zbMATH — the first resource for mathematics

Modeling stochasticity in gene regulation: Characterization in the terms of the underlying distribution function. (English) Zbl 1298.92068
Summary: Intrinsic stochasticity plays an essential role in gene regulation because of a small number of involved molecules of DNA, mRNA and protein of a given species. To better understand this phenomenon, small gene regulatory systems are mathematically modeled as systems of coupled chemical reactions, but the existing exact description utilizing a Chapman-Kolmogorov equation or simulation algorithms is limited and inefficient. The present work considers a much more efficient yet accurate modeling approach, which allows analyzing stochasticity in the system in the terms of the underlying distribution function. We depart from the analysis of a single gene regulatory module to find that the mRNA and protein variance is decomposable into additive terms resulting from respective sources of stochasticity. This variance decomposition is asserted by constructing two approximations to the exact stochastic description: First, the continuous approximation, which considers only the stochasticity due to the intermittent gene activity. Second, the mixed approximation, which in addition attributes stochasticity to the mRNA transcription/decay process. Considered approximations yield systems of first order partial differential equations for the underlying distribution function, which can be efficiently solved using developed numerical methods. Single cell simulations and numerical two-dimensional mRNA-protein stationary distribution functions are presented to confirm accuracy of approximating models.

92D10 Genetics and epigenetics
Full Text: DOI
[1] Ackers, G.K., Johnsson, D., Shea, M., 1982. Quantitative model for gene regulation by phage {\(\lambda\)} repressor. PNAS 79, 1129–1133.
[2] Arkin, A., Ross, J., McAdams, H.H., 1998. Stochastic kinetic analysis of develometal pathway bifurcation in phage {\(\lambda\)}-infected escherichia coli cells. Genetics 149, 1633–1648.
[3] Barken, D., Wang, C.J., Kærns, J., Cheong, R., Hoffman, A., Levchenko, A., 2005. Comment on ”Oscillations in NF-{\(\kappa\)}B signaling control the dynamics of gene expression.” Science 308, 52a.
[4] Blake, W.J., Kærn, M., Cantor, C.R., Collins, J.J., 2003. Noise in eukaryotic gene expression. Nature 422, 633–637.
[5] Bobrowski, A., 2006. Degenerate convergence of semigroups related to a model of stochastic gene expression. Semigroup forum, in press. · Zbl 1114.47068
[6] Cook, D.L., Gerber, A.N., Tapscott, S.J., 1998. Modeling stochastic gene expression: Implications for haplosufficiency. PNAS 95, 15641–15646.
[7] Femino, A.M., Fay, F.S., Fogarty, K., Singer, R.H., 1998. Visualization of single RNA transcripts in situ. Science 280, 585–590.
[8] Gardiner, C.W., 2003. Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag, New York. · Zbl 0862.60050
[9] Gillespie, D.T., 1976. A general method for numerically simulating the stochatic time evolution of coupled chemical reactions. J. Comp. Phys. 22, 403–434.
[10] Gillespie, D.T., 1977. Exact stochastic simulation of coupled chemical reactions. J. Phys. Chem. 81, 2340–2361.
[11] Gygi, S.P., Rochon, Y.B., Franza, R., Aebersold, R., 2003. Correlation between protein and mRNA abundance in yeast. Mol. Cell. Biol. 19, 1720–1730.
[12] Hat, B., Paszek, P., Piechor, K., Kimmel, M., Lipniacki, T., 2006. How number of alles influences gene expression. J. Stat. Phys., under revision. · Zbl 1115.92028
[13] Kierzek, A.M., Zaim, J., Zilemnikiewicz, P., 2001. The effect of transcription and translation initiation frequencies on the stochastic fluctuations in prokaryotic gene expression, J. Biol. Chem. 276, 8165–8172.
[14] Kepler, T.B., Elston, T.C., 2001. Stochasticity in transcriptional regulation: Orgins, consequences, and mathematical representations. Biophys. J. 81, 3116–3136.
[15] Ko, M.S.H., 1991. A stochastic model for gene induction. J. Theor. Biol. 153, 181–194.
[16] Ko, M.S.H., 1992. Induction mechanism of a single gene molecule: Stochastic or deterministic? Bioassays 14, 341–346.
[17] Lipniacki, T., Paszek, P., Brasier, A.R., Luxon, B., Kimmel, M., 2004. Mathematical model of NF-{\(\kappa\)}B module. J. Theor. Biol. 228, 195–215.
[18] Lipniacki, T., Paszek, P., Marciniak-Czochra, A., Brasier, A.R., Kimmel, M., 2006a. Transcriptional stochasticity in gene expression. J. Theor. Biol. 238, 348–367.
