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# Kinetics of Morphogen Gradient Formation

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Science  26 Jan 2007:
Vol. 315, Issue 5811, pp. 521-525
DOI: 10.1126/science.1135774

## Abstract

In the developing fly wing, secreted morphogens such as Decapentaplegic (Dpp) and Wingless (Wg) form gradients of concentration providing positional information. Dpp forms a longer-range gradient than Wg. To understand how the range is controlled, we measured the four key kinetic parameters governing morphogen spreading: the production rate, the effective diffusion coefficient, the degradation rate, and the immobile fraction. The four parameters had different values for Dpp versus Wg. In addition, Dynamin-dependent endocytosis was required for spreading of Dpp, but not Wg. Thus, the cellular mechanisms of Dpp and Wingless spreading are different: Dpp spreading requires endocytic, intracellular trafficking.

Although the molecular and cellular mechanisms controlling morphogen transport have received much attention, many questions remain open (17). This might be due in part to the existence of only a few quantitative studies of the steady-state gradients and the kinetics of morphogen transport. To address this, we studied quantitatively two key morphogens during development of the fly wing: Decapentaplegic (Dpp) and Wingless (Wg).

Dpp is produced at the anterior-posterior compartment boundary in the center of the wing imaginal disc of Drosophila (8) (Fig. 1, A and B). Dpp spreads nondirectionally, is degraded while spreading, and forms a gradient of concentration in the plane of the wing epithelium (2, 7). Regardless of the actual transport mechanism, these facts imply that Dpp spreading can be captured by the physics of molecules that are produced in a localized source, which generates a current j0 [molecules/(μm × s)] at the source boundary; that are degraded with a rate k (s–1); and that spread in a nondirectional manner with an effective diffusion coefficient D (μm2/s). Thus, the rate of change of Dpp concentration in the x-y plane, C(x,y,t), is described by the equation: $Math$(1) where t is time, x > 0 is the distance to the source in the target tissue, ▽2 is the Laplace operator, and δ is the Dirac delta function [see supporting online material (SOM)].

The steady-state solution for Eq. 1 is a single exponential gradient: $Math$(2) where the Dpp concentration C(x) depends only on the distance x from the source, the concentration C0 at the source boundary, and the decay length λ. The decay length corresponds to the distance at which the concentration decays by a factor 1/e of C0 [C(x) = C0 (1/e) at x = λ]. The shape of the gradient therefore depends on two key parameters: λ and C0, which are determined by D, k, and j0.

In the scenario of nondirectional morphogen spreading with degradation, λ in the steady state is related to the diffusion coefficient and the degradation rate by the expression: $Math$(3) In turn, C0 depends on the current j0, and on diffusion and degradation, which occur both in the receiving tissue and in the source: $Math$(4) Indeed, the experimental Dpp distribution in the target tissue was well described by a single exponential (Fig. 1C) [correlation index <R2> = 0.92 ± 0.05, n = 26) with a decay length λ = 20.2 ± 5.7 μm, corresponding to 7.7 ± 2.1 cells (see materials and methods section in SOM).

To measure the kinetic parameters (D, k, and j0) that determine the steady-state shape (characterized by λ and C0) of the Dpp gradient, we developed an experimental strategy based on imaging a functional green fluorescent protein–Dpp fusion (GFP-Dpp) produced at the endogenous wing source (2) and fluorescence recovery after photobleaching (FRAP) (9, 10). The FRAP assay consisted of irreversibly photobleaching the GFP-Dpp fluorescence in a rectangular region of interest (ROI) with dimensions 10 μm by 200 μm adjacent to the source (Fig. 1, D to K). Subsequently, we monitored the fluorescence recovery in the ROI for about 60 min, which occurred at the expense of the nonbleached GFP-Dpp molecules in the neighboring areas, until the fluorescence intensity was close to a plateau (Fig. 1L). To quantify the recoveries, we imaged and projected five z-confocal planes encompassing the most apical 5 μm of the epithelium and measured the average fluorescence intensity of GFP-Dpp in the ROI (see materials and methods and fig. S3).

We first controlled several conditions: (i) we limited photodamage and photobleaching during imaging so that they were negligible, (ii) we imaged most of the GFP-Dpp molecules in the tissue, (iii) we estimated the detection inaccuracy, and (iv) we calibrated detection of fluorescence intensity to GFP-Dpp concentration by using GFP-tagged rotavirus particles (11). These important controls are summarized in section 2 of the supporting online material and in figs. S1 and S2. After verification of these conditions, we used our calibrations to estimate the concentration of GFP-Dpp at the source boundary C0 = 802 ± 312 molecules/μm2 (n = 8 discs), which corresponds to 4379 ± 1741 molecules per cell (see materials and methods). In addition, we estimated the fraction of “extracellular” GFP-Dpp, which was equal to or smaller than 15 ± 3.4% (n = 8 discs) of the total pool. The latter measurement, together with the fact that our detection inaccuracy was less than 2%, showed that the extracellular pool was not a dominant pool (section 2 of SOM).

