A general model for allometric covariationin botanical form and functionCharles A. Price*†, Brian J. Enquist*‡, and Van M. Savage§*Department of Ecology and Evolutionary Biology, University of Arizona, Tucson, AZ 85721;‡Sante Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501;and§Department of Systems Biology, Harvard Medical School, Boston, MA 02115Edited by Karl J. Niklas, Cornell University, Ithaca, NY, and accepted by the Editorial Board June 14, 2007 (received for review March 11, 2007)The West, Brown, and Enquist (WBE) theory for the origin ofallometric scaling laws is centered on the idea that the geometryof the vascular network governs how a suite of organismal traitscovary with each other and, ultimately, how they scale withorganism size. This core assumption has been combined with othersecondary assumptions based on physiological constraints, such asminimizing the scaling of transport and biomechanical costs whilemaximally filling a volume. Together, these assumptions givepredictions for specific ‘‘quarter-power’’ scaling exponents in bi-ology. Here we provide a strong test of the core assumption of WBEby examining how well it holds when the secondary assumptionshave been relaxed. Our relaxed version of WBE predicts thatallometric exponents are highly constrained and covary accordingto specific quantitative functions. To test this core prediction, weassembled several botanical data sets with measures of the allom-etry of morphological traits. A wide variety of plant taxa appear toobey the predictions of the model. Our results (i) underscore theimportance of network geometry in governing the variability andcentral tendency of biological exponents, (ii) support the hypoth-esis that selection has primarily acted to minimize the scaling ofhydrodynamic resistance, and (iii) suggest that additional selectionpressures for alternative branching geometries govern much of theobserved covariation in biological scaling exponents. Understand-ing how selection shapes hierarchical branching networks providesa general framework for understanding the origin and covariationof many allometric traits within a complex integrated phenotype.allometry 兩 fractal 兩 metabolism 兩 scaling 兩 traitsSince the pioneering work of Julian Huxley (1), questions con-cerning how selection influences specific traits within integratedphenotypes have been a prominent focus in comparative biology (2,3). The phenotype is a constellation of traits that often covary witheach other during ontogeny. Further, organism size is a central traitthat influences how most biological structures, processes, anddynamics covary with each other (4, 5). The dependence of a givenbiological trait, Y, on organismal mass, M, is known as allometry.Allometric relationships are often characterized by power laws (1)of the form, Y ⫽ Y0M␪, where␪is the scaling exponent and Y0is anormalization constant that may be characteristic of a given taxon.A sampling of intra- and interspecific data reveals that the centraltendency of␪often approximates quarter-powers (4, 5) (e.g., 1/4,3/4, 3/8, etc.), although for any given relationship considerablevariability may exist (6).West, Brown, and Enquist (7–9) hypothesized that the value of␪for many biological allometrie s arises from the geometry ofvascular networks and resource-exchange surfaces (e.g., cardiovas-cular or plant vascular systems). The core assumption of this theoryis that many organismal, anatomical, and physiological traits arelinked mechanistically by allometric scaling of the vascular network.For example, for plants, whole-plant carbon assimilation or grossphotosynthesis, P, vascular fluid flow rate, Q0, and the number, nL,and mass, ML, of leave s are all assumed to vary or scale propor-tionally with one another, and ultimately with plant mass (M), to thepower␪as P ⬀ Q0⬀ nL⬀ ML⬀ M␪. In addition, several otherhydrodynamic and anatomical attributes of plant hydraulics andbranching morphology are also determined by␪(7, 8, 10–14):specifically, the value of␪⫽ 1/(2a ⫹ b), where a and b characterizethe geometry of the vascular network within a given species. Withinthe model, network geometry and gross morphology are charac-terized by three parameters:␥k⬅lk⫹1lk⬅ n⫺b␤k⬅rk⫹1rk⬅ n⫺a␤៮k⬅rT,k⫹1rT,k⬅ n⫺a៮ /2,[1a–c]which respectively refer to the ratios of branch lengths, externalbranch radii, and internal tube radii (e.g., xylem vessels or trachearyelements) between k adjacent branching levels (Fig. 1). The valueof n is the furcation ratio or number of daughter branches perparent branch. Note that we differ in notation from the originalmodel (8) that defined the branch radii ratio as n⫺a/2, the internaltube radii as a៮ , and the lengths ratio as n⫺1/3. Importantly, here weallow the length and the radii ratios to be continuous traits definedby arbitrary powers b and a, re spectively.The origin of quarter-power scaling emerges when a and b takespecific values, a ⫽ 1/2 and b ⫽ 1/3, thus yielding␪⫽ 3/4. This␪⫽3/4 rule emerges from four assumptions: (i) that the network isvolume-filling; (ii) that the minimum total work to move fluidthrough the network corresponds to minimizing the hydrodynamicresistance in a single tube; (iii) that properties of an individual leaf(e.g., ML, PL, QL) are independent of plant size; and (iv) thatbiomechanical constraints are uniform (8). Extensions of The West,Brown, and Enquist (WBE) theory for the origin of allometricscaling laws model have shown how factors secondary to organismsize, such as temperature and stoichiometry (15), can influenceresidual variation or the normalization (Y0) of allometric scalingrelationships.Recently, several prominent critics (6, 16) have argued that theWBE model cannot explicitly account for the range and origin ofinter- and intraspecific variability in allometric exponents. Further,the assumption of the primacy of the vascular network, althoughimplicit in the original work, has not been explicitly tested (17).Indeed, although detailed treatments of WBE have been applied tothe origin of the␪⫽ 3/4 rule in the vascular networks of animals(7) and plants, including stem branching (8, 12) and leaf venous (13)networks, it is not clear whether the WBE model can account forall, or even most, of the observed variation in allometric exponentsor for how traits covary allometrically within integrated phenotypes.Author contributions: