Geometry and dynamics of tip morphogenesis in Acetabularia

J. theor. Biol. (1988) 131, 461-475

Geometry and Dynamics of Tip Morphogenesis in Acetabularia CHRISTIAN

BRI f~RE

Laboratoire de Biologie Quantitative-CNRS ENSAT-145 Av. de Muret, F 31076 Toulouse, France AND BRIAN G O O D W I N

Developmental Dynamics Research Group, Department of Biology, The Open University, Walton Hall, Milton Keynes, MK7 6AA, U.K. (Received 22 May 1987, and in revised form 27 November 1987) A detailed study of the relationship between the geometry of the regenerating tip in Acetabularia rnediterranea and the dynamics of a mechanochemical model of the process is presented. The tip is described as a viscoelastic shell having two layers: an underlying cytogel with calcium-regulated strain fields, and an external cell wall, treated as an elastic body. When parameter values are chosen to satisfy the bifurcation condition in the cytogel field, spatially non-uniform solutions occur whose patterns depend upon geometry. When growth is included in the model by allowing plastic as well as elastic deformation, tip formation is initiated. 1. Introduction

In two recent papers (Goodwin & Trainor, 1985; Trainor & Goodwin, 1986), a mechanochemical model for cellular morphogenesis was proposed and investigated. This model was first applied to the particular case of the regenerative events following cap removal in the alga Acetabularia mediterranea. The well-defined shape changes during regeneration of apical structures make this unicellular alga a prime organism for studies in plant development. It is important to stress that these changes of shape occur in a single cell and not at the level of a multicellular tissue, where differentiation processes and cell-cell interactions occur. In the present case, the origin of the morphological changes can lie only in cell wall deformations. This implies the existence o f a morphogenetic field which controls structural changes and synthesis of the cell wall. What are the constitutive factors of this morphogenetic field ? An important factor to consider is the turgor pressure, which arises from the difference'of osmotic potential between the vacuolar and the external medium, and is a prerequisite for cell expansion (Green et al., 1971; Ray et al., 1972). The direct effect of this turgor pressure is to establish a stress and strain field in the cell wall and in the cytoskeleton. As a secondary effect, the pressure-induced strain field could affect chemical and transport processes. There is evidence that the maintenance of turgor pressure is dependent upon pressure-sensitive ion pumps in the ptasmalemma (Zimmermann, 1977; Zimmermann & Steudle, 1978). A relationship between acid-secretion, increase 461 0022-5193/88/080461 + 15 $03.00/0

~) 1988 Academic Press Limited

462

C. B R l l ~ R E

AND

B. G O O D W I N

in wall extensibility and cell expansion has also been proposed (Tepfer & Cleland, 1974). In Chara and Nitella, a direct correlation between external acidification and localized cell growth was observed (Lucas & Smith, 1973; Taiz et al., 1981). These observations suggest a causal relationship between local mechanical strain and proton pump activation. But other mechanisms of wall loosening, pressure-dependent or not, may be involved in the process, such as hydrolase secretion. Research on Acetabularia and selected systems indicated that two other important variables are implicated in localized tip growth, namely electrical currents and calcium localization. Studies of different plant species have shown that growth occurs at the point of greatest density of a steady inward-directed current (Jaffe, 1981) and that one of the ions carrying the current is calcium. In Acetabularia, a current flux into the rhizoid and out o f the stalk, a small component of which consists of calcium ions, has been measured with a vibrating probe (O'Shea et al., 1988). The occurrence of increased levels of calcium at the tip of the growing lily pollen tube has been demonstrated (Reiss & Herth, 1979), while Cotton and Vanden Driessche (1987) have made similar observations on the regenerating tip of Acetabularia. These observations all suggest a connection between mechanical, electrical and ionic variables (primarily calcium and protons) in the control of cellular growth and morphogenesis. This hypothesis led Goodwin & Trainor (1985) to derive field equations describing the state of the cytoskeleton in the cortical cytoplasm and the kinetics of cytosolic free calcium, with which it interacts. Later on, the model was enhanced by adding a field equation for the cell wall, to account for pressure effects and mechanical strain-induced wall loosening. Numerical investigations of this model have been carried out and have shown, in a simulation of a one-dimensional version of the model, that growth of a periodic mode does indeed occur for parameter values satisfying the bifurcation conditions (Goodwin & Brirre, 1987). The present paper deals with the study of axially symmetric solutions of this model on two-dimensional surfaces. The question of describing the cell wall expansion in a proper manner will also be discussed. But first, we present a few facts about the morphogenesis of Acetabularia mediterranea. 2. The Morphogenesis of Acetabularia mediterranea Acetabularia mediterranea is a unicellular green alga. At the adult stage, this giant cell consists of a basal foot with a few rhizoids, a stalk about 3 cm long and 0.5-1 mm wide and an apical cap (Fig. 1). Before cap formation, one or several whorls may form at the apical growing end of the stalk. These whorls are radial structures, composed of fast growing branching hairs, which later die leaving a scar on the stalk (for a comprehensive descript.ion, see Puiseux-Dao, 1970). When the stalk is cut, the basal part which contains the nucleus is able to regenerate and form a new cap. The sequence of morphological events which occurs during regeneration can be summarized as follows (Fig. 2): a few hours after cap removal, a new thin apical wall has been formed and some turgor recovered. The apical dome is more or less hemispherical. One or two days later, a small growing tip appears

