Title: Morpheus Unbound: Reimagining the Morphogen Gradient
Abstract: The theory that the spatial organization of cell fate is orchestrated by gradients of diffusing molecules was a major contribution to 20th century developmental biology. Although the existence of morphogens is no longer in doubt, studies on the formation and function of their gradients have yielded far more puzzles than answers. On close inspection, every morphogen gradient seems to use a rich array of regulatory mechanisms, suggesting that the tasks carried out by such systems are far more extensive than previously thought. The theory that the spatial organization of cell fate is orchestrated by gradients of diffusing molecules was a major contribution to 20th century developmental biology. Although the existence of morphogens is no longer in doubt, studies on the formation and function of their gradients have yielded far more puzzles than answers. On close inspection, every morphogen gradient seems to use a rich array of regulatory mechanisms, suggesting that the tasks carried out by such systems are far more extensive than previously thought. Few long-standing problems in biology have been the focus of more curiosity, or the source of greater frustration, than embryonic pattern formation. The problem, simply put, is one of determining how the large-scale organization of cell types in space is dictated by a set of instructions—the genes—that is the same in every cell. This question has captivated biologists, physical scientists, and mathematicians who have infused the field with viewpoints from their own disciplines: physicists see the emergence of large-scale properties from small-scale elements, computer scientists see distributed processing of information; engineers see robust control systems, and mathematicians see coupled partial differential equations with interesting behaviors. In over a century of study, the most influential concept to emerge in the field of pattern formation has been that of the morphogen. In the oldest sense of the word, a morphogen is a substance that is produced by cells and organizes pattern by spreading to other cells. Most modern biologists adopt narrower definitions in line with particular theories about how morphogens work. The view prevalent in the experimental literature comes from Wolpert, 1969Wolpert L. Positional information and the spatial pattern of cellular differentiation.J. Theor. Biol. 1969; 25: 1-47Crossref PubMed Scopus (1874) Google Scholar, who proposed that smoothly declining gradients, formed by the diffusion of morphogens from sources to sinks, assign positional values to cells, which then adopt different fates depending on the values they were assigned. In this view, a morphogen is not just an instructive molecule but one that gives qualitatively different instructions depending on its concentration. What ultimately determines pattern, therefore, is where morphogen gradients cross threshold values at which genes are turned on or off. A different conception of morphogens comes from the theoretical work of Meinhardt and Gierer (Gierer and Meinhardt, 1972Gierer A. Meinhardt H. A theory of biological pattern formation.Kybernetik. 1972; 12: 30-39Crossref PubMed Scopus (1982) Google Scholar, Meinhardt and Gierer, 2000Meinhardt H. Gierer A. Pattern formation by local self-activation and lateral inhibition.Bioessays. 2000; 22: 753-760Crossref PubMed Scopus (440) Google Scholar), which extended the earlier efforts of Turing, 1952Turing A.M. The chemical basis of morphogenesis.Phil. Trans. Roy. Soc. Lond. 1952; B237: 37-72Crossref Google Scholar. This work showed how two morphogens that influence each other's synthesis could trigger the spontaneous emergence of stable, long-range patterns of morphogen activity. In numerical simulations, Meinhardt-Gierer mechanisms produce patterns of repeated stripes and spots that bear an uncanny resemblance to some of those found in nature. The impact of such theories on developmental biology has been great, yet the relationship of experimental biologists to them has been rocky, at best. For years, a failure to identify any animal