Morphogenesis and chemical dissipative structures

Morphogenesis and chemical dissipative structures

J. theor. Biol. (1972) 36, 479-501 Morphogenesis and Chemical Dissipative Stn~ctures A Computer Simulated Case Study HUGO M. MARTINEZ Department of B...

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J. theor. Biol. (1972) 36, 479-501

Morphogenesis and Chemical Dissipative Stn~ctures A Computer Simulated Case Study HUGO M. MARTINEZ Department of Biochemistry and Biophysics, University of California at Smr Francisco, San Francisco, Calif: 94122, U.S.A. (Received 14 September 1971, and in revisedform 20 December 1971) A basis for the chemical prepattem concept of morphogenesis in multicellular organisms is examined in terms of a computer simulated casestudy of a compartmentalized, reaction-diffusion system. The simulation provides a non-linear extension of Turing’s linear analysis of similar systems. Using a chemical reaction type posed by Prigogine, it is shown how spatially non-homogeneous and stable, steady-state concentration distributions result when the system is maintained under conditions far from thermodynamic equilibrium. Interpreting these stable, non-homogeneous, steady states as chemical prepattems for the spatial structuring of growth and/or differentiation, time structuring can also be obtained in the form of a sequence of prepattems generated by the rule that a prepattem is converted into its successor when the growth it is directing results in conditions which render it unstable. The physical simplicity of such a scheme for obtaining spatial and temporal structuring suggests it as a primitive ordering method in the morphogenesis of multicellular organisms. Examples are also given of how complicated spatial prepattems can be viewed as the piecing together of independent, simple prepattems.

1. Introduction The chemical prepattem concept in the form of spatially, non-homogeneous concentration distributions has previously heen advanced (Son&i, 1963) to account for the origin of patterns of cell differentiation in morphogenesis. There is, however, some question regarding the physical plausibility of such a concept because it is not clear that the Turing-type mechanism invoked as a possible basis has the necessary stability. The primary purpose of this study is to show that the required stability can be achieved if one uses the full non-linear version of Turing’s approach instead of relying on linear 479

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approximations. In addition, it is also shown how time structuring can be obtained in the form of a self generating sequence of chemical prepatterns and which is suggestive of a primitive developmental scheme. Turing’s proposal (1952) of a theory of morphogenesis used the idea that patterns of chemical concentrations could arise as a result of instabilities in homogeneous equilibria. In his words: “It is suggested that a system of chemical substances, called morphogens, reacting together and diffusing through a tissue, is adequate to account for the main phenomena of morphogenesis. Such a system, although it may originally be quite homogeneous, may later develop a pattern or structure due to an instability of the homogeneous equilibrium, which is triggered off by random disturbances.” The investigations he reported on in support of this thesis were chiefly concerned with the onset of instability for a hypothetical reaction-diffusion system in a ring and in a spherical configuration of cells. By confining himself to near homogeneous equilibrium conditions (and thus to the employment of linear equations for the reactions), he was able to demonstrate analytically that instabilities could occur under realistic conditions and that these could result in a variety of patterns, the most interesting of which are standing waves. He accordingly suggested that in the case of a ring these periodic waves might account for whorled leaves in certain plants and for the tentacle patterns in hydra, while in the case of a hollow sphere they could account for gastrulation. But he was nevertheless scrupulously careful to remind the reader that such results depended “essentially on the assumption that the reaction rates are linear functions of concentrations, an assumption which is justifiable in the case of a system just beginning to leave a homogeneous condition,” and that “such systems certainly have a special interest as giving the first appearance of a pattern, but they are the exception rather than the rule. Most of an organism most of the time, is developing from one pattern into another, rather than from homogeneity into a pattern.” This more general process to which Turing alluded allows a certain freedom of interpretation, for although he intended to give a computationally oriented case study for a morphogen theory of phyllotaxis in which the elements of the process would be made explicit, the study was never reported on (at least not to our knowledge), with the result that more often than not Turing’s theory of a chemical basis for morphogenesis is regarded as synonymous with the findings of his linear analysis. Viewed as such, one may justifiably object to it when invoked, for instance, as a basis for chemical prepatterns. Thus, Goodwin & Cohen (1969) in discussing gradient based theories of embryonic periodic structures write : “Turing-type periodic waves remain a possibility, but the problem of regulation for such a model is severe. So also are the problems of the time required for the standing wave to be established and its

