Analisa Neraca Massa

Journal of Theoretical Biology 227 (2004) 283–297

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doi:10.1016/j.jtb

Topological analysis of mass-balanced signaling networks:a framework to obtain network properties including crosstalk$

Jason A. Papin, Bernhard O. Palsson*

Department of Bioengineering, University of California, San Diego, 9500 Gilman Drive, La Jolla, CA 92093-0412 USA

Received 8 May 2003; received in revised form 23 October 2003; accepted 5 November 2003

Abstract

Signal transduction networks have only been studied at a small scale because large-scale reconstructions and suitable in silico

analysis methods have not been available. Since reconstructions of large signaling networks are progressing well there is now a need

to develop a framework for analysing structural properties of signaling networks. One such framework is presented here, one that is

based on systemically independent pathways and a mass-balanced representation of signaling events. This approach was applied to a

prototypic signaling network and it allowed for: (1) a systemic analysis of all possible input/output relationships, (2) a quantitative

evaluation of network crosstalk, or the interconnectivity of systemically independent pathways, (3) a measure of the redundancy in

the signaling network, (4) the participation of reactions in signaling pathways, and (5) the calculation of correlated reaction sets.

These properties emerge from network structure and can only be derived and studied within a defined mathematical framework. The

calculations presented are the first of their kind for a signaling network, while similar analysis has been extensively performed for

prototypic and genome-scale metabolic networks. This approach does not yet account for dynamic concentration profiles. Due to

the scalability of the stoichiometric formalism used, the results presented for the prototypic signaling network can be obtained for

large signaling networks once their reconstruction is completed.

r 2003 Elsevier Ltd. All rights reserved.

Keywords: Signal transduction; Network-based pathways; Constraint-based modeling; Extreme pathways; Crosstalk

1. Introduction

Signal transduction is important for essentially allmulti-cellular functions in higher-level organisms. In-creasing financial and human resources in addition togrowing infrastructure are being allocated to signaltransduction research (Gilman et al., 2002). The resultis a wealth of data, and signaling networks are beingdescribed in molecular detail at a rapidly increasing rate.Signaling networks are complex with many levels ofinterconnectivity of many different molecular compo-nents. In order to understand the integrated functionsthat arise from such complex interactions, there is a needfor mathematically based in silico approaches that canbe scaled to account for large cellular-sized networks.

tary data associated with this article can be found, in

on, at doi: 10.1016/j.jtbi.2003.11.016

ing author. Tel.: +1-858-534-5668; fax: +1-858-822-

ess: [email protected] (B.O. Palsson).

e front matter r 2003 Elsevier Ltd. All rights reserved.

i.2003.11.016

The reconstruction of such large networks should nowbecome possible based on vast amounts of data that areaccumulating for signaling networks.Various modeling approaches have been successfully

used to investigate particular features of small-scalesignaling networks. These include: kinetic and spatialanalyses of cell surface receptor mechanisms (Haugh,2002; Lauffenburger and Linderman, 1993; Shvartsmanet al., 2001; Woolf and Linderman, 2003), analyses ofsignaling events within the framework of metaboliccontrol analysis (Kholodenko et al., 1997; Schuster et al.,2000b), kinetic analyses of modular signaling systems(Hoffmann et al., 2002; Schoeberl et al., 2002) andkinase cascades (Heinrich et al., 2002), the incorporationof stochastic mechanisms (Arkin et al., 1998), andmethods for deciphering previously unknown interac-tions in signaling networks (Kholodenko et al., 2002).However, large-scale analyses of signaling networkshave not yet appeared, due in part to a paucity of valuesfor kinetic parameters, concerns regarding the accuracyof existing values for kinetic data (Teusink et al., 2000),

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Single-Input, Single-Output

Single-Input, Multiple-Output Multiple-Input, Multiple-Output

Multiple-Input, Single-Output

complex signaling eventsignal pleiotropy

classical case signal concatenation(A) (B)

(C) (D)

Fig. 1. Classification of signal transduction input–output relation-

ships. The classical case of a transduced signal relates a single input

to a single output (A). Some outputs require the concatenation of

multiple inputs (B). Other signaling interactions occur in which the

transduction of a single input generates multiple outputs, a type of

signaling pleiotropy (C). Complex signaling events arise as multiple

inputs trigger interacting signaling cascades that result in multiple

outputs (D).

J.A. Papin, B.O. Palsson / Journal of Theoretical Biology 227 (2004) 283–297284

strong computational demands of kinetic analyses,limited scalability from small signaling modules usingkinetic models, and a lack of available large-scalereconstructions of signaling networks. Signaling net-works have thus not yet been modeled nor analysed ata scale comparable to that of metabolic and regulatorynetworks (Carlson et al., 2002; Covert and Palsson,2002; Price et al., 2003a; Stelling et al., 2002; Van Dienand Lidstrom, 2002). However, with the emergingreconstruction of large signaling networks (Gilmanet al., 2002), there is a growing need for the developmentof a new framework to study emergent properties ofsignaling networks.In this study, a mass-balanced approach for a

topologically based in silico modeling framework oflarge signaling networks is presented. First, a prototypicsignaling network and its underlying chemical reactionsare presented. Previously, biologically meaningful pro-totypic metabolic and regulatory systems have provento be important tools in the development of methods forin silico analyses of biological processes (e.g. Allen andPalsson, 2003; Bonarius et al., 1997; Covert et al., 2001),a path we now follow with signaling networks. Second,systemically independent signaling pathways were cal-culated and the resultant topological features arediscussed, following established procedures for metabo-lism (Papin et al., 2003). Third, these systemicallyindependent signaling pathways allowed for quantitativemeasures of crosstalk, the degree of interconnectivity ofsignaling pathways. Additionally, complex input/outputrelationships were classified and analysed. This topolo-gically based modeling framework will allow forcharacterizations of properties emerging from the net-work structure of large-scale signaling systems. Thisanalysis focuses on the structure of signaling networks;future work will need to account for compartmentaliza-tion of signaling events, as well as systemic descriptionsof dynamic properties.

