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Paleobotany |
Division of Science, Truman State University, Kirksville, Missouri 63501-4221 USA; Division of Mathematics and Computer Science, Truman State University, Kirksville, Missouri 63501-4221 USA
Received for publication November 3, 2006. Accepted for publication June 26, 2007.
ABSTRACT
Pollen grains of many wind-pollinated plants contain 1–3 air-filled bladders, or sacci. Sacci are thought to help orient the pollen grain in the pollination droplet. Sacci also increase surface area of the pollen grain, yet add minimal mass, thereby increasing dispersal distance; however, this aerodynamic hypothesis has not been tested in a published study. Using scanning electron and transmission electron microscopy, mathematical modeling, and the saccate pollen of three extant conifers with structurally different pollen grains (Pinus, Falcatifolium, Dacrydium), we developed a computational model to investigate pollen flight. The model calculates terminal settling velocity based on structural characters of the pollen grain, including lengths, widths, and depths of the main body and sacci; angle of saccus rotation; and thicknesses of the saccus wall, endoreticulations, intine, and exine. The settling speeds predicted by the model were empirically validated by stroboscopic photography. This study is the first to quantitatively demonstrate the adaptive significance of sacci for the aerodynamics of wind pollination. Modeling pollen both with and without sacci indicated that sacci can reduce pollen settling speeds, thereby increasing dispersal distance, with the exception of pollen grains having robust endoreticulations and those with thick saccus walls. Furthermore, because the mathematical model is based on structural characters and error propagation methods show that the model yields valid results when sample sizes are small, the flight dynamics of fossil pollen can be investigated. Several fossils were studied, including bisaccate (Pinus, Pteruchus, Caytonanthus), monosaccate (Gothania), and nonsaccate (Monoletes) pollen types.
Key Words: aerodynamic efficiency • anemophily • computational model • fossil pollen • mathematical model • palynology • pollen • saccate pollen • wind pollination
Pollination in seed plants can be accomplished by either biotic or abiotic means. Biotic pollination can be facilitated by a variety of animals; abiotic pollination typically involves pollen transfer by either wind or water. Wind pollination is commonly thought to be a random and metabolically wasteful process, in which copious amounts of pollen are produced and released (Niklas, 1985b
). The pollen-to-ovule ratio for wind-pollinated taxa can be orders of magnitude greater than that for insect-pollinated plants (Niklas, 1992
). This larger energy investment may be offset by a lack of showy petals, a lack of nectaries, and no consumption of pollen by an animal vector (Dowding, 1987
). A range of studies undertaken in the last twenty-five years has shown that wind pollination is more efficient than traditionally thought (Niklas, 1987
). These studies have demonstrated that the efficiency of wind pollination is influenced by the aerodynamics of various plant parts that work together to increase the probability of pollen entrapment on the ovulate organs.
The Coniferophyta is one group that has been extensively studied for aerodynamic efficiency, likely due to the prevalence of wind pollination in conifers (Crane, 1986
). The morphology of ovulate cones is such that airborne pollen is aerodynamically directed through the scales to the pollination droplet (Niklas and Paw U, 1983
). Through this process, ovulate cones are able to partially filter-out pollen of non-conspecific taxa (Niklas and Paw U, 1982
; Niklas, 1984
, 1985b
). In addition, the fascicles surrounding ovulate cones of some Pinus species deflect airborne pollen toward the cones by creating turbulent wakes (Niklas, 1984
, 1985a
). Moreover, the harmonic motion of branches bearing ovulate cones may increase the efficiency of wind pollination (Niklas, 1985a
).
The morphology of pollen grains may also affect the aerodynamics of wind pollination. Pollen vectored by animals is often structurally different than pollen dispersed by wind (Osborn et al., 1991
). In conifer pollen, the gametophytic cells are housed in a spheroidal main body, protected by a two-layered pollen wall. The inner layer, or intine, is composed mainly of cellulose. The outer layer, or exine, is composed of sporopollenin. In many conifers, the exine expands outward from the main body to form one to three air-filled bladders, or sacci. A long-standing perspective is that sacci help to orient pollen grains in the pollination droplet to facilitate pollination (Doyle and O'Leary, 1935
), and recent studies have supported this early hypothesis (Tomlinson, 1994
; Runions et al., 1999
; Salter et al., 2002
). Although sacci appear to aid pollen orientation within the pollination droplet, Wodehouse (1935)
proposed that sacci play a role in the aerodynamics of pollen as well. Sacci increase the surface area of the grains, while ideally adding a minimum of mass. This, in turn, increases the amount of drag on pollen grains. Increased drag reduces the settling speed of pollen, causing dispersal distances to increase. Because settling speeds are proportional to mass and density, smaller pollen grains will also travel farther. However, because of the low momentum of such small pollen grains, such grains may have difficulty breaking through the boundary layer of air to land on ovulate cones (Whitehead, 1983
; Niklas, 1985a
). No published studies, however, have investigated the aerodynamic properties of saccate pollen grains.