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stability in the presence of cell movement.” And Wolpert (1969), in proposing his positional information concept, writes: “For example, the prepattem is viewed by Maynard Smith & Sondhi (1961) and Tokunga & Stem (1965) as possibly being represented by variation in the concentration of some substance along an axis, the concentration curve having well defined singularities such as peaks at specific points. The specification of this pattern seems no less easy than that of the original pattern, though Maynard Smith & Sondhi (1961) have attempted to account for the origin of such a prepattern in terms of waves generated by a Turing-like (Turing, 1952) system. This is not very satisfactory and all the difficulties disappear when the concept of prepattem is interpreted in terms of positional information.” The justifiability of these objections of Wolpert and Goodwin & Cohen would be removed if we are careful to interpret the meaning of a morphogen pattern in a stricter sense, namely, that they correspond to a stable, non-homogeneous steady state rather than to those which may arise in the process of leaving an unstable homogeneous steady state and which may or may not be a true indication of the former type. This is no doubt what Turing really had in mind, and what such writers as Rosen (1968) interpret to be the case in identifying Turing’s approach with that of Rashevsky’s (1960) as a means of accounting for the origin of concentration asymmetries. But while such an interpretation may be correct, it imposes the burden of establishing that these stable, nonhomogeneous steady states are in fact realizable. Hence the relevance of the recent work on dissipative structures (Prigogine, 1969). In contrast to equilibrium structures, like that of crystals which do not require the exchange of energy or matter with the environment for their maintenance, dissipative structures are, by definition, highly dependent on such an exchange. The often quoted example of this is in reference to the B&nard problem of classical hydrodynamics in which a sufEciently large temperature gradient across a thin body of fluid can induce ordered circulation in contrast to mere thermal turbulence. For a reactiondiffusion system the induced structure would correspond to a stable, non-homogeneous steadystate distribution of the substances involved or else to a stable time-periodic variation of these. The first is referred to as spatial-ordering and the second as time-ordering. The recent work of Prigogine and co-workers has shown the existence of realizable reaction-diffusion systems in which spatial-ordering can occur provided they are far removed from thermodynamic equilibrium. This would thus seem to relieve the above stated burden were it not for the disturbing fact that even though a system may possess an unstable homogeneous state and a stable non-homogeneous state, this is still not sufficient to guarantee that passage from the former to the latter is reliably possible under ordinary random fluctuations. What may occur, for example, is that the

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system possesses a time-ordering structure as well as a spatial one and that the time-ordering one is preferred. Thus, the system can get caught in a limit cycle should such exist. This is, in fact, what happened in our recently attempted computer solution of some reaction diffusion equations posed by Prigogine (1969). Consideration of this factor and the foregoing remarks on the existence of spatial ordering, would thus seem to indicate that a valid non-linear interpretation of Turing’s theory should involve a reactiondiffusion system for which at least the following is true : (i) the homogeneous steady state is unstable; (ii) it possesses a stable non-homogeneous steady state; (iii) passage from the homogeneous to the non-homogeneous state is reliably possible under ordinary fluctuations. In the sequel we shall present a case study of a system for which these three conditions hold and which has consequently enabled us to explore the possibilities of a developmental scheme implicit in Turing’s approach and which we would like next to describe briefly. The developmental scheme we have in mind is almost entirely described by the above quoted statement of Turing that most of an organism most of the time is developing from one pattern into another, rather than from homogeneity into a pattern, and by the interpretation of a pattern as corresponding to a stable steady-state morphogen distribution. We need only further assert that “developing from one pattern to another” means that the growth response to a morphogen pattern gradually results in conditions which render the inducing morphogen pattern unstable; that at the point of instability a new morphogen pattern sets in which is stable under the new conditions; that this new pattern then directs the growth until it becomes unstable, etc. Conceptually this is a very simple scheme which, in one form or another, has appeared in the developmental literature, but which, to our knowledge, has never been subjected to a serious analysis or stated exactly in these terms. For one thing, until the above mentioned work of Prigogine there has been no real physico-chemical basis for the existence of such patterns, and further, the theoretical investigation of any model system almost certainly requires extensive calculation-at least initially until some insight can be gained into the essential features involved. We were nevertheless led to consider the possibilities of such a scheme as a consequence of a preliminary attempt to simulate patterns of growth and differentiation using growing automata nets as effective models in the manner suggested and pursued by Apter (1966) and Lindenmeyer (1968). Because of its easily programmable nature on a digital computer and because of the precision with which questions can be posed, the growing automata net approach for simulating growth and differentiation patterns seemed at first sight to be a very promising theoretical tool. It was hoped,

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in particular, that it would yield an automata theoretic interpretation of Wolpert’s positional information concept; but it soon became apparent that important headway could not be made in this direction unless rules of interaction for the composing automata were incorporated which had a more reasonable physico-chemical basis than is possible with just a small number of states for the automata. What was consequently realized is that a compartmentalized, reaction-diffusion system (as might correspond to a collection of interacting cells) did in fact consitute an automata net when it was simulated on a digital computer. Candidates for rules of interaction were then readily at hand in the cited works of Prigogine and Turing which, when supplemented with rules of growth, began to reveal an unsuspected, but quite natural, kind of time structuring suggestive of the above developmental scheme. Incorporation of the growth property into a compartmentalized, reactiondiffusion system required a decision as to what would constitute the state of a compartment (cell) that would initiate its replication. Considering that in a reaction-diffusion system involving n substances the state of a cell at any time is specified by the concentration of these, it is natural to identify an event as characterized by a set of inequalities among the n concentrations. For instance, if just two substances X and Y are involved, then satisfaction of the concentration inequality X> Y, say, could correspond to the event that was necessary and sufbcient for replication to commence. (The use of regions of concentration values instead of a specitic concentration value for each of the substances would appear to be the better way of achieving reproducibility of the event.) At the same time, however, since an important event should not be triggered by chance fluctuations, a further requirement would be that the inequality be satisfied for a time significantly larger than achievable by such fluctuations and hence suggestive of a stable steady state of concentrations. And even further, since in a developing organism there is differential replication according to position, the steady-state concentration distribution of the substances must be one which is non-homogeneous in the sense, for example, that for some of the cells the concentration inequality X> Y holds and for others it does not. Finally one must allow for the fact that growth patterns change during developm&. This can be interpreted as a change in the underlying chemical concentration pattern caused by the very growth which it induced, and which can be realized by imagining that the inducing pattern, as a steady state of concentrations, becomes unstable after the growth it induces occurs and switches to a new distribution which is stable under the new cell contlguration. Implied, then, is the development scheme posed above, which will now be illustrated by an example in order to l?x the relevant ideas.