2. Conceptual framework and methods

Signal transduction events can be divided into fourbasic categories (Fig. 1). The ‘‘classical case’’ of asignaling event is when a single external signal generatesa single response, typically a transcriptional event(Fig. 1A). Some events require the concatenation ofmultiple external signals to generate a single response(Fig. 1B). A single external signal can generate multipleoutputs, often referred to as signaling pleiotropy(Fig. 1C). A more complex signaling event occurs asmultiple input signals enter a system comprised ofinteracting, shared components that can generate multi-ple outputs (Fig. 1D). This intermingling of inputs andoutputs arises when the networks grow in size, leadingto fundamentally different behavior from the other three

categories in Fig. 1. There can be multiple ways toconnect inputs and outputs. As shown below, thesecomplicated interconnections in signaling networks canbe decomposed into underlying, irreducible input/output relationships, thus allowing for a topologicalanalysis of all possible types of signaling events in anetwork.

2.1. Prototypic signaling network

The prototypic signaling network used in this study(Fig. 2) represents several characteristics of real signal-ing networks. There are 24 reactions in this prototypicnetwork. Five reactions depict binding events betweenfive ligands and three receptors. Six reactions representthe activation of intracellular molecules by the activatedreceptor. There are three additional reactions in whichintracellular molecules phosphorylate each other. Sevenreactions involve the activation of transcription factors.The final three reactions depict the formation oftranscription factor complexes. There are an additional19 exchange reactions for components that enter andexit the network (e.g. ATP, ADP, transcription factors).This prototypic signaling network incorporates three

types of multifunctionality, or pleiotropy, that are foundin real signaling networks:

* Multiple ligands can bind to multiple receptors(Fig. 2A). For instance, six different ligands havebeen identified which can bind to the epidermalgrowth factor receptor (Carpenter, 2000).

* One activated receptor–ligand complex can activatemore than one secondary molecule (Fig. 2B). Forinstance, an activated transforming growth factor-b(TGF-b) tetrameric receptor can activate the

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Fig. 2. A prototypic signaling network. The compounds that only exist internal to the system and those that can be exchanged across system

boundaries are indicated. Although all the reactions shown are simultaneously considered in the analysis herein, the dashed lines indicate groups of

reactions that have similar properties. The name of the reaction is indicated above the line that connects substrates to products. Reaction groups A,

B, C, D, and E are described in the text, and individually detailed in Table 1.

J.A. Papin, B.O. Palsson / Journal of Theoretical Biology 227 (2004) 283–297 285

Smad-family of transcription factors and the TGF-b-activated kinase 1 (TAK 1) signaling pathway (Klooset al., 2002).

* An intracellular signaling molecule can activatemultiple additional signaling molecules (Fig. 2C).An example of this interaction takes place with theactivation of the signaling molecule Ras. Evidenceexists for the activation of PI3K and the MAPKproteins by the activated Ras protein (Pouysseguret al., 2002).

These three types of multifunctionality are repre-sented in the prototypic signaling network.Two additional features of signaling networks are

represented in the prototypic network. First, the abilityof a combination of signaling molecules to activate acombination of additional signaling molecules is ac-counted for (Fig. 2D). An actual example of this type ofinteraction is the Wnt/b-catenin signaling pathway that

involves a large protein complex and is important indevelopmental processes (Miller et al., 1999). Second,the prototypic signaling network contains representa-tions of the formation of transcription factor complexes(Fig. 2E). Transcription of the human interferon-bgene, for example, requires the presence of a trimerictranscription factor complex of STAT1, STAT2, andp38 proteins (Stark et al., 1998). Although not includedin the prototypic signaling network, signaling eventsthat involve conformational changes can be easilyrepresented within this framework.

2.2. Stoichiometric formalism: describing signaling

networks as systems of chemical reactions

The prototypic signaling network (Fig. 2) is com-prised of a series of chemical transformations as listed inTable 1. These chemical reactions represent binding,dimerization (or trimerization), and phosphorylation

