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Generalizing Gillespie’s Direct Method to Enable Network-Free Simulations

  • Special Issue: Gillespie and His Algorithms
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Abstract

Gillespie’s direct method for stochastic simulation of chemical kinetics is a staple of computational systems biology research. However, the algorithm requires explicit enumeration of all reactions and all chemical species that may arise in the system. In many cases, this is not feasible due to the combinatorial explosion of reactions and species in biological networks. Rule-based modeling frameworks provide a way to exactly represent networks containing such combinatorial complexity, and generalizations of Gillespie’s direct method have been developed as simulation engines for rule-based modeling languages. Here, we provide both a high-level description of the algorithms underlying the simulation engines, termed network-free simulation algorithms, and how they have been applied in systems biology research. We also define a generic rule-based modeling framework and describe a number of technical details required for adapting Gillespie’s direct method for network-free simulation. Finally, we briefly discuss potential avenues for advancing network-free simulation and the role they continue to play in modeling dynamical systems in biology.

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Notes

  1. Most objects in rule-based modeling can be represented visually (and formally) as graphs. Rules then become transformations (i.e., rewriting operations) on these graphs, and much of the jargon relating to rule-based modeling has its origin in graph theory. We will occasionally mention these terms, but will not rely on them.

  2. To our best knowledge, BioNetGen is the only framework that allows molecules with multiple identically named sites. These sites are treated as equivalent. Molecule types must have unique names.

  3. Rules may also define synthesis and degradation of molecules.

  4. A labeled bond links two sites, meaning that both partners in a bond can be determined by the bond label.

  5. In chemistry and ecology, the term species refers to a class of things (molecular configurations or organisms). We retain this convention and refer to specific instances of a species when discussing an individual object that conforms to the features that define a particular species.

  6. In some cases, patterns may be defined as involving unconnected molecules, but we do not adopt this convention

  7. The anchoring molecule simply serves as an arbitrary starting point for traversing the species instance (i.e., multiple starting points may be possible).

  8. Nonproductive (null) events are described in Sect. 6.6.

  9. Depending on how patterns are specified, a brute force approach may be the only way to correctly update the system. See Sect. 6.5.

  10. This traversal, and that in the positive update phase, allows the algorithm to accommodate rules with nonlocal constraints. See Sect. 6.5.

  11. When representing patterns as graphs, this number is the order of the automorphism group of the graph, roughly the number of ways a graph can be mapped to itself.

  12. The BioNetGen language follows the number-of-reactions convention for rate calculation, whereas the Kappa language follows the number-of-matches convention.

  13. The reaction path degeneracy is 2 because binding to either y site on the B dimer pattern results in the same species instance.

  14. This is termed the influence map in the Kappa language and associated simulation engines.

  15. For each site in one of the pattern molecules, the corresponding site in the other pattern molecule must either be equivalent or less specific and consistent.

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Acknowledgements

We thank Jim Faeder for providing helpful feedback on the manuscript. This work was supported by the National Institute of General Medical Sciences (NIGMS) and the National Cancer Institute (NCI) of the National Institutes of Health (NIH) through Grants R01GM111510, P50GM085273 and R01CA197398; by the US Department of Energy (DOE) through contract DE-AC52-06NA25396; and by the Joint Design of Advanced Computing Solutions for Cancer (JDACS4C) program established by DOE and NCI/NIH. Additionally, RS, YTL and SF gratefully acknowledge support from the Center for Nonlinear Studies (CNLS), which is funded by the Laboratory Directed Research and Development (LDRD) program at Los Alamos National Laboratory.

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Authors and Affiliations

  1. Theoretical Biology and Biophysics Group, Theoretical Division, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Ryan Suderman, Eshan D. Mitra, Yen Ting Lin, Keesha E. Erickson, Song Feng & William S. Hlavacek

  2. Center for Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, NM, 87545, USA

    Ryan Suderman, Yen Ting Lin, Song Feng & William S. Hlavacek

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  1. Ryan Suderman
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  2. Eshan D. Mitra
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  6. William S. Hlavacek
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Correspondence to William S. Hlavacek.

Appendix: PDGFR Activation Model

Appendix: PDGFR Activation Model

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Suderman, R., Mitra, E.D., Lin, Y.T. et al. Generalizing Gillespie’s Direct Method to Enable Network-Free Simulations. Bull Math Biol 81, 2822–2848 (2019). https://doi.org/10.1007/s11538-018-0418-2

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