Intrinsic Information Carriers Intrinsic Information Carriers in Combinatorial Dynamical Systems

Many proteins are composed of structural and chemical features—“sites” for short—defined by definite interaction capabilities, such as non-covalent binding or covalent modification of other proteins. This modularity allows for varying degrees of independence, as the behavior of a site might be controlled by the state of some but not all sites of the ambient protein. Independence quickly generates a startling combinatorial complexity that characterizes most biological networks, such as mammalian signaling systems, and effectively prevents their study in terms of kinetic equations—unless the complexity is radically trimmed. Yet, if combinatorial complexity is key to the system’s behavior, eliminating it will prevent, not facilitate, understanding. A more adequate representation of a combinatorial system is afforded by a graph-based framework of rewrite rules where each rule specifies only the information that an interaction mechanism depends on. Unlike reactions, rules deal with patterns, i.e. sets of molecular species, rather than molecular species themselves. Although the stochastic dynamics induced by a set of rules on a mixture of molecules can be simulated, we aim at capturing the system’s average or deterministic behavior. However, expansion of the rules into differential equations at the level of molecular species is not only impractical, but conceptually indefensible. If rules describe patterns of interaction, fully-defined molecular species are unlikely to constitute appropriate units of dynamics. Rather, we must seek aggregated variables reflective of the causal structure laid down by the mechanisms expressed by the rules. We call these variables “fragments” and the process of identifying them “fragmentation”. Ideally, fragments are aspects of the system’s microscopic population that the set of rules can actually distinguish on average; in practice, it may only be feasible to identify an approximation to this. Most importantly, fragments are self-consistent descriptors of system dynamics in that their time evolution is governed by a closed system of kinetic equations. Taken together, fragments are endogenous distinctions that matter for the dynamics of a system, and this warrants viewing them as the carriers of information. Although fragments can be thought of as multi-sets of molecular species (an extensional view), their self-consistency suggests treating them as autonomous aspects cut off from their microscopic anchors (an intensional view). Fragmentation is a seeded process and plays out depending on the seed provided, which leaves open the possibility that different inputs cause distinct fragmentations, in effect altering the set of information carriers that govern the behavior of a system, even though nothing has changed in its microscopic constitution. We provide a mathematical specification of fragments, but not an algorithmic implementation. We have done so elsewhere in rather technical terms with specific biases that, although effective, were lacking an embedding into a more general conceptual framework. Our main objective in this contribution is to provide that framework.