A biomechanical model of anther opening reveals the roles of dehydration and secondary thickening

Summary Understanding the processes that underlie pollen release is a prime target for controlling fertility to enable selective breeding and the efficient production of hybrid crops. Pollen release requires anther opening, which involves changes in the biomechanical properties of the anther wall. In this research, we develop and use a mathematical model to understand how these biomechanical processes lead to anther opening. Our mathematical model describing the biomechanics of anther opening incorporates the bilayer structure of the mature anther wall, which comprises the outer epidermal cell layer, whose turgor pressure is related to its hydration, and the endothecial layer, whose walls contain helical secondary thickening, which resists stretching and bending. The model describes how epidermal dehydration, in association with the thickened endothecial layer, creates forces within the anther wall causing it to bend outwards, resulting in anther opening and pollen release. The model demonstrates that epidermal dehydration can drive anther opening, and suggests why endothecial secondary thickening is essential for this process (explaining the phenotypes presented in the myb26 and nst1nst2 mutants). The research hypothesizes and demonstrates a biomechanical mechanism for anther opening, which appears to be conserved in many other biological situations where tissue movement occurs.

: Geometry of the anther segment under consideration, and associated coordinate systems.
The composite layer comprises epidermal and endothecial cell layers, which we assume to always remain strongly adhered to one another.
We denote tangential stress resultants in each layer by F ± * , and normal stress resultants in the endothecium by N * ; N * generates a bending moment M * . Between the endothecium and the epidermis act a frictional stress Q * and normal reaction R * , as shown in Figure S2. Under these assumptions, a balance of forces gives (Nelson et al., 2011) A balance of moments upon the same element, assuming that the contribution of F − * is negligible, to leading order, where 2h * is the endothecial thickness. The Q * h * term was omitted from the model of Nelson et al.; we estimate its magnitude below.
Taking a discrete representation of the upper cell layer, and assuming θ varies slowly between adjacent cells, tangential and normal force balances yield (Nelson et al., 2011) Introducing the composite in-plane tension F * = F − * + F + * , (2-4) give force and moment equations for the composite layer: We prescribe the following constitutive assumptions, governing the extensions of the cell layers, Figure S2: Distribution of forces along an element of the substrate, the resultant stresses and bending moments. Clockwise arrows indicate bending moments acting the negativeẑ-direction.
in which λ ± are the in-plane stretches of the two layers, λ ± 0 represent the resting strains of the layers, and k ± * are extensional stiffness parameters. In the endothecium, we consider these quantities to be evaluated on the centreline. Endothecial secondary thickening is incorporated into the model via an increase in k − * . Assuming that the frictional force, Q * , is sufficient for perfect adhesion between the epidermis and the endothecium, and taking κ * h * ≪ 1, we may set λ ≡ λ + = λ − so that the net in-plane tension is given by In (7), the first term represents the composite extensibility while the second term may be influenced by the hydration of both cell layers through λ ± 0 .
We make a further constitutive assumption for the endothecium, assuming its bending moment is related to its curvature according to where D * is the endothecial resistance to bending (a quantity which we estimate below) andκ * 0 represents the preferred curvature of the endothecium in the absence of the epidermis; we assumeκ * 0 is spatially uniform.
It is convenient to rewrite the governing equations in terms of the Lagrangian arc-length variables * , related to s * via d s * ds * = λ.
We nondimensionalise the problem by scaling lengths against the resting length of the composite layer (L * 0 ), curvatures against 1/L * 0 and stress resultants against k − * . Writing N = N * /k − * , equations (5a,b), (9) and (1) then become with F given by The above system depends upon three dimensionless parameters: which respectively capture the endothecial resistance to extension relative to its resistance to bending, the extensional stiffness of the epidermis relative to that of the endothecium, and the extent to which dehydration of the epidermis generates changes in the preferred curvature of the anther, resisted by the bending resistance of the endothecium.