Saccate pollen occurs in three families of extant conifers: Pinaceae, Podocarpaceae, and Phyllocladaceae. The sacci in members of the Phyllocladaceae are thought to possibly be vestigial and nonfunctioning. Saccate pollen is also found in numerous extinct groups (Osborn and Taylor, 1994
; Tomlinson, 2000
), including seed ferns (Callistophytales, Glossopteridales, Corystospermales, Caytoniales) and conifers (Cordaitales, Voltziales).
The objectives of this study were to empirically and computationally verify the effects of sacci in the wind dispersal of saccate pollen grains and to develop a valid, mathematical model of pollen settling speeds that can substitute for empirical testing. The mathematical model, based on structural characters, permits flight properties to be measured without physically testing pollen, allowing the flight dynamics of fossil pollen to be investigated. The pollen grains used in this study were chosen because of their structural variability in overall size, saccus size, amount of endoreticulations, and surface ornamentation.
MATERIALS AND METHODS
Pollen material
Pollen of Pinus strobus L. and Pinus mugho Turra (Pinaceae) was gathered from trees located on the campus of Truman State University. Pinus sylvestris L. pollen was collected from a private residence in Kirksville, Missouri, USA. Pollen of Zea mays L. (Poaceae) was gathered from the Truman State University greenhouse. Pollen from Dacrydium franklinii Hook. and Falcatifolium taxoides (Brongn. & Gris) de Laub. (Podocarpaceae) was gathered from herbarium specimens on loan from the Missouri Botanical Garden. Structural data from all fossil pollen and extant Abies were obtained from the published literature: Gothania lesliana (Hirmer) Daghlian et Taylor (Cordaitales; Taylor and Daghlian, 1980
), Caytonanthus arberi (Thomas) Harris (Caytoniales; Osborn, 1994
), Monoletes pollen from Dolerotheca formosa Shopf (Medullosales; Taylor, 1978
), Pteruchus fremouwensis Yao, Taylor, and Taylor (Corystospermales; Osborn and Taylor, 1993
), a fossil Pinus sp. (Coniferales; Phipps et al., 1995
), and Abies concolor (Gord. & Glend.) Lindl. ex Hildebr. (Pinaceae; Kurmann, 1989
).
Microscopy
Dehiscent and near-dehiscent pollen cones of Pinus strobus were cut into sections using a razor blade. Fragments of dehiscent Dacrydium franklinii and Falcatifolium taxoides cones were removed from herbarium sheets and encased in agar blocks before sample preparation.
For studying pollen ultrastructure, the samples were dehydrated in a series of ethanol and acetone rinses and were then gradually infiltrated and embedded in Spurr's epoxy resin (Electron Microscopy Sciences, Fort Washington, Pennsylvania, USA). The embedded specimens were thick- and thin-sectioned with an RMC MT-6000XL ultramicrotome (Tucson, Arizona, USA) using glass and diamond knives. Thick-sections (850 nm) were collected on glass slides, stained with Richardson's stain (azure II and methylene blue), and viewed with bright-field microscopy on an Olympus (Lake Success, New York, USA) BHS compound light microscope to determine whether the pollen grains were fully mature. Thin-sections (70–90 nm) were collected and dried on formvar-coated copper slot grids and were stained with 1% uranyl acetate, lead citrate, and 1% potassium permanganate. Pollen was examined and imaged using a JEOL (Peabody, Massachusetts, USA) JEM-100SX transmission electron microscope (TEM) at 60–80 kV.
For documenting external pollen characters, dissected cone samples were gradually dehydrated in ethanol and critical point dried. Dried cone material was transferred to a glass microscope slide where the sample was macerated with a syringe needle to expose the pollen. Pollen grains were then mounted on aluminum stubs using double-sided tape. The stubs were sputter-coated with gold-palladium and examined and imaged using a JEOL JSM-6100 scanning electron microscope (SEM) at 5 kV.
