Exact model reduction of combinatorial reaction networks

Abstract

Background: Receptors and scaffold proteins usually possess a high number of distinct binding domains inducing the formation of large multiprotein signaling complexes. Due to combinatorial reasons the number of distinguishable species grows exponentially with the number of binding domains and can easily reach several millions. Even by including only a limited number of components and binding domains the resulting models are very large and hardly manageable. A novel model reduction technique allows the significant reduction and modularization of these models.

Results: We introduce methods that extend and complete the already introduced approach. For instance, we provide techniques to handle the formation of multi-scaffold complexes as well as receptor dimerization. Furthermore, we discuss a new modeling approach that allows the direct generation of exactly reduced model structures. The developed methods are used to reduce a model of EGF and insulin receptor crosstalk comprising 5,182 ordinary differential equations (ODEs) to a model with 87 ODEs.

Conclusion: The methods, presented in this contribution, significantly enhance the available methods to exactly reduce models of combinatorial reaction networks.

Figure 1

The three basic scenarios that will be discussed in the following. Figure A depicts a receptor or scaffold protein with single protein ligands, i.e. each binding domain can recruit single proteins which do not possess further binding domains. A scaffold with multiprotein ligands is shown in Figure B. Some of the ligands are scaffolds themselves. The last scenario additionally includes receptor homodimerization. Heterodimerization on the other site corresponds to the scenario depicted in Figure B.

Figure 2

Examples for multiprotein ligand systems. Figure A depicts a chain of signaling proteins without any post-translational modifications such as phosphorylations. All bindings are assumed to interact unidirectionally with each other (black unidirectional arrows). Figure B shows a similar system including domain phosphorylation. Thereby, it is assumed that phosphorylation and subsequent effector binding interact via an all-or-none reaction. Since all-or-none interactions are always bidirectional they are depicted by bidirectional arrows. The last example is a small part of the insulin signaling pathway.

Figure 3

Exemplification of the developed reduced order modeling technique. The considered example is very similar to the previously discussed insulin example. Only the interaction pattern is a bit different. The depicted steps of the reduced order modeling technique are explained in the text.

Figure 4

The shown part of the EGF and insulin receptor network is modeled. The process interactions are depicted by arrows. Black arrows represent uni- and bidirectional interactions, while grey arrows describe all-or-none interactions. A complete mechanistic model of this network comprises 5,182 ODEs, while the exactly reduced one consists of only 87 ODEs.

Figure 5

Simulation results of the generated crosstalk model. The kinetic parameters of the model have been chosen such that the system qualitatively shows the expected behavior. All quantities are depicted in relative concentrations. The overall concentrations of all involved components have been set to 100. The displayed curves show the chosen input signals [EGF], [insulin] and [ERK] as well as the output concentrations [IR(*, SOS, *)], [IR(*, *, SOS)], [EGFR(*, SOS, *).*] and [EGFR(*, *, SOS).*].