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18
Many diseases are caused by molecular changes that affect signal- transduction systems, and some of these diseases, such as chronic myelogenous leukemia, can now be treated by drugs that target signaling proteins, such as kinases (1). Thus, in addi- tion to our curiosity about the fascinating mechanisms that cells use to respond to signals, there is practical motivation to better understand the processes of cellular signaling, in which protein- protein interactions play a central role. Indeed, we would like to make accurate predictions about the functional roles of proteins and the effects of modifying the interactions of proteins in par- ticular systems. Our ability to make such predictions is a mea- sure of our basic understanding of molecular cell biology and is likely to have practical consequences in drug discovery (2, 3). The behavior of a signal-transduction system depends on dy- namic interactions among its proteins (4, 5). The combined ef- fects of these interactions are difficult to predict from intuition alone. When intuition is insufficient, a mathematical model is often useful for acquiring a quantitative and predictive under- standing of a complex dynamical system, and mathematical modeling is being increasingly used to aid in studies of cellular signaling (6, 7). Models have now been developed and tested for signaling events mediated by several well-studied cell sur- face receptors, including the epidermal growth factor receptor (EGFR) (8) and two of the antigen-recognition receptors of the immune system (9). However, current models are still far from capturing all of the relevant mechanistic details of signal-transduction systems that must be considered to provide realistic and complete pictures of how these systems work (10). In particular, models often fail to account for the complexities of protein-protein interactions, such as how these interactions depend on contextual details at the lev- el of protein sites. Also, few models account for the enormous number of possible posttranslational covalent modifications of proteins and for formation of all the possible protein complexes (11). A major reason for this shortcoming is the strain on con- ventional modeling approaches caused by the combinatorial po- tential of protein-protein interactions. New modeling approaches that address this problem involve the use of rules to represent protein-protein interactions. (Rules are also useful for represent- ing other types of biomolecular interactions, but we will focus on protein-protein interactions for purposes of discussion.) The introduction of rules greatly eases the task of specifying a model that incorporates details at the level of protein sites. A rule— such as “ligand binds receptor with rate constant k whenever lig- and and receptor have free binding sites”—describes the features of reactants that are required for a particular type of chemical transformation to take place. Rules simplify the specification of a model when the reactivity of a component in a system is deter- mined by only a subset of its possible features. Here, we review rule-based modeling of signal-transduction systems. What do we expect from a model? A model should incorporate enough details to make testable predictions about quantities that can be measured and controlled in experiments. Whenever pos- sible, the parameters of a model should have a physical rather than phenomenological basis, such that parameters are indepen- dent of system behavior. A model should provide insights and guide experimentation by revealing the logical consequences of knowledge and assumptions about the mechanistic details of a system, including the proteins in the system and their binding sites, enzymatic activities, and sites of posttranslational modifi- cation. The development of models at this level of resolution is

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Many diseases are caused by molecular changes that affect signal-transduction systems, and some of these diseases, such aschronic myelogenous leukemia, can now be treated by drugsthat target signaling proteins, such as kinases (1). Thus, in addi-tion to our curiosity about the fascinating mechanisms that cellsuse to respond to signals, there is practical motivation to betterunderstand the processes of cellular signaling, in which protein-protein interactions play a central role. Indeed, we would like tomake accurate predictions about the functional roles of proteinsand the effects of modifying the interactions of proteins in par-

ticular systems. Our ability to make such predictions is a mea-sure of our basic understanding of molecular cell biology and islikely to have practical consequences in drug discovery (2, 3).

The behavior of a signal-transduction system depends on dy-namic interactions among its proteins (4, 5). The combined ef-fects of these interactions are difficult to predict from intuitionalone. When intuition is insufficient, a mathematical model isoften useful for acquiring a quantitative and predictive under-standing of a complex dynamical system, and mathematicalmodeling is being increasingly used to aid in studies of cellularsignaling (6, 7). Models have now been developed and testedfor signaling events mediated by several well-studied cell sur-face receptors, including the epidermal growth factor receptor(EGFR) (8) and two of the antigen-recognition receptors of theimmune system (9).

However, current models are still far from capturing all of therelevant mechanistic details of signal-transduction systems thatmust be considered to provide realistic and complete pictures ofhow these systems work (10). In particular, models often fail toaccount for the complexities of protein-protein interactions, suchas how these interactions depend on contextual details at the lev-el of protein sites. Also, few models account for the enormousnumber of possible posttranslational covalent modifications ofproteins and for formation of all the possible protein complexes(11). A major reason for this shortcoming is the strain on con-ventional modeling approaches caused by the combinatorial po-tential of protein-protein interactions. New modeling approachesthat address this problem involve the use of rules to representprotein-protein interactions. (Rules are also useful for represent-ing other types of biomolecular interactions, but we will focuson protein-protein interactions for purposes of discussion.) Theintroduction of rules greatly eases the task of specifying a modelthat incorporates details at the level of protein sites. A rule—such as “ligand binds receptor with rate constant k whenever lig-and and receptor have free binding sites”—describes the featuresof reactants that are required for a particular type of chemicaltransformation to take place. Rules simplify the specification ofa model when the reactivity of a component in a system is deter-mined by only a subset of its possible features. Here, we reviewrule-based modeling of signal-transduction systems.

What do we expect from a model? A model should incorporateenough details to make testable predictions about quantities thatcan be measured and controlled in experiments. Whenever pos-sible, the parameters of a model should have a physical ratherthan phenomenological basis, such that parameters are indepen-dent of system behavior. A model should provide insights andguide experimentation by revealing the logical consequences ofknowledge and assumptions about the mechanistic details of asystem, including the proteins in the system and their bindingsites, enzymatic activities, and sites of posttranslational modifi-cation. The development of models at this level of resolution is

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justified by current and emerging capabilities for characterizingsystems at the level of protein sites and for monitoring the dy-namics of multiple readouts of protein interactions (12–16).Thus, although other levels of abstraction may prove useful, weare most interested in models with physicochemical parametersthat capture details about proteins and their interactions at thelevel of protein sites, which we take to include motifs, domains,and subunits.

General features of proteins and protein-protein interactions.Protein-protein interactions triggered by a signal can lead tovarious covalent posttranslational protein modifications, such asphosphorylation, ubiquitination, acetylation, or methylation (4,5, 17). The effect of such modification, which is often re-versible, may be a change in a protein’s enzymatic or bindingactivity, location, or rate of turnover. Signal-regulated protein-protein interactions also mediate the assembly of heterogeneousprotein complexes. A common effect of complex formation isan increase in the activity and specificity of an enzyme throughcolocalization of the enzyme with a substrate (18, 19). Signal-induced protein complexes are also known to regulate enzymat-ic activity through allosteric mechanisms (20, 21).

The enzymatic and binding activities of proteins involved insignaling tend to be localized to modular domains (19, 22),such as the protein tyrosine kinase (PTK) and Src homology 2(SH2) domains of a Src family kinase. Proteins may also con-tain smaller parts, like immunoreceptor tyrosine-based activa-tion motifs (ITAMs) (23), that are sites of modification or bind-ing or both (24). A signaling protein generally has multiple do-mains (25, 26) and binding sites (and binding partners, whichmay have overlapping specificities), as well as multiple sitessubject to posttranslational modifications (which may includethe binding sites themselves) (27) (Fig. 1). Such multiplicitymay permit many combinations of modification and bindingevents, which can be difficult to track in a model (11, 28, 29).

Combinatorial complexity. As a modeler considers moreproteins and protein sites of a signal-transduction system, thenumber of possible protein complexes and combinations ofprotein modifications tends to increase exponentially. More-over, hundreds to thousands of chemical species may be gen-erated by the interactions of only a few proteins (28, 30–33).Thus, a signal-transduction system can be quite large whenviewed as a chemical reaction network. We refer to this po-tential for large size as “combinatorial complexity.”

