## Introduction

Land plants inhabit diverse environments and are characterized by an impressive range of morphological, architectural, physiological and life-history adaptations. While fascinating, this wide range of adaptive strategies presents difficulties for biologists (Givnish 1986; Bazzaz & Grace 1997). In particular, such functional complexity appears to challenge the development of general predictive models built on shared principles governing plant form and function. Nevertheless, theoretical work has indicated that the evolution of diverse branching morphologies can be generated by selection acting to optimize plant function within biomechanical and physical constraints (Horn 1971; Niklas 1994). Recently, the fractal branching model of West, Brown and Enquist (WBE or fractal model) has suggested that general principles governing the scaling of biological resource-distribution networks result in many predictable attributes of biological form and function (West, Brown & Enquist 1997, 1999a, 1999b; Enquist & Niklas 2002; Niklas & Enquist 2002).

The WBE model assumes that evolution by natural selection has acted to maximize the scaling of surfaces (such as leaf, root, lung or gut area) where resources are exchanged with the environment, while simultaneously minimizing the scaling of internal transport distances or resistance. The model also assumes that these surface areas ultimately supply energy-harvesting units (such as the leaf, mitochondria or chloroplast) that are invariant with changes in plant size (West *et al*. 1999b). For many major clades (multicellular animals, vascular plants, etc.) such selection has resulted in a fractal-like, hierarchical vascular distribution network. The model predicts many functional relationships governing variability in organism form and function in the form of a power-law or allometric relationship, *Y* = *b*_{0}*M*^{b}, where *Y* is a trait of interest, *M* is the mass or size of the plant, *b*_{0} is a normalization constant that may vary with the trait of interest and taxonomic level, and *b* is a scaling exponent. The WBE model predicts values of *b* will be multiples of a quarter-power (1/4, 3/4, 3/8, 11/12, etc.: West *et al*. 1997; 1999b; West, Brown & Enquist 2000). Most importantly, a central prediction of the WBE model indicates that, for three-dimensional networks, resource-exchange areas (such as photosynthetic surface area of plants), *A*_{E}; and the total number of chloroplasts, *N*_{chloro} (or mitochondria, *N*_{mito}), will scale as *M*^{3/4}.

The general model has been extended to predict a suite of specific whole-plant physiological and morphological characteristics (West *et al*. 1999a; Enquist, West & Brown 2000). An implicit assumption of the WBE plant model is that external plant morphology parallels the internal resource-distribution network, and consequently that this morphology is volume-filling (West *et al*. 1999a). This assumption works well for trees and shrubs, particularly angiosperms and conifers (West *et al*. 1999a). However, many plant clades contain taxa for which the above-ground architecture does not parallel the internal vasculature (Fig. 1). Usually these plants lack an obvious fractal-like branching morphology (e.g. succulents, palms: Gibson 1973, 1976; Gibson & Nobel 1986). Here we demonstrate that, with minor modification, the general allometric model proposed by West, Brown and Enquist can be extended to predict a suite of physiological and morphological scaling relationships in plants that do not exhibit volume-filling external branching. We then test the extended model's predictions utilizing data from a biometric database on Sonoran Desert succulent plants (Table 1).

Family | Genus | Species | Author | Sample number |
---|---|---|---|---|

Cactaceae | Carnegia | gigantea | (Engelm.) Britt. & Rose | 67 |

Cactaceae | Echinocereus | engelmanii | Engelm. | 33 |

Cactaceae | Ferocactus | wislizenii | (Engelm.) Britt. & Rose | 57 |

Cactaceae | Mammilaria | microcarpa | Engelm. | 149 |

Cactaceae | Opuntia | acanthocarp | Engelm. & Bigelow | 31 |

Cactaceae | Opuntia | arbuscula | Engelm. | 17 |

Cactaceae | Opuntia | engelmanii | Salm-Dyck | 27 |

Cactaceae | Opuntia | fulgida | Engelm. | 23 |

Cactaceae | Opuntia | leptocaulis | DC. | 23 |

Agavaceae | Agave | chrysantha | Peebles | 18 |

The quarter-power exponents derived in the WBE model stem from the assumption that the fractal-like network is space- or volume-filling, and the cross-sectional area of the network is preserved across branching generations (West *et al*. 1997, 1999a). This can be expressed in terms of three variables: the ratio of daughter to parent branch lengths, γ ≡ *l*_{k+1}*/l*_{k}; the ratio of daughter to parent branch radii, β ≡ *r*_{k+1}*/r*_{k}; and the number of daughter branches per parent branch *n*, typically 2 in plants [terminology follows West *et al*. 1997 with the exception that the subscript for terminal units is *p* (petiole) as opposed to *c* (capillaries)]. For a volume-filling fractal object, the relationship between the branch-length ratio, and the branching ratio is γ ≈ *n*^{−1/3}, independent of the branching level. Similarly, the relationship of the branch radii ratio β to the branching ratio *n* is β ≈ *n*^{−1/2}, again independent of branching level.

