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A model generating the pattern of cartilage skeletal elements in the embryonic chick limb
.I. theor. Biol. (1975) 52, 199-217
A Model Generating The Pattern of Cartilage Skeletal Elements in the Embryonic Chick Limb? 0. K. WILBY AND D. A. EDE
Department of Zoology, University of Glasgow, Scotland, U.K. (Received 10 June 1974, and in revised form 30 August 1974) The development of the vertebrate limb involves the production of a specific external form arising from cell division and other growth processes at the cellular level, and the origin within it of specific patterns of tissues arising by cellular differentiation, of which the pattern of cartilages which pm-figure the limb skeleton is the most striking. In this paper we propose a model for the differentiation (or the preceding determination) process that, using only localized cell to cell interactions, can approximate the cartilage pattern in any limb shape. The model requires cells to modify their metabolism irreversibly at critical threshold levels of a diffusible morphogen which may be made or destroyed by these cells. Restrictions inherent in the successful development of a total limb pattern using this systemlead to theprediction that the process is con6ned to a distal band which has no sign&ant interaction with more proximal regions but within this band the characteristic features of the anterior-posterior axis of the limb develop without additional interactions. Cartilage elements are initiated as single “cells” and expand centrifugally to their tinal size; these elements developing sequentially along the anteriorposterior axis, showing a distinct polarity of size. The model also predicts that equivalent cartilage elements in all vertebrate limbs will be roughly the same relative size at determination, the extensive range of adult structures arising by differential growth and fusion, possibly controlled by global aspects of the model. It must be emphasized that this model only satisfactorily simulates the anterior-posterior patterning of cartilage elements, the disto-proximal pattern being externally imposed. The fhral cartilage pattern is shown to be a function of (1) the developing shape of the limb, (2) the position of an initiator region that starts the patterning process and (3) the rate of production of the diffusible morphogen. Using parameters selected with as much realism as possible the model gives a good approximation to the pattern of cartilages found in the normal chick limb; modifying the shape of the limb to that of the talpid3 mutant t A shortened version of this paper was presented at the Conference on Biologically Motivated Automata Theory, June 19-21 at the MITRE Corporation, McLean. Virginia, U.S.A. 199
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produces the characteristic features of the cartilage pattern found in that mutant and modifying the rate of morphogen production simulates patterns resembling those found in some ancestral vertebrate fossil forms. 1. Introduction
The limb in vertebrate embryos develops from a small limb bud; at its first appearance this is a simple hillock of cells arising on the flank, which becomes elongated by outward growth, then paddle-shaped by expansion at its tip. At the earliest stages the bud consists predominantly of undifferentiated mesoderm cells, covered by a thin layer of ectoderm which is modified distally to form a more rigid apical ectodermal ridge (AER). As outgrowth takes place, cell differentiation occurs within the mesoderm to produce a pattern of cartilages which form the basis of the bony skeleton of the limb (Ede, 1971). This developing system constitutes a remarkably clear-cut example of what Arbib (1972) has suggested is the crucial problem for theoretical embryology: How can a single cell in a continually active tissue, where all cells may be growing, dividing, differentiating, or dying, use only information from nearby cells to so grow as to contribute properly to the overall form of the organism? For simplicity we have treated the limb bud as a two-dimensional object, ignoring its dorsal-ventral axis. The remaining axes are the proximal-distal, from the base of the bud to its tip, and the anterior-posterior axis at right angles to it. In both these axes the cartilage pattern exhibits two features which are characteristic of biological pattern in general: polarity and periodicity. In the proximal-distal axis growth is polarized along this axis and a developmental polarity distinguishes the sets of cartilage elements that appear periodically along the limb: thus there appears a basal band (e.g., the upper arm in the human forelimb), a second band (the lower arm), a third (the metacarpals) and a fourth (the digits). We omit, for simplicity, the band of more complex carpal rudiments. Experimental observations indicate a complex interaction between the distal mesoderm and the overlying AER in determining this axis (Zwilling, 1961). Summerbell, Lewis & Wolpert (1973) have developed a plausible model for generating this axial pattern, utilizing what is essentially a clock based on the cycles of cell division in a narrow band of distal mesoderm cells. In our work reported here, pattern generation in this axis arises primarily from the subdivision necessary for the realistic proximal-distal orientation of each cartilage element, rather than from a periodic control mechanism. The structure of the joints between cartilage elements, the relative lengths of each element and their proximo-distal relationship are beyond the scope of the present
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model and so appear as externally imposed restrictions. Qualitatively, however, these restrictions are identical to those of Summerbell et al. (1973). Our model was developed primarily to account for the remaining anteriorposterior axis. Here again the pattern exhibits polarity in the characteristic size of each cartilage element and periodicity in the arrangement of these elements: a single element in the basal band (e.g., the humerus in the upper arm), two elements in the second (radius and ulna in the lower arm), four
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FIG. 1. The normal development of chick at stages 23 to 35 (Hamburger & Hamilton,
limb, showing 1951).
areas
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in the third (metacarpals) and the fourth (digits). This developing pattern is shown in Fig. 1 for the chick limb (generalized limb stages, following Hamburger & Hamilton, 1951). Computer simulations have been performed for stages 24-29 only because of difficulties in successfully simulating differential growth. The control of the anterior-posterior axis appears to lie in the posteriorlateral margin of the limb bud, the zone of polarizing activity (ZPA), discovered by Gasseling & Saunders (1964). As shown in Fig. 2, a graft of ZPA material to the anterior edge of a wing bud, causes limb duplication. In theory, this could be due to a diffusion gradient emanating from the ZPA
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FIG. 2. The ZPA graft and one interpretation. (b) The diffusion gradient interpretation.
(a) The transplant experiment and its result.
as shown, coupling growth and differentiation to particular levels. This basic experiment suggested to us the lateral edge of the ZPA as the location of the “initiator” site in our simulations. 2. Development of Models fn attempting to model this type of embryonic development it is possible to adopt one of two alternative basic approaches: (1) Positional Information (see Wolpert, 1969) where pattern determination is regarded as a two-stage process involving (a) an assignment of a specific positional value to each cell, followed by (b) the interpretation of this positional information in terms of molecular differentiation. (2) Automata Theory (see Arbib, 1971) where each cell is regarded as an automaton, the state of the cell being controlled by a metabolic mechanism sensitive to critical levels of input, producing a particular level or type of output. The position of the cell is strictly irrelevant, its differentiation depending only on its own internal state and an awareness of what its neighbours are doing. Our own work is closer to the second approach. Positional information models most often use a monotonically decreasing gradient, e.g., of some diffusible substance, to specify the positional value, as shown in Fig. 3(a); thus cells might interpret gradient levels between “thresholds” Tl and T2
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Fro. 3. Two alternative ways of determining a periodic pattern: (a) via a monotonic gradient and many “thresholds”, (b) via a periodic gradient and one “threshold”.