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Table 1

Reaction listing for the prototypic signaling network in Fig. 2

Name Chemical equation

bindLR L+R-LR

LRpS LR+ATP+S-ADP+S p+LR in

SpT S p+T-T p+S

TpW T p+W-W p+T

bindL2R2 L2+R2-L2R2

L2R2pS2 L2R2+ATP+S2-ADP+S2 p+L2R2 in

S2pT2 S2 p+T2-S2+T2 p

T2pW2 T2 p+W2- T2+W2 p

bindL3R3 L3+R3-L3R3

L3R3pS3 L3R3+ATP+S3-ADP+S3 p+L3R3 in

S3pT3 S3 p+T3-S3+T3 p

T3pW3 T3 p+W3-T3+W3 p

bindL4R L4+R-LR

bindL5R2 L5+R2-L2R2

L3R3pS L3R3+ATP+S-ADP+S p+L3R3 in

LRpS2 LR+ATP+S2-ADP+S2 p+LR in

L2R2pT L2R2+ATP+T-ADP+T p+L2R2 in

S2pW S2 p+W-S2+W p

T3pW T3 p+W-T3+W p

S2T3pWW2 S2 p+T3 p+W+W2-S2+T3+W2 p+W p

SWpW2W3 S p+W p+W2+W3-W2 p+W3 p+W+S

formW2W3 W2 p+W3 p-W2 W3 pp

formWW2W3 W p+W2 p+W3 p-WW2W3 ppp

form2WW3 2 W p+W3 p-2W W3 ppp

Table 2

System inputs and outputs

Compound Input/

Output

Group

L Input Signaling inputs

L2 Input

L3 Input

L4 Input

L5 Input

R Input Components of signaling

network

R2 Input

R3 Input

W Input

W2 Input

W3 Input

ATP Input Interaction with energy

metabolism

ADP Output

W p Output Signaling outputs

W2 p Output

W3 p Output

W2 W3 pp Output

WW2W3 ppp Output

2W W3 ppp Output

LR in Output Degradation of inactive

receptor-ligand complex

L2R2 in Output

L3R3 in Output


events that are typical chemical transformations foundin signaling networks. The prototypic network has12 inputs and 10 outputs of different types (Table 2).There are five signaling inputs and six signalingoutputs. In addition, components of the network aresynthesized and degraded, and metabolic energy tooperate the network is needed. Prototypic signalingnetworks alternative to that in Fig. 2 could readilyincorporate other types of signaling reactions (e.g.G-protein reactions) using appropriate reactionstoichiometries.The reaction names used in Table 1 characterize the

chemical transformations that are occurring and thecomponents that are involved. For example, reaction‘‘L2R2pS2’’ involves the phosphorylation of moleculeS2 by the receptor–ligand complex L2R2. The reaction‘‘bindLR’’ is the binding of the ligand L to the receptorR to form the receptor–ligand complex LR. The reaction‘‘formW2W3’’ is the dimerization of molecule W2 withthe molecule W3. Signaling reactions, like thosedescribed here, are subject to mass balance andthermodynamic constraints such as those that havebeen used for the analysis of metabolic networks(Burgard and Maranas, 2003; Palsson, 2000), andconsequently can be analysed with in silico analysismethods that have been previously developed (Papinet al., 2003; Schuster et al., 2000a).A stoichiometric formalism enforces explicit and

chemically consistent accounting of the underlying

chemical reactions that constitute a signaling network.Each ‘‘state’’ of a component in the network (e.g.phosphorylated, dimerized) must be represented as anindependent chemical entity. This explicit definitionallows for a chemically accurate accounting of everypart of the network. For example, each active andinactive molecule and each bound and unboundcomplex can be precisely accounted for and the relativeeffects can be assessed. A modified signaling moleculecan be represented as distinct molecules before and afterthe conformational change. Analogous and familiaroccurrences are found in isomerizations occurring inmetabolic networks, where fructose does not equatewith glucose although they have the same chemicalformula, and consequently need to be represented asseparate metabolites.Another important network consideration made

explicit using stoichiometry is the actual connectivitybetween components. For example, if molecule Aphosphorylates molecule B, and molecule B phosphor-ylates molecule C, both of these interactions need to berepresented. The interaction cannot simply be repre-sented as the presence of molecule A results in theactivation of molecule C. If molecule B is additionally

ARTICLE IN PRESSJ.A. Papin, B.O. Palsson / Journal of Theoretical Biology 227 (2004) 283–297 287

connected to other components of the network, itspresence in this chain of reactions is described. Thus,representing a signaling network within a stoichiometricformalism not only forces the explicit accounting of thetrue chemical features of the reactions that comprise thenetwork, but also allows for the application of a welldeveloped set of in silico analysis methods that are basedon reaction stoichiometry (Edwards et al., 1999; Papinet al., 2003). Extreme pathway analysis is one suchmethod.

2.3. Extreme pathway analysis

Topological analyses that characterize fundamental,time-invariant states of a given network have been called‘‘network-based pathways’’ (see reviews in Schilling et al.(1999) and Papin et al. (2003)). Analyses of network-based pathways include extreme currents (Clarke, 1988),elementary modes (Schuster et al., 2000a), and extremepathways (Schilling et al., 2000). For all of theseapproaches, convex analysis (Rockafellar, 1970) is usedto generate a set of systemic pathways that can fullycharacterize a network while ensuring that all networkcomponents are conserved, the network balanced, andreaction reversibility/irreversibility constraints are re-spected. In other words, a network-based pathwaybecomes a network property. It does not represent an‘‘isolated’’ pathway that functions independently of therest of the network, but represents a functional state ofthe entire network.To generate the set of extreme pathways for a

network, a stoichiometric matrix is created whichrepresents the stoichiometry between compounds (asrows) and reactions (as columns). The algorithm ofSchilling et al. (2000) generates a set of extremepathways with the following characteristics: (1) theyare the minimal and irreducible set of systemicallyindependent pathways for a given network, (2) allpermissible routes through a network can be describedby nonnegative linear combinations of the extremepathways, and (3) time-invariant topological propertiesare described with extreme pathway analysis (Papinet al., 2003). Thus, extreme pathway analysis is atopological approach that characterizes systemic, oremergent, properties of a given network based onnetwork structure alone.

2.4. Crosstalk classification

Extreme pathway analysis can be used to quantita-tively analyze the interconnection of multiple inputs andmultiple outputs of signaling pathways (which has beenloosely called crosstalk (Schwartz and Baron, 1999)).Herein, crosstalk is the nonnegative linear combinationof extreme pathways of a signaling network. The

pair-wise combination of extreme pathways is thus thesimplest form of crosstalk. As such, crosstalk insignaling networks can be classified into nine differentcategories based on extreme pathways (Fig. 3). Eachcircle in Fig. 3 represents a set of pathway inputs or a setof pathway outputs. From left to right, each pair ofpathway input sets is classified as disjoint, overlapping,or identical (e.g. if extreme pathway A has input ligandsL and L2 and extreme pathway B has input ligands L2and L3, then extreme pathways A and B have over-lapping inputs). From top to bottom, each pair ofpathway output sets is classified as disjoint, overlapping,or identical. For example, the representation in themiddle of the figure corresponds to two extremepathways with shared (but not identical) sets of inputsand shared (but not identical) sets of outputs. Thus, thetwo independent extreme pathways in this instance haveoverlapping but not identical functionality. Theseclassifications are topological, and thus do not accountfor changes in the activity level of a reaction, butwhether or not a particular set of outputs or inputs ispresent or absent.These nine classification schemes provide a succinct

description of fundamental properties of signalingnetworks:

* In Fig. 3A, a pair-wise comparison of this typerepresents completely independent functions of anetwork; completely separate inputs generate com-pletely separate outputs.

