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Mechanisms for Positional Signalling by Morphogen Transport: a Theoretical Study
J. theor. Biol. (1998) 191, 103–114
Mechanisms for Positional Signalling by Morphogen Transport: a Theoretical Study M K*‡ L W† *Neurobiologie Mole´culaire, Institut Pasteur, 25, rue du Docteur Roux, F-75724 Paris Cedex 15, France; and the †Department of Anatomy, University College London, Medawar Building, Gower Street, London WC1E 6BT, UK (Received on 12 May 1997, Accepted on 23 October 1997)
Gradients of cellular activities are ubiquitous in embryonic development. It is widely believed that the inhomogeneous spatial distribution of a morphogen would be able to set up such gradients. But how then does the morphogen propagate in the first place? Straightforward molecular diffusion is often proposed as a possible mechanism. We first show that, surprisingly, the mere binding of the diffusing morphogen to its membrane receptors suffices to prevent the establishment of a concentration-based positional signalling system. Instead, a flat, saturated distribution of receptor-bound morphogen builds up. Because the distribution spreads gradually from the morphogen source, however, cells may still know their position if they are able to integrate the morphogen signal in time. The irregularities of diffusion in the complex extracellular medium would in fact be partially compensated for by such time summation. Another, non-exclusive possibility is that morphogen transport does not occur by simple diffusion only. We put forth a novel model of receptor-aided, directed diffusion that achieves a spatial distribution of morphogen. Our model is based, as an illustration, on the properties of members of the TGFb family of molecules. We show that two simple hypotheses regarding the kinetics of TGBb binding to its receptors suffice to establish a remarkable transfer mechanism whereby a morphogen such as activin could be both propagated along cell membranes, and transferred between cells that are in contact. The model predicts that morphogen propagation properties depend strongly on the closeness of cell-cell appositions, does not necessitate protein synthesis, accumulation or slow degradation (in contrast to the diffusion/time integration model), and that the morphogen is localised mostly on or close to cell membranes. 7 1998 Academic Press Limited
Introduction In the course of embryonic development, the formation of spatial patterns of cellular differentiation is of fundamental significance. This process can involve induction—interactions between neighbouring cells (Slack, 1993)—and longer-range positional signalling by morphogens (Wolpert, 1969). There is quite good evidence from the development of both Drosophila wings and vertebrate limbs
‡Author to whom correspondence should be addressed. E-mail: mkersz.pasteur.fr 0022–5193/98/050103 + 12 $25.00/0/jt970575
that positional signalling mechanisms are indeed involved in patterning (Nellen et al., 1996; Tickle, 1996). Experimental (O’Reilly et al., 1995) and theoretical (Kerszberg & Changeux, 1994; Kerszberg, 1996) results indicate that membrane receptors and cellular transduction mechanisms can indeed operate so as to read the concentration of signalling molecules with striking precision. But how is such a graded concentration set up in the first place? To what degree is the build up of the morphogen distribution a process independent from its interpretation into differentiation patterns? This is the question we want to examine here. 7 1998 Academic Press Limited
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A commonly invoked mechanism for long-range signalling is based on the diffusion of a signalling molecule. Crick (1970) pointed out that diffusion has just the right characteristics for a cellular signal required to be propagated over less than one millimeter during a period of several hours. Clearly, the concentration gradient that would be set up by release of a diffusible substance could provide cells with positional information. However, gradients, if they are to pattern the embryo, must be constructed with a high degree of reliability; yet it is unlikely that a substance freely diffusible in the extracellular space could provide a very reproducible long-range distribution in the face of screening effects by cells themselves, dependence on available intercellular space, or extraneous perturbations such as blood flow in the vertebrate limb. Assume however these objections are ignored: we show here that simple molecular diffusion cannot establish a graded morphogen distribution. This is because of the binding of a diffusing morphogen to its membrane receptors, which was not included in previous theoretical treatments (Crick, 1970). Binding leads to trapping: morphogen cannot move beyond a point before receptor there has been maximally occupied. The result is that a narrow front advances from the morphogen source, leaving behind a flat, saturated distribution of receptor-bound morphogen. We suggest that there are two (non-exclusive) ways out of this situation. The first possibility is that the morphogen signal transduction cascade allows for time integration or summation. The transduced morphogen signal will then grow in time when receptor is occupied. Close to the morphogen source, this growth can start as soon as morphogen is injected; further away, the growth starts later and the integrated signal lags behind: this gives cells a way to read their position. The second possibility is that morphogen transport does not take place by diffusion only. (It is of course possible that morphogen is not transported at all, and that cell position is determined by successive inductions—we shall discuss this in the Conclusions). Here we propose a novel mechanism for the progressive spread of a morphogen. Our scheme eschews simple physical diffusion, but nevertheless implies the actual transport of the morphogen molecules: this now happens through cell-biological, active, and possibly energy-dependent processes. The transport is done through the agency of those very membrane receptors which render simple diffusion inoperant. The importance of morphogen receptors is usually seen to lie in their pivotal role as triggers of the intracellular signalling cascade. We demonstrate
however that membrane receptors could quite possibly have acquired another significance, upstream of this, by contributing to the cell- and system-wide spatial repartition of their ligand. This would mostly involve the cell membranes and, to a much lesser degree, the intercellular medium. Our model is loosely based on the properties of members of the TGFb family of molecules. These have been implicated in a wide variety of morphogenetic processes (Hogan, 1996). Among the best studied are decapentaplegic (in Insect development) and activin-like molecules (in early Xenopus embryogenesis) (Nellen et al., 1996; Gurdon et al., 1995; Jones et al., 1996). We do not propose however to model here in detail signalling by any of these individual molecules or receptors. Rather, we use what is known of them as hints for inferring plausible general molecular mechanisms which may fulfill the requirements for efficient morphogen transport. We then show by computer simulation that the proposed mechanisms may indeed function as envisaged. The biochemical mechanisms needed for receptoraided, directed transport fall well within the range of complexity exhibited by membrane receptors. In short, we find that two simple hypotheses regarding the kinetics of ligand binding to TGFb-like receptors suffice to establish a remarkable transfer mechanism whereby a morphogen would be propagated along cell membranes, and transferred between cells. It is possible that this active transport needs to be driven by receptor transphosphorylation as happens during ligand binding (which transphosphorylation is instrumental in initiating the intracellular cascade itself). In any case, a graded, stable distribution is quickly achieved, with its maximum close to the morphogen source. In what follows, we present the model’s biological premises and its formal definition; we then use computer simulations to show (1) the failure of diffusion to establish a graded morphogen distribution, and (2) to demonstrate our own model’s functioning. Model The biological premises which underlie the model originate in the molecular biology of activin, BMP-4 and other TGFb receptors (Massague´ & Polyak, 1995; Derynck, 1994). We introduce no receptor configurations other than those known from the TGFb context: our only additional hypotheses concern the kinetic coefficients governing the transitions among some of these configurations.
: (a) R II
RI
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TbR-I, and that transphosphorylation of TbR-I by TbR-II takes place. The intracellular signalling cascade is thus initiated. This description of the upstream activin pathway certainly corresponds to a highly simplified view, but we shall proceed on the assumption that it provides sufficient basis for understanding the kinetics of ligand-receptor interactions. We now state our kinetic hypotheses, as follows: (a) the TbR-I–TbR-II heterodimer has a nuch lower affinity for activin than the TbR-II monomer has, so that, upon binding of TbR-I, the activin-TbRII complex quickly dissociates, leaving behind the intact TbR-I–TbR-II heterodimer (see Fig. 1). (b) the TbR-I–TbR-II heterodimer has a relatively long lifetime, compared with the rate of propagation of activin in the system (as discussed below).
