# A Critical Comparison of Rejection-Based Algorithms for Simulation of Large Biochemical Reaction Networks

## Abstract

The rejection-based simulation technique has been applying to improve the computational efficiency of the stochastic simulation algorithm (SSA) in simulating large reaction networks, which are required for a thorough understanding of biological systems. We compare two recently proposed simulation methods, namely the composition–rejection algorithm (SSA-CR) and the rejection-based SSA (RSSA), aiming for this purpose. We discuss the right interpretation of the rejection-based technique used in these algorithms in order to make an informed choice when dealing with different aspects of biochemical networks. We provide the theoretical analysis as well as the detailed runtime comparison of these algorithms on concrete biological models. We highlight important factors that are omitted in previous analysis of these algorithms. The numerical comparison shows that for reaction networks where the search cost is expensive then SSA-CR is more efficient, and for reaction networks where the update cost is dominant, often the case in practice, then RSSA should be the choice.

## Keywords

Computational biology Stochastic simulation Rejection-based simulation technique## Notes

### Acknowledgements

This work has been partially done when VHT was at the Microsoft Research, University of Trento Centre for Computational and Systems Biology (COSBI), Italy. The research supported by Academy of Finland grant 311639, “Algorithmic Designs for Biomolecular Nanostructures (ALBION)”.

