J. theor. Biol. (1990) 143, 233-250

Modelling of the Duplication of Cortical Units along a Kinety of Paramecium using Reaction-diffusion Equations

HERVI~ LE GUYADERt AND CLAUDE HYVER~

t Laboratoire de Biologie Cellulaie 4 (URA 1134 CNRS), B~timent 444, Universit@ de Paris-Sud, 91405 Orsay Cedex, France and

~. Service de Biophysique, D@partement de Biologie, Centre d'Etudes Nucldaires de Saclay, 91191 Gif-Sur-Yvette Cedex, France

(Received on 26 June 1989, Accepted in revised form on 13 September 1989)

The cortex of Paramecium consists of cortical units arranged longitudinally in kineties. At the centre of each unit is the basal body of a cilium or kinetosome. During mitosis, the number of kinetosomes must double if the cells are to remain identical from one generation to the next.

Duplication of the cortical units takes place by means of a wave propagated from the oral groove to the poles (Iftode et al., 1989, Development 105, 191-211). In a curious way, however, the wave does not travel over the whole cortex but delimits two invariant regions at whose boundary it stops abruptly.

The propagation of such a duplication wave and its abrupt termination have been modelled by a system of reaction-diffusion equations at two levels. The first level is that of an autocatalytic oscillating system, while the second incorporates a cusp type of singularity as occurs in catastrophe theory.

This model provides a complete explanation for the duplication of cortical units, and for the propagation of the wave and its termination.

A biochemical interpretation of the model is suggested in the form of a CaZ+/cal - modulin-dependent kinase. Biological and theoretical data supporting the model are given in detail and the common features that exist between this system and the kinases involved in controlling the cellular cycle are stressed.

Finally, mention is made of the biological significance of the autocatalytic activa- tion of kinases in systems generating spatial or temporal oscillations.

1. Introduction

Reaction-diffusion equations have been widely used to provide a physico-chemical interpretation of structuring in many living systems ever since their theoretical proposition by Turing (1952). They have been applied, for example, to the morpho- genesis of organisms as different as Dictyostelium discoi'deum (Lacalli & Harrison, 1978), Acetabularia mediterranea (Harrison et al., 1984), Hydra littoralis (Macauley Bode & Bode, 1984), Drosophila melanogaster (Kauffman, 1977) as well as to higher plants where they have been used to explain phyllotaxis (Marzec & Kappraff, 1983).

These various individual applications have gone hand in hand with some remark- able technical and conceptual developments (see, e.g. Gierer, 1981; Meinhardt, 1982). Until recently, the only adverse feature was the relative failure to provide any experimental evidence for the morphogens. It was then necessary to regard the use of the term "morphogen" applied to a particular peptide in Hydra (Schaller &

233

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234 H . L E G U Y A D E R A N D C. H Y V E R

Bodenmuller, 1981) or to the differentiation inducing factor in Dictyostelium dis° co~deum (Morris et al., 1987) as amounting to a misuse of terminology in relation to the theory of reaction-diffusion equations. Fortunately, two recent discoveries have mitigated the pessimism: retinoic acid for vertebrate limbs (Thaller & Eichele, 1987; Maden et al., 1988) and the bicoid protein in Drosophila embryos (Driever & Niisslein-Volhard, 1988a, b; Lawrence, 1988) appear to satisfy the requirements of the theory.

As a result, Turing's ideas currently form one of the most frequently cited theoretical sources of explanation in biology for the appearance of a pattern or an increase in structural complexity, both in space and time (Nicolis & Prigogine, 1977; Harrison, 1987).

Nevertheless, apart from some suggestions by Frankel (1984), the reaction- diffusion equations do not appear to have been applied to the study of Ciliates, probably due to a lack of reliable physiological or biochemical data, yet these unicellular animals show a profusion of forms and patterns that has long attracted the attention of biologists.

