Spatial variations of growth within domes having different patterns of principal growth directions

Growth rate variations for two paraboloidal domes: A and B, identical when seen from the outside but differing in the internal pattern of principal growth directions, were modeled by means of the growth tensor and a natural coordinate system. In dome A periclinal trajectories in the axial plane were given by confocal parabolas (as in a tunical dome), in dome B by parabolas converging to the vertex (as in a dome without a tunica). Accordingly, two natural coordinate systems, namely paraboloidal for A and convergent parabolic for B, were used. In both cases, the rate of growth in area on the surfaces of domes was assumed to be isotropic and identical in corresponding points. It appears that distributions of growth rates within domes A and B are similar in their peripheral and central parts and different only in their distal regions. In the latter, growth rates are relatively large; the maximum relative rate of growth in volume is around the geometric focus in dome A, and on the surface around the vertex in dome B.


INTRODUCTION
Principal directions of growth (PDG) can be determined in symplastically growing plant organs (Hejnowicz and Romberger 1984). They are indicated by eigenvectors of symmetric part of the growth tensor. We can associate these directions with each point within the organ, thus there is a pattern of PDG trajectories. Trajectories of PDG are mutually orthogonal (Hejnowicz 1984). In practice, their pattern can be recognized from the cell-wall network (under the condition that the growth is not isotropic) because the line elements oriented along PDG and thus mutually ortho gonal, preserve orthogonality during growth. The curvilinear coordinate system, for which coordinate lines are tangent at any point to the trajectories of PDG is called a natural coordinate system (Hejnowicz 1984). This system is the most appropriate for dynamic descriptions and analysis of the growing organ.
In an apical dome there are three PDG: periclinal, anticlinal and lati tudinal (Fig. la). How do the trajectories of these directions run, if the dome maintains a steady-state shape during growth? Since PDG appear in the cell-wall pattern, the periclines and anticlines can be drawn from these patterns but sometimes they are not well visible in some regions of the organ. From the inspection of the surface layer of cells in the dome it appears that at a given point on the surface, the anticlinal trajectory is normal, and that the two remaining are tangent to the surface. If the dome is a figure of revolution around the axis, and if the tip of the dome does not rotate during growth, the periclinal trajectory on the surface is repre sented by the dome profile, whereas the latitudinal trajectories are always circular. Such a pattern certainly occurs at the surface, but what happens inside? If the dome grows symplastically then the trajectories of PDG must be continuous, and the pattern of trajectories, known at the surface, can be extrapolated into the dome interior (Hejnowicz 1984), However, we can obtain two different patterns in this way, with two types of periclinal trajectories: Aconverging to the segment on the dome axis (as the periclines in the axial plane of the tunical dome), Bconverging to one point at the vertex (as in the dome without a tunica). We denote these cases as A and B, respectively. It can be expected that domes A and B with the two different patterns of PDG have different variations of growth rates inside. The question is: how different are the distributions of growth rates within domes A and B, if the domes are identical from the outside, i.e., they have the same shape, size, and the same rate of growth in area on the dome surface? In the present paper an attempt is made to answer this question by relating it to a paraboloidal-dome-shaped meristem.

