## The Annals of Applied Probability

### A constrained Langevin approximation for chemical reaction networks

#### Abstract

Continuous-time Markov chain models are often used to describe the stochastic dynamics of networks of reacting chemical species, especially in the growing field of systems biology. These Markov chain models are often studied by simulating sample paths in order to generate Monte-Carlo estimates. However, discrete-event stochastic simulation of these models rapidly becomes computationally intensive. Consequently, more tractable diffusion approximations are commonly used in numerical computation, even for modest-sized networks. However, existing approximations either do not respect the constraint that chemical concentrations are never negative (linear noise approximation) or are typically only valid until the concentration of some chemical species first becomes zero (Langevin approximation).

In this paper, we propose an approximation for such Markov chains via reflected diffusion processes that respect the fact that concentrations of chemical species are never negative. We call this a constrained Langevin approximation because it behaves like the Langevin approximation in the interior of the positive orthant, to which it is constrained by instantaneous reflection at the boundary of the orthant. An additional advantage of our approximation is that it can be written down immediately from the chemical reactions. This contrasts with the linear noise approximation, which involves a two-stage procedure—first solve a deterministic reaction rate ordinary differential equation, followed by a stochastic differential equation for fluctuations around those solutions. Our approximation also captures the interaction of nonlinearities in the reaction rate function with the driving noise. In simulations, we have found the computation time for our approximation to be at least comparable to, and often better than, that for the linear noise approximation.

Under mild assumptions, we first prove that our proposed approximation is well defined for all time. Then we prove that it can be obtained as the weak limit of a sequence of jump-diffusion processes that behave like the Langevin approximation in the interior of the positive orthant and like a rescaled version of the Markov chain on the boundary of the orthant. For this limit theorem, we adapt an invariance principle for reflected diffusions, due to Kang and Williams [Ann. Appl. Probab. 17 (2007) 741–779], and modify a result on pathwise uniqueness for reflected diffusions due to Dupuis and Ishii [Ann. Probab. 21 (1993) 554–580]. Some numerical examples illustrate the advantages of our approximation over direct simulation of the Markov chain or use of the linear noise approximation.

#### Article information

Source
Ann. Appl. Probab., Volume 29, Number 3 (2019), 1541-1608.

Dates
Revised: April 2018
First available in Project Euclid: 19 February 2019

https://projecteuclid.org/euclid.aoap/1550566837

Digital Object Identifier
doi:10.1214/18-AAP1421

Mathematical Reviews number (MathSciNet)
MR3914551

#### Citation

Leite, Saul C.; Williams, Ruth J. A constrained Langevin approximation for chemical reaction networks. Ann. Appl. Probab. 29 (2019), no. 3, 1541--1608. doi:10.1214/18-AAP1421. https://projecteuclid.org/euclid.aoap/1550566837