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2. Example of Development Scheme

Consider the following hypothetical

organism.

(i) It consists of a linear array of cells (lilamentous algae, say). (ii) A cell in the filament proceeds to divide if and only if the concentration 2 of a substance S in the cell reaches a threshold level 2,. (iii) In each cell there are two other substances M, and M, called the morphogens, of concentrations X and Y whose difference in concentration influences the rate of change of 2 according to the relation dZ a(X- Y)-bZ if X>Y dt= 1 -bZ if x < Y. (iv) The concentrations X and Y in a cell are in turn determined primarily by other reactions within the cell and by allowing M2 to diffuse between cells. Substances M, and S cannot pass from one cell to another. Starting from a single cell, suppose that it possesses steady-state values X,, and Y, for its morphogens and a Z value Zi < Z,. Assumption (iii) then says that the cell will divide if a *(X,, - YJb > Z,. We suppose this not to be the case, meaning that a cell will not normally divide by itself. However, if two of these cells come together and form a junction which permits morphogen M2 to be exchanged, there is the possibility that a non-homogeneous distribution of M, and Mz will occur which makes it favorable for at least one of the two cells to divide. Thus, if we label the cells A and B and let X,, YA, X,, Y, be their respective morphogen concentrations, then it may be, for instance, that (u/b) { X,(co) - YA(co) > > Z, and (a/b) { X,( oo) - Y,(a)} < Z,, i.e. cell A becomes competent to divide. As will be shown numerically in the sequel this can occur, with the result that we now have three cells. If we assume that each of the daughter cells takes on the same concentration values X and Y of the parent cell and that in the act of division the morphogen concentrations of a cell do not change, the situation that holds upon inception of the three cell configuration is that two have morphogen concentrations favorable for division and one does not. Conceivably, then, the two daughter cells could begin to divide, depending on their Z values. Assuming that upon birth, a cell takes on a Z value Z,, the same as an isolated cell, it then follows from (iii) that it will take time for the Z concentrations of the daughter cells to reach threshold values. And they would reach threshold values were it not for the possibility that the inherited morphogen distribution might be unstable, and this because we now have three cells instead of two. Instability is in fact the case when using the reaction-diffusion system discussed in the

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sequel. A computer simulation of the scheme using this system is sholvn in Fig. 1 to which we now refer. (In this figure, as in subsequent ones, cells with X > Y have been distinguished by an underscoring of X.) What happens to the three cell case arising from the two cell stage is that its morphogen distribu-

: 0.322 5.679 5-679 O-322 "'*4,479 --l-800 1,800 4.479 0.322 5.679 5.679 5.679 5,679 0.322 0.322 5.679 5.619 5.679 ... : 4.479 ---I.800 I.800 I.800 I.800 4.479 4.479 i?i66 1800 i%% i.

O-315 8.366 0.315 0.315 8.366 0.315 0.315 8.366 0.315 0.315 8.366 0.315 *3.926 i?% 3.926 3.926 = 3.926 3.926 1242 ,3.926 3.926 = 3.926

Fro. 1. Developmental scheme based on a two morphogen reaction4Susion system. Growth transition as response to the inducing morphogen concentration pattern is indicated by +, while the morphogen redistribution from an unstable to a stable, steady-state pattun is indicated by * -* 6. -+. Growth is a slow process compared to morphogen redistribution. Upper and lower numbers in each cell are X and Y concentration values rupectively. Xis undencored if X> Y. See text.

