Scaling of morphogen gradients by an expansion-repression integral feedback controlDanny Ben-Zvia and Naama Barkaia,b,1

Departments of aMolecular Genetics and bPhysics of Complex Systems, Weizmann Institute of Science, Rehovot 76100, Israel

Edited by José N. Onuchic, University of California, La Jolla, CA, and approved February 26, 2010 (received for review November 12, 2009)

Despite substantial size variations, proportions of the developingbody plan are maintained with a remarkable precision. Little isknown about the mechanisms that ensure this adaptation (scaling)of pattern with size. Most models of patterning by morphogengradients do not support scaling. In contrast, we show that scalingarises naturally in a general feedback topology, in which the rangeof the morphogen gradient increases with the abundance of somediffusible molecule, whose production, in turn, is repressed bymorphogen signaling. We term this mechanism “expansion–repression” and show that it can function within a wide rangeof biological scenarios. The expansion-repression scaling mecha-nism is analogous to an integral-feedback controller, a key conceptin engineering that is likely to be instrumental also in maintainingbiological homeostasis.

development | patterning | theoretical biology

Multicellular organisms develop through a sequence of pat-terning events in which uniform fields of cells differentiate

into patterned tissues and organs. Positional information iscommonly encoded by morphogen gradients, whereby a signalingpathway is activated in a spatially graded manner over a field ofcells and induces distinct gene expression domains in a concen-tration-dependent manner. In the standard paradigm, a signalingmolecule—the morphogen—is secreted from a local source anddiffuses through the tissue, establishing a gradient that peaks atthe source. The gradient is shaped further by factors that impacton morphogen diffusion or degradation. The activity and abun-dance of these regulators is often influenced by the morphogensignaling itself through a variety of feedbacks (1–6).Developing individuals of the same species vary in size.

However, the proportions of their body plans are kept remark-ably constant. To achieve this proportionate patterning,morphogen gradients ought to scale with the size of the tissue.Despite intense research, little is known about the means bywhich field size is measured and how this information feeds backto shape the morphogen gradient (7–15).Theoretical studies have shown that scaling is not a general

property of morphogen models but requires specialized mecha-nisms. Proposed mechanisms include (i) two diffusible moleculesthat emanate from opposing poles and define the activationprofile through their ratio; (ii) a mechanism that maintains aconstant, size-independent receptor number; and (iii) a topologythat restricts morphogen degradation to only the distal edge ofthe field (a “perfect sink”) (9–11, 13, 15). Whereas thosemechanisms likely apply in some cases, they invoke fine-tunedinteractions, posing significant constraints on the design of thepatterning network (SI Text).Here we show that scaling emerges as a natural consequence

of a feedback topology, which we term “expansion repression”.In this case, patterning is defined by a single morphogen, whoseprofile is shaped by a diffusible molecule, the “expander”. Theexpander functions, directly or indirectly, to facilitate the spreadof the morphogen by enhancing its diffusion or protecting it fromdegradation. In turn, expander production is repressed bymorphogen signaling. Mathematical analysis reveals that scalingis achieved provided that the expander is stable and diffusible.

The generality of the expansion-repression mechanism allowsits implementation within a wide variety of systems. In particular,the expander can have other roles besides its function in scaling: Adiffusible inhibitor, for example, can provide for scaling along withits inhibitory function; similarly, in a dual morphogen system, oneof the morphogens can function as the expander. As we discuss,this generality of the expansion-repression mechanism for scalingreflects its analogy with an integral-feedback controller.

ResultsExpansion-Repression Feedback. We consider a single morphogenM that can induce several cell fates in a concentration-dependentmanner. We assume that M is secreted from a local source anddiffuses in a naive field of cells to establish a concentrationgradient that peaks at the source. We denote the morphogendiffusion coefficient by DM and its degradation rate by αM. Theshape of the morphogen is determined by the diffusion equation

∂½M�∂t

¼ DM ∇2 ½M�− αM ½M�; [1]

with a constantflux ηM at the origin (x=0).Boundary conditions atx = L have a negligible effect on the system under most conditions(SI Text), and we assumed either reflecting or absorbing bounda-ries at x=L or, alternatively,M=0 at infinity (SI Text).We furtherassume an additional diffusible molecule, the expander E, whichinfluences (directly or indirectly) the degradation rate or the dif-fusion of the morphogen, so that

