design and constraints of the drosophila segment polarity

Design and Constraints of the Drosophila SegmentPolarity Module: Robust Spatial Patterning Emergesfrom Intertwined Cell State Switches

GEORGE VON DASSOW1,2nAND GARRETT M. ODELL1,2

1Department of Zoology, University of Washington, Seattle, Washington 981052Friday Harbor Laboratories, Friday Harbor, Washington 98250

ABSTRACT The Drosophila segment polarity genes constitute the last tier in the segmentationcascade; their job is to maintain the boundaries between parasegments and provide positional ‘‘read-outs’’ within each parasegment for the entire developmental history of the animal. These genesconstitute a relatively well-defined network with a relatively well-understood patterning task. In aprevious publication (von Dassow et al. 2000. Nature 406:188–192) we showed that a computer modelpredicts the segment polarity network to be a robust boundary-making device. Here we elaboratethose findings. First, we explore the constraints among parameters that govern the network model.Second, we test architectural variants of the core network, and show that the network tolerates awide variety of adjustments in design. Third, we evaluate several topologically identical models thatincorporate more or less molecular detail, finding that more-complex models perform noticeablybetter than simplified ones. Fourth, we discuss two instances in which the failure of the networkmodel to behave in a life-like fashion highlights mechanistic details that need further experimentalinvestigation. We conclude with an explanation of how the segment polarity network can beunderstood as an interwoven conspiracy of simple dynamical elements, several bistable switches anda homeostat. The robustness with which the network as a whole maintains a spatial regime of stablecell state emerges from generic dynamical properties of these simple elements. J. Exp. Zool. (Mol.Dev. Evol.) 294:179–215, 2002. r 2002 Wiley-Liss, Inc.

A decade ago most developmental biologistsworked in the context of ‘‘developmental path-ways,’’ ordering developmental control genes intochains of regulatory events contributing to theeventual phenotype. The prevailing image of thedevelopmental process evolves to keep up withnew information, and increasingly, biologiststhink about developmental and molecular geneticsin terms of networks of dynamic gene interactionscharacterized by extensive internal crosstalk andfeedback. While this subtle change is surelytoward a more realistic conception of gene func-tion, it poses a potentially serious problem, sincehighly integrated dynamical networks are muchharder to understand, mechanistically, than aretop–down processes like pathways and cascades.This is emphasized by a contrast between twosimple maps of gene interactions in Figure 1. InFigure 1A, published earlier (von Dassow et al.,’93), it is trivial to evaluate what the map, to theextent that it reflects reality, does: either thesignals at the top are strong enough, and theconnections sufficiently strong, too, to generate anoutput at the bottom, or they are not. On the other

hand, in a slightly later paper, the same authors(Schmidt et al., ’96) presented Figure 1B, in whichit is essentially impossible to tell, just from themap alone, what the network does. One correctlysuspects that the behavior of this circuit dependson dynamical parameters, such as the relativestrengths of the various connections composingthe map, or the initial state of the components, etc.

Another subtle shift epitomized in Figure 1 isthat the phenotypic consequences of gene activityseem to have evaporated in Figure 1B. Instead ofcharacterizing the functions of genes and theirproducts in terms of the production of a develop-mental event or a specific tissue type (as inFig. 1A), there is an implicit focus on the genenetwork as a device in and of itself (Hartwell et al.,’99; von Dassow and Munro, ’99). The morebiologists know about a particular mechanism,

Grant sponsor: National Science Foundation; Grant numbers:MCB-9732702, MCB-9817081, MCB-0090835.

nCorrespondence to: George von Dassow, Friday Harbor Labora-tories, Friday Harbor, WA 98250. E-mail: [email protected].

Received 19 September 2001; Accepted 10 June 2002.Published online in Wiley InterScience (www.interscience.wiley.

com). DOI: 10.1002/jez.10144

r 2002 WILEY-LISS, INC.

JOURNAL OF EXPERIMENTAL ZOOLOGY (MOL DEV EVOL) 294:179–215 (2002)

the more the functions of individual genes andtheir products get re-defined in terms of the waythey contribute to intrinsic dynamical behaviors ofthe network, whose function is in turn related tohow that intrinsic behavior is deployed in thethree-dimensional context of the developing em-bryo. But again, it is not possible to answer, usingwords and intuition alone, what is the intrinsicdynamic behavior of this network? An obviouscorollary is that it is also impossible to say, usingintuition alone, whether the map in Figure 1Bactually accounts for any real phenomenon. In thispaper we describe a theoretical study in which weuse a computer simulation model of the segmentpolarity network in early Drosophila development

to explore a set of hypotheses about the nature andfunction of that network. Brief reports on thiswork appeared previously (von Dassow et al.,2000) and in a companion paper (Meir et al.,2002b, this issue); here we expand and extendthose results to suggest that the segment polaritynetwork is a boundary maintaining module thatexhibits remarkable robustness of its steady-statepattern forming behavior to structural, dynamic,and architectural perturbations.

PARAMETER SPACE AS A BIOLOGICALPROBLEM

Consider a simplistic example: two genes acti-vating each other (Fig. 2A). Figure 2B exhibits thesimplest plausible formulation of this circuit’sdynamics: the differential equations say that therate of change in concentration of each geneproduct depends on a dose–response curve inwhich the other gene product appears and afirst-order decay term. In this simple case we canassess the behavior of the network by setting theleft-hand sides to 0, and plotting the resultingcurves (‘‘nullclines’’; Fig. 2C). Where the null-clines cross, the system is at steady state; Figure2C shows that under some conditions there may

Fig. 1. Examples of evolving views of genetic regulatoryarchitecture. (A) Diagram from von Dassow et al. (’93),representing the ‘‘genetic cascade’’ or ‘‘developmental path-way’’ type of thinking necessitated when the task is to orderjust a few known components relative to a particularphenotype. (B) Later diagram from the same authors(Schmidt et al., ’96) captures a nascent ‘‘gene network’’notion, stimulated when enough feedback and cross-talk madethe ‘‘pathway’’ metaphor inappropriate.

Fig. 2. Two-activator switch. (A) Two molecular species Xand Y promote each other’s synthesis, and each exhibits first-order decay. For such a simple system one may solvegraphically for steady states. (C–F) are plots of nullclines forthe equations in B; each curve is obtained by setting thederivative to zero and solving for the state variable governedby that equation. Where the curves cross in the state space,both derivatives are zero. However, such steady-state pointsmay or may not be stable, as determined by the eigenvalues ofthe Jacobian matrix for the differential equations. For certainchoices of parameters, there may be three steady states, onlytwo of which are stable (dashed circles, large graph): one inwhich both species are expressed at high level, and one inwhich neither is expressed. In this case, the system behaveslike a switch: for some initial conditions, it evolves determi-nistically toward one stable steady state, stays there, andreturns after small perturbations; for some other initialconditions, it evolves toward the other stable state and staysthere; and large perturbations can ‘‘switch’’ the system to oneor the other stable state. However, the stability and even theexistence of each state depends on the choice of values for freeparameters. D and E show that switch-like behavior dependscritically on sigmoid dose–response behavior; if the parametern¼1.0, there is no way this system can exhibit switching.Either the nullclines cross, and only the ‘‘on’’ state is stable,or they do not, and only the ‘‘off’’ state is stable. In F, despitethat n 41.0, the parameters KX and KY are such that thenullclines happen not to cross, and only the ‘‘off’’ state isstable.

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G. VON DASSOW AND G.M. ODELL180

exist three steady-state points, two of which arestable (that is, the system tends to return to thosepoints if nudged away from them), in which casethis tiny circuit could function as a switch; ifpushed near one or the other stable steady state(‘‘Off’’ or ‘‘On’’ in this case), it will evolve toward

and remain at that state until perturbed into theneighborhood (attraction basin) of the other stablesteady state. This switch-like behavior exemplifiesthe dependence of the circuit’s behavior on initialconditions.

However, the switch-like behavior itself dependsintimately on the values of several free para-meters. The equations in Figure 2B involve ninefree parameters, which represent empiricallymeasurable values governing the dynamics, suchas the half-life of each molecular species, themaximal transcription rate, etc. Parameters couldthemselves be functions of other genes extrinsic tothe putative module of interest, or they may befunctions of temperature and other environmentalvariables, etc., but with respect to the intrinsicbehavior of the circuit in question they areconstants whose values must be specified some-how. As shown in Figure 2D–F, under someparameter conditions the switch-like behavior iscompletely abolished. Therefore, although thedynamical argument confirms that the circuit inFigure 2B could function as a switch, there is noway to tell, from the network’s topology alone,if it actually does.1 If such a simple device as thetwo-activator switch is so problematic, imaginehow useless topology alone becomes whenconfronted with networks only a little bit morecomplex, such as Figure 1B.

This is not a mere mathematical pet peeve; ithas a genuinely important biological meaning. Wecan think of the free parameters of a network asthe basis for a space (a nine-dimensional space inthe case of Fig. 2) which the network inhabits.Each point in that space is a unique set of valuesfor each free parameter. Given a functionalbehavior for the network, some subset of pointsin parameter space (which for any realistic modulewill have dozens of axes) might confer thatbehavior, and the questions become in whatfraction of parameter space does the circuit‘‘work,’’ and how are the working parameter setsdistributed in that space? Some simple possibilitiesare stereotyped in Figure 3: the fraction could belarge, in which case we would say that thenetwork, or rather the behavior in question, isrobust to parameter variation, whereas if thefraction of working territory is small, we wouldsay it is brittle. Moreover, the evolutionary process

1In fact, the situation is really even a little worse: in general it will beimpossible to use formulations like these to answer whether a given setof known facts about epigenetic relations really does account for aparticular behavior; instead it is only possible to ask whether thereexists any set of parameters for which those facts could do so.

SEGMENT POLARITY NETWORK MODEL 181

itself has to find the working territory, and, havingfound it, it needs to stay within it. Since para-meters are themselves mutable functions of thegene products involved in the network, as well aspossibly others extrinsic to it, mutation constantlydisperses the network through parameter space.Thus the evolutionary process must confront avariety of consequences of how the networkresides in parameter space.

THE SEGMENT POLARITY NETWORK

Dozens of man-millennia have gone into dis-secting segmentation in the fruit fly Drosophila.While understanding of this process remainsincomplete, segmentation in the Drosophila em-bryo ranks among the very best-understood of alldevelopmental mechanisms. The segmentationprocess consists of a series of ever-finer-scalepatterning processes, beginning with morphogens

localized in the egg during oogenesis (summarizedin Fig. 4, following a similar diagram presented byNagy, ’98; the diagram is meant only to convey theflavor of the process). Maternal morphogens likebicoid and nanos form shallow gradients along thefuture A–P axis of the embryo. Gap genes respondto the local concentration of these factors and alsoregulate each other’s production, and the func-tional pattern of expression of these genes is broadbands with moderate-range gradients for eachshoulder. Pair-rule genes respond, in turn, to thelocal concentration of gap gene products, and theirfunctional expression patterns typically corre-spond to alternate segments, with short-rangegradients for shoulders. Many of the gap and pair-rule genes are expressed initially in broaddomains, and their functional expression patternsemerge through repression by other gap andpair-rule genes (Gaul and Jackle, ’87; Edgaret al., ’89; Carroll, ’90; Kraut and Levine, ’91).The segmental register of most pair-rule geneexpression patterns implies that it is, in effect,these genes that map out segments at theblastoderm stage.

Fig. 3. A few possible configurations of functional domainsin parameter space. In each panel the dashed box indicates aboundary within which one might wish to search for‘‘functional’’ sets of parameter values. In (A) a large fractionof parameter choices confer function (the shaded lake). In (B)the model is limited to a tiny puddle. One might find that themodel functions in several disjunct regions of parameterspace, as in C. Finally, in (D), a smallish functional pond issurrounded by a much larger swamp of almost-functionalvalues (darker shading), which might even smoothly gradeinto the fully functional region; in such a case one could useoptimization strategies to find the functional pond as long asone stumbled first into the massive slough surrounding it. InA the working territory occupies about one-third the ‘‘vo-lume’’ of the sampled region, whereas in B the workingterritory occupies more like one-fiftieth. For a realistic genenetwork model, the reader must extrapolate to high dimen-sionalityFsay, 50 dimensions, as for our core segmentpolarity model. As shown in E, if one contiguous half of eachparameter axis within the sampled range were compatiblewith the desired function, in a three-dimensional parameterspace one-eighth of the sampled volume would ‘‘work,’’ butthe equivalent box in 50-D space would occupy a minisculefraction of the sampled volume. (F) For the core segmentpolarity model we found that about one in 200 randomlysampled parameter sets work; the equivalent box in a 3-Dparameter space would occupy nearly the entire sampledvolume. The rectangular prisms in E and F are meant only toexpress the volume fraction, not the complex shape, of thezone that the functional parameter sets occupy in parameterspace.

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However, most of the pair-rule genes are onlytransiently expressed, and it is the next tier in thesegmentation cascade, the segment polaritynetwork, that defines and maintains the bound-aries between segments. The segment polaritygenes respond to the local combination andconcentrations of pair-rule gene products, andthey come to be expressed in narrow stripes inevery segment (reviewed by Martinas-Arias, ’93;DiNardo et al., ’94). Their expression patterns,though more dynamic than this crude character-

ization implies, remain landmarks for the seg-mental boundary in at least a subset of the embryo(especially the ventral ectoderm) throughoutembryogenesis and into the adult (via their rolein maintaining the A–P compartment boundary inimaginal discs). The segment polarity networkincludes: engrailed (en) and its duplicate invected(inv), which encode transcriptional regulatorsexpressed posterior to the parasegmental bound-ary (DiNardo et al., ’88; Martinez-Arias et al., ’88);hedgehog (hh), which encodes a signaling moleculeexpressed under the control of Engrailed (Tabataet al., ’92); wingless (wg), which encodes anothersignaling molecule expressed anterior to theparasegmental boundary (Martinez Arias et al.,’88; van den Heuvel et al., ’89); patched (ptc),which encodes the inhibitory and Hh-bindingcomponent of the Hh receptor (Hooper and Scott,’89; Chen and Struhl, ’96; Marigo et al., ’96;Alcedo and Noll, ’97; Chen and Struhl, ’98);cubitus interruptus (ci), a fascinatingly complextranscriptional regulator that mediates the re-sponse to Hh signaling (Orenic et al., ’90;Schwartz et al., ’95; Alexandre et al., ’96;Dominguez et al., ’96; Aza-Blanc et al., ’97;Hepker et al., ’97; Von Ohlen et al., ’97; Chenet al., ’98; Ohlmeyer and Kalderon, ’98; Wang andHolmgren, ’99); smoothened (smo), fused (fu),costal (cos), Suppressor of fused (Su(fu)), allcomponents of the Hh signaling pathway (Alcedoet al., ’96; van den Heuvel and Ingham, ’96; Sissonet al., ’97; Alves et al., ’98; Monnier et al., ’98);frizzled (fz) and Dfrizzled2 (Dfz2), dishevelled

Fig. 4. A conceptual summary of the Drosophila segmen-tation cascade. Redrawn after a figure of Lisa Nagy’s (’98), andembellished according to various sources, especially the ‘‘bluebook’’ chapters on segmentation (especially Martinez-Arias,’93). This diagram, which is incomplete and inaccurate, is onlyintended to convey the overall gist of the process, as outlinedin the main text. The graphical design is meant to convey thatthe segmentation cascade consists of a series of interwovencassettes. The members of each cassette interact strongly witheach other and have strong influences on members of down-stream cassettes but little if any influence on upstreamcassettes. This diagram is especially misleading with regardto the interface between the gap genes and the primary pair-rule genes, and with regard to interactions among pair-rulegenes. In reality, gap genes exert complex stripe-specificregulatory control over the pair-rule genes, and may repressa given gene in one region while activating it in another. Thisis achieved through the use of quasi-independent enhancerregions for each hairy or eve stripe. Finally, of course, thesegment polarity network is not adequately represented inthis figure at all!