Among this class of Protozoa, Paramecium is one of the most intensely studied species both from the genetic and morphological point of view (see the review in Grr tz , 1988). They are Ciliates with a very elaborate cortex incorporating about 3200 cilia which must double in number at every cell division for the cells to remain identical from one generation to the next. Each cilium is firmly attached by a special organelle, the kinetosome, at the centre of a cortical unit. These must therefore be duplicated before cell division.

The duplication is initiated near the oral groove and progresses to the poles of the cell with a wave-like motion. Surprisingly, this does not involve the whole of the cortex: the wave in fact stops abruptly at the boundary between two cortical regions of Paramecium which are well defined and, as a result, are known as "'invariant fields" (Iftode et al., 1989).

In the absence of any actual experimental data, several hypotheses have been suggested to explain this division of the cortex into such regions. It seemed to us that it would be worthwhile choosing a theoretical method to test these hypotheses and using the reaction-diffusion equations to construct an interpretative model for the propagation of the wave duplicating the cortical units and its abrupt termination at the boundary of the invariant field.

2. The Biological Problem

The Paramecium cortex can be regarded as a mosaic of elementary components each with one cilium, the cortical units. The arrangement of such units in the cortex is not random but exhibits a considerable degree of order. Longitudinally, they are arranged in rows, the kineties. Over the dorsal surface, these are parallel to the major axis of the cell and connect the two poles. Over the ventral surface, they bend and give rise to anterior and posterior sutures which straddle the oral groove (Fig. 1).

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Each cell division creates two daughter cells identical to the parent cell, particularly in relation to the number of cortical units which roughly stays the same in each generation. This occurs by generation of new cortical units immediately anterior to the "old" ones, that is growth occurs only along the longitudinal axes of the cell while the circumferential pattern does not exhibit any growth. It is therefore impor- tant that, before the division, Paramecium duplicates all its cortical units and, as a result, all its kinetosomes.

Duplication of the kinetosomes does not occur haphazardly, but appears to be governed by a wave which is propagated from the oral area towards the poles of the cell (Iftode et al., 1989) and which is associated with the reorganization or duplication of the set of elements forming the cortical unit (epiplasm, kinetodesmal fibres, etc).

It is now definitely accepted that, in Paramecium, two levels of morphogenetic regulation are involved. The local level, that of the cortical unit, has been mainly depicted from the geometrical inversion of ciliary rows (Sonneborn, 1964; Beisson & Sonneborn, 1965) which can be indefinitely propagated in a clone. This kind of observation has generated the concept of cytotaxis, defined as the "ordering and arranging of new cell structure under the influence of pre-existing cell structure" (Sonneborn, 1964). Therefore, the kinetosome assembly, its path of migration within the cortex, the organization of the main associated structures (kinetodesmai fibre, postciliary microtubules,...)mi.e, the polarity and the structure of the cortical unitmare determined by the pattern within the cortical unit and not by any other outside influence (Sonneborn, 1975).

The global level, that of the whole organism has been demonstrated by various microsurgical experiments which seem to confirm that the oral groove acts as a singularity as far as the latter level is concerned (Hanson, 1962; Hanson & Ungerleider, 1973). The duplication wave in particular is initiated in this region. One must notice that this wave gives only the duplication order, but not the polarity which is determined, according to the cytotaxis concept, at the cortical unit level.

However, there is a real surprise when it comes to a detailed description of the wave propagation, which is found to delimit two invariant fields at which boundaries it stops. One of these fields is on the left-anterior section and corresponds to the paratene region; the other is on the median-posterior section and surrounds the cytoproct. The production of the total number of units needed to obtain two entire daughter cells is achieved by an over-duplication of units in certain regions.

All these morphogenetical phenomena are very fast, and appear to be separated in time from the cell growth which starts after th'eir achievement. This remark is important because it allows us to build up a model without growth, simpler than those occuring in plant morphogenesis (Harrison & Kolar, 1988).

There are thus two main classes of kinetics. Dorsal kinetics are homogeneous and have their entire sets of cortical units duplicated. Ventral kinetics, on the other hand, are basically heterogeneous in the great majority of cases. Each of them consists of two sections: one near the oral groove has its cortical units duplicated and therefore behaves like a dorsal kinety; the other near one of the poles is invariant. The boundary between these two regions is very sharp and the duplication process thus stops very abruptly.