as specifying cartilage development. Whilst simple in theory, this model becomes less plausible where the pattern is markedly periodic; thus to produce three bands of cartilage as shown in Fig. 3(a), differentiation must occur between six threshold levels: Tl-T2, T3-T4, T5-T6. This requires a biochemical switch mechanism sensitive to many different triggering levels. As the number of pattern elements increases, so does the number of thresholds, necessitating an extremely complex, sensitive and stable system. The addition or deletion of pattern elements to give, for example, the variety of limb structures seen in fossil, normal and mutant organisms further complicates a monotonic gradient system since both the number of thresholds and the slope of the gradient need regulating. We suggest an alternative system using a periodic gradient as indicated in Fig. 3(b) and a single threshold for cartilage differentiation. Providing such a gradient can be simply set up, the system does not become more complex with an increasing number of elements, nor is extreme stability or sensitivity required, since slight variations in concentration do not markedly alter the pattern. The modscation of this pattern in terms of element number can be regarded simply as a control of overall size as will be explained later, a polydactylous limb requiring no more complex a control system than a
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single digit. It must also be remembered that the regulative patterns of evolutionary primitive and early embryonic organisms are lost in favour of mosaic non-regulative patterns in advanced and more mature organisms. Periodic gradient models have previously been proposed for the control of developmental events. Turing (1952), proposed a model in which two interacting morphogens would, from uniform initial conditions and random perturbations, develop a stable chemical wave pattern. In a recent elegant analysis of this model, Bard & Lauder (1974) have demonstrated that this system is too unreliable to serve as the generating mechanism for features such as digits but is more than adequate for leaf and hair patterns. The “phase gradient” model of Goodwin & Cohen (1969), essentially a system for supplying polarity and positional information on a temporal basis, lends itself easily to the production of periodic structures. The resultant pattern is, however, one based on a “saw tooth” gradient; within each segment the gradient is monotonically decreasing and all the above arguments apply; the basic model is also biochemically complex. In a search for simple patterning mechanisms, in the belief that biological complexity results from the interaction of several simple control processes rather than a single complex one we have looked at a system in which a stationary chemical wave is reliably established by an oscillating travelling wave front. Such a stable periodic gradient can be set up as indicated in Fig. 4. This model system has the added advantage that the patterning process and the differential interpretation of the pattern proceed simultaneously, unlike the basic positional information model, in which the information must be stabilized in its final form before being interpreted. The basic requirements of the model are as follows: (1) cells are sensitive to their internal concentration of a freely diffusible morphogen M ; (2) at concentrations of M below a lower threshold Tl cells are inactive; (3) at concentrations of M above Tl, cells synthesize M; (4) at concentrations of M above a higher threshold T2 cells actively destroy M ; (5) the transformations “inactive to synthetic” and “synthetic to destructive” are irreversible. This can be summarized as follows: [Inactive] [MlrT! [Synthetic] CMl>TZ [Destructive] These “rules” have been incorporated into a series of programs written in Fortran IV for the IBM 3701155 computer of the Edinburgh Regional Computing Centre. Diffusion is simulated as a flux F of M between adjacent
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Fro. 4. A semi-diagrammatic representation of the periodic gradient development of the model systemin one dimension. Small arrowsindicate direction of concentration cw. (1) Induction of synthesis by initiator. (2) & (3) Induction of destruction at second threshold. (4) & (5) Development of second peak and destruction area. (6) Stable residual gradient.
cells where F = d x ([Ml1 - [Ml,) per time unit, d being an arbitrary diffusion constant and [MJ and [Ml, the concentrations of M in adjacent cells. The constant d includes the term l/h’ required by Fick’s first law, where h is the (fixed) distance between cells. Synthesis is considered a constant rate process + S, implying a “saturated” system with a large substrate pool and destruction is considered a concentration dependent process - D x [M], where the value of the constants S and D depend on the maximum value that M has reached in the past. Concentration changes are considered stepwise for simplicity, therefore the total change in concentration of M in the ith cell in a one-dimensional array, a, b. . . h, i, i. . .y, z is calculated as follows for one time unit At. Diffusion :
AIM: = WNI, - CMln- CMlj)* Synthesis :
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Destruction : A[M]D = -~Cf’[M]~“)([M]i+A[M]:+A[M]S), where the value of f[M];““” is 1 between Tl and T2 and 0 outside these limits and the value off’[M]y”, 0 below T2 and 1 above. Thus ACM-J; = A[M];+A[M];+A[M];+A[M]~. As shown semi-diagrammatically in Fig. 4, these rules result in the propagation of a constant velocity wave of M along the cell array from an initiator region where cells are already synthesizing M. This initiator is essential for the formation of a repeatable pattern, random initiation giving the same periodicity of pattern but a different position and polarity. Behind the wave front, the first gradient peak grows linearly by synthesis and then splits into two as the upper threshold is reached and destruction initiated. The interactions of synthesis destruction and diffusion then cause the “trailing” peak to move backwards and downwards to stabilize between two areas of destruction and the “leading” peak to move forwards and upwards to initiate a new area of destruction. The end result is a periodic pattern and periodic residual gradient. Linking the initiation of destruction of M with the differentiation of cartilage, allows patterning and differentiation to proceed simultaneously but the residual gradient is still available for further patterning. The spread of the wave along the cell array is sufficient to specify a temporal developmental pattern and polarity, if all the cells initially have a common time base, (are, for example, the same age) since each cell will be “switched on” in strict order, This temporal pattern and polarity could then control the subsequent growth and differentiation of the cartilage elements “blocked out” by the primary gradient pattern and could similarly control other elements of the limb. In two dimensions, as shown in Fig. 5, the pattern is more complex. Here the cells are surrounded by “free space” into which M can diffuse to make the simulation more realistic and limit the pattern development. A square array of cells is taken so that each cell has eight neighbours that, for the diffusion calculation, are considered equivalent. This approximation does not seriously distort the pattern development but its presence must be noted. A more satisfactory cell array would be a hexagonal matrix, as we have used in cell sorting simulations (Wilby, unpublished). This is, however, more “expensive” in computing time and space. The diffusion equations used had the form
AEMI!= Js 0% - IMIJ&v where dnbr had the following values, O-11 for cell to cell or space to space
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FIG. 5. Development of model in two dimensions, showing shapes of differentiated areas. Times shown are computer steps. T25, spread of synthesis wave from initiator. T50-52, development of first destruction area. T7040, final shapes of subsequent areas.
neighbours, 0.01 for cell to space neighbours. This constant is equivalent to the At/h’ term of McCormick & Salvador-i’s (1964) requirement for numerical stability and is within their limits of h’/At > number of immediate neighbours. Since the average interaction distance h in the square array is 1.2 units, the implication is that At -n. 0.16 units and the equivalent cell to space interaction distance is four units. This is most simply interpreted as saying that the “epidermal” surface membrane is equivalent to about six “mesodermal” membranes. Synthesis and destruction terms are calculated separately for each cell as in the one-dimensional simulation. Total concentration changes are again found in a series of stepwise iterations. The use of simple numerical algorithms rather than more complex integration techniques result in some loss of accuracy, but the relative speed of computation offsets this as small step lengths can be used. The pattern that develops in two dimensions consists of a curved wave front of synthesis spreading out from a line “initiator” followed by point initiation of destruction. The areas of destruction expand centrifugally as shown in Fig. 5, as do real cartilage elements. This centrifugal expansion is not a basic feature of any “positional information” model. With a monotonic gradient the interpretation must either be simultaneous throughout or must be sequential from one side, giving either the whole rudiment in its final determined form or progressive determination across the rudiment. Successive elements form at regular intervals with a distinct polarity of size, resulting from the gradual steepening of the leading peak of M. This size polarity had not been predicted in the initial development of the model, but is a satisfying and realistic feature, giving the observed posterioranterior polarity with no further interaction. The effect is the combined result of the shape of the cell mass and the localized development of the morphogen gradient -with time.
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In both the one- and twodimensional simulations the “wavelength” and relative width of the pattern elements was found to depend inversely on the rates of synthesis and destruction of M. It was found most convenient to preset the “destruction rate” D high at O-9 (i.e. 90 % of a cell’s contents of M will be destroyed in one time unit) and vary the “synthesis rate” S between O-1 and 0.8. This gave well defined areas of destruction with approximately equal areas of synthesis, the “wavelength” depending inversely on the synthesis rate. It is thus possible to select constants that fit a given number of elements into any size cell mass, or, conversely, select a particular sized cell mass that gives a given number of elements with preset constants. 3. Testing and Medlflcation
of the Model
Since the object of the simulation was to produce the cartilage pattern of the limb, the relative sires and shapes of the cell masses were preset. As shown in Fig. 6, a set of cell arrays were taken using the outlines in Fig. 1 as a guide. Free diffusion into the medium is not realistic for an intact limb, since Loewenstein & Penn (1967) have demonstrated that cell/medium boundaries are characterized by cell membranes with a high electrical resistance. In consequence, to limit the pattern development and prevent cartilage elements forming too close to the limb surface, it is necessary to 28
FIG. 6. First attempt at producing limb cartilage patterns. Whole limb patterned from atippkd ZPA initiator ngion, stages after Fig. 1. Seatext for explanation.
