Vanishing configurations in network dynamics with asynchronous updates

Dinwoodie, Ian

doi:10.1090/S0002-9939-2014-12044-2

Vanishing configurations in network dynamics with asynchronous updates
HTML articles powered by AMS MathViewer

by Ian H. Dinwoodie PDF

Proc. Amer. Math. Soc. 142 (2014), 2991-3002 Request permission

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.

References

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, DOI 10.1016/S0022-5193(03)00035-3
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, DOI 10.1016/S0196-8858(02)00509-2
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, DOI 10.1016/j.jtbi.2005.01.023
David Cox, John Little, and Donal O’Shea, Using algebraic geometry, Graduate Texts in Mathematics, vol. 185, Springer-Verlag, New York, 1998. MR 1639811, DOI 10.1007/978-1-4757-6911-1
Ian H. Dinwoodie, Sequential importance sampling of binary sequences, Stat. Comput. 22 (2012), no. 1, 53–63. MR 2865055, DOI 10.1007/s11222-010-9205-0
Ian H. Dinwoodie, Conditional tests on basins of attraction with finite fields, Methodol. Comput. Appl. Probab. 16 (2014), no. 1, 161–168. MR 3169081, DOI 10.1007/s11009-012-9304-9
W. Decker, G.-M. Greuel, G. Pfister, and H. Schönemann (2011), Singular 3-1-3 — A computer algebra system for polynomial computations. http://www.singular.uni-kl.de.
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, DOI 10.1007/s11538-010-9582-8
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.
Martin Kreuzer and Lorenzo Robbiano, Computational commutative algebra. 1, Springer-Verlag, Berlin, 2000. MR 1790326, DOI 10.1007/978-3-540-70628-1
L. Mendoza, A network model for the control of the differentiation process in Th cells, BioSystems 84 (2006), 101-114.
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.
Giovanni Pistone, Eva Riccomagno, and Henry P. Wynn, Algebraic statistics, Monographs on Statistics and Applied Probability, vol. 89, Chapman & Hall/CRC, Boca Raton, FL, 2001. Computational commutative algebra in statistics. MR 2332740
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.
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.
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, DOI 10.1371/journal.pcbi.0030163
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.
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.
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, DOI 10.1090/psapm/064/2359649
R. Thomas, Boolean formalization of genetic control circuits, J. Theor. Biol. 42 (1973), 563-585.
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.
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.