DM ¼ DM ð½E�; ½M�Þ; αM ¼ αM ð½E�; ½M�Þ [2]

are monotonically increasing (DM) or decreasing (αM) functionsof the expander concentration [E]. Note that the diffusion anddegradation rates may depend on the level of morphogen [M],accounting for possible nonlinear effects. Finally, we assumethat expander production is repressed by morphogen signaling, sothat the expander distribution is given by the reaction diffusionequation

∂½E�∂t

¼ DE ∇2 ½E�− αE ½E� þ βEThrep

Threp þ ½M�h

; [3]

where Trep denotes the repression threshold, and h is some Hillcoefficient. Without loss of generality, we assume reflectiveboundary conditions (SI Text). Together, Eqs. 1–3 define theexpansion-repression feedback topology (Fig. 1A).To see how this feedback provides for the scaling of pattern with

size, it is instructive to follow thedynamicsof gradient formation.The

Author contributions: D.B.-Z. and N.B. designed research; D.B.-Z. and N.B. performedresearch; and D.B.-Z. and N.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.1To whom correspondence should be addressed. E-mail: naama.barkai@weizmann.ac.il.

This article contains supporting information online at www.pnas.org/cgi/content/full/0912734107/DCSupplemental.

6924–6929 | PNAS | April 13, 2010 | vol. 107 | no. 15 www.pnas.org/cgi/doi/10.1073/pnas.0912734107

Dow

nloa

ded

by g

uest

on

July

26,

202

1

morphogen diffuses and degrades across the field, generating agradeddistribution. The expander is produced in distal regions of thetissue where signaling is low. Accumulation of the expander leads tothe broadening of the morphogen gradient, which, in turn, narrowsdown the expander production domain (Fig. 1B). Steady state isreached when the gradient stops expanding, that is, when theexpander stops accumulating. Assuming that the expander is stable,steady statewill be reachedonlywhen theexpanderproduction stops,namely when its production is inhibited in virtually the entire fieldand specifically in the distal-most point (x=L). This way, the steady-state level of morphogen at the distal-most position is pinned to thelevel required for theexpander repression,Trep, leading to theoveralladjustment of morphogen spread according to the size of the field.Notably, the adjustment of the profile to a specific boundary

value through the regulation of its spread (length scale) is verydifferent from the adjustment achieved by simply specifying a

boundary condition. The latter affects the profile locally andusually has a moderate effect on the overall level of morphogengradient and does not lead to scaling (SI Text).

Analytical Approximation. To better illustrate the scaling mecha-nism, we rewrite Eq. 1 as

τ∂ ½M�∂t

¼ λ2∂2 ½M�∂x2

− f ð½M�Þ [4]

with τ some typical time scale, λ a typical length scale, and f([M])a degradation term. For example, when a morphogen degradeslinearly, τ−1 is given by the degradation rate, λ ¼ ffiffiffiffiffiffiffiffiffiffi

DMτp

andf([M]) = [M]. The expander E functions to modulate λ, with λ =λ([E]) a monotonically increasing function of [E]. Changingvariables to y ¼ x=λ; ρ ¼ ηMλ=DM , we can write

A B

C D E

F G

Fig. 1. The expansion-repression mechanism. (A) Scheme of the expansion-repression mechanism: Morphogen signaling represses the production of theexpander. The expander in turn expands the morphogen gradient. (B) Dynamics of the expansion-repression mechanism. Initially (time = t1), the morphogengradient is sharp, enabling expression of the expander in a wide domain (orange bar). At later times (time = t2), the expander accumulates, the gradientexpands, and consequently the expander production domain shrinks. At steady state, the expander has accumulated and the gradient is wide enough torepress expander production everywhere. Specifically, the profile at x = L is slightly higher than Trep, which corresponds to a complete repression of expanderproduction. (C) Screen for scaling in the expansion-repression topology: Shown is the distribution of parameters corresponding to systems that scaled (total of2,372 sets). Such systems had two main characteristics: (i) flat expander profile (λE > L, with λE the expander decay length) and (ii) the initial profile is sharpenough to induce the expander. Lmin is the length for which M(L) = Trep when [E] = 0. (D and E) Examples of morphogen gradients defined by the expansion-repression topology: A gradient that does not scale is shown in Dwhereas a gradient that does scale is shown in E. The profiles are shown for fields of length L(black curve) and 2L (gray curve). Distance from the source (x axis) is shown as a function of the relative position x/L. Profiles in nonscaled coordinates areshown in the Inset. Red bars mark the change in the positions of three thresholds used to measure scaling. (F and G) Scaling in a growing field: Simulation ofthe expansion-repression system in a field that grew from 100 to 200 μm during patterning is shown in F for four subsequent time points. The same sim-ulations are shown in G but the gradients are scaled with the size of the field. Parameters for numerical solutions are given in the SI Text.

