  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 (Macauley and Mortveit
(2009)). We extend earlier work by Veliz-Cuba and Stigler (2011), Goles et al.
(2014), 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 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. (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 $10! > 3.6 \cdot 10^6$ 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.
