# American Institute of Mathematical Sciences

December  2011, 4(6): 1443-1456. doi: 10.3934/dcdss.2011.4.1443

## Boolean models of bistable biological systems

 1 Interdisciplinary Center for Applied Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0531, United States 2 Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0123

Received  March 2009 Revised  September 2009 Published  December 2010

This paper presents an algorithm for approximating certain types of dynamical systems given by a system of ordinary delay differential equations by a Boolean network model. Often Boolean models are much simpler to understand than complex differential equations models. The motivation for this work comes from mathematical systems biology. While Boolean mechanisms do not provide information about exact concentration rates or time scales, they are often sufficient to capture steady states and other key dynamics. Due to their intuitive nature, such models are very appealing to researchers in the life sciences. This paper is focused on dynamical systems that exhibit bistability and are described by delay equations. It is shown that if a certain motif including a feedback loop is present in the wiring diagram of the system, the Boolean model captures the bistability of molecular switches. The method is applied to two examples from biology, the lac operon and the phage $\lambda$ lysis/lysogeny switch.
Citation: Franziska Hinkelmann, Reinhard Laubenbacher. Boolean models of bistable biological systems. Discrete and Continuous Dynamical Systems - S, 2011, 4 (6) : 1443-1456. doi: 10.3934/dcdss.2011.4.1443
##### References:
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##### References:
 [1] R. Albert and H. Othmer, The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in drosophila melanogaster, J. Theoret. Biol., 223 (2003), 1-18. doi: 10.1016/S0022-5193(03)00035-3. [2] E. Aurell and K. Sneppen, Epigenetics as a first exit problem, Phys. Rev. Lett., 88 (2002), 048101. doi: 10.1103/PhysRevLett.88.048101. [3] M. Cohn and K. Horibata, Inhibition by glucose of the induced synthesis of the $\beta$-galactoside-enzyme system of escherichia coli. analysis of maintenance, J. Bacteriol., 78 (1959), 601-612. [4] C. Conradi, J. Stelling and J. Raisch, Structure discrimination of continuous models for biochemical reaction networks via finite state machines, Proc. IEEE Int. Symposium on Intelligent Control (Mexico City, Mexico), 2001, 138-143. [5] R. de Boer, Theoretical biology,, p. 9., (). [6] R. Edwards, H. T. Siegelmann, K. Aziza and L. Glass, Symbolic dynamics and computation in model gene networks, Chaos, 11 (2001), 160-169. doi: 10.1063/1.1336498. [7] N. Friedman, M. Linial and I. Nachman, Using bayesian networks to analyze expression data, Journal of Computational Biology, 7 (2000), 601-620. doi: 10.1089/106652700750050961. [8] F. Jacob and J. Monod, Genetic regulatory mechanisms in the synthesis of proteins, J. Mol. Biol., 3 (1961), 318-356. doi: 10.1016/S0022-2836(61)80072-7. [9] S. A. Kauffman, Metabolic stability and epigenesis in randomly constructed genetic nets, Journal of Theoretical Biology, 22 (1969), 437-467. doi: 10.1016/0022-5193(69)90015-0. [10] R. Laubenbacher and A. Jarrah, Dvd - discrete visualizer of dynamics.,  , (). [11] L. Mendoza and I. Xenarios, A method for the generation of standardized qualitative dynamical systems of regulatory networks, Theoretical Biology and Medical Modelling, 3 (2006), 13+. doi: 10.1186/1742-4682-3-13. [12] A. Novick and M. Weiner, Enzyme induction as an all-or-none phenomenon, Proc. Natl. Acad. Sci. USA, 43 (1957), 553-66. doi: 10.1073/pnas.43.7.553. [13] M. Ptashne, "A Genetic Switch Phage Lambda Revisited," 3rd ed., Cold Spring Harbor Laboritory Press, 2004. [14] M. Santillán and M. C. Mackey, Why the lysogenic state of phage lambda is so stable: A mathematical modeling approach, Biophys. J., 86 (2004), 75-84. doi: 10.1016/S0006-3495(04)74085-0. [15] T. Tian and K. Burrage, Stochastic models for regulatory networks of the genetic toggle switch, Proc. Natl. Acad. Sci. USA, 103 (2006), 8372-8377. doi: 10.1073/pnas.0507818103. [16] P. Wong, S. Gladney and J. D. Keasling, Mathematical model of the lac operon: Inducer exclusion, catabolite repression, and diauxic growth on glucose and lactose, Biotechnology progress, 13 (1997), 132-143. doi: 10.1021/bp970003o. [17] N. Yildirim and M. C. Mackey, Feedback regulation in the lactose operon: A mathematical modeling study and comparison with experimental data, Biophys. J., 84 (2003), 2841-2851. doi: 10.1016/S0006-3495(03)70013-7.
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