Mechanical design of apertures and the infolding of pollen grain

Significance

Pollen carries male plant genetic material encapsulated in a hard protective shell containing flexible, soft regions—apertures. The mechanical design of the shell regulates how the pollen grain folds into itself upon dehydration, which often occurs once it becomes exposed to the environment. We investigate folding pathways of pollen grains by studying elastic deformations of inhomogeneous thin shells. Different pathways are governed by the interplay between the elastic properties of the hard and soft regions of the pollen shell and by the aperture shape, number, and size. We delineate regions of mechanical parameters of the pollen grain which lead to complete closure of all apertures, thus reducing water loss and presenting evolutionary viable solutions to the infolding problem.

Abstract

When pollen grains become exposed to the environment, they rapidly desiccate. To protect themselves until rehydration, the grains undergo characteristic infolding with the help of special structures in the grain wall—apertures—where the otherwise thick exine shell is absent or reduced in thickness. Recent theoretical studies have highlighted the importance of apertures for the elastic response and the folding of the grain. Experimental observations show that different pollen grains sharing the same number and type of apertures can nonetheless fold in quite diverse fashions. Using the thin-shell theory of elasticity, we show how both the absolute elastic properties of the pollen wall and the relative elastic differences between the exine wall and the apertures play an important role in determining pollen folding upon desiccation. Focusing primarily on colpate pollen, we delineate the regions of pollen elastic parameters where desiccation leads to a regular, complete closing of all apertures and thus to an infolding which protects the grain against water loss. Phase diagrams of pollen folding pathways indicate that an increase in the number of apertures leads to a reduction of the region of elastic parameters where the apertures close in a regular fashion. The infolding also depends on the details of the aperture shape and size, and our study explains how the features of the mechanical design of apertures influence the pollen folding patterns. Understanding the mechanical principles behind pollen folding pathways should also prove useful for the design of the elastic response of artificial inhomogeneous shells.

Pollination is a crucial process in the life cycle of plants. For it to proceed, pollen grains must leave the anther, which exposes them to rapid desiccation as they cannot actively control their hydration status (1–3). The grains thus require some sort of protective mechanism against desiccation in the period before they land on the stigma of a flower, where they germinate upon rehydration to complete the fertilization. The near-universal protective mechanism against desiccation in pollen during presentation and dispersal is harmomegathy: a characteristic infolding of the grain in response to a decreasing cellular volume upon dehydration (4–8).

The shell of a pollen grain encapsulates the male plant genetic material it carries. It consists of two biopolymeric layers: intine, a soft cellulosic interior layer, and exine, a hard exterior layer composed of sporopollenin and impermeable to water (2, 5, 7). The two layers are not homogeneous throughout the shell, and regions which significantly differ structurally and morphologically from the rest of the shell wall and where the exine layer is either absent or reduced in thickness are termed apertures (5, 7–10). Not only do apertures play a role in the harmomegathic accommodation of grain volume changes, they also function as sites for water uptake and the initiation of pollen tubes (5, 11, 12). Despite these shared functions of apertures, their number, shape, and size vary greatly among and within pollen species (5, 13). Most commonly, apertures occur in the form of a spherical lune (colpus or sulcus), a circular region (porus or ulcus), or a combination of the two (colporus). Monosulcate pollen with a single distal aperture is ancestral in angiosperms (flowering plants) (14–16) and remains a distinctive feature of a large evolutionary line of monocots. In eudicots, a very large clade comprising ∼75% of the extant angiosperm species, pollen grains are most often characterized by three apertures (16) (Fig. 1). In general, many aperture patterns can be observed in angiosperm pollen, and some of them may not easily fall in a predetermined category. Angiosperm pollen can also be inaperturate or omniaperturate, and evolutionary trends appear to favor an increasing number of apertures (9, 11, 16). Nonetheless, the vast majority of angiosperms produce pollen with one or three apertures (13, 15–21).

Fig. 1.

(A) Tricolpate pollen of Betonica officinalis (common hedgenettle). (B) Interapertural thickness of both intine and exine together is $d\approx 1.2\hspace{0.17em}\mathrm{\mu }$m, while the thickness of the intine in the apertural region (where the exine is absent) is approximately half of that value. (C) The hydrated pollen shape in A is spheroidal with a (mean) diameter of $2R_0\approx 35$ to $40\hspace{0.17em}\mathrm{\mu }$m. (D) The pollen elongates upon desiccation when the apertures close and fold inward. Pollen images reprinted with permission from ref. 22; images courtesy of PalDat (2000 onwards, www.paldat.org)/H. Halbritter and S. Ulrich.

During desiccation, apertures often fold inward, and in many pollen species the edges of each aperture eventually touch each other and effectively close off, thus reducing the rate of water loss (23–25) (Fig. 1D). The precise effects of the aperture number or shape on harmomegathic volume accommodation are, however, still not completely understood (8, 9). Empirical studies have shown that harmomegathy depends on the aperture pattern, but it also appears to be influenced by other characteristics such as the size or shape of the grain (9, 26). Some flexibility of the exine wall is also required—since even pollen without apertures is able to accommodate volume changes to a degree (12)—and exine ornamentation and its elastic properties have been suggested to be involved in the process as well (24, 25). Overall, the entire suite of structural adaptations of angiosperm pollen grain seems to be tailored to favor large-scale, mostly inextensional bending of its wall, allowing the apertures to fold inward and in this way reduce the rate of desiccation (1, 25–27).

