A mathematical model for embryonic cell division based on a surface “cleavage field”

J. theor. Biol. (1978) 75, 123-137

A Mathematical Model for Embryonic Cell Division Based on a Surface “Cleavage Field” GIORGKO CATALANO Ist. Matematico Castelnuovo, Univ. di Roma, Italy, and Stazione Zoologica, Napoli, Italy AND J. CHRIS EILBECK Department

of Mathematics, Heriot- Watt University, Edinburgh, EH14 4AS, Scotland

(Received 6 March 1978, and in revisedform

23 June 1978)

The process whereby a fertilized egg divides to give rise to an embryo, i.e. the process of cleavage-which can be considered, in some sense, as the early phase of embryonic differentiation-exhibit in many species a precise geometry. Such a geometry may be altered within certain limits, as was done in various classical experiments, and yet normal differentiation may occur. However, since the pattern of cleavage is clearly under genetic control, any model of cleavage should incorporate some device apt to produce a specific geometry. In this paper, a model for embryonic cell division based on a surface “cleavage field” is described. This surface field may be interpreted, for instance, as the surface density of sources of active transport of ions which diffuse into the cell, although other interpretations -such as the surface density of specific binding sites or functional membrane receptors, etc.-are possible. Assumptions relating the geometry of cleavage to the geometry of the level surfaces of the ionic concentration are given together with a discussion of the change in the surface field due to cleavage. Finally, a simplified two-dimensional version of the model is presented which develops interesting patterns of “cleavage”, calculated by computer, similar in many ways to those of real threedimensional embryos. 1. Introduction Any theory concerning the development

of the embryo has to deal with the

problem of ceI1 division: why, when and how does the process of cleavage occur? In well studied cases, such as that of the sea-urchin, the embryo cleaves with a precise geometry (Hdrstadius, 1973) and by the fourth stage 123

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of division the cells show clear evidence of differentiation: some of the cells are smaller than the rest and develop into the skeleton of the sea urchin. The biologists’ and biochemists’ approach to the problem is to investigate the detailed chemical and physical processes involved in the dividing cell in the hope that eventually some pattern will emerge which will tie together the accumulation of experimental data. The theoretical biologist, on the other hand, takes a completely different approach: all the fine details are temporarily ignored and a search is made for the simplest physically reasonable theory which will demonstrate the general features of cell division as seen by the experimentalist. This is the approach followed in this paper. In an earlier paper (Catalano, 1977), one such model was put forward. During the process of cell division of the fertilized egg, different regions of the cytoplasm are segregated to different cells (blastomeres). In order to provide a differential feedback by the various cytoplasmic regions to the nuclei, a three-dimensional scalar field associated with the cytoplasm of the egg was postulated. The detailed development of this model depends on a knowledge of the fluid movement in a dividing cell, and some progress has been made in this direction (Eilbeck, 197%~). In this paper we discuss an additional possibility-not necessarily an alternative one-that there exists a two-dimensional scalar field, a “surface cleavage field”, associated with the membrane of the egg. For simplicity we assume the internal cleavage field depends only on the surface cleavage field. The internal field determines the cleavage plane, and after cleavage a new surface field will be present, related to the old surface field by the deformation induced in the membrane during cleavage. Thus each cleavage modifies the surface field and hence the position of subsequent cleavage planes may be different. In this simple model a knowledge of the deformation of the surface during cleavage and of its effect on the original surface field of the egg, assumed to be known, is sufficient, in principle, to determine the whole pattern of cleavage. It may be worth noting that various types of interaction between the cell membrane and the cytoplasm have been considered by many authors (cf. Cuatrecasas, 1974; Marx, 1976; Raff, 1976; Winkelhake, 1976). The paper is set out as follows. In section 2 we give a more detailed description of the surface field based on pIausible physical processes, together with the prescription for calculating the associated internal field and cleavage plane. The details of the cleavage process are discussed in section 3. Finally in section 4 we apply the theory to an even more idealized model-a twodimensional “cell” undergoing division in two dimensions. The results of the two dimensional calculation show features suggestive of experimental observed three dimensional developments, giving hope that a full three dimensional calculation will produce worthwhile results.

