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J R Soc Interface. 2009 December 6; 6(41): 1179–1191.
Published online 2009 March 11. doi:  10.1098/rsif.2009.0015
PMCID: PMC2817156

Analysis of dynamic morphogen scale invariance

Abstract

During the development of some tissues, fields of multipotent cells differentiate into distinct cell types in response to the local concentration of a signalling factor called a morphogen. Typically, individual organisms within a population differ in size, but their body plans appear to be scaled versions of a common template. Similarly, closely related species may differ by three or more orders of magnitude in size, yet common structures between species scale to have similar proportions. In standard reaction–diffusion equations, the morphogen range has a length scale that depends on a balance between kinetic and transport processes and not on the length or size of the field of cells being patterned. However, as shown here for a class of morphogen-patterning systems, a number of conditions lead to scale invariance of the morphogen distribution at equilibrium and during the transient approach to equilibrium. Equilibrium scale invariance requires conservation of the total binding site number and total input flux. Dynamic scale invariance additionally requires sufficient binding to slow the diffusion of ligand. The equations derived herein can be extended to the study of other perturbations to gain further insight into the processes regulating the robustness and scaling of morphogen-mediated pattern formation.

Keywords: scale invariance, bicoid, bone morphogenetic protein, systems biology, development, morphogen

1. Introduction

Morphogens are secreted molecules that are distributed in a spatially non-uniform pattern over a field of responding nuclei or cells that read the local concentration of the molecules and react accordingly (Wolpert 1969). The patterning of tissues by morphogens is a dynamic process with kinetics that evolve on multiple time scales, in environments with multiple length scales and with chemical species that span different concentration scales (Lander et al. 2002; Reeves et al. 2006; Umulis et al. 2008). For instance, in some contexts of bone morphogenetic protein (BMP) patterning, the time to pattern a tissue is of the order of hours, receptor equilibration is of the order of minutes, the concentrations of ligands, receptors and other regulators range from 10−1 nM to hundreds of nM and the lengths range from 5 μm for individual cell sizes to hundreds of micrometres for the overall system length (Lander et al. 2002; Umulis et al. 2008). Owing to the multiscale nature of morphogen patterning, it is difficult to intuitively understand the short- and long-term behaviours of the system, and also understand how the processes that occur on different scales balance to control the dynamic evolution of morphogens.

Scale invariance of morphogen-mediated patterning occurs in many different contexts and across different length and time scales. For instance, scale invariance of morphogen patterning occurs during anterior/posterior (A/P) patterning of Drosophila syncytial blastoderm embryos by bicoid (Gregor et al. 2005), during A/P patterning of the Drosophila wing disc by decapentaplegic (Dpp) (Teleman & Cohen 2000), and also during dorsal/ventral (D/V) patterning of Xenopus (Ben-Zvi et al. 2008) and Drosophila embryos by BMPs (D. M. Umulis, M. B. O'Connor & H. G. Othmer, unpublished data). The processes are very different since embryonic A/P patterning relies on the transport of the transcription factor Bcd through the syncytial cytoplasm before binding to nuclei at the embryonic cortex, whereas BMPs are transported around and within a cellularized tissue that makes up the wing disc in Drosophila or the embryo in Xenopus.

Previously, it was suggested that regulation of binding site density could lead to scale invariance of a morphogen distribution at equilibrium (Gregor et al. 2007; Umulis et al. 2008). However, morphogen patterning is a dynamic process in a number of different contexts (Umulis et al. 2006; Bergmann et al. 2007; Kicheva et al. 2007), and scale invariance of the spatial distribution of the morphogen is needed not only at steady state, but also during the transient approach to steady state. In this paper, mathematical analysis and computer simulations are used to investigate the role of morphogen-binding sites in the regulation of dynamic scale invariance. Specifically, the analysis is based on the following biological observations.

  •  Nuclei located at the embryonic periphery of Drosophila embryos (approximately a two-dimensional sheet of nuclei wrapped around a three-dimensional embryo) have been implicated to regulate the dynamics of Bcd transport (Gregor et al. 2005, 2007) along the A/P embryo axis. Additional evidence suggests (i) that nuclei rapidly absorb Bcd protein, and (ii) that the distribution of Bcd is measurably shallower in unfertilized embryos that lack nuclei located at the periphery (Gregor et al. 2007).
  •  Nuclei have been shown to dictate the shape of dpERK signalling (Coppey et al. 2008), which is consistent with nuclear binding and release of morphogen analogous to receptor binding and release. By measuring dpERK signalling during the successive nuclear division cycles of Drosophila development, Coppey et al. showed that (i) the range of dpERK is shortened after each successive nuclear division cycle, which doubles the number of nuclei, and (ii) the dpERK distribution is sensitive to nuclear density, which was demonstrated in shackleton mutant embryos that locally disrupt nuclear density.
  •  The Bcd gradient is scale invariant between related species in the order Diptera, and the number of nuclei in the blastoderm embryo between the species is constant even though the embryo size varies threefold.
  •  The embryos between different species of Diptera differ in size but have similar proportions (suggesting constant geometry/shape; Gregor et al. 2005).
  •  To demonstrate the scale invariance of wing imaginal disc patterning by Dpp, Teleman & Cohen (2000) increased wing disc size by ectopic expression of phosphoinositide 3-kinase family members in the posterior compartment. These mutations affect cell size but do not significantly change the number of cells (suggesting a constant number of binding sites). Furthermore, the ectopic expression in the posterior compartment did not affect the Dpp-secreting cells, which suggests that the total flux of Dpp in the system was unchanged (Teleman & Cohen 2000). Lastly, it was noted by the authors that they did not observe an increase in receptor density on the cell surface in the PI3K mutants.