Ben-Zvi and Barkai PNAS | April 13, 2010 | vol. 107 | no. 15 | 6925

DEV

ELOPM

ENTA

LBIOLO

GY

ENGINEE

RING

Dow

nloa

ded

by g

uest

on

July

26,

202

1

τ∂½M�∂t

¼ ∂2 ½M�∂y2

− f ð½M�Þ; ∂½M�∂y y¼0

¼ − ρ;∂½M�∂y y¼L=λ

¼ ε→y→∞

0:

[5]

For simplicity (and without loss of generality, SI Text), weassume L >> λ. λ then affects the morphogen profile onlythrough y (the scaled coordinate) and ρ (the scaled concen-tration), implying that the steady-state profile can be written as

MðxÞ ¼ Mλ

�x

λstð½E�Þ; ρst ðλstð½E�ÞÞ

�[6]

with ρst and λst the steady-state values of ρ and λ.Consider now the dynamics of gradient formation. As the

expander accumulates, both λ and ρ increase in value, narrowingthe region where the expander is expressed. We further assumethat the expander diffuses rapidly and degrades slowly so that itis approximately uniform across the field and continues toaccumulate as long as it is produced. Under these conditions, thesystem will arrive at a steady state only when the expanderproduction is repressed throughout the field. In particular, at thedistal pole x = L, which is the lowest point of the gradient, thesteady-state profile must satisfy

Mλ

�Lλst

; ρst�

¼ Trep: [7]

The solution to Eq. 7 defines a steady-state length scale that is,to a good approximation, proportional to system size:

λst ¼ Lα0;   α0 ¼ M − 1

λ ðTrep; ρstÞ: [8]

Substituting in Eq. 6 implies the scaling of the full gradient:

MðxÞ ¼ Mλ

� xL; ρ�: [9]

The gradient is scaled because its profile depends on the spatialcoordinate x only through the relative position, x=L.Note that scaling in Eq. 9 is not precise, because the effective

production rate ρ may still be a function of λ and hence of L.Perfect scaling occurs only in systems that are completelyinsensitive (robust) to morphogen production rate ρ. However,we find that deviation from scaling due to this dependency istypically small.

Numerical Analysis. To further examine the generality of scalingwithin the expansion-repression topology, we performed numer-ical analysis, screening systematically different parameters anddegradation schemes of Eqs. 1–3 (Experimental Procedures). Foreach parameter set, we derived the position boundaries defined bythree concentration thresholds and evaluated these boundariesfor fields of normal (L) and double (2L) sizes (Fig. 1 C and D).Parameter sets for which all position boundaries scaled with thesize of the field were considered positive for scaling (ExperimentalProcedures).Scaling was observed for a large class of networks (Fig. 1E

and SI Text). As expected from the analytical treatment, theexpander in these networks was widely diffusing (λE ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiDE=αE

p>L) and continued to accumulate throughout most of

the dynamics. Small, yet finite degradation of the expander (αE)was compensated by residual production that was maintainedalso at steady state. Scaling was observed in a wide variety ofdegradation schemes, including linear degradation (Fig. 1 F andG), quadratic degradation (Fig. S1, Table S1), or combinationsof the two (Fig. 1E). The numerical analysis further verified the

analytically derived link between scaling and robustness: 76% ofthe systems that scaled were relatively robust to fluctuations inmorphogen production rate (Experimental Procedures) (SI Text).Scaling was maintained also if we allowed slow growth of the

tissue (Fig.1 F and G). We included the interaction of morph-ogen with receptors and the possibility of receptor-mediatedendocytosis and verified that scaling is maintained (Fig. S2).Finally, extending the analysis to related topologies that similarlyrely on a single morphogen and a diffusible molecule that mod-ulates its spread revealed that scaling is unique to the expansion-repression feedback (SI Text).

