Research articlesSteady-state invariant genetics: probing the role of morphogen gradient dynamics in developmental patterning
Abstract
Morphogen-mediated patterning is the predominant mechanism by which positional information is established during animal development. In the classical view, the interpretation of positional signals depends on the equilibrium distribution of a morphogen, regardless of the dynamics of gradient formation. The problem of whether or not morphogen dynamics contribute to developmental patterning has not been explored in detail, partly because genetic experiments, which selectively affect signalling dynamics while maintaining unchanged the steady-state morphogen profile, are difficult to design and interpret. Here, I present a modelling-based approach to identify genetic mutations in developmental patterning that may affect the transient, but leave invariant the steady-state signalling gradient. As a case study, this approach is used to explore the dynamic properties of Hedgehog (Hh) signalling in the developing wing of the fruitfly, Drosophila melanogaster. This analysis provides insights into how different properties of the Hh gradient dynamics, such as the duration of exposure to the signal or the maximum width of the transient gradient, can be genetically perturbed without affecting the steady-state distribution of the Hh concentration profile. I propose that this method can be used as an experimental design tool to investigate the role of transient morphogen gradients in developmental patterning and discuss the generality of these ideas in other problems.
1. Introduction
A classical paradigm in developmental biology is that cells in a developing embryo or tissue acquire information about their relative spatial location as a function of the local concentration of chemical signals in their environment called morphogens. The key idea is that the interpretation of positional information results in the establishment of gene expression patterns whose boundaries correspond to concentration thresholds of the morphogen gradient [1–3]. A common assumption of the classical morphogen model is that concentration gradients are interpreted approximately at the steady state in order to produce stable patterns of gene expression [4,5]. However, recent studies in some systems have proposed that morphogens are dynamic during much of the patterning process and have raised criticisms to the classical morphogen concept [6–9]. For instance, if gradient formation evolves in time, how and when are concentration thresholds interpreted? Or, more generally, how do morphogen dynamics contribute to positional information? One possibility is that patterns evolve as the gradient develops, giving rise to transient patterns that converge to stable gene expression domains as the gradient approaches the equilibrium [10]. This scenario supports a ‘no-role’ model for transient gradients (figure 1a), but this seems not to be always the case. For example, the identity of the most posterior digits in the vertebrate limb is not solely determined by the steady-state concentration of a morphogen, but it also depends on the duration of the signal [11–13]. Furthermore, a similar interplay between signal strength and duration of exposure has been recently shown to operate in the specification of distinct neural fates in the vertebrate spinal cord [14]. These studies propose a model of patterning in which cell fate decisions are determined by integrating the strength of the morphogen signal over time (figure 1b). Alternatively, different patterns of gene expression may result from the dynamic properties of a morphogen gradient. For example, we recently proposed that the specification of different spatial domains of expression in response to Hedgehog (Hh) signalling in the Drosophila wing disc depend on a spatial shift of the Hh gradient prior to reaching the equilibrium [15]. In particular, we showed that cells exposed only transiently to Hh express a different combination of genes compared with cells constantly receiving the signal or cells that never receive it at all (figure 1c). Thus, in contrast to the classical morphogen model (figure 1a), patterning is often not exclusively dependent on the steady-state readout of the morphogen concentration, but also on the full history of exposure to the signal (figure 1b,c). Nonetheless, the problem of how the dynamics of gradient formation contribute to morphogen interpretation has been difficult to demonstrate by classical genetic experiments because mutants that affect gradient dynamics are also likely to perturb the steady-state morphogen distribution. Despite the recent introduction of techniques to quantify and measure the dynamic properties of morphogen gradients in living embryos or tissues [16–18], the identification of mutants that selectively affect gradient dynamics with little or no effect on the equilibrium profile remains challenging.
Figure 1. Static versus dynamic models of morphogen-mediated patterning. (a) In the classical morphogen model, gradients are assumed at equilibrium and cellular states (blue, white or red) depend on morphogen concentration thresholds (T1 and T2). In this view, the dynamics of the gradient prior to reaching the equilibrium (grey region) do not have a role in patterning. For example, cells exposed to high morphogen levels transiently, but to reduced levels at the steady state, acquire the same state (red) as cells always exposed to low signalling levels (trajectories 1 and 3). (b) In the temporal integration model, cells consider not only the concentration of the equilibrium gradient, but also the overall duration of signalling exposure. As in (a), the determination of cell fates in this model depends on thresholds, but, in this case, these thresholds are given by integrating the signal concentration over time (area under the curve). (c) In the gradient dynamics model, the dynamics of the gradient are essential for pattern formation. Cells receiving the morphogen will change from a red (OFF) to a blue (ON) state. As in (a), those cells that keep receiving the signal will maintain their blue fate, but cells that lost their transient signal (trajectory 3) will adopt a different state (white) from cells that never reach the switching threshold T (trajectory 1). In (c), the red and white domains are not simple-connected, i.e. trajectories cannot cross the red–white boundary.
For over 50 years, mathematical modelling in developmental biology has provided powerful tools to identify potential mechanisms and properties of morphogen gradient formation and their interpretation in developmental patterning [19–24]. However, much of this literature has been built on the assumption that morphogens are interpreted at the steady state and transient gradients are often ignored. Reducing a dynamical system to its steady state is usually mathematically convenient as the equations of the model often simplify considerably and theoretical tools are readily available to study the stability of solutions. In contrast, less analytical tools exist to investigate the temporal evolution of a gradient and studies that have taken morphogen dynamics into consideration usually rely on numerical simulations [25,26]. In this paper, I present a general theoretical approach to study the role of the dynamics of morphogen gradients in developmental patterning. Given a mathematical model of a morphogen-based pattern formation system, we consider parameter perturbations, referred to here as steady-state invariant perturbations, that may change the transient shape of a morphogen without effectively affecting its steady-state distribution. We introduce a theoretical framework to study these perturbations and discuss how they could be used as a design tool for genetic experiments that permit us to assay the role of gradient dynamics in the specification of patterns of gene expression. As a case study, the dynamic properties of the Hh gradient in the Drosophila wing disc are systematically investigated to illustrate how this approach can provide insights into the roles of signalling dynamics during development.
2. Results
2.1. Theoretical framework: definitions and examples
In order to introduce the concept of steady-state invariant perturbations, consider a simple example, namely a naive model of a one-dimensional morphogen produced at a point source (x = 0) that establishes a concentration gradient by diffusion and linear degradation,
In equation (2.2), A0 and λ represent the amplitude and characteristic length of the steady-state gradient, respectively, and their values determine uniquely this solution. As A0 and λ are defined in terms of the parameters of the system, a perturbation on the wild-type parameter values will maintain the steady-state solution invariant if and only if the values of A0 and λ remain unchanged. To formalize, let 
be the wild-type parameter values and consider a mutant that perturbs the system such that the effective parameter values of the system change to (α′, β′, D′). We say that the mutant preserves the steady-state gradient or is steady-state invariant if the following equations hold:
The set of all steady-state invariant perturbations can be represented geometrically in parameter space and will be referred to as the steady-state invariant set of the system. In general, a steady-state invariant perturbation Δ can be denoted as a vector in parameter space that is based on the point of wild-type parameter values and ends on another point of the steady-state invariant set (figure 2a). In this example, it follows that every perturbed parameter vector (α′, β′, D′) satisfying equation (2.3) must be of the form
Figure 2. Analysis of steady-state invariant perturbations in the case of a morphogen established by diffusion and uniform linear degradation. (a) Steady-state invariant sets can be generally described as subsets of the parameters’ space (blue region). In parameters' space, a steady-state invariant perturbation (Δ) is represented by a vector based on the point of wild-type parameter values (black dot) and ending on another point within the steady-state invariant set (red dot). In the space of states, the dynamics of a system are represented by trajectories starting at an initial state (circle) that may approach equilibrium states (denoted by x). Steady-state invariant parameter perturbations always approach the same steady state, but the dynamics may be qualitatively different (black versus red curves). In contrast, parameter perturbations that are not steady-state invariant may approach a different steady state (green curve), or have other dynamic behaviour. (b) The steady-state invariant set of the morphogen modelled by equation (2.1) is the straight line defined by equation (2.4) with δ > 0. (c–e) Temporal dynamics of the morphogen concentration (from equation (2.1)) when parameters values α, β and D are perturbed along the steady-state invariant set by varying the value of δ. (c) δ = 1 (unperturbed case); (d) δ = 0.5; (e) δ = 2. For comparison purposes, gradients in (c–e) are plotted using the same colour-coded time scale and the (invariant) steady-state profile is displayed in all three panels (dotted line).