Depending on the species, pollen changes its shape during desiccation in surprisingly different ways, from a regular infolding to a seemingly random, irregular fashion—even if the type of exine ornamentation and the aperture condition appear to be almost identical (2, 5, 26). Several theoretical studies have recently explored the role that apertures in pollen grains (25, 27) or, more generally, local soft spots in elastic shells (28–30) play in their folding pathways. Studies of instabilities and buckling in thin shells (31–35) have shown that a perfectly homogeneous spherical shell exposed to uniform pressure or undergoing a change in volume will develop depressions in unpredictable positions due to the high symmetry of the problem (35, 36). The buckling behavior can, however, be altered and guided by creating local weak spots in an otherwise uniform shell (28–30), and this elastic inhomogeneity of the shell can also be realized by varying its thickness (37–39). Similar considerations can be applied to pollen grains (40), where the high sporopollenin content makes exine a very stiff material (41–43) while the apertures can be seen as elastic soft spots in the pollen wall (25). Using a thin-shell elastic model of the pollen grain, Katifori et al. (25) have shown that apertures contribute to harmomegathy by reducing the necessity of the pollen wall to significantly stretch and bend to accommodate volume changes, guiding the pollen to fold in a regular fashion by closing the apertures. While these models thus confirm that various morphological and structural adaptations enable adjustments of pollen grains to sudden volume changes by influencing their mechanical properties, not much is known about the precise nature of this relationship.

A detailed knowledge of the elastic properties of the grain and their spatial variation is required to formulate a model of pollen infolding: A soft spot in an overall soft shell will lead to a different folding pathway than the same soft spot in an otherwise hard shell. Yet the mechanical properties of pollen remain largely unexplored (41–43), and little is known about how these properties and their variation influence folding pathways of pollen grains. Our work addresses this important question by using an elastic model which is versatile enough to investigate different modes of pollen folding based on the estimated elastic properties of pollen walls. This allows us to explore phase diagrams of folding pathways with respect to the overall elastic properties of pollen walls and the degree of weakening provided by the apertures. In this way, we are able to distinguish between regions where the apertures successfully close and the (possibly irregular) pathways where they do not close properly. We also show how the aperture size, shape, and their number all play a role in determining the folding pathways of pollen grains.

Results

Elasticity of Pollen Grains.

A significant amount of monad pollen, shed as a single pollen grain, is approximately spherical in its hydrated form (13, 44) (Fig. 1). For the purposes of the mechanical modeling of pollen grains, we effectively treat the entire interior of the grain as a liquid which acts only to constrain the total volume enclosed by the pollen wall. Such a model could also be formulated in terms of internal pressure imparted on the elastic spherical shell by the water inside it (25). Using a discrete elastic model (Materials and Methods), we assign different microscopic elastic parameters (stretching parameter $ϵ$ and bending parameter $\rho$) to the exine region of the pollen wall and to the apertures where the exine is either absent or thinned. Parameter $f<1$ is the ratio of the elastic parameters in the different regions and it sets the elastic difference between the pollen wall and the apertures—the smaller the value of $f$, the softer the apertural region compared to the exine wall. The overall elasticity of the pollen grain is on the other hand determined by the dimensionless Föppl–von Kármán (FvK) number $\stackrel{\sim }{\gamma }$, proportional both to the ratio of the stretching and bending parameters of the pollen wall and to the square of its radius $R_0^2$ (described in Materials and Methods). It is important to stress that the FvK number $\stackrel{\sim }{\gamma }$ is a quantity which applies to the entire grain surface, both in the exine region and in the apertures, since the softness parameter $f$ cancels out in the ratio of the elastic parameters in the aperture regions. While $\stackrel{\sim }{\gamma }$ can be thus understood as a measure of the overall, global elasticity of the grain, $f$ indicates the extent of elastic inhomogeneity due to the presence of the apertures.

Phase Diagrams of Tricolpate Pollen Folding.

Desiccation and the corresponding volume change lead to infolding of pollen grains. We first study folding pathways of tricolpate pollen, both because triaperturate pollen is the most common in angiosperms and because colpi usually bring about a more regular aperture closing than some other aperture shapes, such as pores (6, 25). We model the colpi as spherical lunes extending from pole to pole, whose size is given by their azimuthal opening angle ${\varphi }_c$. In an $n$-colpate pollen, such apertures span a total angle of ${\varphi }_{ap}=n{\varphi }_c$ and represent $A_{ap}/A_0=n{\varphi }_c/2\pi$ of the total grain area $A_0$. While it is difficult to precisely determine the total area that apertures cover in pollen grains, it can still be fairly reliably estimated from the microscopic images of pollen, where available (13, 45). In tricolpate pollen, $A_{ap}/A_0$ typically ranges from 0.1 to 0.4, depending on the species (13), and the value which we use as representative in this section is $A_{ap}/A_0=1/3$. In the next section, we address the effect of varying both the number and size of colpi on the folding pathways of pollen, and since colpi and circular pores can have different harmomegathic patterns (6, 25), we also investigate the role of the aperture shape.