(b) R II*
These are the major hypotheses of the model. In addition, we have examined the possible importance of a specific, active morphogen transfer mechanism at cell-cell contacts:
(c) R IR II
(c) an activin-TbR-II complex on the membrane of a given cell may exchange its activin with a TbR-II molecule located on an immediately adjacent patch of membrane belonging to another cell (Fig. 2); this does not require configurations other than those already discussed, but does indeed necessitate some new, intercellular reaction mechanism. The picture on
F. 1. A simple model for the activin receptors. (a) There are two receptor types, I and II. (b) Activin (black squares) binds to type-II receptor, inducing an allosteric transformation of its intracellular terminal. This enables dimerisation with type-I receptor. (c) The dimer forms, and releases the activin after a time which is short compared to the dimer’s own lifetime. The intracellular signalling pathway is initiated by type-I transphosphorylation.
Several types of TGFb receptors have been discovered. The two major, best studied subtypes are receptor type I (TbR-I) and II (TbR-II). It is not unreasonable to summarise what has been gathered of their properties in the following simplified way (Fig. 1).* Activin probably does not bind to TbR-I as a monomer (or dimer). Instead, binding first occurs with TbR-II (monomer or dimer). Secondly, it seems that the activin-TbR-II complex then reacts with
*For simplicity we use the activin language, but this is just a matter of convenience; also, it is quite possible that TbR-I and TbR-II are not active as monomers but as homodimers (Luo & Lodish, 1996), so that, when we talk of TbR-I, say, it may in fact be that a TbR-I–TbR-I homodimer is the relevant unit.
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Cell 2
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Cell 1
F. 2. Morphogen exchange among receptors on adjacent cells. The drawing makes it clear that even if the exchange is not eenergy-driven (i.e. ATP-dependent), its probability may nevertheless be quite high.
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Fig. 2 makes the chosen mechanism plausible. The behaviour of the model with and without mechanism c will be discussed. Our point is not to present compelling evidence for our hypotheses; but they are rather straightforward, testable, and they overcome very simply the major hurdle faced by diffusion, that of morphogen trapping by the receptors. One may think of other hypotheses which would achieve a similar goal. But the unavoidable conclusion of our paper will be that at least some such hypothesis is needed in order to understand morphogen gradient setup. For instance, if TbR-I and TbR-II function as homodimers, ligand-induced TbR-I–TbR-II interactions may lead to a short-lived tetrameric configuration, with the subsequently released monomers unable to form ligand-binding homodimers again for some time. Our particular model may now be defined formally by the kinetic equations: RII + A _ R* II
(1)
R*II + RI : RI RII + A
(2)
RI RII : RI + RII
(3)
R*II,cell1 + RII,cell2 _ RII,cell1 + R* II,cell2
(4)
where, for simplicity, the receptors are denoted RI and RII , the morphogen (activin or other) by A, and in the last equation the location of molecules is indicated. The basic phenomena that control the model’s behavior are simple (Fig. 3). Two distinct processes
1
control diffusion on a given membrane and morphogen transfer between cells. On any particular cell, morphogen A binds to the closest available RII , then RI displaces the morphogen, which is thus free to migrate to further sites with free RII . (This is the major difference with simple diffusion, where morphogen is not so readily released.) Since the lifetime of the RI RII heterodimer is long, sites already visited on the membrane are ‘‘burned out’’ and the morphogen will become bound at locations more and more distant from its place of origin. When most of the receptors on a given cell are dimerised, A is free (actually, it is forced) to jump to another cell with free RII . Membrane receptors in the model thus effectively operate as a ‘‘bucket brigade’’ carrying over what can be a very limited amount of morphogen. Because the transfer is not complete, and the lifetime of A itself is not infinite, ultimately the morphogen’s effect will decay at sufficient distance from its source (Gurdon et al., 1994, 1995). We demonstrate these results by computer simulations of the model.
Computer Simulations For the sake of the computations, cells are assumed to exist in a restricted two-dimensional space (Fig. 4). This is artificially divided in a honeybee lattice of ‘‘bins’’, with 124 bins in the vertical dimension, and 248 bins horizontally. These bins are analogous to pixels in a digitised image: their location is fixed, but
2 3
0
F. 3. How receptors function as a ‘‘bucket brigade’’. Free receptors are present on a given cell membrane (0); type-II receptor binds the impinging activin molecules (1); this enables type-I receptor to join the complex. The receptor dimer quickly releases activin, but remains bound (2). The bound type-I/type-II receptor complex does not bind activin with sufficient affinity, so that the released activin diffuses preferentially to free type-II receptor located further on the membrane. At a cell-cell contact or close apposition (3), transfer to another cell occurs either by receptor-receptor exchange or by short-range extracellular diffusion, and the process goes on.
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allow for membrane binding and unbinding however, some extracellular species may also be allowed a non-zero concentration at the membrane. Diffusion is described by probabilities per time step of hopping betwen adjacent bins. Reactions occur locally among the various molecular species. We now proceed to describe in more detail the parameters characterising these processes and the course of a computer simulation.
Parameters A limiting parameter in solving the model numerically is morphogen diffusion, because orders of magnitude for the diffusion coefficients are known and must be respected. From data on membranebound molecular diffusion, one can gather that DS 2 10−9 cm2 s−1 (Jacobson et al., 1987; Kusumi et al., 1993). As to volume diffusion, one may estimate (Crick, 1970; Jain, 1990) DV 2 10−7 cm2 s−1. Since DV is much the larger, we have to set our length and time units to be able to accomodate it. Now, F. 4. The simulations’ underlying geometry. Two-dimensional space is partitioned into a beehive array of pixels or ‘‘bins’’. Closed loops of such bins represent cell membranes. In (a), we show one such cell (m) and part of another (m') in contact with it. Molecules are able to diffuse on a membrane and between membranes (arrows). In (b1), (b2) it is seen how membranes are able to move (arrows).
their color or luminosity reflect the presence, locally, of objects such as molecules, or parts of objects such as cell membranes. Thus, 180 closed strings of adjacent bins compose the ‘‘membranes’’ of 180 cells; membrane bins as well as extracellular ones are occupied by molecules in given concentration. The membranes are mobile; membrane molecules are carried along with their membrane (convection), and extracellular molecules are displaced as membranes move (volume convection). In addition, membrane molecules diffuse on the membrane, and extracellular molecules may diffuse in the extracellular medium; to
*This is in arbitrary units; conversion to actual concentrations may be done approximately by noting that our volume unit is about D2.1 mm = 4 mm3, so that a receptor concentration of 2.0 in arbitrary units corresponds to 2.0[N ·0.25·1015] particles litre−1 where N must be adjusted to yield the actual experimental value, and will then be used to convert other concentrations and the binding constants (see below). †These constants can be compared with experimental values by noting that they are expressed in reactions per t (s) and per [N ·0.25·1015]2 (particles litre−1)2 (see previous footnote).