Implementation of the Expansion-Repression Feedback by a Morphogen-Inhibitor Circuit. The expansion-repression model is general in thesense that it does not specify the molecular mechanism by which theexpander increases the range of the morphogen gradient. In fact,the expander can be associated with different molecular functionsbesides its role in scaling. To illustrate this, we analyze a commonmorphogen-inhibitor circuit and show that it can provide scaling byimplementing the expansion-repression feedback.In a simple morphogen-inhibitor system (Fig. 2A), the mor-

phogen signaling is prevented by an extracellular inhibitor I thatdirectly binds the morphogen and forms an inert diffusiblecomplex, MI. This form of inhibition is quite common, exem-plified, e.g., by the inhibition of bone morphogenic protein(BMP)s by Noggin (16). We further assume that the productionof the inhibitor is repressed by morphogen signaling.The expansion-repression mechanism is realized when the

binding of the morphogen to the inhibitor protects it from deg-radation. This will be the case, for example, when a morphogenis degraded through receptor-mediated endocytosis. The samemechanism can also be realized when the paired morphogeninhibitor diffuses more rapidly than the free morphogen itself(17, 18). In both cases, the inhibitor facilitates the spread of themorphogen, an effect that was recently described for Wnt ligandsand their Frizzled related inhibitors (19). If the inhibitor is alsorepressed by morphogen signaling, it can promote scalingthrough the expansion-repression feedback.To see this, consider the simplest situation in which inhibitor

binding simply prevents the morphogen from binding to itsreceptor R, limiting endocytosis. Using numerical simulations,we followed the dynamics of gradient formation. As wasdescribed for the general expansion-repression feedback, alsohere the gradient is initially narrow, allowing inhibitor pro-duction in a wide region, but it subsequently widens following theaccumulation of the inhibitor. Steady state is achieved when theinhibitor expression is repressed in virtually the entire field andin particular in the distal-most region (SI Text, Fig. S3).Indeed, the steady-state distribution of the bound morphogen

in this system,MR can also be solved analytically (SI Text), giving

½MR�ðxÞ ¼ ρe− μ xL; [10]

with coefficients ρ and μ that depend on different parametersand μ being a weak function of L. Thus, the concentration profileis largely a function of the relative position x/L and not of theabsolute position x. Note that also here, scaling is not precise,because the maximal morphogen level ρ still depends on L.However, this deviation is typically small outside of the proximal-most regions, as we verified by numerical simulations (Fig. 2 B–Dand SI Text). Using a similar analysis we confirmed that scalingcan also be implemented by an inhibitor that competes with themorphogen over interaction with the receptors, (e.g., Dkk1 andWnt ligands, ref. 20) (Fig. 2 E and F and SI Text).

Scaling of the BMP Gradient in the Xenopus Dorsal–Ventral Axis.Another model that implements the expansion-repression feed-back is the one we proposed recently to explain the scaling of the

6926 | www.pnas.org/cgi/doi/10.1073/pnas.0912734107 Ben-Zvi and Barkai

Dow

nloa

ded

by g

uest

on

July

26,

202

1

BMP activation gradient in the early Xenopus embryo (17, 21).Although this system appears very different from the generalexpansion-repression topology, reanalysis of the scaling mecha-nisms revealed that it is based on the same concept. The BMPactivation gradient is established by a molecular network whosemain components include two BMP ligands, Bmp4 and Admp,and an extracellular inhibitor of BMP ligands, Chordin. As weshow, within the model Admp plays a dual role in being both amorphogen and the expander.The model postulates that the BMP ligands, and in particular

Admp, accumulate at the ventral-most region, irrespectively ofwhere they are produced. This accumulation is due to theireffective transport by the inhibitor Chordin, whose localizedproduction at the dorsal pole is the source of asymmetry in this

system. Chordin, coming from the dorsal pole, binds the BMPligands and facilitates their diffusion and disposition at the ventralside. This effective “shuttling” is particularly important for Admp,which, similarly to Chordin, is produced at the dorsal side. Thus,the Admp gradient peaks at the ventral pole, although it is pro-duced dorsally.The basic gradient can be established by Bmp4 alone. How-

ever, as Admp accumulates at the ventral side, it leads to aneffective widening of the activation gradient. This way, it playsthe role of the expander. admp is repressed by morphogen sig-naling (22), closing the expansion-repression feedback loop byfixing the steady-state signaling level at the dorsal-most pole tothe threshold required to repress admp. Scaling follows through

