For biologists, modularity usually refers to the concept that physiological and cell biological regulatory mechanisms can be described as being composed of more or less well-defined functional modules, with sparse connectivity across the boundaries of such modules [1]. We generalize this approach to address the combinatorial complexity that often arises when detailed quantitative models of intracellular networks and pathways are being sought. Similar to describing metabolism as modules that can be reused in different pathways [2], one can view proteins that are composed of multiple domains as functional modules composed of many elements - e.g., Src homology 2 (SH2) binding sites and tyrosines phosphorylation sites [3]. This is a typical situation that generates combinatorial complexity in signaling pathways. For example, in the case of Epidermal Growth Factor Receptor, EGFR [4], a receptor with 10 tyrosine phospho-sites can exist in 2¹⁰=1024 different phosphoforms, and dimerization and interaction with other molecules then leads to millions of possible distinct complexes. At present, kinetics models accounting for dozens of different molecular species are a norm [5], and models accounting for hundreds of species and reactions are no longer rare [6, 7]. Visualization of such networks is difficult at best, and manually specifying the list of species and reactions becomes error-prone and slow. A solution for this challenge can be provided by the modular approach, in the form of (i) defining smaller reusable model components for quantitative models (modeling modules), and (ii) specifying rules of interaction, be it at the protein/molecular complex level, or arbitrary functional level (e.g. kinetic of ligand-receptor binding is independent of receptor phosphoforms). Quantitative models of complex networks can then be assembled from separately constructed and validated components, either directly or via rules.

Several repositories of mathematical models exist that provide validated, annotated models of cellular processes. The BioModels database [15] is one such fast growing repository, providing several hundred of models in simulation-ready SBML format, which is the emerging community standard for kinetic modeling [13]. VCell has a public repository of models in a proprietary VCML format (a richer format than SBML, that includes among others, spatial descriptions for models), which can be exported into the SBML format. Most software tools now provide a facility to translate into SBML working models, or models from their own repository, if they have one. However, often the translation has a significant flaw, that all elements (species and reactions) can not be easily compared and exchanged between the models without manual intervention. Even the well-annotated models stored in BioModels repository that are compliant with MIRIAM (Minimum information requested in the annotation of biochemical models) standard [16] present challenges. For example, currently there is no standard way to distinguish between different states of the same molecules. Thus, all phosphoforms of the same receptor will be annotated with the same reference identifier (GO term, UniProt key, etc.). This means that there still will be often the case that it is impossible to automatically tell whether species X of model A and species Y of model B, which have the same reference identifier, are indeed identical and should be mapped into the same species in a merged model. We address this by adding compatibility with the BioPAX standard, which has an advantage of a fine grained unique identification of all elements of interactions and signaling pathways across multiple databases.

The rule-based approach to define models [9, 10] has the advantage of inherently handling combinatorial complexity. It allows one to account comprehensively and precisely for the possible molecular species implied by the specified interactions, activities, and modifications of the molecules in a system. The model is built by defining the rules that govern how molecules interact (to form complexes, modify internal states, or degrade), as illustrated in Figure 2a. These rules can thus specify how new molecular species can be produced and are used to automatically generate the reaction network, freeing the user from the intense bookkeeping that would be required to enumerate such a network by hand. A general-purpose software package, BioNetGen ([9, 10]), was developed that implements this approach. This tool was used to build several models, including the models of receptor-mediated signaling events stimulated by antigen [20] and epidermal growth factor [4], and the function of Shp2 phosphotase in intracellular signal transduction [21]. The model is usually based on assumptions about modular interactions of proteins and the mechanistic details of protein-protein interactions [3, 22]. BioNetGen can also solve the resulting system of ordinary differential equations, simulating the time courses for each species. The software can also determine time courses using the Gillespie algorithm, which may become important when the concentrations are low. Moreover, species and reactions in a network can be generated only as needed during the course of a network simulation [23, 24], which is important when the number of potential chemical species is too big for any reasonable computations (e.g. ~10⁸ species in the example of independent proteins binding to EGFR tyrosines considered above). Since many of the quantities measured in experiments are properties of ensembles, rather than distinct microscopic species, BioNetGen allows the modeler to calculate functions of microscopic ensembles. For example, the level of phosphorylation of a protein is a function of the concentrations of all microscopic species containing the protein. The appropriate sums of species concentrations that correspond to a defined measurable property of the system are computed automatically.

The BioPAX@VCell framework provides reading of pathway data from BioPAX-supported databases with explicit specification of each and every species and reactions. BioNetGen@VCell provides generation of VCell models from user-specified rules. Both approaches can be used synergistically. Indeed, the data may allow multiple interpretations. For example, a given interaction can be applied to several phosphoforms of a protein, thus providing a rule instead of a single interaction. These efforts should be concurrent with the development of the next level of BioPax ontology (that will better describe protein modifications), as well as the SBML level 3 extension (that will allow description of multi-component multi-state species).

A highly desirable feature for modeling tools is the automated data retrieval and verification using external web resources. After the user picks elements to be included in a model, an intelligent framework should try to infer additional elements (interactions, modifications, kinetic constants) to make a kinetic model complete, and then request this information from available public resources. Qualitative information, such as reaction catalysts, can be requested from databases like Reactome that provide an API for querying and retrieving BioPax data over the web. Quantitative information, such as kinetic constants, can be requested from emerging databases of reaction kinetics, such as SABIO-RK (http://sabio.villa-bosch.de/SABIORK/). When fully implemented, such capabilities will provide a powerful data-driven modeling environment.