi: 10.1007/s11071-006-9068-2.

[12]

M. JensenK. Sneppen and G. Tiana, Sustained oscillations and time delays in gene expression of protein hes1, FEBS Lett., (2003), 176-177. doi: 10.1016/S0014-5793(03)00279-5.

[13]

J. M. Mahaffy and C. V. Pao, Models of genetic control by repression with time delays and spatial effects, J. Math. Biol., 20 (1984), 39-57. doi: 10.1007/BF00275860.

[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. doi: 10.1007/978-3-540-45204-1_5.

[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. doi: 10.1216/rmjm/1181072468.

[17]

H. L. Smith, Oscillations and multiple steady states in a cyclic gene model with repression, J. Math. Biol., 25 (1987), 169-190. doi: 10.1007/BF00276388.

[18]

P. SmolenD. A. Baxter and J. H. Byrne, Effects of macromolecular transport and stochastic fluctuations on the dynamics of genetic regulatory systems, Am. J. Physiol., 277 (1999), C777-C790. doi: 10.1152/ajpcell.1999.277.4.C777.

[19]

P. SmolenD. A. Baxter and J. H. Byrne, Modeling transcriptional control in gene networks-methods, recent results, and future, Bull. Math. Biol., 62 (2000), 247-292. doi: 10.1006/bulm.1999.0155.

[20]

M. SturrockA. J. TerryD. P. XirodimasA. M. Thompson and M. A. J. Chaplain, Spatio-temporal modeling of the Hes1 and p53 Mdm2 intracellular signaling pathways, J. Theor. Biol., 273 (2011), 15-31. doi: 10.1016/j.jtbi.2010.12.016.

[21]

J. Tyson and 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. doi: 10.1023/A:1026161513363.

[23]

——, Smoothness properties of semiflows for differential equations with state-dependent delays, J. Math. Sci., 124 (2004), 5193-5207. doi: 10.1023/B:JOTH.0000047253.23098.12.

[24]

A. Wan and X. Zou, Hopf bifurcation analysis for a model of genetic regulatory system with delay, J. Math. Anal. Appl., 356 (2009), 464-476. doi: 10.1016/j.jmaa.2009.03.037.

[25]

B. WuC. EliscovichY. J. Yoon and R. H. Singer, Translation dynamics of single mRNAs in live cells and neurons, Science, 352 (2016), 1430-1435. doi: 10.1126/science.aaf1084.

Figure 1.  Hes1 regulatory system in a cell: a. inhibition of mRNA transcription in nucleus from protein diffused from cytoplasm, b. translation of mRNA for protein synthesis in cytoplasm
Figure 2.  (a) Equilibrium $(r^*, \, \xi^*) = (11.97050076, \, 2992.625189)$ is stable with $\epsilon = \epsilon_0-\delta$, $c = 0.01 < c_0$ with $\delta = 0.1$; (b) periodic solution appears at $\epsilon = \epsilon_0+\delta$, $c = 0.01 < c_0$
Figure 3.  (a) Equilibrium $(r^*, \, \xi^*) = (11.97050076, \, 2992.625189)$ is asymptotically stable with $\epsilon = \epsilon_0-\delta < \epsilon_0$, $c = c_0+0.001$ (see the solid curve); when initial value is far enough from the equilibrium $(r^*, \, \xi^*) = (11.97050076, \, 2992.625189)$, solution may converge to another equilibrium (see the dashed curve). Subcritical bifurcation occurs at $\epsilon_0$ with $0 < c < c_0$. (b) If $\epsilon>\epsilon_0$ and $c = c_0+0.001$, the equilibrium $(r^*, \, \xi^*)$ is unstable
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