While the details between morphogen-mediated patterning mechanisms are different, they all rely on three basic processes: (i) production/secretion from a source, (ii) transport by diffusion, convection or transcytosis, and (iii) reaction with receptors, other regulators and/or DNA (Lander et al. 2002; Reeves et al. 2006; Umulis et al. 2006, 2008; Coppey et al. 2007, 2008; Gregor et al. 2007; Ben-Zvi et al. 2008). Consider the simple reaction–diffusion equation that describes morphogen transport by diffusion with chemical reactions

equation image
1.1

Here, u is the chemical morphogen [M/L3] (quantities in [ ] denote units: M, quantity; L, length; T, time); D is the diffusion coefficient [L2/T]; Δ[equivalent][big down triangle, open]2 is the Laplace operator [1/L2]; τ is a characteristic time scale [T] for the chemical reactions; and R is the vector of chemical reactions [M/L3] that regulate the morphogen. Even in a very simple description of a morphogen-patterning process as shown by equation (1.1), it is clear that the solution will depend on the balance of time scales associated with diffusion and chemical reactions, specific boundary conditions and geometry of Ω.

To identify conditions of morphogen scale invariance, suppose that the solution of equation (1.1) for the dynamic morphogen distribution can be represented as a product of independent functions of the form An external file that holds a picture, illustration, etc.
Object name is rsif20090015e251.jpg, as commonly occurs in the solution of linear systems. Many morphogen systems can be represented as the product of these functions, even if one cannot find an exact solution for each function. Here, u is the concentration; x is the position; t is the time; p is the vector of parameters; L is a scalar characteristic length for the system; and Λn, ϕn and ψn represent the amplitude, shape and dynamics, respectively, for each component of the series solution. Specific examples of Λn, ϕn and ψn are given later herein. Here, one can determine the sensitivity of each of Λn, ϕn and ψn to perturbations of p and L to gain insight into the dominant parameters that affect each characteristic of morphogen patterning. With the product solution, it is clear that to ensure dynamic scale invariance, three conditions must be met: (i) each amplitude (Λn) is independent of L, (ii) each shape function (ϕn) stretches appropriately in the x-direction with respect to changes in the length L, and (iii) the time-dependent functions (ψn) are independent of L.

Herein, the specific choices for concentration, time and length scales were varied to analyse three stages of dynamic morphogen-mediated patterning: (i) boundary-layer dynamics near the source for very early times, (ii) intermediate receptor–ligand dynamics of binding and release, and (iii) long-time dynamics of morphogen patterning at the tissue scale. Each regime is nonlinear; however, for the boundary-layer and long-time regimes, the system is approximately linear under appropriate assumptions, which allows for more detailed analysis and the identification of conditions for dynamic scale invariance. To explore the validity of the approximate analytical solution, the results were compared with finite-element numerical solutions of the full system of partial differential equations.

By using the product solution of the linear approximations for a basic model of morphogen production, transport and reaction, a number of experimentally testable conditions for equilibrium and dynamic scale invariance were identified for a class of morphogen-mediated patterning mechanisms. The main result is summarized in box 1 and many additional details and explanations are provided throughout the text to account for numerous additional insights into the dynamics of morphogen-mediated pattern formation. The conservation of the total number of binding sites, total input flux of morphogen molecules and relatively high receptor density can lead to automatic scale invariance of the morphogen at steady state and during the transient approach to steady state.

Box 1.

Summary of main result for dynamic morphogen scale invariance

A remarkable feature of organism development is the ability of patterning mechanisms to reliably produce consistent proportions between individuals that vary greatly in size. The preservation of proportion or ‘scale invariance’ is manifest in numerous different biological contexts and has been directly demonstrated in one of the earliest patterning events in Drosophila: A/P patterning by the morphogen bicoid (see main text for references). Most morphogens (including bicoid) develop a spatially non-uniform distribution in the course of a few hours; however, there is increasing evidence that many morphogen distributions never reach steady state and cells continuously respond to the dynamic evolution of the morphogen. Dynamic scale invariance means that the regulation of gene expression occurs at the same relative position and at the same time between tissues/organisms that differ in size. For example, dynamic scale invariance would ensure that the boundary of gene expression for the Bcd target hunchback would occur at 50 per cent embryo length (L) after the same duration of development regardless of the individual organism sizes.

Preservation of proportion is not a property of reaction–diffusion systems in general, but the analysis herein suggests that it may be a property of numerous morphogen-patterning mechanisms that involve the diffusion (with rate D) of a morphogen (C) that binds to sites (R) with rates k1 and k−1 to form An external file that holds a picture, illustration, etc.
Object name is rsif20090015e10.jpg, which undergoes endocytosis/decay with rate ke as shown in the following equations:

equation image

equation image

equation image

and

equation image

Here An external file that holds a picture, illustration, etc.
Object name is rsif20090015e15.jpg is time; x is position; j is the input flux at the source; and RT is the total binding site density. Substituting scaled position ξ=x/L into equation (B 1), and solving for the long-time behaviour of the system, leads to ((2.33) in text)

equation image

For convenience, conditions for steady-state scale invariance are identified first and the results are extended to analysis of the dynamic problem. At steady state, the solution of equation (B 5) is approximately exponential (for the range exhibited by most morphogens), which gives

equation image

and

equation image

The steady-state shape of the distribution will be scale invariant if λ is independent of the length L. Since D, ke and Km are constants, this requires RT[proportional, variant]L−2, which occurs automatically in patterning contexts that: (i) conserve the total number of binding sites, receptors or nuclei (constant total number NT), and (ii) have similar geometries so that RT=σNT/L2 where σ is a geometric proportionality constant. The amplitude C0 in equation (B 6) additionally requires that: (iii) the total morphogen production at the boundary (molecules/time) is constant ((3.8) in text).