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(dsh), shaggy (sgg), naked (nkd), armadillo (arm),and pangolin (pan), all components of winglesssignal transduction (Brunner et al., ’97; Cadiganand Nusse, ’97; van de Wetering et al., ’97; Bhat,’98; Bhanot et al., ’99); and other genes includinggooseberry (gsb) (Li and Noll, ’93) and sloppy-paired (slp) (Grossniklaus et al., ’92; Cadigan et al.,’94b). The en- and wg-expressing domains areinitially set up by the pair-rule genes (Inghamet al., ’88), but as pair-rule expression fades out, enand wg become dependent on one another, beforeeventually locking into various autoregulatoryloops (DiNardo et al., ’88; Heemskerk et al., ’91;Vincent and O’Farrell, ’92; Li and Noll, ’93;Hooper, ’94; Vincent and Lawrence, ’94; Yoffeet al., ’95) that, in a subset of embryonictissues, maintain the compartmentalization ofsegments and the expression of the segmentpolarity genes throughout development. To sum-marize, gap and pair-rule genes encode transcrip-tion factors that are transiently expressedduring segmentation; in contrast, the segmentpolarity genes encode a variety of transcriptionfactors, secreted cell signaling factors, receptors,transducers, cytoskeletal components, etc., andthey are stably expressed throughout develop-ment.

We became interested in the segment polaritynetwork because of the tempting possibility thatthis group of genes might qualify as a develop-mental ‘‘module’’ (von Dassow and Munro, ’99)that the evolutionary process has repeatedlydissociated from its inputs and outputs as it hasbeen redeployed in various developmental con-texts. Current evidence tentatively suggests on thebasis of the conserved expression of engrailedhomologues (Patel et al., ’89; Patel, ’94; Rogersand Kaufman, ’96; Grbic et al., ’98; Peterson et al.,’98), and a few experiments testing the regulatoryrelationship between wingless and en (Nagyand Carroll, ’94; Oppenheimer et al., ’99) suggestthat the segment polarity network may beinvolved in segmentation in all insects, perhapsall arthropods.

The same is apparently not true of the gap andpair-rule genes that, in Drosophila, provide theinitial inputs to the segment polarity network.Essentially all insects (excepting certain polyem-bryonic forms) begin embryogenesis as a syncy-tium in which the hundreds or thousands of nucleishare the cytoplasm and thus interact without theneed (or possibility) of cell–cell communication(Anderson, ’73). The gap and pair-rule networks,as we know them in Drosophila, as well as the

maternal morphogens, depend on the syncytialcontext of segmentation. However, only long-germinsects, including the flies, bees, and some beetles,specify all body segments while nuclei share acommon cytoplasm. Short-germ insects such asthe locusts specify only a few segments whilesyncytial; the rest emerge sequentially from acellular growth zone at the posterior of theembryo. Perhaps the ancestral condition wassomething like the intermediates of today, suchas the dragonflies and many of the bugs andbeetles, in which about half the segments form inthe syncytium, and the rest in a cellular context(Patel, ’94; Tautz et al., ’94). Homologues of gapand pair-rule genes exist in extreme short-germinsects like grasshoppers, and some, but not all, ofthese are expressed in patterns that imply roles insegmentation even during post-cellularizationsegment formation (e.g., Patel et al., ’92, 2001;Sommer and Tautz, ’93; Dawes et al., ’94; Brownet al., ’94, ’97; Dearden and Akam, 2001). How-ever, it seems unlikely that the gap and pair-rulenetworks, to whatever extent they are involved inpost-cellularization segmentation, function thesame way as they do in Drosophila to provide theupstream pre-patterning for the segment polaritynetwork, simply because, again, the known me-chanism by which these two groups of genes’expression patterns emerge depends on the ab-sence of cell boundaries. Thus, it appears that thesegment polarity network has retained its rolein maintaining the segmental boundary despite adramatic change in the very nature of the up-stream patterning mechanisms from which ittakes its prepattern.

Moreover, the segment polarity network, orsubsets thereof, is used in a variety of differentorganisms to do a variety of different patterningtasks. In the most dramatic example, the primor-dium of the butterfly eyespot expresses several ofthe core segment polarity genes in a spatial regimeat least superficially distinct from their segmentalpatterns (Keys et al., ’99). Since the eyespot is anovel adaptation within the Lepidoptera, it is hardto escape the conclusion that the evolutionaryprocess somehow managed to re-deploy the seg-ment polarity network at some point in theinvention of the eyespot.

Thus, the questions we sought to answer, usinga computer simulation model, were as follows: isour map of the segment polarity network completeenough to explain how, in the developing fly,it ‘‘remembers’’ a transient patterning imprintconferred by the pair-rule genes? Do the known


interactions among segment polarity genes ac-count for the apparent modularity of the segmentpolarity network? What properties emerge fromthe conspiracy of these genes, that we could nothave anticipated from descriptions of the partsalone?

SYNOPSIS OF OUR PRIOR WORK ON THESEGMENT POLARITY MODEL

In a previous report (von Dassow et al., 2000) weshowed that a model encompassing core interac-tions of the segment polarity network (Fig. 5B)could mimic patterns of gene expression observedin living embryos (Fig. 5A). Not only does this corenetwork have this capacity, it is highly robust tovariation in both parameter values and initialprepattern. In the jargon of dynamical systemstheory, the network is both structurally anddynamically stable. We thus concluded that thefacts embodied in the model suffice to explain thecanonical phenomenology of the segment polaritynetwork: namely, the stable maintenance of twostable cell states on either side of the paraseg-mental boundary, with a ground state occupyingthe rest of the segment. Furthermore, because the

core network can accomplish this task without anypersistent, extrinsic spatial or temporal biases onany of its components, and because it can do sostarting from a wide variety of initial conditions,we concluded that the segment polarity networkqualifies as a developmental module.

That is, the segment polarity network is a deviceunto itself. It has certain intrinsic behaviors.Those behaviors may be selected among eitherby treating kinetic parameters as tuning dials, orby triggering the module with different prepat-terns. Most remarkably, despite this tunability,the network is usually so robust that only verylarge turns of the metaphorical dials tune thenetwork to different behaviors. But on the otherhand, it is this very robustness that allows thenetwork to move around in parameter space,varying its behavior upon the fundamentaltheme. Another way of looking at it is that whilethe core topology of the segment polarity networkinsulates its dynamical behaviors from subtle,quantitative perturbations, gross qualitativeperturbations alter its behavior, often (nearly)discontinuously.

This simple model (hereafter the ‘‘core net-work,’’ ‘‘core model,’’ or ‘‘spg1’’), because of the

Fig. 5. Core segment polarity network and its task.Redrawn from von Dassow et al. (2000). (A) By the time theDrosophila embryo gastrulates, the segmentation cascade hasmapped out all the borders between the developmental unitsknown as ‘‘parasegments’’; boundaries between paraseg-ments are defined by the expression of wg to the anterior(green) and en to the posterior (blue). As development

proceeds the expression patterns of other segment polaritygenes emerge, and we have stereotyped some of the centralplayers as show. (B) Core segment polarity network describedin von Dassow et al. (2000), which we found was the minimalrealistic network capable of mimicking the stereotypedpattern shown in A.


brevity of its treatment to date, raises a variety ofquestions. Testing these issues and the develop-ment of more complex models provided severalintriguing surprises. This paper tells the rest ofthe story, divided into four sections intended to bereadable on their own. First, we analyze thedistribution of working parameter sets in para-meter space, and show that within a single, largesub-region working parameter sets are verycommon. Second, we test several topologicalvariants of the original segment polarity modelto show that the robustness of the model is notcontingent on a particularly lucky subset of knowninteractions. Third, we contrast versions of themodel with greater or lesser complexity, and showthat, to our surprise, increasing the level of detailactually improves the robustness of the model.And, finally, we show that challenging the modelwith a dynamical patterning test imposes seriousconstraints on either details of the mechanism orthe relative values of certain parameters.

RESULTS AND DISCUSSION

Section I: Constraints among parameters

For the core segment polarity model, we showedpreviously (von Dassow et al., 2000) that foressentially any value of most parameters, somechoice of values for the other parameters allowsthe core network to perform its basic pattern-holding task (such a set of values we call a‘‘solution’’). However, it remains unclear whetherthere exist trade-offs among parameters or biaseson the distribution of working values in parameterspace. To explore this we tested 235,000 randomlychosen parameter sets among which we found1,120 solutions, chosen from the same boundsused in our earlier work (see SupplementaryInformation for von Dassow et al., 2000), andthen plotted solution frequency versus value forall parameters (Fig. 6). While for many para-meters (most half-lives and cooperativity coeffi-cients, primarily) the search returns an essentiallyflat distribution, for an important subset thedistributions are moderately to strongly biased.This is the case for most of the parametersgoverning thresholds (half-maximal coefficients)for the activities of regulators with respect to theirtargets.

For example, Figure 6, column 1, row 4, showsthat solutions are most abundant when the full-length Ci protein is only a weak activator ofwingless transcription. Figure 6, column 1, row 5,shows that solutions are far more common if the

Fig. 6. Parameter distributions for the core segmentpolarity network model. Each cell of the table is a histogram;the horizontal axis is the entire range for the namedparameter, and the vertical axis is the number of solutionsthat have a particular value. Parameter names follow thefollowing conventions: ‘‘k’’ indicates a half-maximal activitycoefficient; ‘‘n’’ a cooperativity coefficient; ‘‘H’’ a half-life(which is actually a time constant); ‘‘r’’ indicates a first-orderrate constant and ‘‘k’’ a second-order constant; the subscriptindicates which interaction the parameter in question gov-erns, so ‘‘kXy’’ means the half-maximal activity of X withrespect to activation of y transcription; ‘‘X3’’ refers to themaximum number of molecules of X per cell (only relevant formolecular species that undergo explicit stoichiometric reac-tions such as hetero-dimerization); and the subscripts ‘‘Endo’’and ‘‘Exo’’ refer to endo- and exo-cytosis, respectively,whereas ‘‘Lmxfer’’ and ‘‘Mxfer’’ refer to exchange betweenadjacent and apposite cell faces, respectively. Ranges arediscussed in the Supplement to von Dassow et al. (2000), butbriefly, half-maximal coefficients, flux rates, transformationrates, and most other parameters range over 3 orders ofmagnitude on a log scale. All half-maximal coefficients rangefrom 10�3 to 1.0, where all concentrations are normalized tothe maximal possible steady-state level for the relevantmolecular species. Half-lives range linearly from 5 to100 min, cooperativity coefficients range linearly from 2.0 to10.0 (except nPTC-CID), and saturability coefficients rangelinearly from 1.0 to 10.0. These histograms result from asearch of 235,875 randomly chosen parameter sets fromwithin those ranges, in which 1,120 ‘‘working’’ sets werefound. In several identical searches, the biases shown here arereproducible, but for the most part the small wiggles are not.


N-terminal repressor form of the Cubitus inter-ruptus protein is an avid repressor of winglesstranscription. These observations of the modelmay correspond to empirical evidence. In particu-lar, in ptc�;wgts embryos, Wg cannot contribute toits own expression, but Ci presumably remainsuncleaved in the absence of Ptc; only weak wgexpression is observed in these embryos, showingthat full-length Ci by itself is a poor activator of wg(Hooper, ’94). Also, Von Ohlen and colleaguesshowed that, in cultured cells, Ci is a weakactivator of the large wg enhancer fragments,although it strengthens substantially when smal-ler pieces concentrating candidate Ci binding sitesare used as the test substrate (Von Ohlen andHooper, ’97; Von Ohlen et al., ’97). Their findingssuggest that other, perhaps generic, factors limitthe effectiveness of Ci in the context of the fullwingless transcription unit.

Other parallels are evident between our model’sparameter preferences and the available empiricalevidence. For instance, the core model prefers thatEngrailed not be too good a repressor of cubitusinterruptus transcription (Fig. 6, column 1, row 9).At the compartment boundary in imaginal discs,en is expressed in anterior compartment cellsclosest to the boundary, but does no more thanslightly diminish the expression of ci in those cells(Strigini and Cohen, ’97). In contrast, our coremodel does not appear to correspond to empiricalobservations on another target of Ci, patched (Fig.6, column 1, rows 6 and 7). The model prefers Ci tobe a good activator but CN a weak repressor of ptc.Holmgren and colleagues have shown that nearthe anterior–posterior compartment boundary inwing imaginal discs, while decapentaplegic tran-scription is activated at relatively low levels of Hhsignaling, i.e., at a low ratio of full-length Ci to therepressor form, ptc is activated only at muchhigher levels of Hh signaling and thus higherratios of Ci to CN (Wang and Holmgren, ’99).These authors even propose that Ci must behyper-activated at the highest levels of Hh signal-ing in order to activate ptc. We explore somepossible reasons for this discrepancy shortly.

Interestingly, the model prefers low rates of Wgtransport (Fig. 6, column 5, rows 3–5). Among themost striking biases in Figure 6 is toward lowvalues for the rate of cell-to-cell Wg exchange (row5). For many years it was disputed whether Wgever diffused at all from the cells in which it issynthesized (for example see Gonzalez et al., ’91;Vincent and Lawrence, ’94; for review see Marti-nez-Arias, ’93). Although the evidence has re-

cently become fairly conclusive that Wg proteindoes indeed travel a considerable distance from itssite of synthesis, there is still much debate abouthow it does so (i.e., by free diffusion or transcy-tosis) and in what form (free or bound to itsreceptors or other proteins), or even if Wgis carried by cells migrating within the plane ofthe epithelium (Hacker et al., ’97; Cadigan et al.,’98; Dierick and Bejsovec, ’98; Moline et al., ’99;Pfeiffer and Vincent, ’99; Sanson et al., ’99; Theand Perrimon, 2000; Strigini and Cohen, 2000).We permitted Wg to be endocytosed and thenre-exocytosed in our models to allow for thepossibility of transcytosis, so the core modelsubsumes transcytosis, ‘‘free’’ diffusion, and othermechanisms.