morphogens led to widespread doubt that such substances exist. By the mid 1990s, this situation changed as a result of studies on bicoid, an intracellular morphogen (Driever and Nusslein-Volhard, 1988Driever W. Nusslein-Volhard C. The bicoid protein determines position in the Drosophila embryo in a concentration-dependent manner.Cell. 1988; 54: 95-104Abstract Full Text PDF PubMed Scopus (679) Google Scholar), and Decaptentaplegic (Dpp), an extracellular morphogen (Ferguson and Anderson, 1992Ferguson E.L. Anderson K.V. Decapentaplegic acts as a morphogen to organize dorsal-ventral pattern in the Drosophila embryo.Cell. 1992; 71: 451-461Abstract Full Text PDF PubMed Scopus (395) Google Scholar, Nellen et al., 1996Nellen D. Burke R. Struhl G. Basler K. Direct and long-range action of a DPP morphogen gradient.Cell. 1996; 85: 357-368Abstract Full Text Full Text PDF PubMed Scopus (781) Google Scholar), both of which contribute to Drosophila development. In the last decade, polypeptides of the fibroblast growth factor (FGF), epithelial growth factor (EGF), Wnt, Hedgehog, and transforming growth factor (TGF)-β families, as well as the vitamin A metabolite retinoic acid, have all emerged as confirmed or likely morphogens (see, for example, Green, 2002Green J. Morphogen gradients, positional information, and Xenopus: interplay of theory and experiment.Dev. Dyn. 2002; 225: 392-408Crossref PubMed Scopus (65) Google Scholar, Tabata and Takei, 2004Tabata T. Takei Y. Morphogens, their identification and regulation.Development. 2004; 131: 703-712Crossref PubMed Scopus (351) Google Scholar, Schier and Talbot, 2005Schier A.F. Talbot W.S. Molecular genetics of axis formation in zebrafish.Annu. Rev. Genet. 2005; 39: 561-613Crossref PubMed Scopus (351) Google Scholar). Despite such progress, a lack of complete comfort with the morphogen concept persists among many biologists. Some prefer to explain pattern formation at the level of gene regulatory networks, attaching minor importance to movements of the molecules that genes encode (e.g., Davidson et al., 2002Davidson E.H. Rast J.P. Oliveri P. Ransick A. Calestani C. Yuh C.H. Minokawa T. Amore G. Hinman V. Arenas-Mena C. et al.A genomic regulatory network for development.Science. 2002; 295: 1669-1678Crossref PubMed Scopus (1136) Google Scholar). Others accept that morphogen gradients exist but question whether diffusion is adequate, or reliable enough, to create them (Kerszberg and Wolpert, 1998Kerszberg M. Wolpert L. Mechanisms for positional signalling by morphogen transport: a theoretical study.J. Theor. Biol. 1998; 191: 103-114Crossref PubMed Scopus (129) Google Scholar, Pfeiffer and Vincent, 1999Pfeiffer S. Vincent J.P. Signalling at a distance: transport of Wingless in the embryonic epidermis of Drosophila.Semin. Cell Dev. Biol. 1999; 10: 303-309Crossref PubMed Scopus (29) Google Scholar). Indeed, the need for reliability—or as engineers call it, robustness—in patterning has become something of an obsession among experimentalists and theoreticians alike, leading many to seek fresh approaches to how morphogens do their jobs (Kerszberg, 1996Kerszberg M. Accurate reading of morphogen concentrations by nuclear receptors: a formal model of complex transduction pathways.J. Theor. 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Rev. Lett. 2005; 94: 018103Crossref Scopus (83) Google Scholar). Although this is a healthy trend, there has been something of a disproportionate emphasis on discovering new molecular mechanisms, rather than thoroughly exploring what known mechanisms do. For example, no less than four separate mechanisms have been proposed as remedies for the (perceived) deficiencies of diffusion in transporting morphogens from one location to another (Kerszberg and Wolpert, 1998Kerszberg M. Wolpert L. Mechanisms for positional signalling by morphogen transport: a theoretical study.J. Theor. Biol. 1998; 191: 103-114Crossref PubMed Scopus (129) Google Scholar, Ramirez-Weber and Kornberg, 1999Ramirez-Weber F.A. Kornberg T.B. Cytonemes: cellular processes that project to the principal signaling center in Drosophila imaginal discs.Cell. 1999; 97: 