Corrections

PHYSICS. For the article ‘‘Anyonic braiding in optical lattices,’’ byChuanwei Zhang, V. W. Scarola, Sumanta Tewari, and S. DasSarma, which appeared in issue 47, November 20, 2007, of ProcNatl Acad Sci USA (104:18415–18420; first published November13, 2007; 10.1073�pnas.0709075104), the authors note the fol-lowing: ‘‘On page 18419, right column, first full paragraph, line8, the statement that the spin–spin correlators T1

xx and T1xy along

a z-link cannot be zero simultaneously is incorrect. They are, infact, both zero for the Kitaev model and, therefore, cannot beused for detecting anyonic statistics. Instead, one should mea-sure the nonzero spin–spin correlator Ti

xx � ��i �D�x �V

x �i� of twoatoms at D� and V along an x-link for detecting anyonic statistics.This error does not invalidate the article’s other conclusions.’’

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INAUGURAL ARTICLE, BIOCHEMISTRY, APPLIED MATHEMATICS. For thearticle ‘‘A mathematical tool for exploring the dynamics ofbiological networks,’’ by Paolo E. Barbano, Marina Spivak, MarcFlajolet, Angus C. Nairn, Paul Greengard, and Leslie Green-gard, which appeared in issue 49, December 4, 2007, of Proc NatlAcad Sci USA (104:19169–19174; first published November 21,2007; 10.1073�pnas.0709955104), the authors note that ref. 23appeared incorrectly. The online version has been corrected.The corrected reference appears below.

23. Aldridge BB, Haller G, Sorger PK, Lauffenburger DA (2006) Syst Biol (Stevenage)153:425–432.

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A mathematical tool for exploring the dynamicsof biological networksPaolo E. Barbano*, Marina Spivak*, Marc Flajolet†, Angus C. Nairn†‡, Paul Greengard†, and Leslie Greengard*§

*Courant Institute, New York University, 251 Mercer Street, New York, NY 10012; †Laboratory of Molecular and Cellular Neuroscience,The Rockefeller University, 1230 York Avenue, New York, NY 10021; and ‡Department of Psychiatry, Yale University School of Medicine,34 Park Street, New Haven, CT 06508

This contribution is part of the special series of Inaugural Articles by members of the National Academy of Sciences elected on May 1, 2007.

Contributed by Leslie Greengard, October 18, 2007 (sent for review September 24, 2007)

We have developed a mathematical approach to the study ofdynamical biological networks, based on combining large-scalenumerical simulation with nonlinear ‘‘dimensionality reduction’’methods. Our work was motivated by an interest in the complexorganization of the signaling cascade centered on the neuronalphosphoprotein DARPP-32 (dopamine- and cAMP-regulated phos-phoprotein of molecular weight 32,000). Our approach has allowedus to detect robust features of the system in the presence of noise.In particular, the global network topology serves to stabilize thenet state of DARPP-32 phosphorylation in response to variation ofthe input levels of the neurotransmitters dopamine and glutamate,despite significant perturbation to the concentrations and levels ofactivity of a number of intermediate chemical species. Further, ourresults suggest that the entire topology of the network is neededto impart this stability to one portion of the network at theexpense of the rest. This could have significant implications forsystems biology, in that large, complex pathways may have prop-erties that are not easily replicated with simple modules.

dimensionality reduction � robustness � systems biology � DARPP-32

Our understanding of the signal transduction machinery usedby cells to provide a coordinated set of appropriate phys-iological responses to multiple diverse stimuli is increasing at anextraordinary rate. A large proportion of studies in the biologicalsciences are now devoted to elucidating such signaling pathways,which are far more complex than had been anticipated. As aresult, it is not possible intuitively to gauge the relative impor-tance of the different components involved.

In this article, we develop a mathematical tool that canelucidate biological network properties by analyzing global fea-tures of the network dynamics. We have chosen as a model theneurotransmitter signaling cascade involving dopamine- andcAMP-regulated phosphoprotein of molecular weight 32,000(DARPP-32) (1), because it is arguably the most thoroughlystudied signal transduction cascade in the mammalian brain andplays an important role in regulating biochemical, electrophys-iological, transcriptional, and behavioral responses to both phys-iological and pharmacological stimuli.