One source of combinatorial complexity, to which we havealready alluded, is multisite protein modification (27). Considerthe EGFR. According to a recent review (34), at least nine ty-rosines in EGFR are phosphorylated during signaling (Fig. 2).Phosphorylation of these tyrosines is mediated by a nonreceptorPTK, Src, or EGFR itself when the PTK domain of one EGFRin a ligand-induced receptor dimer catalyzes phosphorylation ofthe paired EGFR in the dimer. As a simplification, let us as-sume that only one of nine tyrosines of EGFR can be phospho-

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rylated at a time. Under this assumption, there are still 10 dif-ferent phosphoforms of EGFR and 55 distinct combinations ofthese phosphoforms for a receptor dimer. These numbers growto 512 and 131,328 if more than one tyrosine of EGFR can be

phosphorylated at a time. The number of phosphorylation statesrelevant for signaling in particular contexts is unknown, and itseems unlikely that all states are functionally relevant or evenrealized. Indeed, some phosphorylation states may be prohibited(35). However, a modeler may wish to consider the full spec-trum of possible phosphorylation states to assess their relevancein an unbiased manner. The states included in a model will de-pend on the questions asked and the opportunities a modelermay have to make meaningful simplifications.

Another source of combinatorial complexity is multivalentbinding. Consider the death domain (DD), a protein interactiondomain found in many proteins (36, 37). A model of a complexof DDs of the death receptor Fas (also called CD95) and the Fas-associated protein with death domain (FADD) are shown in Fig.3 (38). The model, which is based on interfaces seen in crystalstructures (39, 40), predicts that the Fas and FADD DDs form ahexamer through three interfaces. In fact, interactions throughthe three interfaces can generate a distribution of homo- and het-erooligomers: 6 homodimers, 6 heterodimers, 20 homotrimers,66 heterotrimers, and so on. Without regard for physicochemicallimitations, the reaction network implied by the DD-DD interac-tions is of infinite size. Of course, the actual distribution ofoligomers will be limited by a number of factors, such as proteincopy numbers, binding affinities, steric effects, and cooperativi-ty. However, to elucidate how these factors shape the distribu-tion, a modeler may wish to account for all possibilities.

Dynamical models that account comprehensively for thepossible species in a system can help determine which of thesespecies are relevant for signaling, under what conditions, andwhy. In an analysis of such a model (41), Faeder et al. (42)found that only a small fraction of the possible species in themodel is effectively populated, although there are enough pro-teins to populate all species in the model at a nontrivial level.When parameters that affect system dynamics (such as proteinconcentrations) change, the populated species can shift dramati-cally. These shifts are not predicted by reduced models that omitspecies. Other theoretical analyses also indicate that the popu-lated species in a system are influenced by parameters thataffect system dynamics (43–45). Thus, if certain species of asystem appear to be favored, as in temporal ordering of phos-phorylation events (35), a dynamical explanation might be sug-gested and evaluated with the aid of a model that has the capa-bility to predict how the favored species will be affected byperturbations of system dynamics. A model that has such acapability is one that accounts for all possible species.

Combinatorial complexity raises a series of problems when oneis modeling signal-transduction systems as chemical reactionnetworks. One problem is model specification, the task of stat-ing one’s knowledge and assumptions about a system in a waythat enables mathematical analysis.

Conventional approach to model specification. The textbookapproach to modeling a biochemical system is to draw a reac-tion-scheme diagram depicting the chemical species and reac-tions in the system and then to translate this diagram, which isessentially an organized layout of a list of reactions, manually in-to a set of equations (46), such as a system of coupled ordinarydifferential equations (ODEs). Reaction-scheme diagrams thathave been drawn for signal-transduction systems, some of whichare quite large (47, 48), are based mostly on knowledge and as-

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sumptions about protein-protein interactions. An example of asimple scheme is provided in Fig. 4. A drawback of a reaction-scheme diagram is that it obscures the underlying protein-pro-tein interactions by not explicitly representing them. Still, a dia-gram is far easier to interpret than the corresponding equations,and the readability of a diagram can be improved by usingiconographic annotation. For example, Fig. 5 shows how thescheme of Fig. 4 can be made more readable with the notationsof Kitano et al. (49) and Faeder et al. (50). In some cases, a reac-tion scheme can serve the purpose of model specification well.

Software tools have been developed to help biologists to drawreaction-scheme diagrams, totranslate these diagrams intomathematical equations, and toperform model-based calcula-tions by using standard methodsof scientific computing (http://sbml.org/). Examples includeCellDesigner (51) and the VirtualCell (52). However, these toolsare useful only if the textbookapproach to model specificationcan be applied. In many cases,combinatorial complexity makesthis approach prohibitively time-consuming and error-prone, evenwith the aid of powerful softwaretools for drawing reaction-scheme diagrams (53).

Limits of conventional model-ing. Although the conventionalapproach to model specificationis problematic for signal-trans-duction systems, which aremarked by combinatorial com-plexity, it is nevertheless theapproach most often used tospecify models for these sys-tems, but at a cost. Models de-rived in this way are invariablybased on assumptions, whichmay be difficult to justify, that limit the chemical species andreactions considered to a fraction of those possible. An exampleis the model of Kholodenko et al. (54) for EGFR signaling,which has been extended by a number of researchers (55–58).This model is based on mechanistic assumptions that result in aselective focus on only a fraction of the protein complexes andphosphorylation states that could potentially arise from the pro-tein-protein interactions considered in the model. For example,one assumption is that ligand-induced dimers of EGFR are un-able to dissociate when receptors are phosphorylated, whichseems unlikely. This assumption arises from a description ofsignaling events as an ordered pathway, which is consistent withthe way this system is presented in typical diagrammatic inter-action maps, but inconsistent with rapidly reversible reactionsand multiple branching possibilities. Lifting this and other as-sumptions causes a combinatorial explosion in the number ofpossible reactions and species (59), which makes manual modelspecification impractical.

Rule-based approach to model specification. Given that pro-tein-protein interactions can generate large reaction networks,

what can be done to capture the essence of these interactions with-out ignoring their combinatorial complexity? To address this ques-tion, a number of research groups have suggested, in one form oranother, a new starting point for model specification. The basicidea is to specify protein-protein interactions as rules that serve asgenerators of chemical reactions (or reaction events) and species. Arule specifies the features of proteins that are required for or affect-ed by a particular protein-protein interaction. A rule can be viewedas a definition of a reaction class, a generalized reaction. The rele-vant features of an interaction can often be identified, at least to afirst approximation, because of the modularity of protein catalytic

and interaction domains (19).In one approach to rule-

based modeling (60, 61),which is implemented in theBioNetGen software pack-age, rules are used as genera-tors of chemical species andreactions as follows. A rulecomprises patterns for recog-nizing reactants, a mappingof reactants to products, anda rate law. Given a set of ini-tial chemical species, repre-sented with text strings orgraphs, each rule is used toidentify, through patternmatching, the species thathave features required to un-dergo the transformationfrom reactants to productsspecif ied in the rule. Eachtransformation is performedto obtain product species andthe transformation is as-signed a rate law, the oneassociated with the corre-sponding rule. By conven-tion, it is assumed that theinteraction represented in arule is independent of fea-

tures not explicitly indicated. Thus, multiple species may qualifyas reactants in a type of reaction defined by a rule, and multiplereactions may be generated that have the same characteristic ratelaw, although parameters of the rate law may need to be adjustedfor a variety of contextual reasons (60, 62). The exact number ofreactions generated by a rule depends, in general, on the entireset of rules in which the rule of interest is embedded and also onthe set of species to which rules are initially applied. The speciesand reactions may either be generated in advance of a simula-tion, the “generate-first” approach, or during the course of asimulation, the “on-the-fly” approach (60, 62, 63). With thegenerate-first approach, a network of species and reactions isgenerated through iterative application of the rules until a speci-fied, arbitrary termination condition is satisfied or no new reac-tions are generated (60). With the on-the-fly approach, reactionsare generated as new species become populated, which may beadvantageous when the network is large or unbounded, as is thecase when rule application is nonterminating in the absence ofan arbitrary halting condition (60, 62). An example of a rule forwhich rule evaluation is nonterminating is provided in Fig. 6.