Mathematical model
A structurally based mathematical model of pollen flight was created to study the aerodynamic features of pollen grains. The model was designed to output the settling speed of a single pollen grain, using various structural measurements and physical constants as its input. The model was constructed on the assumption that the main body and sacci of the pollen grain are each a prolate spheroid. The equation used to describe the main body of the pollen grain is that of an ellipsoid:
|
| (1) |
|
| (2) |
is the angle between the long axis of the ellipse and the horizontal axis of the coordinate plane; and a, b, and c are the lengths of the semi-axes of the ellipsoid.
The total force acting on the pollen grain is the sum of the downward force of gravity and the upward drag force. At terminal velocity, where acceleration is zero, the sum of the two forces is zero. Therefore, under these circumstances, the magnitude of the force of gravity is equal to the magnitude of the force of drag. The force applied to the pollen grain due to gravity is equal to the product of the mass m of the grain and the acceleration g due to gravity. In the mathematical model, drag is defined as the product of the grain's velocity US, the viscosity µ of air, the cube root of the volume V, and a drag coefficient CD (Vogel, 1994
). Setting these two forces equal gives:
|
| (3) |
|
| (4) |
).
Structural characters and development of the mathematical model
The variables from Eq. (4) that need to be determined are the volume, mass, and drag coefficient of each pollen grain. Numerous structural characters of the pollen grains were measured to determine these variables (Figs. 1–15). Lengths and angles of these characters were measured from SEM micrographs, and ultrastructural characters were measured from TEM micrographs (Table 1). Micrographs were measured using ImageJ image analysis software (National Institutes of Health, Bethesda, Maryland, USA; ImageJ, 2004
).
|
|
|
), and the overlap was subtracted from the initial, overestimated saccus volume. Expressed mathematically:
|
| (5) |
|
; sporopollenin: 1.27 g/cm3, Southworth, 1988
; cellulose: 1.36 g/cm3, Wayne and Staves, 1991
). For example, the volume of an ellipse is (4/3)
abc, where a, b, and c are the lengths of the semi-axes. The mass of the main body exine would then be approximated by:
|
| (6) |
is used to denote the thickness of layers in the pollen grain and 
represents density. The mass of cellulose was calculated in the same manner. The mass of living cells was not calculated using a measured thickness. For the living cells, the remaining volume of the ellipse not taken up by the exine or intine was used to determine the volume.
The mass of the saccus wall was determined similarly, again using Monte Carlo integration to determine the overlap of the main body with the saccus wall:
|
| (7) |
of endoreticulations in a given area. The average depth endo that the endoreticulations extend into the sacci was then measured. That depth, along with the percentage of endoreticulations in a given section and the volume measurements derived from SEM data, was used to calculate the volume of endoreticulations present. The mass of the endoreticulations was then determined as:
|
| (8) |
Figures 19–21 demonstrate the fit of the modeled pollen grain to the actual measured grain.
|
Modeling fossil pollen
The seed fern Caytonanthus is known from compression fossils, whereas all other fossils in this study are known from permineralizations. Although the exine generally preserves well through the fossilization process, the intine and the once living cells do not. As a result, it was necessary to estimate the thickness of the intine. As such, the intine thicknesses of similar-sized pollen grains from an allied, extant relative were used. For Caytonanthus (Figs. 22, 23), Monoletes (Figs. 32, 33), and Pteruchus (Figs. 27, 30), intine data from Zamia were used (Zavada, 1983
); for fossil Pinus (Figs. 28, 31), data from extant Pinus from this study were used (Table 1); and for Gothania (Figs. 26, 29), intine data from Picea were used (Runions et al., 1999
).
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|
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Mathematical expansion of compressed pollen grains
Caytonanthus pollen (Figs. 22, 23) was "expanded" by measuring the grain's main body and saccus cross-sectional circumferences and using ellipsoids of the same cross-sectional circumference in the model. Because of the resistant nature of sporopollenin, the cross-sectional circumference was assumed to be preserved in compression. These circumferences were measured from SEM micrographs using the program ImageJ (ImageJ, 2004
). Prolate spheroids with the same cross-sectional circumferences were found and used to model the pollen grain using Ramanujan's approximation equation (Almkvist and Berndt, 1988
). Dimensions of the three spheroids were then used in the model to determine the settling speed.