D U P L I C A T I O N OF: P A R A M E C I U M C O R T I C A L U N I T S 2 3 7

Two major sets of assumptions can be made to explain the stoppage in the propagation of the growth wave:

(1) The hypothesis of spatial heterogeneity: it may be assumed that the cortical units in the invariant fields have special properties which make them unrespon- sive to forced motion, or that the underlying cortex or cytoplasm does not conduct the signal.

(2) The hypothesis of spatial homogeneity: constraints of an overall nature arising either from the source or from the nature of the entity being transmitted prevent the signal from being propagated. The medium is a priori isotropic and the cortical units all have the same duplication properties.

A priori, spatial heterogeneity is experimentally the most attractive and easily testable hypothesis. This is because the left-anterior invariant zone has some remark- able intrinsic properties: in particular, an abundant internal cytoskeleton and trans- verse microtulules lie beneath it. Moreover, at the time when cell division takes place, the kinetodesmal fibre in this region shows a slight shortening, which seems to testify that the signal is present in this zone.

On the other hand, hypotheses postulating the isotropy of the substrate space appear to be more easily testable by a system of reaction-diffusion equations. This is an important test, since it could allow us to check whether the hypothesis of spatial hetereogeneity (i.e. of a local difference in structure) must inevitably be adopted.

Thus, the question we ask ourselves is: Can a system of differential reaction- diffusion equations based on Turing's ideas (i.e. postulating the isotropy of the substrate space) explain the existence of invariant fields in Paramecium ?

3. Model Formulation

For the time being, therefore, we are trying to model the behaviour of one kinety on the ventral surface of Paramecium. The various stages in the construction of a model to achieve this are as follows:

(1) Build up a system of reaction-diffusion equations having periodic solutions. Each peak in the concentration of one of the morphogens is then interpreted as fixing the location of one cortical unit.

(2) Test whether the frequency of the wave can be doubled: this would model the duplication of cortical units.

(3) Establish a mechanism enabling the duplication of the peaks to be initiated which could depend on the creation (or disappearance) of a molecule secreted (or absorbed) by the region near the oral groove. The latter would then act as a source (or sink).

3.1. T H E S Y S T E M O F S P A T I A L S T R U C T U R I N G

Many systems satisfying the Turing conditions are available to us. We have chosen one in preference to the others since it has the advantage of using standard biochemical mechanisms that are a priori verifiable. This is:

d A / d t = D1 02A/Or2 + V(r ) - a A B - bA (1)

d B / d t = D2 02B/Or2 + a A B + bA - cB/ (1 +d B) . (2)

2 3 8 H . L E G U Y A D E R A N D C . H Y V E R

The interpretation of the symbols and terms in these equations, in which t is the time and r a space variable, is as follows:

A and B are the concentrations of the two morphogens, one of which, A, is a precursor of the other.

The terms D~ d2A/Or 2 and D2 02B/Or 2 reflect the spatial transfer of A and B by diffusion, according to Fick's law.

The term a A B expresses the autocatalytic t ransformation of A and B according to the reaction A + B ~ B + B.

bA represents a t ransformation of A into B, catalysed or not, necessary to start the autocatalytic process.

c B / ( l + d B ) reflects an enzymatic degradation of B according to standard Michael is -Menten kinetics.

V(r), by producing A, supplies the whole system. This term is constructed a priori as a spatial function but, in any initial study for analysing the system mathematically, will be regarded as constant.

a, b, c and d are coefficients related to the rate coefficients o f the elementary biochemical reactions in the system.

Such a system is dealt with, according to the Turing method, by linearizing around the singular point ( d A / d t = d B / d t = O ; 02A/Or2=O2B/Or2=O). The following relationships are then obtained:

V = c B o / ( l + d B o ) ; Ao = V / ( b + a B o ) .