S1.2 Boundary conditions
We solve (11-12) subject to boundary conditions appropriate to the three cases described in the main text (see Figure S3). In all cases, we assume a rigid support ats = 0 as follows: where x 0 = x * 0 /L * 0 is the locule width scaled against the natural length of the anther segment.
In case I, the anther wall is tighly curled with opposite walls in point contact; we apply boundary conditions on the contact point at the symmetry line (s =s c ) as follows: The third and fourth boundary conditions in (15) respectively enforce that the vertical force and the bending moment vanish at the contact point. In the regions c <s < 1, the anther wall has constant curvature. Epidermal dehydration reduces λ + 0 , causing the layer to gradually uncurl. Whens c = 1 there is a transition to case II, in which the anther remains closed, but it is no longer necessary to track the moving boundary ats =s c . Instead, we impose: As λ + 0 decreases further, and the preferred curvature falls, we monitor the horizontal force exerted upon the symmetry boundary, given by Once this force decays to zero, case III applies, in which the boundary ats = 1 moves away from the symmetry line, mimicking the opening of the anther. Continued epidermal dehydation (reduction of λ + 0 ) generates increasingly open configurations. In case III, all forces and the bending moment vanish at the boundary: In this case, the conditions (18) provide a uniformly valid solution of (11a). and Arabidopsis anthers where these are available. We estimate D * experimentally, as described in Section Notes S2: below; the remaining parameters are estimated via cell-scale models of each layer.

S1.3.1 Cell-scale model of the epidermis
We estimate the epidermal parameters, k + * and λ + 0 , in terms of cell-scale quantities by balancing forces in an epidermal cell. Treating each epidermal cell as a rectangular box of height 2h + * , with elastic walls extending due to turgor pressure p + * , the in-plane stress resultant is given by where E + * is the extensional stiffness of each cell wall. Comparing (19) with (6), we have k + * = 2E + * and We regard p + * as the difference between the hydrostatic pressure and the osmotic pressure; as the epidermis dehydrates, it is therefore possible that P may become negative. We, consider solutions starting in a reference state with λ + 0 > 1 and allow λ + 0 to gradually decrease as the epidermis dehydrates.

S1.3.2 Cell-scale model of the endothecium
In contrast to the epidermis, endothecial cell walls have secondary thickening, comprising stiff lignin fibres which form a helix around the cell. These fibres cause the endothecium to resist stretching and bending. We assume that the lignin fibres provide a dominant mechanical contribution to the endothecial cells, although their interaction with the cell wall will undoubtedly be significant. However, we use the secondary thickening alone to estimate the relative resistances of the endothecium to bending and stretching. Cell-scale parameters are summarised in Table S1.
We denote the Young's modulus of the lignin fibres by E * f , their Poisson ratio by ν f , and their radius by R * f . Costello (1977) gives the bending stiffness of an isolated helix as where γ is the pitch angle. Endothecial cells are approximately 10 − 20 µm long with approximately 5 turns of the helix per cell; we calculate the pitch angle to be in the region of γ = 7 − 14 • . Denoting the extensional stiffness of the helix by k * cell , and using the result of Love (1944), we have  The quantities D * cell and k * cell represent the bending stiffness and extensional stiffness of a single endothecial cell. We expect these values to contribute to the expressions for similar macroscopic quantities for sheets of endothecial cells. From (21) and (22), we estimate α as follows: Equation (23) suggests that endothecial secondary thickening results in a high resistance to stretching and a comparatively low resistance to bending, allowing shrinkage of the epidermis to bend the composite structure. This description of the endothecial cell wall incorporates only the lignin helix and not other cell wall components, nor pre-stress in the cell wall due to endothecial turgor. While there is considerable scope to improve this approximation, we believe the ratio of resistances to bending and stretching predicted above has a sufficient level of accuracy for our purposes.

