1 EXPONENTIAL INCREASE IN DESCRIPTION RESOLUTION Richard H. Zander Technical Report Res Botanica December 2, 2014 Missouri Botanical Garden P.O. Box 299 St. Louis, MO 63166 USA email: richard.zander@mobot.org Modern taxonomy makes use of expensive, dissecting and compound microscopes, spreadsheet coordination of traits and collections, analytic software, giant searchable public databases for literature, nomenclature and specimen archives, and, recently, publication of new names through the World Wide Web. Yet our measurement of taxonomic traits remains primitive. The main metric instrument is essentially a stick with equally spaced notches, or the equivalent in an ocular micrometer. Morphometrics is commonly considered a specialty field focusing on characterizing shapes. Yet there are applications that should be more generally appreciated. Allometric analysis is of particular importance in evolutionary studies, and a systematics that purports to be based on evolutionary principles needs to incorporate relevant techniques and interpretations, or at least describe the important features directly. With animals, many formulae have been developed correlating variation in body size and mass with ecological and physiological limitations. For instance, the square-cube law (originating with Galileo Galilei in 1638) predicts that the doubling of length of an organism isometrically increases the surface area four-fold, and volume and mass eight-fold. Most standard taxonomic measurements in botany are linear, e.g., leaf or petal length, cell length and width, and spore, seed or trunk diameter. But the three spacial dimensions may be treated as linear, areal, and volumetric. If one uses a linear dimension to distinguish an important areal or volumetric difference between species, then resolution is lost. For example, a quadrate organ in one species may average 3 cm per side, while in another species the same organ may average 4 cm per side. The difference in linear measure is not much and may encourage misidentification. Yet the areas average 9 and 16 square cm, quite a difference, while the volumes, if the organ were square, would be 27 and 64 cubic centimeters. The average differences in terms of resolution increase exponentially. There are two reasons why this may prove important. First, increasing differential resolution of an organism s features enhances the distinguishing of taxa. Second, evolution may act preferentially on the area or volume of a trait, and taxonomy reflecting perceived evolutionary differences needs to document such differences directly if possible. Empty resolution is exemplified by considering two linear measurements, 3 and 4. There is a single unit difference. Squaring both yields 9 and 16, and there now is a 7 unit difference. It is empty because there is no anchor in reality. Reality is in part psychological and part epistemological. Although the difference of 1 linear unit and that of 7 square units is essentially the same, converted by an arbitrary arithmetic operation, the anchor is the viewpoint of comparing real squares instead of real lines, and the operation has meaning. A square of size 16 units is nearly twice the size of one of 9 units and that is a real difference.
Linear measures can be made more exact with certain standard techniques. Rather than measuring one item (say a cell) at a time, a linear set can be measured and the length is then divided by the number of items. This avoids compounding mis-measures such as measuring the lumen of a cell, not its actual width including the cell wall. Linear measurements are either averages, i.e., ca. 5 cm, or are ranges, meaning that most measured values may be expected to occur between the low and high measures. This commonly is perceived as an approximation of plus or minus two standard deviations, that is, 0.95 of the distribution of values. Two problems are immediately evident: exponential error and direct measurement. If one has a particular range of error in measuring linear dimensions, then multiplying length by width and perhaps again by depth also multiplies the error for each linear measurement, which must be reflected in the description of the areal or volumetric dimensions. For an example, with no error, a square 3 cm on a side has an area of 9 square cm, while a square 4 cm on a side has an area of 16 cm. The difference in area is 7 square cm. The distinction is exponential, which is the point of this paper. But if one has a total linear error of 0.5 cm in both length and width of squares 3 and 4 cm on a side, the 3 cm square now has a range of (2.75 2 to 3.25 2 ) 7.56 to 10.56 square cm, while the 4 cm square ranges from (3.75 2 to 4.25 2 ) 14.06 to 18.05 square cm. The difference is now (14.06 minus 10.56) 3.5 square cm, which compared to 7 square cm is in fact not itself a severe handicap to estimations of distinction, given