simpler rules for epimorphic regeneration: the polar-coordinate model without polar coordinates

J. theor. Biol. (1981) 88,371-392

Simpler Rules for Epimorphic Regeneration: The Polar-Coordinate Model Without Polar

Coordinates

JULIAN LEWIS

Department of Anatomy, University of London King’s College, Strand, London WC2R 2LS, England

(Received 21 March 1980, and in revised form 1 August 1980)

The polar-coordinate model of French, Bryant & Bryant (1976) describes epimorphic regeneration in insects and amphibians, and correctly predicts some surprising phenomena. Their model rests on two main rules. It is shown here that the tist of these, the Rule of Intercalation, expresses a requirement that the pattern of positional values of the cells shall be continuous, and that the other, the Rule of the Complete Circle, is not an independent hypothesis, but simply a consequence of the continuity requirement. The formulation of the rules in terms of continuity is coordinate-free and applicable in any number of dimensions: polar coordinates do not have the fundamental significance attached to them by French, Bryant&Bryant (1976), nor are there any grounds to think that the system of positional values in an amphibian limb is two-dimensional. Some simple topology helps to clarify the concepts and prove these points. The phenomena codified by French et al. are to be expected in any epimorphically regenerative organ whose pattern of growth and pattern of positional values are defined by a maintained system of diffusible chemical signals.

1. Introduction

Regeneration occurs in many different species of animal. Amputated parts are replaced, intermediate deficiencies are filled in, and duplicate or supernumerary structures are generated in response to certain types of derangement. The consequences are sometimes bizarre and puzzling. For example, if a right leg of a juvenile cockroach is cut off and grafted onto the stump of a left leg, two supernumerary legs will sprout from the site of the graft-host junction; and this phenomenon is all the more remarkable because it is seen also in amphibians. Do such facts imply a profound similarity in the mechanisms controlling regenerative growth and pattern formation in these very different creatures? And if so, what are the underlying general principles of cell behaviour?

French, Bryant & Bryant (1976, 1977) have posed these problems in a most original and illuminating way. They have in effect codified the laws

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governing epimorphic regeneration and related phenomena, and have embodied them in their “polar-co-ordinate model”, presenting a unified account of cockroach limbs, amphibian limbs, and Drosophila imaginal discs. This model is characterised by two main rules-the Rule of Inter- calation and the Rule of the Complete Circle; and these rules are framed in terms of a polar co-ordinate system for the description of the limb or imaginal disc. French, Bryant and Bryant argue that Nature herself uses a polar co-ordinate system to specify the characters of cells. Given the rules, the problem is to discover the mechanisms of cell behaviour that underlie them. The Complete Circle Rule in particular seems baffling: it is phrased globally, in terms of the pattern as a whole, and appears difficult to derive from local principles, governing the individual cells. It has provoked several experiments (e.g. Slack & Savage, 1978; Maden & Turner, 1978; Holder, Tank & Bryant, 1980) and much debate (reviewed by Bryant, French & Bryant, 1980). The experiments have shown that in its original formulation it is partly wrong; but only partly. Bryant, French & Bryant (1980) have now restated it in an amended form, and have suggested a new rule of local cell behaviour, called the Distalization Rule, which entails the amended Complete Circle Rule as a consequence.

I want to present a rather different formulation of the rules embodied in the polar-coordinate model. Those rules can be stated without reference to any particular co-ordinate system, and once they are expressed in this co-ordinate-free manner it becomes plain, not only that the observations give no ground for supposing that Nature’s own co-ordinate system is polar, but also that the Rule of the Commplete Circle (in so far as it is true) is merely a consequence of the Rule of Intercalation. Both rules simply reflect an underlying requirement of continuity. The singularities invoked by Glass (1977) are no more than a mathematical artifact resulting from the use of polar co-ordinates; they do not represent any sort of essential discontinuity in the pattern.

The views expounded here are similar to those expressed, in somewhat different language, by Winfree (1980) in his book, with its many other topological insights into biological pattern, and by Mittenthal (1980), who presents elegant corroborative experiments on crayfish.

2. The Principles of Intercalation Are Seen Most Simply in One Dimension

The obscurities of the polar-coordinate model derive from the mathematical difficulty of discussing patterns in more than one dimension. The polar coordinate model applies the two-dimensional systems; but its authors start by considering the two dimensions one at a time, and then cobble them

EPIMORPHIC REGENERATION 373

together by an additional postulate (the Complete Circle Rule or the Distalization Rule). I shall try to show instead, with the help of some simple ideas from topology, how the rules devised for the one-dimensional case can be framed so as to apply in any number of dimensions. The passage from one to two or more dimensions is thus made easily, without the need for any extra assumptions.

Let us follow French et al. (1976) and take the cockroach leg as our example. It is clothed in a cuticle secreted by the epidermal cells. The patterns of bristles, pigmentation and other markers in the cuticle reveal the characters of the epidermal cells beneath. Thus one can study regeneration and pattern formation in the two-dimensional sheet of cells.