We wondered if Wg transport rates constrainother parameters in the model. To test this weconstrained the Wg ‘‘diffusion’’ rate rMxferWG

(really a cell-surface-to-apposite-cell-surface ex-change reaction rate) to vary only within high,medium, and low sub-ranges and then assessedthe resulting frequency of solutions in a randomsample of parameter sets (Meir et al., 2002b).Solutions are roughly three times more frequentwhen Wg diffusion is slow (with a half-time forequilibration on the order of hours) compared towhen it is fast (half-time on the order of minutes).2

The distributions of all other Wg transport ratesshowed strong responses to the imposed con-straint: if Wg ‘‘diffusion’’ is rapid, then all theseother rates are constrained to be slow, whereas ifWg diffusion is slow, then these other parametersexhibit flat distributions (Fig. 7A). The converse isalso true: if transcytosis of Wg via endocytosis, re-exocytosis, and transport within the cell mem-brane is slowed by constraining the appropriaterates (rEndoWg and rLMxferWG), the constraint uponWg cell-to-cell transport is largely alleviated (Fig.7B). The only other parameter for which wedetected sensitivity to Wg transport rates wasthe half-maximal coefficient governing Wg auto-regulation. In an unconstrained random sampling,this parameter (kWGwg) is distributed in a broad,gentle hump centered around values for whichWg is a moderately weak activator of its own

2The actual parameter used is a dimensionless product of a first-order rate constant and the characteristic time constant (usually1 min). The reaction in question is a concentration-dependentexchange between one cell face and the apposite face of theneighboring cell, in the case of cell-to-cell ‘‘diffusion,’’ or betweenone cell face and the adjacent face of the same cell, in the case of‘‘diffusion’’ within the membrane of an individual cell. The inverseof the first-order rate constant, the time constant, or its relative thehalf-time, is easier to express in words. When the rate constant islarge, the reaction is fast and the half-time small, and vice versa.


production (Fig. 6). If Wg is allowed to diffuserapidly, this hump sharpens into a rather abruptpeak, although there remain solutions for mostportions of the entire range; if Wg diffusion isslow, the hump erodes (Fig. 7D). It likewise erodeswhen we prevent transcytosis.

Originally we omitted cell-to-cell Hh transport.Hh is synthesized as a transmembrane protein butis autolytically processed (Porter et al., 1996a,b) toa cholesterol-tethered form. Processed Hh, at leastin some tissues, travels quite far from its site ofsynthesis (Strigini and Cohen, ’97), but it mayrequire assistance from other proteins to do so(Bellaiche et al., ’98; Burke et al., ’99). However,there is almost no detectable negative effect of Hhcell-to-cell transport when we introduce it to themodel. In fact, this process had a very slight butreproducible positive effect on the frequency ofsolutions (see Meir et al., unpublished observa-tions), but independent of the actual rate of theprocess: the distribution of the Hh diffusion rate isflat (Fig. 7C).

Thus the model exhibits subtle constraints andtrade-offs with regard to parameter distributions,many of which correspond to empirical observa-tions. On the basis of the results described inFigure 6 and other results not shown, we tried toidentify a sub-region of the parameter space inwhich solutions are most frequent. By trial anderror we came up with the region described inTable 1. In this region, the ranges of the half-maximal coefficients were restricted to the 10-foldrange that best matched the peaks in Figure 6.Most cooperativity coefficients were forced to befairly high except (following the results reportedin Meir et al., 2002b) those governing interactionsbetween ptc and ci, which were allowed their fullranges. All half-lives remained unconstrained,

Fig. 7. Parameter distributions as a function of parameterconstraints. Histograms as in Fig. 6. For panels A–D, thesearches are documented in the companion paper by Meir et al.(2002b, this issue); for panel E the results are cited in the text.(A) High rates of Wg cell-to-cell diffusion (governed by theparameter rMxferWG) impose a leftward bias on other Wgtransport rates; slow Wg diffusion rates ameliorate this effect.(B) Constraining Wg endocytosis and intra-membrane diffu-sion to low rates alleviates the bias on Wg cell-to-cell diffusion.(C) There is no bias on the rate of Hh cell-to-cell diffusion. (D)Subtle influence of Wg cell-to-cell diffusion on wg autoregula-tion. (E) Lowering cooperativity of ptc–ci interactions, andproviding ubiquitous early ptc, alleviate constraints on half-maximal coefficients governing Ci/CN regulation of ptc. Leftcolumn of panel E is taken from Fig. 6 to facilitate directcomparison.

3———————————————————————


except for the half-life of the intracellular form ofWg, which we constrained to be low. Finally, ratesgoverning Wg transport were constrained to below. Within this region, the frequency of solutionsis an astounding 4 out of 5 randomly chosen sets.Thus, Table 1 describes a vast, steep-sided canyonin the ‘‘epigenetic landscape’’ for the model,within which the model is almost completelyinsensitive to the exact value chosen for anyparameter, and in which mutual constraintsamong parameters all but disappear. Most strik-ingly, the constraints that define this GrandCanyon are very similar, except for the issue ofptc regulation (see below), to what we would havecome up with had we guessed from the availableempirical data.

The ci–ptc negative feedback loop

It bothered us that the model should seem toprefer Ci to be an avid activator of ptc whenempirical evidence suggests the opposite (seeabove). Furthermore, we had long been puzzledby a persistent defect in the time-varying behaviorof the core network: it seems very difficult toachieve stable patterns of expression for either Ciprotein forms or for ptc. For many parameter setsthat gave stable en, wg, and hh expressionpatterns, the Ci protein levels oscillated in time.3

These defects do not prevent the model frommimicking the patterns of wg, en, and hh, whichare the primary regulators of downstream pro-cesses like cuticle patterning and neurogenesis.

We noticed that when individual cooperativitycoefficients are held fixed at 1.0 (non-cooperative),for three parameters the solution frequencyactually increases slightly (see companion paperby Meir et al., 2002b, this issue). Those threegovern direct links between Ci and ptc. We thusrestricted these three cooperativity coefficients tovary only between 1.0 and 2.0, thus allowing onlyvery mild cooperativity in these links, and re-ranthe random sample. We obtained 1,200 solutionsin 126,525 samples (1 in 105), a significantimprovement. In addition, this restriction slightlyameliorated the bias on repression of ptc by CN(Fig. 7E). Almost all the solutions found conferredstable expression patterns on Ci and CN and ptc,and many exhibited the canonical pattern of Civersus CN distribution (high Ci next to Hh-producing cells, high CN elsewhere) and of ptc(stripes next to the Hh-producing cells). Althoughin the companion paper (Meir et al., 2002b, thisissue) we showed that the global level of coopera-tivity is positively correlated with robustness ofthe model, clearly these results demonstrate that

TABLE1. Canyon in parameter space, inferred from distribution of hits in the standard box1

Parameter Qualitative interpretation Constraint Bounds

kWGen WG activation of en Moderate 0.01^0.1kCNen CN repression of en Strong 0.002^0.02kWGwg WG autoactivation Moderate 0.01^0.1kCIDwg CID activation of wg Weak 0.2^2.0kCNwg CN repression of wg Strong 0.002^0.02kCIDptc CID activation of ptc Strong 0.002^0.02kCNptc CN repression of ptc Weak 0.1^1.0kENcid EN repression of cid Moderate 0.02^0.2kPTCCID PTC stimulation of CID cleavage Strong 0.002^0.02kENhh ENactivation of hh Weak 0.1^1.0kCNhh CN repression of hh Strong 0.002^0.02CCID Max. cleavage rate of CID Rapid 0.1^1.0n (e.g., nCIDptc, nCNptc, nPTCCID) Cooperativity coe⁄cients Steep 5.0^10.0HIWG Half-life of intracellularWG Short 5.0^30.0min.rEndoWG Rate of WG endocytosis Slow 0.001^0.01rExoWG Rate of WG exocytosis Moderately slow 0.005^0.05rMxferWG Rate of WG cell-to-cell exchange Slow 0.002^0.02

1Within the canyon described by the constraints tabulated below, we obtained 1,017 hits in 1,284 random parameter picks, a hit rate of 4 out of 5 (1 in1.26), with an average score of 0.060 (substantially better than the standard box average score of 0.077; the best score possible is approximately 0.035,the cuto¡ above which patterns are rejected is 0.2).

3‘‘Stable’’ versus ‘‘oscillatory’’ is a matter of degree. Nature surelydoes not know the difference between a true stable steady state pointand a small-amplitude limit cycle. However, we assume that large-

amplitude oscillations or chaotic behavior are functionally verydifferent from small limit cycles in many developmental contexts.Our scoring functions take these considerations into account.


cooperativity in regulatory interactions does notin every case foster robustness.

There remained a noticeable bias favoringstrong activation of ptc by full-length Ci. Weshow below that the solution frequency increaseswhen we include initial ubiquitous, moderateexpression of ptc in the prepattern. CN repressionof en is crucial; starting with our standardinitial conditions, ci and en race for control ofthe cell state. Perhaps Ci must be a strongactivator of ptc only so as to quickly get itselfconverted into a repressor and squelch the spreadof Wg-stimulated en expression. We thus con-ducted another sampling with the same con-straints on Ci–ptc-related cooperativity coeffi-cients and with initial ubiquitous expression ofptc mRNA and protein in the prepattern, andfound 1,127 in 68,114 solutions (1 in 60). The biason both Ci activation of ptc and on CN inhibitionof ptc all but disappeared (Fig. 7E).4

Sensitivity to gene dose

Uncoordinated variations in single parametersare a good proxy for a subset of likely mutationsthat one would expect to afflict real genes. Forinstance, a particular mutation might modify asingle nucleotide in the enhancer of some geneX, allowing it to bind with lowered affinity to someessential activator Y; this would be analogous toan increase in the parameter kYx in our modelingformulation. However, many mutations will likelylead to predictably coordinated changes in severalparameters at once. A simple way to impose

coordinated changes in the parameters for eachgene is to vary the copy number. We tested whatfraction of solutions succeed as heterozygotes orwith 1, 2, 4, or 6 additional copies of each locusindividually (Table 2). Notably, for the core modelmost solutions are almost completely insensitiveto extra copies of ptc, hh, and ci. The model seemsrather more sensitive to heterozygosity for hh andto dosage variation in either direction for wg anden. However, this is simply because the goodness-of-fit function that scores the behavior of themodel (see Supplementary Information for vonDassow et al., 2000) is moderately sensitive to thequantitative level of expression of the species itmonitors. We watched the entire test suite for hhand wg heterozygotes, and observed that most ofthe parameter sets that fail do so only because thereduced gene dose, of course, reduces proportion-ally the maximal level to which the gene’sproducts accumulate. In most cases the wg- orhh-heterozygous model makes a qualitativelycorrect pattern which fails merely because someof the concentrations are too low to pass thegoodness-of-fit cutoff. For wg heterozygotes,roughly 75% would have passed had we relaxedthe stringency of the scoring function accordingly;for hh heterozygotes, all but a few would havepassed.

Nevertheless, we stick with the more stringentmeasure, because from the perspective of thesegment polarity network’s downstream targetsthe absolute level of segment polarity geneexpression could be important. Even so, manyparameter sets tolerate quite a lot of variation ingene dose: 61 out of 1,120 solutions toleratedheterozygosity at all loci, and of those 61, the vastmajority (56) also tolerated an extra copy of everylocus. If these were uncorrelated responses, thenon the basis of Table 2 one would expect onlyabout 1% of the 1,120 to survive both hetero-zygosity for and an extra copy of all loci; the factthat 5% survived suggests that these are an

TABLE 2. Sensitivity of the core network to gene dose1

Locus Heterozygote 1 extra copy 2 extra copies 4 extra copies 6 extra copies

En 628 (56%) 872 (78%) 706 (63%) 516 (46%) 361 (32%)Ci 678 (60%) 1,027 (92%) 968 (86%) 886 (79%) 841 (75%)Ptc 901 (80%) 1,104 (99%) 1,101 (98%) 1,098 (98%) 1,091 (97%)Hh 379 (34%) 1,064 (95%) 1,029 (92%) 964 (86%) 872 (78%)Wg 339 (30%) 806 (72%) 646 (58%) 449 (40%) 333 (30%)

1The fraction of the 1,120 good parameter sets (same sets in Fig. 6 and the top line of Table 3 in the companion paper by Meir et al. (2002b, this issue)which also make the right pattern in the presence of di¡erent copy numbers for constituent genes. Percents are given in parentheses.

4This latter observation is unlikely to be relevant to Drosophiladevelopment, because pair-rule gene products persist on the order ofan hour after segment polarity genes first come to be expressed, andprovide a temporary, fading influence that guides the early sorting outof intrasegmental cell states (Martinez-Arias, ’93). We have exploredhow they do so in the context of this model and their influence onrobustness (which is generally positive), but do not report these resultshere. It is, however, unknown how the segment polarity genes acquiretheir initial patterns in other arthropods. Especially in short-germinsect embryos, the segment polarity module may not be able to counton pair-rule genes for guidance.


especially robust subset. Of those, 12 parametersets tolerated all tested gene dose variations. Thisshows, on the one hand, that it is possible to findparameters that make the network robust to 12-fold variation in the dose of every locus. However,those 12 sets came from a search of 235,000random samples, so their frequency is about 1 in2�104. While this is still a high frequency (see thehypothetical comparison in von Dassow et al.,2000, to an engineered circuit), the fact that only afraction of solutions tolerate heterozygosity sug-gests an evolutionary rationale for robustness.While it is hard to imagine that organismsexperience significant selection to tolerate hetero-zygosity or extra copies of genes, certain otherphenomena, from the point of view of the genenetwork, seem very similar. For example, embryosdiffer significantly in volume and can be inducedto differ even more in response to extremeconditions (our unpublished observations). Cellsprobably differ stochastically in the availability ofbiosynthetic machinery, or time spent in inter-phase, or many other factors. Selection forrobustness against such non-heritable variationwould likely render the network robust to geneticvariation that alters gene expression levels.

Summary

In this section we showed that the parameterspace of the core segment polarity model embodiesvarious constraints on individual parameters,including trade-offs among several parameterscontrolling a particular kinetic process. Theseconstraints reveal a sub-region of the parameterspace, still vast, in which the model is almostguaranteed to work. We suggest the visualmetaphor of Grand Canyon, with smaller tribu-taries (within which the model works, but not asrobustly) snaking in from the surrounding pla-teaus to join a wide, steep-walled main canal.