Fig. 2 shows the prototypical, regular infolding process of a tricolpate pollen grain with $\stackrel{\sim }{\gamma }=\mathrm{7,000}$ and $f=0.01$ as its volume decreases from the initial volume $V_0$ to about $V=0.65V_0$ (the volume difference is denoted by $\mathrm{\Delta }V=V_0-V$). The overall grain shape and its changes can be characterized by the dimensions of the minimal bounding box which contain it, $\left(u_x,u_y,u_z\right)$; for a fully hydrated spherical shape, $u_x=u_y=u_z=2R_0$. Colpi are arranged equatorially, and the poles of the pollen grain, i.e., the points where the colpi meet, are positioned on the $z$ axis. As the infolding proceeds, the equatorial dimensions of the bounding box containing the pollen grain shrink (Fig. 2A) and the distance between the poles increases (Fig. 2B). The shape of the grain thus becomes prolate, which can also be quantified by the ratio of the sizes of the bounding box in the $z$ direction and in the equatorial plane (Fig. 2C). The regular infolding of the grain and its elongation along the $z$ axis require an increase in the elastic energy of the shape (Fig. 2D). The apertures close completely once a large enough reduction in volume is reached (in this case, $\mathrm{\Delta }V/V_0\sim 0.35$). Past this point, the edges of the apertures touch, the shell enters a self-collision regime, and further reduction of the volume requires additional deformation of the exine. At the point of the aperture closing, the ratio of the pollen long and short axes exceeds 1.6 in this case. A prolate shape is perhaps the most commonly observed feature in dry pollen grains—this is a direct consequence of a regular infolding in colpate grains, which can otherwise be perfectly spherical in their fully hydrated state, as is also the case in our calculations.

Fig. 2.

Process of regular aperture closing during desiccation of tricolpate pollen with $\stackrel{\sim }{\gamma }=\mathrm{7,000}$, $f=0.01$, and $A_{ap}/A_0=1/3$. In A–D, the plots show the change in different parameters upon a decrease in the relative volume $\mathrm{\Delta }V/V_0$: the scaled average size of the bounding box in the equatorial plane, $\left(u_x+u_y\right)/4R_0$ (A); the scaled size of the bounding box along the $z$ axis, $u_z/2R_0$ (B); their ratio (C); and normalized elastic energy (D). Insets in D show 3D shapes in their bounding boxes (in perspective projection) and their equatorial cross-sections at different values of $\mathrm{\Delta }V/V_0$. The cross-sections contain the points located in the ring around the equator of the mesh, with $z$ coordinates between $-0.04R_0$ and $0.04R_0$; the apertures are indicated in red. Colors in the 3D shapes indicate the mean surface curvature, the brightest yellow and the darkest blue in each individual shape corresponding to its largest and smallest curvature, respectively. In these shapes, the largest mean curvature (yellow) is positive and the smallest one (blue) is negative.

While the apertures of a tricolpate pollen grain with $\stackrel{\sim }{\gamma }=\mathrm{7,000}$ and $f=0.01$ close completely once the volume of the grain is reduced by one-third, such a reduction in the initial volume is in general not necessarily enough for the apertures to close. Whether this happens at a certain $A_{ap}/A_0$ depends both on the FvK number $\stackrel{\sim }{\gamma }$ and on the ratio of the elastic constants of the apertures and the exine $f$. The phase diagram of infolded shapes of tricolpate pollen in the $\left(f,\stackrel{\sim }{\gamma }\right)$ plane, shown in Fig. 3, makes it clear that apertures close completely and in a regular fashion described in Fig. 2 only in a restricted region of the phase diagram. Complete, regular or nearly regular closure of the three colpi requires a pollen grain with $\stackrel{\sim }{\gamma }\gtrsim \mathrm{3,000}$ and $f\lesssim 0.07$. When the apertures are not soft enough compared to the exine (i.e., when $f$ is relatively large, $\sim 0.1$), they do not function well as the regions which dominantly localize the infolding—the pollen grain folds in an asymmetric fashion, with only some of the three colpi closing completely. On the other hand, for sufficiently low values of the FvK number, $\stackrel{\sim }{\gamma }\lesssim \mathrm{1,000}$, the apertures infold symmetrically but do not close completely, even though they are soft compared to the exine.

Fig. 3.