The sheet is stretched out into a shape which can be considered roughly similar to the surface of a cone, with its base corresponding to the base of the leg, and its tip corresponding to the tip of the leg. French, Bryant and Bryant define positions in the leg epidermis by means of an angular co-ordinate, giving the position around the circumference, and a “radial” co-ordinate, indicating the distance from the apex of the cone and so in effect the position along the proximo-distal axis of the limb. To start with, let us forget the circumferential pattern, and focus on the proximo-distal axis. Suppose that a transverse slice is cut out of a segment of the limb, and the distal fragment is grafted back onto the proximal stump (Fig. 1). The animal in due course will

FIG. 1. A transverse slice is cut out of a limb, and the distal fragment is grafted back onto the stump. Intercalary regeneration restores the intermediate structures (shaded). The limb is idealized as a cone, seen from the side.

then regenerate the missing middle part; that is, at the site where the grafted distal cells confront the proximal stump cells, growth occurs, and the newly-grown cells form the pattern of parts which should normally lie between the two levels which have been artificially brought together. [I gloss over the complications arising from homology, when cells from different

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FIG. 2. Two limbs are-amputated, one proximally, the other more distally, and the large distal fragment from the first is grafted onto the large remaining stump of the second. Intercalary regeneration gives rise to a supernumerary set of intermediate structures (shaded), with reversed polarity.

segments are juxtaposed (Bohn, 1976).] A more perverse manifestation of the same behaviour is seen after a slightly different experiment (Fig. 2). One leg is amputated through the proximal part of a segment, while another is amputated through the distal part of that segment, and the long detached part of the first leg is grafted onto the long remaining stump of the second, to make a composite leg in which the middle parts of the pattern are repeated twice over. This monster limb, far from adjusting back to normal, goes on to become even more repetitious. At the graft-host junction, where cells of proximal character confront cells of distal character, growth is again stimu- lated, and again the newly-grown cells form the pattern of parts which should normally lie between the two levels which have been put together. Thus a third set of middle parts is formed, with its proximo-distal polarity reversed, as shown in the figure.

In interpreting these phenomena, French Bryant & Bryant (1976,1977) start from the supposition that cells in different positions in the normal structure have intrinsically different characters, that is, they are non- equivalent (Lewis & Wolpert, 1976). Each cell is presumed to have a


positional value which distinguishes its state from that of cells elsewhere. The Rule of Intercalation then asserts (a) that cell proliferation occurs whenever cells with disparate positional values are juxtaposed, creating a discontinuity in the map of positional value as a function of position; and (b) that the cells of the new tissue formed by this proliferation take on positional values intermediate between those of the cells on either side of the discontinuity. French, Bryant and Bryant then go on to show how the same composite rule applies to regeneration around the circumference of the leg when longitudinal strips are cut out of the epidermis or replaced by strips from other circumferential positions. Only after thus considering each axis singly do French, Bryant and Bryant proceed to the phenomena which are irreducibly two-dimensional.

(A) THE TWO DIMENSIONS OF THE PATTERN ARE INTERDEPENDENT

Suppose, for example, that the distal part of the limb is simply amputated and thrown away. The epidermis around the circumference exposed at the site of amputation grows and heals to form a covering over the cut face of the stump. In the process, cells with different circumferential positional values necessarily come into contact. The Rule of Intercalation predicts growth and the generation of new cells with positional values that are in some sense intermediate between those brought into confrontation. Experiment reveals that the limb regenerates the distal part that has been cut off; in other words, this conformation of different circumferential positional values leads to the establishment of new proximo-distal positional values. The two dimensions of positional value are not separable: the form of the pattern in the circumferential dimension is tied up with its development in the proximo- distal dimension. The Complete Circle Rule couples the two dimensions. It states that “whenever a complete circumference is exposed or generated (by amputation, grafting, wound healing or intercalation) at a given proximo- distal ( = radial) level, then growth occurs, and during this growth, all of the more distal presumptive parts are generated” (Bryant, French & Bryant, 1980). In the original statement of the rule, the converse was also asserted: it was said that where there was not a complete circumference, the distal parts were not generated. It is this latter corollary to the rule that has been refuted (Slack & Savage, 1978; Bryant, French & Bryant, 1980). The rule as given in the above quotation, however, still appears to be true. It correctly predicts the growth of supernumeraries when, for example, the tip of a right leg is grafted onto the stump of a left leg. Bryant, French & Bryant in their latest account (1980) present the Complete Circle Rule as a consequence of a “Distalization Rule”, which supplements the Rule of Intercalation and

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states that: “If a cell which is produced during circumferential intercalation is assigned a circumferential positional value identical. to that of an adjacent cell, . . . the new cell must adopt a radial positional value which is more distal.” The purpose of the present paper is not to show that the Complete Circle Rule is wrong, but to show that it represents a result of the principles of intercalation, without any such supplementary hypotheses.

(B) THE RULE OF INTERCALATION CAN BE APPLIED DIRECTLY IN

TWO DIMENSIONS

Instead of first considering regeneration and intercalation in each of the two dimensions separately, let us try to apply the Rule of Intercalation directly to a two-dimensional disturbance. Suppose a patch-let us say an oval patch-is cut out of the leg epidermis. The cut edges of the epidermis will heal together to cover the wound, the cells in the neighbourhood of the join will proliferate, and the new epidermis thus formed will take on the pattern of positional values appropriate to the interior of the region excised; thus the parts that have been removed will be regenerated (French, 1978). Again, it seems that proliferation occurs when cells with disparate positional values come together, according to clause (a) of the Rule of Intercalation above. The cells of the new tissue formed by this proliferation take on positional values determined by the positional values of the cells around the perimeter of the cut; specifically, they take on the positional values which should normally be enclosed within that perimeter of positional values. In short, the Rule of Intercalation can be said to apply again, if it is slightly reworded. In the original formulation, the Rule stated that the new cells took on values intermediate between those of the cells on either side. This language is essentially one-dimensional. To cope with the two-dimensional example, we say instead that the new cells take on values enclosed by (or “interior to”) those of the cells around the perimeter. The meaning of this statement will be explained in more detail below.

(C) REGENERATION OF AN AMPUTATED LIMB IS AN INSTANCE OF

INTERCALATION

We shall argue that both formulations of the Rule of Intercalation represent a more basic and general principle, a principle of continuity. But it is already easy to see how the Rule of Intercalation in its two-dimensional formulation, without additional postulates, accounts for the regeneration of the distal part of a leg after it has been amputated. For the epithelium forms a continuous sheet covering the tip of the leg as well as its circumference.