One of the remarkable features of molecular biological sys-tems is their ability to respond in a controlled or coherent fashionto external stimuli, despite significant variation in their internalstate. This phenomenon is often referred to as ‘‘robustness’’ andduring the last decade, simulation has begun to play a role inelucidating its molecular basis. Most of this work has relied onusing the fact that some known, simple biological function of anetwork remains coherent over a wide range of conditions. Animportant early example is the work of Barkai and Leibler (2),who showed that the topology of a small network allowed forrobust adaptation of the bacterial chemotaxis system. Theydemonstrated that the system responded appropriately to exter-nal chemical gradients even when kinetic coefficients and chem-ical species concentrations were allowed to vary by orders ofmagnitude. Subsequent experiments (3) confirmed their hypoth-

esis. Important progress has since been made in understandingaspects of cell cycle regulation, developmental control circuits,and signaling (4–8), nicely reviewed in ref. 9.

Broadly speaking, these earlier studies showed that a specificbiological response was generated consistently from a networkdespite significant parameter variation. Suppose, however, thatone is presented with a network but given no clear informationabout the desired output. How does one use the principle ofrobustness in a meaningful way? This is not an artificial situation;current high-throughput technology is continuing to provide vastamounts of information about network topologies with littleknown about the corresponding functionality. Thus, as has beennoted (8–15), it will be essential to develop mathematicalparadigms and computational tools that make use of high-throughput data to reveal the functional components embeddedin complex networks and biochemical pathways. If successful,such tools will allow us to identify some underlying principles ofbiological organization.

In this study, we attempt to extend the use of modeling androbustness to the case of biochemical systems where the outputhas not been or may not be characterized by a simple, welldefined behavior. A critical component of the study has been touse simulation to determine whether there are features whoseresponse remains coherent over a wide range of parametervariation. In the signal transduction setting, our method simu-lates the response of the system to the input in the presence ofnoise and highlights features of the dynamics that remain wellpreserved. Such features are indicative of a form of robustness,and we present evidence that they are likely to correspond tobiologically important functions. Before describing the methoditself, we first provide a brief introduction to our target system.

The DARPP-32 CascadeOver the past two decades, numerous studies have providedstrong support for the conclusion that DARPP-32 is a keyintegrator of various neurotransmitter pathways. In particular,several important striatal signaling pathways responding todopamine and glutamate stimulation have been shown to con-verge on DARPP-32, whose function as a regulatory protein isintimately tied to its phosphorylation state. Four phosphoryla-tion sites are known to alter the activity of this protein (Fig. 1B):Thr-34, Thr-75, Ser-97, and Ser-130 (for mouse DARPP-32). Inthe striatum, increases in dopamine levels in the synaptic cleft

Author contributions: P.E.B., M.S., M.F., A.C.N., P.G., and L.G. designed research; P.E.B. andM.S. performed research; P.E.B. and M.S. analyzed data; and P.E.B., M.S., M.F., A.C.N., P.G.,and L.G. wrote the paper.

The authors declare no conflict of interest.

Freely available online through the PNAS open access option.

§To whom correspondence should be addressed. E-mail: greengard@courant.nyu.edu.

This article contains supporting information online at www.pnas.org/cgi/content/full/0709955104/DC1.

© 2007 by The National Academy of Sciences of the USA

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lead to an increase in intracellular cAMP and activation ofcAMP-dependent protein kinase A (PKA) by postsynaptic D1dopamine receptors (Fig. 1 A). PKA phosphorylates Thr-34 andconverts DARPP-32 into a potent inhibitor of protein phospha-tase-1 (PP1) (16, 17).

Counteracting the effects of PKA, protein phosphatase 2B(PP2B), a Ca2�/calmodulin-activated protein phosphatase, de-phosphorylates Thr-34 in response to cytosolic Ca2� elevationinduced, for example, by glutamate, another crucial neurotrans-mitter. Glutamate also activates cyclin-dependent protein kinase5 (Cdk5), which phosphorylates DARPP-32 at Thr-75, convert-ing it into an inhibitor of PKA (18).

Phosphorylation of DARPP-32 on Ser-97 converts Thr-34 intoa better substrate for PKA, whereas phosphorylation ofDARPP-32 on Ser-130 decreases the rate of Thr-34 dephos-phorylation by PP2B. These sites are targeted by CK2 (caseinkinase 2) and CK1 (casein kinase 1), respectively (19, 20).Compared with PKA, less is known about the signaling pathwaysinvolved in the regulation of these kinases.