(i) in order to have sufficient spatial resolution on cells with diameter 215 mm, we take each bin to have a length of D = 2 mm. (ii) Due to numerical instability, the probabilities for hopping from bin to bin per time step t may not exceed P 2 0.2. By dimensional argument, DV = 10−7 cm2 s−1 2 PD2/t, which yields t 2 0.08 s. In accordance with the estimates quoted above, on-membrane hopping probabilities are then taken as PRI = PRII = 0.002 (see above) and PR*I = PRI RII = 0.001 (to account for larger molecular weight). The absolute concentration of receptor and ligand molecules will not be our main interest in this theoretical study [see however Fig. 5(b)]. The total number of receptors on each cell membrane is assumed constant throughout (the local concentration may vary of course), and starts at a uniform 2.0 per membrane bin.* (We neglect the effects of receptor turnover over the time scale of our simulations.) Certain relative concentrations are important, and are determined by initial conditions and by some of the reaction affinities. The kinetic coefficients k are introduced as reaction probabilities per unit time t or inverse lifetimes, thus: , eqn (1): kbind = 0.1 and kunbind = 10−6: only kbind is significant, as we assume that reaction (2) will be the one actually to deplete the bound state (thus, the receptor affinity kunbind /kbind is essentially irrelevant);†
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F. 5(c). F. 5. Distribution of morphogen when diffusion operates alone (in extracellular space and on cell membranes). (a) A continuous source of morphogen has been operating on the left side for T = 67500 time steps (1 step 1 0.08 seconds, i.e. T = 1.5 h). The concentration of receptor-bound morphogen is shown. (Thus, only the membrane bins are displayed.) The lighter the bin, the larger the concentration. We see that morphogen is sharply restricted to the vicinity of the source: the length d0.5 where concentration drops to half its maximum is indicated by the black arrow. Diffusion in the absence of receptor would lead to an exponential distribution with d0.5 as indicated by the gray arrow. The reason for the considerable discrepancy is that receptor traps the morphogen on cell membranes. (b) Distribution of bound morphogen at T = 0.5 h when the receptor concentration on the cell membranes is reduced to 1% of its previous value in 5a. Notice that transport is much faster, but the receptors are saturated over much of the system. (c) The free morphogen distribution under the same conditions as in (b). The free morphogen has a graded distribution, but is unable to contribute to receptor-mediated signalling.
only, morphogen A is thus generated according to dA = 10−3. dt F. 5(a) and (b).
, eqn (2): kdimer = 0.10; , eqn (3): kdiss = 0.002, i.e. the lifetime of the dimerised receptor is about 500t or 40 s. We assume the morphogen originates from a continuous source located on the left-hand side margin of the system: in the leftmost bins
(5)
Results We perform three types of simulation: first, we examine the case of diffusion alone; we then simulate the behaviour of our full model. Finally, we control our results by modifying in turn some of the model’s basic parameters. When kexch = kdimer = 0, only membrane and extracellular diffusion subsist. The results of a
: representative simulation* show that the morphogen, assumed to diffuse with a coefficient D = 10−7 cm2 s−1, is confined, after one and a half hour, to the first cell layers beyond the source [see Fig. 5(a)]. It has travelled much less than predicted for free diffusion, as shown on the Figure. This is in fact easy to understand, as morphogen, no matter what its rate of injection, must fill statistically available receptor sites before being able to move further in the system. Thus, ultimately, a flat distribution of trapped morphogen is slowly generated, with free morphogen (not bound and therefore not effective) superimposed on that. Swamping the receptors with morphogen does not help, for if transport of morphogen is indeed faster when the receptor is made less abundant [see Fig. 5(b), where receptor concentrations have been divided by 100], the distribution of bound morphogen appears quite uniform, even though free morphogen seems distributed according to an exponential [Fig. 5(c)]. Because of its uniformity, the distribution of bound morphogen cannot be used to impart positional information to cells. There remains of course the possibility that cells have a way to measure the time of arrival of the morphogen concentration front; this could be achieved e.g., if transduction by the activated receptor were cumulative, i.e., integrated over time: this would lead to a signal strength decreasing with the distance from the source, as receptor far away would be activated later. Because of the summation in time, some of the irregularity inherent to diffusion in a complex, random medium would be smoothed out. If the receptor had a very small probability kbind of binding the morphogen, some morphogen would be able to diffuse before being trapped, and trapped morphogen concentration could increase progressively, giving rise indeed to a gradient. (Remember that kbind is not the receptor affinity: this is kunbind /kbind ; kbind measures a dynamic quantity, the probability that the receptor will bind ligand available in its vicinity, while affinity is the equilibrium ratio of unbound to bound ligand, irrespective of whether this is reached or not in the available time.) A small kbind is, however, difficult to justify from what is known of receptor-ligand interactions. In our model, kbind represents the *In a given simulation, the starting cell configuration and subsequent movements are generated in a partially random way; This is controlled by a quasi-random number generator used to ‘‘throw dice’’ each time a ‘‘decision’’ must be made (say, for a membrane to move right or left according to its current environment). This is the so-called Monte–Carlo algorithm (Metropolis & Ulam, 1949). The generator must be ‘‘seeded’’ by a starting number: each seed generates a different quasi-random series, hence a new repeat of the ‘‘experiment’’.
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probability that a ligand molecule located in a 2 mm bin next to a cell membrane will be caught by membrane receptor within the time unit t. Since the bin dimensions were chosen so that diffusion is essentially complete over the bin within t, the probability that ligand and receptor collide during t is essentially 1, and this is usually taken as the factor limiting kbind (Steinfeld et al., 1989). Any significant departure from kbind = 1 must be due to steric or other special accessibility effects. To observe, after a few hours’ time, a non-uniform distribution of bound morphogen with a reasonable spatial range required taking kbind Q 10−5 (data not shown), which seems extraordinarily low indeed: nothing in the molecular biology of, say, the TGFb receptors suggests the plausibility of this unusual value. On the other hand, the dimerisation mechanism included in our model frees morphogen for diffusion automatically, as we now show. : We now turn to simulations of our complete model. Results are presented on Fig. 6. We observe in Fig. 6(a) and (b) the progression of receptor-bound morphogen molecules and of receptor dimerisation further and further away from the morphogen source (on the left). We see that, after 1.5 h, morphogen is distributed over about 15 cell diameters, with a sharply peaked concentration curve, the maximum of which lies at the first cell layer. The dimerised receptor follows at all times a smoother, wider distribution, with a clear tendency to saturation. From these results, it is evident that either receptor-bound morphogen or dimerised receptor, or some combination thereof, could be used to impart positional information to embryonic cells in due time for, e.g., Xenopus development. Dimerisation propagates at greater speed, but also saturates earlier. Note that in our model, propagation can actually proceed faster than in pure diffusion: this is because receptor dimerisation behind the morphogen propagation front largely prevents morphogen backward movement, which is always possible in diffusion. We have plotted on Fig. 6(c) the total concentration of morphogen in the system. Note that much of the morphogen is located close to or on the cell membranes. It is one of our predictions that little morphogen should be found in the extracellular space. This conclusion could, of course, be altered in a model including extracellular matrix morphogen binding sites. Morphogen is transported mostly in the vicinity of the membranes (1) by diffusion away from membrane areas where receptor is dimerised and thus
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Recovery in the face of a variety of perturbations is a fundamental feature of embryogenesis. Here we
assess the various contributions to morphogen transport and the way they contribute to a stable morphogenetic process. We do this by comparing the final (1.5 h) morphogen distributions obtained with the full model (see above) and (a) the model with strongly reduced extracellular diffusion (Fig. 7); (b) the model with a sharp deficit in the dose of injected morphogen (Fig. 8); and (c) the model in the absence of active cell-cell exchange interaction (kexch = 0, see Fig. 9). Figure 7 allows for a comparison of morphogen [part (a)] and dimer [part (b)] distributions in the
F. 6(a).
F. 6(b).
releasing its ligand (which it is unable to bind again for a while) and toward areas with free monomeric type II receptor; and (2) by receptor-receptor exchange (Fig. 2) or simple diffusion again at cell contacts or at points where cells are in close apposition (see Fig. 9 for a discussion of the relative importance of these two cell-cell transfer mechanisms).
