Mathematical model for tissue stresses in growing plant cells and

In this study we propose a simple mathematical model based on the equilibrium equation for the materials de ­ formed elastically. Owing to the turgor pressure of the cells, the peripheral walls of the outer tissue are under ten ­ sion, while the extensible inner tissue is under compression. This well known properties of growing multicellular plant organs can be derived from the equation for equilibrium. The analytic solutions may serve as a good starting point for modeling the growth of a single plant cell or an organ


INTRODUCTION
The notion of growth tensors in developing plant organs has been known for a long time (see Kutschera 1989 for a review).The mathematical description for such systems has been successfully applied to apical meristems where the proliferating cells produce tissue stresses.The newly formed stresses influence the structure of the growing organ and, in particular, the principal directions of growth; also enforces the natural coordinate system which manage the further development of the organ (Hejnowicz et al. 1984;Hejnowicz 1984).A different situation one observes for elongating plant organs, such as coleoptiles in young grasses or elongation zones in roots, where cells division takes place very rarely.Various stresses occur because of different properties of cell walls in the organ growing due to the water uptake driven by the gradient in water poten tial, Kutschera 2000.In particular, walls in the outer tissu es are more rigid and thick, while inner tissues are elastic and thin walled.Thus, in an elongating plant organ, the ou ter tissue in under tension and the inner oneunder com pression (Kutschera 1989(Kutschera , 1995;;Hejnowicz and Sievers 1995).The distribution of the tissue stresses as well as de formation of the particular tissue layers can be found using one of the equilibrium equations for deformed materials.In theory of elasticity, among commonly used constitutive equations (describing mathematical relations between stress and strain, or movement of a body), equations for equilibrium play a significant role in solving the problem of finding the deformation of the bulk which undergoes an external action (Landau and Lifszyc 1993;Atkin and Fox 2005).The equilibrium equation may be derived for mate rials deformed elastically or non-elastically.In the first ca se, the displacement vector a is time-independent, in contrary to the second case.In this article we focus on the elastic properties in order to solve the equation using ana lytic methods.Such approach may be considered as a first step towards further development of the model where the numerical calculations may be unavoidable.

MODEL
Assuming no external volumetric force (like gravity) ac ting on the body, the equilibrium equation for the displace ment vector U takes on the form where V (nabla) is a differential operator, n -Poisson coefficient.Eq. (1) can be analytically solved when the problem exhibits a high degree of symmetry, i.e. cylindri cal or spherical one.A biological argument leads us to consider the cylindrical symmetry, since elongating cells or organs often have such ashape (e.g.internode cell of al gae Nitellopsis obtusa L., or coleoptiles of Graminae).Now, we wish to find an appropriate model reflecting elon gation of (a) asingle plant cell and (b) plant organ.In cylindrical coordinate system vector operators grad, div and curl become (true for arbitrary scalar or vector fields A and I , re spectively) and the problem significantly complicates.Ho wever, if we assume that the displacement field has magnitude depending only on the distance r, then Eq. ( 1) becomes (through Eq. ( 2)) (3) U

b ur =ar +r
Here we propose a model for an elongating plant cell as a hollow cylinder which can be thin or thick (in analogy to thin or thick cell wall, respectively).The cylinder is im mersed in a medium (with pressure pa), and is filled by a homogeneous fluid which plays a role of cell sap (with pressure P-Y, where Yyield threshold).Elastic properties of the cylinder (cell wall) are represented by two physical quantities: Young's modulus e and Poisson coefficient n.
For an elongating plant organ, in turn, it seems that a fil led cylinder (elastic core) covered by a thin layer of quite rigid material can be a good analytic model.The core is an analogue to the inner tissue of the organ, while the layer is an equivalent of the outer tissue.In elongating organs, the inner tissue is an assembly of cells with thin elastic walls, and the outer tissue is composed mainly of epidermal cells which have thick and rigid walls.In our model, these pro perties are reflected by different Young's moduli and Pois son coefficients (e1, n1 for the core, e2, n2 for the overlay).The question is whether such simple models can reflect stresses in the cell wall in (a) a single plant cell or apoplast in (b) plant organ during growth.This interesting problem is discussed in the last section.