Equation (2.4) represents a straight line in parameter space that crosses through the origin and contains the wild-type parameter vector 
(figure 2b). Thus, the line defined by equation (2.4) is the steady-state invariant set of this system. Importantly, a steady-state invariant perturbation may affect the transient evolution of the gradient but always maintains the steady-state morphogen gradient unchanged. In this simple example, the effects on morphogen dynamics along the steady-state invariant set are simple to interpret; for δ < 1, gradient formation is slower than in the wild-type case, while for δ > 1 the steady-state gradient is approached faster than in the wild type (figure 2c–e). In addition, the rates in which the steady state is approached are space dependent; cells adjacent to the source approach equilibrium faster than cells away from it, and this property also holds after steady-state invariant perturbations (see the electronic supplementary material, figure S1). Despite the simplicity of this example, the definition of steady-state invariant sets and the method to compute them can be generalized (box 1).
), D is a vector of diffusion coefficients, f is the reaction function that describes the interactions of gene products, and μ is a vector of kinetic rates or parameters of the system. Note that the systems considered in the text (equations (2.1) and (2.5)) are particular cases of equation (B 1). At the steady state, we set the time derivatives to zero so that the steady-state concentrations, Gss, obey the following equation:
. A parameter perturbation (e.g. a genetic perturbation) of the system is a function Δ of the form Δ(μ) = (δ1μ1, δ1μ2, … , δrμr) ≡ (μ′1, μ′2, … , μ′r) for some positive constants δ1, … , δr. We are interested in the study of perturbations that leave the steady-state solution 
unchanged at least in a region of space S. A steady-state invariant perturbation is a parameter perturbation Δ that satisfies the following property:
and, therefore, we can consider the points (μ′1, μ′2, … , μ′r) in parameter space that satisfy
, of the system (in S) as the set that results from the intersection of Ωx for all 
, i.e.2.2 Hh signalling in the Drosophila wing disc: a case study
With the aim of motivating the applications of the concept of steady-state invariant sets to cases of biological interest, I considered a model of Hh signalling in the Drosophila wing imaginal disc (figure 3a). In the developing wing disc, Hh is secreted by cells in the posterior compartment and forms a concentration gradient into the anterior compartment. In the absence of Hh, the Hh receptor, Patched (Ptc), represses signalling activity in anterior cells. Upon Hh reception, inhibition of Ptc on the signal transduction component Smoothened (Smo) is alleviated. Although the molecular details by which Ptc inhibits Smo and the mechanisms downstream of Smo that result in the interpretation of Hh signalling activity remain largely unclear, experimental data support a phenomenological model which suggests that Hh signal activity is a function of the ratio of bound to unbound Ptc [27].
Figure 3. Analysis of steady-state invariant perturbations for the model of Hh signalling in the Drosophila wing disc. (a) A graphical representation of the Hh signalling network. A Hh gradient emanating from the posterior compartment determines the expression of decapentaplegic (dpp), ptc and engrailed (en) in different domains abutting the AP boundary. (b) Hh-dependent patterning in the wing disc is determined by the dynamics of the Hh gradient. Hh-dependent Ptc upregulation restricts the range of the Hh gradient over time. This property of the Hh signalling network is responsible for a transient expansion of the gradient or ‘overshoot’. The overshoot of the Hh gradient determines patterning of the Drosophila wing disc (see [15] for details). Three gene expression domains are established by this overshoot. Cells that do not receive the signal or that receive sustained signalling levels are denoted by black and blue, respectively. Certain cells that are only transiently exposed to the signal are denoted by the white state and characterized by dpp (but not ptc) expression. The photo displays wild-type expression patterns of ptc and dpp in a third-instar wing disc (with the anterior compartment oriented upwards) carrying a dpp–lacZ reporter immunostained using Ptc (blue) and β-galactosidase (white) antibodies. (c) The steady-state invariant subset described by equation (2.9) represents a line parametrized by the variable δ (black dot, δ = 5; blue dot, δ = 2; green dot, δ = 1; red dot, δ = 0.5; yellow dot, δ = 0.2). (d) Analysis of the overshoot in selected steady-state invariant mutants. Colour coding of the curves corresponds to different values of δ as shown in (c). Note that the amplitude of the overshoot is unaffected by these steady-state invariant perturbations. (e) The duration of the overshoot (in hours) as extracted from (d) (squares) as a function of δ. (f) Representation of the steady-state invariant subset from equation (2.10) in two dimensions. The vertical axis, denoted by μ, represents any other dimensions of parameter space (black dot, δ = 1; blue dot, δ = 0.5; green dot, δ = 0.33; red dot, δ = 0.2; yellow dot, δ = 0.1). (g) Similar analysis to that in (d), but corresponding to the approximate steady-state invariant subset in (f). (h) Steady-state invariant perturbations in (f) considerably affect the amplitude (red), but not the duration of the overshoot (blue).
