On Allometry and Relative Growth in Evolutionary Studies

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y with respect to time (Lande, 1985; Shea,. 1985):. Thus the "allometric coefficient" k is .... change (Gould, 1977). The heterochronic formalism of Alberch et al.
On Allometry and Relative Growth in Evolutionary Studies Department of Ecology and Evolutionary Biology,

University of Arizona, Tucson, Arizona 85721

In his critique of contemporary morphometry, Blackstone (1987) asserts that the conceptual, pattern-oriented framework of allometry ignores developmental processes and is therefore inappropriate for studies in evolutionary morphology. His intent is to emphasize the limitations of allometric studies in the understanding of evolutionary processes. In doing so, however, Blackstone focuses on a descriptive approach to allometry and obscures several fundamental issues. In particular, he claims that information on rates of growth in units of chronological time is necessary to deduce patterns of evolutionary change in developmental processes, and that such information is ignored in allometric analyses. He traces this disregard to a historical departure from the study of underlying processes of growth to the purely descriptive study of size and shape. In contrast, I will argue first that chronological time does not have theoretical or operational priority over other estimates of biological age, such as body size, for the description and comparison of growth patterns; and second that the apparent historical trend he describes corresponds instead to a logical progression from the study of growth rates of individual traits per chronological time, to growth rates of the same traits in terms of more robust estimates of biological time.

can be shown to be the solution of the differential equation relating the specific growth rates of two mensural traits x and y with respect to time (Lande, 1985; Shea, 1985):

Thus the "allometric coefficient" k is equal to the ratio of the specific growth rates of x and y (Reeve and Huxley, 1945). That is, k is an index of the displacement in time of the growth process or "curve" of one trait in relation to that of the other trait (Laird et al., 1968). To the extent that the specific rates remain proportional to one another during growth, k remains constant. Non-proportional rates will produce nonlinear bivariate logarithmic plots, indicating that the allometric coefficient is either a continuous function of age (Creighton and Strauss, 1986) or is multiphasic (Strauss and Fuiman, 1985). That Blackstone has overlooked this basic identity is revealed by his previous study of relative and absolute growth rates in hermit crabs (Blackstone, 1986),in which he recommended the ratio of specific growth rates (calculated across molts) as being more informative than the bivariate ALLOMETRY AND SPECIFIC

allometric coefficient for determining exGROWTH RATES

perimental changes in growth pattern. If As noted by Blackstone, Huxley's (1932) the time intervals over which rates are calderivation of the allometric function was culated are compensated so as not to inbased explicitly on the process of multi- troduce artifactual heterogeneity into the plicative cellular growth (Reeve and Hux- analysis, then the rates-ratio and the biley, 1945; Laird, 1965; Balinsky, 1970). Al- variate allometric coefficient should yield though often claimed to be merely an identical results. His statistical justification empirical description, Huxley's power for the rates-ratio, in terms of demonstrating a "much stronger effect for a given function

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sample size," is based on a faulty pooling of data and assignment of degrees of freedom.