THE NATURAL COORDINATE SYSTEMS FOR A PARABOLOIDAL DOME
Let us consider a paraboloidal dome of the apex, assuming that its shape is a steady-state during growth. The outline of the axial longitudinal section of the dome can be described by x2 = 2pz, where p is the para meter of the parabola (Fig. la). Two natural coordinate systems can be The base of the dome is on the level indicated by the asterisk proposed for this dome, one is paraboloidal coordinate system (A), the other, a convergent parabolic system (B) (see Fig. lb, c). The first represents pat tern A, the second, pattern B of PDG in the axial plane. The dome for which the paraboloidal system is natural, is denoted by A, the second is best described by a convergent parabolic system, B Paraboloidal coordinate system This system was already used in previous models (Hejnowicz et al. 1984a, b). The traces of the coordinate surfaces in the axial plane x, z are confocal parabolas (Fig lb). They represent periclinal and anticlinal trajec tories, while the latitudinal trajectories are circles (they are not marked) because the z is a symmetry axis. The equations defining transformation between rectangular and paraboloidal coordinates, and the scale factors (Spiegel 1959) are as follows: where u > 0, v 0, 0 < <p < 2n, hH= h" = y/u2 + v2, hv = uv.
For <p = 0, we have: u = y/ y/x2 + z2 +z and v = yjyjx2 Fz2 -z. The sur face of dome A is represented by one of the surfaces v. Let us denote it as vs. The focus F divides the axis of the dome into two parts. The dimension of the upper part from F to the tip increases if the dome becomes wider. The relation between v, and p for the parabola which represents the surface of the dome is v, = y/p.

Convergent parabolic system
The system was constructed especially for the modeling of dome growth. The traces of the coordinate surfaces on the x, z plane are shown in Fig. lc. They are parabolas converging to the origin (perclinal trajectories) and ellipses (anticlinal trajectories). The latitudinal trajectories are circles, as previously. The equations for the transformation and scale factors are: For</> = 0 and z^O, we have: « = ?/-x2 + z2, v = The tip of dome B is represented by the origin of the coordinate system, the surface of the dome is represented by the surface v, and, as previously, we have v,= y/p.

GROWTH TENSOR IN ANY COORDINATE SYSTEM WITH ROTATIONAL SYMMETRY
The growth tensor was defined as the covariant derivative of the field V, of displacement velocities of material points in the organ (Hejnowicz and Romberger 1984). This tensor expressed in physical components allows full characterization of the growing organ. Among other things, in a natural coordinate system, diagonal elements of the matrix of the growth tensor represent principal growth rates, i.e., relative elemental rate of growth in length. RERG,. in PDG. The sum of these elements (in the physical compo nents) gives the relative elemental rate of growth in volume, RERGvol.
The growth tensor can be obtained directly in physical components from the dyadic VK (Hejnowicz and Romberger 1984). This method will be used to calculate the general form of the matrix of physical components of the growth tensor for any curvilinear coordinate system with rotational symmetry.
Let us. consider the system (u, v, <p) with rotational symmetry (as it is in paraboloidal and convergent parabolic systems). In this system e", ev, e9 represent the unit base vector and vector V is given by the components: K, K>-Because the differential operator in each curvilinear system has the form (Spiegel 1959): eu 6 t ev 8 9 ( Vu eu + Vv e" + e") = V = -+____ hu du h" dv hv d<p ' where h", h", hv are the scale factors, hence dyadic VK can be written as: e''+Ę'87(Kl,<?*')e''+Tp~Sv^e*)e"+ +TT V(+^7 s7(e,,) + a7(e") • First the components are differentiated partially, then all terms are grouped with respect to the so-called unit dyadics, e, e,-, for i,j = u,v, tp (Spiegel 1959). Thus the sum is obtained which can be expressed as: where Tkj for i,j = u,v,tp are the components of the dyadic. An array of dyadic components, in the form of a 3 by 3 matrix, can be written. This matrix is identical with the sought matrix of the growth physical components. It has the following form: As was mentioned before, the principal growth rates are given by diagonal elements in (3). The element in the upper left hand corner, Tuu, represents RERGt in the periclinal direction (RERG((per)), the element in the bottom right hand corner, Tw, represents RERGt in the latitudinal direction (RERGl(lat)), the element in the center, 7^.,-represents RERG, in the anticlinal direction (RERGl{an)). Specification of the matrix depends on determining the scale factors and the components of the vector field V.