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tion is changed into a stable one in which only the middle cell has X> Y. If we assume that this pattern is established faster than the cells can respond to the one inherited, we then see how it is that the new cell configuration as a whole can determine which cells will divide next-in this case only the center cell. Assuming this one to divide, the next linear array will be of four cells in which the two middle ones have X> Y and the end cells have X< Y. This is essentially a stable pattern, although there is a readjustment of concentrations to actual stable values which do not change the inherited pattern as we have defined it in terms of inequalities. We note that this four cell pattern is just a repeat of the two cell pattern, giving us a clue as to how to guess at patterns when we know them for small numbers of cells. Continuing, the next array will consist of six cells with only the end cells having X< Y. This is unstable and becomes converted into a stable one which is a repeat of the three cell pattern. Growth response to it next gives an eight cell array having a pattern which is essentially stable and which is a repeat of the four cell pattern. The next and final growth response simulated is of twelve cells with an unstable inherited pattern which is converted into a stable one consisting of a sequence of three cell patterns. With this example in mind we now proceed to a justification of the properties attributed to the underlying reaction-diffusion system. 3. The Reaction-Diiusion

System

As noted, we have adopted for our case study some equations posed by Prigogine (1969), in the investigation of dissipative structures. Letting X and Y be concentrations, the kinetic equations of the two morphogens are dX -& = A+X2Y-BX-X, dY = BX-X2Y, dt to the reaction scheme

-

corresponding

(1)

2X 2X+Y

B+X:

:3X

Y+D

X2&, the net result of which is A + B+ E+ D. As pointed out by Prigogine, these reactions are somewhat unrealistic in that they involve a trimolecular

step,

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but we nevertheless adopt them at this stage because of their mathematical simplicity. Equations (1) apply, of course, to a volume element with perfect mixing, and it readily follows that the unique steady-state values of X and Y for such a volume element are X= A and Y = B/A. This is a stable, steady state (as can be shown by the usual technique of linearization), provided B < A2 + 1. If this condition does not hold, limit cycles about the steady state point will result, as verified by automatic computation (Lefever & Nicolis, 1970). Letting D, and Dr be diffusion constants for the two morphogens, equations (1) generalize to the form i?X iYY dr=BX-X2Y+D,V2Y,

(2)

applicable to an arbitrary region for which difision of the morphogens is constant. If we assume that the region consists of II elementary volume elements in each of which there is perfect mixing and between which there is diffusional flow, equations (2) specialize to the set of ordinary differential equations $

= -4+X:&-@+1)X,+

c&xl-x,), J

dY, -iii= -X?&+BX,+ CJ vi&Y,- YJ), of the form with which we will be concerned. With suitable choice of the intercell diffusion constants, one can specify any pattern of interconnection desired as, for instance, a ring of cells which we now proceed to analyze. But 8rst, let us note that equations (3) are of the form

(4) in which the functions

f and g represent the reaction part. 4. ARingofNCells

A collection of cells in which each has exactly two neighbors will be regarded as a ring. Numbering the cells for such a configuration in sequential order and assuming that the diffusion constants do not change along the ring

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will make equations (4) appear as %=.f(xis

Y,)+~L’(Xi+i-2Xi+X*-1),

dx dt=s(xi,B+v'(r,+l-2Yi+Y,-~),

(5)

subject to the boundary conditions X,, 1 = XI, X, = X0, Y,, 1 = Yi and YN= Y,. A homogeneous steady-state solution of these equations will correspond to values Xi = h and Yi = k for which f(h, k) = g(h, k) = 0. Assuming such a solution to exist (which it does for the reaction functions (l)), then letting xi = Xi-h and yi = Yi-k and linearizing by the usual technique of expanding f and g about the values h and k yield the perturbation equations dxi -2Xi+Xi-1)s dt = a%+bYi+&i+l dYi dt = cx,+dY,+v(Y*+,

-2Yi+Y*-l)9

(6)

with a = j;(h, k), b = f,(h, k), c = g,(h, k) and d = g&h, k). As noted by Turing (1952), this collection of pairs of equations can be nicely uncoupled by the introduction of new variables 5 and q defined as: N t, = f$* F exp (-2n*irs)x,, s 1 qr = f * i

s 1

exp (-2n*irs)y,.

Use of the elementary identity

provides the inverse relations N-l

x, = C exp (271.irs/iV)*<,, r=O

N-l

Y, = and then substitution the equations

,go

exp

(2~*WN)-~r~

into equations (6) yields, in terms of the new variables, %={a-411

sin’ (7~- s/N))<, + bqS,

2

sin2 (rrs/N)}q,+c~&.

= {d-4v

(7)