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Applications of rule-based modeling and first software toolsfor model specification. To date, rule-based modeling has beenapplied to only a handful of systems, but there is an increasingawareness of the need for such models. In considering whethera particular application represents an example of a rule-basedmodel, we ask the following question: Are the chemical speciesand reactions in the model drawn from a fixed user-specifiedlist or are they generated automatically in some way by a com-puter algorithm from a set of rules? The latter represents whatwe call rule-based models. The programs used to generate rule-based models can be divided into two categories: those that arespecific to a particular application and those that generalize to arange of problems.

Rule-based modeling has been applied to bacterial chemo-taxis mediated by the Tar-receptor complex [for reviews of thebiology, see (64, 65)]. The complexity of models for receptor-mediated phosphorylation events (66) and the aggregation ofreceptors (67) led Bray and co-workers to develop two general-purpose computer programs, OLIGO (68) and STOCHSIM (69).These programs divide the modeling problem into two parts—formation of multisubunit signaling complexes, and phosphory-lation cascades.

OLIGO uses graphs to represent multiprotein complexes,with each node representing a protein and each edge represent-ing a noncovalent interaction between two proteins. A userdraws a contact graph with the help of a graphical user interface(GUI). OLIGO then generates a reaction network by disassem-bling the complex in all possible ways. For example, the six-subunit complex that is active in Tar-mediated signaling is de-composed into its three protein constituents, and in the process,a network of 14 reversible binding reactions is generated. Asubsequent program, OLIGO-D (43), generates a network in theopposite order, starting with a list of proteins and bindingsites. Both programs generate networks on the basis of theassumption that rings form whenever possible. This simplifi-cation prevents polymerization through chain propagation.Although this assumption may be valid in some cases, thereare cases where extensive oligomerization of signaling pro-teins occurs (70). A further and more severe limitation ofthese software tools is that regulation of protein interactionsthrough covalent posttranslational modif ications, such asphosphorylation, cannot be considered.

Use of OLIGO-D provided a qualitative explanation for howsignaling networks can exhibit a decrease in signaling whencomplex-forming proteins are overexpressed, a phenomenon wewill refer to as high-dose inhibition. Bray and Lay (43) foundthat bridging subunits that connected separate parts of a com-plex were particularly likely to exhibit such effects, but alsoshowed that any subunit with multiple bonds to a complexcould exhibit high-dose inhibition if the binding constants in thenetwork were optimized for such behavior. Levchenko et al.(44) modeled high-dose inhibition for the particular contact ge-ometry of a scaffold protein that binds kinases of a mitogen-ac-tivated protein (MAP) kinase cascade. Although this model wasconstructed manually, a Mathematica-embedded biochemicalmodeling package called Cellerator (71), an early example ofsoftware for rule-based modeling, was later used to generate au-tomatically and to solve the differential equations for a general-ized model of kinase-scaffold interactions (72). Specific func-tions (rules) were written to generate the species and reactionsfor scaffolds and scaffold dimers with various numbers of bind-

ing sites and other properties. The effects of previous assump-tions made to restrict network complexity, such as a rapid sec-ond phosphorylation step for kinase activation, were investigat-ed and found to have a quantitative, but not qualitative, effecton the observed behavior of the system. In particular, the scaf-fold concentration for optimal downstream activation was foundto be insensitive to modeling assumptions.

STOCHSIM (73, 74), the other main modeling package devel-oped by Bray and co-workers, is complementary to OLIGO inthat it allows detailed modeling of the kinetics of covalent mod-ifications, but has fairly severe limitations for modeling com-plex formation. Further discussion of STOCHSIM’s design andcapabilities is provided below. An interesting feature ofStochSim is that it can be used to consider nearest-neighbor in-teractions on regular two-dimensional lattices. This capabilitywas developed to study lateral interactions among receptors in astatic configuration (75, 76).

Goldstein and co-workers have developed a detailed modelof early events in signaling by FcRI (41), the high-affinity re-ceptor for immunoglobulin E (IgE) antibody, which plays animportant role in allergic reactions. The model considers phos-phorylation and binding events initiated by receptor aggregationand encompasses 354 chemical species and 3680 chemical re-actions, but it is based on only 25 parameters (21 rate constantsand 4 protein concentrations). This economy of parameters isachieved by using reaction rules describing conditional proteinbinding and phosphorylation events to generate the species andreactions that arise in the network model. Each reaction gener-ated by a given rule uses the same rate constant. Thus, reactionrates are assumed to depend on a subset of the features of reac-tant species. A typical assumption is that the rate constant forligand-receptor binding is not affected by the cytoplasmic stateof the receptor. Analysis of this model has provided a number ofinsights into how the rate constants and affinities that governparticular interactions within receptor aggregates affect the out-come of signaling (41). For example, it was found that the mul-tiple requirements for transphosphorylation within a receptoraggregate enable the system to discriminate effectively betweenligands that bind with short and long half-lives (9), an effectknown as kinetic proofreading (77).

Chakraborty and co-workers have developed two distinctmodels of early events in T cell receptor (TCR) signaling thathave led to interesting insights and hypotheses about the effectsof spatial organization (45) and cooperative interactions (78) onsignal amplification. In the model of Lee et al. (45), the junctionbetween a T cell and a stimulatory antigen-presenting cell isrepresented by a three-dimensional lattice on which variousproteins involved in early signaling events diffuse and react. Aset of interaction rules based on a detailed reaction scheme isused to determine which biochemical events can occur betweena given pair of particles. At each simulation step, one of threepossible update moves is selected at random with equal probabil-ity: diffusion of a molecule or complex, association or dissocia-tion involving a pair of molecules chosen at random, or a statetransition involving a pair of molecules chosen at random. Thismethod avoids the problem of explicitly generating species andreactions but is restrictive in that the computational cost scaleswith the number of particles squared, limiting simulations torelatively small numbers. A more severe problem that is specificto this model is that although the model includes both reactionand diffusion effects, little attempt is made to simulate the cor-

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rect rates of reaction and diffusion relative to each other or thecorrect absolute rates of reactions. Because quantitative simula-tions can be performed by using physically correct and computa-tionally tractable methods (79), it might be interesting to do so tosee if conclusions inferred from the original model change.

A different, nonspatial model was used by Li et al. (78) tomodel cooperative effects involving the TCR, its coreceptor CD4,

and its ligands [peptide–major histocompatibility complex(MHC) complexes]. The model was designed to test the hypothe-sis that low-affinity binding between MHC molecules bound toendogenous (self) peptides could amplify signaling by a smallnumber of high-affinity peptide-MHC complexes containingantigenic (foreign) peptides. The basic mechanism for coopera-tive enhancement was proposed to be the formation of a pseudo-

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dimer complex, in which a high-affinity bond between antigenicpeptide-MHC and a TCR acts as a scaffold to recruit a CD4molecule, its constitutively associated molecule of the kinaseLck, and a second peptide-MHC molecule, most likely bound toendogenous peptide. The pseudo-dimer complex serves as an ef-ficient machine for TCR activation, because TCRs that transient-ly bind the free peptide-MHC in the complex come into closeproximity with the Lck, which phosphorylates TCR as the initialstep in TCR activation. Reaction rules are given that describe theassociation and dissociation of these components, as well asphosphorylation and dephosphorylation, up to hexameric com-plexes (the size of the pseudodimer loaded with TCR), generatinga network of 741 distinct reactions. A special-purpose programwas used to generate the network, and dynamics were simulatedby using the Gillespie algorithm (80, 81). The model is construct-ed and parameter values are chosen so as to favor the formationof pseudo-dimer complexes only when high-affinity peptide-MHC is present. Under these conditions, cooperative enhance-ment of TCR signaling is observed at low densities of antigenicpeptide, and the strength of this enhancement depends stronglyon the affinity of interactions between TCR and the low-affinity(endogenous) peptide-MHC, which suggests a possible biologicalrole for positive selection of TCRs that exhibit a relatively highaffinity for endogenous MHC. On the basis of new experimentalevidence, Krogsgaard et al. (82) have proposed a modification ofthe pseudodimer model in which the second TCR is recruited tothe complex through its interaction with CD4, and it would be in-teresting to see how this change might affect the behavior of thenetwork model.

