Báo cáo khoa học: ‘Big frog, small frog’ – maintaining proportions in embryonic development Delivered on 2 July 2008 at the 33 rd FEBS Congress in Athens, Greece pot

THE FEBS ⁄ EMBO WOMEN IN SCIENCE LECTURE
‘Big frog, small frog’ – maintaining proportions in
embryonic development
Delivered on 2 July 2008 at the 33
rd
FEBS Congress in Athens,
Greece
Naama Barkai and Danny Ben-Zvi
Department of Molecular Genetics, Weizmann Institute of Science, Rehovot, Israel
Spemann’s experiments and the scaling
of pattern with size in the amphibian
embryo
From the early days of embryology, biologists have
marveled at the remarkable consistency of the develo-
ping body plan. In 1942, Conrad Waddington put
forward the concept of canalization, referring to the
invariance of the wild-type phenotype in the face of
genetic or environmental perturbations [1]. Since then,
extensive research has been devoted to understand the
origins and evolutionary implications of this funda-
mental property of developing organisms [2].
The plasticity of embryonic development, with its
ability to overcome extreme perturbations, was demon-
strated most dramatically in two classic experiments
performed by Hans Spemann at the beginning of the
20th century [3–5] (Fig. 1A,B). In 1903, Spemann used
a thin baby hair to bisect a cleaving newt embryo into
dorsal and ventral halves. Remarkably, dorsal-halved,
but not ventral-halved, embryos healed and developed
into normal, albeit smaller tadpoles. Twenty years
later, in 1924, Hilde Mangold joined Spemann to

perform a second fascinating experiment in which
they transplanted a group of dorsal cells (‘dorsal lip’)
grafted from a donor embryo into the ventral pole of
a recipient embryo. Strikingly, a complete secondary
axis ensued, resulting in Siamese twins. The trans-
planted cells re-specified host tissues to form neural
tissues and somites instead of epidermis and ventral–
posterior mesoderm. The transplanted cells themselves
only contributed to a fraction of the secondary axis,
Keywords
Admp; BMP; Chordin; control theory;
development; dorsal-ventral; feedback;
morphogen gradient; scaling; Xenopus
Correspondence
N. Barkai, Department of Molecular
Genetics, Weizmann Institute of Science,
PO Box 26, Rehovot 76100, Israel
Fax: +972 8 934 4108
Tel: +972 8 934 4429
E-mail:
(Received 10 October 2008, revised 4
December 2008, accepted 11 December
2008)
doi:10.1111/j.1742-4658.2008.06854.x
We discuss mechanisms that enable the scaling of pattern with size during
the development of multicellular organisms. Recently, we analyzed scaling
in the context of the early Xenopus embryo, focusing on the determination
of the dorsal–ventral axis by a gradient of BMP activation. The ability of
this system to withstand extreme perturbation was exemplified in classical
experiments performed by Hans Spemann in the early 20th century. Quan-

titative analysis revealed that patterning is governed by a noncanonical
‘shuttling-based’ mechanism, and defined the feedback enabling the scaling
of pattern with size. Robust scaling is due to molecular implementation of
an integral-feedback controller, which adjusts the width of the BMP mor-
phogen gradient with the size of the system. We present an ‘expansion–
repression’ feedback topology which generalizes this concept for a wider
range of patterning systems, providing a general, and potentially widely
applicable model for the robust scaling of morphogen gradients with size.
Abbreviations
Admp, anti-dorsalizing morphogenic protein; BMP, bone morphogenic protein.
1196 FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS
Fig. 1. Scaling of the BMP gradient along the dorsal–ventral axis in Xenopus embryos (following Reversade & De Robertis [27]). (A, B) The
Spemann experiments. (A) The dorsal half of a Xenopus embryo has the capacity to develop into a complete, though smaller complete
embryo, whereas the ventral half develops into a ‘bellypiece’. (B) When the Spemann Organizer, located at the dorsal–vegetal side of a
donor Xenopus embryo, is transplanted into the ventral–vegetal side of a recipient embryo (arrowhead), two complete axes ensue. (C) Sche-
matic vegetal view of the Xenopus embryo at early gastrula. chordin and admp are expressed on the dorsal side, and bmp4 is expressed
over a wide region centered on the ventral side. The BMP signaling gradient ranges from blue (high) to red (low). (D) The problem of scaling
morphogen gradients. The BMP signaling gradient is can induce at least four cell fates along the dorsal–ventral morphogenic field (0 < x < L,
upper). If the field is shorter (0 < x <L ⁄ 2, lower), the morphogen gradient must also scale. Simply truncating the dorsal half of the field leads
to a patterning defect, for example loss of dorsal fates (lower left). A proper scaling mechanism must change the properties of the gradient
over the entire field and create generally a sharper profile for a smaller animal (lower right). (E) A general model for the BMP patterning
network. Chordin, produced at a constant flux from the dorsal side, inhibits the BMP ligands Admp and Bmp4 by forming a Chordin–ligand
complex. The protease, Xlr, degrades Chordin when it is free, or when it is bound to one of the BMP ligands. Cleaving the complex releases
the active ligand. Both Admp and Bmp4 contribute to BMP signaling which induces the expression of bmp4, and represses expression of
chordin and admp. In the numerical screen, all proteins and complexes were allowed to diffuse freely, and reaction rates were varied over
several orders of magnitude. (F) Inhibition mechanism. Total BMP ligands, free and in complex with Chordin, are spread uniformly along the
dorsal–ventral axis. The protease creates a gradient of the inhibitor, Chordin, leading naturally to an inverse gradient of free ligands. (G) Shut-
tling mechanism. The inhibitor, Chordin, binds the immobile BMP ligands and forms a diffusible complex, effectively carrying the ligands.
The complex is then cleaved by the protease, and the ligands are released. This process allocates the ligands and concentrates them in the
ventral side such that the total BMP ligands distribution, free and in complex, is not uniform but reflects the activation gradient.

N. Barkai and D. Ben-Zvi Maintaining proportions in embryonic development
FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS 1197
primarily the notochord. The embryonic region exhib-
iting this induction capacity, the dorsal blastopore lip
of the early gastrula embryo, was termed the Spemann
Organizer.
Spemann’s experiments showed that a fraction of
the dorsal embryonic cells contain information that is
both required and sufficient to define a full embryonic
axis. Moreover, the experiments demonstrated that the
embryo is able to adjust to an extreme perturbation in
size, and scale its morphological patterns. Indeed, the
ability to scale pattern with size is key to both experi-
ments: successful development of dorsal-half embryos
requires embryos to ‘recognize’ that their ventral half
is missing and scale their pattern accordingly. Simi-
larly, embryos induced to form a secondary axis need
to ‘realize’ that two axes are present, and scale both to
half-embryo size.
The capacity for embryonic self-regulation was subse-
quently demonstrated in additional settings. Subdivid-
ing the blastoderm of the chick embryo into fragments,
for example, can result in multiple axes when cultured
in isolation [6,7]. Armadillo (Dasypus novemcinctus,a
mammal), naturally generates four genetically identical
embryos from a single blastocyst during normal repro-
duction [8]. In humans, identical twins can arise from a
splitting of a single blastocyst [9].
One way to compensate for reduced cell numbers is
by increasing cell proliferation. This, however, is not

the strategy employed by the amphibian embryo. In
1981, Jonathan Cooke generated frog embryos of
reduced sizes by sucking out a significant fraction of
the cells (20–65%) at an early developmental stage
[10]. As expected from the results of Spemann’s experi-
ments, the reduced-size embryos developed into nor-
mal, smaller, tadpoles. Cooke then examined the total
number of mesodermal cells, and their distribution in
the different tissues. Small embryos did not signifi-
cantly compensate for the reduced number of cells.
Still, the cells were properly distributed among the dif-
ferent tissues, with the proportions of cells assigned to
each tissue practically the same as found in wild-type
embryos. Thus, the ability to scale pattern with size
does not stem from increased proliferation, but must
be inherent to the mechanism that guides the pattern-
ing process itself. How this scaling is achieved
remained a mystery.
A BMP activity gradient patterns the
dorsal–ventral axis of early embryos
In many species ranging from cnidarians to mammals,
dorsal–ventral patterning of the early embryo is guided
by a spatial gradient of bone morphogenic protein
(BMP) activity [11–15]. BMP ligands are morphogens,
long-range effectors that induce different cell fates in a
concentration-dependent manner [16]. In vertebrates,
high levels of BMP activity at the ventral side of the
embryo induce ventral fates (e.g. blood and epidermis),
whereas intermediate levels generate pronephros and,
more dorsally, the somites. The absence (or very low