599-607Abstract Full Text Full Text PDF PubMed Scopus (425) Google Scholar, Entchev et al., 2000Entchev E.V. Schwabedissen A. Gonzalez-Gaitan M. Gradient formation of the TGF-beta homolog Dpp.Cell. 2000; 103: 981-991Abstract Full Text Full Text PDF PubMed Google Scholar, Greco et al., 2001Greco V. Hannus M. Eaton S. Argosomes: a potential vehicle for the spread of morphogens through epithelia.Cell. 2001; 106: 633-645Abstract Full Text Full Text PDF PubMed Scopus (332) Google Scholar, Belenkaya et al., 2004Belenkaya T.Y. Han C. Yan D. Opoka R.J. Khodoun M. Liu H. Lin X. Drosophila Dpp morphogen movement is independent of dynamin-mediated endocytosis but regulated by the glypican members of heparan sulfate proteoglycans.Cell. 2004; 119: 231-244Abstract Full Text Full Text PDF PubMed Scopus (233) Google Scholar, Kruse et al., 2004Kruse K. Pantazis P. Bollenbach T. Julicher F. Gonzalez-Gaitan M. Dpp gradient formation by dynamin-dependent endocytosis: receptor trafficking and the diffusion model.Development. 2004; 131: 4843-4856Crossref PubMed Scopus (80) Google Scholar). Of course, the impulse to seek solutions for complicated problems in novel mechanisms is nothing new in science. One is reminded of Schrödinger's conviction that only new quantum physics could explain the permanence of genetic material in living beings (Schrodinger, 1944Schrodinger E. What is Life?. Cambridge University Press, Cambridge1944Google Scholar). With this as a backdrop, we may ask whether it is possible to retool existing theories of morphogens and pattern formation to determine to what extent the findings and concerns of modern experimental biologists can be accommodated and to what extent new mechanisms must be sought. This task can, I believe, be aided by better integrating morphogen theory into the wider context of spatial dynamics problems in biology. It can also be aided by formulating a richer, more complete description of what the performance objectives of patterning systems are. These points are explored below. Life is dynamic on many timescales. Molecules bind and react, cells come and go, organisms are born and die, species evolve. Molecular biologists have built solid frameworks for understanding the dynamic processes underlying life by applying concepts from chemistry, such as kinetics, thermodynamics, and affinity. Using such tools, areas such as metabolism, gene expression, and intracellular signaling have been explored in sophisticated, quantitative ways. Yet life is also dynamic on spatial scales. By this I mean that the stuff of life is not uniform in space—“well-stirred,” to use the chemistry expression—but is arranged in complex forms, from macromolecular assemblies to migratory herds. Questions involving spatial dynamics—how structure and pattern arise and are used—are as important in biology as those involving temporal dynamics. Spatial dynamics can be placed on a firm quantitative footing, too (the tools more often come from physics than chemistry), but it is a vastly more difficult endeavor. This is partly because space has three independent dimensions, but also because objects in it move forward and backward (things only go forward in time) and because interesting spatial questions often involve both space and time. The mathematical, computational, and bookkeeping hurdles associated with describing, analyzing, and simulating spatial phenomena can be formidable, especially when spatial organization cannot be approximated by a few well-stirred compartments. Problems of this type tend to be ones in which there is continuous, rather than stepwise, variation of items in space, e.g., those in which molecular gradients matter. Embryonic pattern formation obviously falls into this category, but so do other areas, such as the control of cell movement and shape by chemoattractants, the exploitation of paracrine and autocrine signaling, and the interaction of intracellular signaling with cell shape and structure. Progress toward quantitative understanding has been accelerating in each of these areas (e.g., Iglesias and Levchenko, 2002Iglesias P.A. Levchenko A. Modeling the cell's guidance