: normal situation (as above) and when the diffusion coefficient in extracellular space is reduced by a factor 10 (to 10−8 cm2 s−1) Clearly, the distributions are affected, but not nearly as severely as one would expect. Volume diffusion is only one of the forces behind morphogen movement in our model. On Fig. 8, we have plotted the effect of a factor 10 reduction in the rate of morphogen injection. We do not display the morphogen distribution, which is dramatically lowered. But the effect of a low morphogen dose is not nearly as marked when one considers the receptor dimer distribution: concentrations are reduced by a factor 2 to 3 at most. This welcome lack of sensitivity stems from the long lifetime assumed for dimerised receptors, which allows for a large buildup of dimerised receptor concentration with little actual morphogen entry into the system. Here is certainly an argument for the
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possible relevance of dimerised receptor signalling in the generation of positional information. Finally, Fig. 9 is an illustration of the relative contribution to morphogen transport of the cell-cell exchange interaction. Setting kexch = 0 actually has little effect on bound morphogen distribution (not shown). But it markedly influences the dimer distribution, by facilitating faster movement of relatively small quantities of morphogen away from the source. When all our data are considered, it seems evident that the major factor for the success of the model is receptor dimerisation. This importance stems from the fact that it allows fast morphogen movement from one cell face to the opposite, while cell-cell morphogen exchange intervenes only punctually, at cell contacts. This punctual transfer can be effected passively (through release by receptor on one cell and reuptake on the next) almost as well as by the kexch –mediated mechanism. Still, close juxtaposition of cells is essential for either process to take place reliably. Conclusions and Perspectives
F. 6(c). F. 6. Time-lapse views of the spatial distribution of morphogen effects in a typical numerical ‘‘experiment’’ involving the full model. (a) Top: distribution of morphogen bound to type II receptor at simulation step 22500 (i.e. 0.5 h); middle: the same after 1.0 h and bottom: after 1.5 h. The morphogen has spread by about 15 cell diameters. A graph of the final distribution is shown below the time-lapse view. Morphogen transport and binding properties of the full model clearly satisfy the conditions required for the establishment of positional information [compare Fig. 5(a)]. (b) The distribution of heterodimerised (type I/type II) receptor, during the same simulation as in 6a. These results demonstrate that receptor dimers may also function as signals of positional information. But note that signs of saturation of dimer concentration already appear at 1.5 h. (c) Total morphogen present in the system at T = 1.5 h. Morphogen is located mostly on membranes and in the immediate vicinity of cells.
We have constructed and solved a model for directed morphogen transport over the extent of an embryonic field, inspired by the properties of activin and other TGFb receptors. We have proved that simple kinetic hypotheses regarding the states of these receptors suffice for them to carry the morphogen over the required distances. This is in contrast with what happens with simple diffusion. Why have we chosen a mechanism which actually carries the morphogen molecules themselves? Might it not have been simpler to assume that cells somehow induced by activin near the latter’s source, would then transmit a further inductive signal to their immediate neighbours, which would then carry the phenomenon over? Some experimental evidence seems to favour such an induction cascade (Reilly & Melton, 1996). We found however that the problem with such a scheme (of which we have tried many variants) is that there is no simple way to prevent the signal from being transmitted arbitrarily far from the source. To function, the scheme requires a cell to receive a signal, read its strength accurately (much more accurately than when it interprets morphogen signals, which is done according to thresholds), and transmit to its neighbours a precisely modulated, slightly weaker excitation, or induction signal. We have not found a transduction mechanism leading to the appropriate attenuation: either the attenuation is too strong, and the signal dies away; or it is too weak, and the signal
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F. 7. The effects of a lowered efficacy of volume diffusion on morphogen gradient buildup. Here extracellular diffusion coefficient has been reduced 10-fold to DV = 10 − 8 cm 2 s − 1 The decrease is not fully reflected in the final graded distributions. (a) Receptor-bound morphogen distribution at T = 1.5 h with diffusion normal (gray dots) and slow (white dots). (b) Same as (a) but for the receptor dimer distribution.
propagates indefinitely. This is impossible with morphogen physically transported away from a source: ultimately, the morphogen decays, and so does the signal. It remains possible to imagine spatially specialised sorts of induction cascades. Assume a strong attenuation of the inductive signal, such that its range is, say, not more than one cell diameter. A first possibility is this: if cells express a battery of receptors, and the first cell row secretes one of the possible ligands, this could induce the second row (but that single row only) to display a second ligand, which would induce in the third row display of yet
another ligand and so on. One can imagine mechanisms for such a change of display, some involving gene transcription and some not. Yet another possibility is that all cells express all ligands, but receptors on cells of the first row, bound with the unique ligand originating at the source, induce receptor on the next cells to adopt a configuration apt to bind another ligand specific of the second row, and so on. (The configuration in question could be e.g. the receptor subunits’ oligomerisation state). At any rate, the fact that we could not build a safe long range communication mechanism without resorting to actual morphogen transport leads us to
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F. 8. Distribution of receptor dimer at T = 1.5 h with a normal rate of morphogen injection (gray) and with a 10-fold reduced injection rate (white). Note that the reduction in dimer concentrations is two to three-fold at most, due to the long lifetime assumed for receptor dimers. In contrast, the receptor-bound morphogen distribution suffers the full 10-fold reduction (not shown).
suggest that the only viable alternative to the sort of transfer process we have introduced (or to diffusion plus signal summation) is a complex variety of membrane molecules playing specialised roles as a function of distance to the morphogen source. Although it does not preclude such complexity, the present model is by far the more parsimonious one. On the one hand, it brings attention to the importance, for morphogenesis, of membrane binding
kinetics, which are experimentally accessible; on the other, it makes a number of testable predictions, namely that most of the morphogen will find itself on cell membranes, not in the extracellular space; that cell-cell contacts are essential for morphogen propagation (see e.g. Wilson & Melton, 1994), even if no specific morphogen transfer reaction occurs there; that the range of propagation is sensitive on morphogen lifetime, faster degradation meaning
F. 9. Receptor dimer distribution, under the same conditions as in Fig. 8 (gray dots), and with the exchange interaction kexch = 0 (white dots). Transport between cells is then left to morphogen re-release and normal diffusion: clearly, morphogen movement is hampered. Note that this would hardly be visible on the bound-morphogen data (not shown). But because the morphogen remains closer to the source and dimerises more receptor there, it leads to a noticeable saturation of the dimer distribution as displayed.
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reduced spatial range; and that the propagation does not depend on continued protein synthesis, although it may depend on energy supply. Our theoretical considerations also suggest that morphogen signalling would be enhanced if the morphogen transduction pathway were to operate, at least to some degree, in such a way as to integrate in time the instantaneous morphogen signal. One way in which diffusion and receptor-aided transport could be distinguished experimentally is by introducing a TbR-I negative-dominant mutant clone (defective in the TbR-II binding domain) in an otherwise normal preparation: our model predicts that the clone should leave a shadow of reduced activin concentration and signalling, while the diffusion model would predict the opposite. In this case, the induction-cascade model makes a prediction similar to our model’s, however. On the other hand, introduction of a TbR-I negative dominant clone defective in the intracellular domain would not affect directed transport or diffusion, but would disrupt some types of induction cascades.