* In Fig. 3B, extreme pathways with disjoint inputsinteract to generate overlapping outputs. This type ofinteraction is conventionally thought of as ‘‘cross-talk’’ (Schwartz and Baron, 1999).

* Pair-wise pathway comparisons of the type in Fig. 3Crepresent a non-discriminate set of signals; twocompletely distinct inputs result in redundant outputsignals.

* A network with a high percentage of extremepathways of the type shown in Fig. 3D wouldrepresent a conservation mechanism; the networkcan use related inputs to generate distinct outputs.This type of mechanism, which arises simply fromstructural properties of the network, allows forsignaling specificity without proportionally increasingthe number of needed inputs.

* In Fig. 3I, the pair of pathways represents completelyoverlapping functions. In this case, the two pathwaysuse the same inputs to generate the same outputs butthrough systemically independent routes.

The definition of these nine categories is possibleusing extreme pathway analysis and they allow forfundamental descriptions, such as those describedabove, of signaling networks and their functions.

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disjoint

disjoint inputs, output overlap

disjoint inputs, redundant outputs

redundant inputs, disjoint outputsinput overlap, disjoint outputs

input overlap, output overlap

inputs outputs

inputs outputs

inputs outputs

inputs outputsinputs outputs

inputs outputs

(A) (D)

(E)(B)

(C) (F)

(G)

(H)

(I)

inputs outputs

inputs outputs

input overlap, redundant outputs

redundant inputs, output overlap

inputs outputs

fully redundant

Fig. 3. Classification scheme for crosstalk in the extreme pathways of a signaling network. From left to right, each pair of input sets is disjoint,

overlapping, and identical. From top to bottom, each pair of output sets is disjoint, overlapping, and identical.


3. Results

The extreme pathways were calculated for theprototypic signaling network in Fig. 2 using establishedmethods (Papin et al., 2003; Schilling et al., 2000). Theseextreme pathways were enumerated and studied. Theresults described below include: (1) a feasibility study ofinput/output relationships, (2) a quantitative analysis ofcrosstalk, (3) pathway redundancy, (4) participation ofreactions in the extreme pathways, and (5) correlatedreaction sets.

3.1. Enumeration of extreme pathways

There are a total of 211 extreme pathways. All thepathways are Type I extreme pathways (Schilling et al.,2000), thus each pathway involves the conversion ofinputs into outputs. Six representative extreme pathwaysfrom the set of 211 are shown in Fig. 4 (the numberingof the extreme pathways is arbitrary and is generated bythe algorithm):

* Pathway 42 is one of the shortest systemicallyindependent routes (a small number of participatingreactions). It results in the activation of a singletranscription factor (W) by a single ligand (L4). This

pathway falls into the classical category (Fig. 1A) ofsignaling events with a direct input–output connection.

* Pathway 133 is another ‘‘classical’’ route (Fig. 1A).Here, the transcription factor W3 is phosphorylated(activated) by the ligand L3.

* Pathway 108 is one of the longest systemicallyindependent routes (a high number of participatingreactions) in the prototypic network. It uses threeligands (L, L2, L3) to activate the transcription factorcomplex (WW2W3). This pathway is an example ofthe signal concatenation described above (Fig. 1B).

* Pathway 37 uses L4 as an input ligand and activatesW2 and 2W W3. This pathway is representative ofsignal pleiotropy with the initiation of multiple eventsfrom a single input in a signaling network (Fig. 1C).

* Pathway 1 uses multiple reactions to connect thebinding of two input ligands, L3 and L5, to theactivation of two transcription factor complexes(2W W3 and W2 W3). This pathway is representativeof the complexity that arises in signaling networkswith multiple input–multiple output relationships(Fig. 1D).

* Pathway 211 uses L3, L4, and L5 to activate W2 andW3 transcription factors. This pathway is anotherexample of the complexity that arises from theinterconnection of components of a signaling net-work (Fig. 1D).

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Fig. 4. Example extreme pathways: systemically independent signaling pathways.


With respect to the classification scheme in Fig. 1,there are 35 pathways that generate single outputs fromsingle inputs (e.g. Pathways 42 and 133 in Fig. 4). Thereare 116 pathways with multiple inputs and a singleoutput (e.g. Pathway 108 in Fig. 4). There are 6pathways that generate multiple outputs from singleinputs (e.g. Pathway 37 in Fig. 4), and there are 54pathways with multiple inputs and multiple outputs (e.g.Pathways 1 and 211 in Fig. 4). All extreme pathways areprovided as supplemental material.The computed set of extreme pathways can be used to

assess the emergent topological properties of theprototypic signaling network. These properties includeinput/output feasibility, crosstalk classifications, net-work redundancy, reaction participation, and correlatedreaction sets. The latter three have been studied inmetabolic networks (Papin et al., 2002a, b; Price et al.,2002). In the subsequent sub-sections we compute andinterpret these emergent properties.