Phase diagram of tricolpate pollen folding in the $\left(f,\stackrel{\sim }{\gamma }\right)$ plane, showing equatorial cross-sections of pollen shapes. The relative change of volume upon desiccation is $\mathrm{\Delta }V/V_0=0.35$ throughout, and the colpi (shown in red in the cross-sections) span one-third of the total area of the pollen surface, $A_{ap}/A_0=1/3$. Numbers next to the pollen shapes denote the elongation of the bounding box of the shapes, $2u_z/\left(u_x+u_y\right)$, indicating the prolateness of the grain. Insets show 3D shapes of desiccated pollen in different parts of the phase diagram, where the arrows indicate the viewpoint with respect to their equatorial cross-section; the colors have the same meaning as in Fig. 2 and the dashed white lines indicate the borders of the apertures. The shaded region of the phase diagram corresponds to the elastic parameters where the apertures close completely and nearly symmetrically, without asymmetric deformation of the exine.

Role of Aperture Shape, Size, and Number.

Inspection of folding pathways of tricolpate pollen points to the importance of the elastic properties of the entire pollen grain and the elastic inhomogeneities provided by the apertures—both of these determine whether or not the apertures close upon desiccation and the manner in which they close. However, at a fixed FvK number of the grain $\stackrel{\sim }{\gamma }$ and softness of the apertures $f$, the pollen folding pathways are further determined by the shape, size, and number of the apertures.

In the results presented thus far, the colpi extended from pole to pole: This is an idealized case (Fig. 1). Even a small reduction in the polar span of the apertures can change the infolded shape of the desiccated pollen, which typically becomes markedly more lobate, and further reduction of the polar span partially invalidates the function of the apertures, which do not close completely anymore. In addition, large exine regions flatten upon infolding. Eventually, the apertures completely fail and the grain buckles in an irregular fashion, as the large intraapertural region become sunken so that the infolded shape resembles a cup. Such a shape is often found in dry inaperturate and porate pollen grains (13). The progressive change of the aperture shape is addressed in SI Appendix, and these results are illustrated in SI Appendix, Fig. S2. Variation of the area covered by the apertures, on the other hand, does not seem to influence much whether the apertures close completely or not. However, changing the aperture area (while retaining its shape) influences the amount of the volume reduction which can be achieved by the full closing of the apertures—the smaller the aperture area, the smaller the volume reduction upon closing (demonstrated in SI Appendix, Fig. S3).

The area covered by the apertures can also be arranged in different ways: It can be united in a single wide colpus or distributed among several ($n$) colpi, each one covering an area of $A_c=A_{ap}/n$. The number of colpi (and apertures in general) can vary significantly among and even within pollen species. It is thus of importance to investigate how this influences the folding pathways of pollen grains. The results are summarized in Fig. 4, which shows phase diagrams of infolded pollen shapes in the $\left(f,\stackrel{\sim }{\gamma }\right)$ plane for grains with $n=1$, 2, 3, and 4 apertures. Analysis presented earlier in Fig. 3 suggests that infolded shapes and pathways can be divided into roughly three categories: grains with apertures that infold symmetrically but do not close (light blue regions in the diagrams in Fig. 4), grains which infold with pronounced deformation of the exine and in which apertures do not close symmetrically and simultaneously (red regions), and grains in which all apertures completely close in a symmetrical or nearly symmetrical fashion (dark gray regions). This categorization of infolded shapes serves well not only for $n=3$, but also for $n=1$, 2, and 4 (cross-sections of the infolded shapes for $n=2$ and $n=4$ are shown in SI Appendix, Figs. S5 and S6). Fig. 4 makes it apparent that regular folding pathways need to be more precisely elastically regulated in grains with larger numbers of apertures, seen as the decrease in the size of the dark gray regions in the phase diagrams as $n$ increases. Only sufficiently soft apertures (small $f$) guarantee a full and symmetrical infolding of the pollen grain, and this condition becomes more stringent with large $n$. Interestingly, in the case of $n=1$, the smallest values of $f$ require grains with large enough FvK numbers ($\stackrel{\sim }{\gamma }\gtrsim \mathrm{7,000}$) to produce apertures which close completely.

Fig. 4.

Phase diagrams of the three main types of infolded pollen shapes in the $\left(f,\stackrel{\sim }{\gamma }\right)$ parameter space for grains with $n=1$, 2, 3, and 4 colpi (as denoted in the diagrams), with $A_{ap}/A_0=1/3$ in all cases. Light blue regions correspond to shapes in which the apertures infold symmetrically but do not close. In the red regions, the exine is significantly deformed and either the apertures close partially or only some of them close fully, and the infolding pattern is either asymmetrical or of lower symmetry than the initial shape. In the dark gray regions, all apertures completely close in a symmetrical or nearly symmetrical fashion. Each region additionally contains a representative equatorial cross-section, positioned at its approximate $\left(f,\stackrel{\sim }{\gamma }\right)$ coordinates.

The case of inaperturate pollen ($n=0$) corresponds in our model to the limiting value of $f=1$ where the apertural and interapertural regions have the same elastic properties and the shell is thus homogeneous, irrespective of $n$. As homogeneous spherical shells undergo a symmetry-breaking buckling instability upon a change in volume (32, 35), this means that the red region in the phase diagrams in Fig. 4 must necessarily appear for all $\stackrel{\sim }{\gamma }$ and $n$ as $f$ approaches 1.