The tip of the leg is just a point in the sheet; and to amputate the distal part of the leg is, from the topological point of view, simply to cut out a patch of epithelium surrounding that point. The regeneration of the amputated part is thus an instance of the regeneration of an excised patch. Note that the reformulated Rule and its predictions do not rest on any particular choice of co-ordinate system. The argument can be taken a step further to explain also, without further assumptions, the generation of supernumerary outgrowths, as we shall show. But first let us see what the Rule of Intercalation has to do with continuity, and set the whole discussion on a more rigorous footing.

(D) CONTINUITY IMPLIES INTERCALATION

The basic idea is obvious in one dimension. The Rule of Intercalation states that whenever a gap is created in the pattern of positional values along a line, cells are intercalated with positional values intermediate between those on either side of the gap. It follows that a pattern without gaps will be re-established; that is, a continuous pattern. Conversely, if the final pattern is continuous, it must necessarily include the full range of positional values intermediate between those which lay on either side of the initial gap. Thus in one dimension, continuity implies intercalation and intercalation implies continuity: the two requirements are equivalent. In two dimensions, the notion of one point being intermediate between two others becomes hazy (Is Paris intermediate between Moscow and Rome?); but we can define intercalation instead in terms of points enclosed by a perimeter. Then again we can say that continuity implies intercalation, though now perhaps this proposition needs more careful explanation and proof.

(E) A PATTERN IS DEFINED BY A MAP FROM THE SET OF POSITIONS TO THE

SET OF POSITIONAL VALUES

Continuity is a property of maps or functions from one space to another. We can apply the concept to the pattern of positional values because that pattern is defined by such a map: we describe the pattern by specifying, for each position in the tissue, the positional value of the cell at that position. For a one-dimensional system it is easy enough thus to represent the pattern by an ordinary graph, giving the positional value (plotted along the y-axis) as a function of position (plotted along the x-axis). For a two-dimensional system, however, it is not so easy. The graph cannot be plotted in the usual way on a sheet of paper, because it is four-dimensional: two dimensions are needed to represent position, and another two to represent positional value as a function of position.

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Another sort of picture can, however, be used instead. Suppose we have a sheet of cells, or epithelium, over which the positional value varies so that in the normal mature system no two points have the same positional value. The set of positional values that are present can then be regarded as a two- dimensional space with the same topology as the epithelium itself: two positional values are taken to be neighbouring points in positional value space if they occur in neighbouring positions in the normal mature epithelium. Figure 3 shows the normal pattern, that is, the normal map of positional value as a function of position. Let us call this map fO. In Fig. 3, the left-hand diagram depicts the limb in ordinary space, with a few selected points labelled by letters a, b, c, etc. The right-hand diagram shows the space of positional values and displays the map fO by marking the values a’ = fo(a), 6’ =fo(b), c’ =f,,(c), etc., corresponding to the positions a, b, c, etc. In a disturbed limb, with a pattern represented by some other map f, the space of possible values would be the same, but the values a’ = f(a), etc., would be differently situated within it.

Dorsal

a

(I->

b .- d o b b

s a”

c

Ventral

P

FIG. 3. The normal pattern of a limb, represented by a map fa from the set of positions to the set of positional values. The left-hand diagram shows the space X of positions in the idealized limb, viewed end-on: the letter o marks the distal tip; the large circle marks a circumference close to the base. The right-hand diagram shows the space P of positional values, indicating the normal values u’ = f,,(a), 6’ = fo(b), etc., corresponding to the positions labelled a, b, etc., in the left-hand picture.

(F) CONTINUOUS PATTERNS IN AN EPITHELIUM SATISFY THE COMPLETE

PERIMETER THEOREM

The formal definition of continuity can be found in any topology textbook (e.g. Patterson, 1959). For present purposes, the idea is well enough expressed by saying that a map f is continuous if it maps points that are neighbours in one space into points that are neighbours in the other; that is, in the present context, a pattern is continuous if neighbouring cells have neighbouring positional values. Continuity imposes powerful restrictions.


Suppose, for example, that we have an epithelial pattern described by a map f, and know that f is continuous. Let S be a simple closed curve surrounding a region R in the epithelium, and let S’ = f(S) be the corresponding curve traced out in positional value space as we travel around S. S’ need not itself be a simple closed curve; but if S’ is a simple closed curve, surrounding a region R’ in positional value space, it follows from the continuity of the function f that all the positional values within the region R’ must be represented among the cells within the region R. In other words, if a region R of epithelium is surrounded by cells with the complete perimeter of positional values that would normally surround a certain set of structures C, and the pattern is continuous, then the set of structures 1 must occur within the region R : the values around the boundary determine that certain values must be included in the interior. Let us call this the Complete Perimeter Theorem. It will be used to derive from the Rule of Intercalation the results which French, Bryant and Bryant derive from the Complete Circle Rule. A formal statement and proof of the theorem are given in Appendix A, together with some other useful theorems provable by the same methods and leading to successful predictions.

(G) FOUR BASIC POSTULATES DESCRIBE INTERCALATION; THE COMPLETE

CIRCLE RULE IS SUPERFLOUS

The Rule of Intercalation is a composite of several clauses, To make the argument precise, and the postulates explicit, I spell them out as follows:

(1) Non-equivalence. Cells in different positions in the normal mature structure have different positional values.

(2) Epimorphic adjustment. Positional values change only in regions of cell proliferation.

(3) Intercdary growth. Cell proliferation occurs where cells with disparate positional values lie improperly close together.

(4) Continuity. Cells adopt positional values such that the pattern will be continuous.