Woolf and Linderman (83) used rules to model binary inter-actions among sets of G protein–coupled receptors (GPCRs),which they took to be in active or inactive states. Using MonteCarlo methods, they determined how rules for GPCR dimeriza-tion and changes of these rules affected the spatial distributionof receptors on a two-dimensional grid and receptor signaling.Using their rule-based models, they were able to make predic-tions consistent with experimental measurements of receptorclustering and second messenger signaling.

Haugh et al. (84) used a series of rule-based models to in-vestigate mechanisms by which protein tyrosine phosphatases(PTPs) might regulate signaling through their association withmembrane-bound receptors. Several cytosolic PTPs containSH2 domains that allow them to bind directly to phosphorylatedreceptors. These interactions can increase PTP activity throughat least three mechanisms: (i) direct allosteric activation, (ii) in-direct allosteric activation through phosphorylation of the PTPby a receptor-associated kinase, and (iii) proximity to receptorphosphotyrosine substrates. Haugh et al. (84) found that the re-lation between PTP activity and receptor activation could betuned by alterations in the relative expression levels of proteinsthat competed with PTPs for binding sites, leading to high-doseinhibition (with respect to the degree of receptor activation) un-der some conditions. The differential equations of the modelwere generated by a special-purpose program. The model alsoincluded algebraic equations, because Michaelis-Menten ratelaws were used to characterize the kinetics of phosphorylationand dephosphorylation reactions with multiple substrates. Theseequations can be avoided by using elementary reactions of theMichaelis-Menten mechanism to model catalytic steps (85), atthe cost of increasing the size of the reaction network (86).

An important application of rule-based modeling has been to

examine the validity and implications of assumptions that have tra-ditionally been made to limit the extent of combinatorial complexi-ty. Conzelmann et al. (87) and Blinov et al. (59) have both devel-oped rule-based versions of the EGFR signaling model originallydeveloped by Kholodenko et al. (54). Because the primary focus of

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Conzelmann et al. (86) was model reduction, which we discusselsewhere, we focus here on the efforts of Blinov et al. (59).

The rule-based version of the EGFR model was developed bygeneralizing the individual reactions in the Kholodenko modelwith the BioNetGen software package (88). Each reaction wasassumed to be a specific instance of an interaction that could beexpressed more generally as a reaction rule by using the rate con-stant of the original reaction. Thus, there is a one-to-one corre-spondence between reactions in the original model and reactionrules in the expanded model, and no new rate parameters are in-troduced. As an example, in the original model, dimerization ofEGFR bound to the epidermal growth factor (EGF) is describedby a single reaction with forward and reverse rate constants k+2and k–2, respectively. In the expanded model, this reaction is con-verted to a reaction rule that specifies that a molecule of EGFRthat is bound to EGF but has a free dimerization site can bind toanother EGFR molecule meeting the same criteria with the sameforward and reverse rate constants as in the original model. Be-cause EGFR has several additional binding and phosphorylationsites, this rule applies to a large number of EGFR-containingspecies and generates a large number of potential dimerization re-actions. In addition, this rule lifts an assumption implicit in theoriginal model that dimers of EGFR could only break up if bothreceptors were unphosphorylated. This assumption precluded theexistence of phosphorylated EGFR monomers, which have beenshown to be significant contributors to signaling through EGFRunder certain conditions (89).

For the same values of the rate parameters, the original and ex-panded models were found to predict nearly identical behavior forthe outputs examined by Kholodenko et al. (54), but the expandedmodel makes new predictions that can be tested experimentally.For example, the expanded model predicts different phosphoryla-tion kinetics for the two primary EGFR tyrosines included in themodel. Recent proteomic measurements indicate that differentphosphorylation sites on EGFR exhibit different kinetics and thatthe kinetics depend on the interaction partners at the various sites(14). These results provide motivation for considering individualtyrosines in models. Further motivation is provided by results indi-cating that small-molecule kinase inhibitors have different effectsat different tyrosines (90). In summary, recasting a heuristic modelof a signaling pathway increased the computational complexity,but did not require additional model parameters and also facilitat-ed understanding of the model by unveiling hidden assumptions inthe original model. The rule-based model better reflects our un-derstanding of protein-protein interactions, is more easily extend-able, and generates additional predictions.

In addition to model specification, a number of other problemsarise from the large number of possible reactions in a signal-trans-duction system. In this section, we briefly review how rules havebeen used to improve the efficiency of procedures for simulatingthe dynamics of a system. We also discuss systematic methods forreducing a model to a simpler one and for verifying that the dynam-ical behavior of a model is consistent with specified properties.

Simulation. The cost of simulating the kinetics of a reactionnetwork depends on the size of the network. The exact scalingof cost with network size depends on a number of factors andvaries from problem to problem, but the scaling is nonlinear forpractical methods. Thus, if a network is large, the simulationcost can be expensive in terms of computation time and/or com-

puter memory. To address this problem, Lok and Brent (62) pro-posed a method of simulation that takes advantage of rules, theon-the-fly method mentioned earlier, which is similar to ap-proaches that have been used to simulate chemical systems(91–93). In this method, rule evaluation is embedded in a dis-crete-event Monte Carlo simulation of reaction kinetics, and reac-tions are generated only when a species is first populated duringa simulation. Thus, parts of a network unconnected to populatedspecies are ignored, which may speed calculations and reducememory requirements. This technique of on-the-fly network gen-eration provides a practical benefit only when the populated frac-tion of a network is sufficiently small and branching is limited.For a highly branched network, the cost of updating the list of re-actions connected to populated species can be an important factoreven if few of the possible species are populated (Fig. 7).

The problem of simulating a highly branched network isaddressed by another Monte Carlo method that takes advantageof rules (69, 94) and is implemented in the STOCHSIM softwarepackage (73). In the STOCHSIM algorithm, rules are used dur-ing a simulation to generate discrete reaction events (that is, toexecute reactions), rather than to generate reactions. A list of re-actions is unnecessary, and the cost of generating one is avoid-ed. Thus, the STOCHSIM algorithm may provide a computation-al cost savings when many reactions are possible but only afraction actually occur. However, its applicability is limited, be-cause STOCHSIM rules explicitly represent only state changes ofproteins. Representing changes of connectivity can be problem-atic. Thus, for example, STOCHSIM cannot be used in a straight-forward way to model the system of Fig. 6.

New simulation capabilities are needed for rule-based mod-eling, because reaction networks derived from rules are unusu-ally large, which is problematic for conventional simulationprocedures. It may be possible to develop additional methods,like those mentioned above, that take advantage of the underly-ing rules of a rule-derived reaction network to speed simulation.For example, it may be possible to extend on-the-fly methodolo-gy from stochastic simulations to ODE-based simulations (95,96), which would likely be more efficient. Current on-the-flymethods essentially consider a species or set of species to berelevant if it is populated by just one molecule, which triggersreaction generation. It would be desirable to refine these meth-ods to incorporate more stringent definitions of relevance suchthat the trigger for reaction generation can be tuned for efficien-cy. Finally, we wonder whether the method of STOCHSIM couldbe made to work with more general rules or could be imple-mented in hardware (97, 98). A general conclusion we drawfrom our experiences with simulation to date is that, becausethere are trade-offs involved in all of the methods that havebeen developed, it is highly desirable for a software platform toprovide access to multiple simulation approaches through thesame interface.