TIP MORPHOGENESIS

IN A C E T A B U L A R I A

463

FIG. 1. Mature Acetabularia mediterranea.

NEW WALL

OLD CELL WALL

FIG. 2. Representation of the morphogenetic changes taking place in the regenerative tip of Acetabularia.

in the polar region and elongates, forming a thin new stalk (100-200 ixm wide). Then the growth stops, the tip flattens and hair primordia appear which grow rapidly and branch. From the centre of the whorl a new tip arises and axial growth resumes. This process may be repeated a number of times, giving a sequence of whorls. The final stage of regeneration is the formation o f a cap primordium, a structure which is considerably more complex than a whorl of hairs but has the same radial symmetry. This regeneration process is strongly dependent on cations such as calcium and magnesium (Goodwin et al., 1983). The plant can produce a normal regenerative sequence only when external calcium concentration is in a specific range. In a whorl, the average distance between hair primordia (wave-length), which is fairly constant for given culture conditions, is a decreasing function of the medium calcium

464

C. B R I I ~ R E A N D

B. G O O D W I N

concentration in both growing (Harrison & Hillier, 1985) and regenerating plants (Goodwin et al., 1987). These results strengthen the idea that calcium is one of the main controlling factors o f plant morphogenesis. 3. Model Formulation (A). VISCOELASTIC P R O P E R T I E S

The first equation describes the displacements of the cytogel (cortical cytoplasm containing the cytoskeleton) near an equilibrium state• The variables to consider are the displacement ~ from this equilibrium and the calcium concentration X (in this theory the cytogel is represented as a continuum). From the displacement field, we can calculate at any point the strain tensor e and the strain rate tensor ~, which account for the elastic deformation and for the viscous motion, respectively• The stress tensor o-, whose elements are the components of the forces per unit area exerted in the three principal space directions, depends on the strain and strain-rate tensors and on the mechanical properties of the cytogel. Since these properties depend on the local calcium concentration, we can write at any point M: ~r(M)

=

cr(e(M), # ( M ) , x ( M ) )

A linear expansion of cr about the equilibrium stress tensor O'o gives: o = O'o+ S. e + A .

(1)

~.

In general, A and S are fourth order tensors (calcium dependent). But for an isotropic material (which we assume for the cytogel), the 81 components o f S can be expressed in terms of just two elastic moduli, the Lame coefficients h and p,; similarly, in the theory of simple liquids, the 81 components of A can be expressed in terms of two coefficients, the shear viscosity ~" and the bulk viscosity, r/. Consider now a very small unit element of cortical material, with a volume density p; it accelerates according to Newton's second law in response to the various elastic, viscous and mechanical forces acting on it such that d2~j div tr + f = p-~7~

(2)

where div o- accounts for the elastic and viscous forces and f describes the external forces. From (1) we have divcr = div O'o+ div (S. e ) + d i v (A. ~).

(3)

But, since tro depends on the calcium concentration OtYo

div O'o= -

c)X

• Vh'

then •

•

aX

. e + - - . #

aX

.Vx.

