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# Scaling of morphogen gradients by an expansion-repression integral feedback control

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

## Abstract

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

Multicellular organisms develop through a sequence of patterning events in which uniform fields of cells differentiate into patterned tissues and organs. Positional information is commonly encoded by morphogen gradients, whereby a signaling pathway is activated in a spatially graded manner over a field of cells and induces distinct gene expression domains in a concentration-dependent manner. In the standard paradigm, a signaling molecule—the morphogen—is secreted from a local source and diffuses through the tissue, establishing a gradient that peaks at the source. The gradient is shaped further by factors that impact on morphogen diffusion or degradation. The activity and abundance of these regulators is often influenced by the morphogen signaling 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 remarkably 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 by which field size is measured and how this information feeds back to shape the morphogen gradient (7–15).

Theoretical studies have shown that scaling is not a general property of morphogen models but requires specialized mechanisms. Proposed mechanisms include (*i*) two diffusible molecules that emanate from opposing poles and define the activation profile through their ratio; (*ii*) a mechanism that maintains a constant, size-independent receptor number; and (*iii*) a topology that restricts morphogen degradation to only the distal edge of the field (a “perfect sink”) (9,10–11, 13, 15). Whereas those mechanisms likely apply in some cases, they invoke fine-tuned interactions, posing significant constraints on the design of the patterning 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, whose profile is shaped by a diffusible molecule, the “expander”. The expander functions, directly or indirectly, to facilitate the spread of the morphogen by enhancing its diffusion or protecting it from degradation. In turn, expander production is repressed by morphogen signaling. Mathematical analysis reveals that scaling is achieved provided that the expander is stable and diffusible.

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

## Results

### Expansion-Repression Feedback.

We consider a single morphogen *M* that can induce several cell fates in a concentration-dependent manner. We assume that *M* is secreted from a local source and diffuses in a naive field of cells to establish a concentration gradient that peaks at the source. We denote the morphogen diffusion coefficient by *D _{M}* and its degradation rate by

*α*. The shape of the morphogen is determined by the diffusion equation

_{M}with a constant flux η* _{M}* at the origin (

*x*= 0). Boundary conditions at

*x = L*have a negligible effect on the system under most conditions (

*SI Text*), and we assumed either reflecting or absorbing boundaries at

*x = L*or, alternatively,

*M = 0*at infinity (

*SI Text*). We further assume an additional diffusible molecule, the expander

*E*, which influences (directly or indirectly) the degradation rate or the diffusion of the morphogen, so that

are monotonically increasing (*D _{M}*) or decreasing (α

*) functions of the expander concentration [*

_{M}*E*]. Note that the diffusion and degradation rates may depend on the level of morphogen [

*M*], accounting for possible nonlinear effects. Finally, we assume that expander production is repressed by morphogen signaling, so that the expander distribution is given by the reaction diffusion equation

where *T*_{rep} denotes the repression threshold, and *h* is some Hill coefficient. Without loss of generality, we assume reflective boundary conditions (*SI Text*). Together, Eqs. **1**–**3** define the expansion-repression feedback topology (Fig. 1*A*).

To see how this feedback provides for the scaling of pattern with size, it is instructive to follow the dynamics of gradient formation. The morphogen diffuses and degrades across the field, generating a graded distribution. The expander is produced in distal regions of the tissue where signaling is low. Accumulation of the expander leads to the broadening of the morphogen gradient, which, in turn, narrows down the expander production domain (Fig. 1*B*). Steady state is reached when the gradient stops expanding, that is, when the expander stops accumulating. Assuming that the expander is stable, steady state will be reached only when the expander production stops, namely when its production is inhibited in virtually the entire field and specifically in the distal-most point (*x = L*). This way, the steady-state level of morphogen at the distal-most position is pinned to the level required for the expander repression, *T*_{rep}, leading to the overall adjustment 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 very different from the adjustment achieved by simply specifying a boundary condition. The latter affects the profile locally and usually has a moderate effect on the overall level of morphogen gradient and does not lead to scaling (*SI Text*).

### Analytical Approximation.

To better illustrate the scaling mechanism, we rewrite Eq. **1** as

with τ some typical time scale, λ a typical length scale, and *f*([*M*]) a degradation term. For example, when a morphogen degrades linearly, τ^{−1} is given by the degradation rate, and *f*([*M*]) = [*M*]. The expander *E* functions to modulate λ, with λ = λ([*E*]) a monotonically increasing function of [*E*]. Changing variables to , we can write

For simplicity (and without loss of generality, *SI Text*), we assume *L >>* λ. λ then affects the morphogen profile only through *y* (the scaled coordinate) and ρ (the scaled concentration), implying that the steady-state profile can be written as

with ρ^{st} and λ^{st} 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, narrowing the region where the expander is expressed. We further assume that the expander diffuses rapidly and degrades slowly so that it is approximately uniform across the field and continues to accumulate as long as it is produced. Under these conditions, the system will arrive at a steady state only when the expander production is repressed throughout the field. In particular, at the distal pole *x = L*, which is the lowest point of the gradient, the steady-state profile must satisfy

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

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

The gradient is scaled because its profile depends on the spatial coordinate *x* only through the relative position, .