A B

C D

E F

Fig. 2. Scaling in a morphogen-inhibitor system. (A) The morphogen-inhibitor system: A morphogen M binds an inhibitor I forming an inert complex MI,which is not capable of binding the receptor R. The morphogen is degraded through receptor-mediated endocytosis. Morphogen–receptor signalingrepresses inhibitor production. (B) Screen for systems that scale: Shown is the distribution of parameter sets corresponding to systems that scaled (total of 309sets). Such systems were characterized by (i) flat profile of the inhibitor λI > L, where λI denotes the inhibitor decay length, and (ii) the system length beinglarge enough to allow induction of the inhibitor. (C and D) Morphogen gradients established by a morphogen-inhibitor system: (C) a gradient that does notscale and (D) a gradient that scales. Notations are as in Fig. 1 D and E. (E) Schematic representation of morphogen and a competitive extracellular inhibitor.The morphogen M competes with the inhibitor I for binding the receptor R. The morphogen–receptor complex is able to signal and represses production ofthe inhibitor. The morphogen is degraded through receptor-mediated endocytosis. (F) Scaling in a competitive-inhibitor system: Shown are solutions forfields of length L (black curve) and 2L (gray curve). Distance from the source (x axis) is a function of the relative position x/L. Nonscaled coordinates are shownin the Inset. Parameters for numerical solutions are given in the SI Text.

Ben-Zvi and Barkai PNAS | April 13, 2010 | vol. 107 | no. 15 | 6927

DEV

ELOPM

ENTA

LBIOLO

GY

ENGINEE

RING

Dow

nloa

ded

by g

uest

on

July

26,

202

1

the expansion-repression feedback described above, leading to asteady-state activation profile,

BMPðxÞ ¼ Trep

ðx=LÞ2; [11]

with Trep the BMP signaling threshold for admp repression (17).

DiscussionScaling of pattern with size is essential for a reliable patterningduring development. Yet, it is not explained by most models ofmorphogen gradients. Here, we show that scaling emerges as anatural property of a feedback topology that we term expansionrepression. The key advantage of this model is that scaling stemsfrom the network structure. Therefore, it is relatively insensitiveto the specific details and parameters of its molecular imple-mentation, allowing integration into a variety of biologicalprocesses.The expansion-repression feedback topology is based on two

diffusible components: a morphogen and an expander. Pat-terning is determined by the spatial distribution of the morph-ogen, but the width of this distribution increases with theaccumulating levels of the expander. Finally, the expander isrelatively stable and its production is repressed by morphogensignaling. The precise implementation may vary: The interactionbetween the morphogen and the expander can be direct orindirect; the expander could affect morphogen spread by con-trolling its diffusion, degradation, or other interactions; andmorphogen degradation may be linear or nonlinear. Theexpander itself may have other biological roles, such as a mor-phogen inhibitor or a morphogen itself, as shown in the exam-ples we provide.It is interesting to note that the expansion-repression top-

ology is analogous to an integral-feedback controller. An inte-gral-feedback controller calculates the error between the actualand desired outputs and adjusts the controlled variable on thebasis of the time integral of this error. In the expansion-repression topology, the controlled variable is the characteristiclength of the morphogen profile, λ, and the objective is to adjustλ with the system size L (Fig. 3, Fig. S4). The desired gradient isone in which the expander is not produced, and thus the currenterror corresponds to the region where the expander is produced.This error is integrated through accumulation of the expanderas it is produced over time. The accumulated expander increasesλ, resulting in an expanded gradient, in which the expanderproduction domain, i.e., the error, is now smaller. The process

ends when the error is zero, that is, when the expander isrepressed in virtually the entire field (although in actualimplementation some residual production is allowed) (SI Text,Table S2).Integral control is a fundamental concept in engineering that

is used in diverse applications. In a biological context, such con-trollers were implicated in the robust sensory adaptation in bac-terial chemotaxis (23), in maintaining fixed levels of ligand-receptor complexes (24), and in yeast homeostasis (25). Our studysuggests that this concept might be effectively used also in facil-itating the scaling of pattern with size during developmentalpatterning.

