Bulletin of Mathematical Biology

, Volume 71, Issue 7, pp 1671–1692 | Cite as

Stochastic Kinetic Modeling of Vesicular Stomatitis Virus Intracellular Growth

  • Sebastian C. Hensel
  • James B. Rawlings
  • John Yin
Original Article


By building kinetic models of biological networks one may advance the development of new modeling approaches while gaining insights into the biology. We focus here on building a stochastic kinetic model for the intracellular growth of vesicular stomatitis virus (VSV), a well-studied virus that encodes five genes. The essential network of VSV reactions creates challenges to stochastic simulation owing to (i) delayed reactions associated with transcription and genome replication, (ii) production of large numbers of intermediate proteins by translation, and (iii) the presence of highly reactive intermediates that rapidly fluctuate in their intracellular levels. We address these issues by developing a hybrid implementation of the model that combines a delayed stochastic simulation algorithm (DSSA) with Langevin equations to simulate the reactions that produce species in high numbers. Further, we employ a quasi-steady-state approximation (QSSA) to overcome the computational burden of small time steps caused by highly reactive species. The simulation is able to capture experimentally observed patterns of viral gene expression. Moreover, the simulation suggests that early levels of a low-abundance species, VSV L mRNA, play a key role in determining the production level of VSV genomes, transcripts, and proteins within an infected cell. Ultimately, these results suggest that stochastic gene expression contribute to the distribution of virus progeny yields from infected cells.