Terminal units (petioles) within the above-ground plant branching network are assumed to be size-invariant across organisms. Thus, within a plant with hierarchical branching, the total number of petioles, *N*_{p}, expected to scale with mass (*N*_{p}∝*M*^{3/4}; West *et al*. 1997). Given the invariance of petiole radius (*r*_{p}), the radius of the basal stem (*r*_{o}) can then be expressed as:

where *N* is the total number of branches and *N*_{p} is the total number of petioles. With *N*_{p}∝*M*^{3/4}, we have *r*_{o}∝ (*M*^{3/4})^{½}*r*_{p}, yielding *r*_{o}/*r*_{p}∝*M*^{3/8}, or *r*_{o}∝*M*^{3/8}, (*D*∝*M*^{3/8}) due to the assumption of invariance in *r*_{p}. Similarly:

or *l*_{o}∝ (*M*^{3/4})^{1/3}*l*_{p}, where *l*_{o} is the length of the basal stem and *l*_{p} is petiole length. This gives *l*_{o}/*l*_{p}∝*M*^{1/4}, or *l*_{o}∝*M*^{1/4} (*H *∝*M*^{1/4}) again due to the assumed invariance of *l*_{p}. Thus the WBE model predicts the scaling of plant height (*H*), basal stem diameter (*D*) and mass (*M*) (Table 2). Statistical analysis of allometric data from trees and shrubs characterized by volume-filling branching generally support these predictions (West *et al*. 1999a; Enquist & Niklas 2002).

Predicted slope | Observed, Cactaceae | ||||||
---|---|---|---|---|---|---|---|

Relationship | Geometric model | Fractal model | Minimal model | Slope | 95% CI | r^{2} | Intercept |

Confidence intervals include predictions from the minimal model in all cases. Bold type, 95% CI of observed data that include the indicated model; underlined, 95% CI that only marginally include the indicated model. Other numbers, empirical data that do not include the indicated model. The observed empirical scaling relationships strongly support the predictions of the minimal branching model. While the predictions of the geometric and minimal models are close in most cases, the empirical data more strongly overlap with the predictions of the minimal than the geometric model.
| |||||||

H∝D^{α} | 1·000 | 0·667 | 1·000 | 0·999 | 0·923–1·075 | 0·704 | −1·187 |

H∝M^{α} | 0·333 | 0·250 | 0·375 | 0·401 | 0·324–0·478 | 0·772 | 1·295 |

D∝M^{α} | 0·333 | 0·375 | 0·375 | 0·411 | 0·331–0·492 | 0·915 | 1·606 |

S∝H^{α} | 1·000 | 1·000 | 1·000 | 1·041 | 0·962–1·120 | 0·826 | −0·012 |

S∝D^{α} | 1·000 | 0·667 | 1·000 | 1·073 | 0·992–1·155 | 0·724 | −1·274 |

S∝M^{α} | 0·333 | 0·250 | 0·375 | 0·415 | 0·335–0·495 | 0·767 | 1·304 |

M_{w}∝M^{α} | 1·000 | 1·000 | 1·000 | 1·069 | 0·864–1·274 | 0·906 | −0·584 |

H∝ | 0·333 | 0·250 | 0·375 | 0·373 | 0·296–0·449 | 0·640 | 1·777 |

D∝ | 0·333 | 0·375 | 0·375 | 0·397 | 0·318–0·476 | 0·876 | 1·980 |

S∝ | 0·333 | 0·250 | 0·375 | 0·388 | 0·309–0·467 | 0·581 | 1·771 |

As stated above, many plant taxa lack an obvious fractal-like branching morphology, with parallel internal vasculature and external morphology (succulents, palms, etc.: Gibson 1973, 1976; Gibson & Nobel 1986). Thus modelling the relationship between the branch-length ratio (ratio of daughter to parent branch lengths, γ ≡ *l*_{k+1}*/l*_{k}) and branching ratio (number of daughter branches per parent branch, typically 2 in plants) as a volume-filling fractal where γ≈*n*^{−1/3} is clearly violated (West *et al*. 1997; West *et al*. 1999a). Consequently the applicability of the WBE model to plants with minimal branching architecture, such as succulents, is unclear (West *et al*. 1999a).

Further, the exchange surfaces (photosynthetic area) in succulents are described by the external surface areas, *A*_{E}, of the entire plant body instead of just the number of terminal branches or petioles, *N*_{p}, multiplied by the average leaf area, <*A>*_{L}, so that *A*_{E} ≈ *N*_{p}*<A>*_{L}. In other words, the WBE model in its current form assumes that the external branching morphology parallels the scaling of vascular-exchange surface areas, and that plants have differentiated photosynthetic and non-photosynthetic tissues in the form of stems and leaves.