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postulate a boundary layer of cells that actively destroy M, the obvious candidate for this is the ectoderm layer, indicated by double boundary lines in the figures. The “simulation” limbs are only approximately the same shape as the real limbs, partly due to the coarseness of resolution. For example, the “stage 24” simulation is 12 cells wide by 9 cells long; in reality this would be about 120 x 90 cells. One “computer cell” is therefore equivalent to a block of 100 real cells. This degree of resolution will be improved in future simulations, eventually to a 1: 1 correlation. With the preset limb shapes, a synthesis rate of O-4 and an initiator region at the posterior edge of the ZPA, cartilage patterns were produced that had, at least in the distal region, both the correct number, and the correct orientation of elements. For example, the “stage 24” simulation produces a “humerus” or “femur” element correctly located and the “stage 25” simulation a “radius and ulna” or “tibia and fibula” with the correct orientation and relative size. In the later proximal areas, however, the pattern does not fit at all, the elements lying transversely across the limb. This is not surprising, since in reality the limb pattern develops proximo-distally, as indicated in Fig. 1 and not simultaneously at a late stage as in, for example, the “stage 29” simulation of Fig. 6. Two modifications were attempted to try to improve the proximal pattern and more closely simulate real development; the first of these [Fig. 7(a)]
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FIG. 7. hkxiifkations to basic model. (a) Whole edge initiator (stippled area). (b) Two step model using proximal pm-patterned cells. ?.a 14
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was simply to extend the initiator to the full length of the limb, maintaining the “whole limb” patterning of the previous simulation. This gives “long bones” parallel to the long axis of the limb, but distal elements are now added successively only at the anterior edge, giving an overall lamina structure that does not resemble the real limb. The second, more major, modification was to combine the fully patterned region from one stage with the unpatterned distal region and initiator from the next developmental stage. The example shown in Fig. 7(b) is of a “stage 26” limb patterned with a lateral edge initiator as in 7(a), combined with the distal region of a “stage 28” limb; this combination being chosen simply for clarity, and being representative of all combinations. The pattern that forms in the distal region consists of two transverse bands, quite unlike the real limb metacarpals, with some ioterdigitation between the proximal “pre-patterned” elements. The reason for this failure is quite simply that the pre-patterned area is a much more effective “initiator” for the unpatterned distal region than the lateral initiator could ever be. This leads us to the conclusion that, for such a model to work in a real limb, the proximo-distal cell to cell communication must be much reduced compared to anterior-posterior. In fact, the proximal patterned areas must be partitioned or isolated from the distal unpatterned areas in some way. We have further modified the model to simulate this proximo-distal partitioning by generating a set of limb shapes representing the regions added by growth in each developmental stage, physically removing the proximal areas. This “growth increment” model is shown in Fig. S(a), the simulation constants and initiator regions being the same as in Fig. 6. This model produces a set of small cartilage elements, orientated along the proximo-distal axis and showing a distinct posterior-anterior polarity. When assembled into the completed pattern, the resemblance to the “real” cartilage pattern is quite striking, especially if the early “condensation pattern” of Fig. 9(b) @de, 1971) is used rather than the older patterns that have undergone differential growth. These condensation patterns are idealized representations of early cartilage development derived from both autoradiographical and histochemical studies of chondrogenesis in the chick limb. A close approximation can be made to the more developed pattern by assuming that each cartilage element grows with the surrounding tissue to maintain a constant relative size initially, and that this growth might be further modified by “global” interactions to produce the sort of pattern that is indicated by the overlays in Fig. 8(c). It must be emphasized that this overlay pattern is not generated by the model in its present form. One substantial objection to the “growth increment” model is the size of the incremental steps and the necessity for growth to be completed before
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(b) (a) Fro. 8. Growth increment model, patterning separate distal limb segments. (a) The separate increments and the pattern each develops. (b) The re-assembled limb showing the total pattern. (c) The total pattern of (b) overlain by proposed growth areas, giving the C--l X-L --.A_-
patterning begins. The step size used was deliberately chosen to give the disto-proximal periodicity of cartilage elements actually found in the real limb. If this periodicity is actually under the control of a separate system, such as the “Progress zone” of Summerbell et al. (1973), then the increment zone can be reduced to a single band of “computer cells”, located four ‘Tcomputer cells” behind the distal tip. In real terms this represents a band ten cells wide, some forty cells behind the AER. The final pattern produced in this way is currently being investigated. 4 Interpretation
and Predictions of the Model
The “growth increment” model thus makes two important predictions about the development of the limb pattern. Firstly that there is little or no proximo-distal interaction and secondly, that all cartilage elements are roughly the same size and shape at determination. Both these conclusions are supported by Wolpert, theoretically and experimentally (personal communication, Summerbell et al., 1973) though interpreted on quite a different model. A further, equally important prediction is that the urea of cells determined as cartilage, and hence the number of cells becoming chondrogenic expands centrifugally. This is currently being tested in our laboratory, both in vitro and in vivo.
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In terms of the “growth increment” model, the essential partitioning could most easily result from a change in the rate of diffusion of M from proximal to distal tissue, raising three possibilities-that the cell membrane permeabilities change, that cell to cell contacts change or that the nature of the intercellular space changes. The first of these could be tested electrophysiologically and would be effective for both direct cell to cell diffusion and for indirect diffusion via the intercellular space. Some evidence for the second possibility has been presented by Searls, Hilfer & Mirow (1972) who found a transition from broad to fine cell to cell contacts, i.e. from whole edge to filopodia, beginning at stage 22 in chondrogenic regions. These authors also find a gradual increase in extracellular material from stage 19, reaching the metachromatic staining threshold at stage 25, which would affect diffusion via the extracellular spaces. These observable changes, however, relate to the cellular differentiation of cartilage and would probably not be effective as required. Observations are needed near the distal tip of the limb that are common to all cells, regardless of differentiation type. The model so far developed predicts that the general cartilage pattern is a function of: (1) the limb shape; (2) the position of an initiator region; (3) the rate of synthesis of M. Changes in (l), i.e. limb shape, are known: the rulpid3 mutant, for example, has broad fan-shaped buds at stage 29. Given a “growth increment” simulation with the taZpid3 shapes and the same constants and relative initiator positions as in Fig. 8, the &a&id3 cartilage pattern, as analysed by Ede & Agerbak (1968), can be produced. This is shown in Fig. 9(d), (e) and (f ). A further correlation here is with the weakening and loss of the distinctive posterior-anterior polarity in size and shape of the cartilage elements which characterizes the mutant; this also appears in the simulation, resulting simply from changing the overall limb shape. Change in (2), i.e. the initiator position, occurs in the ZPA graft shown in Fig. 2. Further experiments are in progress to determine the exact nature of the interactions in these grafts but meanwhile it can be said that, given the duplicate shape, the growth increment model will approximate the duplicate pattern. The time and mode of determination of the ZPA, and hence the initiator region are, as yet, unknown. The two most probable mechanisms are clonal determination at the time of general limb determination, along the lines suggested by Mintz (1970) and position specific determination after preliminary limb growth by any “positional information” type of mechanism (c.f. Wolpert, 1969).