A general property of the Hh signalling pathway is the transcriptional upregulation of ptc in response to Hh activity that results in increased Ptc expression in cells abutting the anterior–posterior (AP) boundary. This feedback property of the Hh signalling pathway is necessary to limit the range of Hh signalling activity and has been implicated in the robustness of the gradient to fluctuations in Hh dosage [23,28]. Furthermore, in a recent study, we provided experimental evidence for a transient expansion (or ‘overshoot’) of the Hh gradient that results from the Hh-dependent Ptc upregulation feedback in the system [15]. We proposed that the Hh overshoot is required to establish three distinct domains of gene expression (figure 3b), supporting the importance of morphogen gradient dynamics in this system. However, the problem of how specific properties of this transient overshoot (e.g. its duration) contribute to patterning has not been studied in detail. Here, I use the concept of steady-state invariant perturbations to make predictions about which genetic perturbations of the system affect the dynamic properties of this overshoot without modifying the equilibrium distribution of the Hh gradient. We consider the mathematical model of the Hh signalling pathway (figure 3a) as originally presented in our previous work [15]:
At the steady state, algebraic substitutions reduce the set of equations (2.5) to the following second-order equation, which is valid exclusively in the anterior compartment (see the electronic supplementary material, appendix):
, 
and 
. This equation is nonlinear and an analytical solution may be difficult (or impossible) to obtain. However, equation (2.6) can be linearized to obtain approximate solutions in regions where Hh is present at sufficiently high (
) or sufficiently low (
) levels, i.e. near and far from the AP boundary, respectively (see the electronic supplementary material, appendix, for details). In each of these regions, equation (2.6) reduces to the form,| near the AP boundary | far from the AP boundary |
|---|---|
| solution | solution |
 |
 |
| parameters | parameters |
 with  |
 with  |
 |
 where xfar is the location of the disc, such that  and LA is the width of the anterior compartment. |
 where LP is the width of the posterior compartment. |
In a similar manner that A0 and λ in equation (2.2) represent the amplitude and characteristic length of the gradient, respectively, it is useful to provide a physical interpretation of the parameters in the solutions of equation (2.7). We note that, assuming [Ptc]SS is approximately constant (
), the steady-state version of the first equation in the set of equations (2.5) can be written in the form of equation (2.7) with 
and C = 0, where 
represents the effective degradation rate of Hh. Therefore, our approximate solutions near and far from the AP boundary are equivalent to assuming that [Ptc]SS is constant in each of these regions and provide an estimate of these constants as a function of the kinetic parameters of the model (table 1). Furthermore, these approximations permit analytical expressions to be obtained that define the steady-state invariant set of the system. In particular, parameter perturbations that maintain each of the solutions of equation (2.7) unchanged will maintain the global steady-state distribution of the Hh morphogen approximately invariant in much of the anterior compartment. Thus, the steady-state invariant set of the linearized model is given explicitly by
denoting the vector of wild-type parameter values and all other parameters kept unperturbed (i.e. at their wild-type values). By substitution into equation (2.8), it is clear that equation (2.9) represents a steady-state invariant subset of the linearized system. In fact, equation (2.9) also leaves invariant the original steady-state equation of the system (equation (2.6)) and, therefore, it is an exact steady-state invariant subset of the system. Geometrically, equation (2.9) represents a line in the αptc0 − αptc − βptc parameter subspace (figure 3c). Numerical simulations suggest that a few-fold changes in δ affect the duration but have little effect on the amplitude of the overshoot (figure 3d,e). In particular, we noted that the duration of the overshoot increases rapidly as δ decreases (δ < 1), but decreases slowly as δ increases (δ >1). Therefore, perturbations along this steady-state invariant subset provide the opportunity to study the role of signal duration in further detail.An even simpler case of a steady-state invariant subset of the linearized system can be obtained by noticing that the system of equations (2.8) does not involve the degradation rate of the Hh–Ptc complex, βHh_Ptc. Therefore, under these approximations, the steady-state solution does not depend on the value of βHh_Ptc. For simplicity, consider the approximate steady-state invariant subset given by