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J, K., I, A. & I, R. (1987). Lateral diffusion of proteins in membranes. Ann. Rev. Physiol. 49, 163–175. J, R. (1990). Physiological barriers to delivery of monoclonal antibodies and other macromolecules in tumors. Cancer Res. 50 (Suppl.), 814–819. J, C. M., A, N. & S, J. C. (1996). Signalling by TGF-b family members: short-range effects of Xnr-2 and BMP-4 contrast with the long-range effects of activin. Curr. Biol. 6, 1468–1475. K, M. & C, J.-P. (1994). A model for reading morphogenetic gradients: autocatalysis and competition at the gene level. Proc. Natl. Acad. Sci. U.S.A. 91, 5823–5827. K, M. (1996). Accurate reading of morphogen concentrations by nuclear receptors: a formal model of complex transduction pathways. J. theor. Biol. 183, 95–104. K, A., S, Y. & Y, M. (1993). Confined lateral diffusion of membrane receptors as studied by single particle tracking (Nanovid Microscopy). Effects of calcium-induced differentiation in cultured epithelial cells. Biophys. J. 65, 2021–2040. L, K. & L, H. (1996). Signaling by chimeric erythropoietinTGF-beta receptors: homodimerization of the cytoplasmic domain of the type I TGF-beta, receptor and heterodimerization with the type II receptor are both required for intracellular signal transduction. EMBO J. 15, 4485–4496. M´, J. & P, K. (1995). Mammalian antiproliferative signals and their targets. Curr. Op. Gen. Dev. 5, 91–96. M, N. & U, S. (1949). The Monte-Carlo Method. J. Am. Statist. Assoc. 44, 338–341. N, D., B, R., S, G. & B, K. (1996). Direct and long-range action of a DPP morphogen gradient. Cell 85, 357–368. O’R, M., S, J. & C, V. (1995). Patterning of the mesoderm in Xenopus: dose-dependent and synergistic effects of Brachyury and Pintallavis. Development 121, (May) 1351–1359. R, K. M. & M, D. A. (1996). Short-range signaling by candidate morphogens of the TGF beta family and evidence for a relay mechanism of induction. Cell 86, 743–754. S, J. (1993). Embryonic induction. Mech. Devel. 41, 91–9107. S, J. I., F, J. S. & H, W. L. (1989). Chemical Kinetics and Dynamics. Englewood Cliffs: Prentice Hall. T, C. (1996). Vertebrate Limb Development. Sem. Cell & Develop. Biol. 7, 137–143. W, P. A. & M, D. (1994). Mesodermal patterning by an inducer gradient depends on secondary cell-cell communication. Curr. Biol. 4, 676–686. W, L. (1969). Positional information and the spatial pattern of cellular differentiation. J. theor. Biol. 25, 1–47.
Mechanisms for Positional Signalling by Morphogen Transport: a Theoretical Study M K*‡ L W† *Neurobiologie Mole´culaire, Institut Pasteur, 25, rue du Docteur Roux, F-75724 Paris Cedex 15, France; and the †Department of Anatomy, University College London, Medawar Building, Gower Street, London WC1E 6BT, UK (Received on 12 May 1997, Accepted on 23 October 1997)
Gradients of cellular activities are ubiquitous in embryonic development. It is widely believed that the inhomogeneous spatial distribution of a morphogen would be able to set up such gradients. But how then does the morphogen propagate in the first place? Straightforward molecular diffusion is often proposed as a possible mechanism. We first show that, surprisingly, the mere binding of the diffusing morphogen to its membrane receptors suffices to prevent the establishment of a concentration-based positional signalling system. Instead, a flat, saturated distribution of receptor-bound morphogen builds up. Because the distribution spreads gradually from the morphogen source, however, cells may still know their position if they are able to integrate the morphogen signal in time. The irregularities of diffusion in the complex extracellular medium would in fact be partially compensated for by such time summation. Another, non-exclusive possibility is that morphogen transport does not occur by simple diffusion only. We put forth a novel model of receptor-aided, directed diffusion that achieves a spatial distribution of morphogen. Our model is based, as an illustration, on the properties of members of the TGFb family of molecules. We show that two simple hypotheses regarding the kinetics of TGBb binding to its receptors suffice to establish a remarkable transfer mechanism whereby a morphogen such as activin could be both propagated along cell membranes, and transferred between cells that are in contact. The model predicts that morphogen propagation properties depend strongly on the closeness of cell-cell appositions, does not necessitate protein synthesis, accumulation or slow degradation (in contrast to the diffusion/time integration model), and that the morphogen is localised mostly on or close to cell membranes. 7 1998 Academic Press Limited
Introduction In the course of embryonic development, the formation of spatial patterns of cellular differentiation is of fundamental significance. This process can involve induction—interactions between neighbouring cells (Slack, 1993)—and longer-range positional signalling by morphogens (Wolpert, 1969). There is quite good evidence from the development of both Drosophila wings and vertebrate limbs
‡Author to whom correspondence should be addressed. E-mail: mkersz.pasteur.fr 0022–5193/98/050103 + 12 $25.00/0/jt970575
that positional signalling mechanisms are indeed involved in patterning (Nellen et al., 1996; Tickle, 1996). Experimental (O’Reilly et al., 1995) and theoretical (Kerszberg & Changeux, 1994; Kerszberg, 1996) results indicate that membrane receptors and cellular transduction mechanisms can indeed operate so as to read the concentration of signalling molecules with striking precision. But how is such a graded concentration set up in the first place? To what degree is the build up of the morphogen distribution a process independent from its interpretation into differentiation patterns? This is the question we want to examine here. 7 1998 Academic Press Limited
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A commonly invoked mechanism for long-range signalling is based on the diffusion of a signalling molecule. Crick (1970) pointed out that diffusion has just the right characteristics for a cellular signal required to be propagated over less than one millimeter during a period of several hours. Clearly, the concentration gradient that would be set up by release of a diffusible substance could provide cells with positional information. However, gradients, if they are to pattern the embryo, must be constructed with a high degree of reliability; yet it is unlikely that a substance freely diffusible in the extracellular space could provide a very reproducible long-range distribution in the face of screening effects by cells themselves, dependence on available intercellular space, or extraneous perturbations such as blood flow in the vertebrate limb. Assume however these objections are ignored: we show here that simple molecular diffusion cannot establish a graded morphogen distribution. This is because of the binding of a diffusing morphogen to its membrane receptors, which was not included in previous theoretical treatments (Crick, 1970). Binding leads to trapping: morphogen cannot move beyond a point before receptor there has been maximally occupied. The result is that a narrow front advances from the morphogen source, leaving behind a flat, saturated distribution of receptor-bound morphogen. We suggest that there are two (non-exclusive) ways out of this situation. The first possibility is that the morphogen signal transduction cascade allows for time integration or summation. The transduced morphogen signal will then grow in time when receptor is occupied. Close to the morphogen source, this growth can start as soon as morphogen is injected; further away, the growth starts later and the integrated signal lags behind: this gives cells a way to read their position. The second possibility is that morphogen transport does not take place by diffusion only. (It is of course possible that morphogen is not transported at all, and that cell position is determined by successive inductions—we shall discuss this in the Conclusions). Here we propose a novel mechanism for the progressive spread of a morphogen. Our scheme eschews simple physical diffusion, but nevertheless implies the actual transport of the morphogen molecules: this now happens through cell-biological, active, and possibly energy-dependent processes. The transport is done through the agency of those very membrane receptors which render simple diffusion inoperant. The importance of morphogen receptors is usually seen to lie in their pivotal role as triggers of the intracellular signalling cascade. We demonstrate
however that membrane receptors could quite possibly have acquired another significance, upstream of this, by contributing to the cell- and system-wide spatial repartition of their ligand. This would mostly involve the cell membranes and, to a much lesser degree, the intercellular medium. Our model is loosely based on the properties of members of the TGFb family of molecules. These have been implicated in a wide variety of morphogenetic processes (Hogan, 1996). Among the best studied are decapentaplegic (in Insect development) and activin-like molecules (in early Xenopus embryogenesis) (Nellen et al., 1996; Gurdon et al., 1995; Jones et al., 1996). We do not propose however to model here in detail signalling by any of these individual molecules or receptors. Rather, we use what is known of them as hints for inferring plausible general molecular mechanisms which may fulfill the requirements for efficient morphogen transport. We then show by computer simulation that the proposed mechanisms may indeed function as envisaged. The biochemical mechanisms needed for receptoraided, directed transport fall well within the range of complexity exhibited by membrane receptors. In short, we find that two simple hypotheses regarding the kinetics of ligand binding to TGFb-like receptors suffice to establish a remarkable transfer mechanism whereby a morphogen would be propagated along cell membranes, and transferred between cells. It is possible that this active transport needs to be driven by receptor transphosphorylation as happens during ligand binding (which transphosphorylation is instrumental in initiating the intracellular cascade itself). In any case, a graded, stable distribution is quickly achieved, with its maximum close to the morphogen source. In what follows, we present the model’s biological premises and its formal definition; we then use computer simulations to show (1) the failure of diffusion to establish a graded morphogen distribution, and (2) to demonstrate our own model’s functioning. Model The biological premises which underlie the model originate in the molecular biology of activin, BMP-4 and other TGFb receptors (Massague´ & Polyak, 1995; Derynck, 1994). We introduce no receptor configurations other than those known from the TGFb context: our only additional hypotheses concern the kinetic coefficients governing the transitions among some of these configurations.