Stochastic kinetics Delayed reactions Langevin Quasi-steady state Genome mRNA Protein Transcription Translation Replication Encapsidation Virus Infection 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Abraham, G., Banerjee, A.K., 1976. Sequential transcription of the genes of vesicular stomatitis virus. Proc. Natl. Acad. Sci. USA 73(5), 1504–1508. CrossRefGoogle Scholar
  2. Arkin, A., Ross, J., McAdams, H., 1998. Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected Escherichia coli cells. Genetics 149(4), 1633–1648. Google Scholar
  3. Ball, L.A., White, C.N., 1976. Order of transcription of genes of vesicular stomatitis virus. Proc. Natl. Acad. Sci. USA 73(2), 442–446. CrossRefGoogle Scholar
  4. Barr, J.N., Whelan, S.P.J., Wertz, G.W., 2002. Transcriptional control of the RNA-dependent RNA polymerase of vesicular stomatitis virus. Biochim. Biophys. Acta, Gene Struct. Expr. 1577(2), 337–353. Google Scholar
  5. Barrio, M., Burrage, K., Leier, A., Tian, T., 2006. Oscillatory regulation of hes1: Discrete stochastic delay modelling and simulation. PLoS Comput. Biol. 2(9), e117. CrossRefGoogle Scholar
  6. Bratsun, D., Volfson, D., Tsimring, L.S., Hasty, J., 2005. Delay-induced stochastic oscillations in gene regulation. Proc. Natl. Acad. Sci. USA 102(41), 14593–14598. CrossRefGoogle Scholar
  7. Delbrück, M., 1940. Statistical fluctuations in autocatalytic reactions. J. Chem. Phys. 8, 120–124. CrossRefGoogle Scholar
  8. Delbrück, M., 1945. The burst size distribution in the growth of bacterial viruses (bacteriophages). J. Bact. 50, 131–135. Google Scholar
  9. E, W., Liu, D., Vanden-Eijnden, E., 2005. Nested stochastic simulation algorithm for chemical kinetic systems with disparate rates. J. Chem. Phys. 123, 194107. CrossRefGoogle Scholar
  10. Flanagan, E.B., Ball, L.A., Wertz, G.W., 2000. Moving the glycoprotein gene of vesicular stomatitis virus to promoter-proximal positions accelerates and enhances the protective immune response. J. Virol. 74(17), 7895–7902. CrossRefGoogle Scholar
  11. Gillespie, D.T., 1976. A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. J. Comput. Phys. 22, 403–434. CrossRefMathSciNetGoogle Scholar
  12. Gillespie, D.T., 1992. A rigorous derivation of the chemical master equation. Physica A 188, 404–425. CrossRefGoogle Scholar
  13. Gillespie, D.T., 2000. The chemical Langevin equation. J. Chem. Phys. 113(1), 297–306. CrossRefGoogle Scholar
  14. Goutsias, J., 2005. Quasiequilibrium approximation of fast reaction kinetics in stochastic biochemical systems. J. Chem. Phys. 122(18), 184102. CrossRefGoogle Scholar
  15. Griffith, M., Courtney, T., Peccoud, J., Sanders, W., 2006. Dynamic partitioning for hybrid simulation of the bistable HIV-1 transactivation network. Bioinformatics 22(22), 2782–2789. CrossRefGoogle Scholar
  16. Haseltine, E.L., Rawlings, J.B., 2002. Approximate simulation of coupled fast and slow reactions for stochastic chemical kinetics. J. Chem. Phys. 117(15), 6959–6969. CrossRefGoogle Scholar
  17. Iverson, L.E., Rose, J.K., 1981. Localized attenuation and discontinuous synthesis during vesicular stomatitis virus transcription. Cell 23(2), 477–484. CrossRefGoogle Scholar
  18. Janssen, J.A.M., 1989. The elimination of fast variables in complex chemical reactions. II. Mesoscopic level (reducible case). J. Stat. Phys. 57(1/2), 171–185. CrossRefGoogle Scholar
  19. Lim, K., Lang, T., Lam, V., Yin, J., 2006. Model-based design of growth-attenuated viruses. PLoS Comput. Biol. 2(9), e116. CrossRefGoogle Scholar
  20. Mastny, E.A., Haseltine, E.L., Rawlings, J.B., 2007. Two classes of quasi-steady-state model reductions for stochastic kinetics. J. Chem. Phys. 127(9), 094106. CrossRefGoogle Scholar
  21. Rao, C.V., Arkin, A.P., 2003. Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. J. Chem. Phys. 118(11), 4999–5010. CrossRefGoogle Scholar
  22. Rose, J., Whitt, M., 2001. Rhabdoviridae: The viruses and their replication. In: Knipe, D., Howley, P. (Eds.), Fields Virology, vol. 1, 4th edn. pp. 1221–1244. Lippincot Williams & Wilkins, Philadelphia. Google Scholar
  23. Salis, H., Kaznessis, Y., 2005a. Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. J. Chem. Phys. 122(5), 054103. CrossRefGoogle Scholar
  24. Salis, H., Kaznessis, Y., 2005b. An equation-free probabilistic steady-state approximation: Dynamic application to the stochastic simulation of biochemical reaction networks. J. Chem. Phys. 123, 214106. CrossRefGoogle Scholar
  25. Samant, A., Vlachos, D.G., 2005. Overcoming stiffness in stochastic simulation stemming from partial equilibrium: A multiscale Monte Carlo algorithm. J. Chem. Phys. 123, 144114. CrossRefGoogle Scholar
  26. Samant, A., Ogunnaike, B., Vlachos, D., 2007. A hybrid multiscale Monte Carlo algorithm (HyMSMC) to cope with disparity in time scales and species populations in intracellular networks. BMC Bioinf. 8(1), 175. CrossRefGoogle Scholar
  27. Simonsen, C.C., Batt-Humphries, S., Summers, D., 1979. RNA synthesis of vesicular stomatitis virus-infected cells: In vivo regulation of replication. J. Virol. 31(1), 124–132. Google Scholar
  28. Spirin, A., 1986. Ribosome Structure and Protein Biosysthesis. Benjamin/Cummings, Redwood City. Google Scholar
  29. Srivastava, R., You, L., Summers, J., Yin, J., 2002. Stochastic vs. deterministic modeling of intracellular viral kinetics. J. Theor. Biol. 218, 309–321. CrossRefMathSciNetGoogle Scholar
  30. van Kampen, N.G., 1992. Stochastic Processes in Physics and Chemistry, 2nd edn. Elsevier, Amsterdam. Google Scholar
  31. Villarreal, L.P., Breindl, M., Holland, J.J., 1976. Determination of molar ratios of vesicular stomatitis virus induced RNA species in BHK21 cells. Biochemistry 15(8), 1663–1667. CrossRefGoogle Scholar
  32. Weinberger, L.S., Burnett, J.C., Toettcher, J.E., Arkin, A.P., Schaffer, D.V., 2005. Stochastic gene expression in a lentiviral positive-feedback loop: HIV-1 Tat fluctuations drive phenotypic diversity. Cell 122(2), 169–182. CrossRefGoogle Scholar
  33. Werner, M., 1991. Kinetic and thermodynamic characterization of the interaction between Q beta-replicase and template RNA molecules. Biochemistry 30(24), 5832–5838. CrossRefGoogle Scholar
  34. Zhu, Y., Yongky, A., Yin, J., 2009. Growth of an RNA virus in single cells reveals a broad fitness distribution. Virology 385(1), 39–46. CrossRefGoogle Scholar

Copyright information

© Society for Mathematical Biology 2009

Authors and Affiliations

  • Sebastian C. Hensel
    • 1
  • James B. Rawlings
    • 1
  • John Yin
    • 1
  1. 1.Department of Chemical and Biological EngineeringUniversity of Wisconsin-MadisonMadisonUSA

Personalised recommendations