Model reduction. Another way to address the issue of combi-natorial complexity is through model reduction. Some researchershave considered ways to obtain a model that is smaller than oneneeded to represent all microscopic details, yet behaviorally equiv-alent to it (42, 87, 99–101). For example, Borisov et al. (99) sug-gested lumping microscopic species of a model together into newvariables to obtain an equivalent dynamical model in terms ofcoarse-grained variables. Recently, Conzelmann et al. (101) havedeveloped a systematic method for obtaining reduced models inthis way. Exact time courses of the macroscopic variables are ob-

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tained by solving a reduced set of differential equations that in-cludes both the macroscopic variables and possibly a set of auxil-iary (mesoscopic) variables. The extent of model compression de-pends on the degree of independence among the protein sites be-ing modeled, because correlation among sites requires tracking ofadditional auxiliary variables. A number of factors, such as coop-erative binding or enzyme-substrate relations within a complex,can give rise to coupling among sites. For example, the phospho-rylation states of two sites in a receptor are correlated if binding ofa cytosolic PTK to one site depends on phosphorylation of thesite, and the PTK, once recruited to the receptor, catalyzes phos-phorylation of the second site. The variable transformation ap-proach has so far only been applied to the case of a single scaf-fold protein containing multiple sites of modification and bind-ing. More work is needed to extend the approach to more com-plex situations. In addition, it would be desirable to develop asystematic method for developing approximate reduced models(for example, by dropping some of the auxiliary variables),which might be helpful when exact compression is not possiblebecause of correlations between sites (100). The problem of

model reduction through aggregation of variables is as difficultas it is necessary, in particular for systems that operate on sever-al spatial and temporal scales.

Model checking. Model checking (102) refers to a suite oftechniques deployed to prove that a model of a system behavesin a specified manner. The techniques originate in computer sci-ence and engineering, where they are also known as programverification. For example, an objective of program verificationis to prove that a highly complex piece of software meets designrequirements. A useful step in this direction is to construct asimplified program (a model) that preserves essential designcharacteristics, and then proving (or disproving) its compliancewith specifications. This is typically done before the actualpiece of software is built, precisely to avoid early design flaws.In the present biological context, the “simplified program thatpreserves essential design characteristics” is a rule-based modelof a signal-transduction system. Various researchers havesought to apply the powerful tools developed in the context ofprogram verification to biological models, particularly whenthey can be cast in the form of rules that are executed stochasti-cally on the fly. Consider a volume in which molecular speciescollide randomly and react when an appropriate rule becomesapplicable, much as in STOCHSIM described above. Unlike theunique deterministic trajectories of ODEs, a stochastic simu-lation will only yield a particular history at each run. Evenwhen many runs are gathered to perform a statistical analysis,observing a time series of concentrations (or molecule num-bers) does not necessarily lead to an understanding of themodel (and the system it is intended to elucidate). Moreover, adetailed time series may not be the most appropriate represen-tation to match against empirical observations that are qualita-tive or expressed in terms of salient events. Here is wheremodel checking becomes useful.

The model checking process requires two inputs: (i) a suit-able representation of the signal-transduction model and (ii) aproperty whose truth within the model we wish to check. In theprocess, a rule-based reaction network is translated (automatical-ly) into an automaton. An automaton is a graph whose nodesrepresent global states of the system (a global state is determinedby the numbers of all molecular species) and whose edges repre-sent transitions between states due to possible reactions. Anautomaton so defined represents, at once, all possible trajecto-ries the system can take and may therefore be very large, eveninfinite. Yet, it does not have to be held in computer memory inits entirety. As with on-the-fly methods described previously, theautomaton graph can be expanded when needed by computingthe partial graph of accessible states (nodes) from the currentstate. As the nodes of the automaton are generated, they are tra-versed systematically (breadth- or depth-first); at the same time,the automaton is checked in a divide-and-conquer fashion (thatis, by recursively breaking the problem down into simpler sub-problems) to determine whether the given property holds. Thisprocedure reflects a strategy similar to that implemented in pro-grams that play chess. Most important, when a property fails tohold, the procedure will reveal why the property fails by exhibit-ing the transition that led to its violation. Model checking is nota simulation, but a clever exhaustive search of the state space ofa model. Naturally, model checking has limitations with regardto the type of properties that can be effectively checked.

Model checking requires properties to be expressed as for-mulas in a logical framework, such as standard first-order logic

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(predicate calculus) augmented with operations that capturetemporal relations between events (“before,” “after”); temporalmodalities (“sometimes,” “always,” “next time,” “until”); con-straints; and perhaps even structural properties (such as graphconnectedness).

The application of model checking to biological systems isin its infancy. At present, biological applications constitutemore an exploration of methods than breakthroughs in biologi-cal insight. An illustrative example is the analysis of Kohn’smap of the cell cycle (103). The properties checked for theKohn map were static assertions, such as, “The activation ofCDC25C is necessary to generate the CDK1-cyclin B com-plex,” as well as dynamic ones, such as oscillations in theabundance of cyclin A (which translates into, “There is a pathsuch that whenever cyclin A is present it eventually disappears,and whenever it is absent, it eventually appears”). These for-mulas do not push the boundaries of biological insight. Yet,they illustrate the potential of queries for consistency checks,behavioral comparisons across models, and compliance withempirical observations.

Rule-based systems may be translated into different types ofautomata that reflect various levels of resolution. The class ofhybrid automata defines a particularly important level. A hybridautomaton represents a mixture of continuous dynamics anddiscrete events, as in a system that is well described by a differ-ential equation model until a condition is reached that triggers adiscrete event altering the dynamics. For example, the net rateof production of a protein may switch from negative to positivewhen the concentration of another protein falls below a thresh-old value. In the class of piecewise affine systems, the dynami-cal equations are linear with an offset. Such automata can serveas qualitative approximations of complicated nonlinear reactionsystems. They effectively transform a nonlinear system into alinear one, but with a structure that can be switched. This typeof model yields a representation of a reaction network that facil-itates model checking. Batt et al. (104) provide an interestingand predictive application of model checking to a hybrid-automaton representation of the nutritional stress response inEscherichia coli. The queries have illuminated the role of mutualinhibition between the transcription factors FIS and CRP, whichare involved in regulating metabolism. Mishra and co-workershave applied hybrid automata techniques and model checking toa stylized Delta-Notch signaling circuit (105). We expect to seeincreasing activity in this area.

General-purpose software is needed to enable rapid rule-basedmodeling of diverse systems. Problem-specific programs can bedifficult to reproduce independently, and the time and effort re-quired to write and debug such a program can be expensive. Al-so, modifications of rules may require reprogramming, whichdiscourages the consideration of alternative hypotheses aboutmechanisms of signaling or extensions of a model. In recentyears, various general-purpose software tools and correspond-ing methods based on rules for protein-protein interactions havebeen developed. Software capabilities and methodology are ad-vancing quickly, and it seems likely that rule-based modelingwill soon be a practical, mature, and accessible method of com-putational systems biology. Below, we discuss several specificsoftware tools and end with a short summary of the lessonslearned from the development of these tools.

STOCHSIM 1.4. The salient feature of this tool (ftp://ftp.cds.caltech.edu/ pub/dbray/) is the representation of proteins ascomputational objects or agents, which have states (73). Modelspecification is accomplished by using a wizard-like GUI,which helps a user create a set of formatted input files definingproteins, their states, individual reactions, and rules for general-ized reactions that change the states of proteins. A protein isrepresented by a list of binary flags encoding the modificationor binding states of its sites. In the state-change rules ofSTOCHSIM, state flags are specified as 0, 1, or “?,” which is awildcard, and plus (+) and minus (–) symbols are used to directthe switching of flags from 0 to 1 and vice versa. A limitation ofSTOCHSIM is its representation of complexes. A complex is ei-ther represented implicitly in terms of protein states, in whichcase the topology of a complex may be difficult to track, or it isrepresented explicitly, in which case the molecular compositionmust be declared manually beforehand in an input file.