levels) of BMP activity in the dorsal region is required
for the formation of notochord and neural tissue.
Indeed, progressively higher levels of Bmp4, an evolu-
tionary conserved BMP ligand, are both sufficient and
necessary for the specification of at least four mesoder-
mal tissue types in Xenopus, notochord, muscle, pro-
nephros and blood, and for the induction of several
distinct domains of gene expression [17].
Direct evidence for the graded distribution of Bmp4
in early embryos is not yet available in Xenopus laevis
due to its opacity and complex genetics. It is possible,
however, to follow the first activation step of the BMP
pathway using antibodies that recognize specifically the
phosphorylated form of the Smad1 protein, pSmad1
[18]. pSmad1 is distributed in a graded fashion along
the dorsal–ventral axis of the embryo, consistent with
a corresponding gradient of BMP activity [19]. Using
dorsal and ventral markers, we have shown that scal-
ing occurs remarkably early, during grastrulation, sup-
porting the scaling of the BMP activation gradient
itself [20].
How is the BMP activation gradient generated? The
key asymmetry is the localized secretion of BMP
antagonists by the Spemann Organizer from the dorsal
side of the embryo. At least one of those inhibitors
(Chordin, see below) can function over long distances
from its point of secretion to generate a spatial activa-
tion gradient along the embryo [21,22].
The polar secretion of BMP antagonists contrasts
with the relative uniform expression of three main

BMP ligands, Bmp7, Bmp4 and Bmp2, which are
widely expressed before gastrulation [23,24] (A. Fain-
sod, personal communication). Paradoxically, a fourth
BMP ligand, anti-dorsalizing morphogenic protein
(Admp), which also participates in the patterning pro-
cess, is expressed at the region displaying lowest BMP
activity: the Spemann Organizer [7,25–27] (Fig. 1C).
Recent experiments suggest that Admp plays an essen-
tial role in the scaling mechanism, as depletion of
Admp abolishes patterning in dorsal-halved embryos
[27]. Admp is expressed dorsally in a pattern overlap-
ping the BMP inhibitors, and is subject to autoregula-
tory transcriptional repression by the BMP pathway.
These features are conserved in many bilateria,
although not in Drosophila which does not have an
Admp homolog.
Maintaining proportions in embryonic development N. Barkai and D. Ben-Zvi
1198 FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS
Computational search for scaled
gradients identifies a ‘shuttling-based’
patterning mechanism
Taken together, evidence suggests that the dorsal–ven-
tral polarity of amphibian embryos is defined by a gra-
dient of BMP activity that extends across the embryo.
This gradient is established by a well-characterized and
evolutionary conserved molecular network [28]. In
standard morphogen models, the scale of the gradient
is determined by intrinsic parameters (e.g. the diffusion
and degradation rates of the morphogen) in a way that
is independent of embryo size. How then can one

account for the clear scaling of pattern with size, as
demonstrated in Spemann’s and Cooke’s experiments
(Fig. 1D)?
As a first step towards an answer, we formulated a
model of the molecular network which functions to
establish the BMP activation gradient [20]. In recent
years, many molecules that refine and reshape the
BMP gradient have been described [5,29–31]. Because
the dorsal–ventral patterning network is conserved
across evolution [11–15], we simplified the model to
consist of the core conserved elements (Fig. 1E): a
BMP ligand (Bmp4), a secreted BMP inhibitor (Chor-
din) produced at a constant flux from the dorsal side,
and the Tolloid-related protease (Xlr) which degrades
Chordin. We also included Admp, as it was shown to
play a key role in the system-level functioning of the
network and to contribute to the ability of halved-
embryos to scale their pattern [27].
The model we considered is rather general, in the
sense that it allows for a range of interactions between
the protein constituents, accounts for the possible dif-
fusion of all components, and does not specify the pre-
cise weight of each process. Clearly, the properties of
the model depend critically on the value of the differ-
ent kinetic parameters (e.g. diffusion coefficients and
rate constants), the majority of which are not known.
In fact, different choice of parameters results in quali-
tatively different mechanisms.
Within this model, we searched for molecular net-
works that enable the scaling of the morphogen gradi-

