Advertisement

Abstract

An important behavioral pattern that can be witnessed in many systems is periodic re-occurrence. For example, most living organisms that we know are governed by a 24 hours rhythm that determines whether they are awake or not. On a larger scale, also whole population numbers of organisms fluctuate in a cyclic manner as in predator-prey relationships. When treating such systems in a deterministic way, i.e., assuming that stochastic effects are negligible, the analysis is a well-studied topic. But if those effects play an important role, recent publications suggest that at least a part of the system should be modeled stochastically. However, in that case, one quickly realizes that characterizing and defining oscillatory behavior is not a straightforward task, which can be solved once and for all. Moreover, efficiently checking whether a given system oscillates or not and if so determining the amplitude of the fluctuations and the time in-between is intricate. This paper shall give an overview of the existing literature on different modeling formalisms for oscillatory systems, definitions of oscillatory behavior, and the respective analysis methods.

Keywords

Model Check Formal Method Circadian Clock Deterministic Model Oscillatory Behavior 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    de Alfaro, L., Roy, P.: Magnifying-lens abstraction for Markov decision processes. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 325–338. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  2. 2.
    Alfonsi, A., Cancès, E., Turinici, G., Di Ventura, B., Huisinga, W.: Exact simulation of hybrid stochastic and deterministic models for biochemical systems. Research Report RR-5435, INRIA (2004)Google Scholar
  3. 3.
    Alfonsi, A., Cancès, E., Turinici, G., Ventura, B.D., Huisinga, W.: Adaptive simulation of hybrid stochastic and deterministic models for biochemical systems. ESAIM: Proc., 14:1–14:13 (2005)Google Scholar
  4. 4.
    Alur, R., Dill, D.L.: A theory of timed automata. Theoretical Computer Science 126(2), 183–235 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Aris, R.: Prolegomena to the rational analysis of systems of chemical reactions. Archive for Rational Mechanics and Analysis 19, 81–99 (1965)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Arkin, A., Ross, J., McAdams, H.H.: Stochastic kinetic analysis of developmental pathway bifurcation in phage lambda-infected escherichia coli cells. Genetics 149(4), 1633–1648 (1998)Google Scholar
  7. 7.
    Arns, M., Buchholz, P., Panchenko, A.: On the numerical analysis of inhomogeneous continuous-time Markov chains. INFORMS Journal on Computing 22(3), 416–432 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Baier, C., Haverkort, B., Hermanns, H., Katoen, J.-P.: Model-checking algorithms for continuous-time Markov chains. IEEE Transactions on Software Engineering 29(6), 524–541 (2003)CrossRefzbMATHGoogle Scholar
  9. 9.
    Baier, C., Katoen, J.-P.: Principles of Model Checking. The MIT Press (2008)Google Scholar
  10. 10.
    Ball, K., Kurtz, T.G., Popovic, L., Rempala, G.: Asymptotic analysis of multiscale approximations to reaction networks. The Annals of Applied Probability 16(4), 1925–1961 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ballarini, P., Guerriero, M.L.: Query-based verification of qualitative trends and oscillations in biochemical systems. Theoretical Computer Science 411(20), 2019–2036 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Ballarini, P., Mardare, R., Mura, I.: Analysing biochemical oscillation through probabilistic model checking. ENTCS 229(1), 3–19 (2009)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Barkai, N., Leibler, S.: Biological rhythms: Circadian clocks limited by noise. Nature 403, 267–268 (2000)Google Scholar
  14. 14.
    Bartocci, E., Corradini, F., Merelli, E., Tesei, L.: Model checking biological oscillators. ENTCS 229(1), 41–58 (2009)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Burrage, K., Hegland, M., Macnamara, S., Sidje, R.: A Krylov-based finite state projection algorithm for solving the chemical master equation arising in the discrete modeling of biological systems. In: Langville, A.N., Stewart, W.J. (eds.) Markov Anniversary Meeting 2006: An International Conference to Celebrate the 150th Anniversary of the Birth of A. A. Markov, pp. 21–38. Boston Books, Charleston (2006)Google Scholar
  16. 16.