Section II: Architectural variants of thecore network model

The core network model lacks certain knownor suspected interactions among its components.An example is that En is claimed to repress ptc(Sanicola et al., ’95), although it is not clearwhether this interaction is direct or mediatedthrough En repression of ci. In addition, since wecould not see why this interaction should benecessary for the function of the network, weinitially chose to leave it out. Again, our goal is toreconstitute, not make a complete re-creation a

priori. The core network model also embodiescertain assumptions about the specific nature ofthe interactions among its components. Forinstance, wg autoregulation is mediated by theintracellular, rather than the extracellular formof the Wg protein (this seemingly bizarre choicewas supported by empirical observations reportedby others; see below). We next test the extent towhich these choices affect the performance of thenetwork. Figure 8 diagrams the core network andthe variants to be explored in this section. Coloredlines indicate additional links; dashed lines in-dicate alternative choices, and no variant embo-dies two dashed lines of the same color. Table 3describes these variants and the results of randomsamplings conducted as above.

Repression of patched by Engrailed

Introduction of a link by which En repressesptc slightly decreased the frequency of solutions.However, any such addition increases the dimen-sion of parameter space. Any bias on the distri-bution of these novel parameters, as for several

Fig. 8. Architectural variants of the core network. Theentire network inhabits each cell; also diagrammed for clarityare the direct effects of one cell on its neighbor (WG¼wingless,EN¼engrailed, SLP¼sloppy paired, HH¼hedgehog,CID¼cubitus interruptus (whole protein), CN¼repressorfragment of cubitus interruptus, PTC¼patched,PH¼Patched–Hedgehog complex). Ellipses represent mRNAs;rectangles, proteins; and hexagons, protein complexes. Arrowsrepresent positive interactions; round-ended lines, negativeones. Solid black lines represent interactions that remain inplace in all variants. Colored lines indicate the variantsdocumented in Table 3. Dashed lines indicate mutuallyexclusive choices; no two dashed lines of the same color existin any one model.


TABLE3.

Architectural

varian

tsof

thecore

netw

ork,an

dtheircons

eque

nces

1

Origina

ltop

olog

y(referen

ceforspgl)

1120

235,875

1in

211

Varian

t:wha

tify

No.

ofhits

No.

oftries

Hitrate

1EN

repressesptctran

scription

344

107,574

1in

313

2EN

repressesptc,plus

ubiquitous

initialp

tc724

101,175

1in

140

3Ubiqu

itou

sea

rlyptcex

pression

896

97,339

1in

109

4CIactivatesen

730

260,94

31in

357

5CIactivateshh

680

260,142

1in

383

6Ci/CN-en/hh

Var.A:C

Iactivates

hh&

en,C

Naglob

alrepressorof

both

enan

dhh

198

122,401

1in

618

7Va

r.B:C

Naco

mpe

titive

inhibitorof

CIw.r.t.h

hregu

lation

(i.e.loc

alhh

inhibitor)

69120,86

11in

1,751

8Va

r.C:C

Naglob

alhh

repressor,bu

taloca

linh

ibitor

ofen

:Ci/CN

syne

rgizes

w/W

gin

enactivation

5121,1235

1in

410

4

9Va

r.D:C

butwithCNaloca

lhhinhibitor

4117,62

61in

410

4

10EN

represseswgtran

scription

538

237,764

1in

442

11eW

Gon

cell’ssu

rface(ins

tead

ofne

ighb

ors’)a

ctivates

entran

scription

486

190,66

11in

392

12iW

G,n

oteW

G,a

ctivates

entran

scription

171

196,66

91in

1,150

13eW

G,n

otiW

G,m

ediateswgau

toregu

lation

456

189,82

41in

416

14SlpVa

r.A:eWGac

tivatesslptran

scription,

enregu

lation

like

line

11ab

ove

347

163784

1in

472

15SlpVa

r.Awithinitialslp

stripe

inwgrow

839

1534

951in

183

16SlpVa

r.B:iWGactivatesslptran

scription,

enregu

lation

like

line

11ab

ove

775

1608

821in

208

17SlpVa

r.Bplus

initialslp

stripe

inwgrow

1538

143798

1in

9318

SlpVa

r.C:sam

eas

Bbu

tiW

Gactivatesen

(likeline

12ab

ove)

281

161840

1in

576

19SlpVa

r.Cplus

initialslp

stripe

inwgrow

751

1588

561in

212

20SlpVa

r.D:A

butwithen

regu

lation

asin

original

mod

el53

31523

511in

286

21SlpVa

r.D

plus

initials

lpstripe

inwgrow

1021

141805

1in

139

22SlpVa

r.E:B

butwithen

regu

lation

asin

original

mod

el1032

147460

1in

143

23SlpVa

r.Eplus

initials

lpstripe

inwgrow

1595

1350

361in

85

1 Top

line

(referen

cemod

el)

isthe

same

data

asTa

ble

3in

the

compa

nion

pape

rby

Meir

etal.

(200

2b,

this

issu

e).

Parameter

range

sas

describe

din

Figure

6;spg1

varian

tsarediag

rammed

inFigure

8.


of the original parameters, will cause a decreasein the frequency in the absence of constraints.Thus the effect of adding the En–ptc link wasunremarkable. We wondered, however, why such alink might exist. Although in our standard teststhe initial conditions consist only of stripes of wgand en and the corresponding proteins, in realitythe prepattern confronted by the segment polaritynetwork in a blastoderm-stage Drosophila embryois much more complex. In particular, ptc isexpressed nearly ubiquitously until mid-germband extension, disappearing from en-expressingcells by stage 9, and finally coming to be expressedonly on each side of the en-expressing stripe atstage 11 (Hooper and Scott, ’89). Indeed, in thecontext of initial background levels of ptc, theEn–ptc link increases the solution frequency.However, even in the absence of the En–ptclink the frequency of solutions increased whenthe initial conditions included a moderatebackground level of ptc. We suspect this isbecause Ptc promotes conversion of basally ex-pressed Ci into the repressor, CN, which thenprevents en expression from spreading beyond itsinitial stripe.

Potential targets of Cubitus interruptus

The core network includes one link whosereality is questionable: the repression of en byCN. We justified this link (von Dassow et al., 2000)on the basis of two published observations: (1)antibodies to Ci stain the polytene chromosomeband in which the engrailed gene resides (Aza-Blanc et al., ’97), and (2) smoothened-dependentHh signaling activates en expression just anteriorto the compartment boundary in imaginal discs(Blair and Ralston, ’97; Strigini and Cohen, ’97)and abdominal epidermis (Lawrence et al., ’99),presumably through the known Smo signalingpathway that terminates in the regulation of Cicleavage. Since other known positive targets ofthe full-length Ci protein are repressed by theN-terminal fragment, we felt justified in hypothe-sizing the CN–en link. Why not then include apositive link between full-length Ci and en? Whenwe do so, the frequency of solutions drops. Thismerely says that if such a positive link exists, itmust be a weak one, and on the basis of thenarrow stripe of imaginal disc cells in which en isactivated by Hh signaling, this must be the case.By analogous reasoning, since in our model CNrepresses hh, should not Ci activate hh? As withen, if it does so the model suffers slightly, but only

to the extent that this added interaction must notbe too strong.5

Even if full-length Ci does not actually regulateeither en or hh in the early embryo, then surelythese links are near-neighbors in the space ofpossible evolutionary changes, since Ci and CNhave the same DNA binding domain. CN probablyacts as a repressor by competing with full-length Cifor binding to consensus sites, and thus ought to bea local repressor of transcription. However, in thecontext of our model we had to assume that CN is aglobal repressor of en and hh. CN could accomplishthis if some factor recruited by full-length Ci (CBPperhaps?) is required for transcription of thesetargets. We tried several different ways of hookingup these links, including models in which CN is aglobal repressor of both en and hh, or of one or theother, or of neither. Table 3 includes the resultsfrom several of these, and shows that these choicescan be very important to the performance of themodel; if indeed these links are easy to establish,then the network is in a dangerous evolutionaryneighborhood with respect to this part of the circuit.

Wingless transport

In our core model we chose to make en sensitiveto the concentration of Wg on apposite cellsurfaces; that is, cells respond to Wg proteinpresented to them by their neighbors. We testedwhether it makes any difference to the model if Wgmust first become associated with the respondingcell’s surface (Table 3). The modest penalty shownin Table 3 (line 11 versus top line) likely resultsfrom the fact that in order for this transfer to takeplace the Wg diffusion rate, which the modelprefers to be slow, must be sufficient to get theprotein from the cell in which it is synthesized tothe cell in which it is sensed. This result under-scores the suggestion that control of Wg traffic islikely to be a critical parameter for optimizing themost basic functions of the segment polaritynetwork. We next tested whether it makes anydifference (to the model) whether wg autoregula-tion is mediated by intracellular or extracellularWg protein. As shown in Table 3, line 13, thefrequency is lower for a model in which extra-cellular Wg stimulates wg transcription.

5It should be noted first that we are aware of no evidence for positiveregulation of hh by Ci. Indeed, the only evidence available is that someproduct of ci must repress hh (Dominguez et al., ’96), and we merelyassumed it is the N-terminal fragment. Full-length Ci binds to (andrequires) the co-activator CBP to activate its targets (Akimaru et al.,97; Chen et al., 2000). In some contexts CBP behaves as a co-repressor(Waltzer and Bienz, ’98), so it is possible that full-length Ci isresponsible for hh (and even conceivably en) repression.


Our choice of intracellular Wg for this role issupported by the analysis of certain wg alleles inwhich abundant Wg protein remains localized tothe cell in which it is synthesized. These allelesblock maintenance of en expression by Wg, but notwg autoregulation (Dierick and Bejsovec, ’98).However, Hooper (’94) showed that the wgautoregulation mechanism is able to spread fromcell to cell in the segment, and Manoukian et al.(’95) showed that porcupine, a membrane proteininvolved in Wg glycosylation and secretion (Tana-ka et al., 2002), is required for wg autoregulation.The observations of Hooper and Manoukian mightbe reconciled with those of Dierick and Bejsovec byassuming that Wg moves among cells by transcy-tosis. Our model shows these hypotheses aredynamically plausible, but clearly we need moredetailed experimental information on Wg trans-port and the autoregulatory pathway.

Sloppy–paired as a candidateintermediate for wingless autoregulation

We considered two candidates for intermediatesteps in wg autoregulation. First, several pieces ofsuggestive evidence implicate the protein kinaseencoded by fused in this process (Hooper, ’94;Therond et al., ’99). The Fu kinase phosphorylatesCi, and perhaps this phosphorylation could hyper-activate Ci (Ohlmeyer and Kalderon, ’98); if inaddition Fu activity is somehow stimulated by Wgsignaling, this could account for wg autoregula-tion. So far, however, we have failed to incorporatea complete, self-consistent set of hypotheses abouthow the action of fused and other components ofthe Hh signal transduction pathway could mediatewg autoregulation.

A better candidate is the transcription factorencoded by sloppy-paired. Slp activates wg andrepresses en (Cadigan et al., 1994a,b). In addition,recent reports suggest that slp is itself activated byWg signaling and repressed by En (Kobayashiet al., ’98; Bhat et al., 2000; Lee and Frasch, 2000).slp is expressed in the Wg-producing cells andtheir immediate neighbors to the anterior andtherefore is an excellent candidate for an inter-mediate in wg autoregulation (see Fig. 8),although from published reports it is not clearfor how long slp is expressed in embryogenesis. Asshown in Table 3, models in which wg isautocatalytic only via slp perform admirably.

We noticed while watching the random sam-pling process work over models using slp tomediate wg autoregulation that many parameter

sets failed because Wg expression declined beforeslp expression ramped up. slp is actually expressedsubstantially earlier than the other segmentpolarity genes (Grossniklaus et al., ’92). Whenwe include an initial stripe of slp mRNA andprotein in the prepattern (an entirely legitimateelement in flies but unknown for other insects) thehit rate improves for obvious reasons. The initialslp stripe also made the model less sensitive tocertain details, such as which compartmental formof Wg activates en. This is because the initial slpstripe keeps en from invading the wg row.6

Summary

The robustness of the segment polarity model isnot due to particular ad hoc wiring choices.Rather, our attempt at a reconstitution resultedin a robust core topology, and that core model isrobust not only to variation in parameters andinitial conditions (von Dassow et al., 2000) but alsoto variation in the architecture of the network, solong as the architectural variants use realisticconnections. We have tested numerous realisticpossibilities that can be addressed within thecontext of the simple model of the core segmentpolarity network, and with respect to its basicboundary-maintaining function. Most of the var-iants we tested have only subtle impacts on theperformance of the model, but Table 3 shows thatclearly one can introduce connections that imposemajor constraints. Before leaving the simplemodel, we emphasize that the map in Figure 8encompasses facts known with varying degrees ofconfidence. In the process of concocting a model ofany biological process, gene network behavior inthis case, one confronts unknowns in which somedecision must be made in the absence of adequateevidence. The modeling approach used here anddescribed in the companion paper (Meir et al.,2002b, this issue), which relies on a stereotypedset of building blocks to express gene regulatoryinteractions, allows us to rapidly test the plausi-bility and mutual consistency of hypotheticalmechanisms and quickly compare models embody-ing different choices.

Section III: Maps at lower and higherresolution

The segment polarity model seeks to express inmathematics certain kinetic processes described

6See the companion paper by Meir et al. (2002b, this issue) fordemonstration that, under relatively tightly specified conditions, asub-network consisting of just slp, wg, and en can do the same job asthe whole network, but not robustly.


by biologists. To what extent do the results dependon the level of detail incorporated in the model?Overly simplistic models fail to incorporate me-chanistic constraints that Nature faces. Forexample, the core segment polarity model ex-presses mathematically the English statement‘‘Wingless activates engrailed transcription’’, butNature has to design a signaling pathway by whichit does so, and the chosen design may haveimportant consequences to the robustness of themechanism as a whole. On the other hand, toocomplex a model becomes useless as an explana-tory tool. In this section we compare models oflesser or greater complexity.

Simpler versions

First, we tested a simplified model that istopologically identical to the core network butcollapses the mRNA and protein products of everylocus into a single entity. The collapsed model hasslightly fewer parameters but, again, exactly thesame topology. We found that it is quite capable ofmaking the desired pattern, but solutions are fivetimes less frequent: the hit rate in a randomsample was roughly 1 in 1,000. Furthermore,somewhat contrary to our naı̈ve expectations, thecollapsed model is computationally less tractable:it takes about an order of magnitude morecomputer effort per parameter set to integratethe collapsed model in searches equivalent tothose performed with the core model.

