Turgor pressure tends to force plant cells towards a spherical form, thus separating them at the angles from adjacent cells. In cooked vegetables containing starch, the swelling pressure of starch gelatinization generates analogous cell separation forces. A theoretical analysis of the relationship between internal pressure and cell separation forces is presented. Apart from the effect of internal pressure, cell separation forces increase with the diameter of the cell and decrease with the number of cell sides. Cell separation forces are reduced by the introduction of intercellular spaces and decrease further as these expand. The relationship between intracellular pressure and cell separation forces provides a basis upon which the strength of intercellular adhesion can be measured by experiment.
An isolated, undifferentiated plant cell distended by turgor pressure would be spherical, because a sphere allows the maximum volume for a given surface area. Cells within intact plants, however, have more complex shapes. They are normally polyhedral, with flat faces and distinct angles, because of their need to remain attached to one another during the development of the plant. They may also elongate due to anisotropy in the deposition of cellulose. It was shown more than a century ago by Kelvin that the solid body approximating most closely to a sphere while maintaining contact with identical surrounding solid bodies was an isotropic tetrakaidecahedron, with eight hexagonal and six four-sided faces (Thomson 1887). Cells approximating to this form, generally with plane faces, are found in plant organs like the potato tuber where growth is nearly isotropic (Nilsson, Hertz & Falk 1958). In rapidly elongating organs, on the other hand, most cells are prismatic and their organization into concentric files leads to a cuboidal form being common. In general three cells meet along one line, called here a tricellular junction if no intercellular space is present.
The more the shape of a cell deviates from spherical, the stronger the tendency of turgor pressure to return it towards a spherical shape and so minimize its surface/volume ratio. This can only be done by separating the cell from its neighbours at the tricellular junctions. In dicotyledonous plants, adhesion between neighbouring cells is the function of the pectic polysaccharides of the middle lamella and the tricellular junctions. The pectic polymers within narrow reinforcing zones at the tricellular junctions take on a distinctive linear, low-ester structure, extensively cross-linked by calcium ions (Knox et al. 1990; Roy, Vian & Roland 1992; Rihouey et al. 1995; Goldberg et al. 1996). It is reasonable to assume that this polymer structure is adapted to resist the turgor-generated cell separation stress, at the point where that stress is concentrated (Fenwick, Jarvis & Apperley 1997).
When adhesion breaks down between cells in the living plant, it does so under strict developmental control (Knox 1992). Thus, controlled enzymatic cell separation at specific tricellular junctions (Roland 1978; Kollöffel & Linssen 1984; Jeffree, Dale & Fry 1986) generates the complex pattern of intercellular spaces that extends continuously throughout most plant organs (Prat et al. 1997). Where a tricellular junction has opened to make way for an intercellular space, reinforcing zones with pectic polysaccharides structurally similar to those at the original tricellular junction are present along the three cell–cell–space junctions where the walls of adjacent cells diverge (Rihouey et al. 1995), and it may be assumed that their function is analogous, preventing further separation of the pair of cells concerned and controlling the size of the intercellular space. In other circumstances, as in abscission, adhesion breaks down along the middle lamellae of two planes of cells and contact between these cells is lost entirely (Morré 1968). Again, loss of intercellular adhesion is restricted to specific locations around specific cells. These examples make it clear that in the presence of turgor pressure, intercellular adhesion is not something that just happens: higher plants have evolved precisely defined and precisely localized polymer systems to prevent cell separation and can, with equal precision, dismantle these systems where cell separation is necessary.
In some cooked vegetables turgor is replaced by the swelling of gelatinized starch inside the cells. There is considerable literature on the ‘rounding-off’ and separation of cooked potato cells by starch swelling pressure, which is the principal factor controlling potato texture [reviewed in Jarvis, MacKenzie & Duncan (1992)]. Mechanically this phenomenon is analogous to the separation of the cells by turgor pressure, but in neither situation has the quantitative relationship between cell separation and internal pressure been studied, either theoretically or experimentally.
This paper presents a theoretical description of the cell separation forces in relation to intracellular pressure, however generated, and to the shape and size of the cell. It is intended to provide a basis for the experimental measurement of intercellular adhesion strength and to clarify the behaviour of those fruit and vegetable tissues (e.g. apple, tomato, potato, pea) in which cell separation is a determining factor in textural quality.
It is simplest to present the analysis for cells of approximately prismatic form, as encountered, for example, in stem tissues. This allows the length of the cell to be assigned a constant value l and allows forces to be calculated per unit length of cell. It also allows the model to be illustrated in two dimensions as the transverse section of the cells. The form of the analysis is similar for cells of other shapes sectioned in a plane normal to a tricellular junction.