Differences in Folding Pathways.

To examine in more detail the differences in the calculated pollen folding pathways, we show in Fig. 5 the changes in pollen shape during desiccation for three cases of tricolpate pollen grains in the three different regions of the phase diagram in Fig. 4: $\stackrel{\sim }{\gamma }=\mathrm{10,000}$ and $f=0.01$ (symmetrical infolding where the apertures close; dark gray region in Fig. 4), $\stackrel{\sim }{\gamma }=\mathrm{7,000}$ and $f=0.1$ (apertures do not close symmetrically and simultaneously; red region), and $\stackrel{\sim }{\gamma }=\mathrm{1,000}$ and $f=0.01$ (symmetrical infolding but the apertures do not close; light blue region). Changes in the pollen shape brought about by the decrease in volume are represented by the variations of the bounding box dimensions of the grains, with Fig. 5 A and B showing the changes in the equatorial plane and along the $z$ direction, respectively. Fig. 5C further shows the changes in the total lengths of the apertures in the equatorial plane, scaled by their corresponding lengths in the fully hydrated pollen. Each of the folding pathways presented in Fig. 5 is a sequence of pollen grain shapes with the lowest elastic energy at each point in a series of progressively decreasing volumes.

Fig. 5.

Different folding pathways for pollen grains with $\stackrel{\sim }{\gamma }=\mathrm{10,000}$, $f=0.01$ (solid lines); $\stackrel{\sim }{\gamma }=\mathrm{1,000}$, $f=0.01$ (dotted lines); and $\stackrel{\sim }{\gamma }=\mathrm{7,000}$, $f=0.1$ (dashed lines) are illustrated by examining the geometry of the grains as they infold and $\mathrm{\Delta }V/V_0$ increases. Shown are the change in the mean equatorial dimensions of the bounding boxes of the grain (A), the half-lengths of the bounding boxes in the $z$ direction (B), and the total lengths of the apertures in the equatorial plane (C). The latter were scaled with the equatorial lengths of the apertures in the fully hydrated state, $R_0{\varphi }_{ap}$. Insets below B show the change in the equatorial cross-sections of a grain with $\stackrel{\sim }{\gamma }=\mathrm{7,000}$ and $f=0.1$ as its volume decreases. The vertical dashed line shows the reduced volume where the folding pathways of the two grains with $f=0.01$ begin to show marked differences. Colored Inset in A shows the positions of the cases represented by the three lines in the $\left(f,\stackrel{\sim }{\gamma }\right)$ plane of the $n=3$ phase diagram from Fig. 4.

The folding pathways of the two cases with soft apertures ($f=0.01$; dotted light blue and solid dark gray lines in Fig. 5) are initially similar as the volume starts to decrease and apertures gradually change their curvatures. At about $\mathrm{\Delta }V/V_0\approx 0.18$, the pathways start to markedly differ and the grain with $\stackrel{\sim }{\gamma }=\mathrm{1,000}$ (dotted light blue line in Fig. 5) almost stops shrinking in the equatorial plane as its volume is reduced further. It also stops elongating along the $z$ axis, after a short acceleration of elongation when $0.2<\mathrm{\Delta }V/V_0<0.23$. This corresponds to a development of two “pinches” at the poles of the grain, as seen in the three-dimensional (3D) shape in Fig. 3, Lower Left. In the case of the grain with $\stackrel{\sim }{\gamma }=\mathrm{10,000}$ (solid dark gray line in Fig. 5), no transitions in the change of the bounding box dimensions are observed as it continuously shrinks in the equatorial plane and elongates along the $z$ axis, and the apertures infold and fully close at around $\mathrm{\Delta }V/V_0=0.35$. Even though the bounding box dimensions of the grain with $\stackrel{\sim }{\gamma }=\mathrm{1,000}$ almost stagnate when $\mathrm{\Delta }V/V_0>0.18$, the grain continues to reduce its volume. Fig. 5C shows that the apertures of this shape start to inflate toward its interior. This process reduces the volume but requires the apertures to stretch, elongating in the equatorial plane by $30\%$ when $\mathrm{\Delta }V/V_0$ increases from $\approx 0.18$ to $\approx 0.35$. No stretching of the apertures is observed in the case of the grain with $\stackrel{\sim }{\gamma }=\mathrm{10,000}$, and this shape reduces its volume by exine bending and a complete closing of the apertures which, once fully closed, take up a much smaller interior volume when compared to the case with $\stackrel{\sim }{\gamma }=\mathrm{1,000}$ ($\mathrm{\Delta }V/V_0=0.35$). Large strains are typical for highly desiccated grains with small $\stackrel{\sim }{\gamma }$ (e.g., with thick walls), which suggests that a nonlinear elastic model may be required to more realistically model such grains in a regime of large deformations (see SI Appendix for details).