Substituting RT=σNT/L2 into equation (B 5) gives

equation image

By inspection of equation (B 8), dynamic morphogen scale invariance additionally requires: (iv) that σNT/L2[dbl greater-than sign]Km in which case f(L)→1. This means that the binding site density RT is greater than the Km value (approximate Km values range from 1 to 1000 nM). How does binding lead to dynamic scale invariance? Binding of morphogens to immobile sites RT dynamically slows diffusion. Lowering the binding site density by increasing the length L increases the effective diffusion by an amount that exactly offsets the longer time it takes for diffusion in a larger domain. The boost in effective diffusion leads to a system that is scale invariant dynamically and at steady state.

2. Results

While the specific biological mechanisms of morphogen-mediated patterning are very different in different contexts (figure 1), the underlying physical processes that establish a non-uniform morphogen distribution are quite similar. Consider three different morphogen processes that occur in Drosophila: (i) diffusion and nuclear import/export of a transcription factor in a syncytium as occurs for Bcd and dpERK signalling (figure 1a; Gregor et al. 2005, 2007; Coppey et al. 2008), (ii) diffusion in a thin fluid layer over a field of cells with receptor binding and endocytosis as occurs as a component of BMP patterning of the dorsal surface (figure 1b; Mizutani et al. 2005; Wang & Ferguson 2005; Umulis et al. 2006), and (iii) diffusion and receptor binding around columnar epithelial cells as occurs in the wing imaginal disc (figure 1c; Reeves et al. 2006). In each of the examples, the mechanism of morphogen removal occurs in a region that can be approximated as a two-dimensional surface: (i) binding to DNA in nuclei located at the periphery of a syncytial embryo, (ii) receptor binding and endocytosis on a cellular membrane, and (iii) binding to receptors located along the basolateral region of columnar cells. Under appropriate geometric simplifications and assumptions (details given elsewhere), each morphogen process can be approximated as a thin sheet that extends in the x-direction. Assume that the surface reactions involve only binding to a receptor and decay of the receptor–ligand complex, and to simplify the analysis, suppose that the total amount of receptors on the surface is constant in time. Here, the term receptor is used loosely to more generally mean binding sites that can represent regions of specific and non-specific binding of a transcription factor to DNA, the binding to and subsequent transport through nuclear pore complexes, or binding to receptors located on a cellular membrane. Let Lx, Ly and Lz be the lengths in the x-, y- and z- directions, respectively, and let C be the concentration of a morphogen in the fluid and RS the concentration of receptor on the surface z=0. Suppose there is a fixed influx of C on the boundary x=0, and zero flux on the remaining faces except z=0, then the governing equations can be written as follows:

equation image
2.1

equation image
2.2

equation image
2.3

equation image
2.4

and

equation image
2.5
Figure 1

Example geometries of morphogen-patterning pathways. (a) Diffusion and nuclear import. (b) Extracellular diffusion with receptor binding and internalization over a field of cells. (c) Diffusion through an extracellular matrix with binding along the basolateral ...

Here, C is the bulk concentration of the morphogen [M/L3]; RS and An external file that holds a picture, illustration, etc.
Object name is rsif20090015e257.jpg are the free and bound binding sites on the surface [M/L2]; An external file that holds a picture, illustration, etc.
Object name is rsif20090015e258.jpg is the time; k1 and k−1 are the forward and reverse binding rates of C to R with units [(M/L3.T)−1] and [T−1], respectively; and ke is the decay/removal rate with units [T−1]. Assuming that the initial condition of free receptors is at equilibrium and the initial levels of C are zero, then RST=ϕR/ke for all time. As a result, the total amount of receptor is constant at every point on the surface z=0, which leads to the conservation condition An external file that holds a picture, illustration, etc.
Object name is rsif20090015e259.jpg (where RST is a constant total level of receptors).

Furthermore, if An external file that holds a picture, illustration, etc.
Object name is rsif20090015e20.jpg, then equations (2.2)–(2.4) can be averaged over Lz, which leads to a volumetric receptor density An external file that holds a picture, illustration, etc.
Object name is rsif20090015e21.jpg and R, where An external file that holds a picture, illustration, etc.
Object name is rsif20090015e22.jpg, An external file that holds a picture, illustration, etc.
Object name is rsif20090015e23.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20090015e24.jpg. In this scenario, the equations for receptor binding on one surface in a thin gap are completely equivalent to a system that has a uniform volumetric binding site density, such as binding sites in the extracellular matrix or import/export in nuclei. In view of the boundary conditions, the solution must be constant in the y- direction since the initial conditions are constant in the Ly-direction. This gives the simplified equations

equation image
2.6

equation image
2.7

equation image
2.8

and

equation image
2.9

To understand the solution over different length and time scales, equations (2.6)–(2.9) are non-dimensionalized by the general scalings α, β, L and τ, which non-dimensionalize the concentration c=C/α, concentration An external file that holds a picture, illustration, etc.
Object name is rsif20090015e29.jpg, position ξ=x/L and time An external file that holds a picture, illustration, etc.
Object name is rsif20090015e30.jpg (i=1, 2, 3 for the early-, intermediate- and long-time behaviour, respectively). Equations (2.6)–(2.9) reduce to

equation image
2.10

equation image
2.11

and

equation image
2.12

2.1. Time-scale analysis

The dynamics of morphogen reaction and transport in equations (2.10)–(2.12) occur on multiple time scales. For instance, one would expect that the time scale for receptor equilibration is much faster than that for transport by diffusion and other processes. To gain insight into the balance of the processes that govern morphogen dynamics on different time scales, one must make appropriate choices for τ, α, β and L in an effort to find a solution that captures the fast, intermediate and slow dynamics. The system has many possible lengths, concentration and time scales, and different combinations can be applied to study the behaviour in different regimes. It is not clear a priori what the appropriate choices for τ, α and β are for the different regimes, so that many possible parameter combinations and a brief description of the parameter group are listed in table 1.