Experimental ProceduresNumerical Analysis. The system was modeled using reaction diffusion equa-tions and was solved numerically for two lengths, L1 = 100 μm and L2 = 200

μm, while scanning systematically a multidimensional parameter space.

Consistent sets were those that displayed a steady-state profile that was

biologically valid, i.e., sharp, graded, with a proximal maximum and distal

minimum for both lengths. Scaling was assessed by measuring the change in

the position boundaries associated with three signalings upon doubling L1.

Position boundaries at L1 were fixed to be x = 25 μm, x = 50 μm, and x = 75

μm. A system was scored positive for scaling if the average relative change in

those three positions was <10%. Simulations were done using a home-

improved version of MATLAB’s PDEPE solver.

Numerical Analysis of Scaling in the Expansion-Repression Topology. Theexpansion-repression topology was modeled using the reaction-

diffusion equations

∂½M�∂t

¼ DM∇2½M�− ð1þ ½E�Þp1 α1M ½M�− ð1þ ½E�Þp1 α2M ½M�2

∂½E�∂t

¼ DE ∇2 ½E�− αE ½E� þ βEðð½M�=Tp2Þ

hÞp21þ ð½M�=Tp2Þh

[12]

with a constant flux ηM of M from x = 0 and reflective boundaries for

all other boundary conditions. Parameters p1 ∈ {−1, 1} and p2 ∈ {0, 1} were

used to define the four possible topologies, with expansion–repression

realized for (p1, p2) = (−1, 0). Eight parameters were screened,

DM ;DE ; α1M ; α2M ;αE ; ηM ; βE ;  and Tp2 ; ranging at least two orders of magni-

tude, and h∈ 2;  4gf . Over 400,000 parameter sets were screened (Table S1).

A parameter set was considered robust if the average relative deviation of

the three positions was <0.1 when the morphogen flux was halved

or doubled.

Fig. 3. The expansion-repression mechanism as an integral feedback controller. Morphogen gradient (system output) is measured by production of theexpander in responding cells (sensor). The region where the expander is repressed (measured output) is compared with L, the region where the expandershould be expressed when the gradient is scaled (reference signal). The expander is produced in the region corresponding to the difference between themeasured error and the reference (current error). Its accumulation is the time integrator (controller). Following production, the elevated expander level(system input) increases the length scale of the gradient (system), which expands the gradient (system output). This process ends when the error is zero; i.e.,the expander is not produced by any cell.

6928 | www.pnas.org/cgi/doi/10.1073/pnas.0912734107 Ben-Zvi and Barkai

Dow

nloa

ded

by g

uest

on

July

26,

202

1

Screen for Scaling in the Morphogen-Inhibitor System. By assuming receptorsare not saturated and that morphogen–receptor interactions are fast (SIText), we obtain the following equations:

∂½M�∂t

¼ DM∇2½M�− kþ½M�½I� þ k− ½MI�− αM ½M�∂½I�∂t

¼ DI∇2½I�− kþ½M�½I� þ k− ½MI�− αI ½I� þ βIT hrep

½M�h þ T hrep

∂½MI�∂t

¼ DMI∇2½MI� þ kþ½M�½I�− k− ½MI�:[13]

Eqs. 13 were solved numerically with reflective boundary conditions andflux ηM of M from x = 0. We screened eight parameters, DI ¼ DMI;

DM ;αM ; αI ; η; βI ;k− ;  and Trep, over at least two orders of magnitude and keptremaining parameters constant (Table S3). Over 63,000 parameter sets weretested. See SI Text, Table S4, for full details of the screens, and Table S5 forparameters of simulations and figures in the text.

ACKNOWLEDGMENTS. We thank Mr. Shlomi Kotler, Dr. Dani Glück, andmembers of our group for fruitful discussions. D.B.-Z. is supported by theAdams Fellowship Program of the Israel Academy of Sciences and Humanities.This work was supported by the European Research Council, the Israel ScienceFoundation, Minerva, and the Hellen and Martin Kimmel award for innovativeinvestigations.