3.2. Input/output feasibility analysis

An assessment of the feasibility of input/outputrelationships can be performed with extreme pathwayanalysis because all possible routes through a networkcan be described by nonnegative linear combinations ofthe extreme pathways (Schilling et al., 2000; Wiback

et al., 2003). A feasible input/output relationshipsignifies that with the available signaling inputs thereexists a valid combination of the extreme pathways thatdescribes the given signaling output. An algorithm forfinding all valid nonnegative extreme pathway combina-tions has been described (Wiback et al., 2003).An ‘‘Input/Output Feasibility Array’’ was generated

and is shown in Fig. 5. The columns represent whichinput ligands were available to the network. The rowsrepresent the different combinations of target signalingoutputs. The dark squares indicate that the givensignaling inputs can produce the given signaling out-puts. The light squares indicate that the given signalinginputs cannot produce the given signaling outputs. Oneimportant result of this analysis is the concise repre-sentation of this complex data set. Scanning the rows ofoutputs for a given combination of interest immediatelyidentifies which combination of inputs can be used toproduce that output. Conversely, scanning the columnsfor a given set of signaling inputs identifies whatcombination of signaling outputs can be generated.A number of noteworthy results are obtained from

this analysis. Two obvious results are that the redundantligands do not allow for the generation of additionaloutputs (e.g. the inputs of L2 and L5 have the samefeasibility columns) and that the presence of all ligandsenables the generation of all the signaling outputs (far

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Fig. 5. Input/output feasibility array. There are 25 input combinations and 26 output combinations (the row and column are not shown where all

inputs and outputs are zero). The dark squares indicate an output that can be generated by the corresponding input. The white squares indicate an

output that cannot be generated by the corresponding input. The numbers to the right of the array indicate the number of inputs that can generate

the corresponding output. The numbers at the bottom of the array indicate the number of outputs that the given input can generate.


right columns). One example of an interesting and non-obvious result is that there are certain input combina-tions that cannot make the transcription factor W3

along with any of the transcription factor complexes.This result shows that the synthesis of the transcriptionfactor W3 is not a flexible function of the prototypicsignaling network. This feature is particularly interesting

in comparison to the very high reaction participation ofthe W3 exchange reaction in the 211 extreme pathways(see Fig. 8 and section below).At the right of the array in Fig. 5 is the number of sets

of signaling inputs that can generate the given signalingoutput. At the bottom of the array is the number of setsof signaling outputs that a given signaling input can


generate. The ligand L3 is particularly important forgenerating many of the signaling outputs (compare thenumber of dark entries in the first three columns). Theseinput/output feasibility features are not apparent from avisual inspection of the prototypic network in Fig. 2.

3.3. Crosstalk analysis

The 211 extreme pathways were compared in a pair-wise fashion resulting in 22,155 [=(2112�211)/2] com-binations. Each comparison was classified according tooverlapping inputs or outputs (Fig. 2). The percentagesof pair-wise comparisons in each category were calcu-lated (Fig. 6). Surprisingly, nearly 95.3% of all pair-wisecombinations of extreme pathways led to overlappinginputs in the signaling network. Additionally, nearly62.0% of the pathway pair-wise combinations corre-sponded to disjoint outputs. This high degree of overlapof inputs is not due to the redundant ligand:receptorinteractions (e.g. R2 can bind to either L2 or L5).Removing all pathways that use L4 or L5 (or both) gavesimilar results. The percentages of combinations withoverlapping inputs (after removing all pathways thatused L4 and L5 as inputs) were 58.6, 21.0, and 10.8 fordisjoint, overlapping, and identical outputs, respectively.Pair-wise pathway combinations with disjoint inputs

and identical outputs suggest a lack of specificity in thesignaling network (e.g. two separate sets of ligandsgenerate the same output). The high percentage ofpathway combinations that have overlapping inputssuggests that the prototypic signaling network hastremendous variability in producing distinct outputsfrom similar inputs. The low percentage of pathwaycombinations with disjoint inputs further supports thischaracterization. Indeed, there are no pair-wise pathwaycombinations with disjoint inputs that have overlappingor identical outputs. This result suggests that when the

1.013.40.0

0.723.00.0

1.658.91.5

Inputs

Outputs

62.0

23.7

14.4

3.395.31.5

Total

Total

Fig. 6. Crosstalk analysis of the prototypic signaling network

following the classification scheme in Fig. 3. With 211 extreme

pathways, there are a total of 22,155 [=(2112�211)/2] pair-wise

comparisons.

inputs for this prototypic signaling network are disjoint,there are only disjoint outputs. Most of the variabilityin function of this prototypic signaling network comesfrom the generation of distinct outputs from similarinputs. The lower percentage of combinations withidentical inputs suggests uniqueness in the prototypicsignaling network; there is not a significant amount ofpathway redundancy, as one set of inputs will in generalgenerate a unique set of outputs.

3.4. Pathway redundancy

Two extreme pathways with identical inputs and/oroutputs represent two systemically independent routesby which a network can be utilized to reach the sameobjective. Analyses of such pathway redundancy ingenome-scale metabolic networks have led to severalinteresting results (Papin et al., 2002a; Price et al., 2002).The pathway redundancy of the prototypic signalingnetwork was evaluated in a similar fashion.

3.4.1. Identical inputs and outputs

There were 135 distinct input/output states of the 211extreme pathways. This result suggests that on averagethe prototypical signaling network can convert anidentical set of inputs into an identical set of outputsusing two systemically independent routes. For example,P 88 and P 108 both use the ligands L2, L3, andL4 to synthesize the transcription factor complexWW2W3 ppp, yet both use different internal reactions(Fig. 7).

