Pollen work with David Smith

Doubling Volume

Consider doubling the volume of rodlike pollen by doubling the length. Our conclusion: it's a bit better to have two small pollen instead of one big one. The penalty becomes slightly smaller for longer pollen, but as the volume increases the curve eventually turns downward (shortly above pseudovolume 2000). Differences between various sizes of pollen are quite small and are unlikely to determine the physical size of pollen.

Also note that for this calculation, we are assuming that it takes as much energy to produce a longer pollen, but that might not be true: the search vehicle pollen-balls-with-filaments, may be able to double its length with only a small cost. In that situation, it would clearly be cost-effective to double the length of the pollen.

Graph: ratio of hit probability of big/small,
Small dimension held fixed at .22,
psuedo-volume ( vol/(4 Pi) ) range is up to 3000,
distance away is a million units.

Hit Likelihood

This graph shows how various pollen shapes compare. Rodlike pollen is for x-coordinate bigger than 1, disklike is less than 1. The minimum fitness is for spherically shaped pollen. A second graph that uses the same miminum dimension for both rod and disk shaped pollen demostrates that if the limiting factor is structural, then pollen should be rodlike because it has a greater likelihood of a hit. Whether pollen that initially evolved to be disklike would later evolve to be rodlike (stability) is a question answered by nonsymmetric calculations: the disk shape can improve by becoming even slightly rodlike. Hence would incrementally become rodlike. These calculations will be shown in a later publication.

distance away is 1000, pseudo-volume is 1,
short dimension ranges over .1 and 10
(shape1.ps), and .5 and 4 (shape2.ps).

Distance Effects

This graph considers distance effects. After rescaling (if distance out increases by a factor of 10, multiply hit probability by 10 to compensate), the graphs look remarkably similar. We conclude that the relative advantage of a particular (ellipsoidal) pollen shape is at most slightly affected by distance.

HOWEVER, the absolute differences are decreasing (just not faster than the probability is decreasing).

Graph: distances 100, 1000, 10000, same other data as shape1.ps

Comparison to Simulation

Results of a comparison to Supercomputer data (hits per 10,000), a random walk simulation:

Warning: this graph doesn't show the differences between the two calculations when considering disklike pollen, but the picture for that case is similar to this one.

Since the simulation used only a finite number of steps, of course that hit probability will be lower than our calculation. Note that the shapes are qualitatively different, with our calculation showing a greater relative advantage for rodlike pollen. This comparison suggests that as the number of steps taken increases (longer pollen survival time), rodlike pollen becomes relatively more effective.

Graph: distance is 75, psuedo-volume is 2, range in singular dimension is 1/32 to 32.

Watch this space for further comments

More detailed comparisons of rodlike and disklike pollen will be covered in a later paper. Our exploration will include ellipsoids without a line of symmetry. Questions about ecology are best addressed to David Smith.