ent with embryo size. Each network corresponded to a
specific choice of parameters. We utilized a numerical
screening approach, searching systematically across the
multidimensional parameter space. The screen identi-
fied two classes of networks that generated morphogen
distribution with a proper dorsal–ventral polarity. The
mechanisms by which pattern was generated differed
qualitatively between these two network classes. The
first class, consisting of the vast majority of networks,
generated polarity strictly through the inhibition of
BMP ligands by the diffusible inhibitor (Chordin).
Within this inhibition-based mechanism, the locally
secreted Chordin diffuses into a region of uniform
BMP, where it is cleaved by a protease (Xlr). A gradi-
ent of inhibition is generated, leading to an inverse
BMP–activation gradient. This mechanism does not
require the redistribution of BMP molecules them-
selves. Rather, the BMP ligand is still mostly uni-
formly distributed, but its activity is graded due to the
graded distribution of its inhibitor (Fig. 1F). The sec-
ond class of networks relies on an alternative, shut-
tling-based mechanism. Here, the BMP ligands are
physically allocated to the dorsal region. The inhibitor,
Chordin, functions as an effective shuttling molecule
carrying Bmp4 and Admp towards the ventral pole.
Within this mechanism, the activation gradient is pri-
marily due to the graded distribution of the BMP
ligands themselves, and not due to their inhibition
by Chordin (Fig. 1G). Only a small fraction of the
screened networks corresponded to this mechanism,

and these networks were all found within a specific
region in the parameter space. Examining the para-
meters of these networks revealed that shuttling is
obtained if the BMP ligands diffuse primarily when
bound by Chordin, and Xlr degrades Chordin primar-
ily when the later is bound to a BMP ligand.
Although both mechanisms establish a graded BMP
activation profile, only the shuttling-based mechanism
was able to scale the profile with the size of the
embryo. Gradients generated by the inhibition-based
mechanism are largely invariant. Consequently, these
gradients do not scale with size, and cannot be used to
maintain proportionate patterning in embryos of dif-
ferent sizes. In sharp contrast, gradients generated by
the shuttling-based mechanism did scale with size:
changing embryo size led to a proportionate modula-
tion of the BMP activation gradient.
The two mechanisms differ also in the sharpness of
the BMP activation profile and in its robustness. Gra-
dients established by the inhibition-based mechanism
are relatively shallow, and are highly sensitive to the
dosage of the inhibitor, activator or protease. By con-
trast, the shuttling-based mechanism defines a much
sharper profile, which is robust to changes in gene dos-
age. In fact, these two features led us to propose previ-
ously that the shuttling-based mechanism is used by
the highly homologous Drosophila network [32,33], a
prediction that was confirmed experimentally [32,34–
38]. Notably, however, the shuttling model based on
the Drosophila network by itself, does not support scal-

ing of pattern with size, whereas the Xenopus-based
model does [20]. Experiments in Xenopus embryo
confirmed the two main predictions of the shuttling
N. Barkai and D. Ben-Zvi Maintaining proportions in embryonic development
FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS 1199
model: Bmp4 is shuttled to the ventral side, and Chor-
din is required for Bmp4 diffusion and redistribution
[20].
The mechanism underlying scaling
How does shuttling ensure the scaling of the BMP
activation profile with the size of the embryo? To try
and answer this question, we have solved analytically
the Xenopus-based model under different limiting con-
ditions. In particular, we considered conditions in
which shuttling is realized, namely when the free BMP
ligands, Admp and Bmp4, do not diffuse, and their
inhibitor, chodrin, is degraded only when in complex
with a ligand [20]. This solution pointed at the follow-
ing three features that enable scaling.
The first feature is the use of the shuttling-based
mechanism, where the BMP ligands are effectively
transported by a common BMP inhibitor to the ven-
tral-most part of the embryo. This establishes a robust,
power-law decaying activation profile. The resulting
profile is invariant to gene dosages and most parame-
ters of the system.
The second critical feature is the presence of two
BMP ligands, with the affinity of Admp to the inhibi-
tor Chordin much lower than the Bmp4–Chordin
affinity. The presence of two ligands competing for the