The irregular folding pathway pertaining to stiffer apertures or weaker exine (red dashed lines in Fig. 5) is very different from the other two, as can also be observed by inspecting the cross-sections of the infolded shapes from this pathway and comparing them with the shapes in Fig. 2 (regular closing). The geometry of the pollen shape now changes nonmonotonously during infolding. Both shape descriptors in Fig. 5 A and B feature a kink at $\mathrm{\Delta }V/V_0\approx 0.253$, indicated by an arrow. At this transition point, the ${\mathrm{C}}_3$ rotational symmetry around the $z$ axis is broken, and past this transition, each of the three apertures continues to infold in a different fashion. In general, the irregular folding pathways of the red regions in the phase diagrams in Fig. 4 are more prone to be influenced by noise, and the absolute minima of energy are more difficult to determine compared with the shapes where the apertures close symmetrically (see SI Appendix for a detailed discussion). The minimization procedure must thus be carried out carefully to determine the solution pathway rather than some deformation pathway (35), which additionally signifies a failure of the apertures to strictly guide the infolding pathways of pollen grains whose elastic properties fall into the red regions of the phase diagrams.

The different folding pathways in Fig. 5 can be explained by considering the energetics of the infolding. A complete infolding of the grain apertures requires some bending of the exine regions as well, which adopt a smaller effective radius in the equatorial plane upon infolding (Fig. 2). Bending deformation of the exine is also required for the elongation of the grain. The exine regions must thus deviate from their spontaneous curvature, characteristic of the fully hydrated state of the grain. While the contribution of stretching in the exine deformation is small, exine bending becomes prohibitively expensive energy-wise for sufficiently small $\stackrel{\sim }{\gamma }$ (i.e., for grains where the ratio $Y/\kappa$ is small). Consequently, instead of a complete infolding of the apertures and the accompanying bending of the exine, the pollen grain reduces the volume by stretching (inflating) its apertures toward its interior. This decreases the bending energy of the exine and lowers the total elastic energy of the grain for small $\stackrel{\sim }{\gamma }$ and sufficiently large $\mathrm{\Delta }V/V_0$. The process, however, requires very soft apertures (small values of $f$) and does not take place when the apertures are stiff enough—in that case, we typically observe an irregular infolding (red regions in Fig. 4), as the apertures are not soft enough to localize the deformation of the pollen wall.

Discussion

Having in mind the various mechanical influences to which pollen grains are exposed on their journey from the anther to the stigma, a robust mechanical design of the grain infolding patterns that can resist various forces and preserve the regularity of the folding pathway should provide an evolutionary advantage. According to our simulations, both the overall elastic properties of the pollen grain (as given by its FvK number $\stackrel{\sim }{\gamma }$) and the softer regions provided by the apertures (and described by the softness parameter $f$) play a key role in determining the folding pathway of pollen grains upon desiccation. For instance, soft apertures are required to close the grain symmetrically—but too soft apertures will not pull in and bend the exine and will instead yield to stress and stretch to reduce the volume. This will not close the grain and could result in a rupture in materials which cannot sustain large stretching deformations ($\gtrsim 30\%$). On the other hand, if the apertures are too hard, they will fail to act as sites of localized deformation but will instead induce a deformation of the entire grain. If the apertures are to guide the pollen folding reliably, a delicate elastic balance is thus required which depends not only on the softness of the apertures $f$ but also on the elasticity of the entire grain, i.e., its FvK number. Small differences in the elastic properties of pollen grains are thus one possible explanation for the observed differences in folding pathways of pollen grains which otherwise appear practically the same (13, 26).

Folding pathways of pollen grains can be furthermore drastically influenced by the shape, size, and number of apertures. It is perhaps not surprising that circular pores do a much worse job at guiding pollen folding along a regular pathway compared to elongated colpi, as has already been observed before (6, 25), while the area covered by the apertures appears to mainly influence the achievable reduction in the volume of the grain before the apertures close. The largest influence, however, on the available folding pathways of pollen grains seems to be provided by the number of apertures: The region of elastic parameters where desiccation leads to a regular, complete closing of the apertures shrinks with the increasing number of apertures (Fig. 4). While there is a benefit to many apertures arising from simple geometrical considerations—the probability of a grain falling on a stigma with one of its apertures touching it increases for grains with more apertures, approaching $100\%$ for sufficiently large $n$ (16), and experiments have observed a positive correlation between germination speed, number of apertures, and pollen water content at dispersal (7, 11)—the majority of angiosperm pollen has either one or three apertures, indicating that there must be factors selecting against too large a number of apertures as well. Our work shows that a possible reason favoring smaller numbers of apertures could be the requirement for a robust mechanical design of pollen grains where the apertures close completely and in a regular fashion and thus slow the rate of water loss.