    Burrage, K., Tian, T.: Poisson Runge-Kutta methods for chemical reaction systems. In: Lu, Y., Sun, W., Tang, T. (eds.) Advances in Scientific Computing and Applications, pp. 82–96. Science Press, Beijing (2004)Google Scholar
  17. 17.
    Burrage, K., Tian, T., Burrage, P.: A multi-scaled approach for simulating chemical reaction systems. Progress in Biophysics and Molecular Biology 85(2-3), 217–234 (2004)CrossRefGoogle Scholar
  18. 18.
    Cao, Y., Gillespie, D.T., Petzold, L.R.: The slow-scale stochastic simulation algorithm. The Journal of Chemical Physics 122(1), 014116 (2005)Google Scholar
  19. 19.
    Cardelli, L.: Artificial biochemistry. Technical report, Microsoft Research (2006)Google Scholar
  20. 20.
    Cardelli, L.: Artificial biochemistry. In: Algorithmic Bioproceses. LNCS. Springer (2008)Google Scholar
  21. 21.
    Casagrande, A., Mysore, V., Piazza, C., Mishra, B.: Independent dynamics hybrid automata in systems biology. In: Proceedings of the First International Conference on Algebraic Biology, pp. 61–73. Universal Academy Press, Tokyo (2005)Google Scholar
  22. 22.
    Chabrier-Rivier, N., Chiaverini, M., Danos, V., Fages, F., Schächter, V.: Modeling and querying biomolecular interaction networks. Theoretical Computer Science 325, 25–44 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Chellaboina, V., Bhat, S., Haddad, W., Bernstein, D.: Modeling and analysis of mass-action kinetics. IEEE Control Systems Magazine 29(4), 60–78 (2009)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Clarke, E.M., Emerson, E.A.: Design and synthesis of synchronization skeletons using branching-time temporal logic. In: Logic of Programs, pp. 52–71. Springer, London (1982)CrossRefGoogle Scholar
  25. 25.
    Cloth, L., Katoen, J.-P., Khattri, M., Pulungan, R.: Model-checking Markov reward models with impulse rewards. In: DSN, Yokohama (2005)Google Scholar
  26. 26.
    Crudu, A., Debussche, A., Radulescu, O.: Hybrid stochastic simplifications for multiscale gene networks. BMC Systems Biology 3(1), 89 (2009)CrossRefGoogle Scholar
  27. 27.
    D’Argenio, P.R., Jeannet, B., Jensen, H.E., Larsen, K.G.: Reachability analysis of probabilistic systems by successive refinements. In: de Luca, L., Gilmore, S. (eds.) PAPM-PROBMIV 2001. LNCS, vol. 2165, pp. 39–56. Springer, Heidelberg (2001)Google Scholar
  28. 28.
    Dayar, T., Mikeev, L., Wolf, V.: On the numerical analysis of stochastic Lotka-Volterra models. In: IMCSIT, pp. 289–296 (2010)Google Scholar
  29. 29.
    de Jong, H.: Modeling and simulation of genetic regulatory systems: A literature review. Journal of Computational Biology 9(1), 67–103 (2002)CrossRefGoogle Scholar
  30. 30.
    Didier, F., Henzinger, T.A., Mateescu, M., Wolf, V.: Fast adaptive uniformization of the chemical master equation. In: Proc., HIBI 2009, pp. 118–127. IEEE Computer Society, Washington, DC (2009)Google Scholar
  31. 31.
    Elowitz, M.B.: Stochastic gene expression in a single cell. Science 297(5584), 1183–1186 (2002)CrossRefGoogle Scholar
  32. 32.
    Elowitz, M.B., Leibler, S.: A synthetic oscillatory network of transcriptional regulators. Nature 403(6767), 335–338 (2000)CrossRefGoogle Scholar
  33. 33.
    Ferrell, J.E., Tsai, T.Y.-C., Yang, Q.: Modeling the cell cycle: Why do certain circuits oscillate? Cell 144(6), 874–885 (2011)CrossRefGoogle Scholar
  34. 34.