binding of Chordin allows for a range of possible
steady-state profiles, depending on the relative abun-
dance of the two ligands. This is in contrast to the case
of a single ligand, where the gradient approaches a
unique profile that is independent of the total ligand
level.
Third, the auto-repression of the BMP-ligand Admp,
is used to effectively sense embryo size and tune the
gradient spread accordingly. Together, these three
features lead to a robust and sharp gradient that is
properly scaled with embryo size.
To understand how scaling is achieved, consider the
following dynamics of gradient formation. Suppose,
for example, that Admp is initially absent so that the
activation gradient is controlled by Bmp4 only. This
gradient is relatively sharp and narrow, because Chor-
din has a high affinity for binding Bmp4, thus leading
to its efficient shuttling and localization at the ventral
pole. The narrow gradient allows for admp expression
in dorsal regions. Admp is then shuttled by Chordin
throughout the embryo, and contributes to overall
BMP signaling. The accumulation of Admp tips the
balance in the competition between Admp and Bmp4
over the interaction with Chordin. Because Admp has
a lower binding affinity to Chordin, its shuttling is less
effective, leading to a wider gradient, expanding
dorsally. However, admp is repressed by BMP signal-
ing. Hence expansion of the gradient eventually leads
to the repression of admp expression in the entire
embryo and specifically at the dorsal pole, where the

gradient is lowest. Once admp is repressed, the gradient
ceases to expand.
More rigorously, these dynamics are conveyed by
the mathematic solutions of the model. We find that
the overall profile of BMP signaling in the embryo,
BMP(x)=Admp(x)+Bmp4(x), can be written as
[20]:
BMPðxÞ%T
rep
k
2
x
2
; ð1Þ
where T
rep
denotes the threshold of BMP activity at
which admp expression is repressed, and k, the scale of
the gradient, is not fixed but depends on the relative
total levels of Admp and Bmp4 [20]. admp is repressed
by BMP signaling with its total level increases as:
dAdmp
tot
dt
¼ b
Admp
ðL À kÞð2Þ
where b
Admp
is the Admp production rate per unit

length, and L the length of the morphogenic field.
Consequently, Eqn (2) determines the steady state
value of k and the sharpness of the profile. Steady
state is obtained only when k =L. Thus, the width of
the gradient is defined precisely by the size of the
embryo. Substituting this back into Eqn (1), we obtain
the steady-state gradient:
BMPðxÞ%
T
rep
ðx=LÞ
2
ð3Þ
Hence the activation profile is a function of the ratio
x ⁄ L, implying the scaling of pattern with size. For
example, a gene that is induced at 50% embryo length
(x ⁄ L = ½) will be expressed at mid-embryo irrespec-
tive of embryo size, and in particular will be found in
the middle of a half-embryo. Note also that the shape
of the profile depends only on T
rep
and is independent
of the other parameters in the system. Accordingly, the
profile is robust to fluctuations in most parameters.
Analogy with integral-feedback
controller
The scaling mechanism described above can be viewed
as an implementation of an integral-feedback control-
ler, a widely used concept in engineering. An integral-
feedback controller is used to adjust the current output

of a system to a desired output, using a three-step
Maintaining proportions in embryonic development N. Barkai and D. Ben-Zvi
1200 FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS
procedure. First, the error between the actual output
of the system and a desired, predefined output is calcu-
lated. Second, this error is integrated over time.
Finally, the controlled variable is adjusted based
on the time-integral of this error. Integral feedback
control is fundamental to many control systems. In a
biological context, integral feedback was shown to
underlie the robust sensory adaptation in bacterial che-
motaxis [39] and was proposed to be instrumental also
for maintaining fixed levels of ligand-receptor com-
plexes [40].
In the present implementation of the integral-feed-
back controller, the controlled variable is the scale
(spread) of the morphogen profile, k in Eqn (1), and
the objective is to adjust it with the size of the embryo,
L. The error is thus the difference between k and L.
This error is integrated over time through the accumu-
lation of Admp (Eqn 2). Finally, this integrated error,
captured by the levels of Admp, feedbacks to control
the gradient spread, k. Note that within this analogy,
Admp plays a dual role as a morphogen and the con-
trol element.
Scaling of pattern with size in other
developmental contexts
The ability to coordinate tissue pattern with tissue size
is not unique to the Xenopus embryo, but is a pro-
found property found in many developmental systems.