remains valid near the AP boundary. In contrast to the subset defined by equation (2.9), perturbations along the subset described by equation (2.10) will affect the amplitude of the overshoot, but its duration will be maintained relatively constant (figure 3g,h). The simplicity of this subset permits the design of genetic steady-state invariant perturbations in the system.2 The ptc14 allele has been characterized as a mutant that is defective in endocytosis-dependent internalization and degradation of the Hh–Ptc complex, suggesting that the stability of the Hh–Ptc complex is increased in Ptc14 mutants [29]. Importantly, other properties of the Ptc protein appear unaffected in ptc14 mutant clones. For example, Ptc14 proteins bind Hh and repress Hh signal transduction in a similar manner to wild-type Ptc [29]. Furthermore, Ptc14 is not overexpressed in anterior clones away from the AP boundary, suggesting that unbound Ptc14 degradation is normal. This genetic evidence suggests that βHh_Ptc decreases in ptc14 mutant mosaics, while other parameters approximately maintain their wild-type values. Therefore, we predict that ptc14 mutants satisfy equation (2.10) and can be considered as a steady-state invariant perturbation of the system. Based on our modelling results, perturbations along this steady-state invariant subset affect the amplitude of the overshoot, but not the overall shape and duration of the transient response, suggesting that more anterior cells would be transiently exposed to the signal (figure 3g,h). Experiments using ptc14 mutant clones abutting the AP show that the expression domains of Hh target genes expand in the region of these clones [29], consistent with the predictions of this work (figure 3g).Other subsets can be similarly obtained from equations (2.8) and their effects on transient dynamics can be systematically analysed. However, sets involving many parameters are difficult to visualize and perturbations along these subsets may not be easily achieved by experimental manipulations (see below).
3. Discussion
3.1. Analysis and computation of steady-state invariant sets
The problem of computing the steady-state invariant set from a given model is in general non-trivial and, with the exception of very simple models (e.g. equation (2.1)), cannot be obtained analytically. Nonetheless, there are different manners to overcome this problem and get a partial or approximate set of steady-state invariant perturbations. The simplest way to obtain at least a steady-state invariant subset of a system is to inspect the steady-state equation and ask which parameter perturbations leave the equation invariant. For example, perturbations that satisfy equation (2.9) leave the steady-state problem (equation (2.6)) unchanged. In these cases, there is no need to obtain the solution of the steady-state equation, but this approach will only provide a subset of steady-state invariant perturbations. We will say that a steady-state invariant set is direct if it can be deduced from the steady-state equation. Note that equation (2.4) is a direct subset of equation (2.1), and, from the steady-state solution (2.2), we can show that it contains all the steady-state invariant perturbations of the system. However, this is not necessarily true in general. Therefore, an interesting theoretical challenge for the future will be to find conditions to determine in which cases the full steady-state invariant set is direct, i.e. when we can obtain all steady-state invariant perturbations of the system without needing to solve the equations. Alternatively, steady-state invariant sets can be obtained approximately by linearization (as we did for equation (2.6)), or by spatially discretizing the system, i.e. turning the problem into a system of ordinary differential equations (as described in box 1). Although these approaches do not always result in exact steady-state invariant perturbations, they are generally applicable and often provide perturbations that are apt for experimental design (e.g. equation (2.10)). In addition, in these approaches, the property of steady-state invariance needs not to be global, i.e. it can be considered in a desired subset of the whole developmental field, and this might often be convenient in practice (e.g. when studying local patterning effects using genetic mosaics).
Another technical challenge that deserves further discussion is how to analyse and visualize steady-state invariant sets. This problem is common in practice because steady-state invariant sets are usually contained in high-dimensional spaces. For example, the dimension of the parameter space of our highly simplified model of Hh signalling (equation (2.5)) is 16. More realistic models may involve an even larger number of parameters, making the resulting steady-state invariant set very difficult to analyse and visualize. Our analysis in this paper reduces to the study of subsets (equations (2.9) and (2.10)) that are contained in particular parameter subspaces and result in tractable manipulations, but an interesting future direction is to use theoretical tools (e.g. nonlinear extensions of principal component analysis; see [30] for a review) to reduce the dimensions of steady-state invariant sets in a way that simplifies their use for experimental design.