The eigenvalues of the system are obtained by putting:

A = Ao+ a. e ~lt+iwr)

B = Bo+ b. e (n+i'r~.

I f w = O, the second degree equation in (1) leads to negative eigenvalues; however, for particular values of (w) that are easy to calculate, these eigenvalues can turn out to be positive, a sign that indicates a structure with spatial periodicity.

It is of fundamental importance to note here that: (1) the range of acceptable w values depends on V; (2) correspondingly, for certain choices of the values given to the system para-

meters, w can vary by a factor of 2 or more. These are significant points. Remember that V is an input to the system and can

therefore be used as a control for the doubling of the wave frequency.

3.2. C O N T R O L O F T H E D U P L I C A T I O N

The next problem is to choose the structure of the function V(r) in such a way that it possesses all the characteristics expected of a control function.

Consider a molecule U, the precursor of V. Let us write:

d U / d t = D3 02 U/Or 2 + K - j U , (3)

D U P L I C A T I O N O F P A R A M E C I U M C O R T I C A L U N I T S 2 3 9

0 3 02U/Or 2 is a term representing the diffusion of U; K expresses a constant input (or endogenous production) of the element U and j U represents the t ransformation of U into V.

When the cell is in a stationary phase (G1 phase), it is assumed that there is no supplementary source or sink of U, so that we can put U = K/ j . On the contrary, at the time when the kinetosomes are duplicating in preparat ion for mitosis, the oral groove (or contractile ring region) acts as a sink for U.

To simplify matters, let us first consider a semi-infinite medium, i.e. one that has a boundary at the oral groove and is infinite in extent towards one pole of the cell. The solution of (3) then takes the form:

U = A - B e - q r.

This solution leads to an equation showing a monotonic variation with r. If U(r) and V(r) are identified with each other to within a constant of proportionality, it can be seen that the monotonic variations of this function do not entail an abrupt change from a mode of wavelength d to one of wavelength 2d, either for A(r, t), B(r, t) or any associated function.

As a result, it seemed essential to introduce the possibility of a threshold type behaviour by a well thought-out use of catastrophe theory. Catastrophes in metabolic systems have previously been described, for example in a cooperative effect in membranes (Changeux et al., 1967) or during kinetics catalysed by an enzyme capable of being inhibited by an excess of its substrate (Hyver, 1980a, b). Let us choose the latter description and add to (3) a term reflecting the disappearance of V catalysed by such an enzyme according to:

d V / d t = j U - f V - ( p V + qV2)/(1 + rV + sV2), (4)

in which j U represents an input coming from U, f V a linear disappearance of the molecule V and ( p V + q V 2 ) / ( l + r V + s V 2) the disappearance of V under the influence of an enzyme inhibited by an excess of its substrate (for the derivation of this form of the expression, see Hyver, 1980a).

For suitable chosen parameters, the function U = f ( V ) obtained for d V/d t = 0 has three solutions, of which two are stable. The characteristic hysteresis effect of the cusp occurs when, as t ime progresses, U evolves by decreasing from a relatively high starting value to the first solution (Fig. 2). V(r) then undergoes a sharp discontinuity corresponding to the jump from one stable solution to the other. The position of the latter in space changes in such a way as to tend asymptotically towards a certain fixed level (Fig. 3).

3.3. T H E O B S E R V A B L E

I f the theory is to be consistent, it seemed to us necessary to introduce a variable related to the observability of the morphogenesis. This was because it appeared unreasonable to put forward the hypothesis that the duplication of cortical units takes place immediately after the increase in morphogen concentration. To take this into account, we simply write:

d F / d t = kB - zF. (5)

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H. LE G U Y A D E R A N D C. HYVER

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FIG. 2. Graph of the function U=f(V) near the singularity obtained from eqn (4) for dV/dt=O. The scales show that the catastrophe is only just registering. This was a deliberate choice, made in order to show that the hypothesis is biologically plausible. The following figures show that such "'gentle catastrophes" can be extremely significant. The numerical values of the parameters in eqn (4) are: j=2.10-5; f = 1; p =8.02; q = r = 0 ; s =3-6.

