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

Yun Li; Fuke Wu; George Yin

doi:10.3934/dcdsb.2019125

Previous Article
Some monotone properties for solutions to a reaction-diffusion model
DCDS-B Home
This Issue
Next Article
The hipster effect: When anti-conformists all look the same

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

Yun Li ^1, , Fuke Wu ^1,, and George Yin ^2,

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.

Keywords: Weak convergence, two-time scales, time delay, gene expression, chemical Langevin equation.

Mathematics Subject Classification: Primary: 60H20, 65C30, 60H30, 34K50; Secondary: 62L20, 92C45, 93C70, 34E10.

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

[1]	Qiuying Li, Lifang Huang, Jianshe Yu. Modulation of first-passage time for bursty gene expression via random signals. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1261-1277. doi: 10.3934/mbe.2017065
[2]	Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete & Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108
[3]	Pavol Bokes. Maintaining gene expression levels by positive feedback in burst size in the presence of infinitesimal delay. Discrete & Continuous Dynamical Systems - B, 2019, 24 (10) : 5539-5552. doi: 10.3934/dcdsb.2019070
[4]	Anouar El Harrak, Amal Bergam, Tri Nguyen-Huu, Pierre Auger, Rachid Mchich. Application of aggregation of variables methods to a class of two-time reaction-diffusion-chemotaxis models of spatially structured populations with constant diffusion. Discrete & Continuous Dynamical Systems - S, 2021, 14 (7) : 2163-2181. doi: 10.3934/dcdss.2021055
[5]	Yunfei Peng, X. Xiang. A class of nonlinear impulsive differential equation and optimal controls on time scales. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1137-1155. doi: 10.3934/dcdsb.2011.16.1137
[6]	Małgorzata Wyrwas, Dorota Mozyrska, Ewa Girejko. Subdifferentials of convex functions on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 671-691. doi: 10.3934/dcds.2011.29.671
[7]	Christophe Cheverry, Thierry Paul. On some geometry of propagation in diffractive time scales. Discrete & Continuous Dynamical Systems, 2012, 32 (2) : 499-538. doi: 10.3934/dcds.2012.32.499
[8]	Sung Kyu Choi, Namjip Koo. Stability of linear dynamic equations on time scales. Conference Publications, 2009, 2009 (Special) : 161-170. doi: 10.3934/proc.2009.2009.161
[9]	Yunfei Peng, X. Xiang, W. Wei. Backward problems of nonlinear dynamical equations on time scales. Discrete & Continuous Dynamical Systems - S, 2011, 4 (6) : 1553-1564. doi: 10.3934/dcdss.2011.4.1553
[10]	Petr Hasil, Petr Zemánek. Critical second order operators on time scales. Conference Publications, 2011, 2011 (Special) : 653-659. doi: 10.3934/proc.2011.2011.653
[11]	Massimiliano Berti, Philippe Bolle. Fast Arnold diffusion in systems with three time scales. Discrete & Continuous Dynamical Systems, 2002, 8 (3) : 795-811. doi: 10.3934/dcds.2002.8.795
[12]	B. Kaymakcalan, R. Mert, A. Zafer. Asymptotic equivalence of dynamic systems on time scales. Conference Publications, 2007, 2007 (Special) : 558-567. doi: 10.3934/proc.2007.2007.558
[13]	Lin Zhao, Zhi-Cheng Wang, Liang Zhang. Threshold dynamics of a time periodic and two–group epidemic model with distributed delay. Mathematical Biosciences & Engineering, 2017, 14 (5&6) : 1535-1563. doi: 10.3934/mbe.2017080
[14]	Marek Bodnar. Distributed delays in Hes1 gene expression model. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2125-2147. doi: 10.3934/dcdsb.2019087
[15]	Xin-Guang Yang, Jing Zhang, Shu Wang. Stability and dynamics of a weak viscoelastic system with memory and nonlinear time-varying delay. Discrete & Continuous Dynamical Systems, 2020, 40 (3) : 1493-1515. doi: 10.3934/dcds.2020084
[16]	Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693
[17]	Gongwei Liu, Baowei Feng, Xinguang Yang. Longtime dynamics for a type of suspension bridge equation with past history and time delay. Communications on Pure & Applied Analysis, 2020, 19 (10) : 4995-5013. doi: 10.3934/cpaa.2020224
[18]	Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577
[19]	Zbigniew Bartosiewicz, Ülle Kotta, Maris Tőnso, Małgorzata Wyrwas. Accessibility conditions of MIMO nonlinear control systems on homogeneous time scales. Mathematical Control & Related Fields, 2016, 6 (2) : 217-250. doi: 10.3934/mcrf.2016002
[20]	Jan L. Cieśliński. Some implications of a new approach to exponential functions on time scales. Conference Publications, 2011, 2011 (Special) : 302-311. doi: 10.3934/proc.2011.2011.302

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

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

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors