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Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks

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Abstract

We explore the use of Array-RQMC, a randomized quasi-Monte Carlo method designed for the simulation of Markov chains, to reduce the variance when simulating stochastic biological or chemical reaction networks with \(\tau \)-leaping. The task is to estimate the expectation of a function of molecule copy numbers at a given future time T by the sample average over n sample paths, and the goal is to reduce the variance of this sample-average estimator. We find that when the method is properly applied, variance reductions by factors in the thousands can be obtained. These factors are much larger than those observed previously by other authors who tried RQMC methods for the same examples. Array-RQMC simulates an array of realizations of the Markov chain and requires a sorting function to reorder these chains according to their states, after each step. The choice of sorting function is a key ingredient for the efficiency of the method, although in our experiments, Array-RQMC was never worse than ordinary Monte Carlo, regardless of the sorting method. The expected number of reactions of each type per step also has an impact on the efficiency gain.

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References

  • Anderson WJ (1991) Continuous-time Markov chains: an applications-oriented approach. Springer, New York

    Book  MATH  Google Scholar 

  • Anderson DF (2008) Incorporating postleap checks in tau-leaping. J Chem Phys 128(5):054–103. https://doi.org/10.1063/1.2819665

    Article  Google Scholar 

  • Anderson D, Higham D (2012) Multilevel Monte Carlo for continuous-time Markov chains, with applications in biochemical kinetics. Multiscale Model Simul 10(1):146–179. https://doi.org/10.1137/110840546

    Article  MathSciNet  MATH  Google Scholar 

  • Anderson DF, Kurtz TG (2011) Continuous time Markov chain models for chemical reaction networks. In: Koeppl H, Densmore D, Setti G, di Bernardo M (eds) Design and analysis of biomolecular circuits, vol 117. Springer, New York, pp 3–42

    Chapter  Google Scholar 

  • Beentjes CHL, Baker RE (2019) Quasi-Monte Carlo methods applied to tau-leaping in stochastic biological systems. Bull Math Biol 81:2931–2959

    Article  MathSciNet  MATH  Google Scholar 

  • Ben Abdellah A, L’Ecuyer P, Puchhammer F (2019) Array-RQMC for option pricing under stochastic volatility models. In: Proceedings of the 2019 Winter Simulation Conference. IEEE Press, pp 440–451. https://www.informs-sim.org/wsc19papers/429.pdf

  • Breiman L (2001) Random forests. Mach Learn 45(1):5–32

    Article  MATH  Google Scholar 

  • Cao Y, Gillespie DT, Petzold LR (2005) The slow-scale stochastic simulation algorithm. J Chem Phys 122(1):014–116. https://doi.org/10.1063/1.1824902

    Article  Google Scholar 

  • de Boor C (2001) A practical guide to splines, 2nd edn. Springer, New York

    MATH  Google Scholar 

  • Demers V, L’Ecuyer P, Tuffin B (2005) A combination of randomized quasi-Monte Carlo with splitting for rare-event simulation. In: Proceedings of the 2005 European simulation and modeling conference. EUROSIS, Ghent, Belgium, pp 25–32

  • Dick J, Pillichshammer F (2010) Digital nets and sequences: discrepancy theory and quasi-Monte Carlo integration. Cambridge University Press, Cambridge

    Book  MATH  Google Scholar 

  • Dick J, Sloan IH, Wang X, Woźniakowski H (2006) Good lattice rules in weighted Korobov spaces with general weights. Numer Math 103:63–97

    Article  MathSciNet  MATH  Google Scholar 

  • Dion M, L’Ecuyer P (2010) American option pricing with randomized quasi-Monte Carlo simulations. In: Proceedings of the 2010 Winter Simulation Conference, pp 2705–2720

  • El Haddad R, Lécot C, L’Ecuyer P (2008) Quasi-Monte Carlo simulation of discrete-time Markov chains on multidimensional state spaces. In: Keller A, Heinrich S, Niederreiter H (eds) Monte Carlo and quasi-Monte Carlo methods 2006. Springer, Berlin, pp 413–429