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 completely insensitive (robust) to morphogen production rate ρ. However, we find that deviation from scaling due to this dependency is typically small.

### Numerical Analysis.

To further examine the generality of scaling within the expansion-repression topology, we performed numerical analysis, screening systematically different parameters and degradation schemes of Eqs. **1**–**3** (*Experimental Procedures*). For each parameter set, we derived the position boundaries defined by three concentration thresholds and evaluated these boundaries for fields of normal (*L*) and double (2*L*) sizes (Fig. 1 *C* and *D*). Parameter sets for which all position boundaries scaled with the size of the field were considered positive for scaling (*Experimental Procedures*).

Scaling was observed for a large class of networks (Fig. 1*E* and *SI Text*). As expected from the analytical treatment, the expander in these networks was widely diffusing () and continued to accumulate throughout most of the dynamics. Small, yet finite degradation of the expander (α* _{E}*) was compensated by residual production that was maintained also at steady state. Scaling was observed in a wide variety of degradation schemes, including linear degradation (Fig. 1

*F*and

*G*), quadratic degradation (Fig. S1, Table S1), or combinations of the two (Fig. 1

*E*). The numerical analysis further verified the analytically derived link between scaling and robustness: 76% of the systems that scaled were relatively robust to fluctuations in morphogen 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 morphogen with receptors and the possibility of receptor-mediated endocytosis and verified that scaling is maintained (Fig. S2). Finally, extending the analysis to related topologies that similarly rely on a single morphogen and a diffusible molecule that modulates 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 the sense that it does not specify the molecular mechanism by which the expander increases the range of the morphogen gradient. In fact, the expander can be associated with different molecular functions besides its role in scaling. To illustrate this, we analyze a common morphogen-inhibitor circuit and show that it can provide scaling by implementing the expansion-repression feedback.

In a simple morphogen-inhibitor system (Fig. 2*A*), the morphogen signaling is prevented by an extracellular inhibitor *I* that directly binds the morphogen and forms an inert diffusible complex, *MI*. This form of inhibition is quite common, exemplified, e.g., by the inhibition of bone morphogenic protein (BMP)s by Noggin (16). We further assume that the production of 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 degradation. This will be the case, for example, when a morphogen is degraded through receptor-mediated endocytosis. The same mechanism can also be realized when the paired morphogen inhibitor diffuses more rapidly than the free morphogen itself (17, 18). In both cases, the inhibitor facilitates the spread of the morphogen, an effect that was recently described for Wnt ligands and their Frizzled related inhibitors (19). If the inhibitor is also repressed by morphogen signaling, it can promote scaling through the expansion-repression feedback.

To see this, consider the simplest situation in which inhibitor binding simply prevents the morphogen from binding to its receptor *R*, limiting endocytosis. Using numerical simulations, we followed the dynamics of gradient formation. As was described for the general expansion-repression feedback, also here the gradient is initially narrow, allowing inhibitor production in a wide region, but it subsequently widens following the accumulation of the inhibitor. Steady state is achieved when the inhibitor expression is repressed in virtually the entire field and in 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

with coefficients ρ and μ that depend on different parameters and μ being a weak function of *L*. Thus, the concentration profile is largely a function of the relative position *x/L* and not of the absolute 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*–*D* and *SI Text*). Using a similar analysis we confirmed that scaling can also be implemented by an inhibitor that competes with the morphogen over interaction with the receptors, (e.g., Dkk1 and Wnt 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 feedback is the one we proposed recently to explain the scaling of the BMP activation gradient in the early *Xenopus* embryo (17, 21). Although this system appears very different from the general expansion-repression topology, reanalysis of the scaling mechanisms revealed that it is based on the same concept. The BMP activation gradient is established by a molecular network whose main components include two BMP ligands, Bmp4 and Admp, and an extracellular inhibitor of BMP ligands, Chordin. As we show, within the model Admp plays a dual role in being both a morphogen and the expander.

The model postulates that the BMP ligands, and in particular Admp, accumulate at the ventral-most region, irrespectively of where they are produced. This accumulation is due to their effective transport by the inhibitor Chordin, whose localized production at the dorsal pole is the source of asymmetry in this system. Chordin, coming from the dorsal pole, binds the BMP ligands and facilitates their diffusion and disposition at the ventral side. 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 produced dorsally.