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2. The Surface “Cleavage Field” We assume that there is a stationary surface density (r of sources of active transport of ions of a certain undetermined species. These sources regulate the ionic concentration u on the inner side of the egg membrane, and we assume that the ionic flux diffusing into the cytoplasm is responsible for triggering cell division. In order to simplify our model as much as possible, we assume the geometry of the division is determined by the steady state concentration gradient and not by the dynamics of the diffusion process. No doubt the time taken to reach this steady state is related to the timing of the division process, and a more elaborate model would include such a caIculation. In a slightly different context this question has been discussed by Munro & Crick (1971), who showed that (almost) steady concentration gradients can be set up in physically reasonable time periods. Thus, despite the possibility that the charged particles be involved in an enormous pattern of chemical reactions, we suppose that the geometry of cleavage can be correlated in a relatively simple manner to the geometry of the steady state concentration gradient, which in turn is related directly to the surface concentration and hence to the scalar field 0. In more mathematical detail the model is as follows: We assume the surface of the cell is a sphere A, of radius r = ao, and the surface of the nucleus is a concentric sphere B, of radius Y = bo(ao > b, > 0). Defined on A, we have a scalar field go = ~~(8, 4) simulating a distribution of sources of active transport of ions across the membrane of the egg, maintaining a stationary ionic corzcentrarion z&B, 4) on A, [where (Y, 19,4) are the usual spherical coordinates]. In what follows we shall assume a simple linear relationship between u. and eO. Another possibility, which we shall not consider further in this paper, is that the field rrO determines the concentration gradient au/&z normal to A,. We define, in the usual manner, the vector of the diffusion current J = -DVu,

(1) where Vu = grad u and D is the diffusion coefficient, normalized to unity for convenience. In addition we define the total tlux 4 of J through the surface A, in the usual way

~JA,,(J)= j J . n ds,

(2)

A0 where II is the inward unit vector normal to A, and the integral is taken over the surface A,. We make the simplest possible assumption about the surface of the nucleus B,: it is composed of a constant distribution of sinks maintaining the ionic concentration at zero.

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Inside the spherical shell bounded by A, and B0 and assumed to be filled with a uniform and isotropic substance, the concentration U(Y, 8, 4, t) is modelled by the linear diffusion equation U* = v2u. (3) Here u, = au/at and V2 is the Laplace operator in spherical co-ordinates. As explained above, we further assume that a steady-state solution has been reached, so that the left-hand side of equation (3) vanishes, and u satisfies the Laplace equation v2u = 0 (4) with the boundary conditions 4%

0, 4) = uote, 44

(54

and u@o, 8, 4) = 0. (5’9 Once u@,~) is known, the internal concentration can be calculated numerically using standard numerical techniques (c.f. Ames, 1977). In some simple cases exact solutions can be found, for example if ~~(0, 4) = c, a constant, then the solution to (4) and (5) is now spherically symmetric:

u(r)= 2

0 0(

l-t!? r

(6)

>

(c.f. Jost, 1960). In this case the level surfaces S’ of constant gradient c’ are given by concentric spheres of radii r,

=

bo

l-(ao-boY

{

-I*

UOC

(7)

>

This solution is of little interest for our theory since its radial symmetry does not single out a specific cleavage plane. In general, however, the level surfaces of the solution of equation (4) will not be spheres, but more complicated open or closed surfaces. Note that without the assumption of the central sink at the nucleus, all level curves would not close, but begin and end on the surface A,. This latter results depends on the well-known mathematical properties of the solution of the Laplace equation (4). The next problem is the relation between the geometry of the solution u (or equivalently the geometry of its level surfaces S’) and the geometry of cleavage. We assume that during cleavage the ends of the spindle are separated by some mutually repulsive force, and in addition there is some inwards force on each centriole due to the local concentration or concentration gradient. For example, if the level surfaces were ellipsoidal the spindle would orientate itself parallel to the major axis of the ellipsoid. For more complicated surfaces this assumption is insufliciently precise, and we limit