#### References

• [1] Anderson, D. F., Higham, D. J., Leite, S. C. and Williams, R. J. (2019). On constrained Langevin equations and (bio)chemical reaction networks. Multiscale Model. Simul. 17 1–30.
• [2] Anderson, D. F. and Kurtz, T. G. (2011). Continuous time Markov chain models for chemical reaction networks. In Design and Analysis of Biomolecular Circuits (H. Koeppl, G. Setti, M. d. Bernardo and D. Densmore, eds.) 3–42. Springer, New York.
• [3] Bossy, M., Gobet, E. and Talay, D. (2004). A symmetrized Euler scheme for an efficient approximation of reflected diffusions. J. Appl. Probab. 41 877–889.
• [4] Dupuis, P. and Ishii, H. (1990). On oblique derivative problems for fully nonlinear second-order elliptic partial differential equations on nonsmooth domains. Nonlinear Anal. 15 1123–1138.
• [5] Dupuis, P. and Ishii, H. (1993). SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 554–580. Correction 36(5) (2008) 1992–1997.
• [6] Ethier, S. N. and Kurtz, T. G. (2005). Markov Processes: Characterization and Convergence, 2nd ed. Wiley, New York.
• [7] Feinberg, M. (1979). Lectures on chemical reaction networks. Delivered at the Mathematics Research Center, Univ. Wisconsin-Madison. Available at https://crnt.osu.edu/lectures-chemical-reaction-networks.
• [8] Gillespie, D. T. (1976). A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22 403–434.
• [9] Gillespie, D. T. (1977). Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81 2340–2361.
• [10] Gillespie, D. T. (1992). A rigorous derivation of the chemical master equation. Phys. A 188 404–425.
• [11] Gillespie, D. T. (2000). The chemical Langevin equation. J. Chem. Phys. 113 297–306.
• [12] Gillespie, D. T. (2001). Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115 1716–1733.
• [13] Higham, D. J. (2001). An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43 525–546.
• [14] Higham, D. J. (2008). Modeling and simulating chemical reactions. SIAM Rev. 50 347–368.
• [15] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd ed. Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 288. Springer, Berlin.
• [16] Kang, W. and Ramanan, K. (2017). On the submartingale problem for reflected diffusions in domains with piecewise smooth boundaries. Ann. Probab. 45 404–468.
• [17] Kang, W. and Williams, R. J. (2007). An invariance principle for semimartingale reflecting Brownian motions in domains with piecewise smooth boundaries. Ann. Appl. Probab. 17 741–779.
• [18] Karatzas, I. and Shreve, S. E. (1991). Brownian Motion and Stochastic Calculus, 2nd ed. Graduate Texts in Mathematics 113. Springer, New York.
• [19] Komlós, J., Major, P. and Tusnády, G. (1975). An approximation of partial sums of independent $\mathrm{RV}$’s and the sample $\mathrm{DF}$. I. Z. Wahrsch. Verw. Gebiete 32 111–131.
• [20] Kurtz, T. G. (1972). The relationship between stochastic and deterministic models for chemical reactions. J. Chem. Phys. 57 2976–2978.
• [21] Kurtz, T. G. (1976). Limit theorems and diffusion approximations for density dependent Markov chains. In Stochastic Systems: Modeling, Identification and Optimization, I (R. J.-B. Wets, ed.). Mathematical Programming Studies 5 67–78. Springer, Berlin.
• [22] Kurtz, T. G. (1977/78). Strong approximation theorems for density dependent Markov chains. Stochastic Process. Appl. 6 223–240.
• [23] Kurtz, T. G. (1981). Approximation of Population Processes. CBMS-NSF Regional Conference Series in Applied Mathematics 36. SIAM, Philadelphia, PA.
• [24] Lépingle, D. (1978). Sur le comportement asymptotique des martingales locales. In Séminaire de Probabilités, XII (Univ. Strasbourg, Strasbourg, 1976/1977). Lecture Notes in Math. 649 148–161. Springer, Berlin.
• [25] Manninen, T., Linne, M. and Ruohonen, K. (2006). Developing Itô stochastic differential equation models for neuronal signal transduction pathways. Comput. Biol. Chem. 30 280–291. DOI:10.1016/j.compbiolchem.2006.04.002.
• [26] Maruyama, G. (1955). Continuous Markov processes and stochastic equations. Rend. Circ. Mat. Palermo (2) 4 48–90.
• [27] McAdams, H. H. and Arkin, A. (1997). Stochastic mechanisms in gene expression. Proc. Natl. Acad. Sci. USA 94 814–819.
• [28] Mélykúti, B. (2010). Theoretical advances in the modelling and interrogation of biochemical reaction systems: Alternative formulations of the chemical Langevin equation and optimal experiment design for model discrimination. Ph.D. thesis, Dept. Statistics, Univ. Oxford. Available at: http://ora.ouls.ox.ac.uk/objects/uuid:d368c04c-b611-41b2-8866-cde16b283b0d.
• [29] Platen, E. (1999). An introduction to numerical methods for stochastic differential equations. Acta Numer. 8 197–246.
• [30] R Core Team (2016). R: A Language and Environment for Statistical Computing. R Foundation for Statistical Computing, Vienna, Austria.
• [31] Rogers, L. C. G. and Williams, D. (2000). Diffusions, Markov Processes, and Martingales. Vol. 2. Itô Calculus. Cambridge Univ. Press, Cambridge. Reprint of the second (1994) edition.
• [32] Schnoerr, D., Sanguinetti, G. and Grima, R. (2014). The complex chemical Langevin equation. J. Chem. Phys. 141 024103. DOI:10.1063/1.4885345.
• [33] Schwalb, B., Tresch, A., Torkler, P., Duemcke, S. and Demel, C. (2015). LSD: Lots of superior depictions. R package version 3.0.
• [34] Scott, M. and Ingalls, B. P. (2005). Using the linear noise approximation to characterize molecular noise in reaction pathways. In Proceedings of the AIChE Conference on Foundations of Systems Biology in Engineering (FOSBE), Santa Barbara, California.
• [35] Stroock, D. W. and Varadhan, S. R. S. (2006). Multidimensional Diffusion Processes. Springer, Berlin. Reprint of the 1997 edition.
• [36] Szpruch, L. and Higham, D. J. (2009/10). Comparing hitting time behavior of Markov jump processes and their diffusion approximations. Multiscale Model. Simul. 8 605–621.
• [37] Turner, T. E., Schnell, S. and Burrage, K. (2004). Stochastic approaches for modelling in vivo reactions. Comput. Biol. Chem. 28 165–178.
• [38] van Kampen, N. G. (1961). A power series expansion of the master equation. Can. J. Phys. 39 551–567.
• [39] Wallace, E. W. J., Petzold, L. R., Gillespie, D. T. and Sanft, K. R. (2012). Linear noise approximation is valid over limited times for any chemical system that is sufficiently large. IET Syst. Biol. 6 102–115.
• [40] Wang, J. G. (1995). The asymptotic behavior of locally square integrable martingales. Ann. Probab. 23 552–585.
• [41] Wilhelm, T. (2009). The smallest chemical reaction system with bistability. BMC Syst. Biol. 3 90.
• [42] Wilkie, J. and Wong, Y. M. (2008). Positivity preserving chemical Langevin equations. Chem. Phys. 353 132–138.
• [43] Wilkinson, D. J. (2006). Stochastic Modelling for Systems Biology. Chapman & Hall/CRC, Boca Raton, FL.