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

‘Big frog, small frog’ – maintaining proportions inembryonic development Greece Naama Barkai and Danny Ben-Zvi Department of Molecular Genetics, Weizmann Institute of Science, Rehovot, I

‘Big frog, small frog’ – maintaining proportions in

embryonic development

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: naama.barkai@weizmann.ac.il

(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.

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.

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 scalpattern-ing 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, durscal-ing 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

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 coefficidiffer-ents 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 activamodula-tion 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

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 scalfollow-ing

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 binChor-ding Bmp4, thus leaChor-ding

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  Trep

k2

where Trep 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:

dAdmptot

dt ¼ bAdmpðL  kÞ ð2Þ

where bAdmp 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  Trep

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 Trep 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

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 pro-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 variaaddi-tion, 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-sophilawing 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.

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 stanstan-dard

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 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-xrep, where xrep 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:

dEtot

dt ¼ bE ðL  a0kÞ; k¼ kðE

totÞ with dk

dE > 0: ð4Þ

Here bE> 0 is the rate by which E is produced per unit length, and a0 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 reducenhanc-ing 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 xrep= 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 xrepequals 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 expanExpan-der everywhere in the field (lower) The expanExpan-der may interact with the heparan sulfate proteoglycans, receptors or any other elements to increase the spread of the gradient.

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 = a0kst, 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

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 interestpattern-ing 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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