This is evident, for example, from the fact that body
plans of most organisms remain largely invariant
despite large variations in size. Classic examples
include the variation in the size of the eggs in amphibi-
ans, birds and invertebrates [10,41–44] (Fig. 2), and
the ability of separated blastomeres to develop into
smaller twin embryos, first shown by Driesch and
Morgan at the end of the 19th century [45,46]. In addi-
tion to natural variation, size is regulated also by the
availability of nutrients or other growth conditions.
Drosophila larvae reared in crowded conditions, for
example, produce adults that are much smaller than
those produced by a well-fed larvae, but the resulting
flies, albeit smaller, are perfectly proportionate [47].
Similarly, removal of the hind-wing imaginal disc in a
caterpillar results in a butterfly with larger than nor-
mal, but perfectly patterned, forewings and forelegs
[48].
Tissue size can also be modulated by genetic manip-
ulations, often without consequences for tissue pattern.
This was demonstrated most extensively in the Dro-
sophila wing imaginal disc. Scaling is maintained upon
mutations in components of the insulin-signaling path-
way, which alter tissue size by affecting either cell
number or cell size [47,49–52]. Similarly, proper scaling
is maintained when disc size is manipulated through a
change the activity of the nitric oxide synthase which
alter cell proliferation [53]. Tissue pattern was shown
to be independent also of mutations that alter cell size,
or cell growth rate, without affecting the overall size of

the wing disc [54].
Direct evidence that scaling is achieved at the level
of the morphogen gradient itself was also provided
[55]. Teleman and Cohen took advantage of the fact
that the developing disc is subdivided into compart-
ments whose size can be controlled independently
[51,52,56], and genetically manipulated only the size of
the posterior compartment. In this way, they estab-
lished a situation in which the Dpp morphogen was
produced along a line source separating two, differ-
ently sized compartments. Remarkably, the Dpp activ-
ity gradient was asymmetric and exhibited precise
scaling with compartment size. Size compensation thus
Fig. 2. Natural variation in the size of
Xenopus embryos. Two wild-type embryos
are shown at the blastula stage and at the
tadpole stage. The larger embryo is twice
the volume of the smaller one.
N. Barkai and D. Ben-Zvi Maintaining proportions in embryonic development
FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS 1201
occurs early, at the level of receptor activation. Under-
standing the mechanisms that could ensure size com-
pensation is a central contemporary challenge.
‘Expansion–repression’ feedback
topology
A general scaling mechanism implementing
integral-feedback control
We asked if the scaling mechanism we identified in
Xenopus can be generalized to explain scaling in other
biological contexts. The patterning mechanism used in

the early amphibian embryo is distinct from the stan-
dard morphogen gradient paradigm. The standard
model assumes a morphogen that is secreted from
a localized source and diffuses away to generate a
gradient that peaks at the source and decays gradually
away from it. This, for example, is the paradigm used
to establish the three key morphogen gradients, Dpp,
Wingless and Hedgehog, in the Drosophila wing
imaginal discs and the dorsal–ventral patterning of the
vertebrate neural tube [57,58]. We reasoned that the
concept of integral-feedback control could be broadly
utilized. To examine this possibility, we conducted
several numerical screens, searching for scaling mecha-
nisms that will function within the standard paradigm
of morphogen gradient formation (D. Ben-Zvi, D.
Gluck & N. Barkai, unpublished results).
We identified a single feedback topology that pro-
vides robust scaling for a wide range of parameters.
This feedback is defined by the following three proper-
ties: (a) diffusion or degradation of the morphogen is
modulated by some molecule, E, the ‘expander’, such
that high levels of E lead to an expanded (wider) gradi-
ent, this can be realized, for example, through interac-
tions with extracellualr proteoglycans; (b) production
of E is repressed by the morphogen, leading to produc-
tion of E in cells subject to a (relatively) low morpho-
gen concentration; (c) E is widely diffusible and
degrades slowly. Consequently, it accumulates during
most of the dynamics and its distribution across the
field is approximately uniform. Notably, we find that