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The clonal determination mechanism requires that the clone of initiator cells, or its active component, remains always near the posterior end of the AER but, given such regular development as has been shown in melanoblastic clones (Mintz, 1971), this should not prove an insurmountable problem. Some evidence for a mesodermal clonal basis and against any position specitic determination based on ectodermal structures, comes from the work on “reaggregate” limbs, produced by packing limb mesoderm cells which have been disaggregated and subsequently randomly reaggregated into ectodermal jackets and grafting to host embryos. Such limbs consist chiefly of the more distal elements, e.g. digits and metacarpals; they do not exhibit any anterior-posterior polarity unless ZPA material is included (Ma&&e & Saunders, 1971) but they do show a distinct periodic pattern, as indicated in Fig. 9(h) from Pautou (1973). In the disto-proximal axis this is presumably a function of the re-establishment of the normal control processes, based on the AER of the ectodermal jacket (this also gives the dorso-ventral polarity). In the anterior-posterior axis, the existence of a few, randomly placed initiator cells could be sufficient to trigger the patterning process. A new initiator clone developed from these cells would stabilize the phase of the reaggregate limb pattern. A change in (3), i.e. in rate of synthesis of M, is shown in Fig. 9(g), producing a polydactylous limb the same size as the “normal”, Fig. 9(a) but with a shorter anterior-posterior “wavelength”. Thus the model predicts two distinct classes of polydactylous mutants-those that have the “normal” digit frequency but a broader paddle and those with an increased digit frequency and a paddle of normal width at the time of determination. The tuZpid3 mutant belongs to the first type but other less extreme examples are common (see Grlineberg, 1963). No clear examples of the second type have been described, suggesting that genetic control of this parameter is rather stable in existing vertebrates. It is extremely difficult to develop a simple and realistic model in which synthesis rate is a variable function controlled by limb size. It is possible therefore that a trial-and-error system operates during evolution and that the successful survivors are those with a functional balance between limb size and digit frequency. One postulate that can be made here is that all vertebrate limbs, no matter what their final size, may be the sume size when cartilage determination occurs. This is a logical extension of Wolpert’s (1969) proposal that all embryonic fields are the same sizeabout 100 cells wide. The model produces a rather crude sketch of the skeletal limb pattern rather than the final form it takes in existing highly evolved land vertebrates. In this it might well resemble the control system in the limb of primitive
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FIG. 9. A comparison of real and simulated limb patterns. (a) The simulated “normal” pattern. (b) Precartilage condensation pattern of normal chick. (c) Cartilage pattern of a stage 32 chick leg. Simulated tuZpida pattern using the same constants as (a) but a fan-shaped paddle. (e) T&W prec&ilage condensation pattern. (f) Stage 32 talpida leg. (g) Polydactylous simulation using the “normal” shape but increased synthesis rate (see text). (h) Cartilage pattern developed in a reaggregated limb. (i) Skeletal structure of a primitive tetrapod limb. (j) Structure of a Saurrpterus fin.
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tetrapod types, e.g. in Fig. 9(i), and in the fin of the Devonian crossopterygian Fig. 9(j) (both after Romer, 1933), from which ,land vertebrates may have evolved. As far as it is reasonable to do so, the simple periodic gradient model may be applied to all limb development with fair accuracy, though producing increasing complexity will require additional refinements in the model just as it did in evolution. It is probable that the real patterning process is more complex than any of the models we present here,. but we believe that the basic periodic gradient is a vital component of limb patterning. A more accurate simulation of the growth process, giving the specific limb shape essential for correct patterning, is needed before it would be profitable to make the basic model more complex. We are developing such a model from that of Ede & Law (1969). Sauripterus,
5. Experimental
Testing of the Model
At this stage the model is obviously incomplete and cannot be expected to give an exact fit to complex interactions produced in experimental grafting operations. In fact, a degree of regulation is possible, and it must be borne in mind that the regulatory capacity of the real system is very limited. Removal or addition of patterned material by extirpation or grafting, or by growth processes, would disturb the residual gradient, possibly enough to influence further differentiation. Thus, abnormal growth might separate two “destructive cartilage” areas sufficiently to allow secondary initiation in the intervening region, and it may well be that the extra interpolated (“ectopic”) digits observed in the taZpid3 mutant (Ede & Kelly, 1964) and also in reaggregate limbs (Pautou, 1973) arise in this way. A further capacity for regulation in our model is shown by the size polarity developed along the anterior-posterior axis, the final size and shape of the cartilage element being a function of the variable fine structure of the gradient. A comparison can be drawn here between the basic regulatory behaviour to be expected from a periodic and a monotonic gradient model. With a periodic gradient, allowing for the size variations mentioned above and assuming that changes in synthesis rate do not occur, regulation will occur as the addition or deletion of whole pattern elements with little or no change in the remaining pattern. On the other hand, regulation of a monotonic gradient along the lines discussed by Wolpert (1969) would change the relative size and spacing of the existing elements but neither add nor subtract any. Thus, excepting possibly the cases where a second initiator or organizer region is produced, giving a “mirror image” type of duplication (as found by Gasseling & Saunders, 1964, for example), the monotonic, regulatory gradient model would not be expected to produce polydactylous limbs of any sort.
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The predictions and assumptions of our model can be empirically tested by measuring the physical parameters of dateloping normal, mutant and grafted limbs. The criteria by which this model then stands or falls are, does cartilage determination in the limb (i) occur centrifugally in blocks of approximately constant size; (ii) occur sequentially across the anteriorposterior axis of the limb with a constant “wave length”. It would seem reasonable to apply this model to all situations in which periodic differentiation occurs, perhaps the most obvious being the regionalization of somitic mesoderm and subsequent formation of somites (Hamilton, 1969; Lanot, 1971; Kieny, Mauger & Sengel, 1972). The linkage of the destruction phase to differentiation will obviously vary with cell type. While not eliminating the desirability of constructing alternative models it is noteworthy that by applying it in limb shapes which have been studied in normal and mutant development, a fair approximation to the cartilage patterns arising in real embryos is produced. The model is susceptible of a number of interpretations at the biochemical level; perhaps the simplest is to postulate an allosteric enzyme system, the enzyme being inactive at concentrations of M below Tl, transforming irreversibly to an active form by binding M when the concentration exceeds Tl and now catalyzing the production of M from a substrate S, finally undergoing a second irreversible transformation by binding more M when the concentration exceeds T2 and catalyzing the reverse reaction M to S. At the present time it is impossible to test the model at this level; in essence it is a set of rules whereby the individual cells interpret a signal from their environment (the concentration of a diffusible morphogen) with respect to their internal state, thereby altering their state and their output, or, in other words, their differentiation and their production/destruction of the morphogen. We gratefully acknowledge support from the Agricultural Research Council and the Science Research Council. REJ?ERENCES of MuthematicuZ Biology, vol. 2 (R. Rosen, ed.). New York and London: Academic Press. BARD, J. & LAUDER, I. (1974). J. theor. Biol. 45, 501. EDE, D. A. (1971). Symp. Sot. exp. Bill. 25,235. EDE, D. A. & AGE-AK, G. S. J. Embryol. exp. Morph. 20, 81. EDE, D. A. & KELLY, W. A. (1974). L Ekbryol. exp. Morph. 12,339. EDE, D. A. & LAW, J. T. (1969). Nature, Land. 221,244. GASSLING, M. T. & SAUNDERS,J. W. JR (1964). Am. Zool. 4,303. &DDWlN, B. & COHEN, M. H. (1%9). J. fheor. Biof. 25,379. GRONEBERO,H. (1963). The Patho&y of Development. Oxford: Blackwell. I-brmmm, V. & HAMILTON, H. L. (1951). J. Morph. 88.49. ARBIE,
M.
(1972).
In Foundations
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HAMILTON, L. (1969). J. Embryol. exp. Morph. 22,253. KIENY, M., MAUGER, A. & SENGEL, P. (1972). Devl Bfol. 28,142. LANOT, R. (1971). J. Embryol. exp. Morph. 26, l-20. bEWEmrEIN, W. R. & PENN, R. D. (1967). J. Ceil. Bill. 33,235. MACCABE, J. A. & SAUNDERS, J. W. (1971). Amt. Rec. 169,372. MCCORMICK, J. M. & SALVADORI, M. (1964). Numerical Metho& in Fortran. New Jersey: Prentice-Hall. MINI-Z, B. (1970). Symp. int. Sot. Cell B&l. 9, 16. MINTZ, B. (1971). Symp. Sot. exp. B&l. 25,345. PATOU, M. P. (1973). J. Embryol. exp. Momh. 29,175. ROMW, A. B. (1933). In Vertebrate Palaeontology. Chicago: University of Chicago Press. SJIARLT, R. L., HILFEX, S. R. Bi MIROW, S. M. (1972). Devl B&l. 28,123. SUMM~RBELL, D., LEWIS, J. H. & WOLPERT, L. (1973). Nature, Lmd. 244,492. TURING, A. M. (1952). Phil. Trans. R. Sot. B 237, 32. WOLPWT, L. (1969). J. theor. Bzbl. 25, 1. ZWILLING, EL (1961). A& Morphogen. 1,301.