## References

- Anderson DF (2007) A modified next reaction method for simulating chemical systems with time dependent propensities and delays. J Chem Phys 127(21):214107CrossRefGoogle Scholar
- Arkin A, Ross J, McAdams HH (1998) Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected
*Escherichia coli*cells. Genetics 149(4):16331648Google Scholar - Blue J, Beichl I, Sullivan F (1995) Faster Monte Carlo simulations. Phys Rev E 51(2):867–868CrossRefGoogle Scholar
- Cai X (2007) Exact stochastic simulation of coupled chemical reactions with delays. J Chem Phys 126(12):124108CrossRefGoogle Scholar
- Cao Y, Li H, Petzold L (2004) Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J Chem Phys 121(9):4059CrossRefGoogle Scholar
- Chylek LA, Holowka DA, Baird BA et al (2014) An interaction library for the \(\text{ Fc }\epsilon \text{ RI }\) signaling network. Front Immunol 5(172):1664–3224Google Scholar
- Devroye L (1986) Non-uniform random variate generation. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Faeder JR, Hlavacek WS, Reischl I et al (2003) Investigation of early events in \(\text{ Fc }\epsilon \text{ RI }\)-mediated signaling using a detailed mathematical model. J Immunol 170:3769–3781CrossRefGoogle Scholar
- Funel N et al (2008) Laser microdissection and primary cell cultures improve pharmacogenetic analysis in pancreatic adenocarcinoma. Lab Invest 88:773–784CrossRefGoogle Scholar
- Gibson M, Bruck J (2000) Efficient exact stochastic simulation of chemical systems with many species and many channels. J Phys Chem A 104(9):1876–1889CrossRefGoogle Scholar
- Gillespie D (1976) A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J Comput Phys 22(4):403–434MathSciNetCrossRefGoogle Scholar
- Gillespie D (1977) Exact stochastic simulation of coupled chemical reactions. J Phys Chem 81(25):2340–2361CrossRefGoogle Scholar
- Gillespie D (1992) A rigorous derivation of the chemical master equation. Physica A 188(1–3):404–425CrossRefGoogle Scholar
- Gillespie D (2001) Approximate accelerated stochastic simulation of chemically reacting. J Chem Phys 115:1716–1733CrossRefGoogle Scholar
- Gupta A, Mendes P (2018) An overview of network-based and-free approaches for stochastic simulation of biochemical systems. Computation 6(1):9CrossRefGoogle Scholar
- Hepburn I, Chen W, Wils S, De Schutter E (2012) STEPS: efficient simulation of stochastic reaction-diffusion models in realistic morphologies. BMC Syst Biol 6:36CrossRefGoogle Scholar
- Hörmann W, Leydold J, Derflinger G (2004) Automatic nonuniform random variate generation. Springer, BerlinCrossRefzbMATHGoogle Scholar
- Kahramanogullari O, Fantaccini G, Lecca P, Morpurgo D, Priami C (2012) Algorithmic modeling quantifies the complementary contribution of metabolic inhibitions to gemcitabine efficacy. PLoS ONE 7(12):e50176Google Scholar
- Lin MY, Lindsay HM, Weitz DA et al (1989) Universality in colloid aggregation. Nature 339:360–362CrossRefGoogle Scholar
- Liu Y, Barua D, Liu P et al (2013) Single-cell measurements of IgE-mediated \(\text{ Fc }\epsilon \text{ RI }\) signaling using an integrated microfluidic platform. PLoS ONE 8(3):60159CrossRefGoogle Scholar
- Marchetti L, Priami C, Thanh VH (2016) HRSSA—efficient hybrid stochastic simulation for spatially homogeneous biochemical reaction networks. J Comput Phys 317:301–317MathSciNetCrossRefzbMATHGoogle Scholar
- Marchetti L, Lombardo R, Corrado P (2017a) HSimulator: hybrid stochastic/deterministic simulation of biochemical reaction networks. Complexity. https://doi.org/10.1155/2017/1232868 zbMATHGoogle Scholar
- Marchetti L, Priami C, Thanh VH (2017b) Simulation algorithms for computational systems biology. Springer, BerlinCrossRefGoogle Scholar
- Mauch S, Stalzer M (2011) Efficient formulations for exact stochastic simulation of chemical systems. IEEE/ACM Trans Comput Biol Bioinform 8(1):27–35CrossRefGoogle Scholar
- McAdams HH, Arkin A (1997) Stochastic mechanisms in gene expression. PNAS 94(3):814–819CrossRefGoogle Scholar
- McAdams HH, Arkin A (1999) It’s a noisy business! genetic regulation at the nanomolar scale. Trends Genet 15(2):65–69CrossRefGoogle Scholar
- McCollum J, Peterson GD, Cox CD et al (2006) The sorting direct method for stochastic simulation of biochemical systems with varying reaction execution behavior. Comput Biol Chem 30(1):39–49CrossRefzbMATHGoogle Scholar
- Meakin P (1988) Models for colloidal aggregation. Annu Rev Phys Chem 39:237–267CrossRefGoogle Scholar
- Ramaswamy R, Sbalzarini IF (2010) A partial-propensity variant of the composition–rejection stochastic simulation algorithm for chemical reaction networks. J Chem Phys 132(4):044102CrossRefGoogle Scholar
- Ramaswamy R, Gonzlez-Segredo N, Sbalzarini IF (2009) A new class of highly efficient exact stochastic simulation algorithms for chemical reaction networks. J Chem Phys 130(24):244104CrossRefGoogle Scholar
- Raser JM, O’Shea EK (2005) Noise in gene expression: origins, consequences, and control. Science 309:2010–2013CrossRefGoogle Scholar
- Schulze T (2008) Efficient kinetic Monte Carlo simulation. J Comput Phys 227(4):2455–2462MathSciNetCrossRefzbMATHGoogle Scholar
- Slepoy A, Thompson AP, Plimpton SJ (2008) A constant-time kinetic Monte Carlo algorithm for simulation of large biochemical reaction networks. J Chem Phys 128(20):205101CrossRefGoogle Scholar
- Thanh VH (2013) On efficient algorithms for stochastic simulation of biochemical reaction systems. PhD thesis, University of Trento, Italy. http://eprints-phd.biblio.unitn.it/1070/. Accessed 4 July 2018
- Thanh VH (2017) Stochastic simulation of biochemical reactions with partial-propensity and rejection-based approaches. Math Biosci 292:67–757MathSciNetCrossRefzbMATHGoogle Scholar
- Thanh VH, Priami C (2015) Simulation of biochemical reactions with time-dependent rates by the rejection-based algorithm. J Chem Phys 143(5):054104CrossRefGoogle Scholar
- Thanh VH, Zunino R (2012) Tree-based search for stochastic simulation algorithm. In: Proceedings of ACM-SACGoogle Scholar
- Thanh VH, Zunino R (2014) Adaptive tree-based search for stochastic simulation algorithm. Int J Comput Biol Drug Des 7(4):341–357CrossRefGoogle Scholar
- Thanh VH, Corrado P, Roberto Z (2014) Efficient rejection-based simulation of biochemical reactions with stochastic noise and delays. J Chem Phys 141(13):134116CrossRefGoogle Scholar
- Thanh VH, Zunino R, Priami C (2015) On the rejection-based algorithm for simulation and analysis of large-scale reaction networks. J Chem Phys 142(24):244106CrossRefGoogle Scholar
- Thanh VH, Priami C, Zunino R (2016) Accelerating rejection-based simulation of biochemical reactions with bounded acceptance probability. J Chem Phys 144(22):224108CrossRefGoogle Scholar
- Thanh VH, Zunino R, Priami C (2017a) Efficient stochastic simulation of biochemical reactions with noise and delays. J Chem Phys 146(8):084107CrossRefGoogle Scholar
- Thanh VH, Zunino R, Priami C (2017b) Efficient constant-time complexity algorithm for stochastic simulation of large reaction networks. IEEE/ACM Trans Comput Biol Bioinform 14(3):657–667CrossRefGoogle Scholar
- Thanh VH, Marchetti L, Reali F et al (2018) Incorporating extrinsic noise into the stochastic simulation of biochemical reactions: a comparison of approaches. J Chem Phys 148(6):064111CrossRefGoogle Scholar
- Veltkamp SA, Beijnen JH, Schellens JHM (2008) Prolonged versus standard gemcitabine infusion: translation of molecular pharmacology to new treatment strategy. Oncologist 13(3):261–276CrossRefGoogle Scholar