: (a) R II
RI
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TbR-I, and that transphosphorylation of TbR-I by TbR-II takes place. The intracellular signalling cascade is thus initiated. This description of the upstream activin pathway certainly corresponds to a highly simplified view, but we shall proceed on the assumption that it provides sufficient basis for understanding the kinetics of ligand-receptor interactions. We now state our kinetic hypotheses, as follows: (a) the TbR-I–TbR-II heterodimer has a nuch lower affinity for activin than the TbR-II monomer has, so that, upon binding of TbR-I, the activin-TbRII complex quickly dissociates, leaving behind the intact TbR-I–TbR-II heterodimer (see Fig. 1). (b) the TbR-I–TbR-II heterodimer has a relatively long lifetime, compared with the rate of propagation of activin in the system (as discussed below).
(b) R II*
These are the major hypotheses of the model. In addition, we have examined the possible importance of a specific, active morphogen transfer mechanism at cell-cell contacts:
(c) R IR II
(c) an activin-TbR-II complex on the membrane of a given cell may exchange its activin with a TbR-II molecule located on an immediately adjacent patch of membrane belonging to another cell (Fig. 2); this does not require configurations other than those already discussed, but does indeed necessitate some new, intercellular reaction mechanism. The picture on
F. 1. A simple model for the activin receptors. (a) There are two receptor types, I and II. (b) Activin (black squares) binds to type-II receptor, inducing an allosteric transformation of its intracellular terminal. This enables dimerisation with type-I receptor. (c) The dimer forms, and releases the activin after a time which is short compared to the dimer’s own lifetime. The intracellular signalling pathway is initiated by type-I transphosphorylation.
Several types of TGFb receptors have been discovered. The two major, best studied subtypes are receptor type I (TbR-I) and II (TbR-II). It is not unreasonable to summarise what has been gathered of their properties in the following simplified way (Fig. 1).* Activin probably does not bind to TbR-I as a monomer (or dimer). Instead, binding first occurs with TbR-II (monomer or dimer). Secondly, it seems that the activin-TbR-II complex then reacts with
*For simplicity we use the activin language, but this is just a matter of convenience; also, it is quite possible that TbR-I and TbR-II are not active as monomers but as homodimers (Luo & Lodish, 1996), so that, when we talk of TbR-I, say, it may in fact be that a TbR-I–TbR-I homodimer is the relevant unit.
.
Cell 2
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Cell 1
F. 2. Morphogen exchange among receptors on adjacent cells. The drawing makes it clear that even if the exchange is not eenergy-driven (i.e. ATP-dependent), its probability may nevertheless be quite high.
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Fig. 2 makes the chosen mechanism plausible. The behaviour of the model with and without mechanism c will be discussed. Our point is not to present compelling evidence for our hypotheses; but they are rather straightforward, testable, and they overcome very simply the major hurdle faced by diffusion, that of morphogen trapping by the receptors. One may think of other hypotheses which would achieve a similar goal. But the unavoidable conclusion of our paper will be that at least some such hypothesis is needed in order to understand morphogen gradient setup. For instance, if TbR-I and TbR-II function as homodimers, ligand-induced TbR-I–TbR-II interactions may lead to a short-lived tetrameric configuration, with the subsequently released monomers unable to form ligand-binding homodimers again for some time. Our particular model may now be defined formally by the kinetic equations: RII + A _ R* II
(1)
R*II + RI : RI RII + A
(2)
RI RII : RI + RII
(3)
R*II,cell1 + RII,cell2 _ RII,cell1 + R* II,cell2
(4)
where, for simplicity, the receptors are denoted RI and RII , the morphogen (activin or other) by A, and in the last equation the location of molecules is indicated. The basic phenomena that control the model’s behavior are simple (Fig. 3). Two distinct processes
1
control diffusion on a given membrane and morphogen transfer between cells. On any particular cell, morphogen A binds to the closest available RII , then RI displaces the morphogen, which is thus free to migrate to further sites with free RII . (This is the major difference with simple diffusion, where morphogen is not so readily released.) Since the lifetime of the RI RII heterodimer is long, sites already visited on the membrane are ‘‘burned out’’ and the morphogen will become bound at locations more and more distant from its place of origin. When most of the receptors on a given cell are dimerised, A is free (actually, it is forced) to jump to another cell with free RII . Membrane receptors in the model thus effectively operate as a ‘‘bucket brigade’’ carrying over what can be a very limited amount of morphogen. Because the transfer is not complete, and the lifetime of A itself is not infinite, ultimately the morphogen’s effect will decay at sufficient distance from its source (Gurdon et al., 1994, 1995). We demonstrate these results by computer simulations of the model.
Computer Simulations For the sake of the computations, cells are assumed to exist in a restricted two-dimensional space (Fig. 4). This is artificially divided in a honeybee lattice of ‘‘bins’’, with 124 bins in the vertical dimension, and 248 bins horizontally. These bins are analogous to pixels in a digitised image: their location is fixed, but
2 3
0
F. 3. How receptors function as a ‘‘bucket brigade’’. Free receptors are present on a given cell membrane (0); type-II receptor binds the impinging activin molecules (1); this enables type-I receptor to join the complex. The receptor dimer quickly releases activin, but remains bound (2). The bound type-I/type-II receptor complex does not bind activin with sufficient affinity, so that the released activin diffuses preferentially to free type-II receptor located further on the membrane. At a cell-cell contact or close apposition (3), transfer to another cell occurs either by receptor-receptor exchange or by short-range extracellular diffusion, and the process goes on.
:
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allow for membrane binding and unbinding however, some extracellular species may also be allowed a non-zero concentration at the membrane. Diffusion is described by probabilities per time step of hopping betwen adjacent bins. Reactions occur locally among the various molecular species. We now proceed to describe in more detail the parameters characterising these processes and the course of a computer simulation.
Parameters A limiting parameter in solving the model numerically is morphogen diffusion, because orders of magnitude for the diffusion coefficients are known and must be respected. From data on membranebound molecular diffusion, one can gather that DS 2 10−9 cm2 s−1 (Jacobson et al., 1987; Kusumi et al., 1993). As to volume diffusion, one may estimate (Crick, 1970; Jain, 1990) DV 2 10−7 cm2 s−1. Since DV is much the larger, we have to set our length and time units to be able to accomodate it. Now, F. 4. The simulations’ underlying geometry. Two-dimensional space is partitioned into a beehive array of pixels or ‘‘bins’’. Closed loops of such bins represent cell membranes. In (a), we show one such cell (m) and part of another (m') in contact with it. Molecules are able to diffuse on a membrane and between membranes (arrows). In (b1), (b2) it is seen how membranes are able to move (arrows).
their color or luminosity reflect the presence, locally, of objects such as molecules, or parts of objects such as cell membranes. Thus, 180 closed strings of adjacent bins compose the ‘‘membranes’’ of 180 cells; membrane bins as well as extracellular ones are occupied by molecules in given concentration. The membranes are mobile; membrane molecules are carried along with their membrane (convection), and extracellular molecules are displaced as membranes move (volume convection). In addition, membrane molecules diffuse on the membrane, and extracellular molecules may diffuse in the extracellular medium; to
*This is in arbitrary units; conversion to actual concentrations may be done approximately by noting that our volume unit is about D2.1 mm = 4 mm3, so that a receptor concentration of 2.0 in arbitrary units corresponds to 2.0[N ·0.25·1015] particles litre−1 where N must be adjusted to yield the actual experimental value, and will then be used to convert other concentrations and the binding constants (see below). †These constants can be compared with experimental values by noting that they are expressed in reactions per t (s) and per [N ·0.25·1015]2 (particles litre−1)2 (see previous footnote).