A Model Generating The Pattern of Cartilage Skeletal Elements in the Embryonic Chick Limb? 0. K. WILBY AND D. A. EDE
Department of Zoology, University of Glasgow, Scotland, U.K. (Received 10 June 1974, and in revised form 30 August 1974) The development of the vertebrate limb involves the production of a specific external form arising from cell division and other growth processes at the cellular level, and the origin within it of specific patterns of tissues arising by cellular differentiation, of which the pattern of cartilages which pm-figure the limb skeleton is the most striking. In this paper we propose a model for the differentiation (or the preceding determination) process that, using only localized cell to cell interactions, can approximate the cartilage pattern in any limb shape. The model requires cells to modify their metabolism irreversibly at critical threshold levels of a diffusible morphogen which may be made or destroyed by these cells. Restrictions inherent in the successful development of a total limb pattern using this systemlead to theprediction that the process is con6ned to a distal band which has no sign&ant interaction with more proximal regions but within this band the characteristic features of the anterior-posterior axis of the limb develop without additional interactions. Cartilage elements are initiated as single “cells” and expand centrifugally to their tinal size; these elements developing sequentially along the anteriorposterior axis, showing a distinct polarity of size. The model also predicts that equivalent cartilage elements in all vertebrate limbs will be roughly the same relative size at determination, the extensive range of adult structures arising by differential growth and fusion, possibly controlled by global aspects of the model. It must be emphasized that this model only satisfactorily simulates the anterior-posterior patterning of cartilage elements, the disto-proximal pattern being externally imposed. The fhral cartilage pattern is shown to be a function of (1) the developing shape of the limb, (2) the position of an initiator region that starts the patterning process and (3) the rate of production of the diffusible morphogen. Using parameters selected with as much realism as possible the model gives a good approximation to the pattern of cartilages found in the normal chick limb; modifying the shape of the limb to that of the talpid3 mutant t A shortened version of this paper was presented at the Conference on Biologically Motivated Automata Theory, June 19-21 at the MITRE Corporation, McLean. Virginia, U.S.A. 199
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produces the characteristic features of the cartilage pattern found in that mutant and modifying the rate of morphogen production simulates patterns resembling those found in some ancestral vertebrate fossil forms. 1. Introduction
The limb in vertebrate embryos develops from a small limb bud; at its first appearance this is a simple hillock of cells arising on the flank, which becomes elongated by outward growth, then paddle-shaped by expansion at its tip. At the earliest stages the bud consists predominantly of undifferentiated mesoderm cells, covered by a thin layer of ectoderm which is modified distally to form a more rigid apical ectodermal ridge (AER). As outgrowth takes place, cell differentiation occurs within the mesoderm to produce a pattern of cartilages which form the basis of the bony skeleton of the limb (Ede, 1971). This developing system constitutes a remarkably clear-cut example of what Arbib (1972) has suggested is the crucial problem for theoretical embryology: How can a single cell in a continually active tissue, where all cells may be growing, dividing, differentiating, or dying, use only information from nearby cells to so grow as to contribute properly to the overall form of the organism? For simplicity we have treated the limb bud as a two-dimensional object, ignoring its dorsal-ventral axis. The remaining axes are the proximal-distal, from the base of the bud to its tip, and the anterior-posterior axis at right angles to it. In both these axes the cartilage pattern exhibits two features which are characteristic of biological pattern in general: polarity and periodicity. In the proximal-distal axis growth is polarized along this axis and a developmental polarity distinguishes the sets of cartilage elements that appear periodically along the limb: thus there appears a basal band (e.g., the upper arm in the human forelimb), a second band (the lower arm), a third (the metacarpals) and a fourth (the digits). We omit, for simplicity, the band of more complex carpal rudiments. Experimental observations indicate a complex interaction between the distal mesoderm and the overlying AER in determining this axis (Zwilling, 1961). Summerbell, Lewis & Wolpert (1973) have developed a plausible model for generating this axial pattern, utilizing what is essentially a clock based on the cycles of cell division in a narrow band of distal mesoderm cells. In our work reported here, pattern generation in this axis arises primarily from the subdivision necessary for the realistic proximal-distal orientation of each cartilage element, rather than from a periodic control mechanism. The structure of the joints between cartilage elements, the relative lengths of each element and their proximo-distal relationship are beyond the scope of the present
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model and so appear as externally imposed restrictions. Qualitatively, however, these restrictions are identical to those of Summerbell et al. (1973). Our model was developed primarily to account for the remaining anteriorposterior axis. Here again the pattern exhibits polarity in the characteristic size of each cartilage element and periodicity in the arrangement of these elements: a single element in the basal band (e.g., the humerus in the upper arm), two elements in the second (radius and ulna in the lower arm), four
23 >
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FIG. 1. The normal development of chick at stages 23 to 35 (Hamburger & Hamilton,
limb, showing 1951).
areas
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in the third (metacarpals) and the fourth (digits). This developing pattern is shown in Fig. 1 for the chick limb (generalized limb stages, following Hamburger & Hamilton, 1951). Computer simulations have been performed for stages 24-29 only because of difficulties in successfully simulating differential growth. The control of the anterior-posterior axis appears to lie in the posteriorlateral margin of the limb bud, the zone of polarizing activity (ZPA), discovered by Gasseling & Saunders (1964). As shown in Fig. 2, a graft of ZPA material to the anterior edge of a wing bud, causes limb duplication. In theory, this could be due to a diffusion gradient emanating from the ZPA
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FIG. 2. The ZPA graft and one interpretation. (b) The diffusion gradient interpretation.
(a) The transplant experiment and its result.
as shown, coupling growth and differentiation to particular levels. This basic experiment suggested to us the lateral edge of the ZPA as the location of the “initiator” site in our simulations. 2. Development of Models fn attempting to model this type of embryonic development it is possible to adopt one of two alternative basic approaches: (1) Positional Information (see Wolpert, 1969) where pattern determination is regarded as a two-stage process involving (a) an assignment of a specific positional value to each cell, followed by (b) the interpretation of this positional information in terms of molecular differentiation. (2) Automata Theory (see Arbib, 1971) where each cell is regarded as an automaton, the state of the cell being controlled by a metabolic mechanism sensitive to critical levels of input, producing a particular level or type of output. The position of the cell is strictly irrelevant, its differentiation depending only on its own internal state and an awareness of what its neighbours are doing. Our own work is closer to the second approach. Positional information models most often use a monotonically decreasing gradient, e.g., of some diffusible substance, to specify the positional value, as shown in Fig. 3(a); thus cells might interpret gradient levels between “thresholds” Tl and T2
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Fro. 3. Two alternative ways of determining a periodic pattern: (a) via a monotonic gradient and many “thresholds”, (b) via a periodic gradient and one “threshold”.