[19] Lipniacki, T., Paszek, P., Brasier, A. R., Luxon, B., Kimmel, M., 2006b. Stochastic regulation in early immune response. Biophys. J. 90, 725–742.
[20] Louis, M., Holm, L., Sanchez, L., Kaufman, M., 2003. A theoretical model for the regulation of sex-lethal, a gene that controls sex determination and dosage compensation in Drosophila melanogaster. Genetics 165, 1355–1384.
[21] McAdams, H.H., Shapiro, L., 1995. Circuit simulation of genetic networks. Science 269, 650–656.
[22] McAdams, H.H., Arkin, A., 1997. Stochastic mechanisms in gene expression. PNAS 94, 814–819.
[23] McKane, A.J., Newman, T.J., 2005. Predator-prey cycles from resonant amplification of demographic stochasticity. Phys. Rev. Lett. 94, 218101.
[24] Nelson, D.E., Ihekwaba, A.E.C., Elliot, M., Johnson, J.R., Gibney, C.A., Foreman, B.E., Nelson, G., See, V., Horton, C.A., Spiller, D.G., Edwards, S.W., McDowell, H.P., Unitt, J.F., Sullivan, E., Grimley, R., Benson, N., Broomhead, D., Kell, D.B., White, M.R.H., 2004. Oscillations in NF-{\(\kappa\)}B signaling control the dynamics of gene expression. Science 306, 704–708.
[25] Nelson, D.E., Horton, C.A., See, V., Johnson, J.R., Nelson, G., Spiller, D.G., Kell, D.B., White, M.R.H., 2005. Response to comment on ”Oscillations in NF-{\(\kappa\)}B signaling control the dynamics of gene expression.” Science 308, 52b.
[26] Paszek, P., Lipniacki, T., Brasier, A.R., Tian, B., Nowak, D.E., Kimmel, M., 2005. Stochastic effects of multiple regulators on expression profiles in eukaryotes. J. Theor. Biol. 233, 423–433.
[27] Paulsson, J., 2004. Summing up noise in gene networks. Nature 427, 415–418.
[28] Percus, J.K., 2005. Small population effects in stochastic population dynamics. Bull. Math. Biol. 67, 1173–1194. · Zbl 1334.92362
[29] Pirone, J.R., Elston, T.C., 2004. Fluctuations in transcription factor binding can explain the graded and binary responses observed in inducible gene expression. J. Theor. Biol. 226, 111–121.
[30] Raser, J.M., O’Shea, E.K., 2004. Control of stochasticity in eukaryotic gene expression. Science 304, 1811–1814.
[31] Simpson, M.L., Cox, C.D., Sayler, G.S., 2004. Frequency domain chemical Langevin analysis of stochasticity in gene transcriptional regulation. J. Theor. Biol. 229, 383–394.
[32] Stirland, J.A., Seymour, Z.C., Windeatt, S., Norris, A.J., Stanley, P., Castro, M.G., Loudon, A.S.I., White, M.R.H., Davis, J.R.E., 2003. Real-time imaging of gene promoter activity using an adenoviral reporter construct demonstrates transcriptional dynamics in normal anterior pituary cells. J. Endocrinol. 178, 61–69.
[33] Takasuka, N., White, M.R.H., Wood, C.D., Robertson, W.R., Davis, J.R.E., 1998. Dynamic changes in prolactin promoter activation in individual living lactotrophic cells. Endocrinology 139, 1361–1368.
[34] Tao, Y., 2004. Intrinsic and external noise in an auto-regulatory genetic network. J. Theor. Biol. 229, 147–156.
[35] Tao, Y., 2004b. Intrinsic noise, gene regulation and steady-state statistics in a two gene network. J. Theor. Biol. 231, 563–568.
[36] Thattai, M., Oudenaarden, A., 2001. Intrinsic noise in gene regulatory networks. PNAS 98, 8614–8619.
[37] Tomioka, R., Kimura, H., Kobayashi, T., Aihira, K., 2004. Multivariate analysis of noise in genetic regulatory networks. J. Theor. Biol. 229, 501–521.
[38] Washburn, M.P., Koller, A., Oshiro, G., Ulaszek, R.R., Plouffe, D., Deciu, C., Winzeler, E., Yates, J.R., III, 2003. Protein pathway and comlex clustering of correlated mRNA and protein expression analyses in Saccharomyces cerevisiae. PNAS 100, 3107–3112.
[39] White, R.J., 2001. Gene Transcription: Mechanisms and Control. Blackwell Science Ltd., Oxford, UK.
[40] Van Kampen, N.G., 2004. Stochastic Processes in Physics and Chemistry. North-Holland. · Zbl 0511.60038
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.