Postulates (1) and (2) do not differ essentially from those of French, Bryant and Bryant; postulate (2) is the defining characteristic of epimorphic, as opposed to morphallactic development. Postulate (3) represents a slight modification of the French, Bryant and Bryant statement. It does not require that there should be an actual discontinuity in the pattern of positional values for growth to occur, but only that the positional values should be squeezed up closer than they are in the normal mature structure. It thus describes not only the intercalary growth seen in regeneration, but also the growth of small-scale immature structures in the course of normal

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development. Postulate (4) replaces the idea of intermediate values with the idea of continuity. In one dimension, as we have seen, this new formulation is equivalent to the old. But the continuity condition is the more simple and the more general, and can be applied as easily in two or three dimensions as in one; the change in the statement of the rule seems slight, and yet is crucial.

It follows from postulates (2) and (3) that, when the normal organ is disturbed, only a narrow rim of cells surrounding the disturbance will be recruited to give rise to a regenerate or other new structure. Cells initially at more than this short distance from the disturbance will keep their original characters. The reasoning is given in Appendix B. Consider now the case in which a patch is cut out of the epithelium. Before the operation the patch was surrounded by cells whose positional values formed a complete perimeter of the region of positional-value space that was represented in the patch. After the operation the disturbed region of epithelium will still be surrounded by this complete perimenter. At the end of the regenerative process, by postulate (4), the pattern must be continuous. Therefore, by the Complete Perimeter Theorem, the patch of tissue regenerated must include cells with all the positional values that would normally lie within the perimeter of the patch excised: all the structures that have been removed will be replaced. By postulate (3), the regenerated structures will continue to grow until they have attained their normal mature spacing and size. Thus regeneration of the excised patch is complete. As we have seen, the regeneration of the distal part of a limb after amputation is a particular case of the same process.

It is easy now to explain also the production of supernumerary outgrowths. Suppose the left and right legs are amputated at the same level along the proximo-distal axis, and the distal part of the right leg is grafted onto the stump of the left leg. Then the pattern of positional values just after the operation will be as shown in Fig. 4. At two points around the graft-host junction, marked a and c, the positional values match; elsewhere they do not. The points of concordance, a and c, divide the line of junction between graft and host into two parts, L and M. Around the line segment L lie cells with the positional values that would normally surround the stump of a leg from which the distal part had been amputated; and a similar set of positional values surround the line segment M. Thanks to intercalary growth and the condition of continuity it then follows, by just the same argument as before, that a complete set of distal structures should be generated from the cells immediately surrounding the line segment L, and a second similar set from those surrounding the line segment M. Thus a pair of supernumerary outgrowths should be produced, as indeed they are.


- f

FIG. 4. The pattern set up by grafting the tip of a right limb onto the stump of a left limb, represented by a map f, as explained in Fig. 3. The outer circle abed represents the end of the host stump; the inner circle (Y&S represents the base of the grafted tip. At two points, a and c, the positional values a’ and c’ of the stump match the positional values (Y’ and y’ of the graft. The dashed line abcy&, forming a perimeter enclosing the line segment L (dotted), maps into a perimeter u’b’c’y’p’a’ enclosing the set of positional values normally corresponding to the distal part of a limb. By the Complete Perimeter Theorem, intercalary regeneration in the region enclosed by abcy@ must therefore give rise to a supernumerary set of distal structures, displaying the positional values enclosed by the perimeter u’b’c’y’@a. By the same argument, a second supernumerary outgrowth must develop in the region enclosed by a perimeter u&y&~

(H) REGENERATION FROM MIRRO-SYMMETRICAL LIMBS IS NOT

PREDICTABLE FROM THE POSTULATES

What must we predict in those cases, so hotly debated in relation to the polar-co-ordinate model, in which a mirror-symmetrical limb-double- dorsal, say, or double-anterior-is created artificially, amputated some- where along its proximo-distal axis, and left to do what it will? The answer is that the rules give no firm prediction: they do not forbid regeneration, neither do they compel it. This is no fault, but a virtue, for the experimental findings are themselves variable (reviewed by Bryant, French & Bryant, 1980). Figure 5(a) shows the pattern of a double-dorsal limb just after its tip has been amputated. Figures 5(b) and 5(c) show two possible outcomes of the regenerative process, both satisfying our postulates, including the requirement of continuity. In Fig. 5(b), the limb has regenerated less than what was cut off, in Fig. 5(c), more than what was cut off. The mathematical explanation is simply that when the positional values surrounding the defect do not completely enclose any region in positional value space, we no longer have a boundary condition forcing the appearance of certain positional values within the perimeter of the defect.

The condition of continuity requires that where there is a complete perimeter of positional values we must also have the complete set of values interior to that perimeter. But the rule is not totalitarian: positional values which are not compulsory are not necessarily forbidden. Thus when the tip

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FIG. 5. Possible patterns after amputation of a double-dorsal symmetrical limb, represented as explained in Fig. 3. In the left-hand diagram, the horizontal dotted line is the line of symmetry of the limb, and the dashed circle abed marks the level of amputation. (a) The map f a,,,p shows the pattern immediately after amputation; the set of positional values occurring in the limb is shaded. (b), (c) The maps fies and f,,,,, show respectively a continuous pattern in which the structures restored by regeneration amount to less than what was cut off, and a continuous pattern in which they amount to more.

of a limb is cut off, rotated, and grafted back onto its own stump, there is nothing to forbid the formation of supernumerary outgrowths; and in fact they sometimes occur (Bohn, 1972; Maden & Turner, 1978). Likewise, after amputation of a double-dorsal limb, there is nothing in the present rules either to forbid or to compel regeneration. As Holder, Tank & Bryant (1980) and Bryant, French & Bryant (1980) have pointed out, the behaviour in such cases may be expected to depend on the way the cut surfaces heal together.