(4)

TIP MORPHOGENESIS IN ACETABULARIA

465

Replacing S and A by the elastic moduli and viscosity coefficients and expressing e and ~ in terms of the displacement and velocity vectors ~ and ~, we finally get the Goodwin and Trainor equation for viscoelasticity d2~

- F. Vx - R ~ + second order terms

(5)

where F=

- act----2°

(5.1)

aX" F is a calcium-dependent second order tensor and the external force term Re stands, in a linear approximation, for the restoring forces due to structural components (e.g. microtubules) which resist local displacement o f the gel. It can be assumed (Goodwin & Trainor, 1985) that the contribution of the acceleration term is negligible, so that the left-hand side of (5) is zero. (B) CALCIUM KINETICS

The Goodwin and Trainor calcium equation describes the simplest aspect of calcium kinetics. These authors assume a reaction with a stoichiometry n between calcium ions and a macromolecule C. k~

C + nCa 2+ .

" C* k- I

where C* represents the complex of n calcium ions bound to the macromolecule C. Assuming that the total concentrations o f the binding macromolecules and o f calcium are constant, a straightforward derivation leads to the kinetic equation.

dx= k,( K dt

- X ) - k,(/3 + X)X"

(6)

where X represents the concentration of free calcium, and/3, K are constants. (NB: /3 + K = n C (total concentration of calcium-binding macromolecules).) In order to take account of stretching or compression effects on calcium release or calcium binding, the rate constant k_t was assumed to be a function of strain. Expanding this function to first order this gives k_x = a + au 01~i. ax~

(6.1)

Finally, the Goodwin and Trainor equation for calcium kinetics is: -OX -=

at

(

a

+ava~i~(K_x)_k,(/3+X)X.+DV2x ax/

(7)

where a term for the diffusion o f calcium, with diffusion coefficient D, has been added.

466

c. BRII~RE AND

B. G O O D W I N

(C) C E L L WALL D E F O R M A T I O N S

The cell wall is generally considered as a visco-elastic material which can be described by a generalized Maxwell-Voigt model for visco-elastic solids (Fujihara et al., 1978). This leads, in the theory of viscoelasticity, to quite complex stress-strain constitutive relations. But in the present study, we are primarily interested in investigating the solutions of the G - T equations when the shape of the wall and the associated strain field are given. That is we can consider the cell wall as being at equilibrium and neglect the effects of its viscosity. (This is the case just before the beginning of regeneration, when there are no morphological regeneration processes is the cytogel.) We then assume that the tip cell wall is a thin shell (like a dome) with linear elastic properties. It bears the load of pressure, Pi on its internal face and on its external face it bears a pressure, Pe, with P = Pi - Pe > 0. Body forces are neglected. The elastic equilibrium equation for the cell wall is then div trw = 0

(8)

where trw represents the wall stress tensor. We assume a linear stress-strain relationship. But in order to account for a possible anisotropy of the cell wall, we considered in the computations two different kinds of elastic moduli as they are measured along the meridian (longitudinal elasticity) or along a parallel line (transverse elasticity). The basal border of the tip wall is assumed to be fixed to the old rigid wall. Hence the boundary conditions are trw. N = PiN on the internal face trw. N = - P e N on the external face

(9)

ew = 0 on the border. where N is the outward vector to the shell. Connection between the wall state and the cytogel state is effected in two ways: - - T h e cytogel is held against the cell wall by vacuolar pressure on the plasmalemma; this implies that gel displacements are tangential to the wall. Furthermore, the wall strain field provides an initial strain field for the cytogel. ~ T h e elasticity coefficients of the cell wall are functions of the cytogel strain field. This is to account for the possible correlation between wall loosening and mechanical strain, via proton pump activation or some other mechanism. The complete model consists of the three equations (5), (7) and (8), with the boundary conditions (9) for the cell and, at the border, Neuman's conditions (no flux) for calcium and no displacement for the cytogel. These equations are solved for displacements of the wall and o f the cytogel, and for calcium concentration.