Cells without intercellular spaces
In the absence of tissue tension, such as that imposed by a constraining epidermal layer, and if the turgor pressure (or other intracellular pressure) is equal to Π and the cell width in section is d, then the load FW carried by each cell wall in the plane of the section is given by:
Straightforward resolution of forces (Fig. 1) then gives the force FC separating the cell from each of its neighbours at the tricellular junction as:
where n is the number of sides in the cell (Jarvis 1992).
More usefully, the cell separation force per unit length of the cell, FC/l, is given by:
Cells with intercellular spaces
Once an intercellular space has opened up, the turgor-generated force separating the cells further is concentrated at the three cell–cell–space junctions where the cell walls diverge (Fig. 2). It is convenient to define the degree of cell separation, and the consequent enlargement of the intercellular space, by the angle θ subtended at the centre of the cell by the intercellular space. This angle increases from zero when the intercellular space first appears, to π/2 when the cells eventually become round. For cells initially square in transverse section the computed relationship between θ and the length of the cell wall remaining in contact with the neighbouring cell is shown in Fig. 3. Resolution of forces then gives the following equation for FY, the cell separation force at each cell–cell–space junction:
The term d’ is used rather than d to denote the fact that the cell width, and hence the load FW on each cell wall, increase as the cell changes shape. The change in cell width is relatively small but if necessary its effect can be estimated as follows. Assuming that n = 4 as in Fig. 2 and that the length of the cell does not change, the condition that the perimeter of the cell in transverse section remains constant may be expressed as:
When Eqn 5 is solved iteratively for n = 4, the increasing cell width as θ increases is shown in Fig. 3. For cells that initially have more than four sides in transverse section, the increase in cell width is smaller than shown. The effect of the increasing cell width is to increase both FW and FY, other things being equal. However, at the same time the increasing cell volume is likely to reduce the turgor pressure from which these forces arise, unless it is maintained by the metabolism of the cell. Figure 3 shows that the increase in cell width and the increase in cell volume are comparable in magnitude so that their effects on FW and FY approximately cancel out.
As θ increases and the cells become round, the cell separation force FY diminishes to zero according to Eqn 4. The computed relationship between FY and θ is shown in Fig. 4 for a representative plant cell with n=4, turgor pressure=1 MPa and initial width 50 μm.
These equations relating cell separation force to internal pressure will, in suitable circumstances, allow manipulation of the experimentally measurable turgor pressure to be used to calculate the strength of intercellular adhesion, which until now has been experimentally inaccessible. However, because cell separation forces are comparable in magnitude to the forces exerted by turgor on an individual cell wall, the utility of this approach is limited to tissues in which adhesion between cells is not unduly strong. Otherwise, the cells may burst instead of separating (Lin & Pitt 1986). This experimental approach also requires the absence of tissue tension, which can greatly alter intercellular separation forces independently of turgor. The strength of cell adhesion is reduced by ripening or cooking in the case of commercially important species, like tomato and potato, in which cell separation has implications for perceived texture.
In this paper no attempt is made to derive cell separation stresses; that is, force per unit area, rather than force per unit length of intercellular junction. This is because the model presented has to describe two situations: the initiation of cell separation and the further splitting of the middle lamella once cell separation has commenced. It appears that the cell separation force is initially carried by the reinforcing zones, rich in calcium pectate, at the tricellular or cell–cell–space junctions: the cell separation force is presumably spread over the width of the reinforcing zones. After these have fractured, however, the model assumed is that of two flexible sheets (the cell walls) glued together with an adhesive (the middle lamella). During this second phase the relationship of cell separation force to the tensile stress exerted upon the middle lamella is complex, because it depends on the bending rigidity of the cell wall. If the cell wall is rigid in bending mode, it will bend only to a relatively large radius when the cells are pulled apart by a given force. The result is that the cell separation force per unit length will be spread over a correspondingly large area of middle lamella and the tensile stress on the middle lamella will be low. This difficulty does not greatly reduce the utility of the model presented, but it should be borne in mind that an experimental treatment that increases the force needed to separate the cells after the initial stage may do so by making the cell walls more resistant to bending, rather than by increasing the tensile strength of the middle lamella.
It may be unexpected that cell diameter appears in the equations for the cell separation forces. It is possible that this is one factor limiting the size of plant cells in nature. It is notable that the largest plant cells (of the order of 1 mm diameter) are normally circular in transverse section. When the inner face of the cell wall is circular, turgor pressure generates no stress to separate the cells.