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ourselves here to the examples given in section 4. The plane of cleavage of the egg is then assumed to be perpendicular to the spindle axis and pass through its centre: if this plane does not pass through the centre of the egg the cell will cleave unequally. We assume the nucleus cleaves in the same proportion as the cell, and that after cleavage each nucleus moves to the centre of its blastomere. Two points require some additional discussion. First the molecular interpretation of the forces acting on the spindle postulated above is somewhat unclear. As a very naive model we may envisage that the level surfaces act as electrostatic potential barriers that orientate the expanding spindle, and that, at equilibrium, balance the biochemical forces that presumably cause the expansion of the spindle (Maria, 1961). Secondly, we have neglected the effect of neighbouring cells on the membrane ionic transport mechanism. We anticipate that a model of contact inhibition of cell division could be constructed based on the vanishing of the total flux 4(J) of the diffusion current vector J at the cell membrane, such that the necessary concentration gradients inside the cell are never achieved. Investigation of this aspect is encouraged by the recent experimental finding by Parisi et al. (1978) that there is a ring of four cells (the “inner micromeres”) in the sea-urchin embryo that stop dividing from the fXth cleavage until the blastula stage. This completes our discussion of a single cleavage model. In studying the properties of such a model, the main difficulty lies in the solution of the three dimensional Laplace equation (4). Obtaining solutions for different choices of o,(B, #) is relatively time consuming, even on modern computers, and the display and interpretation of the resultant level curves requires complicated graphical facilities. We have chosen here to investigate the behaviour of the model in only two dimensions, rather than the physical three dimensions, since in two dimensions the numerical solutions and level curves can be computed with much less effort. Despite these limitations the results in two dimensions show interesting features which correlate well with threedimensional experimental observations. However before presenting these results we need to discuss the modification of the surface field caused by the deformation of the membrane during cleavage. This will provide the necessary link between different stages of division during the development process. 3. Modification

of the Surface Field During Cleavage

In the previous section the determination of the plane of cleavage in our model cell was discussed. We now proceed to discuss the division process itself, particularly the effect on the surface field.

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From the work of Hiramoto (1958, hereafter referred to as 1) we know that cleavage takes place under rigorous volume constancy. Since we are assuming the cytoplasm to be an isotropic and uniform fluid, this is consistent with a constant total volume condition. If the cell cleaves in two equal halves, the radius of each new cell will be related to the old radius by the ratio 2-*. The total area will be increased by a factor 2*, so that if C,, is the mean density on A,, the new mean density Cl on each of the two blastomeres after division will be ii, = Co .2 -*. However the elongation may not be uniform, so that this relation will not hold except as an average over the whole surface. Extensive studies have been devoted to both the cytoplastmic movement and to the area1 changes in the surface of the egg of various species during the first cleavage (c.f. Dan, 1943; Hiramoto, 1971; Motomura, 1940; Sawai, 1976); for our present purpose we refer mainly to Ref. 1 on the sea-urchin cleavage. A study of Figs 4A and 4G of Ref. 1 (p. 413) shows that, considering the surface only, one can define a mapping H-which will be called here the “Hiramoto mapping”-that maps pointwise the surface of either hemisphere determined on the egg by the cleavage plane onto the surface of the corresponding blastomere at the end of the first cleavage (see Fig. 1.) For convenience we take the plane of cleavage to be the (x, v) plane and the axis of symmetry the z axis. The mapping H has an axial symmetry about the z axis, and a singular point since the great circle in the (x, y) plane maps into a single point in the final blastomere. The experimentally observed mapping (Ref. 1) is somewhat more complicated than our mathematical idealization

FIG. 1. A mathematical idealization of the “Hiramoto mapping” between the hemisphere to the right of the cleavage plane and the resulting blastomere. Grid lines are plotted for constant z and 4 (the axial angle of symmetry) and the corresponding z’ # generated from equation 9b, with E = 0.1. The generic point p is mapped into p’. Note that all the points q-q are mapped onto the singular point q’.

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shown in Fig. 1, since the former has a cusp at the singular point and the remaining part of the blastomere is not quite spherical. We assume that the deformation required to produce a spherical blastomere is not crucial to the theory. Where the distinction is important we shall refer to our mathematical version as the “modified Hiramoto map” H,,. Because of the axial symmetry, Ho will depend only on z, and we write the cleavage transformation as H,: z -+ z’ = H,(z) c/5+ I$’ = (b.

(8)

If we place the origin of our new z’ co-ordinate at the centre of the blastomere, Ho maps the interval (0, ao) in z into the interval (- a,2-3, a,2-*) in z’. If the whole surface is stretched uniformly, the mapping will simply be &(z)

= 2-+(22-a,)

(94

whereas the experimental results in Ref. 1 suggest some extra degree of stretching at both poles of the axis of symmetry. This could be achieved by a map of the form H,(z) = 2-3

(l-E)(22-Lx())+ 1

5 4

.