The displacements of the sacci from the main body were estimated using extant pollen grains as a guide because the actual values could not be deduced from the compression. The degree of overlap of the sacci and the main body varied depending on the estimated values of saccus displacement. For determining the effect that estimating the displacement of the sacci had on the grain, the model was run under two scenarios in which the amount of overlap differed (Figs. 24, 25).
Error propagation
Standard error propagation techniques (Bevington, 1969
) were used to ascertain the uncertainty in the calculated settling speed, which can be the result of uncertainty in the input parameters. Because the mathematical model's predictions are tailored for individual grains, most input parameters are measured from equatorial views of the grain in question. Not every model parameter can be provided by equatorial images. Input parameters unavailable in equatorial views, such as saccus wall thickness and grain width, are derived as an average of proximal SEM or TEM image measurements of other grains from the same species.
Stroboscopic photography
Stroboscopic photography was conducted to validate the mathematical model by empirically measuring the settling velocities of Pinus sylvestris, Pinus mugho, and Zea mays pollen grains. Fresh pollen grains were dropped down a cylindrical glass tube with an inner diameter of 4 cm while being illuminated by a strobe light flashing at a constant frequency of 16.67 Hz. The glass tube was covered with a plastic disk, with a small, centrally drilled hole that was aligned with the center of the tube. Tapping the grains through the hole minimized any electrostatic charges that may have been applied to the grains by the plastic disk at the top of the tube. Grains were photographed as they descended through the tube, and the trail-like trajectories were then used to calculate settling velocities. Straight pollen trails indicated there was no electrostatic interaction between the tube and the pollen grains, and, hence, no charge was transferred to the pollen. Stroboscopic photography tests were conducted at ambient temperature and relative humidity levels.
Digital imaging techniques were used to better-define the pollen trails and to help differentiate between single pollen grains and pollen clumps. First, using a function created in MATLAB (MATLAB, 2005
), a "control" photograph of the empty tube was subtracted from an image of pollen falling through the tube at the same physical point in the tube. This technique removed background information from the image while leaving the pollen trail data intact. The resulting image was then thresholded, using digital imaging software (e.g., Adobe Photoshop), to create a binary image. Trails were clearly defined in the binary image, and the relative sizes of individual dots in the trails were used to separate the single grains from the clumps of pollen grains.
Aluminum stubs with double-sided tape were placed at the bottom of the tube so that pollen grains falling though the tube could be collected for analysis with the SEM. Captured pollen was measured so that its settling speed could be computationally modeled and compared to the empirically derived settling speeds measured by stroboscopic photography.
RESULTS
Structural characters
A variety of pollen structural characters were measured to use as input variables for the computational model (Table 1), which determines the terminal settling velocities of the pollen grains in this study.
Saccus wall
Most of the pollen grains had similar saccus wall thicknesses. Gothania and Falcatifolium, however, possess relatively thick saccus walls (Table 1). Caytonanthus had a relatively thin saccus wall (Table 1).
Exine and intine
The weighted average of exine thickness was similar for Dacrydium and Pinus (Table 1). Of the fossil pollen grains, Pteruchus had an exine thickness closest to the extant grains. Gothania, fossil Pinus, and Monoletes had the thickest exine (Table 1). Intine thicknesses of Pinus and Falcatifolium pollen grains were similar (Table 1).
Endoreticulations
The degree to which the endoreticulations extended into the saccus was quite variable among the pollen grains studied. Falcatifolium differed from the other grains in that its endoreticulations extended throughout the sacci (Table 1). The percentage of endoreticulations vs. air for the saccate grains was also quite variable. Caytonanthus possessed the highest percentage (Table 1).
Validation of the mathematical model
Stroboscopic photography and modeled pollen
Pollen grains of several species of Pinus and Zea were modeled, and the predicted settling speeds were compared to those empirically determined by stroboscopic photography (Table 2). The settling velocities measured using stroboscopic photography supported and validated the mathematical model's predictions. Independent sample t tests compared settling velocities for pollen of P. sylvestris, P. mugho, and Z. mays from stroboscopic photography to predictions made by the model. For each taxon, the mean prediction from the computational model did not differ from the mean empirical settling velocity (P > 0.38 in all cases). Thus, the model's predictions were not statistically different from empirically measured values (Table 2).
|
Comparison of modeled pollen to published settling speeds
The model predicted a settling speed for P. strobus pollen of 3.07 ± 0.31 cm/s, whereas Niklas and Paw U (1983)
reported a value of 2.1 cm/s for this species. Standard deviations were not available for the P. strobus grains reported in Niklas and Paw U (1983)
; therefore, the difference between the two values could not be statistically assessed. However, the model's predictions for two other species of Pinus (P. sylvestris and P. mugho) were validated by stroboscopic data from this study (Table 2).