An interesting feature of STOCHSIM is its method for dis-crete-event Monte Carlo simulation of reaction kinetics. Themethod involves picking two agents and then checking to seewhether they react. The method is approximate because time isincremented in fixed steps (69, 94). Thus, a check for accuracy,against the results of exact Monte Carlo methods (80, 81) or aSTOCHSIM run with a smaller time step, is necessary. The pro-cedure may be slow because, for correct results, the time stepmust be small enough such that pairs of reactants most often donot react. However, as discussed above, the method avoids gen-erating reactions, which can be an advantage essential for com-putability in some situations. Shimizu and Bray (94) discuss thescenario of a protein with n sites of modification and m bindingpartners. This protein has up to 2n modified forms and may par-ticipate in as many as m2n distinct bimolecular association reac-tions. For such cases and cases like that of Fig. 7, methods thatavoid the cost of reaction generation, such as the STOCHSIMmethod, may be essential for computability.

Moleculizer 1.0. As with STOCHSIM, a set of input files isused to specify a model in Moleculizer 1.0 (http://www.molsci.org/~lok/moleculizer/) (62); these files are written inXML. Template input files are provided that correspond to afixed set of rule types (reaction generators), which are codedas separate software modules. The templates are used to de-fine individual rules and their parameters, such as rate con-stants. Although the types of rules available to a modeler arefixed, they provide sufficient flexibility to represent a widearray of protein-protein interactions. A feature of Moleculiz-er, not available in STOCHSIM, is the ability to represent thetopology of a protein complex explicitly in terms of pairwiseconnections of binding sites. Also, the connectivity of a com-plex can be considered in a rule definition.

As can STOCHSIM, Moleculizer can be used to perform adiscrete-event Monte Carlo simulation of reaction kinetics, butan exact method, Gillespie’s direct method (80, 81), is used.This method relies on a list of reactions, which Moleculizergenerates on the fly during a simulation. When a species is firstpopulated, rules are evaluated and new reactions involving thisspecies are generated if necessary (that is, if they are not al-ready generated and stored in memory). Given the parametersthat govern a simulation of network dynamics, Moleculizer pro-vides a principled means for identifying the relevant portion ofthe reaction network (that is, the populated species and reac-tions connected to these species). Moleculizer provides other

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simulation methods as well, such as a -leap method forstochastic simulations (106). A new simulation feature is “lookahead” rounds of reaction generation (107). If the number oflook-ahead rounds is set to a large enough number, then Mole-culizer can generate a network without simulating it. Thismethod is then equivalent to the generate-first method men-tioned earlier.

BioNetGen 2.0 and BNGL. Earlier versions of this tool (1.0and 1.1), BioNetGen (http://cellsignaling.lanl.gov/bionetgen/),which are based on the use of term or string rewriting rules torepresent protein-protein interactions, are discussed elsewhere(60, 88). We will focus on version 2.0, which is based on theuse of graphs to represent proteins and protein complexes andthe use of graph rewriting rules to represent protein-protein in-teractions (50, 61). For historical perspective and a review ofapplications of graph transformation in molecular biology, seeRosselló and Valiente (108). Graph rewriting rules are general-izations of term rewriting rules. Unlike STOCHSIM and Mole-culizer, BioNetGen interprets a formal model-specification lan-guage, called the BioNetGen Language (BNGL). An advantageof introducing a language is that model specification becomesindependent of software implementations and thereforeportable. In subsequent sections, we will discuss several lan-guages for rule-based modeling that have been proposed. Inter-esting languages that we will not discuss include Bioglyphics(http://www.bioglyphics.org/) (109), Dynamical Grammars(http://www.arxiv.org/abs/ cs.AI/0511073), and “little b”(http://www.littleb.org/).

In BNGL, chemical species are represented by usinggraphs, which are encoded as structured strings. The basic unitof representation is a molecule string, which may containcomponent strings enclosed in parentheses. In a model speci-fication, molecule strings actually serve several purposes, oneof which is definition of the types of molecules included ina model. An example of a string used for this purpose isEGFR(ECD,aa1092~Y~pY), which defines a molecule calledEGFR containing components called ECD (ectodomain) andaa1092. The string also indicates, by the prefix “~” for statelabels, that the latter component has two possible internalstates called Y and pY. This definition corresponds to one typeof molecule represented in the reaction network of Fig. 5B,which also shows how molecule strings are concatenated torepresent complexes.

In BNGL, rules for protein-protein interactions are repre-sented by using text-encoded graph rewriting rules, whichcontain graph matching patterns. Patterns composed of(incomplete) molecule strings explicitly indicate the featuresrequired of reactants and implicitly indicate, through omis-sion, the features that are irrelevant for a reaction. An exampleof a rule is Grb2(SH2)+EGFR(aa1092~pY) Grb2(SH2!1).EGFR(aa1092~pY!1). This rule indicates that the adaptor pro-tein Grb2 and EGFR can associate if the SH2 domain of Grb2is free and residue 1092 in EGFR is free and phosphorylated.By omission, the rule also indicates that association of Grb2and EGFR is independent of the bound state of the ECD ofEGFR. The “!” character is a prefix for bond labels, whichindicate how components of proteins are connected in a com-plex. The bond labels in this rule indicate that the SH2 domainof Grb2 binds pY1092 in EGFR. The rule for Grb2-EGFRassociation and others are illustrated in Fig. 8 by using theconventions of Faeder et al. (50).

A model is specif ied as a set of graph-rewriting rules,which are associated with rate laws, and a set of seed-speciesgraphs to which the rules are initially applied. Using the itera-tive algorithm for processing rules outlined above (60, 61),BioNetGen can generate a parameterized reaction network(meaning that the reactions are assigned rate laws) withoutperforming a simulation of the network dynamics. Once anetwork has been generated, it can serve as the basis for ODE-based or stochastic simulations (60). Pregenerating a networkand then performing an ODE-based simulation, if possible,may be more efficient than the stochastic simulation methodsof STOCHSIM and Moleculizer, because ODE-based calcula-tions are usually less expensive than Monte Carlo calculationsfor a “small” model (63). If the size of a network is too large topermit ODE-based calculations, then BioNetGen also providesthe capability to generate a network on the fly during a simulation,as Moleculizer does.

One problem that BioNetGen attempts to solve is how toautomatically adjust the rate law of a rule to account for contex-tual differences among reactions in the class of reactions de-fined by the rule. Contextual effects on rate laws can be speci-

1092~YEGFR EGFR

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EGFR(aa1092~pY) -> EGFR(aa1092~Y) k2

EGFR(aa1092~pY) + Grb2(SH2) <-> EGFR(aa1092~pY!1).Grb2(SH2!1) k3,k4

et al50et al61

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fied explicitly in an expanded set of rules, but this approachmay be laborious, and the problem arises often enough that gen-eral automatic solutions merit pursuit. The contextual factorsthat may necessitate modifications of a rate law for a class ofreactions are numerous: (i) Collision frequency can vary (com-pare A + A and A + B reactions); (ii) reaction path degen-eracy (that is, the number of distinct reaction paths from reac-tants to products) can vary, giving rise to different statisticalfactors (compare A.AA.B and AB reactions); (iii) theturnover frequency of an enzymatic reaction may depend on thenumbers of enzymes and substrates in a complex; (iv) a factorequal to a volume ratio may arise for reactions in differentcompartments; (v) diffusion-limited reactions may be affected bymasses (or equivalently, diffusivities) of reactants (60, 62); and soon. BioNetGen handles statistical factors, for example, by consid-ering the symmetries of graphs in reaction rules and the reactionsgenerated by rules (61). Assigning correct statistical factors isessential for a self-consistent model. Figure 6C illustratesreactions of the same class that have different statistical factors.

BioSPI and other tools of process algebra. BioSPI interpretsa model-specification language, a variant of the -calculus, andprovides the capability to perform a stochastic simulation(http://www.wisdom.weizmann.ac.il/~biopsi/) (110). Similartools, which have also been used to specify models for biologi-cal systems, include SPiM (111) and the PEPA Workbench(112). The -calculus is a general and minimalist model-speci-f ication language originally designed to capture essentialfeatures of concurrent and distributed systems in computerscience (113). It is one of many process calculi (algebras) stud-ied in this field (114, 115). These languages are axiomatic,which allows properties of a model specif ication to beformally proven and facilitates certain types of model check-ing (for example, the observational equivalence of two modelspecifications can be determined).