(22~a,)3

(9b)

>

By suitable modification of (9a) or (9b) we can also treat the case of unequal division. A short calculation shows that, if cO(z, 4) is the value of the field co at the point (z, 4) in cylindrical co-ordinates, then the value cl(z’, 4’) of the new field at the end of the (equal) division at the corresponding point (z’, 4’) under the mapping H,, will be given by 23 iT1(z’, 4’) = Hb(4

- %(Z, 41,

w

where H;(z) = dH,(z)/dz. If the concentration u0 at the boundary A, is taken to be linearly dependent on the field go, then the same formula (10) will apply to find the concentration u1 at the boundary A, after division. Once the new surface concentration is known, we can repeat the calculations given in section 2 to find the new cleavage planes, then apply another Hiramoto map along the new axes of symmetry to generate the second division. This process is repeated for as many divisions as are required, though in view of the gross simplifications made in the theory it would be unwise to calculate too many divisions. One further assumption is required to specify the development of the embryo after the first division. If we assume that the point of contact between the first two blastomeres does not slip, then the Hiramoto transformation,

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sinceit implies a stretching of the surface, will induce a rotation in the plane of cleavage of the second and subsequent divisions. Alternatively we can assume some sort of slip condition so that subsequent cleavage planes will not rotate. More experimental evidence is required to elucidate this point. In the two-dimensional simulations shown in the following section, examples involving both types of assumption are given. 4. Two-Dimensional

Simulation

of Patterns of Cleavage

As explained at the end of section 2, calculations involving the full threedimensional model are lengthy and require a detailed study, and these calculations are currently under development by the authors. However the computation can be much more easily carried out in two dimensions, and we include some examples of these calculations, both to elucidate the model in more detail and to show that even in this limited unphysical form the results show an encouraging diversity of structure. We hope the experimental biologist will bear with us in this excursion into “flatland”. The modifications required in applying the model to two dimensional “cells” are straightforward. The spherical surfaces become circles and the equal volume condition becomes an equal area condition. The factors of 2* in the Hiramoto map become 2*, though since our theory depends only on relative concentrations we drop this factor entirely in our concentration calculations. In the case of unequal divisions we assume that each cell will divide in the ratio of areas given by the ratio of the angles subtended at the centre of the cell by each part before division. We assume that both the dependence of u on cr and the Hiramoto map are linear in the calculations that follow. Finally we take the ratio of the cell to nucleus radius as 10. In three dimensions we would expect that the initial distribution of the field c,, would exhibit an axial symmetry, corresponding for instance with the “animal-vegetal polarity” of the sea-urchin egg (c.f. Horstadius, 1973). In two dimensions this means an initial “plane” of symmetry. After some numerical experiments we selected the initial distribution shown in Fig. 2 for our first example. Figure 2(a) shows the graph of the initial surface distribution as a function of vertical height, and Fig. 2(b) shows the level curves of the solution of the steady-state diffusion equation in two dimensions. (Details of the contour plotting program are given in Eilbeck, 197823). The geometry of the largest closed level curve in Fig. 2(b) suggests that, according to the description in section 2, the points a, a’ can be taken as the ends of the “spindle”. The first cleavage of the egg will then occur along the axis of symmetry which is perpendicular to a-a’. The level concentration curves of one of the first two blastomeres is calculated after the new surface

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FIG. 2(a). Graph of initial surface distribution as a function of y = sin B for the first example described in the text. (b) The level curves for the surface concentration graphed in Fig. 2(a). The spindle axis is assumed to be u-u’ and the cleavage plane perpendicular to this.

concentrations have been determined: these are shown in Fig. 3(a). Now the long axis of the level curves is vertical and we can expect the cleavage axis to lie in the horizontal “plane”. After cleavage, assuming full slip conditions, we have the situation as shown in Fig. 3(b). This simple model exhibits a switching of 90” in the angle of cleavage between the first and second divisions: this reflects precisely the behaviour of the initial development of the embryo which is common to many species.

(b)

FIG. 3(a). The level curves for the “egg” shown in Fig. 2 after the first division. The spindle is now assumed to take up a position along the long axis of the elliptical level curves, giving a horizontal cleavage plane. (b) The first four blastomeres after the second division, calculated from the initial concentration shown in Fig. 2 and assuming the full slip conditions described in the text.