    Gardner, T.S., Cantor, C.R., Collins, J.J.: Construction of a genetic toggle switch in Escherichia coli. Nature 403(6767), 339–342 (2000)CrossRefGoogle Scholar
  35. 35.
    Gibson, M.A., Bruck, J.: Efficient exact stochastic simulation of chemical systems with many species and many channels. The Journal of Physical Chemistry A 104(9), 1876–1889 (2000)CrossRefGoogle Scholar
  36. 36.
    Gillespie, D.T.: A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Journal of Computational Physics 22(4), 403–434 (1976)MathSciNetCrossRefGoogle Scholar
  37. 37.
    Gillespie, D.T.: Exact stochastic simulation of coupled chemical reactions. The Journal of Physical Chemistry 81(25), 2340–2361 (1977)CrossRefGoogle Scholar
  38. 38.
    Gillespie, D.T.: A rigorous derivation of the chemical master equation. Physica A 188, 404–425 (1992)CrossRefGoogle Scholar
  39. 39.
    Gillespie, D.T.: Approximate accelerated stochastic simulation of chemically reacting systems. The Journal of Chemical Physics 115(4), 1716 (2001)CrossRefGoogle Scholar
  40. 40.
    Glass, L., Beuter, A., Larocque, D.: Time delays, oscillations, and chaos in physiological control systems. Mathematical Biosciences 90(1-2), 111–125 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Goldbeter, A.: A model for circadian oscillations in the drosophila period protein (PER). Proceedings of the Royal Society B: Biological Sciences 261(1362), 319–324 (1995)CrossRefGoogle Scholar
  42. 42.
    Goldbeter, A.: Computational approaches to cellular rhythms. Nature 420(6912), 238–245 (2002)CrossRefGoogle Scholar
  43. 43.
    Gonze, D., Halloy, J., Goldbeter, A.: Deterministic versus stochastic models for circadian rhythms. Journal of Biological Physics 28(4), 637–653 (2002)CrossRefGoogle Scholar
  44. 44.
    Grassmann, W.: Finding transient solutions in Markovian event systems through randomization. In: The First International Conference on the Numerical Solution of Markov Chains, pp. 375–385 (1990)Google Scholar
  45. 45.
    Griffith, M., Courtney, T., Peccoud, J., Sanders, W.H.: Dynamic partitioning for hybrid simulation of the bistable HIV-1 transactivation network. Bioinformatics 22(22), 2782–2789 (2006)CrossRefGoogle Scholar
  46. 46.
    Gross, D., Miller, D.R.: The randomization technique as a modeling tool and solution procedure for transient Markov processes. Operations Research 32(2), 343–361 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  47. 47.
    Guerriero, M.L., Heath, J.K.: Computational modeling of biological pathways by executable biology. Methods in Enzymology 487, 217–251 (2011)CrossRefGoogle Scholar
  48. 48.
    Henzinger, T.A., Mateescu, M., Wolf, V.: Sliding window abstraction for infinite Markov chains. In: Bouajjani, A., Maler, O. (eds.) CAV 2009. LNCS, vol. 5643, pp. 337–352. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  49. 49.
    Henzinger, T.A., Mikeev, L., Mateescu, M., Wolf, V.: Hybrid numerical solution of the chemical master equation. In: Proc., CMSB 2010, pp. 55–65. ACM, New York (2010)Google Scholar
  50. 50.
    Higgins, J.: The theory of oscillating reactions. Industrial and Engineering Chemistry 59, 18–62 (1967)CrossRefGoogle Scholar
  51. 51.
    Horn, F., Jackson, R.: General mass action kinetics. ARMA 47, 81–116 (1972)MathSciNetCrossRefGoogle Scholar
  52. 52.
    Horton, G., Kulkarni, V.G., Nicol, D.M., Trivedi, K.S.: Fluid stochastic Petri nets: Theory, applications, and solution techniques. European Journal of Operational Research 105(1), 184–201 (1998)CrossRefzbMATHGoogle Scholar
  53. 53.
