# American Institute of Mathematical Sciences

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  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 & 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.  Google Scholar [2] G. Balázsi, A. van Oudenaarden and J. J. Collins, Cellular decision making and biological noise: From microbes to mammals, Cell, 144 (2011), 910-925.   Google Scholar [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.  Google Scholar [4] D. Bratsun, D. Volfson, L. 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.  Google Scholar [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.  Google Scholar [6] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar [7] D. Denault, J. 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.  Google Scholar [8] T. C. Gard, Introduction to Stochastic Differential Equation, Marcel Dekker Inc, New York, 1988. doi: 10.1002/zamm.19890690808.  Google Scholar [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.  Google Scholar [10] D. T. Gillespie, The chemical Langevin equation, The Journal of Chemical Physics, 113 (2000), 297-306.  doi: 10.1063/1.481811.  Google Scholar [11] R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen, 11 (1966), 240-259.   Google Scholar [12] T. Kurtz, Semigroups of conditioned shifts and approximation of Markov processes, Ann. Probab., 3 (1975), 618-642.  doi: 10.1214/aop/1176996305.  Google Scholar [13] T. Kurtz, Approximation of Population Processes, , CBMS-NSF Regional Conference Series in Applied Mathematics, 36, SIAM, Philadelphia, Pa., 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, Chichester, 2008. doi: 10.1016/B978-1-904275-34-3.50013-X.  Google Scholar [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. Google Scholar [18] J. Miekisz, J. 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.  Google Scholar [19] K. M. Ramachandran, A singularly perturbed stochastic delay system with small parameter, Stochastic Anal. Appl., 11 (1993), 209-230.  doi: 10.1080/07362999308809312.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] M. Turcotte, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar

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.  Google Scholar [2] G. Balázsi, A. van Oudenaarden and J. J. Collins, Cellular decision making and biological noise: From microbes to mammals, Cell, 144 (2011), 910-925.   Google Scholar [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.  Google Scholar [4] D. Bratsun, D. Volfson, L. 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.  Google Scholar [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.  Google Scholar [6] J. C. Cox, J. E. Ingersoll and S. A. Ross, A theory of the term structure of interest rates, Econometrica, 53 (1985), 385-407.  doi: 10.2307/1911242.  Google Scholar [7] D. Denault, J. 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.  Google Scholar [8] T. C. Gard, Introduction to Stochastic Differential Equation, Marcel Dekker Inc, New York, 1988. doi: 10.1002/zamm.19890690808.  Google Scholar [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.  Google Scholar [10] D. T. Gillespie, The chemical Langevin equation, The Journal of Chemical Physics, 113 (2000), 297-306.  doi: 10.1063/1.481811.  Google Scholar [11] R. Z. Khasminskii, On stochastic processes defined by differential equations with a small parameter, Teor. Verojatnost. i Primenen, 11 (1966), 240-259.   Google Scholar [12] T. Kurtz, Semigroups of conditioned shifts and approximation of Markov processes, Ann. Probab., 3 (1975), 618-642.  doi: 10.1214/aop/1176996305.  Google Scholar [13] T. Kurtz, Approximation of Population Processes, , CBMS-NSF Regional Conference Series in Applied Mathematics, 36, SIAM, Philadelphia, Pa., 1981.  Google Scholar [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.  Google Scholar [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.  Google Scholar [16] X. Mao, Stochastic Differential Equations and Applications, 2$^nd$ edition, Horwood, Chichester, 2008. doi: 10.1016/B978-1-904275-34-3.50013-X.  Google Scholar [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. Google Scholar [18] J. Miekisz, J. 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.  Google Scholar [19] K. M. Ramachandran, A singularly perturbed stochastic delay system with small parameter, Stochastic Anal. Appl., 11 (1993), 209-230.  doi: 10.1080/07362999308809312.  Google Scholar [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.  Google Scholar [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.  Google Scholar [22] M. Turcotte, J. 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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar [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.  Google Scholar
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