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The choice of B is arbitrary; since the two variables A and B are closely coupled together in the reaction-diffusion system, A could equally well have been chosen as the input in eqn (5).

4. Results

Equations (1), (2), (3), (4) and (5) can only be solved numerically. It was therefore necessary to check by calculation (and after a suitable choice of parameters which we shall not discuss) that the hypotheses were correctly formulated and lead to the desired results being obtained. Systems of non-linear differential equations can, after all, give paradoxical results which are difficult to foresee intuitively (Delattre, 1977). We might wonder, for example, whether the diffusion mechanisms would not bring about very different behaviour when close to the oral groove from that when remote from it.

We therefore had to check, in turn: that the initial pattern of F is perfectly periodic; that the concentration of V shows a sharp fall whose position in space migrates

as time passes;

D U P L I C A T I O N O F P A R A M E C I U M C O R T I C A L U N I T S 241

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FIG. 3. Mode l l i ng o f the cu rve U = f ( r ) as a funct ion o f t ime: t~ = 300; t 2 = 3000; t 3 = 25 000, t 4 = 50 000, t 5 = 100 000, t 6 = 200 000. The t ime scale, d e t e r m i n e d f rom o the r numer i ca l values , m a y be modif ied for o ther sets o f va lues of the s a m e pa rame te r s .

that the pattern obeys this migration, i.e. that it moves at the same time as the duplication wave front;

that the inauguration of the 2d regime is produced by a doubling of the peaks without appreciable transients.

4.1. PERIODICITY OF INITIAL PATFERN

Figure 4 shows that, according to the calculation, the peaks of the morphogen F are arranged with perfect periodicity and that their amplitude is completely stable.

4.2. E V O L U T I O N O F T H E C O N C E N T R A T I O N O F T H E S I G N A L - M O L E C U L E

When Paramecium is in the GI phase, the concentration of V is constant over the whole cell. The hypothesis involves the assumption that, in preparation for mitosis, the oral groove becomes a sink for U with very clear repercussions on V.

Figure 3 shows that V very quickly experiences a sharp fall in its concentration, corresponding to the folding type of catastrophe introduced into eqn (4). There is therefore an actual moving wavefront. In advance of this wavefront, V is large and stable; behind it, the concentration of V becomes suddenly much lower.

The final position of the wavefront depends on the values of the parameters in eqn (3) and the intensity of the sink for U: it can thus be adjusted very flexibly. In this way, therefore, both the dorsal and ventral kineties can be modelled.

242 H. LE G U Y A D E R A N D C. H Y V E R

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FIG. 4. Initial pattern of the distribution of the observable F before cell division. The curve is plotted by connecting up the calculated points without smoothing, which explains the apparent irregularity in the amplitudes (this comment also applies to the following figures). Numerical values of the parameters in eqns (1) and (2): D~ =21.16, D2=0.02116, values of p, q, r and s: as in Fig. 2.

4.3. REPERCUSSION OF THIS WAVEFRONT ON THE

WAVE FREQUENCY OF THE INITIAL PATTERN

Figure 5 shows three stages in the progression of the singularity in F. Behind the wavefront, the frequency of the spatial wave is doubled. Transients appear to be very small and involve only one or two wavelengths at a time.

4.4. FINE DETAIL OF THE KINETICS OF THE DUPLICATION OF A PEAK

The way in which the doubling of frequency occurs is shown in Fig. 6. It can be seen that the doubling takes place by the evolution of the primary peak and not by total disorder followed by reorganization. The fact that the peak splits up gradually is obviously more acceptable biologically.

5. Discussion

The answer to the question raised initially is therefore: it is possible to reproduce the behaviour of a kinety involved in an invariant zone of the cortex of Paramecium by postulating an isotropic medium and constructing a system of reaction-diffusion equations.