4. Axially Symmetric Solutions of the Goodwin-Trainor Equations A numerical method, based on a modified finite element method, has been developed for solving equations (5), (7) and (8) on thin shells (Goodwin & Brirre,

TIP MORPHOGENESIS IN A C E T A B U L A R I A

467

1987). The wall is described by a network of finite elements, each o f which is assumed to obey, at equilibrium, equation (8). For the cytogel, equations (5) and (7) are solved, in similar ways, on a shell assuming the same shape as the last c o m p u t e d wall, the strain field of which is used as initial conditions. This process is iterated for a n u m b e r of time steps. When the wall is fixed after initial stretching, the cytogel system evolves generally toward a steady-state. For particular values of the parameters, non-stationary solutions m a y be observed. We do not consider this case here, restricting our attention to axially symmetric solutions o f the G - T equations on hemispherical or semi-ellipsoidal shells. This simplifies greatly the numerical computations, since the equations have to be solved only along a meridian line. In order to carry out the computations, we first have to provide suitable functions for the calcium-dependent parameters and for the cell wall elasticity moduli.

(A) PARAMETER FUNCTIONS

The parameter functions were chosen according to the forms suggested by Goodwin & Trainor (1985). But, as we were interested in equilibrium and not transient solutions, the viscosity effects were neglected. (i) Gel elasticity Instead o f Lam6 coefficients, we used Young's modulus, E, and the Poisson ratio. Trial simulations showed that there was no qualitative difference in the solutions whether the Poisson ratio was zero or positive, so for computational simplicity a zero value was used. Young's modulus depends on calcium concentration according to Fig. 3 and equation (10), in order to take into account the solation and contraction

1.6 1.2 t~ 0.8 0-4

,

0

I

.

t

4

.

i

,

i

,

i

.

8

i

12

,

i

,

i

.

i

16

Calcium FiG. 3. Variation of Young's modulus of elasticity with calcium concentration in the cytogel (arbitrary units). Qualitative estimation only.

468

c. BRII~RE A N D B. G O O D W I N

effects of calcium on the cytogel (see Goodwin & Trainor, 1985): E(x)=e*+e2 = e~-~

e5 es -1) q (X_eo)2+e3 (x-e4)2+e6 ' e5

X > eo X
(eo - e4) 2 + e6'

(10)

Since in this case Cro= Ee, the parameter F is given by OE F= ---e. Ox

(11)

(ii) Restoring force Restoring force depends on calcium concentration according to G(x)=ag+bg (

cg - 1 ) . X2 + cg

(12)

(iii) Wall elasticity Young's modulus of elasticity Ew, at a given point of the wall and in a given direction (if we consider an anisotropic wall material), is a non-linear function of the strain, s, of the cytogel in that direction (Fig. 4). The cytogel strain is assumed to induce a rapid wall-softening when it passes above a given threshold: Ew(s) = aw exp (-bw(s - ~w)2) + d.,,

= a.,+dw,

s > c.~ S
(13)

I00 8O

L2 6o 4O

20

0

0-005 0.010

0-015

0.020 0.025 0.050 0.0:$5 Strain

FIG. 4. Variation of the cell wall elasticity modulus (arbitrary units) with cytogel strain. Qualitative estimation only.

TIP

MORPHOGENESIS

IN

ACETABULARIA

469

(B) R E S U L T S

In the following results, the values of the pressure difference P ( = P i - P e ) and of Ew were such that the maximum strain of the wall was of the order of 0.02. (i) Wall strain field The field of the wall was computed, on semi-ellipsoidal domes, for various values of the eccentricity, measured as the ratio between the pole elevation and the radius of the base (Fig. 5). For oblate ellipsoids, the strain is maximum at the pole. Near the base, transversal strain can take negative values due to a compression effect. On elongated ellipsoids, the strain gradient is reversed, with a minimum strain at the pole and a maximum strain near the border. The strain field is nearly uniform on a sphere (the decrease in transversal strain near the base is due to the boundary conditions which impose no displacement on the border). These results are in accord with what is to be expected from thin shell theory: for such shells, with no bending, the strain is an inverse function of the curvature (see Novozhilov, 1964). This relation between the strain and the curvature could play an important role in the spatial distribution of growth activity, as we shall see in the following. (ii) Strain field and calcium gradient in the cytogel Solutions of equations (5) and (7) were computed for various shapes of the wall, the wall strain field being used as initial cytogel strain field. The coefficients of the parameter functions were chosen such that the bifurcation condition (which states that F must be high enough; see Goodwin & Trainor; 1985) was satisfied. In this case, the trivial solution for the cytogel (no displacement, homogeneous calcium concentration), with a uniform wall strain field, is unstable and the system can bifurcate toward a non-uniform state. The results are presented in Fig. 5. We can see that the gel strain field and the corresponding calcium gradient are strongly dependent on the wall strain. For elongated ellipsoids, maximum strain and maximum calcium concentration occur near the base. It can be observed that longitudinal strains are more affected by the gel system bifurcation than the transversal strains which remain close to their values in the wall. For oblate semi-spheroids, the calcium gradient is reversed. In the gel, transversal strains are still similar to the wall strains, but the longitudinal strains, although minimum near the base, may be quite different from those in the wall. Their maximum values are not necessarily localized at the pole. Depending on the values o f the parameters, longitudinal strains may have a maximum in the middle region or closer to the pole. This influence o f parameters such as the diffusion coefficient or the pressure on the form of the strain field is shown in Fig. 6 in the case of hemispherical shell. High values of the diffusion coefficient are required for a monotonic gradient in strain and calcium concentration, with a maximum at the pole. For lower values o f D this maximum lies in the middle region or even nearer the border. When the pressure is low, strain is small and so F is small (see equation