(I) THE REQUIREMENT OF STABILITY UNDERLIES THE RULE OF THE

SHORTEST PATH

The continuity condition leaves rather too much vague. For example, it does not prohibit the development of an endless profusion of supernumerary structures. Let us therefore state a fifth postulate which must surely be satisfied.


(5) Stability. The normal pattern is stable against small derangements; that is, small derangements lead only to small regenerative adjust- ments.

Between any two points of the normal structure there are many possible paths, some short and direct, others long and circuitous. Correspondingly, there are many possible ways to fill in tissue between those points while preserving continuity of the pattern. The stability condition entails, loosely speaking, that if the points are close neighbours, the path of regeneration between them will be short.

This has an important consequence. Consider an epithelial organ that is topologically equivalent to the surface of a sphere, such as a Drosophila imaginal disc that has been experimentally detached from the stalk which connects it with the body of the larva. Suppose we cut a small round patch out of this epithelium, leaving a sphere with a small hole in it. What course will regeneration take? If we try to apply the Complete Perimeter Theorem as given above, we encounter an ambiguity. The theorem states that the part regenerated must include all the structures normally surrounded by the perimeter of the patch. But on a sphere, that perimeter may be said to surround both the patch itself and the large complementary part of the sphere. Thus the Complete Perimeter Theorem on a sphere (see Appendix A) admits either of two possibilities: it allows a duplication of the sphere with the patch cut out, giving a sort of dumb-bell structure, as an alternative to a simple replacement of the patch. There could be, so to speak, a long path for regeneration, or a short path. The stability postulate entails that the short path is taken: since the derangement has been small, the regenerated part must also be small; in particular, it cannot be a duplicate of the initial sphere with the hole in it. The sphere with the small hole will thus simply replace the small patch of tissue cut out. But what then of the small excised patch of tissue itself? What regenerative behaviour will it show? According to our rules, which are local, the regenerated tissue will arise only from cells in the neighbourhood of the cut, and will be determined by the pattern of positional values of the cells in that neighbourhood; it will not depend on the cells more remote from the cut, and in particular it will not depend on whether those other cells constitute an almost entire sphere, or only a tiny cap. Thus the small excised patch will regenerate the same set of structures as are regenerated by the sphere from which it was cut; in other words, it will duplicate itself, while the sphere from which it was cut will complete itself. This is the behaviour that is observed (Bryant, 1975), and is described by French, Bryant & Bryant (1976) by their “Rule of the Shortest Path”. It must be said, however that the stability requirement does not in principle lead to predictions for cases where the excised patch is not very small, but is

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of intermediate size. The simplest possibility would be a switch-over from one path of regeneration to the other as the boundary of the excised patch passes some half-way mark; and this is what is implied by the Rule of the Shortest Path as usually stated. But the conditions of continuity and stability by themselves do not prohibit more complex patterns of behaviour in the intermediate regime.

The argument from stability can be used in a similar way to explain the Rule of the Shortest Path in its application to circumferential regeneration in the cockroach leg. If regeneration could occur by the long path around the circumference, the system would be unstable against the removal or destruction of infinitesimally thin strips: the limb would manufacture whole new circumferences and whole new supernumerary outgrowths if you so much as tickled it.

(J) THE AMPHIBIAN LIMB OBEYS THE SAME RULES IN THREE DIMENSIONS

French, Bryant & Bryant (1976), seeing almost the same phenomena in amphibian limb regeneration as in the regeneration of the epithelia of cockroach legs or Drosophila imaginal discs, and having a two-dimensional theory to explain the latter, have suggested that in some sense the amphibian limb also uses a two-dimensional system for specifying positional information, even though the structure itself is plainly three-dimensional. We can reassess this suggestion in the light of our present formulation of the rules of epimorphic regeneration. We shall argue that the suggestion is ill-founded: there is no reason to think that the amphibian limb uses only a two- dimensional form of positional information.

The rules of epimorphic regeneration, as set out in our propositions (1) to (5), make no mention of the dimensionality of the system. They are all applicable equally in one, two-, three- or n-dimensions. In deriving their consequences and predicting such phenomena as the development of supernumeraries, we did indeed make use of a theorem-the Complete Perimeter Theorem-which was stated and proved in two dimensions. But this theorem can be generalised in an obvious way to any number of dimensions. We might call the generalised version the Complete Boundary Theorem. In the three-dimensional version, three-dimensional regions bounded by simple closed surfaces (that is, by surfaces topologically equivalent to the surface of a sphere) take the place of two-dimensional regions bounded by simple closed curves. The generalised theorem is stated formally in Appendix A; roughly speaking, it asserts that if a region R of tissue is surrounded by cells with the complete boundary set of positional values that would normally surround a certain set of structures C, and the


pattern is continuous, then the set of structures C must occur within the region R. Suppose that the mesodermal component of the amphibian limb (including muscle, bone, dermis and other connective tissues) is a system which obeys the rules in three dimensions, and that a cavity is experimentally scooped out in its interior; it then follows, by arguments analogous to those used in two dimensions, that the contents of that cavity should be completely regenerated. In most operations, however-for example, in a simple amputation-one does not scoop out a cavity from the interior, but one cuts away a region that includes part of the boundary surface. We cannot then apply the Complete Boundary Theorem unless we make some asumption about the characters adopted by the cells at the new boundary surface created by the operation. Any naked new boundary surface rapidly becomes covered by epidermis. To explain the phenomena, we therefore make the following hypothesis.

(6) Boundary condition. Mesodermal cells that lie in the layer just beneath the epidermis adopt positional calues appropriate to that layer, that is, positional values chosen from the set which occur in the sub-epidermal layer in the normal limb.