    Chapter  MATH  Google Scholar 

  • Fox BL, Glynn PW (1990) Discrete-time conversion for simulating finite-horizon Markov processes. SIAM J Appl Math 50:1457–1473

    Article  MathSciNet  MATH  Google Scholar 

  • Gerber M, Chopin N (2015) Sequential quasi-Monte Carlo. J R Stat Soc Ser B 77(Part 3):509–579

    Article  MathSciNet  MATH  Google Scholar 

  • Giles MB (2016) Algorithm 955: approximation of the inverse Poisson cumulative distribution. ACM Trans Math Softw 42:1–22

    Article  MathSciNet  MATH  Google Scholar 

  • Gillespie DT (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361. https://doi.org/10.1021/j100540a008

    Article  Google Scholar 

  • Gillespie DT (2000) The chemical Langevin equation. J Chem Phys 113(1):297–306

    Article  Google Scholar 

  • Gillespie DT (2001) Approximate accelerated stochastic simulation of chemically reacting systems. J Chem Phys 115(4):1716–1733. https://doi.org/10.1063/1.1378322

    Article  Google Scholar 

  • Hellander A (2008) Efficient computation of transient solutions of the chemical master equation based on uniformization and quasi-Monte Carlo. J Chem Phys 128(154):109

    Google Scholar 

  • Hickernell FJ (1998) Lattice rules: how well do they measure up? In: Hellekalek P, Larcher G (eds) Random and quasi-random point sets, vol 138. Lecture notes in statistics. Springer, New York, pp 109–166

    Chapter  Google Scholar 

  • Hickernell FJ (2002) Obtaining \({O(N^{-2+\epsilon })}\) convergence for lattice quadrature rules. In: Fang KT, Hickernell FJ, Niederreiter H (eds) Monte Carlo and Quasi-Monte Carlo methods 2000. Springer, Berlin, pp 274–289

    Chapter  Google Scholar 

  • Hickernell FJ, Hong HS, L’Ecuyer P, Lemieux C (2001) Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM J Sci Comput 22(3):1117–1138

    Article  MathSciNet  MATH  Google Scholar 

  • Higham DJ (2008) Modeling and simulating chemical reactions. SIAM Rev 50(2):347–368. https://doi.org/10.1137/060666457

    Article  MathSciNet  MATH  Google Scholar 

  • Joe S, Kuo FY (2008) Constructing Sobol sequences with better two-dimensional projections. SIAM J Sci Comput 30(5):2635–2654

    Article  MathSciNet  MATH  Google Scholar 

  • Kim JK, Josić K, Bennett MR (2015) The relationship between stochastic and deterministic quasi-steady state approximations. BMC Syst Biol 9(87):1–13. https://doi.org/10.1186/s12918-015-0218-3

    Article  Google Scholar 

  • Kloeden PE, Platen E (1992) Numerical solutions of stochastic differential equations. Springer, Berlin

    Book  MATH  Google Scholar 

  • Koh W, Blackwell KT (2012) Improved spatial direct method with gradient-based diffusion to retain full diffusive fluctuations. J Chem Phys 137(15):154–111. https://doi.org/10.1063/1.4758459

    Article  Google Scholar 

  • Lécot C, Coulibaly I (1998) A quasi-Monte Carlo scheme using nets for a linear Boltzmann equation. SIAM J Numer Anal 35(1):51–70

    Article  MathSciNet  MATH  Google Scholar 

  • L’Ecuyer P (1999) Good parameters and implementations for combined multiple recursive random number generators. Oper Res 47(1):159–164

    Article  MATH  Google Scholar 

  • L’Ecuyer P (2009) Quasi-Monte Carlo methods with applications in finance. Fin Stoch 13(3):307–349

    Article  MathSciNet  MATH  Google Scholar 

  • L’Ecuyer P (2012) Random number generation. In: Gentle JE, Haerdle W, Mori Y (eds) Handbook of computational statistics, 2nd edn. Springer, Berlin, pp 35–71

