When rule-based models need to count

Rule-based modelers dislike direct enumeration of cases when more efficient means of enumeration are available. We present an extension of the Kappa language which attaches to agents a notion of level. We detail two encodings that are more concise than the former practice. Rule-based languages are a well-established framework for modeling proteinprotein interactions. Kappa [2,1] is a rule-based language relying on site-graphs. The nodes of sitegraphs are called agents. Agents interact by binding/unbinding through sites. Sites are binding resources, each site is part of at most one edge. A model in Kappa consists of a set of graph rewrite rules with rates. A rule describes a potential interaction given a context. Rates represent probability to fire. In a biological context, it is often the case that a notion of internal state (such as active, methilated, . . . ) is required in order to describe possible interactions. In Kappa, sites are equipped with internal states, facilitating the modeling efforts of the user. However, as shown in the following sections a more systematic encoding of internal states is possible. Another common practice is to attach a level to agents and make an interaction sensitive on the level of its participating agents. We propose here a language extension to store, test and change levels explicitly. Moreover, we present an encoding of levels that induce a linear (in number of levels) blow up of the number of rules. This is in contrast to previous encodings, which induce an exponential blow up in the number of rules. 1 Email: Pierre [email protected] 2 Email: Ioana [email protected] This paper is electronically published in Electronic Notes in Theoretical Computer Science URL: www.elsevier.com/locate/entcs ar X iv :1 70 8. 02 65 1v 1 [ qbi o. O T ] 7 A ug 2 01 7 Boutillier, Cristescu 1 When enumeration is necessary The following motivating example [4] demonstrates a typical problem in which levels are necessary: KaiC proteins have 6 independent phosphorylation sites. (De)phosphorylation of every site is independent. The more sites are phosphorylated, the bigger the probability that KaiC binds KaiA is. A typical way to deal with this example consist in explicitly encoding rules for the internal states of the sites of interest. However, doing so induces an exponential blow-up in the number of rules (in the number of levels). The BNGL [3] language introduces a notion of indistinguishable sites. i.e. one can define an agent with n sites that all have the same name. Consequently, a single rule to specify that k sites (out of n) are phosphorylated is enough. Moreover, the number of species is also reduced. Still, there are exponentially many ways to go from one level to another and enumeration is necessary to faithfully respect the dynamics of the system. We now show that adding a syntactic layer to Kappa that offers support for counting will avoid the explosion in the number of rules.