(i) in order to have sufficient spatial resolution on cells with diameter 215 mm, we take each bin to have a length of D = 2 mm. (ii) Due to numerical instability, the probabilities for hopping from bin to bin per time step t may not exceed P 2 0.2. By dimensional argument, DV = 10−7 cm2 s−1 2 PD2/t, which yields t 2 0.08 s. In accordance with the estimates quoted above, on-membrane hopping probabilities are then taken as PRI = PRII = 0.002 (see above) and PR*I = PRI RII = 0.001 (to account for larger molecular weight). The absolute concentration of receptor and ligand molecules will not be our main interest in this theoretical study [see however Fig. 5(b)]. The total number of receptors on each cell membrane is assumed constant throughout (the local concentration may vary of course), and starts at a uniform 2.0 per membrane bin.* (We neglect the effects of receptor turnover over the time scale of our simulations.) Certain relative concentrations are important, and are determined by initial conditions and by some of the reaction affinities. The kinetic coefficients k are introduced as reaction probabilities per unit time t or inverse lifetimes, thus: , eqn (1): kbind = 0.1 and kunbind = 10−6: only kbind is significant, as we assume that reaction (2) will be the one actually to deplete the bound state (thus, the receptor affinity kunbind /kbind is essentially irrelevant);†
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F. 5(c). F. 5. Distribution of morphogen when diffusion operates alone (in extracellular space and on cell membranes). (a) A continuous source of morphogen has been operating on the left side for T = 67500 time steps (1 step 1 0.08 seconds, i.e. T = 1.5 h). The concentration of receptor-bound morphogen is shown. (Thus, only the membrane bins are displayed.) The lighter the bin, the larger the concentration. We see that morphogen is sharply restricted to the vicinity of the source: the length d0.5 where concentration drops to half its maximum is indicated by the black arrow. Diffusion in the absence of receptor would lead to an exponential distribution with d0.5 as indicated by the gray arrow. The reason for the considerable discrepancy is that receptor traps the morphogen on cell membranes. (b) Distribution of bound morphogen at T = 0.5 h when the receptor concentration on the cell membranes is reduced to 1% of its previous value in 5a. Notice that transport is much faster, but the receptors are saturated over much of the system. (c) The free morphogen distribution under the same conditions as in (b). The free morphogen has a graded distribution, but is unable to contribute to receptor-mediated signalling.
only, morphogen A is thus generated according to dA = 10−3. dt F. 5(a) and (b).
, eqn (2): kdimer = 0.10; , eqn (3): kdiss = 0.002, i.e. the lifetime of the dimerised receptor is about 500t or 40 s. We assume the morphogen originates from a continuous source located on the left-hand side margin of the system: in the leftmost bins
(5)
Results We perform three types of simulation: first, we examine the case of diffusion alone; we then simulate the behaviour of our full model. Finally, we control our results by modifying in turn some of the model’s basic parameters. When kexch = kdimer = 0, only membrane and extracellular diffusion subsist. The results of a
: representative simulation* show that the morphogen, assumed to diffuse with a coefficient D = 10−7 cm2 s−1, is confined, after one and a half hour, to the first cell layers beyond the source [see Fig. 5(a)]. It has travelled much less than predicted for free diffusion, as shown on the Figure. This is in fact easy to understand, as morphogen, no matter what its rate of injection, must fill statistically available receptor sites before being able to move further in the system. Thus, ultimately, a flat distribution of trapped morphogen is slowly generated, with free morphogen (not bound and therefore not effective) superimposed on that. Swamping the receptors with morphogen does not help, for if transport of morphogen is indeed faster when the receptor is made less abundant [see Fig. 5(b), where receptor concentrations have been divided by 100], the distribution of bound morphogen appears quite uniform, even though free morphogen seems distributed according to an exponential [Fig. 5(c)]. Because of its uniformity, the distribution of bound morphogen cannot be used to impart positional information to cells. There remains of course the possibility that cells have a way to measure the time of arrival of the morphogen concentration front; this could be achieved e.g., if transduction by the activated receptor were cumulative, i.e., integrated over time: this would lead to a signal strength decreasing with the distance from the source, as receptor far away would be activated later. Because of the summation in time, some of the irregularity inherent to diffusion in a complex, random medium would be smoothed out. If the receptor had a very small probability kbind of binding the morphogen, some morphogen would be able to diffuse before being trapped, and trapped morphogen concentration could increase progressively, giving rise indeed to a gradient. (Remember that kbind is not the receptor affinity: this is kunbind /kbind ; kbind measures a dynamic quantity, the probability that the receptor will bind ligand available in its vicinity, while affinity is the equilibrium ratio of unbound to bound ligand, irrespective of whether this is reached or not in the available time.) A small kbind is, however, difficult to justify from what is known of receptor-ligand interactions. In our model, kbind represents the *In a given simulation, the starting cell configuration and subsequent movements are generated in a partially random way; This is controlled by a quasi-random number generator used to ‘‘throw dice’’ each time a ‘‘decision’’ must be made (say, for a membrane to move right or left according to its current environment). This is the so-called Monte–Carlo algorithm (Metropolis & Ulam, 1949). The generator must be ‘‘seeded’’ by a starting number: each seed generates a different quasi-random series, hence a new repeat of the ‘‘experiment’’.
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probability that a ligand molecule located in a 2 mm bin next to a cell membrane will be caught by membrane receptor within the time unit t. Since the bin dimensions were chosen so that diffusion is essentially complete over the bin within t, the probability that ligand and receptor collide during t is essentially 1, and this is usually taken as the factor limiting kbind (Steinfeld et al., 1989). Any significant departure from kbind = 1 must be due to steric or other special accessibility effects. To observe, after a few hours’ time, a non-uniform distribution of bound morphogen with a reasonable spatial range required taking kbind Q 10−5 (data not shown), which seems extraordinarily low indeed: nothing in the molecular biology of, say, the TGFb receptors suggests the plausibility of this unusual value. On the other hand, the dimerisation mechanism included in our model frees morphogen for diffusion automatically, as we now show. : We now turn to simulations of our complete model. Results are presented on Fig. 6. We observe in Fig. 6(a) and (b) the progression of receptor-bound morphogen molecules and of receptor dimerisation further and further away from the morphogen source (on the left). We see that, after 1.5 h, morphogen is distributed over about 15 cell diameters, with a sharply peaked concentration curve, the maximum of which lies at the first cell layer. The dimerised receptor follows at all times a smoother, wider distribution, with a clear tendency to saturation. From these results, it is evident that either receptor-bound morphogen or dimerised receptor, or some combination thereof, could be used to impart positional information to embryonic cells in due time for, e.g., Xenopus development. Dimerisation propagates at greater speed, but also saturates earlier. Note that in our model, propagation can actually proceed faster than in pure diffusion: this is because receptor dimerisation behind the morphogen propagation front largely prevents morphogen backward movement, which is always possible in diffusion. We have plotted on Fig. 6(c) the total concentration of morphogen in the system. Note that much of the morphogen is located close to or on the cell membranes. It is one of our predictions that little morphogen should be found in the extracellular space. This conclusion could, of course, be altered in a model including extracellular matrix morphogen binding sites. Morphogen is transported mostly in the vicinity of the membranes (1) by diffusion away from membrane areas where receptor is dimerised and thus
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Recovery in the face of a variety of perturbations is a fundamental feature of embryogenesis. Here we
assess the various contributions to morphogen transport and the way they contribute to a stable morphogenetic process. We do this by comparing the final (1.5 h) morphogen distributions obtained with the full model (see above) and (a) the model with strongly reduced extracellular diffusion (Fig. 7); (b) the model with a sharp deficit in the dose of injected morphogen (Fig. 8); and (c) the model in the absence of active cell-cell exchange interaction (kexch = 0, see Fig. 9). Figure 7 allows for a comparison of morphogen [part (a)] and dimer [part (b)] distributions in the
F. 6(a).
F. 6(b).
releasing its ligand (which it is unable to bind again for a while) and toward areas with free monomeric type II receptor; and (2) by receptor-receptor exchange (Fig. 2) or simple diffusion again at cell contacts or at points where cells are in close apposition (see Fig. 9 for a discussion of the relative importance of these two cell-cell transfer mechanisms).