As a consequence of phosphorylation of Thr-34 or Thr-75,DARPP-32 is a dual-function inhibitor of either PP1 or PKA,respectively. Through distinct signal transduction cascades, anddifferential regulation of the state of DARPP-32 phosphoryla-tion, dopamine tends to activate PKA and inhibit PP1, whereasglutamate tends to do the reverse. Through inhibition of eitherPP1 or PKA, DARPP-32 regulates the state of phosphorylationand activity of key substrates, including many ion channels,pumps, neurotransmitter receptors, and transcription factorsnecessary for altering the physiological status of striatal neuronsand, in turn, for altering the function of striatal circuits. Notably,there are ‘‘feed-forward’’ loops in both the dopamine andglutamate regulatory cascades. Positive feedback is also present;PKA drives DARPP-32 to the Thr-75-dephosphorylation statethrough activation of PP2A, thus removing its own inhibition. Amore detailed picture of the allowed states of phosphorylationof DARPP-32 (the ‘‘DARPP-32 subnetwork’’) is shown in Fig.1C. In all, there are 102 chemical species in our model, includingthe intermediates involved in the various chemical reactionsillustrated in Fig. 1 (see Appendix for a discussion of thenumerical solution of the dynamical system and supportinginformation (SI) Text for a list of all component species).

Because much is understood about the composition of theDARPP-32 signaling cascade, it is a natural setting for compu-tational studies. Recently, for example, two groups have simu-lated aspects of the network to study the effect of transient Ca2�and dopamine (or cAMP) signals on DARPP-32 phosphoryla-tion (21, 22). In both cases, the authors found interesting,time-dependent features of the pathway. A rather general math-ematical framework for this kind of analysis has recently beendeveloped, based on adapting dynamical systems methods (23).The authors identified phase-space domains showing high sen-sitivity to initial conditions that lead to qualitatively differenttransient activities. Our goal here is rather different (andcomplementary). We would like to understand why such a highlyelaborate network has evolved in the first place. In particular, wewould like to address two questions: (i) whether the complexityof the network plays a role in its functionality, and (ii) how robustfeatures of an arbitrary network can be learned by simulation.

The Simulation ModelAlthough DARPP-32 is known to be affected by multipleneurotransmitter signals, we restrict our attention here to mod-eling the system’s response to the competition between dopa-mine and glutamate. For each choice of the dopamine andglutamate signals, we allow the system to evolve from a specificinitial state to a quasi-steady state by observing the chemicalreactions in the network over time, until they appear to haveachieved a steady state. The chemical concentrations of allcomponent species at that ‘‘quasi-steady state’’ time can beviewed as a quantitative descriptor of the network response. Byvarying simulation conditions, we create an artificial data set thatcan be studied. Before discussing the data exploration step, wefirst describe the mathematical model in more detail.

Both the dopamine and glutamate signals are defined asfunctions of time according to

d�t� � �1 � ��p�t�, g�t� � �p�t�,

where p(t) is a Gaussian pulse with maximum value, Pmax,centered at t0 of variance �:

Fig. 1. DARPP-32 regulatory pathways. (A) Simplified view of DARPP-32pathway. The dopamine and glutamate cascades are shown on Left and Right(red and green), respectively. A partial view of the states of phosphorylationof DARPP-32 is shown in Center (blue). (B) Phosphorylation and dephosphor-ylation states of DARPP-32. DARPP-32 is phosphorylated at Thr-34 by PKA, atThr-75 by cdk5, at Ser-97 by CK2, and at Ser-130 by CK1. Phospho-Thr-34 ispreferentially dephosphorylated by PP-2B (or calcineurin); phospho-Thr-75 ispreferentially dephosphorylated by PP-2A; phospho-Ser-130 is preferentiallydephosphorylated by PP-2C; the phosphatase for phospho-Ser-97 is not yetcharacterized. Green arrow indicates positive effect; red arrows indicatenegative effects. [Adapted from Nairn et al. (1).] (C) Detailed view ofDARPP-32 subnetwork. Shown are the various states of phosphorylationallowed in our model.

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p�t� � Pmaxe��t�t0�2/�4��.