The results of our study thus contextualize and develop the conclusions of the work by Katifori et al. (25). While they have demonstrated the role of both compliant apertures and exine ornamentation in achieving predictable and reversible pollen infolding, our work shows that such a pathway is realized only for a sufficiently tuned mechanical design of the grain, involving a complex interplay of pollen elasticity and the aperture number, shape, and size. A change in any of these parameters can push the pollen grain out of the dark gray regions of the phase diagram in Fig. 4—which were of primary interest in ref. 25—and induce an irregular infolding of the grain (see also SI Appendix for additional information). This has been implicitly demonstrated in ref. 46, where exine-deficient mutants of tricolpate pollen grains of Arabidopsis thaliana, exhibiting a significantly weakened exine, no longer closed properly upon desiccation despite the presence of the apertures. A weakened exine corresponds to larger values of $f$, which again shows that such a change will push the pollen grains away from a regular closing pathway. To connect these observations with our results in a more straightforward fashion, studies aimed specifically at connecting the elastic properties of different pollen to their folding pathways are needed.

The results presented in this work emphasize the importance of a more detailed theoretical and experimental exploration of the mechanical and elastic properties of pollen grains to obtain a better understanding of their folding pathways. The different types of pathways we have observed should be valid for any size class of pollen, as the general range of FvK numbers we considered depends on the ratio of the pollen size and the exine thickness, and larger pollen grains tend to have thicker exines (47), thus constraining the range of FvK numbers. Given the ample variation of pollen types, however, it would not be surprising to find pollen grains which fall outside of this range, and our thin-shell model should be applied with caution to the cases where the thickness of the exine becomes comparable to the size of the pollen grain. Our model should also be applied with some care to pollen grains with pronounced sculpting of the exine or the presence of ornamentation in the apertural region which can, for instance, lead to steric effects influencing the closure of the apertures (25). We have also demonstrated that a more complete modeling of the grain infolding will likely need to account for nonlinearity of the elastic response of the grain and allow for a more nuanced difference between the elasticity of the exine and the apertures, which is in our model accounted for by a single softness parameter $f$.

In addition to providing a mechanical explanation for some evolved features of pollen grain structure, our work should prove useful in the design of inhomogeneous elastic shells which respond reversibly to changes in the osmotic pressure. Colloidal capsules, for example, can sustain an external osmotic pressure up to a critical point after which they buckle. This process can be strongly influenced by structural inhomogeneities in the capsule shells (38), which is consistent with our results, since apertures can be also thought of as designed inhomogeneities—these can thus dramatically change the buckling pathway of the capsule and guide it in a desired direction. Pollen grains themselves can be used as templates for containers for microencapsulation of compounds and as drug delivery vehicles (48–50), and it is of interest to devise strategies to manipulate pollen beyond its natural performance limits (51). One way to achieve this, suggested by the results of our study, is through a manipulation of the elastic properties of the exine and the apertures. This could be done for instance by gene manipulation (20, 46, 52–54) or environmental stimuli (55, 56), which can influence the thickness of the pollen wall, its sporopollenin content, or even the exine pattern, all of which have the potential to change the elastic response of the grain.

Materials and Methods

Elastic Model of Pollen Folding and Desiccation.

We assume that the fully hydrated, spherical shape of the pollen grain has no residual elastic stresses in the wall. The energy of the system $E$ is constructed by formulating the discrete bending and stretching energy contributions on a spherical mesh of triangles approximating the fully hydrated grain (more details regarding the mesh construction are discussed in SI Appendix). The stretching contribution is implemented along the edges $i$ of the mesh, penalizing edge extensions and shortenings $l_i$ from their value in the fully hydrated state $l_i^0$. The bending contribution is implemented between pairs of triangles $I$, $J$ sharing an edge, so that the local bending energy is parameterized in terms of the deviations of the angles between the triangle normals ${\theta }_{I,J}$ from their value in the fully hydrated state ${\theta }_{I,J}^0$:

$$E=\sum _i\frac{{ϵ}_i}{2}{\left(l_i-l_i^0\right)}^2+\sum _{I>J}{\rho }_{I,J}\left[1-\cos \left({\theta }_{I,J}-{\theta }_{I,J}^0\right)\right].$$ [1]

The stretching and bending elastic parameters (${ϵ}_i$ and ${\rho }_{I,J}$, respectively) mirror the elastic inhomogeneities in the grain structure, and their values are set depending on whether the edges $i$ and faces $I,J$ are located in the aperture regions (${ϵ}_{ap},{\rho }_{ap}$) or in the surrounding exine regions (${ϵ}_{ex}$, ${\rho }_{ex}$). (Cases when edge endpoints or neighboring faces are in different domains are explained in SI Appendix.) There are obviously many different ways to set up the inhomogeneity of elastic properties: To make the problem tractable, we introduce a scaling, softness parameter $f<1$ (25) such that ${ϵ}_{ap}=f{ϵ}_{ex}$ and ${\rho }_{ap}=f{\rho }_{ex}$. A smaller value of $f$ simulates a larger elastic difference between the aperture and the exine, while $f=1$ represents the limiting point where the aperture becomes the same as its surroundings in terms of its elasticity.