: normal situation (as above) and when the diffusion coefficient in extracellular space is reduced by a factor 10 (to 10−8 cm2 s−1) Clearly, the distributions are affected, but not nearly as severely as one would expect. Volume diffusion is only one of the forces behind morphogen movement in our model. On Fig. 8, we have plotted the effect of a factor 10 reduction in the rate of morphogen injection. We do not display the morphogen distribution, which is dramatically lowered. But the effect of a low morphogen dose is not nearly as marked when one considers the receptor dimer distribution: concentrations are reduced by a factor 2 to 3 at most. This welcome lack of sensitivity stems from the long lifetime assumed for dimerised receptors, which allows for a large buildup of dimerised receptor concentration with little actual morphogen entry into the system. Here is certainly an argument for the
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possible relevance of dimerised receptor signalling in the generation of positional information. Finally, Fig. 9 is an illustration of the relative contribution to morphogen transport of the cell-cell exchange interaction. Setting kexch = 0 actually has little effect on bound morphogen distribution (not shown). But it markedly influences the dimer distribution, by facilitating faster movement of relatively small quantities of morphogen away from the source. When all our data are considered, it seems evident that the major factor for the success of the model is receptor dimerisation. This importance stems from the fact that it allows fast morphogen movement from one cell face to the opposite, while cell-cell morphogen exchange intervenes only punctually, at cell contacts. This punctual transfer can be effected passively (through release by receptor on one cell and reuptake on the next) almost as well as by the kexch –mediated mechanism. Still, close juxtaposition of cells is essential for either process to take place reliably. Conclusions and Perspectives
F. 6(c). F. 6. Time-lapse views of the spatial distribution of morphogen effects in a typical numerical ‘‘experiment’’ involving the full model. (a) Top: distribution of morphogen bound to type II receptor at simulation step 22500 (i.e. 0.5 h); middle: the same after 1.0 h and bottom: after 1.5 h. The morphogen has spread by about 15 cell diameters. A graph of the final distribution is shown below the time-lapse view. Morphogen transport and binding properties of the full model clearly satisfy the conditions required for the establishment of positional information [compare Fig. 5(a)]. (b) The distribution of heterodimerised (type I/type II) receptor, during the same simulation as in 6a. These results demonstrate that receptor dimers may also function as signals of positional information. But note that signs of saturation of dimer concentration already appear at 1.5 h. (c) Total morphogen present in the system at T = 1.5 h. Morphogen is located mostly on membranes and in the immediate vicinity of cells.
We have constructed and solved a model for directed morphogen transport over the extent of an embryonic field, inspired by the properties of activin and other TGFb receptors. We have proved that simple kinetic hypotheses regarding the states of these receptors suffice for them to carry the morphogen over the required distances. This is in contrast with what happens with simple diffusion. Why have we chosen a mechanism which actually carries the morphogen molecules themselves? Might it not have been simpler to assume that cells somehow induced by activin near the latter’s source, would then transmit a further inductive signal to their immediate neighbours, which would then carry the phenomenon over? Some experimental evidence seems to favour such an induction cascade (Reilly & Melton, 1996). We found however that the problem with such a scheme (of which we have tried many variants) is that there is no simple way to prevent the signal from being transmitted arbitrarily far from the source. To function, the scheme requires a cell to receive a signal, read its strength accurately (much more accurately than when it interprets morphogen signals, which is done according to thresholds), and transmit to its neighbours a precisely modulated, slightly weaker excitation, or induction signal. We have not found a transduction mechanism leading to the appropriate attenuation: either the attenuation is too strong, and the signal dies away; or it is too weak, and the signal
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F. 7. The effects of a lowered efficacy of volume diffusion on morphogen gradient buildup. Here extracellular diffusion coefficient has been reduced 10-fold to DV = 10 − 8 cm 2 s − 1 The decrease is not fully reflected in the final graded distributions. (a) Receptor-bound morphogen distribution at T = 1.5 h with diffusion normal (gray dots) and slow (white dots). (b) Same as (a) but for the receptor dimer distribution.
propagates indefinitely. This is impossible with morphogen physically transported away from a source: ultimately, the morphogen decays, and so does the signal. It remains possible to imagine spatially specialised sorts of induction cascades. Assume a strong attenuation of the inductive signal, such that its range is, say, not more than one cell diameter. A first possibility is this: if cells express a battery of receptors, and the first cell row secretes one of the possible ligands, this could induce the second row (but that single row only) to display a second ligand, which would induce in the third row display of yet
another ligand and so on. One can imagine mechanisms for such a change of display, some involving gene transcription and some not. Yet another possibility is that all cells express all ligands, but receptors on cells of the first row, bound with the unique ligand originating at the source, induce receptor on the next cells to adopt a configuration apt to bind another ligand specific of the second row, and so on. (The configuration in question could be e.g. the receptor subunits’ oligomerisation state). At any rate, the fact that we could not build a safe long range communication mechanism without resorting to actual morphogen transport leads us to
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F. 8. Distribution of receptor dimer at T = 1.5 h with a normal rate of morphogen injection (gray) and with a 10-fold reduced injection rate (white). Note that the reduction in dimer concentrations is two to three-fold at most, due to the long lifetime assumed for receptor dimers. In contrast, the receptor-bound morphogen distribution suffers the full 10-fold reduction (not shown).
suggest that the only viable alternative to the sort of transfer process we have introduced (or to diffusion plus signal summation) is a complex variety of membrane molecules playing specialised roles as a function of distance to the morphogen source. Although it does not preclude such complexity, the present model is by far the more parsimonious one. On the one hand, it brings attention to the importance, for morphogenesis, of membrane binding
kinetics, which are experimentally accessible; on the other, it makes a number of testable predictions, namely that most of the morphogen will find itself on cell membranes, not in the extracellular space; that cell-cell contacts are essential for morphogen propagation (see e.g. Wilson & Melton, 1994), even if no specific morphogen transfer reaction occurs there; that the range of propagation is sensitive on morphogen lifetime, faster degradation meaning
F. 9. Receptor dimer distribution, under the same conditions as in Fig. 8 (gray dots), and with the exchange interaction kexch = 0 (white dots). Transport between cells is then left to morphogen re-release and normal diffusion: clearly, morphogen movement is hampered. Note that this would hardly be visible on the bound-morphogen data (not shown). But because the morphogen remains closer to the source and dimerises more receptor there, it leads to a noticeable saturation of the dimer distribution as displayed.
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reduced spatial range; and that the propagation does not depend on continued protein synthesis, although it may depend on energy supply. Our theoretical considerations also suggest that morphogen signalling would be enhanced if the morphogen transduction pathway were to operate, at least to some degree, in such a way as to integrate in time the instantaneous morphogen signal. One way in which diffusion and receptor-aided transport could be distinguished experimentally is by introducing a TbR-I negative-dominant mutant clone (defective in the TbR-II binding domain) in an otherwise normal preparation: our model predicts that the clone should leave a shadow of reduced activin concentration and signalling, while the diffusion model would predict the opposite. In this case, the induction-cascade model makes a prediction similar to our model’s, however. On the other hand, introduction of a TbR-I negative dominant clone defective in the intracellular domain would not affect directed transport or diffusion, but would disrupt some types of induction cascades.
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