    Chapter  Google Scholar 

  • L’Ecuyer P (2016) SSJ: Stochastic Simulation in Java. http://simul.iro.umontreal.ca/ssj/

  • L’Ecuyer P (2018) Randomized quasi-Monte Carlo: an introduction for practitioners. In: Glynn PW, Owen AB (eds) Monte Carlo and Quasi-Monte Carlo methods: MCQMC 2016. Springer, Berlin, pp 29–52

    Chapter  MATH  Google Scholar 

  • L’Ecuyer P, Buist E (2005) Simulation in Java with SSJ. In: Proceedings of the 2005 Winter Simulation Conference. IEEE Press, Piscataway, NJ, pp 611–620

  • L’Ecuyer P, Lemieux C (2000) Variance reduction via lattice rules. Manag Sci 46(9):1214–1235

    Article  MATH  Google Scholar 

  • L’Ecuyer P, Lemieux C (2002) Recent advances in randomized quasi-Monte Carlo methods. In: Dror M, L’Ecuyer P, Szidarovszky F (eds) Modeling uncertainty: an examination of stochastic theory, methods, and applications. Kluwer Academic, Boston, pp 419–474

    Chapter  Google Scholar 

  • L’Ecuyer P, Munger D (2012) On figures of merit for randomly-shifted lattice rules. In: Woźniakowski H, Plaskota L (eds) Monte Carlo and Quasi-Monte Carlo methods 2010. Springer, Berlin, pp 133–159

    Chapter  Google Scholar 

  • L’Ecuyer P, Munger D (2016) Algorithm 958: lattice builder: a general software tool for constructing rank-1 lattice rules. ACM Trans Mathe Softw 42(2):Article 15

  • L’Ecuyer P, Lécot C, Tuffin B (2006) Randomized quasi-Monte Carlo simulation of Markov chains with an ordered state space. In: Niederreiter H, Talay D (eds) Monte Carlo and Quasi-Monte Carlo methods 2004. Springer, Berlin, pp 331–342

    Chapter  MATH  Google Scholar 

  • L’Ecuyer P, Demers V, Tuffin B (2007) Rare-events, splitting, and quasi-Monte Carlo. ACM Trans Model Comput Simul 17(2):Article 9, 45 pages

  • L’Ecuyer P, Lécot C, Tuffin B (2008) A randomized quasi-Monte Carlo simulation method for Markov chains. Oper Res 56(4):958–975

    Article  MATH  Google Scholar 

  • L’Ecuyer P, Lécot C, L’Archevêque-Gaudet A (2009) On array-RQMC for Markov chains: mapping alternatives and convergence rates. In: L’Ecuyer P, Owen AB (eds) Monte Carlo and Quasi-Monte Carlo methods 2008. Springer, Berlin, pp 485–500

    Chapter  MATH  Google Scholar 

  • L’Ecuyer P, Munger D, Lécot C, Tuffin B (2018) Sorting methods and convergence rates for Array-RQMC: some empirical comparisons. Math Comput Simul 143:191–201

    Article  MathSciNet  MATH  Google Scholar 

  • L’Ecuyer P, Marion P, Godin M, Fuchhammer F (2020) A tool for custom construction of QMC and RQMC point sets. In: Monte Carlo and Quasi-Monte Carlo methods: MCQMC 2020, submitted manuscript. Available at http://www.iro.umontreal.ca/~lecuyer/myftp/papers/mcqmc20latnet.pdf

  • Lemieux C (2009) Monte Carlo and Quasi-Monte Carlo sampling. Springer, Berlin

    MATH  Google Scholar 

  • Lemieux C, Cieslak M, Luttmer K (2004) RandQMC user’s guide: a package for randomized quasi-Monte Carlo methods in C. Software user’s guide. Available at http://www.math.uwaterloo.ca/~clemieux/randqmc.html

  • Matousěk J (1998) On the \(L_2\)-discrepancy for anchored boxes. J Complex 14:527–556