as specifying cartilage development. Whilst simple in theory, this model becomes less plausible where the pattern is markedly periodic; thus to produce three bands of cartilage as shown in Fig. 3(a), differentiation must occur between six threshold levels: Tl-T2, T3-T4, T5-T6. This requires a biochemical switch mechanism sensitive to many different triggering levels. As the number of pattern elements increases, so does the number of thresholds, necessitating an extremely complex, sensitive and stable system. The addition or deletion of pattern elements to give, for example, the variety of limb structures seen in fossil, normal and mutant organisms further complicates a monotonic gradient system since both the number of thresholds and the slope of the gradient need regulating. We suggest an alternative system using a periodic gradient as indicated in Fig. 3(b) and a single threshold for cartilage differentiation. Providing such a gradient can be simply set up, the system does not become more complex with an increasing number of elements, nor is extreme stability or sensitivity required, since slight variations in concentration do not markedly alter the pattern. The modscation of this pattern in terms of element number can be regarded simply as a control of overall size as will be explained later, a polydactylous limb requiring no more complex a control system than a
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single digit. It must also be remembered that the regulative patterns of evolutionary primitive and early embryonic organisms are lost in favour of mosaic non-regulative patterns in advanced and more mature organisms. Periodic gradient models have previously been proposed for the control of developmental events. Turing (1952), proposed a model in which two interacting morphogens would, from uniform initial conditions and random perturbations, develop a stable chemical wave pattern. In a recent elegant analysis of this model, Bard & Lauder (1974) have demonstrated that this system is too unreliable to serve as the generating mechanism for features such as digits but is more than adequate for leaf and hair patterns. The “phase gradient” model of Goodwin & Cohen (1969), essentially a system for supplying polarity and positional information on a temporal basis, lends itself easily to the production of periodic structures. The resultant pattern is, however, one based on a “saw tooth” gradient; within each segment the gradient is monotonically decreasing and all the above arguments apply; the basic model is also biochemically complex. In a search for simple patterning mechanisms, in the belief that biological complexity results from the interaction of several simple control processes rather than a single complex one we have looked at a system in which a stationary chemical wave is reliably established by an oscillating travelling wave front. Such a stable periodic gradient can be set up as indicated in Fig. 4. This model system has the added advantage that the patterning process and the differential interpretation of the pattern proceed simultaneously, unlike the basic positional information model, in which the information must be stabilized in its final form before being interpreted. The basic requirements of the model are as follows: (1) cells are sensitive to their internal concentration of a freely diffusible morphogen M ; (2) at concentrations of M below a lower threshold Tl cells are inactive; (3) at concentrations of M above Tl, cells synthesize M; (4) at concentrations of M above a higher threshold T2 cells actively destroy M ; (5) the transformations “inactive to synthetic” and “synthetic to destructive” are irreversible. This can be summarized as follows: [Inactive] [MlrT! [Synthetic] CMl>TZ [Destructive] These “rules” have been incorporated into a series of programs written in Fortran IV for the IBM 3701155 computer of the Edinburgh Regional Computing Centre. Diffusion is simulated as a flux F of M between adjacent
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Fro. 4. A semi-diagrammatic representation of the periodic gradient development of the model systemin one dimension. Small arrowsindicate direction of concentration cw. (1) Induction of synthesis by initiator. (2) & (3) Induction of destruction at second threshold. (4) & (5) Development of second peak and destruction area. (6) Stable residual gradient.
cells where F = d x ([Ml1 - [Ml,) per time unit, d being an arbitrary diffusion constant and [MJ and [Ml, the concentrations of M in adjacent cells. The constant d includes the term l/h’ required by Fick’s first law, where h is the (fixed) distance between cells. Synthesis is considered a constant rate process + S, implying a “saturated” system with a large substrate pool and destruction is considered a concentration dependent process - D x [M], where the value of the constants S and D depend on the maximum value that M has reached in the past. Concentration changes are considered stepwise for simplicity, therefore the total change in concentration of M in the ith cell in a one-dimensional array, a, b. . . h, i, i. . .y, z is calculated as follows for one time unit At. Diffusion :
AIM: = WNI, - CMln- CMlj)* Synthesis :
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Destruction : A[M]D = -~Cf’[M]~“)([M]i+A[M]:+A[M]S), where the value of f[M];““” is 1 between Tl and T2 and 0 outside these limits and the value off’[M]y”, 0 below T2 and 1 above. Thus ACM-J; = A[M];+A[M];+A[M];+A[M]~. As shown semi-diagrammatically in Fig. 4, these rules result in the propagation of a constant velocity wave of M along the cell array from an initiator region where cells are already synthesizing M. This initiator is essential for the formation of a repeatable pattern, random initiation giving the same periodicity of pattern but a different position and polarity. Behind the wave front, the first gradient peak grows linearly by synthesis and then splits into two as the upper threshold is reached and destruction initiated. The interactions of synthesis destruction and diffusion then cause the “trailing” peak to move backwards and downwards to stabilize between two areas of destruction and the “leading” peak to move forwards and upwards to initiate a new area of destruction. The end result is a periodic pattern and periodic residual gradient. Linking the initiation of destruction of M with the differentiation of cartilage, allows patterning and differentiation to proceed simultaneously but the residual gradient is still available for further patterning. The spread of the wave along the cell array is sufficient to specify a temporal developmental pattern and polarity, if all the cells initially have a common time base, (are, for example, the same age) since each cell will be “switched on” in strict order, This temporal pattern and polarity could then control the subsequent growth and differentiation of the cartilage elements “blocked out” by the primary gradient pattern and could similarly control other elements of the limb. In two dimensions, as shown in Fig. 5, the pattern is more complex. Here the cells are surrounded by “free space” into which M can diffuse to make the simulation more realistic and limit the pattern development. A square array of cells is taken so that each cell has eight neighbours that, for the diffusion calculation, are considered equivalent. This approximation does not seriously distort the pattern development but its presence must be noted. A more satisfactory cell array would be a hexagonal matrix, as we have used in cell sorting simulations (Wilby, unpublished). This is, however, more “expensive” in computing time and space. The diffusion equations used had the form
AEMI!= Js 0% - IMIJ&v where dnbr had the following values, O-11 for cell to cell or space to space
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FIG. 5. Development of model in two dimensions, showing shapes of differentiated areas. Times shown are computer steps. T25, spread of synthesis wave from initiator. T50-52, development of first destruction area. T7040, final shapes of subsequent areas.
neighbours, 0.01 for cell to space neighbours. This constant is equivalent to the At/h’ term of McCormick & Salvador-i’s (1964) requirement for numerical stability and is within their limits of h’/At > number of immediate neighbours. Since the average interaction distance h in the square array is 1.2 units, the implication is that At -n. 0.16 units and the equivalent cell to space interaction distance is four units. This is most simply interpreted as saying that the “epidermal” surface membrane is equivalent to about six “mesodermal” membranes. Synthesis and destruction terms are calculated separately for each cell as in the one-dimensional simulation. Total concentration changes are again found in a series of stepwise iterations. The use of simple numerical algorithms rather than more complex integration techniques result in some loss of accuracy, but the relative speed of computation offsets this as small step lengths can be used. The pattern that develops in two dimensions consists of a curved wave front of synthesis spreading out from a line “initiator” followed by point initiation of destruction. The areas of destruction expand centrifugally as shown in Fig. 5, as do real cartilage elements. This centrifugal expansion is not a basic feature of any “positional information” model. With a monotonic gradient the interpretation must either be simultaneous throughout or must be sequential from one side, giving either the whole rudiment in its final determined form or progressive determination across the rudiment. Successive elements form at regular intervals with a distinct polarity of size, resulting from the gradual steepening of the leading peak of M. This size polarity had not been predicted in the initial development of the model, but is a satisfying and realistic feature, giving the observed posterioranterior polarity with no further interaction. The effect is the combined result of the shape of the cell mass and the localized development of the morphogen gradient -with time.
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In both the one- and twodimensional simulations the “wavelength” and relative width of the pattern elements was found to depend inversely on the rates of synthesis and destruction of M. It was found most convenient to preset the “destruction rate” D high at O-9 (i.e. 90 % of a cell’s contents of M will be destroyed in one time unit) and vary the “synthesis rate” S between O-1 and 0.8. This gave well defined areas of destruction with approximately equal areas of synthesis, the “wavelength” depending inversely on the synthesis rate. It is thus possible to select constants that fit a given number of elements into any size cell mass, or, conversely, select a particular sized cell mass that gives a given number of elements with preset constants. 3. Testing and Medlflcation
of the Model
Since the object of the simulation was to produce the cartilage pattern of the limb, the relative sires and shapes of the cell masses were preset. As shown in Fig. 6, a set of cell arrays were taken using the outlines in Fig. 1 as a guide. Free diffusion into the medium is not realistic for an intact limb, since Loewenstein & Penn (1967) have demonstrated that cell/medium boundaries are characterized by cell membranes with a high electrical resistance. In consequence, to limit the pattern development and prevent cartilage elements forming too close to the limb surface, it is necessary to 28
FIG. 6. First attempt at producing limb cartilage patterns. Whole limb patterned from atippkd ZPA initiator ngion, stages after Fig. 1. Seatext for explanation.