    Lohmueller, J., et al.: Progress toward construction and modelling of a tri-stable toggle switch in e. coli. IET Synthetic Biology 1(1.2), 25–28 (2007)CrossRefGoogle Scholar
  54. 54.
    Jahnke, T., Huisinga, W.: Solving the chemical master equation for monomolecular reaction systems analytically. Journal of Mathematical Biology 54(1), 1–26 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Jensen, A.: Markoff chains as an aid in the study of Markoff processes. Scandinavian Actuarial Journal 1953(suppl. 1), 87–91 (1953)Google Scholar
  56. 56.
    Kampen, N.V.: Stochastic processes in physics and chemistry. North Holland (2007)Google Scholar
  57. 57.
    Katoen, J.-P., Klink, D., Leucker, M., Wolf, V.: Three-valued abstraction for continuous-time Markov chains. In: Damm, W., Hermanns, H. (eds.) CAV 2007. LNCS, vol. 4590, pp. 311–324. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  58. 58.
    Kolmogoroff, A.: Über die analytischen Methoden in der Wahrscheinlichkeitsrechnung. Mathematische Annalen 104, 415–458 (1931)MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Kowalewski, S., Engell, S., Stursberg, O.: On the generation of timed discrete approximations for continuous systems. MCMDS 6(1), 51–70 (2000)CrossRefzbMATHGoogle Scholar
  60. 60.
    Krishna, S.: Minimal model of spiky oscillations in NF-κb signaling. PNAS 103(29), 10840–10845 (2006)CrossRefGoogle Scholar
  61. 61.
    Kummer, U., Krajnc, B., Pahle, J., Green, A.K., Dixon, C.J., Marhl, M.: Transition from stochastic to deterministic behavior in calcium oscillations. Biophysical Journal 89(3), 1603–1611 (2005)CrossRefGoogle Scholar
  62. 62.
    Kurtz, T.G.: The Relationship between Stochastic and Deterministic Models for Chemical Reactions. The Journal of Chemical Physics 57(7), 2976–2978 (1972)CrossRefGoogle Scholar
  63. 63.
    Kwiatkowska, M., Norman, G., Pacheco, A.: Model checking expected time and expected reward formulae with random time bounds. In: Proc. 2nd Euro-Japanese Workshop on Stochastic Risk Modelling for Finance, Insurance, Production and Reliability (2002)Google Scholar
  64. 64.
    Kwiatkowska, M., Norman, G., Parker, D.: Game-based abstraction for Markov decision processes. In: Proc. QEST, pp. 157–166. IEEE CS Press (2006)Google Scholar
  65. 65.
    Kwiatkowska, M., Norman, G., Parker, D.: PRISM 4.0: Verification of probabilistic real-time systems. In: Gopalakrishnan, G., Qadeer, S. (eds.) CAV 2011. LNCS, vol. 6806, pp. 585–591. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  66. 66.
    Lapin, M., Mikeev, L., Wolf, V.: SHAVE – Stochastic hybrid analysis of Markov population models. In: Proc. HSCC. ACM, New York (2011)Google Scholar
  67. 67.
    Leloup, J.C.: Toward a detailed computational model for the mammalian circadian clock. PNAS 100(12), 7051–7056 (2003)CrossRefGoogle Scholar
  68. 68.
    Leloup, J.C., Gonze, D., Goldbeter, A.: Limit cycle models for circadian rhythms based on transcriptional regulation in drosophila and neurospora. Journal of Biological Rhythms 14(6), 433–448 (1999)CrossRefGoogle Scholar
  69. 69.
    Lewis, J.: Autoinhibition with transcriptional delay: A simple mechanism for the zebrafish somitogenesis oscillator. Current Biology 13(16), 1398–1408 (2003)CrossRefGoogle Scholar
  70. 70.
    Lotka, A.: Elements of mathematical biology. Dover Publications (1956); Reprinted from Lotka, A.J. Elements of physical biology (1924)Google Scholar
  71. 71.