Use of -calculus to model protein-protein interactions wassuggested by Regev et al. (116), who provided, loosely speak-ing, the following translations of biochemistry to . Proteinsare mobile processes (meaning agents that exchange messagesthat can affect their behavior); protein sites are communicationchannels (ports through which messages are passed fromsenders to receivers); and protein-protein interactions are com-munications. The channels of a system and the messages thatcan be sent and received along these channels are essentiallyrules for protein-protein interactions. The proteins in a com-plex are linked by a backbone (that is, by colocation in a com-munication compartment), and the topology of a complex is in-dicated by pair-wise communications. In this framework, sig-nal transduction can be viewed as asynchronous concurrentcomputation and studied by using pertinent methods fromcomputer science (117–119).

Because -calculus is a minimalist language and some of itsnotational features are irrelevant and even inappropriate for bio-logical applications (for example, communication has direction-ality, whereas physical association does not), researchers havesought to develop a more congruent higher-level language formodeling biochemical systems that retains the mathematicalformality of . One proposal is -calculus (120). In this lan-guage, a protein is represented as a collection of sites, whichcan be visualized as a box with nodes, representing sites, on theborder. Coincidentally, the appearance of such a box is muchlike that of a state node in a process diagram (49). Also, protein

complexes are represented as graphs, in which edges connectinteracting sites of proteins. For simulation purposes, the -calculus can be reduced to -calculus.

Although process algebra has been used to specify biologicalprocesses in some detail, such as cell adhesion (121, 122), it isunclear at this time whether methods of process algebra canreach fundamentally beyond the level of understanding provid-ed by ODEs and stochastic simulations. Process algebra may,however, provide a formal foundation for rule-based modelingthat enables principled mathematical reasoning about systembehavior and its dependency on interaction capabilities of sys-tem components.

Pathway Logic Assistant. Pathway Logic Assistant providesan interface to “Pathway Logic” models of signal-transductionsystems (http://www.csl. sri.com/users/clt/PLweb/pl.html)(123) def ined by using the Maude specif ication language(http://maude.cs.uiuc.edu/). Pathway Logic models can for-mally represent protein-protein interactions at different levelsof resolution, ranging from details about abstract proteinstates (124) to details about functional domains and bindingsites (125, 126). Eker et al. (127) proposed conventions forusing Maude to specify term rewriting rules for protein-protein interactions. Later, conventions for specifying graphsand graph rewriting rules were introduced (126), which allow thetopology of a protein complex to be explicitly considered. Themodels obtained with Maude are essentially unparameterizedchemical reaction networks, meaning that rate laws are notspecified for reactions. With the addition of rate information,a reaction network can be converted to ODEs, for example,but this capability is not native to Pathway Logic Assistant.Instead, this tool, through the formalism of Petri nets, enablesa qualitative analysis of a reaction network, such as visualiza-tion and interactive exploration of the network. Also, a model-er can specify a formula in linear temporal logic (LTL) thatdefines a putative property, and then evaluate this formula in amodel-checking query to determine whether the propertyholds for the system. Properties that can be specified in LTLare qualitative and relate to stable states of paths. For example,a query can be specified to determine whether a particularprotein complex is potentially present in a signal-transductionsystem and to find a sequence of reactions leading to it. Ofcourse, the value of qualitative analysis is limited, as thedynamics of protein-protein interactions are important forsystem behavior, for example, dynamics influence whichprotein complexes are populated during signaling (42). Still, itwill be interesting to see what biological insights can beobtained from qualitative analysis alone, because quantitativerate parameters for protein-protein interactions are oftenunavailable or uncertain.

BIOCHAM 2.4. This tool, BIOCHAM 2.4, interprets amodel-specification language in which term-rewriting rules areused to represent protein-protein interactions (http://contraintes.inria.fr/BIOCHAM/) (128). Structured strings are used to repre-sent chemical species. Rules, which may contain string-matchingpatterns, indicate how strings representing reactants are rewrittento obtain strings representing products. As is also true for otherterm-rewriting approaches (60, 88, 127), the topology of aprotein complex cannot be explicitly considered in a rule,which can be a limitation. Unlike the -calculus or the generalMaude programming language, but like BNGL, the -calculus,and Pathway Logic, the BIOCHAM language has been de-

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signed specifically for the purpose of modeling biological sys-tems. The language is simple by design but still expressiveenough to specify meaningful models. For example, a modelhas been specified for the cell cycle–control system describedby Kohn (32) (103).

Like BioNetGen, BIOCHAM can process a set of rulesassociated with rate laws to generate a parameterized reactionnetwork and the corresponding ODEs, and then perform andanalyze ODE-based simulations. In addition, like PathwayLogic Assistant, BIOCHAM can evaluate model-checkingqueries, which can be specified by using Computation TreeLogic (CTL) as well as LTL. CTL and LTL are closely related;however, CTL queries can be specif ied that cannot beexpressed in LTL and vice versa (129). Model checking in theframework of BIOCHAM is provided through an interface tothe NuSMV model checker (http://nusmv.irst.itc.it/).BIOCHAM essentially converts a chemical reaction network toa NuSMV input file. Thus, model-checking capabilities arelikewise accessible, in principle, from any rule-based modelingtool that generates a reaction network. An intriguing feature ofBIOCHAM is a machine-learning system for modifying rulesor parameters to obtain consistency with behavioral constraintsspecified in CTL or LTL (130).

Lessons learned. Many of the software tools discussed aboveare based on a formal model-specification language, which hasthe desirable property of being application-independent. A stan-dard language has yet to emerge, but the idea of using rewritingrules to represent protein-protein interactions has been suggest-ed multiple times. Likewise, in several approaches, graphs areused to represent the contact map of protein complexes. Oneneeds to consider the configuration of a complex when connec-tivity affects reactivity, as exemplified in Fig. 9. Chemical reac-tion network models derived from rules tend to be large. It maytherefore be nontrivial to confirm that the model produced by aset of rules is the one intended. A high standard of software reli-ability, including careful consideration of physical chemistry, isessential. To cope with the complexity of models, we need soft-ware tools that assist in reasoning about a model and that pro-vide means for automatically monitoring system properties at aquantitative and qualitative level. For example, BioNetGen pro-vides output rules for calculating properties of sets of species,such as a sum of concentrations of species containing a particu-lar protein. Finally, rules can be used in different ways to studya system. The different approaches now available appear to becomplementary, and new approaches may be useful.

The Systems Biology Markup Language (SBML) (131), whichis based on XML, is a popular standardized format for the elec-tronic storage, exchange, and reuse of mathematical models ofbiochemical systems. A number of software packages are nowavailable that import or export models specified in SBML(http://sbml.org/), and a public repository for annotated SBML-encoded models has been established (132, 133). However,SBML, like most SBML-aware software, is based on the as-sumption that a model can be specified adequately in terms of areaction scheme, which is not likely to hold for a model of a sig-nal-transduction system because of the context-sensitivity andcombinatorial complexity of protein-protein interactions. Ver-sions of SBML up through Level 2 Version 2, which is presentlybeing finalized, do not provide direct support for compactly

representing multiple protein complexes or multiple states ofproteins. Likewise, there is no support for defining rules forprotein-protein interactions or other generalized reactions. Tospecify a model in strict SBML, one must enumerate all of theindividual chemical species and reactions included in the model.Rules used to derive these species and reactions could actuallybe included in a SBML file as annotation, but such annotationwould be nonstandard. Thus, for interpretability, SBML can onlybe used to represent a rule-based model as a list of rule-derivedreactions, which means that SBML encodings of rule-basedmodels tend to be exceedingly verbose and difficult to compre-hend or modify (50, 88). In light of this limitation, severalextensions of SBML that incorporate rules have been proposed

(1342–137). It is anticipated that such extensions will be avail-able with the SBML update to Level 3 (138, 139).

