Modelers of molecular signaling networks must cope with the combinatorial explosion of protein states generated by posttranslational modifications and complex formation. Rule-based models provide a powerful alternative to approaches that require explicit enumeration of all possible molecular species of a system. Such models consist of formal rules stipulating the (partial) contexts wherein specific protein-protein interactions occur. These contexts specify molecular patterns that are usually less detailed than molecular species. Yet, the execution of rule-based dynamics requires stochastic simulation, which can be very costly. It thus appears desirable to convert a rule-based model into a reduced system of differential equations by exploiting the granularity at which rules specify interactions. We present a formal (and automated) method for constructing a coarsegrained and self-consistent dynamical system aimed at molecular patterns that are distinguishable by the dynamics of the original system as posited by the rules. The method is formally sound and never requires the execution of the rule-based model. The coarsegrained variables do not depend on the values of the rate constants appearing in the rules, and typically form a system of greatly reduced dimension that can be amenable to numerical integration and further model reduction techniques.protein interaction networks ͉ rule-based models ͉ model reduction ͉ distinguishability ͉ information carriers M olecular biology is spectacularly successful in disassembling cellular systems and anchoring cell-biological behaviors of staggering complexity in chemistry. This raises the challenge of reconstituting molecular systems formally, in pursuit of principles that would make their behavior more intelligible and their control more deliberate. This pursuit is as much driven by the practical need to cure disease as it reflects a desire for a theoretical perspective needed to understand the complexity of cellular phenotypes. In achieving such a perspective, we must deal with two broad problems. First, we must be able to represent and analyze molecular interaction systems of combinatorial complexity. Although ubiquitous, such systems are perhaps most notorious in the context of cellular signaling. The posttranslational modification of proteins and their noncovalent association into transient complexes generate an astronomical number of possible molecular species that can relay signals (1). The question then becomes how to reason about system dynamics if we cannot possibly consider a differential equation for each chemical species that can appear in a system. Second, understanding systems requires resisting the temptation of adopting the view of an outside observer. The outside view is indeed appropriate for the chemical analysis of a network, since the experimenter deliberately interacts in specific ways with the network to create measurable distinctions. Yet, the network, as a dynamical system, may not be capable of making these same distinctions. For example, an experi...

Many proteins are composed of structural and chemical features--"sites" for short--characterized by definite interaction capabilities, such as noncovalent 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 shapes 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 provided by a graph-based framework of rewrite rules where each rule specifies only the information that an interaction mechanism depends on. Unlike reactions, which deal with molecular species, rules deal with patterns, i.e., multisets of molecular species. Although the stochastic dynamics induced by a collection of rules on a mixture of molecules can be simulated, it appears useful to capture the system's average or deterministic behavior by means of differential equations. However, expansion of the rules into kinetic equations at the level of molecular species is not only impractical, but conceptually indefensible. If rules describe bona fide patterns of interaction, molecular species are unlikely to constitute appropriate units of dynamics. Rather, we must seek aggregate variables reflective of the causal structure laid down 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, which warrants viewing them as the carriers of information. Although fragments can be thought of as multisets of molecular species (an extensional view), their self-consistency suggests treating them as autonomous aspects cut off from their microscopic realization (an intensional view). Fragmentation is a seeded process that depends on the choice of observables whose dynamics one insists to describe. Different observables can 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. In this contribution, we present a mathematical specification of fragments, but not an algorithmic implementation. We have described the latter elsewhere in rather technical terms that, although effective, were lacking an embedding ...