    Maler, O., Batt, G.: Approximating continuous systems by timed automata. In: Fisher, J. (ed.) FMSB 2008. LNCS (LNBI), vol. 5054, pp. 77–89. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  72. 72.
    Martiel, J.-L., Goldbeter, A.: A model based on receptor desensitization for cyclic AMP signaling in dictyostelium cells. Biophysical Journal 52(5), 807–828 (1987)CrossRefzbMATHGoogle Scholar
  73. 73.
    MATLAB. Version 7.11.0.584 (R2010b). The MathWorks Inc., Natick, Massachusetts (2010)Google Scholar
  74. 74.
    McAdams, H.H., Arkin, A.: Stochastic mechanisms in gene expression. Proceedings of the National Academy of Sciences of the United States of America 94(3), 814–819 (1997)CrossRefGoogle Scholar
  75. 75.
    Menz, S., Latorre, J.C., Schtte, C., Huisinga, W.: Hybrid stochastic–deterministic solution of the chemical master equation. MMS 10(4), 1232–1262 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  76. 76.
    Meyer, K.M., Wiegand, K., Ward, D., Moustakas, A.: SATCHMO: A spatial simulation model of growth, competition, and mortality in cycling savanna patches. Ecological Modelling 209(24), 377–391 (2007)CrossRefGoogle Scholar
  77. 77.
    Mikeev, L., Neuhäußer, M.R., Spieler, D., Wolf, V.: On-the-fly verification and optimization of DTA-properties for large Markov chains. FMSD, 1–25 (2012)Google Scholar
  78. 78.
    Mincheva, M.: Oscillations in non-mass action kinetics models of biochemical reaction networks arising from pairs of subnetworks. Journal of Mathematical Chemistry 50(5), 1111–1125 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  79. 79.
    van Moorsel, A.P.A., Wolter, K.: Numerical solution of non-homogeneous Markov processes through uniformization. In: Proceedings of the 12th European Simulation Multiconference on Simulation - Past, Present and Future, pp. 710–717. SCS Europe (1998)Google Scholar
  80. 80.
    Munsky, B., Khammash, M.: The finite state projection algorithm for the solution of the chemical master equation. The Journal of Chemical Physics 124(4), 044104 (2006)Google Scholar
  81. 81.
    Murray, J.D.: Mathematical Biology. Springer, New York (1993)CrossRefzbMATHGoogle Scholar
  82. 82.
    Nakano, S., Yamaguchi, S.: Two modeling methods for signaling pathways with multiple signals using uppaal. Proc. BioPPN, 87–101 (2011)Google Scholar
  83. 83.
    Parker, D.: PRISM Tutorial - Circadian Clock, http://www.prismmodelchecker.org/tutorial/circadian.php
  84. 84.
    Piazza, C., Antoniotti, M., Mysore, V., Policriti, A., Winkler, F., Mishra, B.: Algorithmic algebraic model checking I: Challenges from systems biology. In: Etessami, K., Rajamani, S.K. (eds.) CAV 2005. LNCS, vol. 3576, pp. 5–19. Springer, Heidelberg (2005)CrossRefGoogle Scholar
  85. 85.
    Pnueli, A.: The temporal logic of programs. In: Proc., SFCS, pp. 46–57. IEEE Computer Society, Washington, DC (1977)Google Scholar
  86. 86.
    Rao, C.V., Arkin, A.P.: Stochastic chemical kinetics and the quasi-steady-state assumption: Application to the Gillespie algorithm. The Journal of Chemical Physics 118(11), 4999 (2003)CrossRefGoogle Scholar
  87. 87.
    Reppert, S.M., Weaver, D.R.: Coordination of circadian timing in mammals. Nature 418(6901), 935–941 (2002)CrossRefGoogle Scholar
  88. 88.
    Salis, H., Kaznessis, Y.: Accurate hybrid stochastic simulation of a system of coupled chemical or biochemical reactions. The Journal of Chemical Physics 122(5), 54103 (2005)CrossRefGoogle Scholar
  89. 89.