scaling does not require fine-tuning of parameters or
the existence of unique signaling machinery. Rather,
scaling is an immediate consequence of the feedback
topology. We denote this feedback topology as ‘expan-
sion–repression’ mechanism (Fig. 3A,C). The ‘expan-
sion–repression’ feedback relies on general building
blocks that were identified in a variety of systems. In
fact, the use of feedback-regulation to shape the diffu-
sion or degradation of morphogens, e.g. through the
regulation of receptors, proteases or components that
interact with heparan sulfate proteoglycans is quite
common, and was described in a large number of
morphogen systems [59–61].
Closer inspection of the ‘expansion–repression’ feed-
back revealed that scaling is indeed achieved by imple-
mentation of an integral-feedback controller (Fig. 3B).
As before, the controlled variable is the decay length
of morphogen profile, k, and the objective is to adjust
this decay length with the system size L. The error cor-
responds to the size of the region where E is expressed,
and is given by L-x
rep
, where x
rep
is the distal-most
position where E is repressed. The accumulation of E
functions as the time-integrator of the error. Because
expander levels define the width of the gradient, this
integrated error is fed back to the system. Mathemati-
cally, we find that the system can be described by the

following equations:
dE
tot
dt
¼ b
E
ÁðL À a
0
kÞ; k ¼ kðE
tot
Þ with
dk
dE
> 0: ð4Þ
Here b
E
> 0 is the rate by which E is produced per
unit length, and a
0
is some dimensionless constant
Fig. 3. Expansion–repression mechanism. (A) Expansion–repression feedback is based on two properties. First, the morphogen represses
an expander molecule. Second, the expander functions to increases the spread of the morphogen, k, by some mechanism such as enhanc-
ing morphogen diffusion or reducing its degradation. The expander must be diffusible and relatively stable. (B) An integral feedback controller
underlies the scaling mechanism. The target of the control circuit is to scale the gradient with the size of the field. The morphogen gradient
(system output) is measured by induction ⁄ repression of the expander in each cell (sensor). x
rep
, the distal most position where the expander
is induced (measure error) is compared with the desired scale, for which the expander is not induced at all, i.e. x
rep
=L(reference). The

region where the expander is induced (measured error) produces the expander, which accumulates in the field. This accumulation turns the
controller into an integral controller. The increase in the expander level (system input) increases the length scale of the gradient (system).
This increase changes the morphogen gradient (system output), and the process is repeated with the induction ⁄ repression of the expander.
This process halts when x
rep
equals the distal-most position in the field, hence the expander levels and length scale stabilize. (C) Schematic
representation of expansion–repression dynamics. High morphogen signaling in shown in green, whereas low signaling is shown in red. The
morphogen is produced and secreted at the proximal region. Initially, its spread is small and the gradient is narrow. Consequently, the expan-
der (purple) is expressed and is secreted over a wide area in the distal region of the field (upper). Accumulation and diffusion of the Expan-
der expands the gradient (middle) until the gradient is wide enough to repress the expander everywhere in the field (lower). The expander
may interact with the heparan sulfate proteoglycans, receptors or any other elements to increase the spread of the gradient.
Maintaining proportions in embryonic development N. Barkai and D. Ben-Zvi
1202 FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS
N. Barkai and D. Ben-Zvi Maintaining proportions in embryonic development
FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS 1203
independent of the field size, L. The directionality of
the feedback ensures that at a steady state, E is prop-
erly adjusted such that L = a
0
k
st
, implying scaling of
the morphogen profile with the size of the field.
Clearly, these equations define an integral-feedback
controller. Thus, any implementation of the ‘expan-
sion–repression’ feedback module, regardless of the
exact molecular details, will lead to robust scaling of
the morphogen pattern with the size of the field.
Comparison with other scaling mechanisms
The main advantage of the ‘expansion–repression’

mechanism for scaling is its robustness. The use of
integral-feedback ensures scaling for a wide range of
parameters without the need to fine-tune rate constants
or the precise functional dependency between the dif-
ferent parameters. Scaling is achieved by the structure
of the network, independent of other aspects of the
morphogen gradient such as degradation and transport
mechanisms. Previous theoretical attempts to explain
scaling have focused on three general paradigms,
described below.
Arguably, the simplest scaling mechanism is the
so-called ‘perfect sink’ solution. A ‘perfect sink’ degr-
ades morphogen rapidly, and consumes all morphogen
molecules reaching its position. If positioned at the
edge of the field, opposing the morphogen source, a
perfect sink will lead to scaling, but only when (a) the
morphogen does not degrade during its motion within
the field and (b) morphogen levels at the source are kept
constant. Biologically, these two conditions rarely hold.
In most cases the morphogen does degrade during its
movement across the tissue through interaction with
inhibitors or with receptors. Moreover, it is probably
the rate of production at the source, rather then the level
of morphogen, which is kept fixed. It is therefore unli-
kely that perfect sink contributes to scaling in most bio-
logically relevant situations.
Several studies have suggested that scaling is
achieved through the integration of two opposing
gradients, e.g. when two morphogen sources are posi-
tioned at two opposing poles [62–64]. In this case,