470

C.

BRII~RE

AND

B. GOODWIN (o)

Wall strain

0.03

Shape

CyJogel strain 0.04

0-0~

I o-0, 0

~ ~ i

i

120 j

4

0-02 ~ , i -.~

t

Elosticily

Calcium

j

8 80

'

~

'

40,

0-2

0-4

0-6 Radius

0"8

I

0

0.4

O-B

I "2 0 0-4 Distance from the pole

O'B

I-2

(b) 0.03

Shape

Cytocjel strain

WOII strain

).OZ ),

0"02

0

,~-J: . . . . . .

\

120

0,2

0.4

0"6 Rodius

0'8

0

Elasticity

0'4

0'8

Calcium

1"2

0

0"4

0"8

I "2

Distance from the pole (c)

Shape

O[

120

~';'"'"

Elasticity

Calcium

80 40

0"2

0.4

0-6

Radius

0" 8

0

4

0-4 0"8 I-2

1.6

0

0.4 0.8

1.2 1 6

Distance from the pole

FIG. 5. N u m e r i c a l s i m u l a t i o n s o f the G - T e q u a t i o n s for v a r i o u s s h a p e s o f the cell wall ( a x i a l l y s y m m e t r i c case). P a r a m e t e r v a l u e s (arbitrary units): Pressure P = 2 , a,,. = 100, e o = 4 , e t = e 2 = 1, es = 0.01, a = k t = 0.0001, a~t = 0.01, K = 6 0 0 0 , / 3 = 1000, D = 0.1, ag = 0-1. - l o n g i t u d i n a l variables; - - transverse variables. ( D i s t a n c e s from the p o l e are m e a s u r e d a l o n g a m e r i d i a n ; elasticity refers to the Y o u n g ' s m o d u l u s o f the wall m e a s u r e d a l o n g a line o f l o n g i t u d e or o f latitude.)

TIP MORPHOGENESIS

471

IN ACETABULARIA (a)

Shape

Wall

Cytogel strain

strain

0-0Z O,

0.0 I

i

I

I

i

I

I

'. i

I

Elasticity

120

I

i

~-.-~

.,%

Calcium

8O 4O

0

0.2 0-4 0.6 0-8

0.4

O'B

1.2

0

Distance

Rodius

from

0.4 0'8 pole

the

I "2

(b) Wall strain

0.03 F

Shape

Cytogel strain 0.04.

I 0-G2

0-01 0

\

-"-. I

I

I

P

I

I

I

0.02" , . ~ , . ~ *,

0

-:1

Elasticity

120 ¸

Calcium 8-

80" 40 • i

L

0-2 0 4 o'~ o~s

'~

0

0-4

0-8

1.2

Radius

Shape

0

Distance

from

0-4 0-8 pole

the

:I °I

0"03

Wall

I-Z

(c)

strain

Cytocjel

strain

0"02 o-

.

i

i

120

i

I

i

i

o.oz

i

',

-----.~. .

,

O[ ......