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It is now apparent that the homogeneous solution leading to the linearization will be unstable if and only if one of the roots of the characteristic equation P+{qp+v) sill2 (m/N)-(u+d)}l+ + (4~ sin2 (m/N) - a}{4v sin’ (ns/N) - d) - bc = 0 (8) has a positive real part. This will certainly be the case if the coefficient of rl in equation (8) is negative, and hence if 4@ + v) sin2 &s/N) < (a + d) for some value of s, or simply that (a+ d) > 0. In the particular case for which f(X,, YJ=A+XfY,-@+1)X, and g(X,, YJ== -X:F+BX,, one readily calculates that h = A, k = B/A, a = B- 1, b = A2, c = -B and d = -A’. Therefore, a+d = B-(1 + A’). Requiring that B < 1 + A2 to avoid having oscillations in an isolated cell, instability for ring will then necessitate having the constant term of equation (8) being negative, or, equivalently, that for some s - BA2 > (4~ sin’ @s/N) -(B - 1)) * (4v sin’ (xs/N) + A’}. (9) This is impossible for s = 0, as this would imply that A2 < 0. But if we assume that B- 1 > 4~ so that the first right hand factor of inequality (9) is negative irrespective of the value of s, then for any s for which sin (zr/N) # 0 we need only take v large enough to insure that inequality (9) holds. Summarizing these elementary considerations: (i) the steady state of a single cell is stable if and only if B < 1 + A’; (ii) assuming a single cell to be stable, a ring of N identical cells has an unstable homogeneous steady state for sufficiently large v if B > 4~+ 1. Here, p and v are, respectively, the intercell diffusion constants for the X and Y substances. While it is true, as just seen, that a collection of cells can have an unstable, homogeneous, steady-state morphogen distribution, this is no guarantee that such a state will be converted to one which is nonhomogeneous. The latter must of course exist, be stable, and there should be no interference with the passage. In our case, choosing A = 3, B = 9.75, p= 0, and v = 1 renders the steady state of a single cell stable and that of a ring of any number of cells unstable. The existence of a non-homogeneous steady state is established by showing that the equations A+X:&-(B+1)XI+p(X1+1-2Xi+Xi-i)

= 0

and -X:y,+BX,+v(y,,,-2y,+y,-,)

= 0,

with A, B, p and v having the specified values, have a real, positive solution other than X, = A and Yi = B/A. This set of non-linear, algebraic equations might’ be amenable to analytical methods of solution, but it was elected instead to solve them by the numerical integration of the corresponding

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differential equations (5), using initial conditions which differed but little from the homogeneous steady state X, = A and Yi = B/A. In effect, the reaction-diffusion system is simulated and one observes what happens to the

lm We] 2(blm 3(a) m 3(b) 1-1 4(a) -1

4(b) -1

5(a)

0.318 4.149

7.029 1469

0.303 2.816

7.031 1468

0.318 4.149

5’bJ

0.303 2.817

7.058 1469

0.318 4.150

0.318 4.149

6(a)

8.368 1242

0.315 3.925

0.315 3.925

8.368 =

0.315 3.925

0.315 3.925

6(b)

J69J 1.795

0.307 3,142

5.695 i%?

0.307 3,142

5.695 1.795

0.307 3.142

7(a)

0.306 3&6

7.320 i?i?

0.317 4.097

0.317 4.097

7,320 1.412

0.306 3.046

5.119 x

7(b)

0.306 3.046

7.320 1412

0.317 4.096

0.317 4.097

7.319 1.413

0.306 3.046

5.119 1.986

8(a)

0.317 4.080

6.942 E

0.303 2.835

6.941 m

0.317 4.080

0.316 3.990

8.547 1.217

0.316 3.990

8(b)

0.303 2.835

6.937 1487

0.317 4.079

0.316 3.989

8.547 =

0.316 3.989

0.317 4.078

6,944 1.486

9(a)

0,305 2.892

7.251 1425

0.318 4.107

0.318 4.107

7.248 1426

0.305 2.992

5,477 i?ii?j

0.307 3.210

5.473 =

g(b)

0.305 2.992

7.248 a

0.318 4.107

0.318 4,107

7.251 I.425

0.305 2.992

5.473 1.864

0.307 3.210

5.477 i%ij

lota)

7.235 iTz

0.317 4.043

0.316 3.974

8.505 1222

0.316 3.994

0.317 4.043

7.237 FE

0.306 3049

5.146 m

0.306 3.049

lo@‘)

0.303 2.817

7.028 1469

0.318 4.150

0.318 4.150

m 1.469

0.303 2.817

7.032 1408

0.318 4.149

0.318 4.148

7.031 1.466

7.030 1468

FIG. 2. Ring arrays. Stable morphogen distributions resulting from perturbed homogeneous condition of X= 3 and Y= 3.25 for each cell. X values are upper and Y values lower numbers. X underscored if X > Y. End cells of each array are joined together to form ring. Suffixes (a), (b) refer to different random perturbations of initial homogeneous condition.

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0.318 7.029 0.303 7.028 0.318 4.150 z 2.817 j%$ 4.151