1. FreemanM (2000) Feedback control of intercellular signalling in development. Nature

408:313–319.2. Crozatier M, Glise B, Vincent A (2004) Patterns in evolution: Veins of the Drosophila

wing. Trends Genet 20:498–505.3. Dessaud E, McMahon AP, Briscoe J (2008) Pattern formation in the vertebrate neural

tube: A sonic hedgehog morphogen-regulated transcriptional network. Development

135:2489–2503.4. O’Connor MB, Umulis D, Othmer HG, Blair SS (2006) Shaping BMP morphogen

gradients in the Drosophila embryo and pupal wing. Development 133:183–193.5. Eldar A, Rosin D, Shilo BZ, Barkai N (2003) Self-enhanced ligand degradation underlies

robustness of morphogen gradients. Dev Cell 5:635–646.6. Perrimon N, McMahon AP (1999) Negative feedback mechanisms and their roles

during pattern formation. Cell 97:13–16.7. GregorT,McGregorAP,WieschausEF (2008) Shapeandfunctionof thebicoidmorphogen

gradient in dipteran species with different sized embryos. Dev Biol 316:350–358.8. Bollenbach T, et al. (2008) Precision of the Dpp gradient.Development 135:1137–1146.9. McHale P, Rappel WJ, Levine H (2006) Embryonic pattern scaling achieved by

oppositely directed morphogen gradients. Phys Biol 3:107–120.10. Howard M, ten Wolde PR (2005) Finding the center reliably: Robust patterns of

developmental gene expression. Phys Rev Lett 95:208103.11. Houchmandzadeh B, Wieschaus E, Leibler S (2005) Precise domain specification in

the developing Drosophila embryo. Phys Rev E Stat Nonlin Soft Matter Phys 72:

061920.12. Aegerter CM, Aegerter-Wilmsen T (2006) Comment on “Precise domain specification

in the developing Drosophila embryo”. Phys Rev E Stat Nonlin Soft Matter Phys 74:

023901.

13. Aegerter-Wilmsen T, Aegerter CM, Bisseling T (2005) Model for the robustestablishment of precise proportions in the early Drosophila embryo. J Theor Biol 234:13–19.

14. Affolter M, Basler K (2007) The Decapentaplegic morphogen gradient: From patternformation to growth regulation. Nat Rev Genet 8:663–674.

15. Umulis DM (2009) Analysis of dynamic morphogen scale invariance. J R SocInterfaceVol 6, pp 1179–1191.

16. Zimmerman LB, De Jesús-Escobar JM, Harland RM (1996) The Spemann organizersignal noggin binds and inactivates bone morphogenetic protein 4. Cell 86:599–606.

17. Ben-Zvi D, Shilo BZ, Fainsod A, Barkai N (2008) Scaling of the BMP activation gradientin Xenopus embryos. Nature 453:1205–1211.

18. Eldar A, et al. (2002) Robustness of the BMP morphogen gradient in Drosophilaembryonic patterning. Nature 419:304–308.

19. Mii Y, Taira M (2009) Secreted Frizzled-related proteins enhance the diffusion of Wntligands and expand their signalling range. Development 136:4083–4088.

20. Mao B, et al. (2001) LDL-receptor-related protein 6 is a receptor for Dickkopf proteins.Nature 411:321–325.

21. Barkai N, Ben-Zvi D (2009) ‘Big frog, small frog’—maintaining proportions inembryonic development: Delivered on 2 July 2008 at the 33rd FEBS Congress inAthens, Greece. FEBS J 276:1196–1207.

22. Reversade B, De Robertis EM (2005) Regulation of ADMP and BMP2/4/7 at oppositeembryonic poles generates a self-regulating morphogenetic field. Cell 123:1147–1160.

23. Barkai N, Leibler S (1997) Robustness in simple biochemical networks. Nature 387:913–917.

24. Hufnagel L, Kreuger J, Cohen SM, Shraiman BI (2006) On the role of glypicans in theprocess of morphogen gradient formation. Dev Biol 300:512–522.

25. Muzzey D, Gómez-Uribe CA, Mettetal JT, van Oudenaarden A (2009) A systems-levelanalysis of perfect adaptation in yeast osmoregulation. Cell 138:160–171.

Ben-Zvi and Barkai PNAS | April 13, 2010 | vol. 107 | no. 15 | 6929

DEV

ELOPM

ENTA

LBIOLO

GY

ENGINEE

RING

Dow

nloa

ded

by g

uest

on

July

26,

202

1