zbMATH — the first resource for mathematics

Formulas for intrinsic noise evaluation in oscillatory genetic networks. (English) Zbl 1410.92042
Summary: The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.
92C42 Systems biology, networks
92C40 Biochemistry, molecular biology
Full Text: DOI
[1] Anou-Kandil, H.; Freiling, G.; Ionescu, V.; Jank, G., Matrix Riccati equations in control and systems theory, (2003), Birkhauser Verlag
[2] Chabot, J.R.; Pedraza, J.M.; Luitel, P.; van Oudenaarden, A., Stochastic gene expression out-of-steady-state in the cyanobacterial Circadian clock, Nature, 450, 1249-1252, (2007)
[3] Coddington, E.A.; Levinson, N., Theory of ordinary differential equations, (1955), McGraw-Hill, (Paperback, McGraw-Hill Education, 1984) · Zbl 0064.33002
[4] Elf, J.; Ehrenberg, M., Fast evaluation of fluctuations in biochemical networks with the linear noise approximation, Genome res., 13, 2475-2484, (2003)
[5] Gillespie, D.T., Exact stochastic simulations of coupled chemical reactions, J. phys. chem., 81, 25, 2340-2361, (1997)
[6] Gillespie, D.T., Approximate accelerated stochastic simulation of chemically reacting systems, J. chem. phys., 115, 1716-1733, (2001)
[7] Goldbeter, A., A model for Circadian oscillations in the drosophila period protein (PER), Proc. R. soc. London, B261, 319-324, (1995)
[8] Gonze, D.; Halloy, L.; Goldbeter, A., Robustness of Circadian rhythms with respect to molecular noise, Proc. natl. acad. sci. USA, 99, 673-678, (2002)
[9] Gonze, D.; Goldbeter, A., Circadian rhythms and molecular noise, Chaos 16, 026110, 1-11, (2006) · Zbl 1152.92312
[10] Kepler, T.B.; Elston, T.C., Stochasticity in transcriptional regulation: origins, consequences, and mathematical representations, Biophys. J., 81, 3116-3136, (2001)
[11] Lestas, I.; Paulsson, J.; Ross, N.E.; Vinnicombe, G., Noise in gene regulatory networks, IEEE trans. autom. control, special issue syst. biol., 189-200, (2008) · Zbl 1366.92045
[12] Miura, S.; Shimokawa, T.; Nomura, T., Stochastic simulations on a model of Circadian rhythm generation, Biosystems, 93, 133-140, (2008)
[13] Montagnier, P.; Spiter, R.J.; Angeles, J., The control of linear time-periodic systems using floquet – lyapunov theory, Int. J. control, 77, 5, 472-490, (2004) · Zbl 1061.93050
[14] Ogawa, K.; Takekawa, N.; Hinohara, T.; Uchida, K.; Shibata, S., On robust stability and sensitivity of Circadian rhythms, Asian J. control, 8, 3, 281-289, (2006)
[15] Paulsson, J., Models of stochastic gene expression, Phys. life rev., 2, 157-175, (2005)
[16] Scott, M.; Ingalls, B.; Kaern, M., Estimations of intrinsic and extrinsic noise in models of nonlinear genetic networks, Chaos 16, 026107, 1-15, (2006) · Zbl 1152.92339
[17] Swain, P.S.; Elowitz, M.B.; Siggia, E.D., Intrinsic and extrinsic contributions to stochasticity in gene expression, Proc. natl. acad. sci. USA, 99, 12795-12800, (2002)
[18] Thattai, M.; van Oudenaarden, A., Intrinsic noise in gene regulatory networks, Proc. natl. acad. sci. USA, 98, 3614-3619, (2001)
[19] Tian, T.; Burrage, K., Binominal leap methods orsimuling chemical kinetics, J. chem. phys., 121, 1035-1036, (2004)
[20] Tomioka, R.; Kimura, H.; Kobayashi, T.J.; Aihara, K., Multivariate analysis of noise in genetic regulatory networks, J. theor. biol., 229, 501-521, (2004)
[21] Tomita, K.; Ohta, T.; Tomita, H., Irreversible circulation and orbital revolution—hard mode instability in far-from-equilibrium situation, Prog. theor. phys., 52, 1744-1765, (1974)
[22] van Kampen, N.G., Stochastic processes in physics and chemistry, (1997), Elsevier Science The Netherlands · Zbl 0974.60020
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.