3.2. Steady-state invariant perturbations as a tool to investigate the roles of morphogen gradient dynamics
The main goal of this work is to use it as a tool to design genetic experiments that separate transient from equilibrium effects in developmental patterning. One problem that arises in practice is that steady-state invariant perturbations often impose constraints on many independent kinetic parameters of the system. For instance, in our example of a free morphogen established by diffusion and linear degradation (equation (2.1)), only those perturbations in which all three parameters are altered in the same proportion remain on the steady-state invariant set (equation (2.4)). Despite the simplicity of this example, a genetic perturbation subject to these constraints (i.e. such that equation (2.4) approx. holds) is difficult to achieve experimentally because kinetic rates, in general, cannot be precisely tuned using standard genetic techniques. This example illustrates a fundamental property of this approach as a tool for modelling-based experimental design: simple models that result in relatively simple steady-state invariant sets do not necessarily predict simple genetic experiments.
In practice, it is convenient to know, based on the steady-state invariant set, how difficult it is to design a genetic experiment that causes a steady-state invariant perturbation in the system. In other words, how likely is it for a random parameter perturbation to be steady-state invariant? A geometric notion that provides insights into this question is the concept of co-dimension.3 For the practical purposes of this study, we define the co-dimension of a set as follows. Let M be a steady-state invariant set that affects s parameters and can be parametrized by r variables; then the co-dimension of M is defined as codim(M) = s − r. For instance, the steady-state invariant subsets defined by equations (2.9) and (2.10) depend on three (αptc0, αptc and βptc) and one (βHh_Ptc) parameters, respectively, and both are parametrized by the single variable δ. Therefore, their co-dimensions are two and zero, respectively. Intuitively, the co-dimension of a set is inversely related to the concept of degrees of freedom, i.e. the number of parameters that can be arbitrarily varied without abandoning the set. In general, steady-state invariant subsets of small co-dimensions are expected to be more useful in practice because, in these cases, perturbations have the property that independent kinetic parameters are loosely coupled (codim(M) > 0) or not coupled at all (codim(M) = 0; e.g. equation (2.10)).
As a modelling-based experimental design tool, the approach introduced in this paper also exhibits some technical limitations. One consideration is that steady-state invariant sets depend on the details of the mathematical models. For example, the model of Hh signalling described by equations (2.5) assumes that the signal is activated as a function of the ratio of bound to unbound Ptc [27]. However, this model has been controversial in the literature as it fails to explain experimental data from a study that strongly support that Ptc inhibits Smo catalytically [31]. In order to test whether the prediction that Ptc14 is approximately a steady-state invariant perturbation depends on the choice of the model; an alternative model that assumes the inhibition of the signal by the catalytic action of Ptc was considered (see the electronic supplementary material, appendix equation (A 8)). Interestingly, using similar assumptions near and far from the AP boundary, it is possible to show that Ptc14 is also a steady-state invariant perturbation in this alternative model (see the electronic supplementary material, appendix, for details). However, this is not necessarily true in general, and these situations need to be considered on a case-by-case basis.
Another technical limitation of these tools is that mathematical models are always based on highly simplified representations of the biological system, so that steady-state invariant perturbations predicted by the theory may not be so experimentally. Therefore, these tools should be used as predictors of steady-state invariant genetic perturbations but appropriate control experiments should be performed to show that the steady-state distribution of the signal is in fact unchanged by the proposed perturbation.
4. Concluding remarks
There is much recent interest in the problem of how the dynamics of morphogen gradients contribute to developmental patterning (reviewed in [32]). This work presented a theoretical approach that may help to identify mutations that affect the transient establishment of morphogen gradients without changing their equilibrium distribution. Although the experimental implementation of these tools to study the role of morphogen gradient dynamics in developmental pattern formation is left to future studies, the details of how they can be used in practical cases were provided. As an example, our analysis of steady-state invariant subsets in a model of Hh signalling in the Drosophila wing provided the opportunity to decouple two properties of the Hh gradient overshoot, namely duration and amplitude. The duration of the overshoot can be modulated in a steady-state invariant manner by varying the rates of ptc production and degradation (figure 3c–e). Furthermore, the amplitude of the overshoot can be affected by reducing the rate of Ptc-dependent Hh internalization and degradation. In fact, previous genetic experiments in which a mutant form of Ptc (Ptc14) that is defective in Ptc-mediated Hh endocytosis, but otherwise appears to function normally, provide an example of an experimental steady-state invariant perturbation in this system.