    MATH  Google Scholar 

  • Niederreiter H (1992) Random number generation and quasi-Monte Carlo methods. In: SIAM CBMS-NSF regional conference series in applied mathematics, vol 63. SIAM

  • Owen AB (1997a) Monte Carlo variance of scrambled equidistribution quadrature. SIAM J Numer Anal 34(5):1884–1910

    Article  MathSciNet  MATH  Google Scholar 

  • Owen AB (1997b) Scrambled net variance for integrals of smooth functions. Ann Stat 25(4):1541–1562

    Article  MathSciNet  MATH  Google Scholar 

  • Owen AB (1998) Latin supercube sampling for very high-dimensional simulations. ACM Trans Model Comput Simul 8(1):71–102

    Article  MATH  Google Scholar 

  • Padgett JMA, Ilie S (2016) An adaptive tau-leaping method for stochastic simulations of reaction-diffusion systems. AIP Adv 6(3):035–217. https://doi.org/10.1063/1.4944952

    Article  Google Scholar 

  • Pollock DSG (1993) Smoothing with cubic splines. Technical report, University of London, Queen Mary and Westfield College, London

  • Rao CV, Arkin AP (2003) Stochastic chemical kinetics and the quasi-steady-state assumption: application to the Gillespie algorithm. J Chem Phys 118(11):4999–5010. https://doi.org/10.1063/1.1545446

    Article  Google Scholar 

  • Sinescu V, L’Ecuyer P (2012) Variance bounds and existence results for randomly shifted lattice rules. J Comput Appl Math 236:3296–3307

    Article  MathSciNet  MATH  Google Scholar 

  • Sloan IH, Joe S (1994) Lattice methods for multiple integration. Clarendon Press, Oxford

    MATH  Google Scholar 

  • Sobol’ IM (1967) The distribution of points in a cube and the approximate evaluation of integrals. USSR Comput Math Math Phys 7(4):86–112

    Article  MathSciNet  MATH  Google Scholar 

  • Strehl R, Ilie S (2015) Hybrid stochastic simulation of reaction–diffusion systems with slow and fast dynamics. J Chem Phys 143(23):108–234. https://doi.org/10.1063/1.4937491

    Article  Google Scholar 

  • Thomas P, Straube AV, Grima R (2012) The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions. BMC Syst Biol. https://doi.org/10.1186/1752-0509-6-39

    Article  Google Scholar 

  • Wächter C, Keller A (2008) Efficient simultaneous simulation of Markov chains. In: Keller A, Heinrich S, Niederreiter H (eds) Monte Carlo and Quasi-Monte Carlo methods 2006. Springer, Berlin, pp 669–684

    Chapter  Google Scholar 

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Acknowledgements

This work has been supported by a Canada Research Chair, an IVADO Research Grant, and an NSERC Discovery Grant Number RGPIN-110050 to P. L’Ecuyer. F. Puchhammer was also supported by Spanish and Basque governments fundings through BCAM (ERDF, ESF, SEV-2017-0718, PID2019-108111RB-I00, PID2019-104927GB-C22, BERC 2018e2021, EXP. 2019/00432, ELKARTEK KK-2020/00049), and the computing infrastructure of i2BASQUE academic network and IZO-SGI SGIker (UPV).

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  1. Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Basque Country, Spain

    Florian Puchhammer

  2. DIRO, Université de Montréal, C.P. 6128, Succ. Centre-Ville, Montréal, H3C 3J7, Canada

    Florian Puchhammer, Amal Ben Abdellah & Pierre L’Ecuyer

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  1. Florian Puchhammer
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Correspondence to Florian Puchhammer.

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Puchhammer, F., Ben Abdellah, A. & L’Ecuyer, P. Variance Reduction with Array-RQMC for Tau-Leaping Simulation of Stochastic Biological and Chemical Reaction Networks. Bull Math Biol 83, 91 (2021). https://doi.org/10.1007/s11538-021-00920-5

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