Abstract
We consider dynamics in a class of piecewise-linear ordinary differential equations and in an electronic circuit that model genetic networks. In these models, gene activity varies continuously in time. However, as in Boolean or discrete-time switching networks, gene activity is driven high or low based only on whether the activities of the regulating genes are high or low (i.e., above or below certain thresholds). Depending on the “regulatory logic”, these models can exhibit simple dynamics, like stable fixed points or oscillation, or chaotic dynamics. The observed qualitative and quantitative differences between the dynamics in the idealized equations and the dynamics in the electronic circuit lead us to focus attention on the analysis of the dynamics as a function of parameter values. We propose new techniques for solving the inverse problem – the problem of inferring the regulatory logic and parameters from time series data. We also give new symbolic and statistical methods for characterizing dynamics in these networks.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
B. Alberts A. Johnson J. Lewis et al. (2002) Molecular Biology of the Cell fourth edition Garland Science New York
J. Monod F. Jacob (1961) ArticleTitleGeneral conclusions: Teleonomic mechanisms in cellular metabolism, growth, and differentiation Cold Spring Harb Symp. Quant. Biol. 26 389–401
TS. Gardner CR. Cantor JJ. Collins (2000) ArticleTitleConstruction of a genetic toggle switch in Escherichia coli Nature 403 339–342 Occurrence Handle2000Natur.403..339G
M.B. Elowitz S. Leibler (2000) ArticleTitleA synthetic oscillatory network of transcriptional regulators Nature 403 335–338 Occurrence Handle10.1038/35002125 Occurrence Handle2000Natur.403..335E
J. Hasty D.D McMillen JJ. Collins (2002) ArticleTitleEngineered gene circuits Nature 420 224–230 Occurrence Handle10.1038/nature01257 Occurrence Handle2002Natur.420..224H
SA. Kauffman (1969) ArticleTitleMetabolic stability and epigenesis in randomly constructed genetic networks J. Theor. Biol. 22 437–467 Occurrence Handle10.1016/0022-5193(69)90015-0 Occurrence Handle39 #7465
SA. Kauffman (1993) Origins of Order: Self-Organization and Selection in Evolution Oxford University Press Oxford
U. Bastolla G. Parisi (1997) ArticleTitleA numerical study of the critical line of Kauffman networks J. Theor. Biol. 187 117–133 Occurrence Handle10.1006/jtbi.1997.0423
S. Bilke F. Sjunnesson (2001) ArticleTitleStability of the Kauffman model Phys. Rev. E 65 016129 Occurrence Handle2002PhRvE..65a6129B
B. Derrida Y. Pomeau (1986) ArticleTitleRandom networks of automata – a simple annealed approximation Europhys. Lett. 1 45–49 Occurrence Handle1986EL......1...45D
H. Flyvbjerg (1988) ArticleTitleAn order parameter for networks of automata J. Phys. A: Math. Gen. 21 L955–L960 Occurrence Handle10.1088/0305-4470/21/7/031 Occurrence Handle1988JPhA...21L.955F Occurrence Handle0657.68059 Occurrence Handle968303
X. Qu M. Aldana L.P. Kadanoff (2002) ArticleTitleNumerical and theoretical studies of noise effects in the Kauffman model J. Stat. Phys. 109 967–985 Occurrence Handle10.1023/A:1020416308456
L. Glass SA. Kauffman (1973) ArticleTitleThe logical analysis of continuous, nonlinear biochemical control networks J. Theor. Biol. 39 103–129 Occurrence Handle10.1016/0022-5193(73)90208-7
L. Glass (1975) ArticleTitleCombinatorial and topological methods in nonlinear chemical kinetics J. Chem. Phys. 63 1325 Occurrence Handle10.1063/1.431518 Occurrence Handle1975JChPh..63.1325G