In a way, this positive result is frustrating on its own: because it invalidates none of the initial hypotheses, it leaves the question completely open. The model is in fact only one of many that are possible and it therefore cannot claim to provide a

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D U P L I C A T I O N O F P A R A M E C I U M C O R T I C A L U N I T S

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FIG. 5. Modelling of the progression ofthe wave duplicating the cortical units. Values ofthe parameters: length L = 200; (a) t = 25 000, (b) t= 50 000, (c) t= 100 000. (c), Gives an aspect of the model near the asymptotic state.

definitive representa t ion o f reality. Consequent ly , the result does not enable us to reject the hypothes is o f spatial heterogenei ty in the cortical units.

Nevertheless, the mode l is impor tan t in several respects: (1) It has the novel feature o f reminding us that the hypothes is o f spatial

homogene i ty (i.e. the most difficult one to test, exper imental ly speaking) should not be forgotten.

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FiG. 6. Detail of the duplication of a peak. The observation is made by taking a more limited spatial scale (L = 50) over much shorter times: (a) t = 4650, (b) t--9450, (c) t = 19 050. This makes it possible to describe the evolution of a primary peak which splits up gradually.

(2) It demons t ra tes the possibili ty o f a dynamic spatial " m e m o r y " which can easily be dupl ica ted unde r the effect o f a sequence o f controls .

(3) A l though the mode l is a priori comple te ly hypothe t ica l in mos t respects, we are nevertheless justified in specula t ing about its interpretat ion, especially in o rder to test its biological consistency.

D U P L I C A T I O N O F P A R A M E C I U M C O R T I C A L U N I T S 245

We may wonder what interpretations we are to put on: the oscillating A / B system and the variable F, the observable, with which it is

directly correlated; the variable V which is used to initiate the duplication of the peaks of the

morphogen; and lastly, the variable U representing the general control variable, in relation

to which the oral groove (or the contractile ring area) serves as a sink. Above all, however, it must be remembered that the system was constructed for

the purpose of replying to a precise theoretical question and not in order to model given biochemical characteristics. This is an important point since some features of the system were described arbitrarily, particularly: (i) the contractile ring area is a sink for U (it could have been considered as a source, with the same result as far as the model is concerned); (ii) the total output of U is incorporated in V, just as the output of V is in A (we could have considered only a part of it); (iii) the observable F is directly related to the morphogen B (the molecule A could equally well have been chosen),

5.1. H A S T H E D Y N A M I C S O F A C A L M O D U L I N D E P E N D E N T K I N A S E

B E E N M O D E L L E D ?

Phosphorylation and dephosphorylation of proteins catalysed by kinases and phosphatases are currently classified among the major mechanisms by which cell functions are regulated (Edelman et al., 1987), to such an extent that, by analogy with electronics, kinases have been likened to biological "chips" (Hunter, 1987). It is also known that many kinases are activated by second messengers such as cAMP, cGMP or Ca 2+, according to a sequential system as suggested above.

Moreover, although an analysis of the biochemical mechanisms involved in the morphogenesis of Paramecium has not been carded out, several partial experimental results may serve as a guide during our consideration of this question.

(1) It is now well known that Ca 2+ plays a determining role during ciliary beating (Evans et al., 1987). Various indications lead us to think that this ion is also an important agent in morphogenesis during cell division.

One of these indications is the existence of the mutant Kin 241, which produces many morphogenetic abnormalities (excess kinetics, reversed kinetics, abnormal division of cell nuclei, etc). Most of these abnormalities are found to be lessened or even prevented by the addition of EDTA to the culture medium (Rossignol & Beisson, personal communication). Although this cannot be regarded as irrefutable proof, it can nevertheless be reasonably assumed that the deficiency in the mutant kin 241 affects its calcium metabolism.

But, it is known that the cortical alveoli of Paramecium form a calcic repository that is both large (Stelly et al., 1987) and continuous (Allen, 1971). This Ca 2+ could be involved during cell division.

(2) It has proved possible to locate calmodulin accurately by using specific antibodies (Momayazi et al., 1986). Its presence has been detected particularly around the kinetosomes.