system.Sci. STKE. 2002; 2002: RE12PubMed Google Scholar, Wiley et al., 2003Wiley H.S. Shvartsman S.Y. Lauffenburger D.A. Computational modeling of the EGF-receptor system: a paradigm for systems biology.Trends Cell Biol. 2003; 13: 43-50Abstract Full Text Full Text PDF PubMed Scopus (278) Google Scholar, Reas and Ballaro, 2004Reas P.G. Ballaro B. Reaction-diffusion equations for simulation of calcium signalling in cell systems.Riv. Biol. 2004; 97: 443-468PubMed Google Scholar, Meyers et al., 2006Meyers J. Craig J. Odde D.J. Potential for control of signaling pathways via cell size and shape.Curr. Biol. 2006; 16: 1685-1693Abstract Full Text Full Text PDF PubMed Scopus (153) Google Scholar), in part because of improvements in computing speed and power. Examination of the literature suggests that general themes and strategies, common to many spatially dynamic biological systems, are beginning to emerge. Accordingly, work being done on morphogens and morphogen gradients is increasingly relevant to the interests of biologists of all kinds. A common thread among many spatial dynamics problems in biology is that interesting behaviors arise out of the randomly directed movement of molecules. In the macroscopic world, the kind of motion we typically encounter is what physicists call ballistic: objects stay at rest until acted upon by force and then move in the direction of the force for some time until, usually through friction, they come to rest. In contrast, the molecules in and around cells are in constant motion at extremely high velocities (the result of thermal energy) but travel only miniscule distances before colliding with other objects and randomly changing direction (Berg, 1993Berg H.C. Random Walks in Biology.2nd ed. Princeton Univ. Press, Princeton, NJ1993Google Scholar). Because the trajectory of any individual molecule approximates a random walk, a set of statistical rules can describe their collective behavior. For example, if a group of molecules is placed at one location, it will spread out in a predictable way to fill the remaining space. That such molecules are driven from high to low concentration prompts a view of a concentration difference as a sort of force (“driving force”). This seems to suggest that, in thinking about aggregate molecular motion, everyday ballistic intuitions can be applied. Nothing could be less true. Consider that objects moving ballistically have a speed, whereas sets of objects spreading randomly do not. If it takes 10 min for a set of randomly moving molecules to travel an average of 10 μm, it will take 40 min to go 20 μm and 90 min to go 30 μm (a quadratic relationship between time and distance is a cardinal feature of random walks). What captures how molecules spread out is not a velocity but a diffusion coefficient (or diffusivity). Extracting this number from experimental data can be tricky, especially when molecules are not just moving but also undergoing binding, degradation, or chemical modification. In the case of morphogen gradients, the problem is not just that transport can be difficult to measure but that the application of ballistic thinking to experimental data so easily leads to misapprehensions. A striking example is shown in Figure 1A. The solid curve is the spatial gradient that would be formed by a freely diffusing morphogen with the diffusivity of a typical protein that is continuously produced for 200 s in a 20-μm-wide domain. The dashed curve shows what would happen under the same conditions if morphogen diffusivity were decreased by a factor of five. If this morphogen induces a particular gene at a concentration threshold of 8, then slowing its diffusion will reduce the width of the domain of gene induction from about 100 μm to 65 μm. But if the threshold for gene expression is 20, the domain of gene induction will increase from 15 μm to 35 μm. How can making a morphogen move slower cause it to act farther away? The explanation is that lowered diffusivity allows the morphogen to accumulate to much higher levels near its source. In a recent model of the sonic hedgehog (Shh) gradient that patterns the ventral neural tube of the chick embryo (Saha and Schaffer, 2006Saha K. Schaffer D.V. Signal dynamics in Sonic hedgehog tissue patterning.Development. 