3.4.2. Identical outputs from different inputs

The number of redundant output states with differentinputs was also calculated. There were 17 distinct outputstates in the set of extreme pathways for the network(Table 3). The most redundant output state is thesynthesis of the transcription factor complexWW2W3 ppp, which can be synthesized by 50 systemi-cally independent extreme pathways. Because extremepathways are systemically independent, the combinator-ial effect of the multiple pathways that produce W p,W2 p, and W3 p cannot explain the redundancy in theoutput of WW2W3 ppp. Rather, the redundancy is aresult of emergent uses of the network to produce theparticular transcription factor. The least redundantoutput is the transcription factor W3 p, which issynthesized by only one extreme pathway (Pathway133 in Fig. 4), although it is also an output incombination with other transcription factors. Thus,there are no alternative ways to activate only W3 in thisprototypic signaling network. It is particularly interest-ing to note that output vectors with multiple outputs canbe more redundant than output vectors with singleoutputs. It would be expected that the more specific theoutput vector, the less redundancy in routes to

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Fig. 7. Completely redundant extreme pathways. Pathways 88 and 108 have identical inputs and outputs and yet use different internal reactions.

Table 3

The number of extreme pathways with equivalent output states

Distinct output states Number of pathways

W p W2 p W3 p W2 W3 pp WW2W3 ppp 2W W3 ppp

0 0 0 0 1 0 50

0 0 0 1 0 0 26

0 1 0 0 0 1 22

0 1 1 0 0 0 22

0 4 0 0 0 1 10

0 2 0 0 1 0 10

0 2 1 0 0 0 10

0 1 0 1 0 0 10

0 0 0 2 0 0 10

0 0 0 0 0 1 10

1 0 0 0 0 0 10

0 2 0 0 0 1 4

0 0 0 2 0 1 4

0 1 0 0 0 0 4

1 1 0 0 0 0 4

1 0 0 1 0 0 4

0 0 1 0 0 0 1


synthesize the product. However, as these resultsillustrate, the level of interaction of components in asignaling network may lead to complex emergentproperties.

3.5. Reaction participation

The number of extreme pathways that a particularreaction participates in can be computed (Papin et al.,2002b). Disrupting or regulating the activity of highlyconnected reactions would influence a large number ofextreme pathways, or functional network states. Thepercentage (of a total of 211) of extreme pathways that

use each individual reaction in the network wasdetermined (Fig. 8). Of the internal reactions,‘‘bindL3R3,’’ which characterizes the binding of theligand L3 to the receptor R3, has the highest participa-tion; a total of 151 of the 211 pathways (72%) utilize thisreaction. The reaction ‘‘T3pW,’’ which characterizes thephosphorylation of the transcription factor W by theintracellular signaling molecule T3, has the lowestreaction participation; a total of 18 pathways (9%)utilize this reaction.The participation of reactions in extreme pathways

has been suggested as a metric for evaluating the relativeimportance of reactions in allowing for the possible

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Internal Reaction Percent of PathwaysbindL3R3 72%

SWpW2W3 65%L3R3pS3 58%

S3pT3 58%LRpS 46%

L2R2pS2 40%bindLR 34%

bindL4R 34%S2T3pWW2 33%

LRpS2 31%T3pW3 31%

TpW 29%formWW2W3 28%

bindL2R2 27%bindL5R2 27%

L3R3pS 27%formW2W3 26%

S2pW 25%form2WW3 24%

L2R2pT 16%S2pT2 15%

T2pW2 15%SpT 13%

T3pW 9%

Exchange Reaction Percent of PathwaysATP 100%ADP 100%W3 91%W2 90%L3 72%R3 72%

L3R3_in 72%R 67%

LR_in 67%W 61%

R2 55%L2R2_in 55%

W2_p 45%L 34%

L4 34%WW2W3_ppp 28%

L2 27%L5 27%

W2_W3_pp 26%2W_W3_ppp 24%

W3_p 16%W_p 9%

Reaction Participation

0%

10%

20%

30%

40%

50%

60%

70%

80%

90%

100%

0 10 20 30 40

Reaction Number

Per

cen

tag

e o

f P

ath

way

s

Exchange ReactionsInternal Reactions

Fig. 8. Reaction participation in the extreme pathways of the prototypic signaling network of Fig. 2. The internal and exchange reactions are sorted

independently in the inset table according to descending reaction participation.


phenotypes of an organism (Papin et al., 2002b). Ofparticular note in the reaction participation values isthe high participation of reaction ‘‘SWpW2W3,’’ whichinvolves the simultaneous phosphorylation of thetranscription factors W2 and W3 by the two moleculesS and W. This reaction participates in 138 pathways(65%). Although the exchange reactions of W2 p andW3 p (products of the reaction ‘‘SWpW2W3’’) are high(Fig. 8), they can be formed by other reaction processes.Thus, this result suggests that the reaction‘‘SWpW2W3’’ is an important contributor to the rangeof possible functions of the prototypic signalingnetwork.The reaction participation values for the exchange

fluxes (the inputs and outputs) were also evaluated(Fig. 8). Since phosphate is the common ‘‘currency’’ inthis prototypic signaling network, an expected result isthat exchange reactions for ATP and ADP have 100%reaction participation. This result reflects the fact thatthe phosphorylation potential of an ATP is needed todrive every functional state of this prototypic signalingnetwork. Signaling networks of this type are thus tightlycoupled to energy metabolism. The relative participa-tions of the exchange reactions for the three individualactivated transcription factors, W p, W2 p, and W3 p

are 9%, 45%, and 16%, respectively. There is a greaterdegree of variability in the synthesis of the transcriptionfactor W2 p than the other two transcription factors.This type of analysis can suggest which signaling

functions are more robust to changes in networkstructure (e.g. gene deletions or knock-downs).