Table 1
Time and concentration scales. ((bl) denotes boundary layer. Magnitude is a qualitative estimate of the relative rate, concentration or length, and cross-comparisons should only be made between those in the same regime: either boundary layer (bl) or for ...

The concentration scale Km is equivalent to the Michaelis–Menten constant and is defined by Km=(k−1+ke)/k1. The subscript δ denotes that the time scale is defined for the boundary layer δ.

If the concentration of ligand and receptor is initially zero everywhere, the early dynamics should be governed by processes a short distance (lx) from the source at the boundary. Specifically, the early dynamics will be governed by the input flux (τ), short-range diffusion (τDδ) and receptor binding (τB). A short time after that, other processes, such as receptor equilibration (τR), endocytosis (τE) and long-range diffusion (τD), contribute to the dynamics over the full length Lx. In the following sections, a number of different time-scale hierarchies are selected to analyse the boundary-layer, intermediate- and long-time regimes, which are applicable to numerous biological pathways.

2.2. Part I: boundary-layer dynamics

It is difficult to select a characteristic length for the boundary layer (lx) a priori, so it is assumed to be sufficiently small so that the boundary-layer dynamics are faster than all the other processes. Choosing L=lx for the length scale, τ=Cδlx/j, so that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e34.jpg for the time scale, α=Cδ and β=RTCδ/(Km+Cδ) for the concentration scales and substituting them into equations (2.10)–(2.12) leads to the following equations:

equation image
2.13

equation image
2.14

and

equation image
2.15

The very early dynamics of ligand (c) and bound receptor An external file that holds a picture, illustration, etc.
Object name is rsif20090015e38.jpg can be easily understood by the analysis of the relative time scales in equations (2.13)–(2.15). First, assume that in this scale, An external file that holds a picture, illustration, etc.
Object name is rsif20090015e39.jpg. All the kinetic terms in equations (2.13) and (2.14) disappear and the very early dynamics are determined solely by the flux at the boundary (2.15) and diffusion away from the source (2.13). If the flux time scale is much shorter than the diffusion time scale An external file that holds a picture, illustration, etc.
Object name is rsif20090015e40.jpg, the ligand accumulates at the source without diffusing away. If the diffusion time scale in the boundary is shorter than the input flux time scale An external file that holds a picture, illustration, etc.
Object name is rsif20090015e41.jpg, then the ligand immediately diffuses away from the source, the flux term is essentially zero and nothing appears to happen. When the flux and diffusion time scales are of the same order of magnitude (τDδτ), the boundary-layer equations give rise to a diffusion equation with a constant source at x=0 and diffusion coefficient τ/τDδ. These three scenarios represent the very short-time behaviour at the source before any appreciable receptor binding.

Shortly after production at the boundary begins, receptor binding (τB) occurs close to the source. Early during patterning, there will be an excess of binding sites relative to ligand levels (i.e. An external file that holds a picture, illustration, etc.
Object name is rsif20090015e42.jpg), which means that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e43.jpg. Suppose that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e44.jpg, defining ϵδ=τB/τRδ and noting that τRδ<τCδ,τU (by definition), equations (2.13)–(2.15) lead to the following equations, with a source at ξδ=0:

equation image
2.16

equation image
2.17

and

equation image
2.18

In the t1 time scale, this simplifies to a linear reaction–diffusion equation on a semi-infinite interval in the limits (lim lx→0, lim ϵδ→0)

equation image
2.19

and

equation image
2.20

Here, the diffusion coefficient is An external file that holds a picture, illustration, etc.
Object name is rsif20090015e50.jpg, the decay term is An external file that holds a picture, illustration, etc.
Object name is rsif20090015e51.jpg and the flux is An external file that holds a picture, illustration, etc.
Object name is rsif20090015e52.jpg. Solving equation (2.19) with the boundary condition (2.20) gives

equation image
2.21

Solutions of the very short-time dynamics given by equation (2.21) are plotted along with the finite-element solution of equations (2.10)–(2.12) in figure 2a.

Figure 2
(a) Solutions from the finite-element method (FEM) simulation of equations (2.6)–(2.9) (solid line), boundary-layer equation (2.21) (dashed line) and equation (2.39) (dotted line). (b) Plot of the L norm for the boundary-layer (solid ...

2.3. Part II: intermediate- and long-time dynamics

2.3.1. Intermediate dynamics

Following the short-time dynamics in the boundary layer, other processes determine the intermediate dynamics. Two important parameter groups that relate the different time scales are ϵ1=τR/τE and ϵ2=τB/τD. The first (ϵ1) relates the ratio of the time scale of receptor equilibration (τR) to the time scale for the total loss of C by endocytosis/decay of the An external file that holds a picture, illustration, etc.
Object name is rsif20090015e54.jpg complex (τE), and ϵ2 relates the time scale that C is absorbed by binding sites (τB) to the time scale for diffusion (τD). As will be shown later, for a reasonable morphogen profile, An external file that holds a picture, illustration, etc.
Object name is rsif20090015e55.jpg and more evidence for this is given later herein. Substituting α=C0, β=RTC0(Km+C0)−1 and τ=τR so that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e56.jpg, ϵ1 into equations the following and rearranging gives the following equations for the intermediate dynamics:

equation image
2.22

equation image
2.23

equation image
2.24

equation image
2.25

and

equation image
2.26

It appears that on the intermediate time scale, equation (2.22) is governed principally by the receptor equilibration kinetics; however, it is not clear at this point whether the ratio of time scales for receptor equilibration to receptor binding (τR/τB) is large or small. If small, the receptor-binding kinetics do not play a major role in the dynamics of c, even on the intermediate time scale. As expected, for equation (2.23), the receptor-binding kinetics dominate the evolution of An external file that holds a picture, illustration, etc.
Object name is rsif20090015e62.jpg. Equations (2.22)–(2.24) for the intermediate time scale are nonlinear and provide limited additional information about the dynamic evolution of c. However, the intermediate scale leads naturally to the slower time scale that corresponds to the long-time and long-range morphogen patterning.

2.3.2. Long-time dynamics

After the initial boundary-layer formation and initial receptor-binding kinetics when An external file that holds a picture, illustration, etc.
Object name is rsif20090015e63.jpg or An external file that holds a picture, illustration, etc.
Object name is rsif20090015e64.jpg, the slower processes of long-range diffusion and ligand endocytosis/decay dictate the dynamics of the morphogen distribution.

Defining t3=ϵ2t2 and substituting this into equations (2.22) and (2.23) rescales the equations for An external file that holds a picture, illustration, etc.
Object name is rsif20090015e65.jpg and c for large times,

equation image
2.27

and

equation image
2.28

Adding equations (2.27) and (2.28) gives the following equation for the total amount of morphogen (free+receptor bound):

equation image
2.29

Taking the limit ϵ2→0 of equations (2.27)–(2.29) with the condition that ϵ1/ϵ2→const. leads to an algebraic equation for An external file that holds a picture, illustration, etc.
Object name is rsif20090015e69.jpg and the long-time behaviour for the total amount of ligand. Solving for An external file that holds a picture, illustration, etc.
Object name is rsif20090015e70.jpg, calculating the derivative of An external file that holds a picture, illustration, etc.
Object name is rsif20090015e71.jpg with respect to t3 and substituting the solution into equation (2.29) gives

equation image
2.30

equation image
2.31

and

equation image
2.32

2.4. Solution of linear approximation for long-time dynamics

The resulting equation for the long-time behaviour of unbound c is a nonlinear PDE with no general solution. However, the equation is linear if the level of ligand C0 is small, so that the total amount of bound ligand An external file that holds a picture, illustration, etc.
Object name is rsif20090015e75.jpg is far from saturation. Specifically, this requires An external file that holds a picture, illustration, etc.
Object name is rsif20090015e76.jpg, so that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e77.jpg, and equation (2.31) becomes a linear non-homogeneous PDE,

equation image
2.33

equation image
2.34

equation image
2.35

and

equation image
2.36

By introducing v(ξ,t3)=c(ξ,t3)−cE(ξ), where cE(ξ) is the equilibrium solution, the above non-homogeneous PDE for c(ξ,t3) is converted into a simpler homogeneous PDE in the displacement variable v(ξ,t3). The equilibrium solution for equations (2.33)–(2.36) is

equation image
2.37

and

equation image
2.38

Also, the exact transient solution is

equation image
2.39

equation image
2.40

and

equation image
2.41

The solution for the linear approximation with n=100 is shown along with a finite-element solution of equations (2.6)–(2.9) for very early times in figure 2a,b and the equilibrium solution (2.37) is shown in figure 3a for An external file that holds a picture, illustration, etc.
Object name is rsif20090015e87.jpg. As shown in figure 2a,b, the long-time behaviour better approximates the finite-element method (FEM) solution after the first 100 s (parameters are listed in the caption to figure figure3).3). Figure 3b shows the transient evolution of c(ξ,t3) at ξ=0, 0.2, 0.4 for the linear approximation versus the FEM solution for long times. Figure 3a,b demonstrates that the linear approximation captures the equilibrium and dynamic behaviour of the full nonlinear system when An external file that holds a picture, illustration, etc.
Object name is rsif20090015e88.jpg. Note that C0 is determined by setting ξ=0, cE(0)=1 in equation (2.37). As expected, the linear approximation breaks down as C0Km both at equilibrium (figure 3c) and dynamically (figure 3d).

Figure 3
(a) Example equilibrium solution for the linear approximation (solid line, cE(ξ)) and finite-element solution of the full problem (open circles, cFEM(ξ)). (b) Comparison of dynamics in the linear approximation (solid line, Clinear) versus ...

3. Conditions for scale invariance

The solutions for the early- and long-time dynamics of equations (2.6)–(2.9) provide a framework to investigate plausible mechanisms of biological scale invariance. To make the analysis more straightforward, conditions for scale invariance of the equilibrium solution are identified first, followed by a brief analysis of the boundary layer and a more extensive analysis of the long-time behaviour of the system. Since the time scale of many morphogen-mediated pathways is of the order of tens to hundreds of minutes, the long-time and equilibrium behaviours are most relevant for real biological morphogen pathways.

3.1. Conditions for scale invariance of the equilibrium solution

The equilibrium solution (2.37) can be represented as a product of an amplitude and shape function, cE(ξ,p,Lx)=Λ(p,Lx).ϕ(ξ,p,Lx).