When � � 0, the signal is entirely dopamine, and when � � 1,the signal is entirely glutamate. The full state of the network attime t is defined by the vector C(t) � (C1(t), C2(t), C3(t), . . . ,C102(t)), where each component Ci(t) corresponds to a singlechemical species (DARPP-32, PKA, cAMP, PP1, DARPP-32-Thr-34-P, etc.). The initial conditions for the ith species aredefined by Ci(0). We assume each network connection describeseither a binding reaction (causing an activation or deactivation)or an enzymatic reaction. The binding of dopamine to the D1receptor (D1R), for example, which releases the active subunitdenoted by G�, is modeled as

dopamine � D1R -|0k1

k�1G�, G� � AC -|0

k2

k�2AC-G�.

Thus, the differential equation for [G�] is

d�G�]dt

� k1�dopamine��D1R� � k�1�G�� � k2�AC��G��

� k�2�AC-G�� .

We do not follow strict Michaelis–Menten kinetics, because weare computing the dynamics of the enzyme concentrations. Thephosphorylation of PP2A by PKA, for example, is modeled asfollows:

PP2A � PKA-cAMP -|0k1

k�1PP2A-PKA-cAMP

OBk2

PKA-cAMP�PP2A-P

PP2A-POBk3

PP2A

The first equation describes the forward enzymatic reaction andthe second allows for the (nonenzymatic) dephosphorylation ofPP2A-P. The ordinary differential equation for [PP2-A] takesthe form

d�PP2A�dt

� � k1�PP2A��PKA-cAMP�

� k�1�PP2A-PKA-cAMP� � k3�PP2A-P� .

The complete set of components can be found in the SI Text.In short, given initial concentrations Ci(0) for each chemical

species and the kinetic coefficients ki for each reaction, thechanges in concentration are determined as a function of time bysolving a system of ordinary differential equations. The values ofthe initial concentrations and kinetic coefficients are chosenfrom a random distribution, with kinetic coefficients an order ofmagnitude larger than the concentrations. No subsequent fittingor refinement was used (see Appendix for further details). Theequations are evolved until a quasi-steady-state time teq at whichpoint the component chemical species have stopped fluctuating.The steady-state concentrations of the various species are thenthe components of a single point

C� teq� � �C1� teq� , C2� teq� , C3� teq� , . . . , C102� teq�� .

in 102-dimensional space. Because the simulation depends onthe input signals d(t), g(t), and therefore the value of theparameter �, we make this explicit and write

C�� , teq� � �C1�� , teq� , C2�� , teq� ,

C3�� , teq� , . . . , C102�� , teq�� .

Suppose now that we have fixed the initial concentrations andkinetic coefficients and carried out a sequence of 1,000 simulationswith � varying from 0 to 1. Assuming that the solution of the kineticequations depends smoothly on the parameter �, the points C(�,teq) will lie on a curve in 102-dimensional space. We refer to thiscollection of 1,000 points as a noise-free data set. To mimic thefluctuations of a real biological system, we repeat the 1,000 simu-lations just discussed, but for each new value of �, we add noise. Bynoise, we mean that we add a random perturbation to each initialvalue and kinetic coefficient. We consider two cases. When theperturbations are drawn from a uniform distribution with maxi-mum relative magnitude 30%, we refer to the perturbation asmodest. When the perturbations are drawn from a uniform distri-bution with maximum relative magnitude 50%, we refer to theperturbation as large. The two new sets of 1,000 data points sogenerated are referred to as the modest perturbation and largeperturbation data sets, respectively. (More thorough randomizationprocedures could obviously be used, but the preceding procedureis sufficient for the present study.)

In this study, we are interested in whether the modest and largeperturbation data sets C(�,teq) still have any coherent behavior inresponse to the input signal parameter �. Because the state ofphosphorylation of DARPP-32 reflects neurotransmitter signals,we are particularly interested in looking at the coherence of theDARPP-32 subnetwork and the regulatory cascades separately. Forthis, we want to explore the data sets visually, but unfortunately, thatis difficult to do in 102 dimensions. We therefore have made use ofdimensionality reduction methods to project the data onto athree-dimensional space, where direct inspection is feasible.