that the areas of the squares average only 9 and 16 square cm. Thus, if the squares were measured linearly, as per standard taxonomic practice, the difference between their, say, lengths would be 1 cm, while if converted to areas, the difference would be 7 square cm. With an error of 0.5 cm for each length, the difference between the two squares would be (3.75 minus 3.25) 0.5 cm for linear measure but 3.5 square cm for areal measure. Thus, simply multiplying length times width can result in a better distinction even with some reasonable error. Error is also magnified when averages are used. A range of measurements of a spore diameter from 8 to 10 µm translates to an average volume (where V = 4/3πr 3 ) of 396 µm 3, yet a spore diameter with a range of 9 to 11 µm gives an average volume of 539 µm 3. Although the measurements overlap in range, the difference between the averages 396 and 539 must be held to be not particularly significant when the measured difference between the averages of the two spore diameters is only 1 µm, a difficult distinction to make with common tools. There is no clear gap between the overlapping ranges of the two spore sizes. Clearly some judgment is necessary. Problematically, measuring area or volume directly is technically not particularly easy (or we would have been doing such by now as standard practice); see, for example, discussion by Reponen et al. (2001). Assume we are measuring leaf cells, as is often the case with bryophytes. Placing a square grid (ocular micrometer grid) over the image of many leaf cells is possible if one counts the number of cells completely included, then adds half the number of cells partially included. An old way to measure area is to copy the object, e.g., a leaf, onto paper, then cut out the paper and weight it in reference to the weight of a square of the same paper of known area, see also the graph paper method of Pandey and Singh (2011). Another way to measure area is by occlusion. For example, one may fill with color 10 of the cells in an image of areolation in a graphics program. Various areas covered by that color might be measured directly by the computer. A variant of this is used by the ImageJ scientific image analysis system (Ferreira & Rasband 2012). Different methods, of course, introduce different kinds of error. 2
What about measuring the volume of a small organ? Displacement is the usual method of measuring volume directly (e.g. Huxley 1971), yet for small objects this is not (now) feasible. Perhaps if a fluid like electricity or radiative gas were displaced, a reduction in activity might be translated to volume displaced, see also the automated particle sizer (Chapella a991.). One way to measure thickness of a transparent object (e.g., cell) under a microscope is to focus on the near, then the far surfaces, and gauge the distance traveled by the objective using the focusing knob (Travis et al. 1997). It may be that area can be measured directly and then thickness measured separately and multiplied by the area to get the volume. Studies of fungi often use spore volume measurements or at least estimations, but the techniques are complex (Foster & Bills 2011: 330). Spore size ranges in the moss genus Tortula (Pottiaceae) in North America north of Mexico (Flora of North America Editorial Committee 2007) are presented in Table 1. Given that the species of this genus occur in often patchy microhabitats, spore size might be of critical importance in reproduction. There is a great range of sizes. Precinctiveness, the restriction of diaspores to the immediate area of the parental plant (Carlquist 1966), is of importance in patchy environments, and implies K-selection. Species of Tortula with larger spores, these being less likely to disperse far from the originating sporangium, may well correlate with patchy environments, but the appropriate field work has not yet been done. One may note that species of Tortula with larger spores generally have the peristome reduced or absent, which would correlate with more restricted dispersal. It is clear in Table 1 that the maximum in the range of spore diameter of most species is matched rather well with the minimum value for other species when species are arranged by spore size. There are no clear size classes. If the size ranges are generally meant to represent second standard deviations on both sides of the average, that is, a range of 95 percent of the variation, then comparisons of diameter ranges must be between species with non-overlapping size ranges. For example, as in Table 1, micrometer ranges of 8 10 versus 11 15, or 11 15 versus 18 23, or 18 23 versus 23 30, or 23 30 versus 33 40. There are effectively only five size classes following this line of reasoning. The internal limits are variable since they are established on a sliding scale, but the ranges of non-overlap for spore diameter remain about