Both the lowered solution frequency and theincreased stiffness of the equations are due to thefact that the collapsed model is much more proneto large-amplitude temporal oscillations than isthe core model. Collapsing the mRNA and proteininto a single species eliminates a major source ofkinetic delays (e.g., when an mRNA declinestemporarily, the protein declines a little later).While delays are generally thought of as a sourceof oscillations, in this case the existence of anintermediate step between stimulus (input into agene’s enhancer region) and response (accumula-tion of the protein product to equilibrium level)tends to damp out oscillations. The desiredbehavior for the segment polarity model is basedon positively reinforcing but mutually exclusivestable cell states. A balance of positive feedbackand negative crosstalk promotes this behavior.The model includes at least one prominentnegative feedback loop (between Ptc and Ci) thattends to generate oscillations. Whether or notthese oscillations propagate to the positive feed-

back loops that stabilize cell states depends bothon the relative time scale with which eachcomponent responds and also on how well non-linear thresholds buffer each component fromoscillations in input stimuli. In the core model,all transcriptional responses involve ramping upor down the level of two different species whichcould have very different half-lives. Since the half-lives determine how long it takes each species toreach equilibrium, the response to an oscillatoryinput is damped by the slowest-equilibrating stepin the chain. In contrast, in the collapsed model,there are half as many opportunities to damposcillations, and thus a much greater fraction ofthe parameter space succumbs to oscillations thatprobably originate in the Ci/Ptc negative feedbackloop. It is not yet clear to us whether this findingis specific to this particular network, but it ispossible that gene networks are ‘‘designed’’ toexploit dynamical rules like these to stabilize geneexpression levels.

Second, we attempted to concoct a different andmuch simpler ‘‘engineered’’ model that wouldaccomplish the same task of stabilizing twodifferent cell states right next to one another.After many failures, we accomplished this usingtwo loci encoding diffusible proteins, each of whichsomehow activates itself, suppresses the other,and hetero-dimerizes with the other to form asubstance that activates expression of both genes(Fig. 9A). This toy model can work but is far lessrobust to parameter variation than is the segmentpolarity network model (Fig. 9B). With 27 freeparameters, the solution frequency was on theorder of o1 in 104 (n¼29 hits). If the kinetics ofthe two loci’s products are constrained to besymmetric (e.g., if A and B have identical half-lives, diffusion coefficients, avidity for targets,etc.), then the solution frequency becomes moreamenable (1 in E150, n¼646 hits, 16 freeparameters), but this is a very stringent con-straint! If the mRNAs and proteins are collapsedinto single nodes as above, this design breaksdown completely; there must exist some scarcesolution somewhere in the parameter space of thecollapsed version, but we have not yet discoveredits hiding place, even with hand tuning.7 Perhapswe are just poor engineers, and some other elegant

7The reason is essentially the same as in the collapsed model of thesegment polarity network: cut out the intermediate steps, and themodel becomes much more susceptible to oscillations driven bynegative feedback. Thus our experience with the engineered toysubstantiates our explanation for the segment polarity network, andhints at the possibility of a general rule for designing robust geneticmechanisms.


device might do better. However, it is not the casethat the network in Figure 9A represents anintrinsically poor design; we managed, indeed, tobuild it up until it became quite robust, at whichpoint it had concomitantly grown almost ascomplex as the core segment polarity network(not shown).

A third simplification is discussed in thecompanion paper by Meir et al. (2002b, this issue).We found that a sub-network consisting of wg, en,and slp is capable of accomplishing the same

boundary-maintenance task as the core segmentpolarity network, but it is not nearly as robust toinitial conditions or to parameter variation. Thusthe core segment polarity model is not, as weoriginally reported (von Dassow et al., 2000), thesimplest realistic topology capable of accomplish-ing this task; rather, adding another component,slp, allows one to break the network into twopieces, one of which can do the basic task required,but which becomes more robust when coupled tothe other sub-network.

Incorporating a signal transductionpathway for Wg

The core model omits the dynamics of the signaltransduction pathway between Wg and en. A briefsummary of the real pathway would begin withsecretion of Wg from cells that synthesize it. Oncesecreted, Wg interacts with a number of cellsurface proteins, including receptors in theFrizzled family (Bhanot et al., ’96, ’99; Bhat, ’98;Sato et al., ’99; Sivasankaran et al., 2000), thetransmembrane receptor Notch (Wesley, ’99), andheparan sulfate proteoglycans (Hacker et al., ’97;Lin and Perrimon, ’99; Tsuda et al., ’99). We arenot aware of any role for Notch in segmentboundary maintenance in the early embryo.Heparan sulfate proteoglycans may restrict Wgtraffic across the parasegment boundary, but thisrole may or may not affect segment boundarymaintenance. The products of three related genes,frizzled (fz), Dfrizzled2 (Dfz2), and Dfrizzled3(Dfz3) are the receptors involved in en regulationby Wg. fz was originally discovered as a tissuepolarity gene (Vinson et al., ’89), and it seeminglyretains a sociological association with that func-tion. Dfz3 exhibits little signaling activity and mayserve primarily as an attenuator of Wg signaling(Sato et al., ’99). Thus Dfz2 assumes the mantle of‘‘the Wingless receptor.’’ Nevertheless all threeare expressed in early fly embryos, and at leastfz and Dfz2 can substitute for one another duringsegmentation (Bhanot et al., ’99; Chen and Struhl,’99). Hereafter, for simplicity, we use ‘‘Fz’’ to referto all frizzled-family Wg receptors simultaneously.

Wg binding to Fz somehow causes the product ofdisheveled to suppress activity of the kinase GSK3,encoded by shaggy. GSK3 phosphorylates severalproteins, among them the product of armadillo(arm, the Drosophila homolog of b-catenin),Daxin, and a Drosophila homologue of APC(reviewed by Cadigan and Nusse, ’97). These threeproteins form a complex which targets Arm for

Fig. 9. Simple network that does a similar task to thesegment polarity network. (A) Topology of the concoctednetwork; A and B are secreted proteins that promote theirown expression but repress each other; they dimerize toproduce a diffusible complex which functions as an activator ofboth loci. (B) Wheel plot of parameter sets that succeed inholding a boundary with an A-expressing cell state to theanterior and a B-expressing cell state to the posterior. Clearlythere are sharp biases on certain parameters. Not apparent isthe strong preference for A and B to be symmetric, i.e., to begoverned by nearly identical values for every parameter (seemain text).


ubiquitin-dependent degradation involving theproduct of the gene supernumerary limbs (Jiangand Struhl, ’98). Thus Wg signaling somehowsuppresses GSK3 activity, reducing Arm’s affinityfor Daxin and APC, allowing Arm to accumulate inthe cell. Arm also binds to the Drosophilahomologue of TCF, encoded by pangolin (pan).Arm and Pan enter the nucleus and act togetheras a transcriptional activator of Wg target genes,including en (Brunner et al., ’97; van de Weteringet al., ’97). In the absence of sufficient Arm, Panmay bind to the transcriptional corepressor en-coded by groucho (Cavallo et al., ’98); thus Panmay constitute a transcription switch by bindingalternately to different partners, much as Ciachieves through its cleavage.

In the model ‘‘spg2’’ (depicted in Fig. 10), therates at which cleavage destroys Arm varies withthe occupancy of Fz by Wg, Arm binds toconstitutively produced Pan, and the complexactivates en. There is also strong evidence thatWg signaling results in the repression of Dfz2transcription, both in the wing imaginal disc andduring segmentation, probably via Arm and Pan(Cadigan et al., ’98). Variants of spg2 includedseveral combinations of choices (Fig. 10; Table 4),which, at the time we composed the model, couldnot be conclusively differentiated on the basis ofempirical observations. Our interpretation of theliterature tentatively leans us toward spg2 VariantE or G as the most realistic. We report the resultsfrom a handful of other variants to show (Table 4)that none of these details makes more than amodest difference in the solution frequency,although in the model if Pan represses en it mustnot do so too avidly. It is worth noting thatconstraints manifest in spg1, on whether Wg ispresented to cells or must become associated withthem in order to act, are abolished with the

incorporation of even such a simple representationof the Wg signal transduction pathway: variants Eand F are identical to B and C except for thisdetail. However, most startling is the fact that it isunexpectedly easier to find solutions for spg2 thanfor the original model, spg1. Whereas the fre-quency for spg1 and its variants tends to bearound 1 in 200 or somewhat worse, the frequencyfor spg2 is E1 in 100, despite the greatercomplexity of the latter model (7572 parameters,depending on the variant, versus around 50 forspg1, 8 loci versus five loci, and 21 species versus13 species).

Fig. 10. Topology of spg2. Same diagram conventions as inFig. 8, and again no two dashed lines of the same color exist inany one model (WG¼wingless, FZ¼frizzled, FW¼Frizzled-Wingless complex, ARM¼armadillo, PAN¼pangolin, EN¼engrailed, SLP¼sloppy paired, HH¼hedgehog, CID¼cubitusubitus interruptus (whole protein), CN¼repressor fragment ofcubitus interruptus, PTC¼patched, PH¼Patched–Hedgehogcomplex). Ground symbol represents constitutive destructionof Arm at a rate regulated by Fz occupancy.

TABLE 4. Solution frequency for spg2 variants

Variant Distinguishing feature No. of hits No. of samples Hit rate

A Pan and Arm^Pan compete to regulate fz, Arm cleavage stimulated by free Fz 912 98,628 1 in 108B Same as A but Arm cleavage inhibited byWg-bound Fz 1,165 129,845 1 in 111C Same as A but fz depends on Pan only 925 100,827 1 in 109D Same as B but fz depends on Pan only 1,217 124,961 1 in 103E Same as B butWg binds to Fz on the same cell surface rather than opposite

surfaces1,265 130,713 1 in 103

F Same as E but fz depends on Pan only 1,214 129,370 1 in 107G Same as E but Pan represses en 475 139,411 1 in 293H Same as F but Pan represses en 501 133,721 1 in 267

1Searches within a parameter space box as in Figure 6 (see also the Supplement to von Dassow et al., 2000) but with all cooperatively coe⁄cientsranging between 1.0 and 10.0.Variants are diagrammed in Figure 10.


Clearly we require a metric by which to comparemodels with different numbers of parameters. Apriori we expect the ease of finding solutions to godown as the parameter count rises. As complexityincreases, the null hypothesis is that the morethings there are to break the more likely one willfail. The null hypothesis should hold if the greatercomplexity comes about by intercalating featuresthat do not inherently synergize to make eachother less likely to fail, that is, if the addedcomponents are just links in a chain. Since thedifferences between the models spg1 and spg2consist essentially of components added in series,we expected the null hypothesis to be validated,which would predict a lower frequency of solutionsaccording to the chain of reasoning that follows.

The frequency is roughly the product of theprobabilities that the dice throw yields a ‘‘good’’value for each parameter (it is a little more subtleif parameters are correlated). For the originalmodel with 48 free parameters and no restrictionson cooperativity (see companion paper by Meiret al., 2002, this issue), a hit rate of 1 in 250corresponds to an average probability of about89% per parameter of landing on a good value(48th root of 1/250 E0.89). For spg2 with 75 or soparameters, a hit rate of 1 in 100 corresponds toan average probability of 94%. This does not seeman impressive difference until one works it out theother way around: if the average per parameterprobability of a good pick were to remain constantat 89% when going from spg1 to spg2, then the hitrate for the variants of spg2 should be near 0.89 tothe 73rd to 77th power, or 1 in 4,000–8,000,depending on the variant. This makes the differ-ence between spg1 and spg2 look more dramatic.In the future we hope to devise better measures ofrobustness, but to date we have not been able to doso without a large investment of computation. Forexample, we have developed two functional mea-surements (Meir et al., 2002a): one in which weassess how much ‘‘mutation’’ the model cantolerate, and another that involves assessing thesuccess rate of recombinants between workingparameter sets. However, these methods require agreat deal of computational trials and oftencorrelate with the conclusions of the ‘‘nth root’’formula described above, so for this paper we stickwith that measure.8

In contrast, the simplified version of spg1 inwhich mRNA and protein were collapsed together(42 free parameters) should have exhibited asolution frequency of 0.89 to the 43rd power, or1 in 150, had it been equivalent to spg1; instead ityielded only about 1 in 1,000. If our engineeredtwo-locus model were as robust as the coresegment polarity model, we might expect 1 in 25,not o1 in 104, or 1 in 6 for the version with strictsymmetry constraints instead of 1 in 150. For thearchitectural variants of the core model, it turnsout that in most cases the decrease in solutionfrequency is nearly commensurate with the slightincrease in parameter number, although somelinks (like the introduction of slp) appear to confera real positive advantage on robustness; that is,those few in the latter group, as well as theincreased detail represented in spg2, representgenuine improvements in the design of the net-work.

Toward more complete networks

The core network model incorporates only atrivial representation of wg autoregulation. As wenoted above, there is as yet too little informationabout this process for us to do much better, but slpseems an excellent candidate for an intermediatetranscriptional regulator. slp has the additionaland very interesting property that it engages inboth a positive feedback loop with wg and anegative feedback loop with en (see discussionabove in the context of spg1). Another possibilityis gooseberry, which appears to be a target of Hhsignaling through Ci (Bhat, ’96; Von Ohlen et al.,’97) and also a target of Wg, and which positivelyregulates wg during mid- to late embryogenesis(Li et al., ’93; Li and Noll, ’93). However gsb isknown not to be required for early wg autoregula-tion (Hooper, ’94). It is not clear at the presentwriting whether slp is stably expressed through-out embryogenesis; published reports end withmid-embryogenesis and we can’t detect strongsegmental expression beyond approximately stage12 (V. Rich and G von Dassow, unpublished), butslp is expressed in the appropriate region in atleast some segments in the adult (Struhl et al.,’97). It could be that gsb takes over wg auto-regulation during mid-embryogenesis. Clearlythere remain significant empirical questionsabout how wg autoregulation evolves throughout8The ‘‘nth root’’ metric will deceive greatly when the parameter

space encompasses many distinct basins of ‘‘working territory’’. Insuch a case, where there are in effect many qualitatively distinctkinetic ‘‘recipes’’ for solving the problem, the mutational or recombi-national methods will provide a much more accurate measure of therobustness of the network because they tell us how robust is a given

recipe. For all variants of the segment polarity network, it appearsthere is a single major solution basin in parameter space, albeit thatbasin is fed by many rivulets and side canyons and wash-gullies. Thusfor this network the ‘‘nth root’’ seems adequate.


embryogenesis, questions which need to be re-solved before we can confidently incorporate anycredible solution in a model.

In the core network we modeled Ptc as a directmodulator of Ci cleavage, which it is not. In realityPtc is a stoichiometric inhibitor of the transmem-brane protein encoded by smoothened (Alcedoet al., ’96; Alcedo and Noll, ’97; Chen and Struhl,’98). In the absence of Ptc, Smo transduces aconstitutive signal that stabilizes Ci by regulatingits association in a complex with several otherproteins including the Fu protein kinase, theproduct of Suppressor of fused, and the kinesin-like protein encoded by costal-2 (Robbins et al.,’97; Sisson et al., ’97). This complex both preventsfull-length Ci (but not CN) from entering thenucleus and also directs Ci to proteolysis whichconverts it to CN. Ptc binding to Smo preventsSmo from signaling; Hh binding to Ptc preventsPtc from binding Smo; thus, Hh titrates awayPtc, allowing Smo to send its signal, resultingin the accumulation of full-length Ci. ProteinKinase A also regulates Ci: phosphorylation byPKA may modulate Ci binding to Fu, Su(fu), andCos2, may modulate Ci cleavability, and mayinfluence the ability of Ci to enter the nucleusand activate its targets (Chen et al., ’98; Wangand Holmgren, ’99). Unfortunately, it is notapparent yet whether or how Hh (or anythingelse during segmentation) might influence PKA;the current suggestion places Smo and PKA onparallel pathways.