3.6. Correlated reaction sets

From the set of extreme pathways for a network,correlated reaction sets can be calculated (Papin et al.,2002b). Correlated reaction sets are a collection ofreactions that are either always present or always absentin all of the extreme pathways (this property is relatedto enzyme subsets in elementary modes (Pfeiffer et al.,1999)). Effectively, these sets of reactions functiontogether in a given network, although the reactionsthemselves may not be adjacent in a network map. Forinstance, two reactions may belong to a correlatedreaction set in which one reaction involves the phos-phorylation of a cytoplasmic protein by a receptor andthe other reaction involves the transport of a transcrip-tion factor into the nucleus.The correlated reactions for the prototypic signaling

network are listed in Table 4. Reaction set 13 is theexpected grouping of the exchanges of ATP and ADP.Reaction set 4 consists of the input of the ligand L3, thereceptor R3, the reaction that represents the binding ofL3 to R3, and the output of the inactivated ligand–receptor complex. Reaction sets 1, 2, 6, and 7 include thereaction for the binding of the ligand to the respectivereceptor and the input of the corresponding ligand.However, these four sets do not include the input of the

ARTICLE IN PRESS

Table 4

Systemically correlated reaction sets for the prototypic signaling

network in Fig. 2

Reaction set Reaction names

1 bindLR, L

2 bindL2R2, L2

3 S2pT2, T2pW2

4 bindL3R3, L3, R3, L3R3 in

5 L3R3pS3, S3pT3

6 bindL4R,L4

7 bindL5R2, L5

8 formW2W3, W2 W3 pp

9 form2WW2W3, WW2W3-ppp

10 form2WW3, 2W W3 ppp

11 R, LR in

12 R2, L2R2 in

13 ATP, ADP


respective receptors because these receptors are notexclusive to the particular ligand (e.g. R2 can bind toeither L5 or L2). Reaction sets 8, 9, and 10 include thereactions for the formation of each of the transcriptionfactor complexes and the output of the respectiveproduct. Reaction sets 11 and 12 include the input ofthe receptors R and R2 and the outputs of the ligand–receptor complexes. Although the correlated reactionssets described above include some obvious groupings,they readily demonstrate what parts of the network areisolated from each other (e.g. the receptor R3 can onlybind to the ligand L3). However, some non-obviouscorrelated reaction sets indicate more complex networkstructure. Reaction set 3 includes the reactions ‘‘S2pT2’’and ‘‘T2pW2.’’ After an initial inspection, ‘‘L2R2pS2’’might be grouped with this set, or ‘‘SpT’’ might begrouped with ‘‘TpW’’ in a parallel fashion. However,these other reactions have compounds or products thatcan connect to other reactions in the network. Similarly,reaction set 5 includes reactions ‘‘L3R3pS3’’ and‘‘S3pT3’’ which are isolated from the rest of thenetwork.The appearance of correlated reaction sets 3 and 5 is

an indication of a degree of isolation of thesecomponents from the rest of the network. For actuallarge-scale signaling networks, such correlated reactionswould be near impossible to isolate with a simple visualinspection of a reaction map. Furthermore, since suchcorrelated reaction sets correspond to the coordinatedfunctions of different reactions, it has been suggestedthat these reaction sets may correlate with proteinswhose genes are co-regulated (Papin et al., 2002b).

4. Discussion

This study presents a stoichiometrically based analysisof signaling networks using methods that have been

established and extensively used for the analysis ofmetabolic networks. This type of analysis is enabled bydescribing the chemical reactions that comprise asignaling network with chemical equations that in turncan be described by a stoichiometric matrix. Stoichio-metric formulation allows the computation of extremepathways that in turn enable the study of emergentproperties of the signaling network. The emergentnetwork properties that can be analysed in this wayinclude: (1) feasible input/output relationships, (2)crosstalk, (3) pathway redundancy, (4) reaction partici-pations in the systemic pathways, and (5) correlatedreaction sets. These are integrated network propertiesthat are non-obvious and impossible to derive from asimple visual inspection of reaction maps. The threelatter properties have been studied for metabolicnetworks (Papin et al., 2002a, b; Price et al., 2002).Topological analysis of biological reaction networks

based on their node and link connectivities haveappeared (Fell and Wagner, 2000; Jeong et al., 2000).This type of topological analysis leads to concepts like‘‘scale-free’’ and ‘‘small world’’ networks (Barabasi,2002; Watts, 1999). The analysis presented herein,although topologically or structurally based, differsfundamentally from these studies. The present analysisis focused on functional and integrated network proper-ties. The input/output relationships are based onbalanced network functions and thus can representactual physiological signaling functions. Extreme path-ways are systemically independent functional units,derived strictly from network topology. Mathematically,extreme pathways are the minimal and irreducible set ofconically independent basis vectors (Schilling et al.,2000), and all allowable routes through a given networkcan be represented by nonnegative linear combinationsof the extreme pathways. These precise mathematicalcharacterizations allow for unambiguous descriptions ofa signaling network and definition of their emergentstructural properties.The first two emergent properties defined and studied,

feasible input/output relationships and crosstalk, repre-sent emergent properties that have not been previouslystudied using extreme pathway analysis. The array offeasible input/output relationships concisely representedwhat possible signaling output states the signalingnetwork could reach with a given set of inputs(Fig. 5). These input/output signaling relationships canbe decomposed into systemically independent extremepathways. This decomposition leads to a definition ofseveral categories of crosstalk (Fig. 6). Experimentalevidence for this ability of signaling networks tomaintain specificity by merely controlling topologicalproperties is emerging (Park et al., 2003; Ptashne andGann, 2003). Thus, defining crosstalk with precisemathematical terms may become important to under-standing the systems biology of signaling networks.


