Abstract
Regulatory gene networks contain generic modules, like those involving feedback loops, which are essential for the regulation of many biological functions (Guido et al. in Nature 439:856–860, 2006). We consider a class of self-regulated genes which are the building blocks of many regulatory gene networks, and study the steady-state distribution of the associated Gillespie algorithm by providing efficient numerical algorithms. We also study a regulatory gene network of interest in gene therapy, using mean-field models with time delays. Convergence of the related time-nonhomogeneous Markov chain is established for a class of linear catalytic networks with feedback loops.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Abramov, V., Liptser, R., 2004. On existence of limiting distributions for time-nonhomogeneous countable Markov processes. Queueing Syst. 46, 353–361.
Bellman, R., Cooke, K.L., 1963. Differential-Difference Equations. Academic Press, New York.
Billingsley, P., 1968. Convergence of Probability Measures, vol. XII. New York, Wiley. 253 p.
Bratsun, D., Volfson, D., Tsimring, L., Hasty, J., 2005. Delayed-induced stochastic oscillations in gene regulation. Proc. Natl. Acad. Sci. U.S.A. 102(41), 14593–14598.
Burrage, K., Tian, T., Burrage, P., 2004. A multi-scale approach for simulating chemical reaction systems. Prog. Biophys. Mol. Biol. 85, 217.
Cao, Y., Li, H., Petzold, L., 2004. Efficient formulation of the stochastic simulation algorithm for chemically reacting systems. J. Chem. Phys. 121, 4059–4067.
Cao, Y., Gillespie, D., Petzold, L., 2005. The slow scale stochastic simulation algorithm. J. Chem. Phys. 122, 014116.
Coppel, L.W.A., 1965. Stability and Asymptotic Behavior in Differential Equations. Heath Mathematical Monographs. Boston.
Darvey, I.G., Ninham, B.W., Staff, P.J., 1966. Stochastic models for second-order chemical reaction kinetics. The equilibrium state. J. Chem. Phys. 45, 2145.
Dill, K., Bromberg, S., 2003. Molecular Driving Forces. Garland, New York.
Fournier, T., Gabriel, J.-P., Mazza, C., Pasquier, J., Galbete, J.L., Mermod, N., 2007. Steady-state expression of self-regulated genes. Bioinformatics 23(23), 3185–3192.
Gabriel, J.-P., Hanisch, H., Hirsch, W.M., 1981. Dynamic equilibria of helminthic infections? In: Chapman, D.G., Gallucci, V.F. (Eds.), Quantitative Population Dynamics, vol. 13, pp. 83–104. International Cooperative Publishing House, Fairland. Stat. Ecol. Ser.
Gadgil, C., Lee, C., Othmer, H., 2005. A stochastic analysis of first-order reaction networks. Bull. Math. Biol. 67, 901–946.
Gillespie, D., 1977. Exact stochastic simulation of coupled chemical reactions. J. Chem. Phys. 81, 2340–2361.
Gillespie, D., 2001. Approximate accelerated stochastic simulation of chemically reacting systems. J. Chem. Phys. 115, 1716–1733.
Goutsias, J., 2005. Quasi-equilibrium approximation of fast reactions kinetics in stochastic biochemical systems. J. Chem. Phys. 122, 184102.
Goutsias, J., Kim, S., 2006. Stochastic transcriptional regulatory systems with time delays: a mean-field approximation. J. Comput. Biol. 13, 1049–1076.
Guido, N., Wang, X., Adalsteinsson, D., McMillen, D., Hasty, J., Cantor, C., Elston, T., Collins, J., 2006. A bottom-up approach to gene regulation. Nature 439, 856–860.
Halanay, A., 1966. Differential Equations: Stability, Oscillations, Time Lags. Academic Press, New York.
Hirsch, W.M., Hanisch, H., Gabriel, J.-P., 1985. Differential equation models of some parasitic infections: methods for the study of asymptotic behavior. Commun. Pure Appl. Math. 38, 733–753.
Hornos, J., Schultz, D., Innocentini, G., Wang, J., Walczak, A., Onuchic, J., Wolynes, P., 2005. Self-regulating genes: An exact solution. Phys. Rev. E 72, 051907-1–051907-5.
Imhof, M., Chatellard, P., Mermod, N., 2000. A regulatory network for the efficient control of transgene expression. J. Gene Med. 2, 107–116.
Karlebach, G., Shamir, R., 2008. Modelling and analysis of gene regulatory networks. Nat. Rev. Mol. Cell Biol. 9, 770–780.
Kepler, T., Elston, T., 2001. Stochasticity in Transcriptional Regulation: Origins, Consequences and Mathematical Representations. Biophys. J. 81(1), 3116–3136.
Lipan, O., Wong, W.H., 2005. The use of oscillatory signals in the study of genetic networks. Proc. Natl. Acad. Sci. U.S.A. 102, 7063–7068.
Nasell, I., Hirsch, W.M., 1972. A mathematical model of some helminthic infections. Commun. Pure Appl. Math. 25, 459–477.
Nasell, I., Hirsch, W.M., 1973. The transmission dynamics of schistosomiasis. Commun. Pure Appl. Math. 26, 395–453.
Randrianarivony, A., 1997. Fractions continues, q-nombres de Catalan et q-polynômes de Genocchi. Eur. J. Comb. 18, 75–92.
Paulsson, J., 2005. Models of stochastic gene expression. Phys. Life Rev. 2, 157–175.
Peccoud, J., Ycart, B., 1995. Markovian modelling of gene product synthesis. Theor. Popul. Biol. 48(2), 222–234.
Pedraza, J., van Oudenaarden, A., 2005. Noise propagation in gene networks. Science 307, 1965–1969.
Visco, P., Allen, R., Ewans, M., 2008a. Exact Solution of a model DNA-inversion switch with orientational control. Phys. Rev. Lett. 101, 118104.
Visco, P., Allen, R., Ewans, M., 2008b. Statistical physics of a model binary switch with linear feedback. arXiv:0812.3867.
Zhang, Q., Yin, G., 1997. Structural properties of Markov chains with weak and strong interactions. Stoch. Proc. Appl. 70, 181–197.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Fournier, T., Gabriel, JP., Mazza, C. et al. Stochastic Models and Numerical Algorithms for a Class of Regulatory Gene Networks. Bull. Math. Biol. 71, 1394–1431 (2009). https://doi.org/10.1007/s11538-009-9407-9
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11538-009-9407-9