This is roughly equivalent to saying that mesodermal cells close to the epidermis are guaranteed to have positional values corresponding to dermis rather than to some deep tissue such as bone or muscle. The set of sub-epidermal positional values is two-dimensional. The sub-epidermal layer of mesodermal cells therefore behaves in precisely the same way as the epithelium of the cockroach leg. Just as the cockroach regenerates the epithelium of a new limb after amputation, so also the amphibian must regenerate at least the sub-epidermal layer of mesoderm of a new limb. But it follows then from the Complete Boundary Theorem in three dimensions that the amphibian must regenerate also the complete set of interior mesodermal structures of a new limb. By similar arguments, the amphibian must sprout supernumerary limbs, complete with all their interior structures, in the same circumstances as the cockroach. By postulating the boundary condition (6), we thus explain the similarities between cockroach and amphibian without denying that the amphibian limb is essentially three-dimensional in its patterning. Indeed, we could in the same way account for the regeneration of the three-dimensional pattern of muscles and other internal tissues of the cockroach leg, as well as the regeneration of the two-dimensional pattern of its epidermis.

It should, however, be said that the set of postulates (1) to (6) for amphibian limb regeneration are somethirg of an oversimplification. Inter- calary regeneration in amphibian limbs does not always occur (Iten & Bryant, 1975), and even regeneration after amputation can be prevented if

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mature epidermis is grafted onto the face of the amputation stump in place of wound epidermis, or if the limb is denervated (Stocum, 1977). Evidently discontinuities in the pattern of mesodermal positional values are not always enough by themselves to trigger regeneration; other conditions, depending, for example, on interactions between epidermis and mesoderm, as in the embryonic chick limb (Rubin & Saunders, 1972; Summerbell, Lewis & Wolpert, 1972), must also be satisfied.

(K) THE CONTINUITY POSTULATE MAY HAVE A SIMPLE CHEMICAL BASIS

At the heart of my account lies the postulate of continuity. The special advantage in formulating the rules in terms of continuity is that it admits of a particularly simple molecular explanation. Continuity is assured if positional values are defined by the concentrations of chemicals which diffuse from cell to cell: the concentrations in neighbouring cells must then be closely similar, and so the neighbouring cells must have neighbouring positional values. It does not matter how many such diffusible controlling substances there may be. The argument requires only that in the final stable structure produced at the end of the regenerative process, there should not be cells with freak combinations of concentrations, not corresponding to any positional value of the normal set. There may be other possible interpretations of the continuity condition, but this seems to me the most plausible.

The postulate of intercalary growth can also be seen as the mark of a very simple mechanism for the control of growth in normal development, as follows. Cell division is governed by influences from surrounding cells with different positional values. Cell division causes the distance from those cells with different positional values to grow, and the magnitudes of the influences to change. Eventually the influences cease to be of such a magnitude as to provoke cell division: the structure has then locally reached its mature size.

Viewed in this light, it is not so surprising that the behaviour described by the polar-coordinate model should recur in many very different species, and that amphibians and insects alike should show such oddities as the production of supernumerary limbs at sites of graft-host junction. These regenerative phenomena may be expected in almost any organ whose pattern and size are controlled epimorphically throughout life by a system of diffusible signals that pass from cell to cell. Conversely, when, as in the chick limb, we find that these regenerative phenomena do not occur, we may take it as evidence that the pattern and its growth are not determined by a maintained system of diffusible signals; the implication then is that the cells in such organs are autonomous, and behave according to their individual


memories of signals that were transient, and operated only in early development (Lewis, Slack & Wolpert, 1977; Lewis & Wolpert, 1976).

3. Conclusion

This paper presents no new observations. It accepts the rules of epimorphic regeneration embodied in the polar-coordinate model, and does not deny them. Its purpose is rather to reveal their basis. It revises the account given by French, Bryant and Bryant in three important respects: (1) it denies that polar co-ordinates have any fundamental significance, and declares them an arbitrary choice, a system used by man and not necessarily by Nature; (2) it shows that the most mysterious of the rules, the Rule of the Complete Circle, is merely a consequence of the banal requirement of continuity; (3) it shows that there is no reason to suppose that the system of positional information in the amphibian limb is two-dimensional. It brings us perhaps a little closer to understanding the cellular mechanism of regeneration, and the meaning of the similarities and differences between species. It shows, lastly, how topology can help us to derive from simple postulates about the local properties of a pattern, some strong statements about its global organisation.

I am very grateful to Margarida Barros, Gavin Swanson, Susan Bryant, Nigel Holder, Art Winfree and Jay Mittenthal for discussion and comments.

REFERENCES

BOHN, H. (1972). J. Embryol. exp morph. 28,185. BOHN, H. (1976). In InsectDevelopment (P. A. Lawrence, ed.) pp. 170-185. Symp. Roy. Ent.

Soc.8. BRYANT, P. J. (1975). Ciba Found. Symp. 29,71. BRYANT, P. J., BRYANT, S. V. & FRENCH, V. (1977). Sci. Am. 237,66. BRYANT, S. V., FRENCH, V. & BRYANT, P. J. (1980). (Submitted for publication.) FRENCH, V. (1978). J. Embryol. exp Morph. 47,53. FRENCH, V., BRYANT, P. J. & BRYANT, S. V. (1976). Science 193,969. GLASS, L. (1977). Science 198,321. HOLDER, N., TANK, P. W. & BRYANT, S. V. (1980). Dev. Biol. 74,302. ITEN, L. E. & BRYANT, S. V. (1975). Dev. L”iol. 44, 119. LEWIS, J. H. & WOLPERT, L. (1976). J. theor. Biol. 62,479. LEWIS, J. H., SLACK, J. M. W. & WOLPERT, L. (1977). J. theor. Biol. 65, 579. MADEN, M. & TURNER, R. N. (1978). Nature 273,232. MM-TJZNTHAL, J. (1980). Paper in preparation. PA~RSON, E. M. (1959). Topology, 2nd edn. Edinburgh: Oliver and Boyd. RUBIN, L. & SAUNDERS, J. W. (1972). Dev. Biol. 28,94. SLACK, J. M. W. & SAVAGE, S. (1978). Nature 271,760.