2006; 133: 889-900Crossref PubMed Scopus (86) Google Scholar), a version of just this situation arose, in which it was calculated that reducing Shh diffusion should increase the range of Shh action. This is a finding of immediate practical importance: current wisdom is that heparan sulfate proteoglycans promote the transport of Drosophila hedgehog (Hh), because removing them markedly decreases the range of Hh action (Bellaiche et al., 1998Bellaiche Y. The I. Perrimon N. Tout-velu is a Drosophila homologue of the putative tumour suppressor EXT-1 and is needed for Hh diffusion.Nature. 1998; 394: 85-88Crossref PubMed Scopus (414) Google Scholar, Lin, 2004Lin X. Functions of heparan sulfate proteoglycans in cell signaling during development.Development. 2004; 131: 6009-6021Crossref PubMed Scopus (500) Google Scholar). The findings of Saha and Schaffer, 2006Saha K. Schaffer D.V. Signal dynamics in Sonic hedgehog tissue patterning.Development. 2006; 133: 889-900Crossref PubMed Scopus (86) Google Scholar tell us that these very observations could just as easily mean that proteoglycans inhibit Hh transport! Another counterintuitive feature of random transport is the way it responds to obstacles. Consider an experimental manipulation that causes morphogens to accumulate at a discrete location, such as at the edge of a mutant cell clone facing a morphogen source. The usual interpretation is that the morphogen accumulates because it cannot pass through the clone (e.g., Entchev et al., 2000Entchev E.V. Schwabedissen A. Gonzalez-Gaitan M. Gradient formation of the TGF-beta homolog Dpp.Cell. 2000; 103: 981-991Abstract Full Text Full Text PDF PubMed Google Scholar). This is certainly how ballistic motion behaves: if I stand in a field and hit tennis balls in all directions, balls end up scattered about with a density that declines with distance; if a wall is placed at one point in the field, balls accumulate in front of it (Figure 1B). But it is not how molecules behave: if randomly moving molecules are released from a point source and encounter a barrier, they simply move away from, and around, the obstacle (Figure 1C). The ability to sidestep obstacles also explains why freely diffusing molecules will traverse any random maze in not much more time than it takes them to cross the same distance in free space (Rusakov and Kullmann, 1998Rusakov D.A. Kullmann D.M. Geometric and viscous components of the tortuosity of the extracellular space in the brain.Proc. Natl. Acad. Sci. USA. 1998; 95: 8975-8980Crossref PubMed Scopus (147) Google Scholar). This result helps us accept what otherwise may seem counterintuitive: that the labyrinth of tortuous intercellular spaces in most tissues poses little impediment to the free diffusion of morphogens. Limited awareness of this fact probably explains why morphogens are so often depicted moving exclusively within relatively unobstructed spaces on the apical surfaces of epithelia (Pfeiffer and Vincent, 1999Pfeiffer S. Vincent J.P. Signalling at a distance: transport of Wingless in the embryonic epidermis of Drosophila.Semin. Cell Dev. Biol. 1999; 10: 303-309Crossref PubMed Scopus (29) Google Scholar, Christian, 2000Christian J.L. BMP, Wnt and Hedgehog signals: how far can they go?.Curr. Opin. Cell Biol. 2000; 12: 244-249Crossref PubMed Scopus (68) Google Scholar, Belenkaya et al., 2004Belenkaya T.Y. Han C. Yan D. Opoka R.J. Khodoun M. Liu H. Lin X. Drosophila Dpp morphogen movement is independent of dynamin-mediated endocytosis but regulated by the glypican members of heparan sulfate proteoglycans.Cell. 2004; 119: 231-244Abstract Full Text Full Text PDF PubMed Scopus (233) Google Scholar, Lin, 2004Lin X. Functions of heparan sulfate proteoglycans in cell signaling during development.Development. 