Dimensionality Reduction and Data ExplorationThe detection of stable structures in noisy, high-dimensionaldata are a recurring problem in many areas of science. Fortu-nately, very effective approaches for this have been developed inthe past few years, including locally linear embedding (LLE) (24)and Laplacian eigenmaps (LEs) (25). These have received a lotof attention in machine learning and pattern recognition, and wewill not seek to review the literature here, except to note thatLLE and LEs are beginning to be used in bioinformatics (26).

The intuition underlying LLE and LE is that, if the data pointslie on a curve or surface (or other ‘‘low-dimensional manifold’’),then a specific pattern of neighbor relations must hold. On acurve in 102-dimensional space sampled at a sequence of points,for example, each point has two nearest neighbors: the one justbehind and the one just ahead. The way in which LLE and LEfunction is (i) to find the nearest neighbors in the original (in ourcase, 102-dimensional) space, and (ii) to replicate the neighborrelation in a lower dimensional ‘‘embedding’’ space (in our case,three-dimensional). We refer the reader to the original articlesfor a description of this mathematically sophisticated and highlynonlinear procedure (24, 25). Unfortunately, it should be notedthat the coordinates of the points in the embedding space bearno clear relation to any of the coordinates in the original space.Nevertheless, simple visualization can be used to inspect thegeometry of the data set. Curves should map to curves, two-dimensional surfaces to surfaces, etc. Details of our implemen-tation can be found in Appendix, as can a short discussion of theunderlying mathematics. Here, we illustrate the method with asimple example. For this, consider the set of points in threedimensions defined by

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C��� � �� sin(8.2�)cos(11.2�), � cos(8.1�),4�),

sampled at 300 points with � on the interval [�3, 3]. They clearlylie on a space curve (Fig. 2A). Imagine, however, that we wereonly able to visualize two-dimensional graphs. A straightforwardlinear projection onto a two-dimensional space is obtained bysimply plotting the first two coordinates (Fig. 2B). Notably, thestructure of the original data is lost. Use of LE to project ontotwo dimensions yields the curve shown in Fig. 2C. The fact thatthe data lie on a curve has been clearly and faithfully replicated.

Our data exploration procedure is now easy to describe. We takethe data set (noise-free, modest perturbation, or large perturbation)and project it onto a three-dimensional space by using LE. If theresponse to variation in the parameter � is coherent and approx-imates a curve in 102-dimensional space, then it will be visible as acurve in three dimensions. Furthermore, we can explore the dataset at will. The same method can be used to project only a subsetof the components onto the three-dimensional embedding space. Inthe present context, for example, one can project only theDARPP-32 subnetwork or only the regulatory cascades.

It should be noted that signal transduction cascades are naturaltargets for this approach, because the dimensionality of the systemresponse being sought is naturally controlled by the input param-eters (here the dopamine/glutamate ratio). We briefly discuss theextension to higher-dimensional surfaces in the Appendix.

Simulation ResultsUsing the data generated by our simulations, we are able to identifynetwork components that maintain coherence in the presence ofnoise. We are also able to identify specific topological features ofthe larger network that appear to be crucial for the robust behaviorof those components.

In Fig. 3 A–C, we plot the projection of the DARPP-32subnetwork and regulatory cascades in the noise-free, modestperturbation and large perturbation cases. In the noise-free case,as expected, clean curves are traced out for both the DARPP-32subnetwork and the regulatory cascades in response to varyingthe neurotransmitter signals by letting � range from 0 to 1 inincrements of 0.005. In the modest perturbation case, theDARPP-32 subnetwork displays a remarkably stable response,despite the noise introduced into the system, whereas theregulatory cascades have lost coherence. In the large perturba-tion case, the fluctuations in initial concentrations and kineticcoefficients are sufficiently large that they overcome the inher-ent stability/robustness of the DARPP-32 subnetwork response.

To confirm that the topology of the network plays an importantrole, we have broken it in three different ways (Fig. 4). In the firstcase, cAMP is assumed to activate PP2A directly, rather thanthrough the intermediary PKA. Under this artificial modification,

both PKA and PP2A are still activated in a coordinated fashion, andthe DARPP-32 subnetwork retains its coherence under perturba-tions (Fig. 3D). In the second case, PP2A molecules are completelyremoved from the system (by setting their values to zero during thesimulation) (Fig. 4); as a result, the DARPP-32 subnetwork loses itscoherence (Fig. 3E).