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STOCUM, D. L. (1977). In Vertebrate Limb and Somite Morphogenesis (D. A. Ede, J. R. Hinchliffe & M. Balls, eds), pp. 347-371. Cambridge: Cambridge University Press.

SUMMERBELL,D.,LEWIS,J. H. & WOLPERT,L. (1973).Nature 244,492. WINFREE, A. T. (1980). The Geometry of Biological Time. New York: Springer.

APPENDIX A

Some Topological Theorems

We consider first the two-dimensional case, in which we have a pattern of positional values in an epithelial sheet. Let X be the two-dimensional manifold of points in the sheet, and P the set of positional values. For any point x EX, let f&z) be the corresponding positional value in the normal pattern. We asume that in the normal pattern (i) different points in the sheet are associated with different positional values, and (ii) all the possible positional values are represented. Then we can assign to P a topology like that of X, such that the map fO:X + P defining the normal pattern is a homeomorphism. (For an explanation of the topological terms used here, see any introductory textbook, e.g. Patterson, 1959.) This is equivalent to supposing that a pair of positional values are close neighbours in P if and only if the normal corresponding pair of positions are close neighbours in X. We assume (iii) that X and consequently P are both homeomorphic to ordinary two-dimensional Euclidean space lR*; that is, that the epithelium has a straightforward topology without free edges or toroidal “handles” for example. An arbitrary pattern can be described by a map f :X + P, such that the positional value at the point x E X is f(n). A simple closed curve S in X is defined as a curve homeomorphic to the perimeter of a circle S1 in W2; that is, S is a simple closed curve if and only if there exists a homeomorphism g : S1 + S. The set of positional values S’ = f(S) into which f maps S may or may not be itself a simple closed curve. The theorem we want to prove can be stated as follows.

Complete perimeter theorem

Let X and P be manifolds homeomorphic to R2. Let f :X + P be continuous. Let S be a simple closed curve in X described by a homeomorphism g : Sr + S. Let R be the region of X that is enclosed by S. Let S’ = f(S) be the image of S in P. Let g’ : S1 + S’ be the map defined by the rule g’(a) =fk(a>l f or all a E S1. Then if g’ is itself a homeomorphism, so that S’ is a simple closed curve homeomorphic to S and enclosing a region R’ in the space P, it follows that R’ sf(R); or in other words, that for every point p in the region of P enclosed by S’, there is at least one point x in the region of X enclosed by S, such that f(x) = p.


Proof .

The gist of the proof is as follows. The perimeter S of the region R can be contracted within R, via a continuous family of progressively smaller perimeters, down to a single point r E R. The corresponding family of curves in P, if f is continuous, must then likewise constitute a continuous contrac- tion of S’ down to a single point f(r). If S’ is a simple closed curve homeomorphic to S, this latter family of curves must sweep through all the points enclosed by S’ (and possibly through some other regions as well). Therefore every point enclosed by S’ must be the image under f of some point on one of the family of perimeters in R ; hence every point enclosed by S’ must be represented via f in the region of X that is enclosed by S.

To state the argument more rigorously, we have to make precise the notion of a continuous family of maps. Let go: S1 +A0 and gl : S1 + Al, where A0 and Al are both subsets of X, be two continuous maps. Then the family of maps {go : S1 + A, c XI0 I u 5 l} is a continuous deformation of go into gl if and only if the map H : Sr x [0, l] + X defined by H(a, v) = g,(a), for a E St, u E [0, 11, is continuous. H is then called a homotopy between go and gl. We make use of two lemmas, ,Cor which we give no proof since both seem intuitively obvious:

Lemma 1. Let X be a manifold homeomorphic to R*. Let go : S1 + S c X be a homeomorphism and let gl : Si + {r} c X map the whole of S1 into a single point r in the region R enclosed by S. Then there exists a homotopy H : S1 x [0, l] + X from go to gl which lies entirely within the region R enclosed by S; that is, H(& x [0, 11) c R.

Lemma 2. Let P be a manifold homeomorphic to I@. Let gb : S1 + S’ c P be a homeomorphism and let g; : S1+{r’}c P map the whole of S1 into a single point r’ anywhere in P. Then any homotopy H’ : S1 x [0, l] + P that may exist between gb and g; must sweep through the whole of the region R’ enclosed by S’; that is, R’ c H’(S1 x [0, 11).

We apply Lemma 1 directly to the case in which go is simply the map g : S1 + S referred to in the statement of the theorem, and gl : S1 + {r} is defined by choosing an arbitrary point r in the region R enclosed by S. Then there is a homotopy H’ : Si X [0, l] +X between g and gl such that H(S1 x [0, 11) E R. We define the map H’ : S1 x [0, l]+ P by the rule H’(a, v) = f[H(a, u)] for a E S1, v E [O, 11. Since f and H are continuous, so also is H’. Furthermore, H’(u, 0) = f [H(a, O)] = f [g(u)] = g’(u), and H’(u, 1) = f[H(u, l)] =f[gl(u>] =f(r>. Therefore H’ is a homotopy between g’ : S1 + S’C P and gi : S,+Cf(r)}c P. It follows from Lemma 2 that if g’ is a homeomorphism, so that S’ encloses a region R’, then R’s H’(S1 x [0, 11). But H’(S1 x [0, 11) =f(H(Sl x [0, 1-j)) cf(R). Therefore R’sf(R). This completes the proof of the theorem.

