August  2019, 24(8): 4417-4443. doi: 10.3934/dcdsb.2019125

Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations

1. 

School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan, Hubei, 430074, China

2. 

Department of Mathematics, Wayne State University, Detroit, MI 48202 USA

* Corresponding author: Fuke Wu

Received  July 2018 Revised  November 2018 Published  August 2019 Early access  June 2019

Fund Project: Yun Li is supported by in part by the National Natural Science Foundations of China (Grant Nos. 61873320 and 11761130072). Fuke Wu is supported by the National Natural Science Foundations of China (Grant Nos. 61873320 and 11761130072) and the Royal Society-Newton Advanced Fellowship. George Yin is supported by the National Science Foundation under grant DMS-1710827.

Gene regulatory networks, which are complex high-dimensional stochastic dynamical systems, are often subject to evident intrinsic fluctuations. It is deemed reasonable to model the systems by the chemical Langevin equations. Since the mRNA dynamics are faster than the protein dynamics, we have a two-time scales system. In general, the process of protein degradation involves time delays. In this paper, we take the system memory into consideration in which we consider a model with a complete memory represented by an integral delay from $ 0 $ to $ t $. Based on the averaging principle and perturbed test function method, this work examines the weak convergence of the slow-varying process. By treating the fast-varying process as a random noise, under appropriate conditions, it is shown that the slow-varying process converges weakly to the solution of a stochastic differential delay equation whose coefficients are the average of those of the original slow-varying process with respect to the invariant measure of the fast-varying process.

Citation: Yun Li, Fuke Wu, George Yin. Asymptotic behavior of gene expression with complete memory and two-time scales based on the chemical Langevin equations. Discrete and Continuous Dynamical Systems - B, 2019, 24 (8) : 4417-4443. doi: 10.3934/dcdsb.2019125
References:
[1]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, 4$^nd$ edition, New York: Garland, 2002. doi: 10.1097/00024382-200209000-00015.

[2]

G. BalázsiA. van Oudenaarden and J. J. Collins, Cellular decision making and biological noise: From microbes to mammals, Cell, 144 (2011), 910-925. 

[3]

M. Bodnar, General model of a cascade of reactions with times: Global stability analysis, J. Differential Eqs., 259 (2015), 777-795.  doi: 10.1016/j.jde.2015.02.024.

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D. BratsunD. VolfsonL. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. National Academy of Sci. USA, 102 (2005), 14593-14598.  doi: 10.1073/pnas.0503858102.

[5]

C. Clayton and M. Shapira, Post-transcriptional regulation of gene expression in trypanosomes and leishmanias, Molecular and Biochemical Parasitology, 156 (2007), 93-101.  doi: 10.1016/j.molbiopara.2007.07.007.

[6]

J. C. CoxJ. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

[7]

D. DenaultJ. Loros and J. C. Dunlap, WC-2 mediates WC-1-FRQ interaction within the PAS protein-linked circadian feedback loop of Neurospora, EMBO J., 20 (2001), 109-117.  doi: 10.1093/emboj/20.1.109.

[8]

T. C. Gard, Introduction to Stochastic Differential Equation, Marcel Dekker Inc, New York, 1988. doi: 10.1002/zamm.19890690808.

[9]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, Journal of computational physics, 22 (1976), 403-434.  doi: 10.1016/0021-9991(76)90041-3.

[10]

D. T. Gillespie, The chemical Langevin equation, The Journal of Chemical Physics, 113 (2000), 297-306.  doi: 10.1063/1.481811.

[11]

R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen, 11 (1966), 240-259. 

[12]

T. Kurtz, Semigroups of conditioned shifts and approximation of Markov processes, Ann. Probab., 3 (1975), 618-642.  doi: 10.1214/aop/1176996305.

[13]

T. Kurtz, Approximation of Population Processes, , CBMS-NSF Regional Conference Series in Applied Mathematics, 36, SIAM, Philadelphia, Pa., 1981.

[14] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.  doi: 10.1137/1028023.
[15]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0.

[16]

X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, Chichester, 2008. doi: 10.1016/B978-1-904275-34-3.50013-X.

[17]

B. Mélykúti, K. Burrage and K. C. Zygalakis, Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation, The Journal of chemical physics, 132 (2010), 164109.

[18]

J. MiekiszJ. Poleszczuk and M. Bodnar, Stochastic models of gene expression with delayed degradation, Bull. Math. Bio., 73 (2011), 2231-2247.  doi: 10.1007/s11538-010-9622-4.

[19]

K. M. Ramachandran, A singularly perturbed stochastic delay system with small parameter, Stochastic Anal. Appl., 11 (1993), 209-230.  doi: 10.1080/07362999308809312.

[20]

K. M. Ramachandran, Stability of stochastic delay differential equation with a small parameter, Stochastic Anal. Appl., 26 (2008), 710-723.  doi: 10.1080/07362990802128271.

[21]

M. Thattai and A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. USA, 98 (2001), 8614-8619.  doi: 10.1073/pnas.151588598.

[22]

M. TurcotteJ. Garcia-Ojalvo and G. M. Süel, A genetic timer through noise-induced stabilization of an unstable state, Proc. Natl. Acad. Sci. USA, 105 (2008), 15732-15737.  doi: 10.1073/pnas.0806349105.

[23]

F. Wu, T. Tian, J. B. Rawlings and G. Yin, Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations, J. Chemical Phy., 144 (2016), 174112. doi: 10.1063/1.4948407.

[24]

G. Yin and K. M. Ramachandran, A differential delay equation with wideband noise perturbations, Stochastic Process. Appl., 35 (1990), 231-249.  doi: 10.1016/0304-4149(90)90004-C.