The basic gradient can be established by Bmp4 alone. However, as Admp accumulates at the ventral side, it leads to an effective widening of the activation gradient. This way, it plays the role of the expander. *admp* is repressed by morphogen signaling (22), closing the expansion-repression feedback loop by fixing the steady-state signaling level at the dorsal-most pole to the threshold required to repress *admp*. Scaling follows through the expansion-repression feedback described above, leading to a steady-state activation profile,

with *T*_{rep} the BMP signaling threshold for *admp* repression (17).

## Discussion

Scaling of pattern with size is essential for a reliable patterning during development. Yet, it is not explained by most models of morphogen gradients. Here, we show that scaling emerges as a natural property of a feedback topology that we term expansion repression. The key advantage of this model is that scaling stems from the network structure. Therefore, it is relatively insensitive to the specific details and parameters of its molecular implementation, allowing integration into a variety of biological processes.

The expansion-repression feedback topology is based on two diffusible components: a morphogen and an expander. Patterning is determined by the spatial distribution of the morphogen, but the width of this distribution increases with the accumulating levels of the expander. Finally, the expander is relatively stable and its production is repressed by morphogen signaling. The precise implementation may vary: The interaction between the morphogen and the expander can be direct or indirect; the expander could affect morphogen spread by controlling its diffusion, degradation, or other interactions; and morphogen degradation may be linear or nonlinear. The expander itself may have other biological roles, such as a morphogen inhibitor or a morphogen itself, as shown in the examples we provide.

It is interesting to note that the expansion-repression topology is analogous to an integral-feedback controller. An integral-feedback controller calculates the error between the actual and desired outputs and adjusts the controlled variable on the basis of the time integral of this error. In the expansion-repression topology, the controlled variable is the characteristic length of the morphogen profile, λ, and the objective is to adjust λ with the system size *L* (Fig. 3, Fig. S4). The desired gradient is one in which the expander is not produced, and thus the current error corresponds to the region where the expander is produced. This error is integrated through accumulation of the expander as it is produced over time. The accumulated expander increases λ, resulting in an expanded gradient, in which the expander production domain, i.e., the error, is now smaller. The process ends when the error is zero, that is, when the expander is repressed in virtually the entire field (although in actual implementation 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 controllers were implicated in the robust sensory adaptation in bacterial chemotaxis (23), in maintaining fixed levels of ligand-receptor complexes (24), and in yeast homeostasis (25). Our study suggests that this concept might be effectively used also in facilitating the scaling of pattern with size during developmental patterning.

## Experimental Procedures

### Numerical Analysis.

The system was modeled using reaction diffusion equations and was solved numerically for two lengths, *L*_{1} = 100 μm and *L*_{2} = 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 *L*_{1}. Position boundaries at *L*_{1} 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.

The expansion-repression topology was modeled using the reaction-diffusion equations

with a constant flux η* _{M}* of

*M*from

*x*= 0 and reflective boundaries for all other boundary conditions. Parameters

*p*

_{1}∈ {−1, 1} and

*p*

_{2}∈ {0, 1} were used to define the four possible topologies, with expansion–repression realized for (

*p*

_{1},

*p*

_{2}) = (−1, 0). Eight parameters were screened, ranging at least two orders of magnitude, and . 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.

### Screen for Scaling in the Morphogen-Inhibitor System.

By assuming receptors are not saturated and that morphogen–receptor interactions are fast (*SI Text*), we obtain the following equations:

Eqs. **13** were solved numerically with reflective boundary conditions and flux η* _{M}* of

*M*from

*x*= 0. We screened eight parameters, , over at least two orders of magnitude and kept remaining parameters constant (Table S3). Over 63,000 parameter sets were tested. See

*SI Text*, Table S4, for full details of the screens, and Table S5 for parameters of simulations and figures in the text.

## Acknowledgments

We thank Mr. Shlomi Kotler, Dr. Dani Glück, and members of our group for fruitful discussions. D.B.-Z. is supported by the Adams Fellowship Program of the Israel Academy of Sciences and Humanities. This work was supported by the European Research Council, the Israel Science Foundation, Minerva, and the Hellen and Martin Kimmel award for innovative investigations.

## Footnotes

^{1}To whom correspondence should be addressed. E-mail: naama.barkai{at}weizmann.ac.il.Author contributions: D.B.-Z. and N.B. designed research; D.B.-Z. and N.B. performed research; and D.B.-Z. and N.B. wrote the paper.

The authors declare no conflict of interest.

This article is a PNAS Direct Submission.

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

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