Elasticity

~. , -

Calcium

40

0.2 0.4 0"6 0.8 Radius

0

0"4

O-B

I-2 Distance

0 from

the

0~4 0.8 pole

I-2

FIG. 6. Numerical solutions of the G - T equations on a hemispherical shell for various values of the pressure P and of the diffusion coefficient D. Parameter values (arbitrary units): (a) P = 1.5, D - - 0 . 1 ; (b) P = 2.4, D = 0.1; (c) P = 2, D = 1 (to be compared with Fig. 5. ( ) longitudinal variables; (- - -) transverse variables.

472

C. B R I I ~ R E

AND

B. G O O D W I N

(12)) and no bifurcation can occur. On the other hand, when the pressure is too high, no bifurcation can occur because of the positive slope of the E-curve for high values of calcium (see Fig. 3), which occur with elevated cytogel strain (Fig. 6b). In fact bifurcation can occur only when the wall strain lies in a given range, which depends on the other parameter values. From a biological point of view, these results show that growth initiation and growth localization are certainly strongly dependent on the shape o f the apical dome. Polar regeneration seems unlikely to occur on elongated ellipsoids: an oblate shape should be more efficient. On the contrary, tip flattening could be initiated from an elongated dome, with the maximum of strain and calcium concentration lying in a sub-polar ring. It is also demonstrated that turgor pressure is a bifurcation parameter for this model, as the occurrence of a strong calcium concentration and strain at the pole depends, in part, on its value. Therefore, it can be imagined that apical regeneration cannot occur while the turgor pressure remains below the bifurcation threshold, but is triggered as soon as the pressure rises above it, allowing calcium and strain gradients to arise. In the above results, the wall deformation was assumed to be maintained constant after the initial stretching. No growth process was involved. In the following, we investigate the behaviour of the model when the wall elasticity varies with the cytogel strain and when plastic deformations are allowed. 5. Growth Simulation

The growth of an element of cell wall can be defined as the process by which its initial size is increased regardless of any elastic deformation. Let Lo be the length of a small element o f cell wall and L its length when it is stretched under turgot pressure effect. We will say that growth has occurred if its initial length Lo has increased after release of pressure. This definition is, o f course, only legitimate assuming that all the conditions required for wall synthesis are fulfilled. We do not consider that growth is just a passive plastic deformation. But it is reasonable to think that growth begins with a slippage of microfibril layers. Therefore we can consider two kinds of wall deformations: a purely elastic deformation due to the elastic properties of cellulose microfibrils, without rupture of bonds, and a plastic one which results from breaking of specific bonds between different layers, allowing a slippage of these layers. The approach we used for modelling these processes is based on the following points: (a) the variation of the wall elasticity with the cytogel strain; (b) the increase of the rest length of a wall element when the wall strain is too high. Point (a) is justified by the fact that growth is initiated by a local wall softening. This softening, which corresponds to a decrease of the elasticity moduli of the cell wall, could be controlled by the cytogel, via strain-dependent proton pumps. Point (b) supposes that the wall softening and the corresponding increase in wall strain (under the effect of turgot pressure) result in a breakage of specific bonds which allows layers o f microfibriis to slip. It is generally accepted that elementary

MORPHOGENESIS

TIP

IN

473

ACETABULARIA

growth rate is proportional to the difference b e t w e e n the turgor pressure and a threshold value, characteristic o f the m e c h a n i c a l properties o f the wall material. The important fact is that there is a pressure threshold under w h i c h no growth can occur ( R a y et al., 1972). For c o l l e n c h y m a cells, the wall d e f o r m a t i o n is elastic for a pressure l o w e r than about 5 x 105 Pa and plastic for higher values ( R o l a n d & Pilet, 1974). T h e s e e x p e r i m e n t a l results support the h y p o t h e s i s o f a m i n i m u m wall strain under w h i c h there is no plastic d e f o r m a t i o n o f the wall. A c c o r d i n g to this, the rest length Lo o f a small e l e m e n t o f cell wall varies with the wall strain s as dLo dt

- a L o ( s - so)

s > So

=0

S
w h e r e So is a strain threshold and a is a growth coefficient. The cell wall elasticity varies with the cytogel strain a c c o r d i n g to the function defined in Section 3. Figure 7 s h o w s the first steps o f a s i m u l a t i o n o f apical growth for a h e m i s p h e r i c a l d o m e . The w h o l e m o d e l (equations (5), (7), (9), (11), (13) and (14)) w a s used. This first positive result was o b t a i n e d for specific values o f the parameters. For another set o f parameter values, the result c o u l d be quite different: overall e x p o n e n t i a l growth or, on the contrary, very s l o w and limited growth, for e x a m p l e . It appears that the c h a n g e o f f o r m o f the wall, w h i c h modifies the strain field, has a strong influence on the result. T h e range o f parameter values where "realistic" growth can occur, r e m a i n s to be e x p l o r e d .