A final, less obvious network modification consists of trying toremove as much of the regulatory cascades as possible. For this,dopamine is assumed to convert PKA to its active form (PKA-cAMP) directly and glutamate is assumed to convert PP2B to itsactive form (PP2B-Ca) directly (Fig. 4). The result of this is that theDARPP-32 subnetwork becomes much more sensitive to pertur-bations (Fig. 3F). In other words, the regulatory cascades cannot beremoved from the signal transduction pathway while retaining arobust response.

ConclusionsIn the present study, we have found convincing numerical evidencethat the topological structure of the DARPP-32 pathway serves auseful purpose—to faithfully translate neurotransmitter signals atthe cell surface into the net state of phosphorylation of DARPP-32in the cell’s interior, despite the presence of noise. What we foundparticularly striking is the fact that the other components of thepathway (in the regulatory cascades) are themselves poorly con-trolled in response to the dopamine/glutamate signal, but that theyare crucial to the overall functioning of the signaling system. Thisobservation suggests that attempts at capturing the full networkdynamics in a small module (the goal of some efforts in systemsbiology) may be hampered by subtle but critical topologicalconsiderations.

The tool we have developed here will allow scientists to analyzethe robust response of a network to an input signal in a new way:by creating artificial data that simulate biological noise, and byexploring those data through dimensionality reduction. The mostcompelling feature of the method is, perhaps, that it is unsuper-vised—if the network is effectuating a coherent response in somesubset of its components, that response should be visible to the user.

AppendixNumerical Solution of System of Differential Equations. The dynamicsof the DARPP-32 network is modeled by means of a set ofenzymatic and binding reactions. Each chemical species is given aninitial concentration drawn from a uniform random distribution onthe interval [0, 0.1]. The kinetic coefficients (ki) in the reactions aredrawn from a uniform random distribution on the interval [0, 1].The full system of ordinary differential equations is solved for eachdopamine and glutamate input signal pair (d(t), g(t)), using theCVODE package available from the Sundials site (http://

Fig. 2. Dimensionality reduction of a one-parameter family of data (a space curve). In A, a three-dimensional data plot reveals that the points seem to lie on a curve.In B, a two-dimensional ‘‘visualization’’ of the data is obtained by linear projection onto the x-y plane. The geometry of the data is difficult to see. In C, Laplacianeigenmaps are used to project the data onto a two-dimensional space in a much more revealing manner. The fact that the data lie on a curve is quite apparent. Forrealistic systems, like the DARPP-32 network, where the original data are high-dimensional and inaccessible to simple visualization, this is a critical tool.

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acts.nersc.gov/sundials/) at Lawrence Livermore National Labora-tory.

The specific integration method used is designed for stiffsystems, based on BDF (backward differentiation formula) withNewton iteration.

As indicated in the text, the modest and large perturbation setsare generated as follows. As the parameter � varies from 0 to 1, aperturbation is added to each of the original initial concentrationsand kinetic parameters. These perturbations are also drawn from auniform distribution, but with maximum magnitude equal to either30% or 50% depending on each of the original values. These definethe modest and large perturbation data sets.

Implementation of Laplacian Eigenmaps. Our visualization methodrelies on the method of Laplacian Eigenmaps developed by Belkinand Niyogi (25). The details of our implementation follow, whereN denotes the number of high-dimensional data points.

1. For a fixed parameter n, add an edge between nodes i and j,if i is one of the n nearest neighbors of j, or j is one of the nnearest neighbors of i.

2. Let W be an N�N matrix. Set W(i, j) � 1, if there is an edgebetween nodes i and j and zero otherwise.

3. Let D be a diagonal N�N matrix with entries D(i, i) � jW(j,i) (column sums).

4. Define L � D � W, the ‘‘graph Laplacian.’’

5. Solve the generalized eigenvalue problem Ly � �Dy. Let (�0,y0), (�1, y1), (�2, y2), and (�3, y3) denote the lowest foureigenvalue/eigenvector pairs. The lowest eigenvalue �0 iseasily seen to be zero from the definitions of L, D, and W andthe corresponding eigenvector is identically one. It carries nouseful information.

6. For each high-dimensional data point Xi, compute its pro-jection onto three-dimensional space according to the for-mula: Xi 3 Yi � (y1(i), y2(i), y3(i)).