[25]

G. Yin and H. Q. Zhang, Singularly perturbed Markov chains: Limit results and applications, Ann. Appl. Probab., 17 (2007), 207-229.  doi: 10.1214/105051606000000682.

[26]

G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Two-time-scale Approach, 2$^nd$ edition, Springer, Now York, 2013. doi: 10.1007/978-1-4614-4346-9.

show all references

References:
[1]

B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts and P. Walter, Molecular Biology of the Cell, 4$^nd$ edition, New York: Garland, 2002. doi: 10.1097/00024382-200209000-00015.

[2]

G. BalázsiA. van Oudenaarden and J. J. Collins, Cellular decision making and biological noise: From microbes to mammals, Cell, 144 (2011), 910-925. 

[3]

M. Bodnar, General model of a cascade of reactions with times: Global stability analysis, J. Differential Eqs., 259 (2015), 777-795.  doi: 10.1016/j.jde.2015.02.024.

[4]

D. BratsunD. VolfsonL. S. Tsimring and J. Hasty, Delay-induced stochastic oscillations in gene regulation, Proc. National Academy of Sci. USA, 102 (2005), 14593-14598.  doi: 10.1073/pnas.0503858102.

[5]

C. Clayton and M. Shapira, Post-transcriptional regulation of gene expression in trypanosomes and leishmanias, Molecular and Biochemical Parasitology, 156 (2007), 93-101.  doi: 10.1016/j.molbiopara.2007.07.007.

[6]

J. C. CoxJ. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.

[7]

D. DenaultJ. Loros and J. C. Dunlap, WC-2 mediates WC-1-FRQ interaction within the PAS protein-linked circadian feedback loop of Neurospora, EMBO J., 20 (2001), 109-117.  doi: 10.1093/emboj/20.1.109.

[8]

T. C. Gard, Introduction to Stochastic Differential Equation, Marcel Dekker Inc, New York, 1988. doi: 10.1002/zamm.19890690808.

[9]

D. T. Gillespie, A general method for numerically simulating the stochastic time evolution of coupled chemical reactions, Journal of computational physics, 22 (1976), 403-434.  doi: 10.1016/0021-9991(76)90041-3.

[10]

D. T. Gillespie, The chemical Langevin equation, The Journal of Chemical Physics, 113 (2000), 297-306.  doi: 10.1063/1.481811.

[11]

R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen, 11 (1966), 240-259. 

[12]

T. Kurtz, Semigroups of conditioned shifts and approximation of Markov processes, Ann. Probab., 3 (1975), 618-642.  doi: 10.1214/aop/1176996305.

[13]

T. Kurtz, Approximation of Population Processes, , CBMS-NSF Regional Conference Series in Applied Mathematics, 36, SIAM, Philadelphia, Pa., 1981.

[14] H. J. Kushner, Approximation and Weak Convergence Methods for Random Processes, with Applications to Stochastic Systems Theory, MIT Press, Cambridge, MA, 1984.  doi: 10.1137/1028023.
[15]

H. J. Kushner, Weak Convergence Methods and Singularly Perturbed Stochastic Control and Filtering Problems, Birkhäuser, Boston, MA, 1990. doi: 10.1007/978-1-4612-4482-0.

[16]

X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, Chichester, 2008. doi: 10.1016/B978-1-904275-34-3.50013-X.

[17]

B. Mélykúti, K. Burrage and K. C. Zygalakis, Fast stochastic simulation of biochemical reaction systems by alternative formulations of the chemical Langevin equation, The Journal of chemical physics, 132 (2010), 164109.

[18]

J. MiekiszJ. Poleszczuk and M. Bodnar, Stochastic models of gene expression with delayed degradation, Bull. Math. Bio., 73 (2011), 2231-2247.  doi: 10.1007/s11538-010-9622-4.

[19]

K. M. Ramachandran, A singularly perturbed stochastic delay system with small parameter, Stochastic Anal. Appl., 11 (1993), 209-230.  doi: 10.1080/07362999308809312.

[20]

K. M. Ramachandran, Stability of stochastic delay differential equation with a small parameter, Stochastic Anal. Appl., 26 (2008), 710-723.  doi: 10.1080/07362990802128271.

[21]

M. Thattai and A. van Oudenaarden, Intrinsic noise in gene regulatory networks, Proc. Natl. Acad. Sci. USA, 98 (2001), 8614-8619.  doi: 10.1073/pnas.151588598.

[22]

M. TurcotteJ. Garcia-Ojalvo and G. M. Süel, A genetic timer through noise-induced stabilization of an unstable state, Proc. Natl. Acad. Sci. USA, 105 (2008), 15732-15737.  doi: 10.1073/pnas.0806349105.

[23]

F. Wu, T. Tian, J. B. Rawlings and G. Yin, Approximate method for stochastic chemical kinetics with two-time scales by chemical Langevin equations, J. Chemical Phy., 144 (2016), 174112. doi: 10.1063/1.4948407.

[24]

G. Yin and K. M. Ramachandran, A differential delay equation with wideband noise perturbations, Stochastic Process. Appl., 35 (1990), 231-249.  doi: 10.1016/0304-4149(90)90004-C.

[25]

G. Yin and H. Q. Zhang, Singularly perturbed Markov chains: Limit results and applications, Ann. Appl. Probab., 17 (2007), 207-229.  doi: 10.1214/105051606000000682.

[26]

G. Yin and Q. Zhang, Continuous-time Markov Chains and Applications: A Two-time-scale Approach, 2$^nd$ edition, Springer, Now York, 2013. doi: 10.1007/978-1-4614-4346-9.

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