An additional factor to consider with stroboscopic photography is that the drag force experienced by objects with low Reynolds numbers is increased by the proximity of walls (Vogel, 1994
). For example, a cylinder with a 2-cm radius has a 0.67% impact on the actual settling velocity. This correction was considered when evaluating results from the stroboscopic experiments in this investigation, and the empirical data continue to validate those of the computational model. However, the degree of this ‘‘wall effect'' on the reported settling speeds in other published stroboscopic studies is unknown.
The settling velocities for Z. mays grains predicted by the computational model and measured using stroboscopic photography differed from the settling velocities previously reported (Aylor, 2002
) using similar methods (Table 2). However, this difference was due to differences in grain sizes. The pollen grains modeled in this study averaged 73.9 x 60.4 µm. When additional pollen was modeled using the grain dimensions reported by Aylor (2002; 90 µm in diameter), the computational model predicted a settling speed of 26.3 cm/s, which is statistically similar to Aylor's (2002) findings.
Flight orientation
Micrograph analysis of the pollen grains from the stroboscopic photography experiments indicate that saccate pollen grains fall with their sacci pointing upward (i.e., backward relative to wind direction). Of the 103 pollen grains that were not in clumps, 92 landed with their sacci pointing upward. This validates the model's assumption of orientation and its choice of drag coefficients.
When modeling flight characteristics of a pollen grain main body without sacci, questions of flight orientation are more difficult to resolve. Objects with symmetric density distributions have no preferred orientation in low Reynolds number flows (Vogel, 1994
), but protrusions or asymmetries create a preferred orientation. The main body without sacci is modeled by a symmetric ellipsoid, but few real pollen grains have a completely symmetric main body. Main body settling speeds predicted by the model (Table 3) represent the average of speeds calculated using the long axis of the main body both perpendicular and parallel to the direction of motion. In no case did the orientation of the main body affect the qualitative distinction as to whether the main body alone would fall faster or slower than the pollen grain with sacci.
|
). Because of their low Reynolds numbers and small mass, pollen grains reach their terminal velocity almost immediately. The drag coefficients used in this computational model are valid only at low Reynolds numbers (less than 1.0). The Reynolds numbers of all of the pollen grains studied here are sufficiently small, except for Monoletes (Table 3). However, the Reynolds number for Monoletes pollen is still within the range for which the mathematical model approximates a correct terminal settling velocity for this grain type (Clift et al., 1978
).
Model predictions
Extant pollen
The computational model predicts that a Pinus grain would fall faster without sacci, at 3.40 cm/s (Table 3), thereby tending to decrease its dispersal distance. The model shows the same result for pollen grains of Dacrydium, although the difference between the settling speeds is smaller. When modeled, the settling speed is 1.22 cm/s with sacci and 1.29 cm/s without (Table 3). This pattern does not hold for Falcatifolium pollen grains. For Falcatifolium, the modeled settling speed of a grain is 2.57 cm/s with sacci and 2.20 cm/s without (Table 3), which indicates that the pollen would be dispersed farther without sacci.
Fossil pollen
Three permineralized taxa with saccate pollen were studied using the computational model (Figs. 26–31, Table 3). The settling speed for fossil Pinus grains, similar to extant Pinus pollen, increased from 2.67 to 3.24 cm/s when the sacci were removed in the model (Table 3). This supports the idea that sacci help to increase dispersal distance of Pinus pollen. Bisaccate pollen of Pteruchus gave the same results as that of fossil Pinus sp. (Table 3); sacci helped to increase dispersal distance. For Pteruchus, the settling speed of pollen was 1.88 cm/s with sacci and 2.16 cm/s without sacci. The settling speed of monosaccate pollen of Gothania increased from 8.45 to 9.98 cm/s when the saccus was removed (Table 3).
Nonsaccate Monoletes pollen grains are large (Figs. 32–33), which contributes to their rapid settling speed of 140 cm/s (Table 3). The model-predicted settling speed of the Monoletes grains is more than one order of magnitude greater than the largest of the other settling speeds calculated in this study.
