ABSTRACT
Mathematical models rooted in network representations are becoming increasingly more common for capturing a broad range of phenomena. Boolean networks (BNs) represent a mathematical abstraction suited for establishing general theory applicable to such systems. A key thread in BN research is developing theory that connects the structure of the network and the local rules to phase space properties or so-called structure-to-function theory. While most theory for BNs has been developed for the synchronous case, the focus of this work is on asynchronously updated BNs (ABNs) which are natural to consider from the point of view of applications to real systems where perfect synchrony is uncommon. A central question in this regard is sensitivity of dynamics of ABNs with respect to perturbations to the asynchronous update scheme. Macauley & Mortveit [Nonlinearity 22, 421-436 (2009)] showed that the periodic orbits are structurally invariant under toric equivalence of the update sequences. In this paper and under the same equivalence of the update scheme, the authors (i) extend that result to the entire phase space, (ii) establish a Lipschitz continuity result for sequences of maximal transient paths, and (iii) establish that within a toric equivalence class the maximal transient length may at most take on two distinct values. In addition, the proofs offer insight into the general asynchronous phase space of Boolean networks.
ABSTRACT
We present mathematical techniques for exhaustive studies of long-term dynamics of asynchronous biological system models. Specifically, we extend the notion of [Formula: see text]-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 (Macauley and Mortveit in Nonlinearity 22(2):421, 2009). We extend earlier work by Veliz-Cuba and Stigler (J Comput Biol 18(6):783-794, 2011), Goles et al. (Bull Math Biol 75(6):939-966, 2013), 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 Macauley and Mortveit (Proc Am Math Soc 136(12):4157-4165, 2008. https://doi.org/10.1090/S0002-9939-09-09884-0 ; 2009; Electron J Comb 18:197, 2011a; Discret Contin Dyn Syst 4(6):1533-1541, 2011b. https://doi.org/10.3934/dcdss.2011.4.1533 ; Theor Comput Sci 504:26-37, 2013. https://doi.org/10.1016/j.tcs.2012.09.015 ; in: Isokawa T, Imai K, Matsui N, Peper F, Umeo H (eds) Cellular automata and discrete complex systems, 2014. https://doi.org/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 (2011), 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 Weinstein et al. (BMC Bioinform 16(1):1, 2015). 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 [Formula: see text] 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.