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For our second example we selected the slightly more complicated symmetric initial distribution shown in Fig. 4, and assumed no-slip conditions at all cleavage points during subsequent divisions. Due to the locally unsymmetric form of the maxima and minima in the original distribution, after the first division all the cleavage planes lay at an angle to the initial division. Where the level curves were elliptical, and exact determination of the angle of the major axis was difficult, we took the criteria that the spindle passed through the centre of the cell and the point on the circle with half the

ho / Y

! ‘0 i /’

,&--La-----

.*

..- _e-----__ --------
-. -

‘\

( 1 i

0

\ : I

-I’

:!. : : : : : : : : (a)

FIG. 4. The initial surface distribution and corresponding level curves of the second example describedin the text.

cell radius for which the concentration was a minimum. Due to the rotations induced at each cleavage by the no-slip condition, results are rather sensitive to the criteria used for predicting the spindle axis. With the above criteria the results after 3 divisions are shown in Fig. 5. In this case there is an intriguing semicircular structure which could be identified with the partial development of a two-dimensional “blastula”. Additional examples, including the possibility of unequal cleavage even at the first division (as in the case of mammalian eggs), can be easily obtained by variating the initial field on the boundary. These two-dimensional examples show an encouraging amount of structure which can be correlated directly with well-known three-dimensional features. There is no reason to suppose that calculations on the full three-dimensional model will not reveal similar features and thus go some way towards

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FIG. 5. The pattern of cleavage developing from the initial distribution assuming a no-slip condition.

133

shown in Fig. 4,

simulating real cleavage patterns, obtaining theoretical predictions for the hypothetical initial field that can be experimentally tested. 5. Discussion

One of the most important tests for a “pattern of cleavage” simulated by computer is, of course, a comparison with a real pattern of cleavage. Is it possible, for instance, to select an initial distribution such that the resulting pattern of cleavage be symmetrical and equal? In a two-dimensional simulation this can be certainly obtained, as it is shown by the particular choice of u&) of section 4 in the case of Fig. 3, and such a possibility should

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survive also in three dimensions. Now, is it possible to make a dzJerent choice of z&y) so as to obtain the .sflPnekind of cleavage of Fig. 3(b)? As a matter of fact, there are many different choices of u,(y) that would produce the same kind of cleavage, at least in two dimensions, although we are not reporting here samples of those graphs. Again, this elasticity in the selection of the initial distribution is likely to be available also in three dimensions. In a three-dimensional simulation, however, it could happen that an initial distribution producing a given kind of cleavage turns out to be completely unrealistic when one tries to envisage a molecular mechanism whereby such a distribution is built up in the egg (and we are not studying this aspect of the problem in the present paper). In this case, one has to switch to a different choice of the initial distribution which would produce the same type of cleavage and yet be sufficiently realistic. Properties of the egg, such as the existence of an axial symmetry, of a vegetal-animal polarity, etc., should help in selecting an appropriate class of initial distributions. As long as the simulation proceeds beyond the first few divisions, the class of different “permissible” initial distributions shrinks considerably, so that we are left very soon with a limited range of different choices. In the case of the sea-urchin, for instance, we could start with an axially symmetric initial distribution ~~(8, 4) = u,(4), independent of 19 (on a sphere of radius r = 1). Whether or not such an ~(4) exists so as to reproduce the geometry of cleavage typical of the sea-urchin embryo (say, at least, for the first four divisions, so that the formation of micromeres would be included) cannot be ascertained until a full three-dimensional simulation will be performed. For other types of cleavage, such as spiral cleavage, incomplete cleavage, etc., a three-dimensional calculation is also unavoidable and must take into account, in general, the combined effects of a non-uniform asymmetric initial distribution u,(B, 4) and of the Hiramoto mapping. The latter, by itself, introduces space rotations when it is simultaneously applied to a system of blastomeres. When cleavage is not total (i.e. only a portion of the egg cleaves, as, for instance, in amphibians) our mathematical model should be adjusted, since in that case the Hiramoto mapping looses its axial symmetry and a more accurate study of cytoplasmic movement is needed. The fact that different initial distributions may produce the same pattern of cleavage does not contain enough information about the sensitivity of the qualitative pattern of cleavage to changes in a given initial distribution. However, this question is automatically answered by the mathematical properties of Laplace equation: in fact, as it is well known, the solution of the boundary value problem for Laplace equation depends continuously on the data of the problem, so that such a solution has a certain stability and it