five. Given the different size classes determined for diameter, one may note that diameters do not overlap in about five general ranges while when converted to spherical spore volume, such volumes exhibit eight non-overlapping size classes, an increase in resolution. The fact that with reasonable error simple calculation of area from length and width still increases resolution, one can expect that size classes of spore volumes calculated from diameters retains a practical value in both taxonomic distinction and evolutionary theory involving spore volume. A fact of concern is that organic spheres have various deformations from the ideal, for instance, moss spores may have a trilete ridge, or they may be elliptical or reniform in shape. As pointed out by Kiklas and Spatz (2012: 362) the ideal relationship between area and volume is that the surface area of an object scales as the 2/3 power of the object s volume. The allometry of organisms, however, in practice scale the area to the 3/4 power of the object s volume. The volume is thus less than that calculated for the perfect sphere. Comparisons using actual volumes may be less distinguishing than such using ideal spheres, but the increase in resolution is doubtless valuable in both. Needed now are imaginative technicians and inventors who might devise new, inexpensive, and simple tools to measure area and volume directly in the course of taxonomic 3
4 work. In the past, a taxonomist was content with microscope, herbarium cabinet, and a stick with equally spaced notches. An inexpensive replacement for the linear scale would increase resolution of descriptions exponentially and enhance direct evaluation of the evolutionary significance of taxonomically important traits. BIBLIOGRAPHY Carlquist, S. 1966. The biota of long-distance dispersal. III. Loss of dispersibility in the Hawaiian flora. Brittonia 18: 310-335. Chapella, I. H. 1991. Spore size revisited: Analysis of spore populations using an automated particle sizer. Sydowia 43: 1 14. Ferreira, T. & W. Rasband. 2012. Image J User Guide. IJ 1.46r. http://rsbweb.nih.gov/ij/docs/guide/user-guide.pdf. Foster, M. S. & G. F. Bills. 2011. Biodiversity of Fungi: Inventory and Monitoring Methods. Academic Press, Burlington, Massachusetts. Huxley, P. A. 1971. Leaf volume: A simple method for measurement and some notes on its use in studies of Leaf growth. Journal of Applied Ecology 8: 148 153. Niklas, K. J. & H.-C. Spatz. 2012. Plant Physics. University of Chicago Press, Chicago. Pandey, S. K. & H. Singh. 2011. A simple, cost-effective method for leaf area estimation. Journal of Botany 2011(Article ID 658240): 1 6. Reponen, T., S. A. Grinshpun, K. L. Conwell, J. Wiest & M. Anderson. 2010. Aerodynamic versus physical size of spores: Measurement and implication for respiratory deposition, Grana 40: 119 125, DOI: 10.1080/00173130152625851 Travis, A. J. & S. D. Murison, P. Perrry & A. Chesson. 1997. Measurement of cell wall volume using confocal microscopy and its application to studies of forage degradation. Annals of Botany 80: 1 11. Table 1. Spore sizes in Tortula (Pottiaceae) of North America north of Mexico. Measurements in micrometers, where 1000 µm = 1 centimeter, and cubic micrometers, where (1000 3 ) 1,000,000,000 µm = 1 cubic centimeter. The larger spore sizes correlate with capsule or peristome reduction, implying K-selection and precinctiveness. Spores are not in clear size classes by absolute measurements but diameters do not overlap in about five general ranges while volumes exhibit eight non-overlapping size classes. Species Spore diam FROM Spore diam TO Ave diam Diam size class Spore vol FROM Spore vol TO Ave vol Vol size class Tortula plinthobia 8 10 9 1 268 523 396 1 Tortula porteri 8 10 9 1 268 523 396 1 Tortula obtusifolia 9 11 10 1 382 697 539 1 Tortula muralis 8 12 10 1 268 905 586 1 Tortula amplexa 9 13 11 1 382 1150 766 1 Tortula brevipes 10 13 11.5 1 523 1150 837 2 Tortula californica 11 14 12.5 2 697 1436 1067 2 Tortula inermis 11 15 13 2 697 1767 1232 2 Tortula cuneifolia 12 15 13.5 2 905 1767 1336 2 Tortula bolanderi 13 15 14 2 1150 1767 1458 3
5 Tortula mucronifolia 12 18 15 2 905 3053 1979 3 Tortula subulata 12 18 15 2 905 3053 1979 3 Tortula guepinii 13 18 15.5 2 1150 3053 2101 3 Tortula atrovirens 15 18 16.5 2 1767 3053 2410 4 Tortula deciduidentata 15 18 16.5 2 1767 3053 2410 4 Tortula lanceola 15 18 16.5 2 1767 3053 2410 4 Tortula laureri 18 23 20.5 3 3053 6369 4711 5 Tortula hoppeana 20 23 21.5 3 4188 6369 5278 5 Tortula protobryoides 20 25 22.5 3 4188 8179 6183 5 Tortula leucostoma 23 25 24 3 6369 8179 7274 6 Tortula systylia 22 30 26 3 5574 14134 9854 6 Tortula nevadensis 23 30 26.5 4 6369 14134 10251 6 Tortula truncata 25 30 27.5 4 8179 14134 11156 7 Tortula cernua 25 35 30 4 8179 22444 15311 7 Tortula modica 30 35 32.5 4 14134 22444 18289 8 Tortula acaulon 33 40 36.5 5 18812 33502 26157 8