As steps toward a more realistic framework forfurther exploring the segment polarity architec-ture we tested the models ‘‘spg3’’ and ‘‘spg4,’’extensions of spg2 which we do not discuss indetail here. Spg3 attempted to account for wgauto-regulation via Fused and Ci instead ofthrough slp, and was unsuccessful. Spg4 incorpo-rates smo and drops the assumption of irreversiblehetero-dimerizations; all complexes are allowed todissociate. Spg4d, the variant most comparable tothe original model, is governed by 84 free para-meters. In 40,326 random samples we found 386solutions (1 in 104). The null hypothesis wouldhave predicted roughly 1 in 20,000 if the greatercomplexity of spg4 compared to spg1 did nothingspecific to improve the outcome.

Summary

Figure 11 shows graphically how the variousmodels compare; to summarize, whereas a nullhypothesis would predict that increasing the level

of detail in the model would make it harder to findworking parameter sets in a random search, infact, for this network, adding known details makesit easier to find solutions in parameter space, withrespect to the boundary-maintenance task that weconsider to be the fundamental biological role ofthe segment polarity network. However, spg4(and more complex versions not discussedhere) begins to show signs of a certain complexityburden. In a way, in spg4 the various cell statesthat make up the desired pattern are too stable.It seems that spg4 embodies, in effect, manydifferent kinetic mechanisms for stabilizingthe target pattern, all of which overlap in para-meter space. On the one hand, this makes thisnetwork even more robust than simpler approx-imations, but on the other hand, it also means

Fig. 11. Fragility of segment polarity network models as afunction of complexity. Plotted here is the natural log of 1 overthe solution frequency for several models, versus the numberof free parameters in each model. The quantity on the verticalaxis gives a rough answer to the question, ‘‘how hard is it tofind parameter sets that make the model work?’’ The stippledline represents the null hypothesis, discussed in the main text.The solid line plots the actual results for segment polaritymodels that incorporate a progressively greater level ofintermediate detail, as reflected in the parameter count. Alsoincluded is the engineered network (described in Fig. 9) whichis the most fragile of all, despite its simplicity.


it is a much less dynamic network. Cell states inspg4 appear to be so much more self-reinforcingthan in spg1 that it is very difficult to perturb themodel once it locks in on a particular spatialpattern of cell states.

Section IV: Stripe sharpening,a dynamic function

Segment polarity genes are expressed in dy-namic patterns. For example, stripes of expressionof both wg and en are widely reported to narrow asthe germ band extends in the 3.5- to 5.0-hr oldDrosophila embryo. Also, in the early Drosophilaembryo cephalic segments differ from the trunksegments in the detailed patterns of segmentpolarity gene expression and in their mutualdependencies (Gallitano-Mendel and Finkelstein,’97); even between dorsal, lateral, and ventralregions of the trunk segments there may be majordifferences, since wg expression only persistsbeyond germ-band extension in the ventral neuro-genic ectoderm. In this section we challenge themodel to mimic the reported narrowing ofwg- and en-expressing stripes during germ-bandextension.

Sharpening the wingless stripe

All models described above failed to account forhow the stripe of wg expression sharpens duringgerm band extension. As the germ band extends,cell rearrangement causes each segment to expand

along the anterior–posterior axis from 4 cells wideto roughly 8 cells wide; by stage 12 each segment,after the combined effects of cell division, rear-rangement, and neuroblast ingression, is about 12cells wide in the epidermal layer. Throughout, thewg stripe remains only one cell wide. Thus, cellsthat once expressed wg must turn it off as theymove too far from the influence of Hh-producingcells. None of our models could account for howwg, once activated, could turn off, since wgautoregulation plays such an important role inthe most fundamental behavior of the network. Inall the models described so far, Wg autoregulatoryand Ci/CN pathways effect wg expression addi-tively. That is, to be conservative we assumed thatthese pathways were completely different and hadno positive or negative influence on each other.Hooper (’94) interpreted some of her results tomean that these pathway synergize, but did notspecifically distinguish between additive and sy-nergistic models, so in the absence of a molecularmechanism for wg autoregulation we stuck withan additive model as the simplest possibility.

Analysis of nullclines in Figure 12 shows theproblem. Figure 12A and B show generic equa-tions for a self-activating gene product and thenullclines for those equations, respectively. For wgautoregulation to provide the asymmetry forwhich it was introduced to the model (von Dassowet al., 2000), the differential equations governingwg products must, in the absence of any otherfactors, have two stable steady states: wg off andwg on, corresponding to the lower- and uppermost

Fig. 12. Simple equations and nullclines for an autoregu-latory gene. (A) Boilerplate equations in which the product ofgene x activates further x mRNA synthesis according to asigmoid dose–response curve. X protein is translated accord-ing to a simple unregulated formula. Both species exhibit first-order decay. (B) Nullclines for the system of equations inpanel A. Nullclines are made by setting the derivatives to zeroand plotting the formula when solved for x mRNA (black line)as a function of X protein, or X protein as a function of xmRNA (dashed line). Where the nullclines cross, bothderivatives are zero and the system is at steady state. Analysisof eigenvalues of the Jacobian matrix determines whether thesteady state is stable, and it is a trivial matter to show that, inpanel B, the ‘‘off’’ state is stable and the ‘‘on’’ state is stable.The state in between is unstable and lies along the line (calleda ‘‘separatrix’’) dividing the basins of attraction for the twostable steady states. Above the separatrix, the system willevolve inevitably unless perturbed across the separatrix to the‘‘on’’ state, below it to the ‘‘off’’ state. The position of thesecond crossing-point along the X nullcline is an easy gaugefor the ‘‘switching threshold.’’ Such a system is bistable. (C)

Effect of varying the half-maximal coefficient. Such an effect,an increase in kXx, might be accomplished by some synergistictranscriptional coactivator that participates by binding to Xand altering its conformation to improve its affinity forbinding sites in the x enhancer. (D) Effect on the system inpanel A of varying the top line. Such an effect might beachieved by a cofactor that regulates the ability of X to recruittranscriptional machinery to the promoter. In C and D the‘‘off’’ state is always preserved. In contrast, an additivelyautoregulating gene is probably very hard to turn off once itgets turned on. (E) Simple additive model of an autoregula-tory enhancer region in which both X and Y contribute,independently, to transcriptional activation of x. See Appen-dix A to the companion paper by Meir et al. (2002b, this issue)for a comparison with other formulas. (F) Nullclines for asystem involving the equation for the derivative of x mRNA inpanel E, and the equation for X protein in panel A,showing the effect of varying the concentration of Y given theparameters shown. Only in a narrow window, and only forcarefully chosen parameter values, is the whole system everbistable, and even then the ‘‘off’’ state is not completely off.

3———————————————————————————————————————————————


crossings in Figure 12B. The middle crossing is anunstable steady state that lies on the line separat-ing basins of attraction for the two stable steadystates. This picture is the basis for the absoluterequirement that the response of wg to Wgsignaling must be cooperative (see companionpaper by Meir et al., 2002b, this issue). If Ci actsadditively, it can only shift upward the baseline onthe solid curve in Figure 12F, eventually destabi-lizing the ‘‘off’’ state completely. Considering theformula in Figure 12A, there are several otherpossible ways that Ci/CN could interact with Wg(see Fig. A4 in Appendix A to the companion paperby Meir et al., 2002b, this issue): Ci could

synergize with Wg either by shifting up the topline (Fig. 12C), or by shifting left the inflectionpoint (Fig. 12D).

In order for the boundary to be asymmetric,whatever happens in the presence of high Wg onthe anterior side of the compartment boundarymust not happen on the posterior of the en-expressing row (where Ci is also elevated) in theinitial absence of high Wg. That is fine as long asWg can sustain itself by autoregulation. However,this condition prevents stripe sharpening for anadditive model, and furthermore it may be wrong:in fly embryos Ci is absolutely required for wgexpression although by itself it is only able toweakly activate wg (Hooper, ’94). In the modelspresented so far, this is not so; Ci is usually onlyrequired in the models (and not even in everycase) to keep wg on during an initial period.Indeed, given the additive model it is difficult tofind parameters such that Ci would be required forwg expression and yet not be able to activate wgposteriorly (especially if Wg diffuses at all), hencethe preference that Ci be a poor activator and CNa good inhibitor of wg (Fig. 6).

The reason this poisons stripe sharpening isthat, if Wg is a good activator of itself, as cellsmove away from the Hh-producing region theywill experience no deficit if CN only acts as acompetitive antagonist of full-length Ci. Thus,with a purely additive model of the wg enhancerregion, once wg turns on it stays on no matterwhat in all but a narrow crevice of parameterspace. That crevice, part of which is illustrated inFigure 12F, requires exquisitely fine tuning. CNcannot turn off wg unless Wg is just barely tooweak to keep itself on in the absence of an assistfrom Ci. Even if it is possible, we could never havefound it in a random search: the right conditionsare exquisitely sensitive to half a dozen or moreparameters.

These problems evaporate if Ci and Wg syner-gistically activate wg. Many synergistic mechan-isms are plausible: Wg could promote its ownexpression, for example, by stimulating Ci tobecome a better activator of wg; alternatively,Wg signaling could result in the expressionor activation of some other factor such as Slp;Slp could synergize with Ci at the level ofenhancer binding, or transcriptional activation ofwg could require some collaboration betweenseveral proteins recruited by various indepen-dently binding transcription factors. We tested aversion of spg1 using a simple synergistic formulafor the regulation of wg transcription. In a search

Fig. 13. Synergistic model allows wg stripes to narrow. (A)Initial state for stripe-sharpening test. This pattern is astereotype of what each segment might look like immediatelyafter germ band extension: cell rearrangements make thesegment almost twice as wide along the anterior–posterioraxis, causing en-expressing cells and wg-expressing cells tomove away from the parasegmental boundary either poster-iorly or anteriorly, respectively. Starting with this initialpattern we test whether parameter sets that succeed at thestandard boundary-maintenance task enable the model toturn off wg or en expression in cells that ‘‘no longer’’ abut theparasegment boundary directly. Not shown, but also includedin the test pattern, are Wg, En, and Hh levels corresponding tothe mRNA concentrations shown here, and also modest levelsof ptc mRNA and protein in cells 1, 2, 5, and 6, numbered fromthe left. (B–E) Four out of 21 parameter sets we found (seetext) that enable wg stripe sharpening. Some parameter setsallow both the wg and en stripes to sharpen (as in B and C).Some parameter sets (as in C) only enable the model toreduce, but not completely eliminate, wg expression in cellsnot immediately adjacent to Hh-producing cells. Most of thesuccessful sets accomplish wg sharpening by achieving aquasi-stable difference in the relative abundance of full-lengthCi (CID) and the N-terminal repressor (CN), as in B–D. A few,however, do not: in E, the steady-state pattern of Ci and CN isuniform in all non-en-expressing cells, but the wg stripesharpens because, during the initial transients, Ci accumu-lates too slowly in the distal cells to ‘‘save’’ wg expression.


of 163,875 random parameter sets, we found 276parameter sets for which the model meets thestandard test of holding the boundary; for 21 ofthose parameter sets, the model was able toeffect the narrowing of a two-cell-wide wg stripeto only one cell wide. That is, under these 21parameter sets, when challenged with an initialprepattern intended to stereotype the state ofgene expression just at the end of the fast phase ofgerm-band extension (see Fig. 13), the modelcompletely shut down wg in the row of cells 1 celldiameter away from the Hh-producing cells,leaving only a one-cell-wide wg stripe. This is nota remarkable showing; it corresponds to only E1in 104. However, this was a very crude test, and weexpect that as details of the wg autoregulationmechanism emerge, we will be able to improve themodel’s performance on this test.

For many of the results described in the firstthree sections of this chapter we have testedwhether our conclusions depend on the choiceof an additive or a synergistic model, and theydo not. It is a matter for empirical investiga-tion whether Wg and Ci synergize, and how. Thereare various reasons to favor an additive model.In particular, Ci drives transcription from thewg enhancer in cultured cells, presumably inthe absence of whatever factors are involvedin wg autoregulation (Von Ohlen et al., ’97). Inembryos in which Wg signaling is prevented,low levels of wg are expressed, presumably underthe control of Ci (Hooper, ’94). From Figure 12Cand D it is evident that a simple synergisticmodel could not account for this. On the otherhand, in embryos the fact that wg expressionis absolutely dependent on Ci suggests a synergis-tic mechanism of some sort. Our model showsthat there are major functional consequences forthis choice, at least with respect to stripe sharpen-ing, and that if Ci and Wg do not synergizethen something big is likely to be missing fromthe model. A possibility, if Ci and Wg do notsynergize but instead act additively, is that Wgsignaling may become highly localized to thecompartment boundary vicinity through localexpression or biased intracellular localization ofco-receptors and other proteins required for Wgsignal transduction.

Sharpening the engrailed stripe

Based on an elegant cell-marking study, Vincentand O’Farrell (’92) showed that cells lose enexpression as they move away from the paraseg-

ment boundary during germ-band extension.Vincent and Lawrence (’94) showed that this isbecause the Wg signal travels only over a veryshort range, as little as 1 cell diameter. The corenetwork (spg1) had little difficulty with the task ofsharpening the en stripe in the same testsdescribed above. As many as one-third of thesolutions for spg1-grade models were capable ofsharpening the en stripe from two cells to one cellwide. To our dismay, we found that more complexmodels (spg2 and spg4 and others) have a muchharder time with this test. We watched spg4d runwith hundreds of different solutions, only one ofwhich enabled this model to sharpen a 2-cell-wideen stripe to one cell wide. It was very rare for themodel to achieve uneven distributions of Panversus the Arm–Pan complex. In fact, despitelocalized Wg production, only a minority ofsolutions enabled the model to achieve spatiallyvarying distributions of the Fz–Wg complex;without spatially varying Fz occupancy, no spatialvariation in Arm cleavage will arise, and no spatialregime of Pan versus Arm–Pan complex levels ispossible. How, then, is en expression confined to astripe at all in spg4, never mind how it sharpens?

Consider the interactions of Wg and Fz and Armand Pan: they bind to each other stoichiometri-cally. We allow the maximum level of proteins inour models to range from 103 to 106 copies per cell(see Supplementary Information to von Dassowet al., 2000, for the rationale). Considering twoproteins that bind to each other, in the standardparameter space box we use, only a small sliverwill exist where those proteins have the samemaximum expression level. If the two proteinshave widely different concentrations in the cell,the excess of one may completely swamp the other.This is the basis for ligand buffering between anykind of interacting entities, from protons topeople. Imagine that Arm, for instance, canaccumulate to a tenfold higher level than Pan,and that Arm and Pan bind with reasonably highaffinity. Then, even if the concentration of Arm ishalf maximal in one cell but maximal in theothers, the concentration of the complex will benearly the same in both cells.