2 4 6 H. LE G U Y A D E R A N D C. H Y V E R

(3) It has been demonstrated (Krryer et al., 1987) that phosphorylated proteins are continually associated with the set of microtubular organizing centres present in the cortex; protein phosphorylations are also involved when, during the cell cycle, the cortical cytoskeleton undergoes reorganization other than that associated with microtubules (kinetodesmal fibre, epiplasm, etc). Such phosphorylations are mostly produced by kinases.

We can therefore reasonably put forward the hypothesis that there exists in Paramecium a morphogenetic dynamics controlled by a Ca2÷/calmodulin dependent kinase.

We now have to test the consistency of such a hypothesis.

(i) Can the A/B system model the behaviour of a kinase?

Several features appear to confirm such an identification. When we wrote eqns (1) and (2), it was assumed that the transformation of the A morphogen into B occurs autocatalytically with initiation by a different mechanism, and that B is degraded enzymatically. Now it is known that autophosphorylation, which is either initiated spontaneously or by another kinase, is one of the most remarkable properties of kinases (Edelman et al., 1987; Honegger et al., 1989; Schulman & Lou, 1989). This property could even be the basis of a mechanism for dynamic memory storage (Lisman, 1985).

With this interpretation, the degradation of B expressed by eqn (2) would then simply be its dephosphorylation by a phosphatase.

The observation variable F could then represent the proteins phosphorylated by the kinase. Given that the elements controlling the duplication of cortical units are unknown and that the mechanism of the biosynthesis of kinetosomes is still a mystery (Bornens & Karsenti, 1984), the interpretation can only be taken further with great difficulty. We should nevertheless stress the regulatory significance of the phosphory- lation of the proteins related to the cytoskeleton (MAPs, MTOCs) as the initiator of their assembly and disassembly, as demonstrated for the nuclear lamina (Gerace & Blobel, 1980).

(ii) Can the variable U represent the Ca2+?

Equation (3) models the behaviour of a molecule capable of diffusion whose homeostasis is ensured by the equilibrium between a constant input (or endogenous production) and an active output whose level is proportional to the concentration of U. It is known that the concentration of Ca 2+ is perfectly controlled in the cytoplasm, with pumps rejecting or sequestring the Ca 2+ which enters across the membrane (Carafoli, 1987).

( iii) Can the variable V represent calmodulin ?

If A and B are the dephosphorylated and phosphorylated forms of a kinase and U is the Ca 2+ ion, then V must model calmodulin. Equation (4) ascribes three properties to it: (i) it is activated by Ca2+; (ii) it activates phosphorylation of the

D U P L I C A T I O N O F P A R A M E C I U M C O R T I C A L U N I T S 247

kinase; (iii) it is deactivated by an enzyme capable of being inhibited by an excess of its substrate.

The first two properties agree exactly with the suggested interpretation; the third, on the other hand, appears to be completely hypothetical in the present state of our knowledge.

The interpretations of the variables suggested above seem to be consistent. The final point to be checked is the initiation of the duplication.

(iv) Is the dynamics of the duplication plausible?

It was postulated that the contractile ring area played the role of a sink for U, i.e. for the Ca 2÷, and that the strength of this sink altered the position at which the duplication wavefront stopped. However, the calcium mechanoreceptor channels are not distributed uniformly over the whole cortex. There is an anterior-posterior gradient as well as a very pronounced dorsal-ventral variation: the dorsal surface gives a greater response than the ventral surface (Ogura & Machemer, 1980). It is clear that, according to all the data concerning both the homeostasis of Ca 2÷ (Carafoli, 1987) and the role of Ca -,÷ during mitosis (Keith, 1987), it would un- doubtedly be more consistent to postulate the existence of a source in this case.

Now we know that the model works and gives comparable results using either a sink or a source for U. tn view of this, we could have carried out the calculations again so as to make the analogy more plausible. We rejected this idea because such an artificial process has little importance for the consistency of the model. Moreover, the model becomes attached to an interpretation that may evolve in the future. We should stress that, in the light of the structure of the model and the experimental data summarized above, it is impossible to make an absolutely certain identification.