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The assessment of "body size," in relation to which we can estimate rates of change of particular characters, is a controversial problem that has been apOTHER ESTIMATES OF BIOLOGICAL AGE proached in a number of ways. Although Data on absolute developmental rates can various measured attributes (e.g., lengths be important in evolutionary studies, but or weights) have been used as estimates of we cannot assume that they represent size, and therefore of biological age, it is "true" rates of biological change. Specific generally acknowledged that composite rates are scaled to chronological time, measures which estimate "robustness" are which is only one of a variety of opera- preferable to any particular traits. This is tional estimates of biological time, the ac- so for two reasons: because individual tual scale onto which ontogenetic events characters are generally "noisier" than should be mapped. The observations that composite traits, exhibiting more individdevelopmental rates are highly nonlinear, ual or random variation that is averaged are temperate-dependent, and can be mod- out in the composite; and because any parified by a host of other environmental vari- ticular character is likely to be allometric ables and metabolic states (e.g., Ursin, 1979; with respect to overall body size. For exTaylor, 1981), indicate that chronological ample, "standard length" in fishes has long time has only an approximate relationship been used by ichthyologists as a measure to biological age. If such confounding vari- of body size, but is generally negatively ables are experimentally standardized or allometric with respect to overall size due observationally accounted for, then rela- to coordinated increases in body depth and tive growth rates of mensural traits may width (Chernoff and Miller, 1982; Humbe informative. However, time-dependent phries, 1984). absolute rates are approximations. The need for a composite measure of A reasonable criticism of bivariate allo- body size has led directly to the application metric coefficients estimated from men- of multivariate statistics, particularly prinsural data is that, because these ratios are cipal components analysis. For ontogenetic independent of the absolute rates, direct data taken from a growth series of a single comparisons of specific growth rates pro- sample or population, Jolicoeur (1963a) vide more complete information about demonstrated that the eigenvector (first growth-related changes (Blackstone, 1986). principal component) extracted from the For example, if the allometric coefficient of covariance matrix of logarithmic values detrait x with respect to y is less than 1, then scribes relative changes in the measured the rate of change of x may be decreasing characters during growth. Thus the during growth, or that of y-maybe increas- "scores" (projections) of individuals on the ing, or the rates may be simultaneously but first component are measures of their overdifferentially changing in either direction. all body size, while the loadings (direction Without some estimate of the relationship cosines) are estimates of the rates of change of the traits to biological time, we have no of individual characters with size (Leamy way of knowing which of these is the case. and Bradley, 1982; Lande, 1985). Note in However, for situations in which reliable particular that the information about relchronological data are lacking, as for most ative bivariate growth is not lost among samples from natural populations, this is the multivariate coefficients. For any two precisely the advantage of multivariate al- characters, the ratio of their loadings on lometric coefficients over bivariate ones. the component is proportional to the coefMultivariate allometries are rates of growth ficient k in Huxley's function (Jolicoeur, estimated with respect to overall body size, 196313). The pairwise coefficients are simwhich is a different estimate of biological ply rescaled by an appropriate constant of age than chronological time, but one more proportionality, chosen so that the overall directly tied to growth. rate of change is taken to be isometry (Hills,

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1982; Shea, 1985). Nor is the analysis re- events (Kluge and Strauss, 1985). Use of duced to relative rates only. Principal com- the model allows specification of develponent loadings are regression coefficients opment in terms of three basic control paof characters on components, therefore the rameters: age at onset, age at offset, and unscaled loadings on the first component mean rate of change. Ha11 (1975, 1984; see of ontogenetic data are absolute rates of also Katz, 1980) has considered mechanisms that might govern these parameters change with respect to size. The multivariate context of allometry during skeletal development, and suggests encompasses naturally the multiple pat- that the number and mitotic activity of the terns of differential growth (i.e., growth cells in the initial skeletal condensation, gradients and allometric shape-change), regulated by inductive tissue interacleading to the characterization of allome- tions, may provide a fundamental level of try as the "study of the consequences of control. The important point is that the size for shape" (Bookstein et al., 1985). The quantitative extrapolation of this basic fact that these patterns often differ system- ontogenetic model of heterochronic evoatically among even closely related species lutionary change (Bookstein et al., 1985; accounts for the extension of the basic Creighton and Strauss, 1986) follows natmodel to "static" or "evolutionary" scal- urally from multivariate methods for the ings of form among species, which unfor- analysis of changing form. Sophisticated mathematical and statistitunately have also been termed allometry (Cheverud, 1982; Lande, 1985). Thus it is cal descriptions of growth may be as "renot true that "allometry has become the ductionist" and as related to the dynamics study of size and shape" to the exclusion of developmental systems as were Huxof growth processes (Blackstone, 1987).The ley's (Skalak et al., 1982; Gordon, 1983), study of allometry has greatly increased in though the descriptions are developed at scope over the past 50 years to encompass an appropriate level of morphogenetic a much wider range of scaling phenomena. complexity (Alberch, 1980). Blackstone's But the slight confusion that still exists in view that "the sizelshape framework for the literature as to the relationships of scal- allometry is based on an explicit rejection ing at various taxonomic levels does not of deductive interpretations of growth" is change the fact that ontogenetic allometry based on an overly simplistic interpretais still very much grounded in the study tion of the role of morphometry in evoof cellular behavior, as is the large and lutionary studies. parallel literature on the scaling of physACKNOWLEDGMENTS iological processes (Platt and Silvert, 1981; The use of multivariate factors as estimates of biCalder, 1983; Peters, 1983). HETEROCHRONY

ological time and the extension to higher levels of scaling phenomena have recently been discussed by Bookstein et al. (1985), and many of the points emphasized here are treated in more detail in that volume. I thank Barry Chernoff and Marilyn Houck for comments on the manuscript.

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