Equation (2.37) can be rewritten as

equation image
3.1

Now substituting the parameters An external file that holds a picture, illustration, etc.
Object name is rsif20090015e90.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20090015e91.jpg into An external file that holds a picture, illustration, etc.
Object name is rsif20090015e92.jpg gives

equation image
3.2

Earlier, it was speculated that the ratio An external file that holds a picture, illustration, etc.
Object name is rsif20090015e94.jpg for a reasonable morphogen profile. Since An external file that holds a picture, illustration, etc.
Object name is rsif20090015e95.jpg in this context, it follows that θ−1≈1 and the decay length of the morphogen distribution at equilibrium is determined solely by μ≈(ϵ1/ϵ2)1/2. If An external file that holds a picture, illustration, etc.
Object name is rsif20090015e96.jpg, the equilibrium solution decays rapidly from the source, and the range of the morphogen would be too short to effectively pattern a field of cells in a concentration-dependent manner. If An external file that holds a picture, illustration, etc.
Object name is rsif20090015e97.jpg, the shape of the morphogen distribution will be essentially flat over the field of cells and thus neighbouring cells would adopt identical fates. For An external file that holds a picture, illustration, etc.
Object name is rsif20090015e98.jpg, it constrains the maximum of the ratio of time scales to An external file that holds a picture, illustration, etc.
Object name is rsif20090015e99.jpg. What does An external file that holds a picture, illustration, etc.
Object name is rsif20090015e100.jpg mean? Substituting the parameters back into μ2 gives

equation image
3.3

Equation (3.3) is a ratio of a measure of the diffusion time against a grouping of binding, equilibration and removal kinetic time scales. In essence, An external file that holds a picture, illustration, etc.
Object name is rsif20090015e102.jpg means that the kinetic and diffusion time scales are of the same order of magnitude and since the distribution of morphogen is determined by a balance between the time scales, it leads to an overall length scale of morphogen appropriate for having sufficient range and sufficient slope for patterning.

The shape ϕ will be scale invariant if and only if μ is independent of Lx. It is readily apparent that μ is independent of the system length (Lx) if An external file that holds a picture, illustration, etc.
Object name is rsif20090015e103.jpg, since D, ke and Km are constant parameters. Substituting RT=NT/LxLyLz into equation (3.3) makes the requirements for scale invariance more transparent in the following equation:

equation image
3.4

Given equation (3.4), and since ke, D and Km are constant, μ2 will be independent of Lx and the equilibrium shape function (ϕ) will be scale invariant if An external file that holds a picture, illustration, etc.
Object name is rsif20090015e105.jpg is also constant. A number of scenarios can hypothetically ensure a constant ratio and the two most biologically tenable options are the following: (i) the total number of binding sites (NT) scales in proportion to An external file that holds a picture, illustration, etc.
Object name is rsif20090015e106.jpg with fixed Ly, Lz, or (ii) NT remains constant and the product of LxLyLz is proportional to An external file that holds a picture, illustration, etc.
Object name is rsif20090015e107.jpg. While the first possibility cannot be ruled out since there may be mechanisms that carefully regulate NT in proportion to An external file that holds a picture, illustration, etc.
Object name is rsif20090015e108.jpg, the indirect evidence (detailed in §1) suggests that the second possibility of constant NT and appropriate scaling of LxLyLz may contribute to scale invariance for a number of morphogen-patterning mechanisms.

These observations suggest that (i) NT may be constant between organisms within and between species at the same developmental stage (Teleman & Cohen 2000; Gregor et al. 2005), (ii) different-sized organisms have similar proportions so that Ly=ρLx, where ρ is the proportionality constant (Gregor et al. 2005), and (iii) the total input flux may be constant (specifically for wing disc patterning by Dpp; Teleman & Cohen 2000). Lastly, it is assumed that the height of the gap for diffusion (Lz) is constant, which is expected since the distance between cells, the thickness of the cortical layer or the thickness of the perivitelline space are intrinsic to the cell biology that develops those regions and not the scale of the system. With these conditions, μ2 can be rewritten as

equation image
3.5

where

equation image
3.6

In other words, the shape function ϕ is independent of Lx under conditions of (i) constant total number of binding sites, (ii) constant gap height Lz, and (iii) uniform dilations in tissue size.

If μ is independent of Lx, the amplitude Λ will be invariant if J scales appropriately for changes in the system length. However, J=τD/τj=jLx/(DC0) and amplitude invariance can only be achieved if An external file that holds a picture, illustration, etc.
Object name is rsif20090015e111.jpg.

One plausible mechanism that ensures the proper scaling of j occurs when the total input number flux Φ is constant instead of the flux density j. On a two-dimensional sheet as analysed here, this leads to

equation image
3.7

that is,

equation image
3.8

Substituting equation (3.8) into J and noting that Ly/Lx=ρ, a constant, it follows that J=Φ(ρLzDC0)−1 and so Λ is independent of Lx. As expected, when the conditions for equilibrium scale invariance are met, the distribution of morphogen shifts in proportion to the length of the system, as shown in figure 4a in x-coordinates. When remapped onto ξ[set membership][0,1], the profiles are superimposed over each other as shown in figure 4b.

Figure 4
(a) Equilibrium solution for the linear approximation versus position for systems with characteristic lengths (Lx) of 10 μm (solid line), 100 μm (open circles) and 1000 μm (dotted line). (b) Same as in ( ...

3.2. Analysis of boundary-layer dynamics

For the very early dynamics, the characteristic time scale is related to the flux of molecules into the system at the boundary (τ=Cδlx/j). The An external file that holds a picture, illustration, etc.
Object name is rsif20090015e114.jpg time scale inherently depends on the choice of the characteristic length used to study the boundary-layer dynamics (lx). To determine whether the very early dynamics are scale invariant, the solution is remapped to actual time (An external file that holds a picture, illustration, etc.
Object name is rsif20090015e115.jpg) and scaled position ξ[set membership][0,1]. Equation (2.21) then becomes

equation image
3.9

In equation (3.9), it is clear that the very early dynamics in the boundary layer are not scale invariant since the time (An external file that holds a picture, illustration, etc.
Object name is rsif20090015e117.jpg) is scaled by the overall length Lx.