Shape

Wall strain

0.02 "

O. t [

Cytogel strain

0-81 0.01 •

".-I

0

\

\

120

'\, \

',\

t

f

I

I

I

I

I

Elasticity I

"

'~

0,6 i 0,4 i I

Calcium

8 ~, "\'-, \'~

80"

40"

r[ t

//

i 0.2

0.4

0-6

Radius

0.8

I

0.4

0"8

1,2

1.6

0

0,4

0.8

1.2.

1.6

Distance from the pole

FIG. 7. Wall shape, strain fields and calcium gradient in a growing (initially hemispherical) shell. Parameter values (arbitrary units): P = 2, a,. = 100, b,. = 0.02, d,. = 10, a = l, so = 0' 1, ag = 0.1, e0 = 4, e t = e2 = 1, e3 = 0.01, a = k I = 0-0001, b = 0.01, K = 6000,/3 = 1000, D = 0.1.

474

c. BRll~RE AND B. GOODWIN 6. Conclusion

The present model can simulate the first stages of apical regeneration o f A c e t a b u l a r i a cells. In a range of parameter values, solutions of the G o o d w i n and

Trainor equations can be obtained which provide expected cytogel strain fields and calcium gradients necessary for growth initiation. By adding to the initial model the growth equation (14) and the cytogel strain dependence of the wall elasticity (13), effective growth can be simulated. However, several difficulties still remain. Depending on the value of the growth coefficient, wall expansion either stops quickly or, on the contrary, extends to the whole apical dome, leading to a bursting growth. In these two cases, the changes of wall shape play an important role in the determination of its behaviour. As noted before, in a thin elastic shell, the strain field is dependent on the curvature. Polar growth implies: (i) a decrease of the curvature and then an increase in stress and strain in the sub-polar zone; and (ii) an increase of the curvature at the pole, with a lower strain and stress. These two p h e n o m e n a could explain either the arrest o f polar expansion (due to the decrease of tension at the pole) or the spreading of the growing zone (due to increase of tension in the sub-polar area). This means that in real plants regulations exist which overcome the negative effects o f the changes in curvature and keep the expansion localized near the pole to produce an axial growth. Such control processes could have various origins: good candidates might be changes in wall structure with time and calcium exchanges between the cytogel and the external medium. It is well known that the cell wall can be d e c o m p o s e d into a primary wall, with randomly distributed microfibril layers, and a secondary wall (formed later) with a multinet structure. In the polar growing zone, the wall has a primary structure which can be assumed as homogeneous, but in the subpolar zone, the wall is progressively thickened and its mechanical resistance increased. This wall thickening could then be a possible factor in control o f logitudinal growth. The other factor to consider is the influx/effiux of calcium in the zone where growth occurs. In the present model, these fluxes of calcium have been neglected. They were considered as second order processes which should not have an effect on the initial cytogel pre-pattern and so could not enter in a first-order model. But in effective growth, such second-order processes could play an important role and therefore have to be integrated in a complete model of cellular growth. Part of this work was done during a one year fellowship spent by C.B. at the Department of Biology of the Open University as part of the exchange programme between the Royal Society of London and the Centre National de la Recherche Scientifique. REFERENCES COTTON, G. & VANDEN DRIESSCHE, T. (1987). J. Cell Science 87, 337. FUJIHARA, S., YAMAMOTO,R. & MASUDA, Y. (1978). Biorheology 15, 63. GOODWIN, B. C. & PATEROMICHELAKIS,S. (1979). Planta 145, 427. GOODWlN, B. C., SKELTON,J. L. & KIRK-BELL,S. M. (1983). Planta 157, 1. GOODWlN, B. C. & TRAINOR, L. E. H. (1985). J. theor. Biol. 117, 79.

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