We then studied the recovery profiles in the FRAP experiments to determine D, k, and j0, as well as the immobile fraction ψ. ψ is the fraction of molecules that did not recover in the ROI during the experiment. The standard procedure (12) to solve the diffusion equation, which neglects production and degradation and is commonly used in FRAP studies of single cells (13), is not suitable for our FRAP assay in tissues. The time span of recovery in tissue FRAP was 30 to 90 min (Fig. 1), so that production and degradation could not be neglected. Thus, we solved Eq. 1 for the particular geometry and conditions of our FRAP experiments in the tissue (SOM, fig. S3D).

From the resulting time-dependent concentration profile C(x,y,t) (fig. S3D), we calculated the average concentration f(t) in the ROI as a function of time. We optimized the two parameters D and ψ, which determine the shape of this theoretical recovery curve, to obtain the best agreement with the experimental curves (see SOM). The parameters k and j0 were not independent, because we measured two key quantities before bleaching that impose further constraints: (i) the decay length λ, which allows us to determine k from the fitted D via Eq. 3: $Math$, and (ii) the concentration at the source boundary C0, which enabled us to determine j0 via Eq. 4: $Math$. The resulting theoretical curves were in excellent agreement with the experimental recovery data (R2 = 0.95 ± 0.03) (see also table S1), which provided sufficient constraints to confidently determine the actual values of D and ψ. Thus, analysis of the experimental recovery curves, together with Eqs. 3 and 4, allowed us to determine the kinetic parameters D, k, j0, and ψ of morphogen spreading.

The effective diffusion coefficient of GFP-Dpp was D = 0.10 ± 0.05 μm2/s (n = 8 experiments). This number is about three orders of magnitude less than the coefficient of free diffusion in water of molecules with the size of the mature Dpp homodimer [similar to that of GFP, which has D = 87 μm2/s (14)], but consistent with (i) restricted diffusion of Dpp in the extracellular space [e.g., hindered by binding to immobile extracellular matrix molecules or membrane receptors (1517)] and (ii) endocytosis plus recycling of Dpp through a fast recycling pathway (18). Dpp was degraded at a rate k = 2.52 × 10–4 ±1.29×10–4 s–1 corresponding to a GFP-Dpp half-life of about 45 min, which is consistent with the turnover time of Dpp molecules in the developing wing determined biochemically (7). In addition, the majority of the GFP-Dpp molecules in the target tissue were immobile or moved very slowly (ψ = 62 ± 8%), i.e., with recovery kinetics too slow compared with the 60-min time scale of the FRAP experiments. This immobile pool was stored in intracellular compartments, because the extracellular pool was equal to or smaller than 15% of the total pool. We found j0 = 3.98 ± 2.34 molecules/(μm × s), which implies that the Gal4-driven GFP-Dpp production rate from the endogenous source was ν = 2.69 ± 1.58 molecules per cell per second. The effective production rate ν per cell with diameter a can be obtained from j0 by ν ≈ 2a2j0/w for small values of the width of the source w (μm) (3).

We tested the validity of the diffusion and degradation description for the FRAP recoveries by performing FRAP experiments in different geometries (fig. S4 and table S1). The results of these experiments were consistent with the independence of the four kinetic parameters on position in the tissue (see SOM).

After studying the GFP-Dpp kinetics, we analyzed the dependence of gradient formation on endocytosis by performing the FRAP assay in animals where the target tissue was mutant for the thermosensitive shibirets1 allele and in which the source was rescued by a functional shibire+ transgene [“shibire-rescue” animals; see materials and methods and (2)]. The role of endocytosis has previously been studied using shibire-rescue animals in qualitative assays (1, 2) (SOM and fig. S5). The FRAP assay allowed us to determine separately D, k, ν, and ψ in each experimental condition and to measure sensitively the kinetic effects in conditions of partial or complete endocytic block [in the wing, shibirets1 is a tight thermosensitive mutant at 34°C, but leaky at 32°C (2) (fig. S5)]. In addition, during the FRAP experiment, the effects could be detected immediately, after only a few minutes of block. In control animals, D, k, ν, and ψ were not significantly different at 25°C, 32°C, and 34°C (figs. S6 and S7). In shibire-rescue animals, which were raised at the permissive temperature, the tight endocytic block at 34°C during the FRAP experiment impaired Dpp spreading completely, and no recovery could be detected into the ROI (Fig. 2, A and F to M). When the block was released by shifting the temperature down to 25°C, fluorescence recovered into the ROI, which indicated that the effect was reversible and that the lack of recovery was not due to unspecific tissue damage (fig. S8).