cells can extract information about the size of the
field by effectively comparing the two gradients. If
the morphogen degrades linearly, scaling is guaran-
teed for a single position within the field. This mech-
anism cannot be used to scale multiple threshold
positions, in sharp contrast to the situations
described above for the ‘expansion–repression’ topol-
ogy, where nearly all threshold positions scaled with
the size of the field, maintaining proportions. The sit-
uation somewhat improves if both morphogens are
degraded in the exact same nonlinear manner through-
out the field. In this case, scaling holds over a wide
domain of the field (N. Barkai & D. Ben-Zvi, unpub-
lished results). However, even under these conditions,
scaling requires the ‘fine-tuning’ of the reactions of the
two molecular gradients, a fact which might limit its
biological application.
The ‘expansion–repression’ feedback is more related
to a third class of mechanisms, which assumes the exis-
tence of a chemical species, analogous to the proposed
‘expander’, whose concentration affects the length scale
(spread) of the morphogen gradient. In the context of
self-organized patterning (‘turing-like’ mechanism), a
similar chemical species alters the wavelength of the
activator profile [65–67]. The level of this secreted
species is assumed to be proportional to some power
of the field size, depending on specific assumptions
and boundary conditions. No feedback, however, is
assumed between the morphogen signaling and the
production of this secreted species. The proposed

‘expansion–repression’ feedback topology thus extends
and generalizes this approach, by introducing feedback
on the production of the expander molecule and
applying it upon the standard morphogen gradient
paradigm. This feedback results in an effective
integral-feedback controller, enabling a robust scaling
of morphogen spread with the system size, in a manner
that does not depend on parameters or on the details
of the interactions in the system.
Other successful attempts to model scaling in specific
systems [68,69] considered scaling of a single position
of the field and not the entire gradient, and relied on
the unique properties of those systems.
Concluding remarks
The development of multicellular organisms is charac-
terized by extensive changes in size and morphology.
Growth and patterning must be coordinated, and the
ability to scale pattern with size is one manifestation
of this coordination. Coordination can be achieved if
size is defined by the patterning process itself, e.g. if
tissue size is controlled by precisely the same morpho-
gen gradient that defines tissue pattern. External
factors governing size may also take effect through
changing the physical properties and the length scale
of the morphogen gradient. Alternatively, size can be
defined independently, and scaling of pattern achieved
at the level of the patterning process itself. This review
has focused on the latter paradigm, which appears to
hold for early developmental processes. It is possible
that later processes are governed by a more intricate

interplay between growth and patterning.
Maintaining proportions in embryonic development N. Barkai and D. Ben-Zvi
1204 FEBS Journal 276 (2009) 1196–1207 ª 2009 The Authors Journal compilation ª 2009 FEBS
The scaling mechanism we describe implements, in
molecular terms, the concept of integral-feedback con-
trol. The main advantage of this mechanism is its
robustness: scaling does not require ‘fine-tuning’ of
reaction rate constants but is inherent to the mecha-
nism itself. Moreover, there is no need for precise
adjustment of the molecular interactions. Scaling is the
outcome of the general feedback topology, which can
be implemented in a variety of ways. For example, in
the basic ‘expansion–repression’ topology, all that is
required is for the Expander molecule to be widely dif-
fusible and stable, to be repressed by morphogen sig-
naling and influence (in some unspecified way) the
diffusion or degradation of the morphogen. This
mechanism can be applied in various ways by develop-
ing organisms, as we have shown for the Xenopus
embryo.
The ability to scale pattern with size is highly impor-
tant for normal development. It enables the organism
to compensate for natural variation and overcome
periods of nutrient limitation, which reduce embryo
and tissue size. In addition, such a capacity may also
be important for facilitating the evolutionary adapta-
tion of body size, because the pattern will automati-
cally adjust with any mutation that alters body size,
without the need for further adjustment of the pattern-
ing mechanism. It will be interesting to examine

whether the same scaling mechanisms that function
within a given species, also operate to define the differ-
ence in size between species.
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