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cannot fluctuate radically because of small random disturbances. The Hiramoto mapping, in turn, has an even greater stability. Characteristic changes in the normal geometry of cleavage have been caused-as it is reported in a number of papers about experiments on amphibians, echinoderms, ctenophores, etc., scattered throughout the biological literature-either by mechanical or chemical treatment. Moreover, there are experimental evidences that changes in the localization of the developmental potential are sometimes associated with changes in the pattern of cleavage. The recent work on ctenophores by G. Freeman (1976a,b), in which the effects of compression are compared with those of a chemical treatment of the embryo, can be considered as a typical and important representative of the above mentioned studies. Before we consider Freeman’s experiments, it is perhaps convenient to mention explicitly two properties that are satisfied by our model: (1) Two equal initial distributions, u0 and u,,, always produce the same geometry of cleavage, induce the same internal field and hence the same localization of the developmental potential; (2) Two different initial distributions may (sometimes) produce the same geometry of cleavage, although the induced internal fields are different; two different internal fields, in turn, may be associated, apriori, with the same localization or with different localizations of the developmental potential. In the first type of Freeman’s experiments (1976a) embryos have the second cleavage suppressed by an inhibitor (cytochalasin B) and may either recommence that one or skip it and move straight onto the next one. This could be interpreted as follows: let u1 be the initial distribution on the surface of each blastomere of the 2-cell stage (configuration B-2 in Fig. 1 of Freeman’s paper (1976b, hereafter referred to as F, p. 334); since the second cleavage is inhibited, at the end of the second cycle we still have a 2-cell configuration (B-3, Ref. F). During this cycle, which lasts for a period T, the inhibitor is present only for a time interval between t = 0 and t = to (with 0 < to c T). We assume that u1 = u,(B, 4) becomes a time dependent distribution u1 = ~~(0, 4, t) as soon as t > 0 [with ~~(6, 4) = ~~(0, 4, O)]. Suppressing for brevity the explicit indication of the space variables 0 and 4, one reasonable way of defining the distribution u1 = ul(t) may be the following: q(t) =f(t) when 0 < t < t,, and u,(t) = f(t,)g(tto) when t, < t < T, where f and g are two appropriate functions and g satisfies the condition g(0) = 1 (in particular, if g E 1, then u1 is time dependent only in presence of the inhibitor). Thus, the new distribution u,(T) at the end of the second cycle (during which one cell division has been suppressed), clearly depends on t,. The possibility that a temporal evolution of a surface distribution on the cell membrane be caused by an inhibitor of

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cell division, as we assumed above, is strongly suggested (although in a different context than that of ctenophores) by various experimental evidences when the inhibitor which is used is cytochalasin B as in Freeman’s experiments (see, for instance, T. T. Puck, 1977, and other papers in this series). Now, using the above notations, the results of Freeman’s experiments show that, as long as to varies in the interval between 20 min and 35 min (being T = 1 hr), u,(T) either results substantially unchanged [i.e. u,(T) = q(O)], or it turns out to be substantially equal to the distribution u2 of each blastomere of the normal cleavage at the end of the second division (A - 3, Ref. F). The situation corresponding to u,(T) = u,(O)-which is reported to occur in 25 per cent of the cases-could be easily explained in terms of a very weak action of cytochalasin B on the cell membrane (and the possible existence of a threshold depending on the value of to). The other situation, corresponding to u,(T) = u2, occurs in 75 per cent of the cases and, although possible mechanisms may be envisaged that could cause u,(t) to evolve towards u,based on the fact that cytochalasin B acts also on the cytoskeleton of the cell, inhibiting its microfilaments, and may change the morphology and the activity of the membrane, we believe that even an attempt at a molecular interpretation is out of each. Therefore, the peculiar evolution of ul(t) towards u2 can be, at the moment, only postulated; on the other hand, the different interpretations that have been proposed, far from our model, are recognized to be only postulated as well. A comparison of the various modified configurations, resulting either by inhibition or compression (Ref. F), with the normal pattern of cleavage provides at least five extremely interesting situations that can be all easily and automatically interpreted [once the postulate on the evolution of al(t) is accepted] in terms of the initial distributions ul, u,(T) and ol. It can be shown that each one of these five si&ations is consistent with properties (1) and (2) above. Here we may conclude that our model appears to be at least compatible with the behaviour of cleavage in response to different treatments (such as compression, centrifugation, separation or fusion of blastomeres, action by chemical agents), as was briefly discussed in the particular but important case of Freeman’s experiments. So far, we are not aware of any experimental evidence incompatible with the model. Of course, compatibility represents only a pre-requisite for the development of the model which depends, in the first place, on full three-dimensional simulations. One of us (JCE) is grateful to the SRC for financial and computational support. The other (GC) wouId like to thank Professor R. J. Knops of the Department of Mathematics, Heriot-Watt University, for his hospitality during a visit in which much of this work was done, and Professor A. Monroy for discussions.

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