This lack of recovery after the tight endocytic block was imposed could be due to decreased diffusion or increased degradation rate. Because no recovery was observed, a theoretical curve could not be fitted to determine the actual values of the kinetic parameters. Thus, we studied the recoveries under conditions of partial block of endocytosis at 32°C. At 32°C, the diffusion coefficient in shibire-rescued animals decreased by a factor of more than 2 compared with that of control animals at 32°C (Fig. 2, B and C) and with shibire-rescued animals at the permissive temperature 25°C (Fig. 2, D and E, and table S1). The degradation rate was not increased, but decreased by a factor of about 2. Finally, the production rate was also affected, whereas ψ was not (see materials and methods). Thus endocytosis is required not only for Dpp degradation, but also for Dpp movement.

It is worth noting here that (i) given our imprecision of measurement (2%, see SOM), the minimal recovery we would have been able to detect is 2% of the total pool (fig. S2, E to G), excluding the possibility that we might have missed a significant extracellular recovery; (ii) other morphogens did recover in this FRAP shibire-rescue assay, which excludes an intrinsic artifact of the treatment as causing the lack of recovery (see below); (iii) the observed effects of endocytic block were acute, because the lack of recovery was observed after a few minutes of block; and (iv) the endocytic block was reversible.

To explore whether this kinetic description of Dpp applied to other morphogens, we performed the same FRAP analysis for another secreted ligand, Wingless (Wg) (Fig. 3, A to H and J). In order to compare Wg and Dpp, we expressed a functional GFP-Wingless fusion (19) at the endogenous Dpp source, instead of the endogenous source region of Wg in wild type (Fig. 3, A to D). Wg also moves in the tissue nondirectionally and is degraded, but had a different profile (Fig. 3I). The Dpp gradient is a long-range gradient with decay length λ = 20.2 μm, whereas, in these in vivo conditions, Wg made a short-range gradient with λ = 5.8 ± 2.04 μm (Figs. 3I and 4A). Which kinetic parameter could account for this difference? Because Dpp and Wg have inherently different properties—Wg is a lipid-modified molecule (20); Dpp is not (21)—they are likely to display different mechanisms and kinetics of spreading through the epithelium.

The shorter decay length of the Wingless gradient was due to a higher degradation rate of GFP-Wingless, by a factor of 5, and to a lesser extent its smaller diffusion coefficient (Fig. 4). Although the Gal4 driver was the same in the Dpp and Wg experiments, the production rate of Wg was about seven times that of Dpp, which implied that their maturation and secretion were controlled differently. In addition, while 62% of the Dpp molecules were immobile, the Wg pool was almost fully mobile at 25°C (ψ = 9.2 ± 13%), although, unlike Dpp, a significant immobile fraction appeared at higher experimental temperatures (Fig. 3, K and N). The different immobile fractions of Dpp and Wg at 25°C validated the specificity of the Dpp immobile fraction. Thus, the immobile fraction was not an artifact of incomplete recovery in sick cells. Finally, in contrast to Dpp, Wg transport and degradation were independent of Dynamin endocytosis (Fig. 3, K to O). Indeed, Wg movement has been suggested to be Dynamin-independent (6, 22). In addition, expression of dominant-negative Dynamin and/or long-term thermosensitive shibire block caused an extension of the gradient in the wing (5, 22, 23), which was attributed to decreased degradation (5, 22). Our FRAP approach studying the results of an acute block suggests that endocytosis is not required for Wg transport and degradation or, alternatively, that endocytosis of Wg is Dynamin-independent.

Altogether, the GFP-Wingless FRAP experiments (i) validated our FRAP assay and shibirerescue experiment; (ii) indicated that different morphogen gradients can be generated by independently fine-tuning D, k, ν, and ψ; and (iii) showed that different morphogens may use different mechanisms of transport and cellular machineries (e.g., Dynamin-dependent versus Dynamin-independent transport) to achieve the formation of morphogen gradients.

Supporting Online Material

Materials and Methods

SOM Text

Figs. S1 to S8

Table S1

Movies S1 to S3

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