SOLUTIONS
For the model of an elongating plant cell the solution is ur(r)=ar+b/r with two parameters a and b determined from the boundary conditions, which are expressed for the stress tensor {s}ij: • srr = -pa for r = Ri (i=1,2 depending on whether we are dealing with thin or thick cylinder, R2>R1) • srr = -(P-Y) for r = R0 (the inner radius is equal in both cases), see Fig. 1.Now, because in considered models we take into account only relatively small elastic deformations, the (7) For the model of an elongating plant organ the solution is more complicated:

r-»Rf r-»R|'
The latter two conditions need a comment.The assump tion that the displacement vector has to be continuous is naturalduring growth the organ shouldn't tear.Radial element of the stress tensor srr should also be continuous, because the discontinuity would affect the appearance of some force acting on the surface r = R1 causing attraction or repulsion between the core and the outlay.We believe that such situation is not the case herein the growing organ such dislocations and additional pressure in the inter face do not occur.The above listed conditions generate expressions for a1, a2 and b2 coefficients and can be found in the Appendix.
The pressure P-Y has been given as 0.3 MPa (as in typi cal elongating plant cell), pa = 0.1 MPa and p= 0.2 MPa.
In Fig. 1 we have visualized both models for elongating (a) plant cell and (b) organ.The main conclusion from the results presented in Figs 2-7 is drastic lowering of the de formation when thickening the cylinder's wall in case (a) orwhen covering the elastic core by a rigid layer in (b).The radial and angular elements of the stress tensor, srr and sff, have been presented in Figs 5-6, respectively.In Fig. 7 one can see that the presence of the outer layer causes de crease of srr within the whole cylinder, however, in the ela stic core srr remains constant, while in the outlay it increa ses up to p (left box).Different response has been obtained for the angular element sff, namely an abrupt discontinuity  at r = R1 occurs.In the elastic core sffis lowered with re spect to the uncovered "bare" core (indicated in Fig. 7 by the solid line), but in the outlay sff is much greater (right box).Both results presented in Fig. 7 we may refer to the compression in the inner tissue and tension in the outer tis sue (see Kutschera 1989Kutschera , 2000)).

DISCUSSION
The aim of the present study was to find the distribution of the tissue stresses and deformation of elongating plant cells and organs using simple mathematical formulae.The equilibrium equation for deformed bodies is undoubtedly Lewicka S. et al.  a good candidate, as it has been derived from principle laws (energy conservation law and continuity equation) and as such is universal.In systems with high degree of symmetry, the equation (for bodies deformed elastically) can be solved analytically, and solutions are easy to inter pret.Nonetheless, the question of its appropriateness still remains.For an elongating plant cell, the solution is exact, if we do not take into account other shapes of the cell than cylindrical, and treat the cell wall as a homogeneous struc ture.However, an elongating organ is a multicellular orga nism, so the model is satisfactory, if we are interesting in general and qualitative description.Such description is also simplified as we divide the organ only into two compart ments: outer and inner tissue.Both are homogeneous, but possess different mechanical properties.Further develop ment should account for the multicellularity, i.e. the more precise model should consider more complicated boundary conditions.Especially, the elements of the deformation and stress tensors depend on all spatial coordinates: r, f, and z.Moreover, wishing to find both elastic and plastic proper ties of the growing organ one should use the equilibrium equation for bodies deformed visco-elastically.It means that since vector i,' depends also on time, the problem complicates and the system may need to be solved using numerical methods.

APPENDIX
After some tedious calculations the expressions for a1, a2 and b2 read:

Fig. 2 .
Fig. 2. Radial coordinate u(r) of the displacement vector U =(wr,0,0) ob tained via theoretical calculations from the equilibrium equation.The left figure presents ur dependence on r in two cases of a hollow thin (dashed line) or thick cylinder (solid line).The right figure presents ur(r) in the fil led elastic cylinder covered by athin layer of rigid material (the model of an elongating organ).As a reference aplot of "bare" elastic organ has be en added (solid line).

Fig. 3 .
Fig. 3. Visualization (in aform of parametric 3D plot obtained for angles 0 < f< p/2) of the single cell analytic model.The figure presents the ra dial coordinate of the displacement vector |7 ; thin cell wallupper plot, thick cell wall -lower plot.

Fig. 4 .
Fig. 4. Visualization (in a form of parametric 3D plot obtained for angles 0 < f< p/2) of the elongating organ model.The figure presents the radial coordinate ur; homogeneous "bare" elastic materialupper plot, elastic material covered by athin rigid layerlower plot.

Fig. 5 .
Fig. 5.The diagonal rr element of the stress tensor {s}ij as a function of radius r and angle f -3D visualization of the model for an elongating organ.

Fig. 6 .
Fig. 6.The diagonal ff element of the stress tensor {s}ij as a function of radius r and angle f -3D visualization of the model for an elongating organ.