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postulate a boundary layer of cells that actively destroy M, the obvious candidate for this is the ectoderm layer, indicated by double boundary lines in the figures. The “simulation” limbs are only approximately the same shape as the real limbs, partly due to the coarseness of resolution. For example, the “stage 24” simulation is 12 cells wide by 9 cells long; in reality this would be about 120 x 90 cells. One “computer cell” is therefore equivalent to a block of 100 real cells. This degree of resolution will be improved in future simulations, eventually to a 1: 1 correlation. With the preset limb shapes, a synthesis rate of O-4 and an initiator region at the posterior edge of the ZPA, cartilage patterns were produced that had, at least in the distal region, both the correct number, and the correct orientation of elements. For example, the “stage 24” simulation produces a “humerus” or “femur” element correctly located and the “stage 25” simulation a “radius and ulna” or “tibia and fibula” with the correct orientation and relative size. In the later proximal areas, however, the pattern does not fit at all, the elements lying transversely across the limb. This is not surprising, since in reality the limb pattern develops proximo-distally, as indicated in Fig. 1 and not simultaneously at a late stage as in, for example, the “stage 29” simulation of Fig. 6. Two modifications were attempted to try to improve the proximal pattern and more closely simulate real development; the first of these [Fig. 7(a)]
(a)
(b)
FIG. 7. hkxiifkations to basic model. (a) Whole edge initiator (stippled area). (b) Two step model using proximal pm-patterned cells. ?.a 14
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was simply to extend the initiator to the full length of the limb, maintaining the “whole limb” patterning of the previous simulation. This gives “long bones” parallel to the long axis of the limb, but distal elements are now added successively only at the anterior edge, giving an overall lamina structure that does not resemble the real limb. The second, more major, modification was to combine the fully patterned region from one stage with the unpatterned distal region and initiator from the next developmental stage. The example shown in Fig. 7(b) is of a “stage 26” limb patterned with a lateral edge initiator as in 7(a), combined with the distal region of a “stage 28” limb; this combination being chosen simply for clarity, and being representative of all combinations. The pattern that forms in the distal region consists of two transverse bands, quite unlike the real limb metacarpals, with some ioterdigitation between the proximal “pre-patterned” elements. The reason for this failure is quite simply that the pre-patterned area is a much more effective “initiator” for the unpatterned distal region than the lateral initiator could ever be. This leads us to the conclusion that, for such a model to work in a real limb, the proximo-distal cell to cell communication must be much reduced compared to anterior-posterior. In fact, the proximal patterned areas must be partitioned or isolated from the distal unpatterned areas in some way. We have further modified the model to simulate this proximo-distal partitioning by generating a set of limb shapes representing the regions added by growth in each developmental stage, physically removing the proximal areas. This “growth increment” model is shown in Fig. S(a), the simulation constants and initiator regions being the same as in Fig. 6. This model produces a set of small cartilage elements, orientated along the proximo-distal axis and showing a distinct posterior-anterior polarity. When assembled into the completed pattern, the resemblance to the “real” cartilage pattern is quite striking, especially if the early “condensation pattern” of Fig. 9(b) @de, 1971) is used rather than the older patterns that have undergone differential growth. These condensation patterns are idealized representations of early cartilage development derived from both autoradiographical and histochemical studies of chondrogenesis in the chick limb. A close approximation can be made to the more developed pattern by assuming that each cartilage element grows with the surrounding tissue to maintain a constant relative size initially, and that this growth might be further modified by “global” interactions to produce the sort of pattern that is indicated by the overlays in Fig. 8(c). It must be emphasized that this overlay pattern is not generated by the model in its present form. One substantial objection to the “growth increment” model is the size of the incremental steps and the necessity for growth to be completed before
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(c)
(b) (a) Fro. 8. Growth increment model, patterning separate distal limb segments. (a) The separate increments and the pattern each develops. (b) The re-assembled limb showing the total pattern. (c) The total pattern of (b) overlain by proposed growth areas, giving the C--l X-L --.A_-
patterning begins. The step size used was deliberately chosen to give the disto-proximal periodicity of cartilage elements actually found in the real limb. If this periodicity is actually under the control of a separate system, such as the “Progress zone” of Summerbell et al. (1973), then the increment zone can be reduced to a single band of “computer cells”, located four ‘Tcomputer cells” behind the distal tip. In real terms this represents a band ten cells wide, some forty cells behind the AER. The final pattern produced in this way is currently being investigated. 4 Interpretation
and Predictions of the Model
The “growth increment” model thus makes two important predictions about the development of the limb pattern. Firstly that there is little or no proximo-distal interaction and secondly, that all cartilage elements are roughly the same size and shape at determination. Both these conclusions are supported by Wolpert, theoretically and experimentally (personal communication, Summerbell et al., 1973) though interpreted on quite a different model. A further, equally important prediction is that the urea of cells determined as cartilage, and hence the number of cells becoming chondrogenic expands centrifugally. This is currently being tested in our laboratory, both in vitro and in vivo.
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In terms of the “growth increment” model, the essential partitioning could most easily result from a change in the rate of diffusion of M from proximal to distal tissue, raising three possibilities-that the cell membrane permeabilities change, that cell to cell contacts change or that the nature of the intercellular space changes. The first of these could be tested electrophysiologically and would be effective for both direct cell to cell diffusion and for indirect diffusion via the intercellular space. Some evidence for the second possibility has been presented by Searls, Hilfer & Mirow (1972) who found a transition from broad to fine cell to cell contacts, i.e. from whole edge to filopodia, beginning at stage 22 in chondrogenic regions. These authors also find a gradual increase in extracellular material from stage 19, reaching the metachromatic staining threshold at stage 25, which would affect diffusion via the extracellular spaces. These observable changes, however, relate to the cellular differentiation of cartilage and would probably not be effective as required. Observations are needed near the distal tip of the limb that are common to all cells, regardless of differentiation type. The model so far developed predicts that the general cartilage pattern is a function of: (1) the limb shape; (2) the position of an initiator region; (3) the rate of synthesis of M. Changes in (l), i.e. limb shape, are known: the rulpid3 mutant, for example, has broad fan-shaped buds at stage 29. Given a “growth increment” simulation with the taZpid3 shapes and the same constants and relative initiator positions as in Fig. 8, the &a&id3 cartilage pattern, as analysed by Ede & Agerbak (1968), can be produced. This is shown in Fig. 9(d), (e) and (f ). A further correlation here is with the weakening and loss of the distinctive posterior-anterior polarity in size and shape of the cartilage elements which characterizes the mutant; this also appears in the simulation, resulting simply from changing the overall limb shape. Change in (2), i.e. the initiator position, occurs in the ZPA graft shown in Fig. 2. Further experiments are in progress to determine the exact nature of the interactions in these grafts but meanwhile it can be said that, given the duplicate shape, the growth increment model will approximate the duplicate pattern. The time and mode of determination of the ZPA, and hence the initiator region are, as yet, unknown. The two most probable mechanisms are clonal determination at the time of general limb determination, along the lines suggested by Mintz (1970) and position specific determination after preliminary limb growth by any “positional information” type of mechanism (c.f. Wolpert, 1969).