7. Plot the data points Yi in three dimensions with coordinatesobtained from step 6.

Plots in the article used n � 10 neighbors. The choice n � 15produced qualitatively similar images.

Developing Quantitative Measures of Robustness. The computa-tional approach described in this article is based on geometricintuition. In its present form, biologists using the visualization toolare responsible for selecting the subnetworks or modules to analyzefor robustness, and their own intuition in declaring that the patternseen in the projected data is of interest. It would, therefore, be ofsignificant interest to both quantify and automate the analyticprocess. For this, we need a more precise mathematical formulationof the problem.

We denote by S � {X(t)} � �M the full data set of trajectories,where M is the dimension of the original data vectors (M � 102 in

Fig. 3. Response manifolds for numerical simulations. (A) In the absence of perturbations, both the DARPP-32 subnetwork and the regulatory cascade follow(exactly) a simple one-parameter path, determined by the dopamine/glutamate ratio. The high-dimensional space curves are visualized in three dimensions byusing Laplacian eigenmaps. (B) With modest perturbation, the response of the DARPP-32 subnetwork continues to follow a one-dimensional (but slightly noisypath). The regulatory cascades, however, no longer reflect a clear signal. (C) With sufficient noise (the large perturbation data set), the DARPP-32 response alsoloses its coherence, at least over some range of the dopamine/glutamate ratio. (D) Modified network 1 (see Fig. 4). cAMP is assumed to activate PP2A directly.Both PKA and PP2A are still activated in a coordinated fashion, and the DARPP-32 subnetwork clearly retains its coherence under modest perturbation. (E)Modified network 2 (see Fig. 4). All PP2A species are removed from the system (by setting their values to zero during the simulation). Clearly, the DARPP-32subnetwork loses its coherence. (F) Modified network 3 (see Fig. 4). The regulatory cascades are eliminated by assuming that dopamine directly converts PKAto its active form (PKA-cAMP) and that glutamate directly converts PP2B to its active form (PP2B-Ca). As a result, the DARPP-32 subnetwork becomes much moresensitive to perturbations. That is, the removal of the regulatory cascades significantly affects the robustness of the signal transduction pathway.

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our study) and we denote by U � {Y(t)} � �K the trajectories of asubset of components (K M). We denote by P(S) or P(U) theprojection of the respective trajectories onto the n-dimensionalembedding space (n � 3 in our study). In essence, we are simplyseeking to identify those subsets U that remain in a well definedneighborhood of the unperturbed trajectory. A measure of robust-ness can, therefore, be obtained by computing some statistics onP(U). In the case of a one-dimensional signal, this could simply bethe radius of the tubular neighborhood around the unperturbedtrajectory that contains 95% of the noisy data. Given such ameasure, visual inspection would no longer be required. A morethorough development of the underlying mathematics is underway.

Automating the search process is unfortunately harder, becauseit is subject to ‘‘combinatorial explosion.’’ That is, given a systemwith M components, the number of potential K-dimensional mod-ules is of the order

M!�M � K�!K!

.

For large values of M, investigating all modules with more thanone or two components becomes rapidly infeasible. We need toinvestigate hierarchical methods to overcome this barrier. Forthe moment, that selection process is in the hands of the user,driven by conjectures concerning the underlying biology.

This work was supported by U.S. Army Medical Research AcquisitionActivity Grant W81XWH-04-1-0307 and NYSTAR Contract C04006 (toP.B.); by Defense Advanced Research Projects Agency Defense SciencesOffice Contract HR0011-04-C-0057 and Department of Energy ContractDEFG-02-00-ER25053 (to L.G.); by the National Science Foundationthrough the IGERT Program in Computational Biology (M.S.); ac-knowledge support from by U.S. Public Health Service GrantsMH40899, MH074866, and DA10044 (to P.G., M.F., and A.C.N.).

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Fig. 4. Three simulated modified networks. In modification 1, cAMP bypasses PKA and is assumed to directly catalyze the phosphorylation of PP2A (indicatedby labels 1). In modification 2, all PP2A species are deleted from the system (indicated by labels 2). In modification 3, dopamine is assumed to convert PKA toits active form (PKA-cAMP) directly and glutamate is assumed to convert PP2B to is active form (PP2B-Ca) directly (indicated by labels 3).

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