5(b) 0.315 8.584 0.318 0.320 5460 3.895 5 4.110 4,329 E

6(a) 0.315 8-367 0.315 0.315 8.367 0.315 3.926 @ 3.926 3.926 1242 3.926 6(b) O-315 8-368 O-315 0,315 8,368 O-315 3-923 m 3.926 3.926 = 3.926 7(a) 0,315 8406 O-316 0.316 7-671 O-309 3.664 3,921 i?% 3.958 3.996 1350 3.375 m 709 0.315 8.410 O-316 O-316 7.672 0.309 % 3.918 1238 3.957 3.995 i?% 3.375 2.709 8(a) 0.315 8.453 O-316 0.317 6.935 0-303 a O-318 3,913 i?i% 3.998 4.084 1488 2,826 1469 4.149 8(b) 0.315 8.457 0.316 0.317 6.935 0.303 7.037 0.318 3.911 E 3.997 4.083 1488 2.826 1467 4.148 9(a) 0.315 8.444 0.316 0.317 7.083 0,304 5.978 0.310 3.932 3.914 E 3.989 4.065 i?% 2.934 i%t 3.472 % 9(b) 0.315 8-443 0.316 0,317 7.083 0.304 5.977 0*310 D 3.915 5 3.989 4065 1.458 2.934 m 3-412 2.540 lo(a) 0,315 8.433 0.316 0.317 7242 0.306 5139 0,305 7*‘0-317 3,916 5 3.981 4045 1.427 3.051 1*98Q 3044 1.414 4097 lo(b) 0.315 8.432 O-316 0.317 m 0.306 m 0.305 7.313 0.317 3,916 G 3.981 4045 l-427 3.051 1.980. 3044 = 4097 Fro. 3. Linear arrays. Stable morphogen distributions resulting from perturbed homogeneous condition of X= 3 and Y = 3-25 for each cell. X vahres upper and Y values lower numbers. X underscored if X> Y. Sufhxes (a), (b) refer to different random perturbations of initial homogeneous condition.

concentration values X,, Y( when the system is perturbed from the homogeneous steady state. Since the latter has been made an unstable one, steadystate values achieved by the simulation will correspond to non-homogeneous T.B.

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ones and, in addition, prove their attainability from the homogeneous state. Attainability from steady states which are non-homogeneous but which become unstable as a result of growth and/or differentiation is another matter to be investigated separately and is perhaps best done relative to specific rules, as in the example used above to illustrate the developmental scheme. The restricted aim here is to show that stable non-homogeneous patterns are possible and that they can be reached from an unstable homogeneous one under relatively mild perturbations. Also, we are interested in seeing if there are simple rules whereby one can guess at the non-homogeneous patterns according to the topology and number of cells of the collection. This will be crucial in later investigations when large numbers of morphogens are considered; computational methods, no matter how fast, will then need to be supplemented by good guesses. In Fig. 2 there is shown the computational results for rings of from two to ten cells relative to the initial condition that all cells in the ring suffer a random perturbation from the homogeneous state. Thus, cell i in a ring is given an initial X value of 3*O+AX, where AXi is a normally distributed random number of mean zero and standard deviation equal to O-03, while its Y value is 3*25+A Y1 with A Yi normally distributed of mean zero and standard deviation of 0.0325. Each ring is represented twice corresponding to different sets of initial perturbations in order to explore the possibility that different stable cotigurations can be obtained from the homogeneous conditions. As in the case of Fig. 1, cells with X > Y have been distinguished by underscoring X. It should also be understood that the ends are joined together to form the ring structure. A number of interesting features pertaining to the stable non-homogeneous states can be noted. (i) The six-cell and the ten-cell ring both yield a second different stable distribution. For the six-cell ring one distribution is a twofold repetition of the two-cell ring and the other is a one-fold repetition of the three-cell ring. For the ten-cell ring, distribution (a) is a joining together of the three-cell and seven-cell distributions, while (b) is a repetition of the five-cell distribution. (ii) The three-cell distribution can be cut at one point to give a stable linear distribution. See Fig. 3. (iii) The four-cell distribution is a repetition of the two-cell distribution. (iv) The five-cell distribution can be cut at one point to give a stable linear distribution. See Fig. 3. (v) The seven-cell distribution can be cut at one point to give a stable linear distribution. See Fig. 3.

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(vi) The eight-cell distribution is essentially a joining together of the three and five-cell distributions. (vii) The nine-cell distribution is not composed of three-cell distributions as might be anticipated, although the latter is probably stable. One cut is possible, but it is not known if the corresponding linear case is stable. (viii) In no case is a cell both a sink and a source for morphogen Y. (ix) Whenever X> Y, the cell is a sink for Y and whenever X< Y it is a source for Y. 5. ALhear

Amy

ofNCells

Equations (5) when subject to the boundary conditions X,, r = YN+ I = 0 and X,, = Y0 = 0 will be the required general equations for a linear array of N identical cells. But since we no longer have circular symmetry, there is no evident way of mathematically uncoupling the pairs of equations, as in the case of a ring, and thus of simplifying the stability analysis. The preliminary study was therefore limited to computer simulations of the type employed for the rings. Figure 3 shows the results of these simulations for arrays of up to ten cells. The initial conditions were once again a random perturbation relative to the homogeneous steady state, and two sets of initial conditions were supplied for each cell array. We note the following features. (i) Only the four-cell and five-cell arrays resulted in different stable non-homogeneous distributions. Cell distribution 4(b) is a repetition of the two-cell distribution, while 5(b) is essentially a joining of the two-cell and three-cell distributions with some disturbance to both. (ii) The six-cell distribution is a repetition of the three-cell distribution. (iii) The seven-cell array is the joining of the 4(a) and three-cell distributions. (iv) The eight-cell distribution is the joining of the 5(a) and three-cell distributions. (v) The nine-cell distribution is the joining of the six-cell and the threecell distributions. (vi) The ten-cell distribution is the joining of the seven-cell and the three-cell distributions. 6. Tmdimensional

Arrays of Four-sided Cells

Computer simulations were also performed with configurations in which a cell could have more than two but not more than four neighbors. Figure 4

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shows the results with rectangular arrays consisting of 2 x 2,2 x 3, 3 x 3,4 x 3 and 9 x 9 cells. Except for the 9 x 9 case each of the cells in an array was given a random perturbation from the homogeneous state as its initial condition. In

3x3

4.362 2x 0.303 2.851 4.363 f2x6 2x3

0.306 3.122 8.349 I.245 0.306 3,122 4x3

FIG. 4. Rectangular arrays. Stable morphogen distributions resulting from perturbed homogeneous condition of X = 3 and Y = 3.25. Xvaluea upper and Y values lower numbers. X underscored if X> Y.

