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Attractor stability in finite asynchronous biological system models. (English) Zbl 1415.92087

Summary: We present mathematical techniques for exhaustive studies of long-term dynamics of asynchronous biological system models. Specifically, we extend the notion of \(\kappa \)-equivalence developed for graph dynamical systems to support systematic analysis of all possible attractor configurations that can be generated when varying the asynchronous update order [M. Macauley and H. S. Mortveit, Nonlinearity 22, No. 2, 421–436 (2009; Zbl 1159.37010)]. We extend earlier work by A. Veliz-Cuba and B. Stigler [“Boolean models can explain bistability in the lac operon”, J. Comput. Biol. 18, No. 6, 783–794 (2011; doi:10\.1089/cmb.2011.0031)], E. Goles et al. [Bull. Math. Biol. 75, No. 6, 939–966 (2013; Zbl 1272.92017)], and others by comparing long-term dynamics up to topological conjugation: rather than comparing the exact states and their transitions on attractors, we only compare the attractor structures. In general, obtaining this information is computationally intractable. Here, we adapt and apply combinatorial theory for dynamical systems from M. MacAuley and H. S. Mortveit [Proc. Am. Math. Soc. 136, No. 12, 4157–4165 (2008; Zbl 1157.05008); Electron. J. Comb. 18, No. 1, Research Paper P197, 18 p. (2011; Zbl 1250.20035); Discrete Contin. Dyn. Syst., Ser. S 4, No. 6, 1533–1541 (2011; Zbl 1232.93080); “An atlas of limit set dynamics for asynchronous elementary cellular automata”, Theor. Comput. Sci. 504, 26–37 (2013; doi:10\.1016/j.tcs.2012.09.015); “Cycle equivalence of finite dynamical systems containing symmetries”, in: Cellular automata and discrete complex systems (2014; doi:10\.1007/978-3-319-18812-6\_6)] to develop computational methods that greatly reduce this computational cost. We give a detailed algorithm and apply it to (i) the lac operon model for Escherichia coli proposed by Veliz-Cuba and Stigler [loc. cit.], and (ii) the regulatory network involved in the control of the cell cycle and cell differentiation in the Caenorhabditis elegans vulva precursor cells proposed by N. Weinstein et al. [“A model of the regulatory network involved in the control of the cell cycle and cell differentiation in the caenorhabditis elegans vulva”, BMC Bioinform 16(1):1 (2015; doi:10\.1186/s12859-015-0498-z)]. In both cases, we uncover all possible limit cycle structures for these networks under sequential updates. Specifically, for the lac operon model, rather than examining all \(10! > 3.6 \cdot 10^6\) sequential update orders, we demonstrate that it is sufficient to consider 344 representative update orders, and, more notably, that these 344 representatives give rise to 4 distinct attractor structures. A similar analysis performed for the C. elegans model demonstrates that it has precisely 125 distinct attractor structures. We conclude with observations on the variety and distribution of the models’ attractor structures and use the results to discuss their robustness.

MSC:

92C42 Systems biology, networks
34D45 Attractors of solutions to ordinary differential equations
92-04 Software, source code, etc. for problems pertaining to biology

Software:

gensim; NetworkX; BoolNet
PDFBibTeX XMLCite
Full Text: DOI

References:

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