2004; 131: 6009-6021Crossref PubMed Scopus (500) Google Scholar). In reality, apical transport incurs far more difficulties than basolateral, due to the potential for massive morphogen loss to the overlying medium. The strange world of randomly moving molecules gets even stranger when we add in effects of production and destruction. In well-stirred systems, if the capacity for destruction of a substance exceeds its rate of production, a steady state can be approached in which the amount of the substance tends toward a constant value. The same is true for spatially dynamic systems, except that if production and destruction occur at different locations, stable gradients form. There are useful rules of thumb about such gradients. For example, between a discrete source and a discrete sink, a linear gradient will form. In contrast, when diffusion from a localized source is balanced by destruction that occurs with constant probability everywhere, the steady-state gradient will have an exponential shape. The length over which such a gradient decays to 1/e of its highest value—a number sometimes referred to as the length scale of the gradient—will be equal to the square root of the ratio of diffusivity to the degradation rate constant (Eldar et al., 2003Eldar A. Rosin D. Shilo B.Z. Barkai N. Self-enhanced ligand degradation underlies robustness of morphogen gradients.Dev. Cell. 2003; 5: 635-646Abstract Full Text Full Text PDF PubMed Scopus (221) Google Scholar, Gregor et al., 2005Gregor T. Bialek W. de Ruyter van Steveninck R.R. Tank D.W. Wieschaus E.F. Diffusion and scaling during early embryonic pattern formation.Proc. Natl. Acad. Sci. USA. 2005; 102: 18403-18407Crossref PubMed Scopus (222) Google Scholar, Reeves et al., 2006Reeves G.T. Muratov C.B. Schupbach T. Shvartsman S.Y. Quantitative models of developmental pattern formation.Dev. Cell. 2006; 11: 289-300Abstract Full Text Full Text PDF PubMed Scopus (76) Google Scholar). The length of any region of interest divided by the length scale of a molecule diffusing in it is its Thiele modulus, a term recently borrowed from engineering (Goentoro et al., 2006Goentoro L.A. Reeves G.T. Kowal C.P. Martinelli L. Schupbach T. Shvartsman S.Y. Quantifying the Gurken morphogen gradient in Drosophila oogenesis.Dev. Cell. 2006; 11: 263-272Abstract Full Text Full Text PDF PubMed Scopus (69) Google Scholar, Meyers et al., 2006Meyers J. Craig J. Odde D.J. Potential for control of signaling pathways via cell size and shape.Curr. Biol. 2006; 16: 1685-1693Abstract Full Text Full Text PDF PubMed Scopus (153) Google Scholar). Historically, most models of morphogen gradients and pattern formation—whether drawn from the “Wolpertian” or “Meinhardtian” perspectives—make the assumption that pattern is driven by steady-state gradients. Indeed, it seems logical that patterning information ought to be something stably maintained, especially because downstream responses of cells (e.g., gene expression) are relatively slow compared with the times required for diffusing molecules to move from one cell to another. Indeed, for some morphogen gradients, such as the Dpp and Wg gradients of the Drosophila wing imaginal disc, experiments show that the time over which gradients develop is short compared with the several days over which patterning occurs (Strigini and Cohen, 1999Strigini M. Cohen S.M. Formation of morphogen gradients in the Drosophila wing.Semin. Cell Dev. Biol. 1999; 10: 335-344Crossref PubMed Scopus (93) Google Scholar). But other cases are less clear: in the zebrafish embryo, only 4–5 hr elapse between late blastula—when localized expression domains of bone morphogenetic proteins (BMPs), Wnts, Nodal, FGFs, and Retinoic acid emerge—and late gastrula stages, by which time an enormous amount of patterning orchestrated by these molecules has taken place (Schier and Talbot, 2005Schier A.F. Talbot W.S. Molecular genetics of axis formation in zebrafish.Annu. Rev. Genet. 2005; 39: 561-613Crossref PubMed Scopus (351) Google Scholar). In the Drosophila embryo, things move even faster: at 25°C, the bicoid gradient forms and does its job in anteroposterior patterning in less than 2 hr (Gregor et al., 2005Gregor T. Bialek W. de Ruyter van Steveninck R.R. Tank D.W. Wieschaus E.F. Diffusion and scaling during early embryonic pattern formation.Proc. Natl. Acad. Sci. USA. 2005; 102: 18403-18407Crossref PubMed Scopus (222) Google Scholar), and the BMP morphogen gradient at the dorsal midline forms and specifies dorsoventral pattern in under 1 hr (Dorfman and Shilo, 2001Dorfman R. Shilo B.Z. Biphasic activation of the BMP pathway patterns the Drosophila embryonic dorsal region.Development. 