HELD V FOR THE ISOTROPIC SURFACE GROWTH, SPECIFICATION OF THE GROWTH TENSOR (PHYSICAL COMPONENTS)
The feature of isotropic RERG in area on the dome surface (we will call it an isotropic surface growth) means that for each point on the surface, RERGt is the same in any direction in the plane tangent to the surface at this point. It is possible, however, that the values of RERGt for the points differing in the distance from the vertex, are different. We know that in the plane tangent to the surface, two PDG exist, namely periclinal and latitudinal, therefore, RERGuper} must be equal to RERGl{la,^ for v = vs. Denoting the components of V on the surface vs by Vu, Vv, Vv, from (3) the following condition is obtained: where h", hv, are the scale factor of the system for v = t>s. For domes A and B. Kt. = F0 = 0 and does not depend on <p because their growth is steady and without rotation, thus there remains: After the integration (5) with respect to u (for <p = const.), PM, i.e., Vu on the surface of the dome, can be obtained. The field Vu for the whole dome, h from the relation Vu (u, v) = Vu (u, vs) (Hejnowicz 1984), one can calcuhu late. By this means, we will determine Vu, and then the growth tensor for both domes A and B Dome A: The variant of isotropic surface growth in a paraboloidal coordinate system was already considered in the previous paper (Hejnowicz et al. 1984b). The following expression for Vu was obtained there: where cI is constant. Accordingly, the growth tensor (3) (physical com ponents) for dome A is: Dome B: For a convergent parabolic system, condition (5) is the following: where in = -j x/4u4 + t>4. Upon integration Pu = c2v/m-l, where c2 is the integration constant. Introducing scaling factor we obtain the displacement velocity for all points in the dome: where w = ~ x/4u4 + r4 and in is the same as previously. The specific form where m and in are as previously, and

m2 (in -1)-in2 (m-1) m(/h+l) H = -------^7=----------------------------1 ----------------. mm(w+l) m(m+l)
It is worth noting that conditions (4) and (5) say nothing about the values of RERG, at different points on the surface vs. In order to have equal values of RERG, in corresponding points on the surfaces of both domes, it must be assumed that velocity (6) is equal to velocity (8) at corresponding points of both domes, for instance at vertices and on base levels. As can be seen, this is satisfied at vertices because Vu -0 there for u = 0 and v = vs. Let us consider the second point. Denoting by u£ and u" the coordinates for which the surfaces of domes A and B are crossed by the base level, from (6) and (8) there is: Let us assume that vs for both domes is vs= 3 and the coordinates uh are as follows: u'b' = 6 in the paraboloidal system and u" as 4.69 in the convergent parabolic system. Thus, for c, = 1 from (10), there must be c2 = 3. For these constants, the velocities Vu on the surfaces of domes A and B are identical in corresponding points, in consequence, Tuu and Tw in (7) are equal to Tuu and in (9). Thus the fields V and the growth tensors in physical components for domes A and B are fully specified.

DISTRIBUTION OF GROWTH RATES
RERG, in different directions and RERGvol for domes A and B, were calculated from the specified growth tensors. The results are shown in the form of computer-made maps in Figs. 2, 3 and 4. In Fig. 2a, b growth rates in the planes PL (see Fig. la), are given. The maps of RERG, for domes A and B are similar there, except for a small region near the focus in the dome A. On the surfaces of both domes there is an isotropy of the relative elemental rate of growth in area. The plots of RERG, around the points on the surface are circles, but values of RERG, decrease with increasing distance from the vertex. In corresponding points on the surfaces of both domes the RERG,'s are equal, as was assumed. Below the surface, within the dome interior, RERG, in the PL planes becomes anisotropic, RERG"per, becomes higher than RERG",al,. Maximum of RERG"per, is in the distal part of both domes, for dome A particularly at the geometric focus.
In axial plane PA (see Fig. la), growth rates for both domes are similar at the base level only (Fig. 3a, b). On the axis of each dome, RERG"per, is more or less twice as large as RERG"ant), but on the way from the axis to the surface RERGl(anl) decreases in dome A, whereas it increases in dome B. In the latter there is the maximum of RERGllanl} in the distal part of the dome. This maximum occurs on the surface around the vertex. In dome A in the segment of the dome axis between the focus and the tip, RERGl(anl} disappears, whereas in dome B in the same region there is RERGHanl}> 0. Figure 4 shows variations in volumetric growth rates: RERGvnl are smallest ai the base level of domes, then they increase as the distance from the tip decreases. The maximum of RERG,.ol for dome A is in the distal part at the focus, and for dome B it is on the surface around the vertex. The area of the higher values of RERGvol, from ranges 4 and 5 in the legend for Fig. 4, is relatively small when compared to the size of the \Vhole dome. The mean RERGvol calculated for the whole dome is more or less similar for domes A and B.