Caytonanthus pollen grains were mathematically expanded to three dimensions in order to accurately study them. The computational model was run under two scenarios to determine how estimating the displacement of the sacci affected the pollen. With a small amount of overlap between sacci and main body (Fig. 24), the settling speed was calculated to be 0.394 cm/s (Table 3). When more overlap was estimated (Fig. 25), the settling speed was calculated to be 0.402 cm/s (Table 3). After theoretical removal of the sacci, the calculated settling speed was 0.368 cm/s (Table 3), indicating that the modeled pollen grain dispersed farther without sacci.
Influence of structural characters on extant pollen
The mathematical model has also been used to determine how key structural characters of pollen grains affect settling speed. Three-dimensional graphs were created by plotting settling speed (the dependent variable) against two structural components as independent variables.
Plotting main body size against saccus size (Figs. 34, 35) shows similar changes in settling speed for Pinus and Dacrydium. In Pinus and Dacrydium, holding the sacci size constant as the main body size increases causes settling speed to increase (Figs. 34, 35), thereby decreasing dispersal distance. If the size of sacci is increased while the main body is held constant, the settling speed decreases. If both are increased, the settling speed still increases (Figs. 34, 35). By contrast, as the sacci of Falcatifolium grains are enlarged, the settling speed increases (Fig. 36), thereby decreasing dispersal distance. This phenomenon may be more clearly seen in Fig. 39.
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DISCUSSION
Role of sacci in wind dispersal
Increasing dispersal distance
For the pollen grains of Pinus, Dacrydium, Gothania, fossil Pinus, and Pteruchus, the presence of sacci decreased the model-predicted settling speed, which would allow grains to disperse farther. Sacci add a significant amount of surface area to these pollen grains, increasing drag, but add a relatively small amount of mass. The small amount of added mass is due both to the relatively small amount of endoreticulations and the thin saccus wall in these five taxa. If the amount of endoreticulations in Pinus pollen was increased, an increased settling speed would result (Fig. 38).
The sacci on pollen grains of Dacrydium were very small and not large enough to influence settling speeds. For Dacrydium pollen, however, a decrease in settling speed does not require a large decrease in the amount of endoreticulations or a large increase in the size of the sacci (Fig. 37). These relationships corroborated results indicating the small difference between modeled settling speeds of the grain with and without sacci (Table 3). This difference was most likely due to the relatively small size of the sacci in Dacrydium pollen.
Model predictions for Gothania grains also indicate that the saccus contributed to a decrease in settling speed and an increase in dispersal distance.
Decreasing dispersal distance
For pollen grains of Falcatifolium and Caytonanthus, the thickness of the saccus wall and/or the large amount of endoreticulations in the sacci cause the pollen grain to fall faster than it would if it did not have sacci. The endoreticulations in Falcatifolium pollen span the entirety of the sacci cavities and are thicker than endoreticulations in the other taxa studied. As a consequence, Falcatifolium sacci are extensively filled with sporopollenin, which adds considerable mass to the pollen. When the amount of endoreticulations in Falcatifolium was computationally decreased, increasing the size of the sacci did reduce the settling speed as in other taxa (Fig. 39). The increase in the saccus size or decrease in the proportion of endoreticulations needed to reduce settling speed were consistent with the large difference between the settling speeds of Falcatifolium grains modeled with and without sacci (Table 3).
Model-predicted results for Caytonanthus pollen also show a decreased settling speed when sacci are removed (Table 3). This result is due to the robust thickness of endoreticulations in Caytonanthus sacci, which collectively contributes to increased grain mass. In Caytonanthus pollen, the endoreticulations do not extend all the way through the saccus cavity as they do in Falcatifolium, and the saccus wall is thinner than that of Falcatifolium.
Monoletes
The pollination syndrome of Monoletes-producing plants has been hypothesized to be wind-pollinated, animal-pollinated (Taylor and Millay, 1979
), or gravity-pollinated (Niklas, 1992
). Although the wind-pollination hypothesis is partially supported by an alveolate pollen wall, the spongy-alveolate exine in Monoletes is ultrastructurally different from the platelike alveolate exine of wind-pollinated conifer and cycad pollen. The rapid settling speed of Monoletes pollen also indicates that this particular pollen grain was unlikely to have been wind-dispersed. The spongy-alveolate wall may have been a repository for chemicals secreted by the tapetum. These chemicals may have been similar to pollenkitt and may have facilitated animal pollination.