Pollen desiccation is modeled by gradually decreasing the volume of the grain, treating the volume enclosed by the elastic shell $V$ as a mechanical constraint (57). The relative change of the grain volume is given by $\mathrm{\Delta }V/V_0$, where $V_0$ is the volume of the fully hydrated pollen and $\mathrm{\Delta }V=V_0-V$. Dehydration in pollen varies to a great extent (6, 44, 58, 59), and the water content of pollen correlates with the volume of the grain. In Cucurbita pepo, for example, pollen volume has been observed to be $\sim 20\%$ higher than the hydration status of the grain (59) (meaning that the volume of the grain with, e.g., 50% water content is about 70% of the volume it has at 100% water content: $V=0.7V_0$ and $\mathrm{\Delta }V/V_0=0.3$ in our notation).

Continuum Thin-Shell Elasticity.

In the continuum theory of thin shells, their elastic behavior is governed by a single dimensionless parameter, the FvK number $\gamma =\left(Y/\kappa \right)R_0^2$ (60), which depends on the ratio of two elastic parameters: 2D Young’s modulus $Y$ and bending rigidity $\kappa$; $R_0$ is the radius of the shell. Continuum elastic moduli $Y$ and $\kappa$ are proportional to the microscopic stretching and bending parameters $ϵ$ and $\rho$ (60), and the constants of proportionality are on the order of one. The precise relationship between the discrete and continuum elastic constants depends, however, on both the nature of the triangulation and the geometry of the shape being triangulated, as demonstrated in refs. 61 and 62. For our purposes, we define the quantity

$$\stackrel{\sim }{\gamma }\equiv \frac{ϵ}{\rho }{R}_0^2,$$ [2]

which we call the FvK number, having in mind that it may differ from the continuum value of the FvK number $\gamma$ by a multiplicative constant on the order of one. The exact value of this constant is irrelevant for our purposes, since the important effects take place in the interval spanning orders of magnitude in $\gamma$ and the elastic constants (and thus the FvK numbers) of pollen grains are largely unknown.

Elastic Properties of Pollen Grains.

While recent experiments have estimated Young’s modulus of different pollen grains [obtaining atypically high values in the process, $Y=10$ to 16 GPa for pollen with 15 to $40\hspace{0.17em}\mathrm{\mu }$m in diameter (43)], elastic constants of pollen remain otherwise poorly known. To estimate the range of $\gamma$ relevant for pollen grains, we thus combine the expressions connecting 2D Young’s modulus and bending rigidity with 3D Young’s modulus and Poisson’s ratio to write the FvK number as (33, 35)

$$\gamma =12\hspace{0.17em}\left(1-{\nu }^2\right){\left(\frac{R_0}{d}\right)}^2,$$ [3]

where $d$ is shell thickness and $\nu$ is 3D Poisson’s ratio. The latter can be estimated to be $\nu \approx 0.35$, a value appropriate for rigid polymers and based on the heavily cross-linked structure of sporopollenin (43). While $\nu$ can vary from 0.2 (very rigid) to 0.5 (rubber-like), the exact number is not particularly important for our estimate of $\gamma$, since the ratio $R_0/d$ plays a more dominant role.

Even though pollen comes in sizes from less than 10 $\mathrm{\mu }$m to more than 100 $\mathrm{\mu }$m in diameter—depending also on the degree of hydration and the preparation method—the majority of monad, spheroidal pollen is in the range of 10 to 100 $\mathrm{\mu }$m in diameter (13). In general, pollen thickness increases with pollen size; several studies have estimated the thickness of the exine wall in different types and species of pollen to be in the range $d\in \left[0.5,2.5\right]\hspace{0.17em}\mathrm{\mu }$m (24, 47, 63). Our analysis of the available data in the online palynological database PalDat (13) indicates that for a large sample of spheroidal pollen with radii $R_0\in \left[\mathrm{5,50}\right]\hspace{0.17em}\mathrm{\mu }$m an exine thickness of $d=1$ to $2\hspace{0.17em}\mathrm{\mu }$m is a very good approximation. This excludes any pronounced variations in the exine ornamentation and sculpturing in the form of, e.g., spikes or other protrusions on the scale of $R_0$. We can thus estimate that $R_0/d\approx 5$ to 30 [similar to what has been reported in the literature (47, 63)] and that consequently the FvK numbers of pollen can span a large range, $\gamma \approx 10^2$ to $10^4$. Fig. 1 shows an example of pollen grain from Betonica officinalis, with an estimated $R_0/d\sim 15$ to 20 and $\gamma \sim 2$ to $4\times 10^3$.

Exine ornamentation on a scale much smaller than $R_0$ is not important for the process of harmomegathy, which takes place on the spatial scale comparable to $R_0$. Such effects can be thus included in some average sense in the elastic parameters of the exine. While the exine is very strong (due to either its thickness or its physical properties) (23, 43), recent atomic force microscopy studies estimate that aperture regions are at least an order of magnitude softer, and the smallest measured values of aperture Young’s modulus are two orders of magnitude smaller than the measured Young’s modulus of exine (20, 64). We can thus estimate that the values of the softness parameter $f$ are in the range of $f=0.01$ to 0.1, which is what we use in our analysis.

Data Availability.

All the necessary data and the methods required to reproduce the results of this study are given in the main text and SI Appendix.