These two emergent properties are of some impor-tance. However, the number of extreme pathwayscomputed for genome-scale networks is large (Papinet al., 2002a; Price et al., 2002; Schilling et al., 2002;Schilling and Palsson, 2000) complicating their evalua-tion. Thus, analysis methods are needed to evaluateimportant features of these large sets of extremepathways and reduce their number to a manageableone so that meaningful biological analysis is enabled(Papin et al., 2003). Recently, several methods have beenused to characterize general features of sets of extremepathways: (1) singular value decomposition has beenapplied to extreme pathway matrices and importantsystemic features have been described including keyregulatory trade-offs (Price et al., 2003b, c), (2) thenumber of extreme pathways for a network have beenreduced with the implementation of transcriptionalregulatory rules (Covert and Palsson, 2003), and (3) analgorithm has been developed that describes the a-cone,the set of all allowable nonnegative linear combinationsthat characterize a given network state (Wiback et al.,2003). The use of known regulation to eliminate extremepathways (Covert and Palsson, 2003), and thus thecorresponding functional states of the network, mayprove to be a biologically meaningful way to study themalleability in and adaptation of signaling networkfunction.The emergent network properties of redundancy,

reaction participation, and correlated reaction sets havebeen previously studied for genome-scale metabolicnetworks (Papin et al., 2002a, b; Price et al., 2002).These properties have similar uses in the analysis ofsignaling networks. Extreme pathway analysis allowsfor a quantitative measure of pathway redundancy insignaling networks, and there is increasing experimentalevidence that pathway redundancy is an integral part ofactual signaling networks (Yu et al., 2003). Extremepathways enable the calculation of correlated reactionsets, which may correspond to network regulatorystructure (Papin et al., 2002b). Interestingly, correlatedreaction sets may be interpreted as ‘‘systemic modules’’because they represent a set of components that alwaysfunction together in a signaling network. This featurewould augment the ongoing efforts to define functionalmodules in signaling networks that are focused oninspection of signaling maps and thus naturally on‘‘adjacent’’ components.Reaction participation is another emergent property

that can be evaluated by extreme pathway analysis, andmay be of particular importance in the experimentalinterrogation of signaling networks. Reactions thatparticipate in a high number of extreme pathways aresignificant in defining the range of signal output states.Controlling or disrupting reactions with high participa-tion numbers would allow for broad control of networkfunctions whereas controlling those with low participa-

tion numbers would have a more specific elimination ofnetwork functions. Such analysis is likely to be ofinterest in medical and metabolic engineering applica-tions where signaling functions are targeted.All interactions amongst molecules inside cells are

fundamentally chemical transformations; chemical reac-tions involving breaking and forming covalent bonds, orassociation reactions between molecules to form com-plexes that are held together with non-covalent bonds.Thus a stoichiometric formalism can fundamentally beused to describe any biochemical reaction network.Since signaling networks are basically sets of biochem-ical transformations, they too can be described by astoichiometric matrix. This approach has not been usedheretofore for signaling networks and thus representssomewhat of a conceptual departure from the existingliterature. Although many think of signaling networksas the transduction of ‘‘information’’ it is clear that allbiological information fundamentally has an underlyingchemical basis. Thus, the endpoint of a signaling cascadethat involves transcriptional regulation ultimately isthe binding state of an upstream regulatory region. Thebinding states of a set of regulatory regions can bethought of as a definition of a phenotypic state sincecollectively they would determine the protein beingmade at a given time. Although it may appear at firstglance that networks like metabolism and signaling aredifferent given their different overall biological func-tions, they are the same when viewed as a set ofunderlying chemical transformations.Given the growing literature of stoichiometric and

constraint-based modeling of metabolism (Papin et al.,2003; Price et al., 2003a), we should now he able toapply these developments to signaling and to otherbiochemical reaction networks. The challenge in doingso is two-fold: first, to represent all the underlingchemical events as chemical transformations and thusstoichiometrically, and second, to interpret the results ofthe analysis with the biological function of the networkin mind. In this study, for instance, we introduce aquantitative definition of crosstalk as a nonnegativelinear combination of extreme pathways, which is acombination state of two underlying irreducible networkstates. Such a combination of extreme pathways doesexist in metabolic networks, but takes on a differentinterpretation for a signaling network. The resultspresented here thus promise to be the first step towardsa new type of topologically based analysis of signalingnetworks. This conceptual advance is based on repre-senting the signaling network with a stoichiometricmatrix that in turn relies on the chemical description ofall the underlying chemical transformations that com-prise the signaling network.There is much interest in dynamically simulating

signaling pathways and describing concentration pro-files of signaling components. The current analysis does

ARTICLE IN PRESSJ.A. Papin, B.O. Palsson / Journal of Theoretical Biology 227 (2004) 283–297296

not give us information about dynamic functionalitiesnor concentrations of network components. To getdynamic properties the kinetic constants have to beknown. However, an increasing number of studies areshowing that the structure of biochemical reactionnetworks may be the key determinant of their overallproperties (e.g. bacterial chemotaxis (Alon et al., 1999),pattern formation in the Drosophila embryo (vonDassow et al., 2000), large prototypic reaction networks(Aldana and Cluzel, 2003)). Thus extreme pathwayanalysis may prove useful as the primary way tocharacterize signaling networks. Extreme pathwayanalysis can also be broadened to include other cellularprocesses.The framework used here falls within the constraint-

based in silico modeling approach (Palsson, 2000). Thisapproach, initially developed for metabolic networks,continues to broaden in scope to include additionalparameters and cellular processes (Allen and Palsson,2003; Covert and Palsson, 2002). This methodology,which continues to be developed, can be applied tosignaling networks that have been described with amass-balanced formalism once the chemical reactionsthat comprise them are elucidated. With large-scalereconstructions of actual signaling networks becomingavailable, we can apply the framework used here todetermine their functional structural properties.

Acknowledgements

We thank Shankar Subramaniam for a critical read-ing of the manuscript. We also acknowledge financialsupport from the National Institutes of Health (GM068837) and the Whitaker Foundation (GraduateResearch Fellowship to JP).

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