Effects of desiccation and pollen shape on drag
The amount of drag that acts on the pollen grain is related to the overall shape of the pollen grain (Niklas, 1992
), and pollen shape can change with the grain's hydration states (Blackmore and Barnes, 1986
). For example, when desiccated, the sacci of most conifer pollen grains close in around the aperture, protecting against dehydration. The longer the pollen grain spends in the air after it has dehisced, the more susceptible the grain is to water loss. Desiccation can cause a pollen grain to change its shape or orientation in flight, affecting the amount of drag on the pollen grain and consequently the settling speed (Vogel, 1994
; Niklas, 1992
). The shape of the grain as it travels through the air and the face of the pollen grain exposed to the moving air both influence the magnitude of the drag acting upon the grain. These shapes and orientations are crucial components of the mathematical model's predictions of pollen settling speeds. By using a coefficient that accurately depicts the amount of drag on the pollen grain under a variety of hydration states, the model can correctly estimate accurate pollen flight as hydration changes.
Surface ornamentation
Surface ornamentation may provide lift for the pollen grain, as is the case for dimples on a golf ball. However, the aerodynamic effects of surface ornamentation are often ignored when pollen grains are discussed because of their low Reynolds numbers and the difficulties with quantifying surface ornamentation. At the low Reynolds numbers of pollen grains, viscous forces of the air are thought to dominate the aerodynamic behavior of the grains (Vogel, 1994
; Niklas, 1992
). Many authors have stated that surface ornamentation can be ignored for objects with low Reynolds numbers (Vogel, 1994
). Other authors, however, have suggested that surface ornamentation should decrease the settling velocity (Gregory, 1973
; Niklas, 1985a
). This decrease could result from increased lift or reduced density of the pollen grain. Bolick (1990)
hypothesized that surface ornamentation can affect pollen settling velocity if the degree of roughness surpasses some theoretical threshold. Even though Vogel (1994)
has stated that the aerodynamic influence of surface ornamentation at low Reynolds numbers should not have an impact according to conventional considerations, he has also stated that there is evidence in the literature (e.g., Zeleny and McKeehan, 1910
) that the difference between measured settling speeds and those predicted by Stokes' Law is large enough to warrant investigating this phenomenon.
The effects of surface ornamentation on pollen flight have not yet been incorporated into the mathematical model. For a modeled pollen grain that has a slower settling speed without its sacci, the surface ornamentation of the grain could dramatically change the conclusions about the flight properties of the grain.
Conclusion
This study is the first to quantitatively demonstrate the adaptive significance of sacci for the aerodynamics of wind pollination. Sacci do function to decrease the settling speed of pollen grains, with some notable exceptions. These exceptions include pollen grains containing robust and/or extensive endoreticulations, as well as those with thick saccus walls.
The mathematical model is also important in paleobotany and in understanding the reproductive biology of key groups. This study showed that Monoletes pollen grains fall much too rapidly to be adequately dispersed by wind. Gothania pollen also settled faster than the other saccate grains. This investigation has demonstrated that compressed pollen grains can be mathematically expanded to allow for more in-depth study and analysis. The new mathematical model has also been shown to yield valid results even when sample sizes are small, which facilitates the study of fossil pollen grains. Error propagation analysis indicates that the unavoidable uncertainties associated with measurements of fossil pollen are not large enough to prohibit drawing meaningful conclusions from their modeled settling speeds.
Finally, it is clear from the current study that the structure and function of pollen sacci are the result of intense selection. Sacci are now known to serve both aerodynamic and buoyancy roles among the extant and extinct seed plants that produce saccate pollen.
FOOTNOTES
1 The authors thank I. M. Lindevald (Truman State University) for contributions to the initial computational model, as well as T. Barcus and T. W. Sorrell (Truman State University) for their help with stroboscopic photography experiments. The authors also thank the Missouri Botanical Garden for a loan of herbarium material. This study was supported in part by Truman State University and the National Science Foundation (DUE-0337760 and DUE-0436348). 
4 Author for correspondence (josborn{at}tcnj.edu
)
5 Current address: Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, Kansas 66045 USA
6 Current address: School of Science, The College of New Jersey, Ewing, New Jersey 08628 USA
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