Combinatorial protein-protein interactions on a polymerizing scaffold.

Scaffold proteins organize cellular processes by bringing signaling molecules into interaction, sometimes by forming large signalosomes. Several of these scaffolds are known to polymerize. Their assemblies should therefore not be understood as stoichiometric aggregates, but as combinatorial ensembles. We analyze the combinatorial interaction of ligands loaded on polymeric scaffolds, in both a continuum and discrete setting, and compare it with multivalent scaffolds with fixed number of binding sites. The quantity of interest is the abundance of ligand interaction possibilities-the catalytic potential Q-in a configurational mixture. Upon increasing scaffold abundance, scaffolding systems are known to first increase opportunities for ligand interaction and then to shut them down as ligands become isolated on distinct scaffolds. The polymerizing system stands out in that the dependency of Q on protomer concentration switches from being dominated by a first order to a second order term within a range determined by the polymerization affinity. This behavior boosts Q beyond that of any multivalent scaffold system. In addition, the subsequent drop-off is considerably mitigated in that Q decreases with half the power in protomer concentration than for any multivalent scaffold. We explain this behavior in terms of how the concentration profile of the polymer-length distribution adjusts to changes in protomer concentration and affinity. The discrete case turns out to be similar, but the behavior can be exaggerated at small protomer numbers because of a maximal polymer size, analogous to finite-size effects in bond percolation on a lattice.

The Kappa platform for rule-based modeling.

Motivation: We present an overview of the Kappa platform, an integrated suite of analysis and visualization techniques for building and interactively exploring rule-based models. The main components of the platform are the Kappa Simulator, the Kappa Static Analyzer and the Kappa Story Extractor. In addition to these components, we describe the Kappa User Interface, which includes a range of interactive visualization tools for rule-based models needed to make sense of the complexity of biological systems. We argue that, in this approach, modeling is akin to programming and can likewise benefit from an integrated development environment. Our platform is a step in this direction. Results: We discuss details about the computation and rendering of static, dynamic, and causal views of a model, which include the contact map (CM), snaphots at different resolutions, the dynamic influence network (DIN) and causal compression. We provide use cases illustrating how these concepts generate insight. Specifically, we show how the CM and snapshots provide information about systems capable of polymerization, such as Wnt signaling. A well-understood model of the KaiABC oscillator, translated into Kappa from the literature, is deployed to demonstrate the DIN and its use in understanding systems dynamics. Finally, we discuss how pathways might be discovered or recovered from a rule-based model by means of causal compression, as exemplified for early events in EGF signaling. Availability and implementation: The Kappa platform is available via the project website at kappalanguage.org. All components of the platform are open source and freely available through the authors' code repositories.