But then, how is it that en is not expressedubiquitously as well under such conditions? In themodel, transcription of en is regulated by both theArm–Pan complex and by CN; meanwhile, Enprevents expression of ci. The CN–En doublenegative requires a positive input if it is to be abistable switch: both arm and pan are expressedubiquitously, and if they are not stoichiometrically


matched then it is possible that they merelyprovide a basal input to en. In that case, once engets turned on, it keeps itself on by repressing itsonly repressor, ci. Wg signaling would be requiredonly initially, as en expression ramps up. Indeedthis may explain two phenomena about thesegment polarity network. There is only a rela-tively short window of time during which en isdependent on Wg signaling, and that is duringgerm band extension as the expression of bothgenes is still below maximum. Second, Heemskerkshowed that heat-shock induced ectopic expres-sion of en after germ-band extension could causean expansion of the en stripe (Heemskerk et al.,’91). Historically this result was taken to indicatethat En activates its own production. However, itcould be instead that the heat-shock induced Enmerely shuts down ci expression, and that by lategerm-band extension there is sufficient Arm andPan that excess En ‘‘throws the switch’’ and ishenceforth clonally inherited (see Fig. 14D).

These considerations suggest how to achievestripe sharpening. Arm and Pan should be basallyexpressed at similar levels and be able to accumu-late to nearly the same maximal cellular concen-tration, and perhaps Wg and Fz should be likewiseconstrained. Furthermore, we wondered whether,in this context, it would help sharpen the stripe iffree Pan were to act as a repressor of en. It turnsout that only stoichiometric balancing of Arm andPan matters. Since automated screening methodshave not yet given us reliable results with thestripe-sharpening challenge, we resorted toscreening the model’s behavior by eye. We beganby collecting automatically, for several variants ofspg4d, several hundred solutions that pass thestandard 4-cell-wide pattern-holding test. We thenchallenged them with the 8-cell-wide ‘‘germ bandextended’’ test (Fig. 13) and watched how themodel behaved over a 1,000-min period whengoverned by each of those parameter sets. Forspg4d itself, we screened a first catch of 4600solutions and found only one that sharpened the enstripe (i.e., the posterior en cell turned off both enand hh completely, replacing them with a cell statein which full-length Ci and ptc were both abun-dantly expressed). In one other case, the posterioren cell merely came to express en at a reduced levelcompared to the cell immediately adjacent to theWg-producing cells. This shows that spg4d iscapable, intrinsically, of accomplishing this task,but the frequency is on the order of 1 in 105 or less.

We conducted identical screenings with variantsof spg4d that introduced either constraints on the

Fig. 14. Elements of the segment polarity network.Individual panels here represent subsets of the connectionsin Figs. 8 and 10, some of them abbreviated. (A) Positivefeedback between slp and wg makes a bistable switch which,like the simplistic example in Fig. 2, could have two stablesteady states, both components on or both off, with a thresholdfor turning on or off. (B) Similar to A, mutual activationbetween wg, en, and hh potentially creates another bistableswitch. (C) Mutual repression between slp and en, whencombined with a positive input to each, makes an exclusive-orswitch in which one or the other can be expressed at a leveldetermined by the input (in this case provided by Wgsignaling). (D) The mutual repression we hypothesize betweenci and en would also make an exclusive-or switch (enclosed inthe dashed box), but in this case the positive inputs to eachside of the switch are independently adjustable, unlessotherwise coupled (as they are in the context of the wholecircuit). This sub-circuit might explain how the en expressingcell state becomes self-sustaining at the end of germ-bandextension (CIT Heemskerk); if sufficient Arm–Pan complexaccumulates in the absence of Wg signaling, then pre-existinghigh-level En in the cell might prevent repressors, such as slpand possible ci, from accumulating in the same cell. Thishypothesis could be tested by replicating Heemskerk’s hs–enexperiments in a genetic background in which the expectedbackground level of the Arm–Pan complex was reduced. (E)Interactions between the products of ptc and ci constitute ahomeostatFthese components participate in a feedback loopwith a single negative interaction: the more Ptc is produced inresponse to full-length Ci protein, the more Ci is converted toCN, not only depleting Ci but also repressing ptc transcription.If ci is basally expressed, as it is in the models discussed here,then negative feedback leads to a stable steady state with someintermediate level of ptc expression, neither fully expressednor fully repressed. Hh protein binds to and titrates away Ptc;thus Hh tunes the homeostat, just as we adjust the householdthermostat. Titrating away Ptc decreases Ci cleavage rate,consequently increasing the steady-state ptc expression level.In addition increased Ptc expression soaks up Hh, as long asPtc can accumulate to a level commensurate with the pool ofHh available for binding.


relative expression levels of Arm and Pan, oradditional similar constraints on Wg and Fz, orthe postulated repressive link between Pan anden. When imposing the first of these constraints,out of 3� 104 random samples, we found 6 thatexhibited some evidence of en stripe sharpening, 3of which completely turned off en in the cellfurther from the Wg source. This is about an orderof magnitude more frequent (at least) than forspg4d (6 in 3� 104 vs. 2 in 6� 105). Otherconstraints on stoichiometry reduced rather thanimproved the frequency, and the Pan–en link hadno detectable effect. Thus, we could patch up ourmodel to make it behave as we desired, but theconstraint introduced is a serious one, and evenwith it the model does not perform exceptionallywell at this test.

Summary

For the model to exhibit sharpening of the wgand en stripes we had to impose, in the firstinstance, a specific mechanism for combining tworegulatory pathways that feed into wg transcrip-tion, and in the second instance, a constraint thatthe two components of the crucial transcriptionalswitch that mediates en activation must beexpressed at nearly equal levels. We are currentlyinvestigating en stripe sharpening in real em-bryos. We find that En expression increases atleast 5-fold during germ-band extension and,furthermore, that En is expressed in one-cell-widestripes only transiently, if ever (V. Rich and G. vonDassow, unpublished). Nevertheless, comparingan En-GFP reporter with En antibody stainingshows that some cells do appear to lose enexpression on the posterior edge of the stripe,confirming Vincent and O’Farrell’s (’92) results(V. Rich and G. von Dassow, unpublished). Thus itmay be that cells that are on their way to, but havenot yet achieved, full en expression fail to maintainit as they move posteriorly. Whatever the answer,from the modeling perspective what is importantis that we had difficulty getting a simple model ofthe Wg pathway simply to turn off. Of coursefuture, more faithful models might erase thisdifficulty. Regardless, the same difficulty withstoichiometric balancing is likely to afflict manysimilar signaling processes that rely on regulatingrelative availability of binding partners.

CONCLUSIONS

In this paper we sought to take known facts orhypotheses, incorporate them into our working

model of the segment polarity gene network, andassess how the model’s behavior alters in re-sponse. That is, we have engaged here in whatLawrence and Sampedro (’93), quoting Crick, call‘‘carpentry.’’ Lawrence and Sampedro criticizedearly attempts to synthesize facts about Droso-phila segmentation in terms of causal chains ofinteractions among individual genes. They remarkthat the carpentry approach to synthesis oftenresults in circular or self-contradictory and ad hoccollections of predicates, and they proposedinstead to conceive of the segment-patterningprocess in terms of gradients, abstracted fromthe specific activities of particular molecules indetermining cell states within the segment. Lawr-ence and Sampedro articulated an aestheticallyand phenomenologically satisfying hypothesiswithout substantial recourse to what even thenwas remarkable knowledge of the molecularnature of segment polarity genes and theirinteractions. However, their account, by its verynature, provides no synthesis of the disparate factsof molecular genetic analysis and thus seemsunsatisfying to someone attempting to fit togetherthe molecular puzzle pieces.

The problem with the carpentry that Lawrenceand Sampedro derided is that it consisted of verbalreasoning only. This paper represents an attemptto formalize carpentry, using computers to solvedifferential equations that characterize the geneticinteractions inferred by molecular geneticists.Whatever else one can say about the usefulnessof models of complex systems, computers are notfooled by circularity and logical contradictions,and thus the computer model lets us conduct akind of reconstitution experiment with the knownfacts about the process of interest. This exerciseyields three pragmatic results: first, a set ofempirically testable predictions about the specificreal network expressed in the models; second,general predictions about the emergence of sys-tems-level properties of gene regulatory networks;and third, a platform from which to explore theevolvability of developmental mechanisms, hope-fully developing a bridge across the conceptual gapbetween mechanistic analysis of developmentand population-level studies of variation andvariability.

In addition, the reconstitution exercise revealshow simple elements, such as positive and nega-tive feedback loops, with readily understooddynamical properties conspire to form a largermechanism. The segment polarity model consistsof several intertwined dynamical elements, as


shown in Figure 14. It includes at least twopositive feedback loops that, separately, couldexhibit bistable on/off switch-like function: wgauto-activation, likely via slp (Fig. 14A), and themutual promotion of wg and en expression (Fig.14B). In addition there are at least two pairwisecross-inhibitions which, given some positiveinput to each component, ought to behave aseither/or switches: mutual inhibition between enand slp (Fig. 14C), and the hypothetical mutualinhibition by ci and en (Fig. 14D). In the formercase, Wg signaling provides a common positiveinput to each gene, and in the latter case eitherbasal expression or, again, Wg signaling providesthe necessary input. Finally, the model incorpo-rates a single negative feedback loop between ciand ptc that could act as a homeostat (Fig. 14E).

The first two dynamical elements in Figure 14promote stabilization of wg-expressing and en-expressing cell states, respectively, assumingpermissive conditions for each state. For example,in the case of Figure 14A, the permissive condi-tions would be that there be too little CN presentto repress wg, and that there be too little Enpresent to repress slp. The next two elementsensure mutual exclusivity of wg- and en-expres-sing cell states; note that a cell state expressingboth wg and en becomes possible if parameters aretuned such that En is a weak repressor of slp, Slpis a strong activator of wg but a weak repressor ofen, and either Ci is not required for wg expressionor En only weakly represses ci. The ci–ptchomeostat maintains the responsiveness of aground state and also, depending on the relativelevel of expression of Hh and Ptc, buffers Hhconcentration within the vicinity of its source.Furthermore, if Arm and Pan concentrations aretuned such that there is little or no basalactivation of en, and if Ci is actually required forwg activation, then the ground state can ‘‘invade’’either the wg- or the en-expressing cell state,should the factors sustaining those states fade(i.e., as the germ band extends). At least, that’show the model works.

This conception of the segment polarity moduleas a system of mutually entrained cell stateswitches explains how a global ‘‘design’’ feature,the level of cooperativity in regulatory interac-tions, confers upon the network robustness tovariation in parameters, to initial conditions, or tonoise. Increasing cooperativity promotes switch-like behavior in the dynamical elements dia-grammed in Figure 14A–D. Looking at it the otherway around, with low cooperativities throughout

these switching elements, the cell states thatdefine the model’s functional behavior become,in a sense, more elastic. Thus, for example, at lowcooperativity the entire network ‘‘rings’’ in re-sponse to oscillations generated in the ci–ptc loop,or in response to noise, thus disrupting themaintenance of a stable spatial regime of cellstates. In the same way, at low cooperativity thenetwork’s steady states ‘‘stretch’’ easily in re-sponse to changes in parameters.

Genetic mutations and environmental variationboth correspond to coordinated changes in somesubset of parameter values. Any mutation of aparticular locus in the network corresponds to amove along some diagonal in parameter space.Similarly, an environmental perturbation such asa change in temperature corresponds to a movealong some (other) diagonal in parameter space.Developmental noise might correspond to a ran-dom walk, over the course of development, alongarbitrary diagonals. (One must also think of anychange in initial conditions as an equivalentperturbation.) Increasing cooperativity withinthe dynamical elements shown in Figure 14A–Dincreases the tolerance of the network to movesalong any diagonal in parameter space. Thissuggests that switching networks such as thisone epitomize a congruence, in the sense of Anceland Fontana (2000), between robustness to thethree major categories of insult that physiologicalmechanisms confront.

Population genetic models show that while itmay be relatively straightforward to select forcanalization against environmental perturbations,it is difficult to concoct an analogous scenario forgenetic canalization (Wagner et al., ’97; Gibson andWagner, 2000). The apparent plastogenetic con-gruence manifest in the segment polarity model,and also in our model of the neurogenic network(Meir et al., 2002a), raises the possibility thatgenetic canalization might emerge as a byproductof selection for robustness to environmental per-turbation. More generally, these considerationsmake us hopeful that models like these can providean interpretive metaphor to explore the mechan-istic origins of systems-level properties of geneticarchitecture, such as epistasis, canalization, anddominance, which until recently have been neces-sarily treated as black boxes in evolutionary theory.

ACKNOWLEDGMENTS

Thanks to Richard Strathmann for commentson an early draft of the manuscript, to Virginia


Rich for comments and proofreading, and to EliMeir and Dara Lehman for suggestions and advicethat contributed to the work described here. Thiswork was supported by the National ScienceFoundation (MCB-9732702, MCB-9817081, andMCB-0090835) to GMO. The main body of thispaper, in modified form, was previously publishedas part of GvD’s doctoral thesis (University ofWashington Department of Zoology, 2000). Thework described here was primarily conductedwhile in residence at Friday Harbor Laboratories,and we thank the director, Dennis Willows, foraccommodating us.

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MATHEMATICAL APPENDIX

Each of the models described in this paper consists of a system of coupled ordinary differentialequations, assembled and solved by the computer program Ingeneue (described in companion paper byMeir et al., ’02, this issue). In this section we use the following short-hand:

� X; �X ; �Xð Þ ¼ X�X

��X

X þ X�X

� �; ð1Þ

ðX; �X ; vXÞ ¼ 1 � �ðX; �X ; vXÞ; ð2Þ

Xi;j ¼ amount of X on cell i; face j;

n i; jð Þ ¼ index of neighbor to cell i at face j;

Xn i;jð Þ;jþ3 ¼ X on cell face apposite to i; j;

Xtoti ¼

X6

j¼1

Xi;j ¼ total X in cell i;

Xtotn i;jð Þ ¼

X6

j¼1

Xn i;jð Þ;jþ3 ¼ total X presented to cell i by neighbors:

All equations in this appendix are presented in the dimensionless form used by Ingeneue andexplained in the supplement to von Dassow et al. (’00). In Ingeneue each process (transcription,cleavage, first-order decay, ligand binding, etc.) must be represented as a single additive term. Below,equations correspond directly to the script used by Ingeneue, except that for brevity and convenience wegroup terms differently.