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The clonal determination mechanism requires that the clone of initiator cells, or its active component, remains always near the posterior end of the AER but, given such regular development as has been shown in melanoblastic clones (Mintz, 1971), this should not prove an insurmountable problem. Some evidence for a mesodermal clonal basis and against any position specitic determination based on ectodermal structures, comes from the work on “reaggregate” limbs, produced by packing limb mesoderm cells which have been disaggregated and subsequently randomly reaggregated into ectodermal jackets and grafting to host embryos. Such limbs consist chiefly of the more distal elements, e.g. digits and metacarpals; they do not exhibit any anterior-posterior polarity unless ZPA material is included (Ma&&e & Saunders, 1971) but they do show a distinct periodic pattern, as indicated in Fig. 9(h) from Pautou (1973). In the disto-proximal axis this is presumably a function of the re-establishment of the normal control processes, based on the AER of the ectodermal jacket (this also gives the dorso-ventral polarity). In the anterior-posterior axis, the existence of a few, randomly placed initiator cells could be sufficient to trigger the patterning process. A new initiator clone developed from these cells would stabilize the phase of the reaggregate limb pattern. A change in (3), i.e. in rate of synthesis of M, is shown in Fig. 9(g), producing a polydactylous limb the same size as the “normal”, Fig. 9(a) but with a shorter anterior-posterior “wavelength”. Thus the model predicts two distinct classes of polydactylous mutants-those that have the “normal” digit frequency but a broader paddle and those with an increased digit frequency and a paddle of normal width at the time of determination. The tuZpid3 mutant belongs to the first type but other less extreme examples are common (see Grlineberg, 1963). No clear examples of the second type have been described, suggesting that genetic control of this parameter is rather stable in existing vertebrates. It is extremely difficult to develop a simple and realistic model in which synthesis rate is a variable function controlled by limb size. It is possible therefore that a trial-and-error system operates during evolution and that the successful survivors are those with a functional balance between limb size and digit frequency. One postulate that can be made here is that all vertebrate limbs, no matter what their final size, may be the sume size when cartilage determination occurs. This is a logical extension of Wolpert’s (1969) proposal that all embryonic fields are the same sizeabout 100 cells wide. The model produces a rather crude sketch of the skeletal limb pattern rather than the final form it takes in existing highly evolved land vertebrates. In this it might well resemble the control system in the limb of primitive
(a)
04
(e)
(4
(Q)
(f-d
(i)
(j)
FIG. 9. A comparison of real and simulated limb patterns. (a) The simulated “normal” pattern. (b) Precartilage condensation pattern of normal chick. (c) Cartilage pattern of a stage 32 chick leg. Simulated tuZpida pattern using the same constants as (a) but a fan-shaped paddle. (e) T&W prec&ilage condensation pattern. (f) Stage 32 talpida leg. (g) Polydactylous simulation using the “normal” shape but increased synthesis rate (see text). (h) Cartilage pattern developed in a reaggregated limb. (i) Skeletal structure of a primitive tetrapod limb. (j) Structure of a Saurrpterus fin.
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tetrapod types, e.g. in Fig. 9(i), and in the fin of the Devonian crossopterygian Fig. 9(j) (both after Romer, 1933), from which ,land vertebrates may have evolved. As far as it is reasonable to do so, the simple periodic gradient model may be applied to all limb development with fair accuracy, though producing increasing complexity will require additional refinements in the model just as it did in evolution. It is probable that the real patterning process is more complex than any of the models we present here,. but we believe that the basic periodic gradient is a vital component of limb patterning. A more accurate simulation of the growth process, giving the specific limb shape essential for correct patterning, is needed before it would be profitable to make the basic model more complex. We are developing such a model from that of Ede & Law (1969). Sauripterus,
5. Experimental
Testing of the Model
At this stage the model is obviously incomplete and cannot be expected to give an exact fit to complex interactions produced in experimental grafting operations. In fact, a degree of regulation is possible, and it must be borne in mind that the regulatory capacity of the real system is very limited. Removal or addition of patterned material by extirpation or grafting, or by growth processes, would disturb the residual gradient, possibly enough to influence further differentiation. Thus, abnormal growth might separate two “destructive cartilage” areas sufficiently to allow secondary initiation in the intervening region, and it may well be that the extra interpolated (“ectopic”) digits observed in the taZpid3 mutant (Ede & Kelly, 1964) and also in reaggregate limbs (Pautou, 1973) arise in this way. A further capacity for regulation in our model is shown by the size polarity developed along the anterior-posterior axis, the final size and shape of the cartilage element being a function of the variable fine structure of the gradient. A comparison can be drawn here between the basic regulatory behaviour to be expected from a periodic and a monotonic gradient model. With a periodic gradient, allowing for the size variations mentioned above and assuming that changes in synthesis rate do not occur, regulation will occur as the addition or deletion of whole pattern elements with little or no change in the remaining pattern. On the other hand, regulation of a monotonic gradient along the lines discussed by Wolpert (1969) would change the relative size and spacing of the existing elements but neither add nor subtract any. Thus, excepting possibly the cases where a second initiator or organizer region is produced, giving a “mirror image” type of duplication (as found by Gasseling & Saunders, 1964, for example), the monotonic, regulatory gradient model would not be expected to produce polydactylous limbs of any sort.
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The predictions and assumptions of our model can be empirically tested by measuring the physical parameters of dateloping normal, mutant and grafted limbs. The criteria by which this model then stands or falls are, does cartilage determination in the limb (i) occur centrifugally in blocks of approximately constant size; (ii) occur sequentially across the anteriorposterior axis of the limb with a constant “wave length”. It would seem reasonable to apply this model to all situations in which periodic differentiation occurs, perhaps the most obvious being the regionalization of somitic mesoderm and subsequent formation of somites (Hamilton, 1969; Lanot, 1971; Kieny, Mauger & Sengel, 1972). The linkage of the destruction phase to differentiation will obviously vary with cell type. While not eliminating the desirability of constructing alternative models it is noteworthy that by applying it in limb shapes which have been studied in normal and mutant development, a fair approximation to the cartilage patterns arising in real embryos is produced. The model is susceptible of a number of interpretations at the biochemical level; perhaps the simplest is to postulate an allosteric enzyme system, the enzyme being inactive at concentrations of M below Tl, transforming irreversibly to an active form by binding M when the concentration exceeds Tl and now catalyzing the production of M from a substrate S, finally undergoing a second irreversible transformation by binding more M when the concentration exceeds T2 and catalyzing the reverse reaction M to S. At the present time it is impossible to test the model at this level; in essence it is a set of rules whereby the individual cells interpret a signal from their environment (the concentration of a diffusible morphogen) with respect to their internal state, thereby altering their state and their output, or, in other words, their differentiation and their production/destruction of the morphogen. We gratefully acknowledge support from the Agricultural Research Council and the Science Research Council. REJ?ERENCES of MuthematicuZ Biology, vol. 2 (R. Rosen, ed.). New York and London: Academic Press. BARD, J. & LAUDER, I. (1974). J. theor. Biol. 45, 501. EDE, D. A. (1971). Symp. Sot. exp. Bill. 25,235. EDE, D. A. & AGE-AK, G. S. J. Embryol. exp. Morph. 20, 81. EDE, D. A. & KELLY, W. A. (1974). L Ekbryol. exp. Morph. 12,339. EDE, D. A. & LAW, J. T. (1969). Nature, Land. 221,244. GASSLING, M. T. & SAUNDERS,J. W. JR (1964). Am. Zool. 4,303. &DDWlN, B. & COHEN, M. H. (1%9). J. fheor. Biof. 25,379. GRONEBERO,H. (1963). The Patho&y of Development. Oxford: Blackwell. I-brmmm, V. & HAMILTON, H. L. (1951). J. Morph. 88.49. ARBIE,
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(1972).
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HAMILTON, L. (1969). J. Embryol. exp. Morph. 22,253. KIENY, M., MAUGER, A. & SENGEL, P. (1972). Devl Bfol. 28,142. LANOT, R. (1971). J. Embryol. exp. Morph. 26, l-20. bEWEmrEIN, W. R. & PENN, R. D. (1967). J. Ceil. Bill. 33,235. MACCABE, J. A. & SAUNDERS, J. W. (1971). Amt. Rec. 169,372. MCCORMICK, J. M. & SALVADORI, M. (1964). Numerical Metho& in Fortran. New Jersey: Prentice-Hall. MINI-Z, B. (1970). Symp. int. Sot. Cell B&l. 9, 16. MINTZ, B. (1971). Symp. Sot. exp. B&l. 25,345. PATOU, M. P. (1973). J. Embryol. exp. Momh. 29,175. ROMW, A. B. (1933). In Vertebrate Palaeontology. Chicago: University of Chicago Press. SJIARLT, R. L., HILFEX, S. R. Bi MIROW, S. M. (1972). Devl B&l. 28,123. SUMM~RBELL, D., LEWIS, J. H. & WOLPERT, L. (1973). Nature, Lmd. 244,492. TURING, A. M. (1952). Phil. Trans. R. Sot. B 237, 32. WOLPWT, L. (1969). J. theor. Bzbl. 25, 1. ZWILLING, EL (1961). A& Morphogen. 1,301.