The theory of steady-state invariant sets introduced here was motivated by the problem of whether or not morphogen dynamics contribute to developmental patterning. Nonetheless, the applications of the tools presented here are not limited to the design of experimental perturbations and may be broadly applicable to other problems. For example, in engineering, it is often desirable to design systems in which the steady state is robust to a certain class of perturbations. Therefore, the converse problem, namely to construct a dynamical system that will maintain a desired set of perturbations invariant, poses an interesting theoretical challenge for the future. The context in which this converse problem has been studied in some detail is robust control theory, and there are several examples in which imposing invariance into the steady-state values of some variables constrains the dynamic properties of the system. For instance, a well-known result in control theory is that the steady-state error in some desired signal remains negligible to external perturbations, or variations in internal components, if and only if the error dynamics are governed by integral feedback, and this has been shown to apply to perfect adaptation in chemotaxis [33]. Finally, it is likely that evolution has faced and solved this converse problem by selecting for optimal mechanisms of developmental patterning; in some systems, natural selection may have minimized the effects of certain classes of genetic perturbations to maintain invariant some essential gene expression patterns. Alternatively, animal evolution could have employed the diversity of morphogen dynamics to enrich the mechanisms underlying developmental patterning.
5. Methods
5.1. Wing disc immunostaining
In figure 3b, a third instar wing disc of genotype dpp10 638/CyO was immunostained with mouse anti-Ptc (Hybridoma Bank Developmental Studies at the University of Iowa) and rabbit anti-β-galactosidase (Invitrogen) antisera following standard techniques. dpp10 638 is a lacZ enhancer trap reporter that expresses nuclear β-galactosidase under the control of the dpp enhancer.
5.2. Analytical solutions
The analytical solution of equation (2.1) was derived in Bergmann et al. [6] and is given by
. This solution was used to generate the gradient profiles in figure 2c–e. Approximate analytical solutions of equation (2.6) near and far from the AP boundary (equation (2.7)) were obtained by linearization, and the full details of these approximations are provided in the electronic supplementary material, appendix.5.3. Numerical simulations
Our model of Hh-dependent patterning of the wing disc is assumed one-dimensional along the AP axis. We used a system of coordinates centred at the AP boundary (x = 0), where the posterior and anterior ends of the disc were assumed at 
and 
, respectively (with LP = LA = 100 µm, the length of each compartment). Equations (2.5) were numerically solved with the parameter values as shown in the electronic supplementary material, table S1, and using a Forward-in-Time-Centred-in-Space algorithm implemented in Matlab. As in previous work [15], we use zero initial conditions, except for [ptc] and [Ptc], which were taken as
Furthermore, we imposed zero-flux boundary conditions at the disc extremes (
; 
), and assumed continuity of the [Hh]SS profile and its derivative at the AP boundary (x = 0; see the electronic supplementary material, appendix, for further details).
Acknowledgements
I thank John Doyle for his advice and insightful discussions about this work. I also thank Angelike Stathopoulos and Arthur Lander for comments on the manuscript. This work was supported by a grant from the National Institutes of Health (R01-GM078992) to John Doyle.
Footnotes
1 In this example, we ignore boundary conditions by assuming that the system is infinitely long and diffusion occurs in both directions from the source. These assumptions allow us to solve the full time-dependent problem (2.1) exactly using the method of Fourier transforms (see §5) and solution (2.2) arises as a limit of the time-dependent solution when t→∞.
2 Strictly, our assumptions hold in each region where the steady-state levels of Ptc are nearly uniform (i.e. near the AP boundary, where Ptc levels are upregulated; and away from the Hh signalling domain, where Ptc is present at basal levels). Thus, steady-state invariant perturbations satisfying equation (2.11) should be valid in much of the anterior compartment; however, there may be a small domain separating these regions where Ptc levels cannot be considered constant and where steady-state invariance cannot be guaranteed.