DISCUSSION
The patterns of distribution of growth rates for domes A and B are different, as was expected, but it appears that the differences are not large. On one hand, PDG (principal directions of growth) are of great importance in connection with the growth variation pattern inside the dome.
Fie. 4 Computer-made maps of volumetric growth rate, RERG,"t. for domes A and B.
RERGu.i for about 120 points within each dome were taken into interpolation. The full range of values of RERGmi was divided into 5 equal parts numbered from 1 to 5 and marked by different graphical symbols, shown in the legend. The program SYMAP for the Riad 32 computer was u& d to obtain the regions on the map on the other hand, even considerable changes in the pattern of PDG trajectories do not matter much for the picture of growth of the dome as a whole. The maps presented in this paper indicate that the mean RERGvol calculated for the whole dome is similar in the cases A and B, moreover, they show that mean RERGvol calculated for the distal part only is also similar, althought in this part, the differences between A and B dome in the pattern of growth variations are the biggest. In both domes, maxima of volumetric growth rates are present in their distal parts. Such maxima are characteristic not only for paraboloidal domes, they were also found in elliptic and hyperbolic domes in the case of an isotropic surface growth on the surfaces of domes (Hejnowicz et al. 1984b). All this cases with the local maximum seem to be unrealistic when con fronted with known empirical facts. The results that have been obtained indicate that the maxima are related either to the type of coordinate system, or to the mode of the growth specified on the surface of the dome. They appear in a small region of the dome around the origin of the coordinate system. In this region, anticlinal distance between periclinal trajectories increases relatively quickly over a small area and it can give a local maximum.
For the description of growth in an organ in terms of the growth tensor, we use appropriate orthogonal coordinate systems. Appropriateit means that the coordinate lines resemble a real pattern of PDG trajectories. Often the adjustment can be done only roughly, however, knowing how different the system is from a real pattern of trajectories, one can indicate in what direction the calculated rates should be corrected to approach the real rates.
Comparing both paraboloidal and convergent parabolic systems proves that the second one is more complicated mathematically, admittedly, a lot of difficulties may be liminated by computer technique.
From the two patterns of PDG used in this paper the first was proposed for a tunical dome and the second for a dome without the tunica. The tunica is defined as a surface layer of cells in the shoot apex without periclinal divisions (Hejnowicz 1980). Hence, there is no anticlinal growth within this layer. Such a feature in relation to the apical dome is well represented in the paraboloidal coordinate system. In this system the assumption: RERGl(ant) = 0 for the segment of the dome axis between the focus and the tip, makes it possible that the segment mentioned does not increase during dome growth and, therefore, the tunica layer can be formed. The absence of the tunica can be interpreted as the decrease of the same segment to 0. Hence the proposition that a convariant parabolic system may be the natural system for the dome without a tunica growing steadily, has been made.
There is an interesting case of a potential tunica (Foster 1939) which occurs in conifers. There are some periclinal divisions in the surface layer of cells in the shoot apex, and so, in the apex there is no tunica in a strict sense. What coordinate system, paraboloidal or convergent parabolic, is more natural for this case? This question cannot be answered explicitly because both systems have some strong and some weak points. For the study of growth variations within the apices of spruce seedlings (Nakielski 1987), the convergent parabolic system was chosen.