2001; 128: 965-972PubMed Google Scholar). Can such gradients achieve a steady state rapidly enough? For most simple scenarios, the dominant factor determining the rate of approach to steady state is the average lifetime of morphogen molecules (i.e., the inverse of their degradation rate constant [Gregor et al., 2005Gregor T. Bialek W. de Ruyter van Steveninck R.R. Tank D.W. Wieschaus E.F. Diffusion and scaling during early embryonic pattern formation.Proc. Natl. Acad. Sci. USA. 2005; 102: 18403-18407Crossref PubMed Scopus (222) Google Scholar, Lander et al., 2005Lander A.D. Nie Q. Vargas B. Wan F.Y.M. Aggregation of a Distributed Source in Morphogen Gradient Formation.Stud. Appl. Math. 2005; 114: 343-374Crossref Scopus (14) Google Scholar]). For bicoid, an intracellular protein, a lifetime on the order of minutes is plausible. Of course, as Gregor et al., 2005Gregor T. Bialek W. de Ruyter van Steveninck R.R. Tank D.W. Wieschaus E.F. Diffusion and scaling during early embryonic pattern formation.Proc. Natl. Acad. Sci. USA. 2005; 102: 18403-18407Crossref PubMed Scopus (222) Google Scholar argue, because degradation rate also affects steady-state length scale (see above), the need to reach a steady state within 1–2 hr should constrain bicoid gradients to a maximum size of 1–2 mm (this, interestingly, seems to be about as big as insect embryos get). For secreted polypeptide morphogens, degradation usually proceeds through sequential steps of binding, receptor-mediated endocytosis, and lysosomal proteolysis (Scholpp and Brand, 2004Scholpp S. Brand M. Endocytosis controls spreading and effective signaling range of Fgf8 protein.Curr. Biol. 2004; 14: 1834-1841Abstract Full Text Full Text PDF PubMed Scopus (91) Google Scholar, Marois et al., 2006Marois E. Mahmoud A. Eaton S. The endocytic pathway and formation of the Wingless morphogen gradient.Development. 2006; 133: 307-317Crossref PubMed Scopus (136) Google Scholar). It is questionable whether such events are fast enough for a steady-state BMP gradient to develop in the early fly embryo in less than 1 hr. Indeed, the idea that this gradient does not pattern under steady-state conditions is supported by experimental data, as there are combinatorial phenotypes of certain mutations that are difficult to explain otherwise (Mizutani et al., 2005Mizutani C.M. Nie Q. Wan F.Y. Zhang Y.T. Vilmos P. Sousa-Neves R. Bier E. Marsh J.L. Lander A.D. Formation of the BMP activity gradient in the Drosophila embryo.Dev. Cell. 2005; 8: 915-924Abstract Full Text Full Text PDF PubMed Scopus (123) Google Scholar). Other cases in which analysis suggests that steady states are not achieved by morphogen gradients include the previously mentioned model of neural tube patterning by Shh (Saha and Schaffer, 2006Saha K. Schaffer D.V. Signal dynamics in Sonic hedgehog tissue patterning.Development. 2006; 133: 889-900Crossref PubMed Scopus (86) Google Scholar) and some models of the Drosophila embryo segmentation network (Gursky et al., 2004Gursky V.V. Jaeger J. Kozlov K.N. Reinitz J. Samsonov A.M. Pattern formation and nuclear division are uncoupled in Drosophila segmentation: comparison of spatially discrete and continuous models.Physica D. 2004; 197: 286-302Crossref Scopus (34) Google Scholar). Why is it so important to know whether patterning by morphogen gradients occurs under transient or steady-state conditions? For one thing, the underlying mathematics tells us that responses of gradients to external manipulations can be very different in the two regimes. Consider the predicted effects of “nonspecific” binding sites: if a diffusing morphogen binds reversibly to immobile sites in its environment, then to the extent it does so, it spreads more slowly. At any time during the approach to steady state, the observed gradient will be narrower (as though diffusivity had decreased). But at steady state