390 J. LEWIS

In an arbitrary number of dimensions, we have the following exactly analogous generalised theorem, in which simple closed boundaries homeomorphic to the surface S,-l of the IZ -dimensional unit sphere play the part of simple closed curves homeomorphic to &.

Complete boundary theorem

Let X and P be manifolds homeomorphic to R”. Let f :X + P be continuous. Let S be a simple closed boundary in X described by a homeomorphism g : SnP1 + S. Let R be the region of X that is enclosed by S. Let S’ = f (S) be the image of S in P. Let g’ : S-1 + S’ be the map defined by the rule g’(a) = f [g(a)] for all a E &-I. Then if g’ is itself a homeomorphism, so that S’ is a simple closed boundary homeomorphic to S and enclosing a region R’ in the space P, it follows that R’ c f (R); or in other words, that for every point p in the region of P enclosed by S’, there is at least one point x in the region of X enclosed by S, such that f(x) = p.

The proof of this theorem for arbitrary IZ is exactly analogous to that already given for the special case where II = 2.

Theorems for Annular Regions of a Two-dimensional Sheet

Related theorems can be stated for other topological conditions. For example, if we have, in a two-dimensional sheet X, an annular region R, bounded by two simple closed curves nested one inside the other, and f is continuous and maps these boundary curves homeomorphically into a similarly nested pair of curves demarcating an annular region R h in P, then again R b G f (R,); that is, all the points of the annular region R h in P must be represented via f in the annular region R, in X(see Fig. Al). If, on the other hand, the curves bounding the annulus R, map under f homeomorphically

I I a a’

d b - d’ b’ fnes,ed

I ! x P

FIG Al. A possible pattern for an annular region of the limb. In the pattern represented by the map fnsti, the boundaries abed and a/3y6 of the annular region correspond to nested boundaries a’b’c’d’ and (~‘p’y’B’ in positional-value space.


into a pair of simple closed curves which do not lie nested one inside the other, but enclose disjoint regions Rb and R:, then we have instead that Rb URI. cf(R,) (see Fig. A2).

X P

FIG. A2. Another possible pattern for an annular region of the limb. In the pattern represented by the map fdisjoint, the nested boundaries abed and a$@ correspond to separate, non-nested boundaries a’b’c’d’ and cu’p’y’S’ in positional-value space. In each case, the shaded annular region of the limb shown in the left-hand diagram must include, at the least, tissues with all the positional values shaded in the companion diagram on the right. Fig. Al is applicable to the case in which a transverse slice has been cut out of the limb; Fig. A2 is applicable to the case in which the tip of one limb has been grafted onto a hole in the side of another.

Proofs by homotopy can again be easily constructed along the lines of the proof given above for the Complete Perimeter Theorem; the details are left to the reader. It is the first of the above pair of theorems for annular regions that guarantees intercalary regeneration when a transverse slice is cut out of a cockroach leg and the distal fragment is grafted back onto the proximal stump. The second theorem of the pair entails that, for example, if a patch is cut out of the side of one limb, and the distal portion of another limb is grafted onto the site thus exposed, an intercalary regenerate will be formed which must include an additional supernumerary set of distal structures. Mittenthal’s (1980) experiments on the crayfish leg provide a beautiful instance of this phenomenon.

Theorems for an Epithelium Homeomorphic to the Surface of a Sphere

If the manifolds X and P are homeomorphic not to the plane R2 but to the surface SZ of a sphere, the conditions of the Complete Perimeter Theorem are no longer satisfied. A simple closed curve on a sphere divides it into two finite regions, and either of these regions may be said to be enclosed by the curve. An analogue of the Complete Perimeter Theorem can, however, be stated as follows.

Let X and P be manifolds homeomorphic to a sphere SZ. Let f :X + P be continuous. Let S be a simple closed curve in X described by a homeomorphism g : S1 + S. Let RI and R2 be the two regions into which S divides X

392 J. LEWIS

Let S’ = f(S) be the image of S in P. Let g’ : S1 + S’ be the map defined by the rule g’(a) =f[g(a)] for all a E S1. Then if g’ is itself a homeomorphism, so that S’ is a simple closed curve homeomorphic to S and dividing the space P into two regions R; and R;, it follows that either R; sf(R1) or Ri Ed, and that either R; cf(R2) or Rk cf(R2).

This again can be easily proved by the same methods as before.

APPENDIX B

The Intercalated Tissue Derives only from a Narrow Rim of Cells Close to the Initial Fault in the Pattern

The argument is essentially the same in any number of dimensions. Consider therefore a line of cells; and suppose that a discontinuity in the pattern of positional values is set up at the origin at time zero. The discontinuity will provoke cells in its neighbourhood to divide [by postulate (3)] and thereupon to take on new positional values [according to postulates (2) and (4)]. Thus a modification of the initial pattern of positional values will spread out from the origin; cells which have been affected will in their turn affect their neighbours, and so on. In other words, a “front” of cell division and adjustment will propagate from the site of the discontinuity at a speed related to the rate of cell division. Let us suppose that this front propagates through the originally quiescent tissue at a speed of not more than c cell diameters per cell division cycle. Then in the course of IZ cell cycles it will have travelled no further than nc cell diameters. Meanwhile the cells that have been recruited into the intercalary region of cell division will be multiplying exponentially [subject to postulate (3)], giving a total number of new cells of the order of 2”~. Thus the number of cells recruited from the originally quiescent tissue to form the intercalary growth will be very small in comparison with the ultimate number of intercalary cells: the two numbers will be roughly in the ratio it : 2”. Thus even though the intercalary regenerate may be large, it will be derived only from relatively few cells in the immediate neighbourhood of the original discontinuity, while cells outside that neighbourhood remain unaffected.

simpler rules for epimorphic regeneration: the polar-coordinate model without polar coordinates

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