L. Glass (1977) Combinatorial aspects of dynamics in biological systems U. Landman (Eds) Statistical Mechanics and Statistical Methods in Theory and Application. Plenum New York 585–611
L. Glass JS. Pasternack (1978) ArticleTitleStable oscillations in mathematical models of biological control systems J. Math. Biol. 6 207–223 Occurrence Handle83a:92002
L.L. Glass (1985) Boolean and continuous models for the generation of biological rhythms J. Demongeot E. Goles M. Tchuente (Eds) Dynamical Systems and Cellular Automata. Academic Press London 197–206
T. Mestl C. Lemay L. Glass (1996) ArticleTitleChaos in high dimensional neural and gene networks Physica D 98 33–52 Occurrence Handle10.1016/0167-2789(96)00086-3 Occurrence Handle97g:58112
R. Edwards L. Glass (2000) ArticleTitleCombinatorial explosion in model gene networks Chaos 10 691–704 Occurrence Handle10.1063/1.1286997 Occurrence Handle2000Chaos..10..691E Occurrence Handle2001k:92013
R. Edwards (2001) ArticleTitleChaos in neural and gene networks with hard switching Diff. Eq. Dyn. Sys. 9 187–220
R. Edwards HT. Siegelmann K. Aziza L. Glass (2001) ArticleTitleSymbolic dynamics and computation in model gene networks Chaos 11 160–169 Occurrence Handle2001Chaos..11..160E
TJ. Perkins MT. Hallett L. Glass (2004) ArticleTitleInferring models of gene expression dynamics J. Theor. Biol. 230 289–299 Occurrence Handle10.1016/j.jtbi.2004.05.022 Occurrence Handle2088017
R. Edwards (2000) ArticleTitleAnalysis of continuous time switching networks Physica D 146 165–199 Occurrence Handle10.1016/S0167-2789(00)00130-5 Occurrence Handle2000PhyD..146..165E Occurrence Handle0986.94051 Occurrence Handle2001h:92007
R. Edwards L. Glass (2006) A calculus for relating the dynamics and structure of complex biological networks R.S. Berry J. Jortner (Eds) Adventures in Chemical Physics: A Special Volume of Advances in Chemical Physics. Vol. 137 John Wiley & Sons, Inc. Hoboken, NJ 151–178
JP. Mason PS. Linsay JJ. Collins L. Glass (2004) ArticleTitleEvolving complex dynamics in electronic models of genetic networks Chaos 14 707–715 Occurrence Handle10.1063/1.1786683 Occurrence Handle2004Chaos..14..707M Occurrence Handle2089493
BC. Goodwin (1976) Analytical Physiology of Cells and Developing Organisms Academic Press London
R. Thomas R. D’Ari (1990) Biological Feedback CRC Press Boca Raton
H.H. McAdams L. Shapiro (1995) ArticleTitleCircuit simulation of genetic networks Science 269 650–656 Occurrence Handle1995Sci...269..650M
H. Jong ParticleDe JL. Gouze C.C. Hernandez et al. (2004) ArticleTitleQualitative simulation of genetic regulatory networks using piecewise-linear models Bull. Math. Biol. 66 301–340
S. Muroga (1979) Logic Design and Switching Theory John Wiley New York
D. Lind B. Marcus (1995) An Introduction to Symbolic Dynamics and Coding Cambridge University Press Cambridge
M. Brin G. Stuck (2002) Introduction to Dynamical Systems Cambridge University Press Cambridge
C. Grebogi E. Ott JA. Yorke (1982) ArticleTitleChaotic attractors in crisis Phys. Rev. Lett. 48 1507–1510 Occurrence Handle10.1103/PhysRevLett.48.1507 Occurrence Handle1982PhRvL..48.1507G Occurrence Handle83i:58064
M. Sipser (1997) Introduction to the Theory of Computation PWS Publishers Boston
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Glass, L., Perkins, T.J., Mason, J. et al. Chaotic Dynamics in an Electronic Model of a Genetic Network. J Stat Phys 121, 969–994 (2005). https://doi.org/10.1007/s10955-005-7009-y
Received:
Accepted:
Issue Date:
DOI: https://doi.org/10.1007/s10955-005-7009-y