3.3. Conditions for long-time dynamic scale invariance

In addition to the steady-state morphogen distribution, the morphogen dynamics are also important in a number of biological contexts. Equations (2.39)–(2.41) were further analysed to determine (i) the rate of approach to the equilibrium solution, and (ii) whether it is possible to achieve scale invariance dynamically, i.e. the morphogen distribution in different-sized domains is proportional for all times during the transient evolution of the distribution.

Since c(ξ,t3)=cE(ξ)+v(ξ,t3) and cE(ξ) is scale invariant by the conditions outlined in §3.1, it follows that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e118.jpg will be scale invariant if each of the Λn, ϕn and ψn is independent of Lx.

Here, the solution is represented by a sum of terms as shown in the following by rearranging equations (2.39)–(2.41):

equation image
3.10

equation image
3.11

equation image
3.12

equation image
3.13

equation image
3.14

and

equation image
3.15

The conditions for scale invariance identified in §3.1 make the analysis of dynamic scale invariance straightforward. First, since An external file that holds a picture, illustration, etc.
Object name is rsif20090015e125.jpg, there is a direct relationship between actual time (An external file that holds a picture, illustration, etc.
Object name is rsif20090015e126.jpg) and scaled time An external file that holds a picture, illustration, etc.
Object name is rsif20090015e127.jpg, which means that if t3 is independent of Lx, then the actual time (An external file that holds a picture, illustration, etc.
Object name is rsif20090015e128.jpg) is also independent of Lx.

It is immediately apparent that the shape functions (ϕn) are scale invariant; however, it is not clear whether the Λn and ψn are independent of Lx. Recalling that θ≈1 and that J and μ are independent of Lx (§3.1), and substituting An external file that holds a picture, illustration, etc.
Object name is rsif20090015e129.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20090015e130.jpg into equations (3.11) and (3.12) for Λn gives

equation image
3.16

which is automatically independent of Lx.

Next, substituting An external file that holds a picture, illustration, etc.
Object name is rsif20090015e132.jpg and An external file that holds a picture, illustration, etc.
Object name is rsif20090015e133.jpg into equation (3.14) (ψn) gives

equation image
3.17

and recalling

equation image
3.18

the conditions for dynamic scale invariance are more transparent. Whether or not the morphogen evolution is scale invariant depends on the ratio of RT to Km. Substituting RT/Km into equation (3.17) gives

equation image
3.19

If receptor density relative to the Km value is small so that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e137.jpg, then equation (3.19) becomes

equation image
3.20

and the time it takes to reach equilibrium increases in proportion to the length of the system squared. In the other limit An external file that holds a picture, illustration, etc.
Object name is rsif20090015e139.jpg

equation image
3.21

and the dynamics of the morphogen distribution are completely independent of the length of the system.

Figure 4c (left axis) shows how the value of the leading term associated with each exponential depends on the ratio of RT/Km, which provides an indication of the time it takes to reach equilibrium. Figure 4c (right axis) shows the time it takes for the FEM solution to reach 90 per cent of the equilibrium values for a given set of parameters. As RT/Km increases, the exponential term becomes more negative, which means the solution approaches equilibrium more rapidly until levelling off at a constant value, which is consistent with the behaviour of the FEM solution of the full model. However, since An external file that holds a picture, illustration, etc.
Object name is rsif20090015e141.jpg, for fixed An external file that holds a picture, illustration, etc.
Object name is rsif20090015e142.jpg, this places limits on the maximum size for which dynamic scale invariance occurs automatically. Specifically, dynamic scale invariance requires that An external file that holds a picture, illustration, etc.
Object name is rsif20090015e143.jpg (figure 4d).

What does this mean for real biological systems? The ratio τR/τB in equation (3.17) represents the relationship between time scales of binding site equilibration to the time scale of ligand binding and removal from the freely diffusing pool. While τR/τB emerged as an important relationship between time scales, it can also be interpreted as a ratio of the concentration scales RT/Km. In this context, the ratio of RT to Km relates the total binding site density to the concentration of ligand necessary to achieve 50 per cent binding site occupancy. A small RT/Km suggests either a very low concentration of receptors or very weak binding (either a slow on-rate or a fast off-rate). With relatively few binding sites or weak binding, diffusion is largely unhindered and changing the binding site density has little effect on the dynamics of transport, which leads to an increase in the time it takes to approach steady state. On the other hand, a large RT/Km suggests either a high receptor concentration or tight binding. The large number of binding sites transiently slows the diffusion process and decreasing the density of binding sites leads to a faster effective diffusion. For a range of parameter values, if the binding site density decreases An external file that holds a picture, illustration, etc.
Object name is rsif20090015e144.jpg as Lx increases, the effective diffusion constant increases in proportion to An external file that holds a picture, illustration, etc.
Object name is rsif20090015e145.jpg. The boost in effective diffusion leads to a system that is scale invariant both dynamically and at equilibrium.

4. Discussion

In this paper, multiple time scales are used to better understand the dynamics and identify conditions for scale invariance of morphogen-mediated patterning. The principal conditions that lead to scale invariance of a morphogen distribution are

  •  conservation of the total number of binding sites for the morphogen;
  •  conservation of the total number of morphogen molecules secreted in a given length of time at the boundary;
  •  for scale invariance of morphogen gradient interpretation, cells (or nuclei) must respond to the total occupancy of receptors and not to the surface density of occupied receptors (appendix A); and
  •  for dynamic scale invariance, there needs to be an excess of binding sites on the surface to slow the effective diffusion.