    Sanft, K., Gillespie, D., Petzold, L.: Legitimacy of the stochastic Michaelis Menten approximation. IET Systems Biology 5(1), 58 (2011)CrossRefGoogle Scholar
  90. 90.
    Schivo, S., et al.: Modelling biological pathway dynamics with timed automata. In: BIBE, pp. 447–453. IEEE (2012)Google Scholar
  91. 91.
    Schuster, S., Marhl, M., Höfer, T.: Modelling of simple and complex calcium oscillations. European Journal of Biochemistry 269(5), 1333–1355 (2002)CrossRefGoogle Scholar
  92. 92.
    Singh, A., Hespanha, J.P.: Stochastic hybrid systems for studying biochemical processes. Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences 368(1930), 4995–5011 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  93. 93.
    Smolen, P., Hardin, P.E., Lo, B.S., Baxter, D.A., Byrne, J.H.: Simulation of drosophila circadian oscillations, mutations, and light responses by a model with VRI, PDP-1, and CLK. Biophysical Journal 86(5), 2786–2802 (2004)CrossRefGoogle Scholar
  94. 94.
    Somogyi, R., Stucki, J.W.: Hormone-induced calcium oscillations in liver cells can be explained by a simple one pool model. Journal of Biological Chemistry 266(17), 11068–11077 (1991)Google Scholar
  95. 95.
    Spieler, D.: Model checking of oscillatory and noisy periodic behavior in Markovian population models. Technical report, Saarland University (2009), Master thesis available at http://mosi.cs.uni-saarland.de/?page_id=93
  96. 96.
    Steinfeld, J., Francisco, J., Hase, W.: Chemical kinetics and dynamics. Prentice Hall (1989)Google Scholar
  97. 97.
    Stiver, J.A., Antsaklis, P.J.: State space partitioning for hybrid control systems. In: American Control Conference, pp. 2303–2304. IEEE (1993)Google Scholar
  98. 98.
    Tang, Y., Othmer, H.G.: Excitation, oscillations and wave propagation in a G-protein-based model of signal transduction in dictyostelium discoideum. Philosophical Transactions of the Royal Society B: Biological Sciences 349(1328), 179–195 (1995)CrossRefGoogle Scholar
  99. 99.
    Tyson, J.J.: Biological switches and clocks. Journal of the Royal Society Interface 5, S1–S8 (2008)Google Scholar
  100. 100.
    van Dijk, N.M.: Uniformization for nonhomogeneous Markov chains. Operations Research Letters 12(5), 283–291 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  101. 101.
    Vilar, J., Kueh, H.-Y., Barkai, N., Leibler, S.: Mechanisms of noise-resistance in genetic oscillators. PNAS 99(9), 5988–5992 (2002)CrossRefGoogle Scholar
  102. 102.
    Volterra, V.: Fluctuations in the abundance of a species considered mathematically. Nature 118, 558–560 (1926)CrossRefzbMATHGoogle Scholar
  103. 103.
    Wagner, H., Möller, M., Prank, K.: COAST: controllable approximative stochastic reaction algorithm. The Journal of Chemical Physics 125(17), 174104 (2006)CrossRefGoogle Scholar
  104. 104.
    Wolkenhauer, O., Ullah, M., Kolch, W., Cho, K.-H.: Modeling and simulation of intracellular dynamics: Choosing an appropriate framework. IEEE Transactions on Nanobioscience 3(3), 200–207 (2004)CrossRefGoogle Scholar
  105. 105.
    Zeilinger, M.N., Farr, E.M., Taylor, S.R., Kay, S.A., Doyle, F.J.: A novel computational model of the circadian clock in arabidopsis that incorporates PRR7 and PRR9. Molecular Systems Biology 2 (2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2014

Authors and Affiliations

  • Alexander Andreychenko
    • 1
  • Thilo Krüger
    • 1
  • David Spieler
    • 1
  1. 1.Saarland UniversityGermany

Personalised recommendations