SBML Level 3 is anticipated to introduce a modular lan-guage extension capability, which will allow different languagefeatures to be added to a common language core, which will bebased on SBML Level 2. Language extensions will add syntaxand semantics for software tools that share a common theme,such as the tools for rule-based modeling reviewed here.Models specified in SBML Level 3 will include a declaration ofthe feature sets (beyond the core Level 3 features) required forproper interpretation. The presence of a feature tag will inform a

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software tool reading the model that the model uses that partic-ular feature and will permit the tool to quit gracefully if it doesnot have the necessary interpretive capabilities.

The latest proposed Level 3 extensions for rule-based mod-eling (136, 137), like several of the software tools reviewedhere, are based on the concept of using graphs to represent pro-teins and graph-rewritingrules to represent protein-protein interactions. As dis-cussed above, such rules areexpressive enough to conveyinformation about the contex-tual constraints on a protein-protein interaction and theparts of proteins responsiblefor and affected by an interac-tion. The same is not true ofthe standardized exchangeformat widely used today tostore information about pro-tein-protein interactions inelectronic databases (140).Thus, extension of SBML toincorporate rules for pro-tein-protein interactions hasthe potential to improve notonly our ability to model thedynamics of protein-proteininteractions, but also our abil-ity to archive basic knowledgeabout them. Rules for largenumbers of protein interac-tions might be specif ied onthe basis of high-throughputassays (141), data extractedwith literature-mining tools (142), or statistical analysis oflarge data sets (143). An important next step in the develop-ment of SBML is software for converting a putative SBMLLevel 3 specification of a rule-based model into Level 2 form.

A mathematical model provides at least two benefits. Its impli-cations, no matter how hidden from intuition, can usually berevealed through computer-aided analysis. A model also con-stitutes a precise statement of a modeler’s understanding andassumptions about a system. Thus, it is a powerful tool forsummarizing knowledge about a system, although a model maybe difficult to comprehend. If it is represented as a list of equa-tions or reactions, the formalism of the model obscures the un-derlying grammar of protein interactions that generates thechemical species and reactions included in the model. A widelyused and more comprehensible way of representing knowledgeabout a signal-transduction system is a diagrammatic interac-tion map in which proteins and their interactions (or the func-tional consequences of these interactions) are indicated by la-beled cartoons and arrows. Most maps are ambiguous and lacka mathematical interpretation. However, formal conventionshave been proposed for drawing interaction maps such thatthey have precise meanings (144, 145). An example of a simplemap drawn according to these conventions is provided in Fig.10. It illustrates interactions of EGFR and Grb2. No automatic

procedures for translating this type of map into a mathematicalmodel are yet available (145), although software intended toprovide such a capability has been developed on the basis ofsimilar diagrammatic formalisms (146–149). Nevertheless, au-tomatically translating a map like that of Fig. 10 into a mathe-matical model remains an open problem when combinatorial

complexity is present.An alternative to represent-

ing a model as a list of reac-tions or equations is to expressa model as a set of rules,which resemble reactions inmany ways but are highlycompressive. If the rules aregraph-based, then they can al-so be visualized, and therefore,they provide a new means ofvisualizing knowledge of pro-tein-protein interactions. Anexample of rules that repre-sent interactions considered inFig. 10 is shown in Fig. 8.Similar illustrations would beobtained by using any of theother graph-based model spec-ification approaches discussedabove. An advantage of visualrules over interaction maps isthat the rules can be evaluatedautomatically to obtain amathematical model. Whenthe rules of Fig. 8 are appliedto a seed set of species com-posed of unphosphorylateddimeric EGFR and cytosolic

Grb2, they generate the reaction network illustrated in Figs. 4and 5. Rules can also be made more readable by enhancingthem with graphical notation developed for annotating pro-tein-protein interactions (49, 53) (http://www.sbgn.org/).

New technologies are enabling an increasingly quantitative char-acterization of signal transduction events at the level of proteinsites. The resultant data streams challenge our capacity to inter-pret on the basis of intuition alone. Understanding the dynamicsof signaling systems hinges on appropriate mathematical meth-ods and models. A mathematical model tells a story that con-verts facts into putative knowledge by animating facts with dy-namical behavior, making assumptions explicit, and enablingformal manipulations whose correctness can be assessed. Pre-sumed knowledge generated in this way is more useful, predic-tive, and empirically testable. A model is a bookkeeping devicethat helps us to assess whether we know enough about a system.

The multiplicity of sites at which a signaling protein can bemodified and a protein’s potential for assembling into diversecomplexes result in a combinatorial explosion of molecularstates that stymies conventional modeling approaches. Modelsof signal-transduction therefore require software tools groundedin a mathematical framework that represents proteins as gram-matical structures with specific context-dependent actions.Well-known challenges to modeling include the problem of

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compositionality, that is, the need for a framework for growingmodels in a cumulative fashion by plugging together indepen-dently developed models of subsystems. A different composi-tionality problem arises from the need to juxtapose systems thatoperate at different spatial and temporal scales, or at vastly dif-ferent molecular densities. Some of these challenges are begin-ning to be addressed with the introduction of rules for protein-protein interactions and the development of methods for rule-based modeling, like those reviewed here.

Rule-based modeling is a fundamental departure from tradi-tional dynamical systems based on ordinary differential equa-tions (117). Dynamical systems require all possible interactionsto be explicitly specified ahead of time. In contrast, rule-basedapproaches can be used to spawn reactions and molecular statesonly if the dynamics of the system generates the appropriatecontext (on-the-fly operation). Unlike differential equations,rule-based approaches represent molecules with an internalstructure that encodes potential behavior that may never be trig-gered in a given system but that could be triggered if the systemwere changed. This aspect of rules is what makes rule-based ap-proaches extensible on the go, much like dropping a newmolecule into a reaction mixture alters the intrinsic possibilitiesof that mixture. (Note that a modeler should avoid specifyingrules that are overly general, because such rules may generateunintended reactions when a model is extended.) Rule-basedapproaches do not jettison differential equations. In their sim-plest mode of operation, they enable the automatic generationof ODE systems that are too large to be written down by hand.Although rules have been used to formalize interactions amongobjects of various types for over two decades (150), they onlyrecently have been applied to signal-transduction systems.

General-purpose tools for rule-based modeling would bene-fit from the adoption of a standard specification language fordefining models in software-independent ways. SBML nowserves as such a standard, but it conceives of a model as anexhaustive list of reactions. Because this view fails to recognizethe linguistic character of complex proteins and protein com-plexes, it becomes quickly impractical. In response to thisshortcoming, extensions of SBML consider the adoption ofrules for representing protein interactions. Much remains to bedone to devise languages capable of expressing more fine-grained aspects of protein structure and contextual constraints,including those of a geometric kind. Early examples of captur-ing spatial effects in a rule-based fashion were phenomenologi-cal models of morphogenesis (151) and models of virus capsidassembly (152, 153).

There is a clear need for making the modeling process moreaccessible to nonspecialists. Likewise, it is highly desirable toinvite exploration of the possible, such as neighboring signalingnetworks that are accessible from actual ones by a few shufflingoperations. Deriving mechanistic network models from empiri-cal observations, evolving or designing networks to behave inspecified ways (154), and predicting the consequences of inter-ventions are all tasks that depend on efficient ways of exploringalternative network models. These objectives require the partialautomation of the modeling process. Such automation would becritically enabled by the adoption of a formal language toexpress properties of proteins and their interactions or propertiesof whole systems, as extracted from models (via model check-ing) or empirical observations. This language could be used toannotate the behavior of a protein or a small system and could

be stored alongside other information in databases. A statementin a formal language has a well-defined semantics, that is, aclear and precise interpretation, which could perhaps be quicklygrasped through visualization. At the same time, such a state-ment would lead a double life as a codelet, a fragment of com-puter code, which can be downloaded and used as a componentin a model of any system in which the corresponding protein orprotein-protein interaction plays a role. Formal statements aboutsystem behavior (originating in empirical observations or trustedmodels) could be used as test-suites for new models or refine-ments. We believe the progress in rule-based modeling summa-rized in this review may lead ultimately to a common languagefor systems biology, much like the four-letter code for DNAsequences provides a common language for bioinformatics.

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