S1.3.3 The composite layer
Due to the presence of endothecial secondary thickening, we expect the epidermal resistance to extension to be much less than that of the endothecium (β ≪ 1) and the endothecial resistance to extension to be much greater than its resistance to bending (α ≫ 1). The formula for the dimensionless parameter Φ can be rearranged (using (23)) to give Since β ≪ 1 and h * ≪ L * 0 , we conclude that it is appropriate to assume Φ = O(1), as this captures the essential balance between epidermal contraction and endothecial bending (see (13)

S1.4 Reduced model in the inextensible limit
For α ≫ 1 and β ≪ 1, such that αβ = Φ/(h * /L * ) ≫ 1, we rescale dependent variables according to To leading order in α −1 , (12) implies that Restricting attention to the case in which λ − 0 = 1 and λ + 0 is spatially uniform, (26) reduces to the classical beam equations dF to leading order in β. Boundary conditions follow directly from (15-18), with the leading order condition on κ now being The system (28,29) is governed by a single material parameter (stretching in (27) decouples from the problem); leading-order configurations are determined by Φ alone, with transitions between cases I, II and III occuring at for O(1) constants C i . In this limit, the shapes of solution branches vary trivially with Φ, as illustrated in Figure 3. Figure S4 illustrates the distributions of stress and strain attained by the configurations shown in Figure 2. The figure confirms that stresses are of order 1/α (see (25)) and variations in stretch are of order β (see (27)), with strain varying from approximately 15% extension to approximately 10% compression for the configurations shown (corresponding to a choice of β = 0.2).

Notes S2: Estimating the forces of anther dehiscence
To determine the forces associated with anther dehiscence, lily anthers were placed upon a flat support and compressed by a prescribed load. The anthers were allowed to dehydrate, and it was recorded whether the force of anther opening was sufficient to overcome the load, or whether the load was sufficient to keep the anther closed.
Mature anthers just prior to opening were removed from the flowers of pink oriental lilies just before testing. Each flower contained six anthers, of which two were used as non-loaded controls. The anthers were placed in the orientation of Figure 1c, on a glass slide (stabilised using blu-tack if necessary) and the load was applied upon the upper surface (equivalent to the x-direction in the model above) by means of another flat plate upon which varying masses were placed. Masses were distributed along the length of the anther. For each set of six anthers, if both controls had opened (i.e. a visible split along the whole length of the anther could be seen) but none of the loaded anthers had opened, then this load was considered great enough to overcome the opening force. A total of 78 anthers were tested and opening times varied from 50 minutes to 6 hours. Mass of the order of 2 g was needed to prevent any loaded anthers opening. Since the mass applied was spread over two locule pairs we can assume that 1 g, i.e. a distributed load on each locule pair of approximately 0.01 N, was sufficient force to keep the anther closed.
Section S1.4 suggests that dimensional stress resultants are of magnitude k − * /α, i.e. D * /L * 2 0 . These stress resultants have dimensions of force per unit length. For a critical force of 0.01 N, distributed over a length of 20 mm (the length of the glass slide), the experimental results suggest stress resultants of approximately 0.5 Pa m. For L * 0 = 2 mm (see Table 1), this gives an estimate for D * of 2 × 10 −6 Pa m 3 . Comparing (20) and (30) suggests that Φ is of magnitude k + * / (2p + * h + * ) when the anther opens.
Noting (13), and assuming that the two cell layers are of approximately equal thickness (h + * ≃ h * ), it follows that For the values of h * and L * 0 estimated in Table 1, and an estimated epidermal turgor pressure of approximately 2 bar (Bonner & Dickinson, 1990), the theoretical model gives D * ∼ 2 × 10 −6 Pa m 3 , which is consistent with the experimental prediction, providing a consistency check for our model assumptions.  Figure S5 shows the critical level of epidermal hydration (λ + 0 ) required to open the anther, as a function of L * 0 , for parameters corresponding to a lily anther as in Figure 2. As L * 0 is varied, α and Φ are updated according to (13), keeping material parameters fixed. The figure illustrates that for smaller values of L * 0 , the anther will open for larger values of λ + 0 , i.e. requiring less epidermal dehydration. The model thus illustrates that dehiscence initially occurs at the tips as a consequence of the shape of the anther wall.