Internal coarse-graining of molecular systems

Abstract

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 coarse-grained 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 coarse-grained 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.

Fig. 1.

Rules and reactions in Kappa. (A) A rule captures a high-level mechanistic statement (empirical or hypothetical) about a protein–protein interaction in terms of a rewrite directive plus rate constant(s). The left-hand side (lhs) of the rule is a pattern of partially specified agents and represents the contextual information necessary for identifying reaction instances that proceed according to the rule. The right-hand side (rhs) expresses the actions that may occur when the conditions specified on the lhs are met in a reaction mixture of Kappa agents. A maximal connected subgraph on the lhs of a rule is called a rule component. (B) The rule in A matches a combination of agents in 2 distinct ways giving rise to 2 possible reactions with different outcomes. Note that because of their local nature, Kappa rules with >1 lhs component may apply in both a unimolecular and bimolecular situation. This is why such rules are given 2 rate constants, a first-order (k₁) and a second-order (k₂) constant. In a textual representation, agents are names followed by an interface of sites delimited by parentheses. Bonds are labeled by superscripts and internal states at a site by subscripts. In the graphical rendition, internal states are indicated as labeled barbs. See SI Appendix, section 1 and the section Kappa: A Language for Molecular Biology for more details.

Fig. 2.

Rules and fragments. The figure provides assistance in establishing criteria that define fragments, as detailed in the section From Rules to ODEs. The top row depicts a (unimolecular) rule whose lhs component is r_lhs. The third row from top shows fully specified molecular species (ground-level objects), numbered 1 to 4. The second row depicts various patterns, A to D. Arrows indicate embedding relations of one pattern (graph) into another (see SI Appendix, section 1.4). The rectangles at the bottom provide a schematic of relationships between sets of molecular species that match the patterns A–D and r_lhs. Note that D embeds into r_lhs; its matching instances are therefore a superset of those of r_lhs. Also, D does not overlap with r_lhs on a site that r modifies. Hence r has no effect on D.

Fig. 3.

The contact map. (A) The contact map is a graph whose nodes are the agents in the model and whose edges are possible bonds between sites. Filled circles indicate sites with modifications of state. The contact map is a fine-grained version of what is known as a protein–protein interaction (PPI) map, in that its edges end in sites of agents and not just agents. (B) The annotated contact map (ACM) after decoration induced by the directives Cov1–Cov3 and Edg1.

Fig. 4.

Examples illustrating the syntactical criteria Cov1 and Cov2 for determining classes in the covering of an agent. See section 3 of the SI Appendix for further details.

Fig. 5.

Comparison between microscopic dynamics and fragment dynamics. Wiggly curves: The microscopic dynamics of the early EGFR example is executed with a Doob–Gillespie simulation (9) while reporting the coarse-grained fragment concentrations. This serves as a proxy for the deterministic microscopic dynamics. Steady curves: The output of the deterministic fragment dynamics. Still, many fragments (and many more molecular species) only acquire tiny concentration values, causing far fewer than 38 curves to be discernible by eye in this plot.