The model referred to as ‘‘spg1’’ is identical to the model described in that paper, and consists of thefollowing equations, instantiated across an arbitrary-size cell grid (enumerated by the index i below):

d eni

d�¼ T�

Hen� EWGtot

n i;jð Þ

�� CNi; �CNen; �CNenð Þ; �WGen; �WGenÞ�eniÞ; ð3Þ

d ENi

d�¼ To

HENeni � ENið Þ; ð4Þ

d wgi

d�¼ T�

Hwg

�CIwg � � CIi � CNi; �CNwg; �CNwg

� �; �CIwg; �CIwg

� �þ �WGwg � � IWGi; �WGwg; �WGwg

� �1 þ �CIwg � � CIi � CNi; �CNwg; �CNwg



� � !

� T�wgi

Hwg; ð5Þ


d IWGi

d�¼ T�

HIWGwgi � IWGið Þ þ T� rEndoWGEWGtot

i � rExoWGIWGi

� �; ð6Þ

d EWGi;j

d�¼T�

rExoWGIWGi

6 � rEndoWGEWGi;jþrMxferWG EWGn i;jð Þ;jþ3 � EWGi;j

� �þ

rLMxferWG EWGi;j�1 þ EWGi;jþ1 � 2EWGi;j

� �0B@

1CA� T�EWGi;j

HIWG; ð7Þ

d ptci

d�¼ T�

Hptcð�ðCIi � ðCNi; �CNptc; �CNptcÞ; �CIptc; �CIptcÞ � ptciÞ ð8Þ

d PTCi;j

d�¼ T�

HPTC

ptci

6� PTCi;j

� �� T�kPTCHH½HH��HHn i;jð Þ;jþ3 � PTCi;j

þ T�rLMxferPTC PTCi;j�1 þ PTCi;jþ1 � 2PTCi;j

� �; ð9Þ

d cii

d�¼ T�

Hcið�ðBi � ðENi; �ENci; �ENciÞ; �Bci; �BciÞ � ciiÞ; ð10Þ

d CIi

d�¼ T�

HCIcii � CIið Þ � T�CCICIi � � PTCtot

i ; �PTC�CI; �PTC�CI

� �; ð11Þ

d CNi

d�¼ T�CCICIi � � PTCtot


� �� T�CNi

HCIð12Þ

d hhi

d�¼ T�

Hhh� ENi � CNi; �CNhh; �CNhhð Þ; �ENhh; �ENhhð Þ � hhið Þ; ð13Þ

d HHi;j

d�¼ T�

HHH

hhi

6� HHi;j

� �� T�kPTCHH½PTC��PTCn i;jð Þ;jþ3 � HHi;j

þ T�rLMxferHH HHi;j�1 þ HHi;jþ1 � 2HHi;j

� �; ð14Þ

d PHi;j

d�¼ T�kPTCHH½HH��HHn i;jð Þ;jþ3 � PTCi;j �

T�PHi;j

HPH: ð15Þ

In Section II we described re-wiring spg1 in several ways. For example, adding inhibition of ptctranscription by En amounts to replacing Eq. (8) above with

d ptci

d�¼ T�

Hptcð�ðCIi � ðCNi; �CNptc; �CNptcÞ; �CIptc; �CIptcÞ � ðENi; �ENptc; �ENptcÞ � ptciÞ: ð16Þ

For another example, to introduce the products of the gene slp, as in line 14 in Table 3, we add theequations

d slpi

d�¼ T�

Hslpð�ðEWGtot

i � ENi; �ENslp; �ENslp

� �; �WGslp; �WGslp

� �� slpiÞ; ð17Þ

d SLPi

d�¼ T�

HSLPslpi � SLPið Þ: ð18Þ

In addition we modify the equations for en and wg, replacing Eqs. (3) and (5) as follows:

d eni

d�¼ T�

Hen� EWGtot

i � CNi; �CNen; �CNenð Þ; �WGen; �WGen

� �� SLPi; �SLPen; �SLPenð Þ � eni

� �; ð19Þ

d wgi

d�¼ T�

Hwg



� �þ �SLPwg � � SLPi; �SLPwg; �SLPwg



� �þ �SLPwg � � SLPi; �SLPwg; �SLPwg

� � !

� T�wgi

Hwg: ð20Þ


Naturally this represents just one of the many ways which one could introduce the new Nodes andlinks. Equations (5) and (20) both express an additive relationship between the positive influences onwg transcription. In Section IV we show that such a format makes it very difficult to find parametersthat enable the stripe of wg-expressing cells to sharpen into a single-cell-wide row. An alternate,synergistic formulation that enables wg stripe sharpening corresponds to replacing Eq. (5) with

d wgi

d�¼ T�

Hwg� CIi � CNi; �CNwg; �CNwg


� �� IWGi; �WGwg; �WGwg

� �� wgi

� �: ð21Þ

More complex versions of the model, such as spg2 and spg4, are derived in the same way, by addingextra Nodes and modifying various additive terms as needed. For example, the model ‘‘spg2’’, Variant E(see Fig. 10 and Table 4), consists of the following equations:

d eni

d�¼ T�

Hen� APi � CNi; �CNen; �CNenð Þ; �APen; �APenð Þ � enið Þ; ð22Þ

d ENi

d�¼ T�


d wgi

d�¼ T�

Hwg







� � !

� T�wgi

Hwg; ð24Þ

d IWGi

d�¼ T�

HIWGwgi � IWGið Þ þ T� rEndoWGEWGtot

i � rExoWGIWGi

� �; ð25Þ

d EWGi;j

d�¼ T�

rExoWGIWGi

6 � rEndoWGEWGi;jþ

rMxferWG EWGn i;jð Þ;jþ3 � EWGi;j

� �þ


� ��

T�kFZWG½FZ��FZi;j � EWGi;j

0BBBBB@

1CCCCCA� T�EWGi;j

HIWG; ð26Þ

d fzi

d�¼ T�

Hfz� PANi � APi; �APfz; �APfz

� �; �PANfz; �PANfz

� �� fzi

� �; ð27Þ

d FZi;j

d�¼ T�

HFZ

fzi

6� FZi;j

� �� T�kFZWG½EWG��EWGi;j � FZi;j þ T�rLMxferFZ FZi;j�1 þ FZi;jþ1 � 2FZi;j

� �; ð28Þ

d FWi;j

d�¼ T�kFZWG½EWG��EWGi;j � FZi;j þ T�rLMxferFW FWi;j�1 þ FWi;jþ1 � 2FWi;j

� �� T�FWi;j

HFW; ð29Þ

d armi

d�¼ T�

Harm� Bi; �Barm; �Barmð Þ � armið Þ; ð30Þ

d ARMi

d�¼ T�

HARMarmi � ARMið Þ

� T�CARMARMi � FWtoti ; �FW �ARM; �FW �ARM

� �� T�kARMPAN½PAN��PANi � ARMi; ð31Þ

d pani

d�¼ T�

Hpan� Bi; �Bpan; �Bpan

� �� pani

� �; ð32Þ

d PANi

d�¼ T�

HPANpani � PANið Þ � T�kARMPAN½ARM��ARMi � PANi; ð33Þ


d APi

d�¼ T�kARMPAN½PAN��PANi � ARMi �

T�APi

HAP; ð34Þ

d ptci

d�¼ T�

Hptc� CIi � CNi; �CNptc; �CNptc

� �; �CIptc; �CIptc

� �� ptci

� �; ð35Þ

d PTCi;j

d�¼ T�

HPTC

ptci

6� PTCi;j

� �� T�kPTCHH½HH��HHn i;jð Þ;jþ3 � PTCi;j þ T�rLMxferPTC PTCi;j�1 þ PTCi;jþ1 � 2PTCi;j

� �; ð36Þ

d cii

d�¼ T�

Hci� Bi � ENi; �ENci; �ENcið Þ; �Bci; �Bcið Þ � ciið Þ; ð37Þ

d CIi

d�¼ T�

HCIcii � CIið Þ � T�CCICIi � � PTCtot


� �; ð38Þ

d CNi



� �� T�CNi

HCI; ð39Þ

d hhi

d�¼ T�

Hhh� ENi � CNi; �CNhh; �CNhhð Þ; �ENhh; �ENhhð Þ � hhið Þ; ð40Þ

d HHi;j

d�¼ T�

HHH

hhi

6� HHi;j

� �� T�kPTCHH½PTC��PTCn i;jð Þ;jþ3 � HHi;j þ T�rLMxferHH HHi;j�1 þ HHi;jþ1 � 2HHi;j

� �; ð41Þ

d PHi;j


T�PHi;j

HPH: ð42Þ

A subset of the network including only en, slp, and wg (described in Fig. 6 of the companion paper) isrepresented by the following equations:

d eni

d�¼ T�

Hen� WGtot

i ; �WGen; �WGen

� �� SLPi; �SLPen; �SLPenð Þ � eni

� �; ð43Þ

d ENi

d�¼ T�


d slpi

d�¼ T�

Hslp� WGtot

i ; �WGslp; �WGslp

� �� ENi; �ENslp; �ENslp

� �� slpi

� �; ð45Þ

d SLPi

d�¼ T�

HSLPslpi � SLPið Þ; ð46Þ

d wgi

d�¼ T�

Hwg� SLPi; �SLPwg; �SLPwg

� �� wgi

� �; ð47Þ

d WGi;j

d�¼ T�

HWG

wgi

6� WGi

� �þ T� rMxferWG WGn i;jð Þ;jþ3 � WGi;j

� ��þrLMxferWG WGi;j�1 þ WGi;jþ1 � 2WGi;j

� ��:

ð48ÞAnother simplified model used Section III of this paper collapses Nodes representing mRNA and

protein into a single Node for each locus:

d ENi

d�¼ T�

HEN� EWGtot

n i;jð Þ � CNi; �CNen; �CNenð Þ; �WGen; �WGen

� �� ENi

� �; ð49Þ


d IWGi

d�¼ T�

HIWG







� � !

� T�IWGi

HIWGþ T� rEndoWGEWGtot

i � rExoWGIWGi

� �;

ð50Þ

d EWGi;j

d�¼ T�

rExoWGIWGi

6 � rEndoWGEWGi;jþ

rMxferWG EWGn i;jð Þ;jþ3 � EWGi;j

� �þ


� �0BB@

1CCA� T�EWGi;j

HIWG; ð51Þ

d PTCi;j

d�¼ T�

HPTC

� CIi � CNi; �CNptc; �CNptc

� �; �CIptc; �CIptc

� �6

� PTCi;j

� �

� T�kPTCHH½HH��HHn i;jð Þ;jþ3 � PTCi;j þ T�rLMxferPTC PTCi;j�1 þ PTCi;jþ1 � 2PTCi;j

� �;

ð52Þ

d CIi

d�¼ T�

HCIð�ð Bi � ENi; �ENci; �ENcið Þ; �Bci; �Bcið Þ � CIiÞ � T�CCICIi � � PTCtot


� �; ð53Þ

d CNi



� �� T�CNi

HCI; ð54Þ

d HHi;j

d�¼ T�

HHH

� ENi � CNi; �CNhh; �CNhhð Þ; �ENhh; �ENhhð Þ6

� HHi;j

� �� T�kPTCHH½PTC��PTCn i;jð Þ;jþ3 � HHi;j þ T�rLMxferHH HHi;j�1 þ HHi;jþ1 � 2HHi;j

� �; ð55Þ

d PHi;j


T�PHi;j

HPH: ð56Þ

Notice that Eqs. (49–56) consist largely of the same terms as Eqs. (3–15). In the original model, thetwo processes of translation and first-order non-specific decay cause each (scaled) protein concentrationto track the mRNA concentration at a rate determined by the protein’s half-life. In the segment polaritymodels, most proteins undergo additional transformations; however En does not. Equation (4),governing EN concentration, is a simple tracking function (likewise for Eq. (23), in spg2). The non-dimensionalization recipe we use reveals that the half-life of each molecular species determines howlong that species takes to decline or accumulate to steady-state concentration. The half-lives thusgovern the timescale on which Nodes respond to any perturbations at the biosynthetic level. Thedifference between the collapsed version in Eqs. (49–56) and our ‘‘normal’’ form, in which mRNA andprotein are separate species, is the elimination of this dampening layer. In the normal form, if theprotein’s half-life is longer than the mRNA’s half-life, separation into two (or more) steps insulates theconcentration of functional product (protein) from fluctuations in the level of transcriptional stimulus.The collapsed network consequently can work, but, more often than the original network, fails to hold astable steady state in the face of oscillations generated within the ci-ptc negative feedback loop. The realprocess of gene expression is broken into many intermediate steps, as are metabolic pathways, and onewonders if evolution has exploited the oscillation-damping effect of intermediate steps with slowerresponse times to insulate biochemical networks against stochastic variability.

Finally, in Section III we compared networks ‘‘inspired by actual events’’ to a simple network of ourown concoction, which is described by the following equations governing the products of twohypothetical genes called A and B:

d ai

d�¼ T�

Hað�ðABtot

i � Btotn i;jð Þ; �Ba; �Ba

� �; �ABa; �ABaÞ � � Ai; �Aa; �Aað Þ � aiÞ; ð57Þ


d Ai;j

d�¼ T�

HA

ai

6� Ai;j

� �þ T�

kAB!AþB½B��½A��

ABi;j � kAþB!AB½B��Bi;j � Ai;j

þrMxferA An i;jð Þ;jþ3 � Ai;j

� �þ rLMxferA Ai;j�1 þ Ai;jþ1 � 2Ai;j

� �0@

1A; ð58Þ

d bi

d�¼ T�

Hbð�ðABtot

i � Atotn i;jð Þ; �Ab; �Ab

� �; �ABb; �ABbÞ � � Bi; �Bb; �Bbð Þ � biÞ; ð59Þ

d Bi;j

d�¼ T�

HB

bi

6� Bi;j

� �þ T�

kAB!AþBABi;j � kAþB!AB½A��Ai;j � Bi;j

þrMxferB Bn i;jð Þ;jþ3 � Bi;j

� �þ rLMxferB Bi;j�1 þ Bi;jþ1 � 2Bi;j

� � !

; ð60Þ

d ABi;j

d�¼ �T�ABi;j

HABþ T�

kAþB!AB½A��Ai;j � Bi;j � kAB!AþBABi;j

þrMxferAB ABn i;jð Þ;jþ3 � ABi;j

� �þ rLMxferAB ABi;j�1 þ ABi;jþ1 � 2ABi;j

� � !

: ð61Þ

This network is entirely fictional. Any resemblance to real networks, living or dead, is accidental.Note that the term for dissociation in the equation governing A is scaled by the ratio of the maximumattainable concentrations of A and B ([A]o/[B]o) to make the non-dimensionalization scheme workout; otherwise the imaginary components A and B are symmetric with respect to their interactionsand transcriptional control, but can differ quantitatively because they are governed by indepen-dent parameters. If their various parameters are likewise forced to be symmetrically valued, then[A]o/[B]o=1.


design and constraints of the drosophila segment polarity

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