The properties for dynamic scale invariance are based on evidence from a number of different pathways and the analysis suggests that the simple mechanism of binding and release may contribute to scale invariance. While supported by observations, a number of theoretical and experimental studies will further delineate mechanisms of scale invariance in different contexts. The model presented here depends on a number of assumptions, and future work should be extended to include the following: (i) non-uniform binding site density as occurs in wing imaginal disc patterning by Dpp (Lecuit & Cohen 1998; Teleman & Cohen 2000; Serpe et al. 2008), (ii) binding site density that changes in time (Umulis et al. 2006; Gregor et al. 2007; Coppey et al. 2008; Serpe et al. 2008), (iii) feedback from receptor signalling (Fujise et al. 2003; Wang & Ferguson 2005; Umulis et al. 2006), (iv) nonlinear extracellular regulation as occurs in dorsoventral patterning of Drosophila and Xenopus embryos (Mizutani et al. 2005; Shimmi et al. 2005; O'Connor et al. 2006; Umulis et al. 2006, 2008; Ben-Zvi et al. 2008), (v) more detailed analysis of nuclear import/export kinetics (Coppey et al. 2007, 2008; Gregor et al. 2007), (vi) transcytosis through columnar epithelial cells (Kruse et al. 2004; Bollenbach et al. 2005; Kicheva et al. 2007), and (vii) more realistic geometries of the underlying tissues. The analysis presented herein serves as a starting point for future analysis, but it also serves as the common theme that links the diverse morphogen pathways.

The conditions for scale invariance are general enough that they can be tested in a number of different contexts. For instance, one could measure the distribution of pMad signalling and extracellular Dpp–green fluorescent protein (Kicheva et al. 2007) in the posterior compartment of wing discs that ectopically express cyclin-dependent kinases to increase or decrease the number of cells. Furthermore, additional experiments such as changing the binding site density (receptors, Cv-2, etc.) during D/V embryonic patterning by BMPs, disrupting the nuclear density during A/P patterning by Bcd, measuring the time it takes for A/P pattern formation by Bcd in different-sized embryos or altering the geometry of a developing tissue would all provide additional insight into the mechanisms of biological scale invariance.

In summary, morphogens mediated by diffusion, binding and endocytosis/degradation can yield scale invariance automatically if the number of binding sites is conserved for changes in system size. As mentioned in Gregor et al. (2007), this may lead to scaling of bicoid. For systems with excess binding sites, scale invariance may occur dynamically as well as at equilibrium.

Acknowledgements

I wish to thank Hans Othmer, Michael O'Connor and the reviewers for their helpful comments on this paper. Supported in part by NIH (GM29123) to H.O.

Appendix A

A.1.  Ligand decay

In addition to endocytosis or decay of morphogen from binding sites, one could envision a related but mechanistically different process where the morphogen has an intrinsic lifetime that may or may not be affected by binding. For instance, if the morphogen is a transcription factor that is diffusing over a field of nuclei, the bound and unbound states do not undergo endocytosis as in receptor ligand systems and one would expect similar decay times for the bound and unbound morphogens. In essence, this adds a decay term to the equation for unbound ligand in equation (2.1). Using analogous analysis to that developed herein, the extra kinetic step introduces another ratio of time scales ϵ3=kc/(k1(Km+C0)). This gives

equation image
A1

If the decay rates in the bound and unbound states are equal, then ϵ1=ϵ3. Scale invariance at equilibrium is assured if An external file that holds a picture, illustration, etc.
Object name is rsif20090015e147.jpg, in which case the majority of morphogens is in the bound state and the contribution of unbound decay is negligible.

A.2.  Scale invariance of receptor–ligand systems

The above analysis demonstrates how scale invariance can be achieved if the binding site density scales as L2; however, it is more important that the interpretation of the signal scales properly as well. In the case of nuclei and transcription factor systems such as A/P patterning in Drosophila, the local concentration at discrete sites is constant for changes in system size so that the scale invariance at the level of occupied binding sites is automatic. For ligand–receptor systems, the extracellular gradient scales by reducing the concentration of binding sites. Since receptor concentrations decrease for increases in size, the surface density of receptor-bound ligand also decreases. Thus, the concentration of ligand-bound receptors is not scale invariant and cells must interpret the signal by another mechanism. One possibility for morphogen interpretation is to integrate the signal over the entire surface of each cell into a downstream response. In essence, cells count the total number of ligand-occupied receptors to initiate a downstream response as has been shown for activin signalling (Dyson & Gurdon 1998). Suppose cells are spaced in a square lattice with dimensions lx=aLx and ly=bLy. Mathematically, for a square surface arranged in a regular array, the total signal is represented by

equation image
A2

Here, An external file that holds a picture, illustration, etc.
Object name is rsif20090015e149.jpg is the total number of occupied receptors on the surface of the cell specified by position n, m, at time t3. The constants a and b correspond to the fractional length of a cell in scaled coordinates. To show that the integral of receptors on the surface scales, apply the first mean-value theorem for integrals, which states that there exists some c[set membership][a,b] such that the following equality is true:

equation image
A3

This leads to the following expression for the number of ligand-occupied receptors per cell:

equation image
A4

Then, the mean-value theorem for integration of a single cell gives

equation image
A5

and

equation image
A6

Since cE(ξ) is independent of Lx, it follows that ξ* is independent of Lx only if v(ξ,t3) is independent of Lx by the conditions shown herein.

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