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Vanishing configurations in network dynamics with asynchronous updates

Author: Ian H. Dinwoodie
Journal: Proc. Amer. Math. Soc. 142 (2014), 2991-3002
MSC (2010): Primary 13P25, 62M86
Published electronically: May 22, 2014
MathSciNet review: 3223354
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Abstract: We consider Boolean dynamics for biological networks where stochasticity is introduced through asynchronous updates. An exact method is given for finding states which can reach a steady state with positive probability, and a method is given for finding states which cannot reach other steady states. These methods are based on computational commutative algebra. The algorithms are applied to dynamics of a cell survival network to determine node assignments that exclude termination in a cancerous state.

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  • [1] Réka Albert and Hans G. Othmer, The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster, J. Theoret. Biol. 223 (2003), no. 1, 1-18. MR 2069236,
  • [2] Eric Babson, Shmuel Onn, and Rekha Thomas, The Hilbert zonotope and a polynomial time algorithm for universal Gröbner bases, Adv. in Appl. Math. 30 (2003), no. 3, 529-544. MR 1973955 (2004h:13030),
  • [3] Madalena Chaves, Réka Albert, and Eduardo D. Sontag, Robustness and fragility of Boolean models for genetic regulatory networks, J. Theoret. Biol. 235 (2005), no. 3, 431-449. MR 2158953,
  • [4] David Cox, John Little, and Donal O'Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811 (99h:13033)
  • [5] Ian H. Dinwoodie, Sequential importance sampling of binary sequences, Stat. Comput. 22 (2012), no. 1, 53-63. MR 2865055,
  • [6] I. H. Dinwoodie, Conditional tests on basins of attraction with finite fields. Methodol. Comput. Appl. Probab. 16 (2014), no. 1, 161-168. MR 3169081
  • [7] W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann (2011),
    SINGULAR 3-1-3 -- A computer algebra system for polynomial computations.
  • [8] Franziska Hinkelmann, David Murrugarra, Abdul Salam Jarrah, and Reinhard Laubenbacher, A mathematical framework for agent based models of complex biological networks, Bull. Math. Biol. 73 (2011), no. 7, 1583-1602. MR 2802438 (2012f:92003),
  • [9] S. Klamt, J. Saez-Rodriquez, J. A. Lindquist, L. Simeoni, and E. D. Gilles, A methodology for the structural and functional analysis of signalling and regulatory networks, BMC Bioinformatics 7 (2006), 1471-2105.
  • [10] Martin Kreuzer and Lorenzo Robbiano, Computational commutative algebra. 1, Springer-Verlag, Berlin, 2000. MR 1790326 (2001j:13027)
  • [11] L. Mendoza, A network model for the control of the differentiation process in Th cells, BioSystems 84 (2006), 101-114.
  • [12] M. K. Morris, J. Saez-Rodriguez, P. K. Sorger, and D. A. Lauffenburger, Logic-based models for the analysis of cell signaling networks, Biochemistry 49 (2010), 3216-3224.
  • [13] Giovanni Pistone, Eva Riccomagno, and Henry P. Wynn, Algebraic statistics, Computational commutative algebra in statistics. Monographs on Statistics and Applied Probability, vol. 89, Chapman & Hall/CRC, Boca Raton, FL, 2001. MR 2332740 (2008f:62098)
  • [14] A. Saadatpour, R-S. Wang, A. Liao, X. Liu, T. P. Loughran, I. Albert, and R. Albert, Dynamical and structural analysis of a T cell survival network identifies novel candidate therapeutic targets for large granula lymphocyte leukemia, PLoS Comp. Biol. 7 (2011), e1002267.
  • [15] K. Sachs, O. Perez, D. Pe'er, D. A. Lauffenburger, and G. P. Nolan, Causal protein-signaling networks derived from multiparameter single-cell data, Science 308 (2005), 523-529.
  • [16] Julio Saez-Rodriguez, Luca Simeoni, Jonathan A. Lindquist, Rebecca Hemenway, Ursula Bommhardt, Boerge Arndt, Utz-Uwe Haus, Robert Weismantel, Ernst D. Gilles, Steffen Klamt, and Burkhart Schraven, A logical model provides insights into T cell receptor signaling, PLoS Comput. Biol. 3 (2007), no. 8, 1580-1590. MR 2369374,
  • [17] J. Saez-Rodriquez, L. G. Alexopoulos, M. Zhang, M. Morris, D. A. Lauffenburger, and P. K. Sorger, Comparing signaling networks between normal and transformed hepatocytes using discrete logical models, Cancer Res. 71 (2001), 5400-5411.
  • [18] R. Schlatter, K. Schmich, I. A. Vizcarra, P. Scheurich, T. Sauter, C. Borner, M. Ederer, I. Merfort, and O. Sawodny, ON/OFF and beyond--A Boolean model of apoptosis. PLoS Comp. Biol. 5 (2009), e1000595.
  • [19] Brandilyn Stigler, Polynomial dynamical systems in systems biology, Modeling and simulation of biological networks, Proc. Sympos. Appl. Math., vol. 64, Amer. Math. Soc., Providence, RI, 2007, pp. 53-84. MR 2359649 (2008i:92019)
  • [20] R. Thomas, Boolean formalization of genetic control circuits, J. Theor. Biol. 42 (1973), 563-585.
  • [21] A. V. Werhli, M. Grzegorczyk, and D. Husmeier, Comparative evaluation of reverse engineering gene regulatory networks with relevance networks, graphical gaussian models, and bayesian networks, Bioinformatics 22 (2006), 2523-2531.
  • [22] R. Zhang, M. V. Shah, J. Yang, S. B. Nyland, X. Liu, J. Yun, R. Albert, and T. P. Loughran, Network model of survival signaling in large granular lymphocyte leukemia, PNAS 105 (2008), 16308-16313.

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Additional Information

Ian H. Dinwoodie
Affiliation: Department of Mathematical Sciences, Portland State University, Portland, Oregon 97201

Keywords: Asynchronous network, basin of attraction, Boolean network, Groebner basis, Markov chain.
Received by editor(s): June 25, 2012
Received by editor(s) in revised form: September 25, 2012
Published electronically: May 22, 2014
Communicated by: David Levin
Article copyright: © Copyright 2014 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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