The main possibilities are given in Table 1. Our preference here is given to hypothesis 1 for the reasons developed above, but also because this solution is in agreement with the point of view put forward by Harrison et aL (1988) as regards the morphogenesis of Acetabularia.

The processes in this unicellular alga appear to conform to a two-level hierarchical system. First of all, the Ca 2÷ is distributed uniformly around a ring surrounding the

TABLE 1

The main interpretations of the variables in the model We have chosen interpretation 1 (see text). MTs: Microtubules; MTOCs: Microtubule

Organizing Centres

P. HYP. 1 HYP, 2 HYP. 3

U Ca -~+ CaZ+/Calmodulin Ca2÷/Calmodulin V Calmodulin Kinase Kinase 1

A/B Kinase MTOCs Kinase 2 F MTOCs MTs MTOCs

2 4 8 H. LE G U Y A D E R A N D C. H Y V E R

end of the tip; a morphogen X, which could be a kinase according to the authors, then becomes distributed periodically along this ring, leading to the formation of a whorl.

This last point, together with the ubiquity of the role of kinases (Hunter, 1987), leads us to assume that a model like the one proposed could have a more general application than that initially intended.

This model was constructed in response to a precise concerning a spatially periodic structure, without any definite hypothesis regarding the initiation of the phenomenon. It is clear that the latter concerns the cell cycle and therefore involves a temporally periodic organization. Recent data obtained on the MPF (maturation promoting factor), particularly through a detailed genetic study of cdc mutants in yeast, have once again revealed the importance of kinases regulated sequentially.

The MPFs of Xenopus (Cyert & Kirschner, 1988), of the clam Spisula (Draetta et al., 1989), of starfish (Labb6 et al., 1989) and of human HeLa cells (Draetta & Beach, 1988) are kinases similar to the p34 cdc2 of Schizosaccharomyces pombe (Simanis & Nurse, 1986) and to the p34 cdc:s of Saccharomyces cerevisiae (Russel et al., 1989) with the following properties:

their activation occurs autocatalytically; a spontaneous activation occurs, at least in vitro; they obey sequential controls and it should be noted that cyclins, molecules

bonding with p34 ¢d~2 (Draetta et al., 1989; Murray & Kirschner, 1989) exhibit a catastrophic type of behaviour with an abrupt proteolytic degradation at the end of mitosis.

In view of these data, Murray et al. (1989) and Labb6 et al. (1989) propose an autocatalytic activation of the MPF, the phosphorylated form being inactive and the dephosphorylated form active in most cases, according to a model similar to that of eqns (1) and (2), leaving aside diffusion phenomena. It is well known that autocatalysis is a condition favouring the appearance of an oscillating system.

As a result of these considerations, it could even be envisaged that the spatial structuring system in Paramecium as described above could be derived from a system oscillating in time like that of the MPF.

A confirmation of this hypothesis could be the observation for the same molecule of both types of oscillation in different cells. Now it has been shown (Krryer et al., 1987) that:

(a) antibodies staining phosphorylated proteins associated with centrosomes of HeLa cells exclusively during mitosis (temporal oscillation) are fixed indepen- dently of the cell cycle in the neighbourhood of Paramecium kinetosomes (spatial oscillation);

(b) this technique has also made it possible to observe in Paramecium at the time of mitosis phosphorylations involving the kinetodesmal fibre and the epiplasm (temporal oscillation).

In this way, through its catalytic properties, a kinase is sufficient to create a system which can oscillate either temporally or spatially, depending on the situation. From an evolutionary point of view, this fact would account for the importance of kinases as initiators.

D U P L I C A T I O N O F P A R A M E C I U M C O R T I C A L U N I T S 249

We wish to t h a n k Professor A. Adout te for careful ly reading the whole manusc r ip t and fruitful suggest ions and F. I f tode for the d rawing of the Fig. 1.

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