495 the 9 x 9 case,four cells were selectedat random and only these received a random perturbation from the homogeneousstate.A numbet of interesting features can be noted. First, as in the case of the one-dimensional arrayt~, MORPHOGENESIS

(see

AND

DISSIPATIVE

STRUCTURES

Stable Fig. 4)

Stable ?

8-E

Stable (see

Stable

Fig. 3)

Stable

rBl!zR =

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~wt. 5. l~wmgular ermys. Steady-state morpho@n p@ttcms built up from rubpattans C&I solidlydrati haveX> Y. Questionmarkafter stablemeansarray wasnot tasted for stability.

4%

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no two cells having X> Y are ever in direct communication, and each cell with X > Y acts as a sink for morphogen Y. Next, except for the 2 x 2 case, there is only one cell in the array which has the largest value of A’. Thus, each of these arrays has a distinguished cell, reminiscent, with considerable interpretive latitude, of the ingredient necessary to the organizing center concept of biology. The patterns of Fig. 4 arise, as noted, from a perturbation of the homogeneous state. On the other hand,, Fig. 5 shows the kinds of stable patterns which can exist even though they were not obtained from the homogeneous state. Whether or not they can be so obtained remains to be investigated. The important matter to observe at this stage, however, is that each is made up of stable subpatterns and that fairly complicated patterns can be obtained

9 cells

81 cells

Fro. 6. Cylindrical arrays. Edges marked A joined together to obtain cylinder. Stable morphogen distributions resulting from perturbed homogeneous condition of X=3 and Y= 3.25 for each cell. X values upper and Y valuea lower numbers. X underscored if x> Y.

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by this piecing together of aheady stable patterns. The compatability of the piecesis simply that there is no net diffusion between them, although in the caseof the one-dimensional arrays dissimilar subpatternscould be compatible with some small amount of net diffusion. In the ilatter case,each subpattern suffers somedistortion. At any rate, what we seehere is perhaps the glimmering of important rules of organization meriting further study. Cylindrical surfaceswere the next logical topological structure to be looked at in a prehminary fashion. Figure 6 shows the resulting pat ms for nine cell and 81-04 arrays obtained from perturbing the horn0”geneous state, while Fig. 7 shows patterns which can be obtained by the piecing together of stable subpatterns as was done for the rectangular arrays; also, as for the latter, they are illustrative rather than exhaustive of the possibilities. Shown with each array is the basic subpattem. It would be well to point out here that in all casestried so far, a steadystate pattern which can be pieced together from stable non-homogeneous subpattems is itself stable. This poses a highly interesting conjecture.

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1

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A Stobb

Fko. 7. Qildricd arrays. Ed#m marked A joined together to form cylinder. Stead+ state morphogca patterns built up from subpattems. Ceils solidly drawn have X> Y.

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9 cell torus

B IO.301 18*50910.300 16.338 0.301 0,310 0,305 19.061 IO.301 1 2.587 3.425 2,968 2623 8,101 9.553 0.300 8.882 i%t~ 2.455 1.172 izfi 0.299 8.664 0.301 0~300 2.487 1.20 1 2.595 0.305 0.302 0.307 3.141 3.007 0.300 A 0.300 2.422 t 2.452 8,323 0.302 0,301 8.206 0.300 6.965 IO.308 IO.308 10. /~/::;;;~~~~~““l~= 2,497 8.467 1228 0 302 2.674

3,025 I.134 0.306 0.301 3,069 2.583 0.303 7.664 2.753 ] =

2.592 4.791 = 0.298 21.247

3.198 0.301 2.593 8.476 lm

2.470 8.850 1.177 0.305 2.988

B 81 cell torus FIG. 8 . Torus and cube surface arrays. Edges marked with same letter are joined together. Stable morphogen distributions resulting from perturbed homogeneous condition of X = 3 and Y = 3 325. X values upper and Y values lower numbers. X underscored if X> Y.

The next topological case treated was that of a closed surface. Of the three arrays simulated, one is a three-cell torus, another is a nine-cell torus and the last is the surface of a cube. The results are shown in Fig. Sconstituting the patterns resulting from a perturbed homogeneous state. 7. Two-dimensional Arrays of Six-sided Cells In none of the foregoing configurations have we been particularly concerned with topologies which might be regarded as typically biological, except in the

MORPHOGENESIS

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499

case of linear simulating Elaments. In using two-dimensional arrays one should really take into account the close packing feature in that what we usually observe is a cell surrounded by six neighbors instead of four. Accordingly, we have simulated an 8 x 9 planar array of six-sided cells. Figure 9

FIQ. 9. Re&mguk array of six-sided cells. Stable morphogcn distribution rcaulting from perturbed homogeneous condition of X= 3 and Y= 3.25. X valucs upper and Y values lower numbers. X undcmcoxed if X> Y.

shows the steady-state pattern resulting from a random perturbation of the homogeneous state for such an array. Once again we note the presence of single cell having the largest value of X. 8. A Four-cell Embryo Stage A biological configuration of some interest is the four-cell embryonic stage which often has the connectivity shown in Fig. 10. Two stable, nonhomogeneous patterns were found for such a cotiguration obtained from different perturbations of the homogeneous condition. At this point it became convenient to ask which of the two patterns would more likely result from the homogeneous condition. A random perturbation of concentrations was therefore introduced continuously (at each integration time step) rather than just at time zero. Pattern (b) resulted.

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0.312 3,634

I I.060 0.947

5.693 I .796

0.312 3.634

0.306 3.142

0.306 3.142 I

0.312 3.634

5694 I.796 /

\

I

(b)

(a)

FIG. 10. Typical four-cell embryonic stage and corresponding stable morphogen distributions resulting from different perturbations of homogeneous condition X=3 and Y = 3.25 for each cell. Xvalues upper and Y values lower numbers. Xunderswred if X > Y.

9. Discussion The significance of the above simulated results depends largely on whether they demonstrate a basis for the chemical prepattern concept. In this regard, it is felt that the results do overcome the usual instability objections to the Turing approach and, as such, establish its candidacy on a firmer footing. But it is quite another matter to argue for significance beyond this supportive level. It is not at all clear, for instance, that important organizational principles at the cellular level are necessarily the consequence of reciprocal interactions -and even if they were, that they are describable in reaction-diffusion terms. The spatial and temporal organizing scheme proposed here was arrived at in terms of Turing-like systems, but it would not be surprising if the essential aspects involved could be realized in other ways. Our simulation is just a particular example of an automata net, any number of which can no doubt be devised that will exhibit stable, non-homogeneous patterns of internal states and, with suitable rules of replication, almost any time structuring desired. The special appeal here is one of simplicity in showing how very stable chemical concentration patterns can be established in an array of identical compartments, and which is indicative thereby of a primitive means of achieving spatial and temporal organization, either by itself or when supplemented by the concept of “competence” (Sondhi, 1963). But to make this indication more conclusive it is of course necessary to use reactions which are more likely to be involved in actual cells. Logical candidates for such reactions

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would be those concerned with repression and derepression. A system of these is presently under study which is similar to the one proposed in the recent and highly relevant work of Tsanev & Sendov (1971) on a possible mechanism for cell differentiation. It is there argued that repression-derepression can explain the reversible functional changes in protein patterns, but that the stable and essentially irreversible protein patterns leading to differentiation require mechanisms which determine transcribability and supersede repressionderepression activity. Blocking-deblocking is the term used for this higher level of transcription control and so interpreted that if a particular genetic unit is deblocked (transcribable) the blocked state can only be achieved by mitotic division. Blocked states are regarded characteristic of differentiation, and hence the thesis, in accord with much experimental evidence, that a cell must undergo divisions before it can reach a differentiated state. Within such a theory it is tempting from our point of view to consider that cell division in morphogenesis is subject to spatial control according to stable, steady-state concentration distributions of derepressors, say, and that the distribution of these is influenced, if not wholly determined, by Turing-like mechanism. This is essentially what was done in our example of a developmental scheme and illustrates, at least in principle, how differential growth can be made subject to the entire spatial configuration of cells in a stable and well determined manner, and hence of indirectly specifying spatial patterns of cell differentiation. ‘Ibis work was aided by the Amelia C. Cook Endowment Fund for resesrch in cancer and by NIH Grant GM-17539. We also gratefully acknowledge the excellent computer progr amming and technical assistance of Miss L. Wabl. REFERENCES APIER, J. J. (1966). “Cybernetics m-l Development”. Oxford: Per-on Press. GOODWIN. B. & C~-JEN, M. (1969). J. &or. Biol. 25. 49. Lmvm, k. & N~com,-G. (i970): J. &or. Biol. 30, -267. LINDENMAYER, A. (1968). J. &or. BioZ. 18,280, 300. PIWOGINE, I. (1969). In Z%eoreticul Physics d&logy. Amsterdam: North-Holland zt.i&sK Y, N. (1960). “Muthetnutical Biophysks”. New York: Dover Publications. Rostm, R. (1968). Int. Rev. cytol. 23, 25. SONDHI, K. C. (1%3). Q. Rev. B&d, 38,289. TSANW, R. & SENDOV, B. (1971). f. &or. BzOl. 30, 337. TURINQ, A. M. (1952). Phil. lkms. R. Sot. B 237, 37. WOLPERT, L. (1969). J. theor. Bill. 25, 1.

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