Here we show that the WBE model can be successfully extended to predict the scaling of morphology in minimally branching plants. We first define an external branching, *E*, and internal vascular, *I*, branching architecture (xylem architecture) where, respectively:

According to equations 3 and 4, and refer to the length and radius of a given external branch; and refer to the length and radius of a given internal vascular bundle that branches from the main vascular bundle into smaller strands (or, following the terminology of Gibson 1976; Gibson & Nobel 1986, vascular axial bundles split into vascular strands; see also Fig. 1b of West *et al*. 1999a). For the evolution of succulent morphology, we assume that natural selection has operated to minimize the volume-filling nature of branching (limiting the number of external branching generations which influences the value of γ^{E}) but not the scaling of photosynthetic surfaces, which are governed by the values of γ^{l} and β^{l}. Thus selection to minimize external branching will yield an architecture that will depart from volume filling as branch number decreases, so that γ^{E}→*n*^{−1/2} (West *et al*. 1999a).

However, the internal vascular network still has the problem of branching in order to supply all living cells within a three-dimensional volume inside the succulent. If selection has also operated to maximize the scaling of the number of terminal energy-harvesting units (*N*_{mito} and *N*_{chloro}), in addition to the exchange surfaces that supply these terminal units, as hypothesized by WBE, then the scaling of exchange surface areas supplied by the internal vascular system should approximate volume filling where γ^{l}→*n*^{−1/3}. Therefore, within succulents, if as governed by the internal vascular anatomy γ^{l} = *n*^{−1/3} and β^{l} *= n*^{−1/2}, then we would predict that the scaling of whole-plant photosynthetic surface areas or *A*_{E} will still scale as *M*^{3/4}, even though external branching is minimized.

The number of chloroplasts (*N*_{chloro}) exposed to radiant light potentially limits metabolic production in plants. Thus the 3/4-power scaling of surface area with mass results from selection to maximize surface area for light interception and metabolic gas exchange, while simultaneously minimizing travel time for biologically important resources (West *et al*. 1997, 1999b). This can be expressed as *N*_{c}∝*A*_{E}∝*M*^{b}, where *b* = (2 + *å*_{A})/3 + *å*_{A} + *å*_{L}). *å*_{A} and *å*_{L} are arbitrary exponents associated with the scaling of area and length, respectively (for detailed explanation see West *et al*. 1999b). Maximization of *b* and thus surface area occurs when *å*_{A} = 1 and *å*_{L} *=* 0, leading to *b* = 3/4.

For plants in arid environments, extensive photosynthetic surface area is a potential liability given the limited availability of water (Hunt & Nobel 1987; Niklas 1994; Niklas 2002). It is generally thought that succulents, particularly cacti, exhibit morphological adaptations that limit water loss via surface areas while increasing the capacity for water storage (Gibson & Nobel 1986). The geometric form that has the least amount of surface area for a given mass (or volume) is a sphere (Niklas 1994), and many cacti approximate spheres or oblate spheroids (e.g. *Ferocactus* or *Mammilaria* spp., Buxbaum 1950; Gibson & Nobel 1986; Anderson 2001). Thus, in contrast to the WBE model, a simple geometric model would predict that succulent surface areas should scale with an exponent closer to 2/3.

Our extension of the WBE model indicates that, in plants with branching that is not volume filling (minimal branching), the surface area will still scale with the 3/4 power of their mass. Because succulents lack petioles, the total photosynthetic surface area is given by *A*_{E} instead of *N*_{p}*A*_{L}, thus *A*_{E}∝*M*^{3/4}. However, the scaling of external morphology differs from plants with both volume-filling branching and branching predicted by the simple geometric model (Table 2). Specifically, substituting *A*_{E} for (*N*_{p}*A*_{L}) in equation 2 yields predictions for the scaling of external morphology as:

where the 1/2 exponent originates from the constraint on external morphology imposed by minimal branching (e.g. when plants do not branch or grow in a volume-filling manner; West *et al*. 1999a). Further, given the assumption of self-similarity between branching levels that is expected to hold even in minimally branching plants, the total length of the plant (or height, *H*) will be directly proportional to the length of the basal stem, , so that *H*∝∝∝*M*^{3/8} (Fig. 3). As both basal stem radius and total length (height) scale with mass to the 3/8 power under minimal branching (∝*M*^{3/8}, ∝*M*^{3/8}, respectively), we expect an isometric scaling of radius and total length, ∝ (*H*∝*D*^{1}) (Table 2; Fig. 3).

We can further extend the minimal branching model to predict overall plant canopy spread (*S*), measured as average crown diameter. Having established that total vessel length is proportional to height (West *et al*. 1999a), if there exists no systematic change in branch angle with plant size (mass), canopy spread should simply scale isometrically with total plant length, *S* ∝ , or *S* ∝ *H*^{1} (Fig. 3), regardless of external branching morphology (minimal or fractal). By substitution, we also have *S* ∝ *D*^{1}, and *S* ∝ *M*^{3/8} (Table 2; Fig. 3).

The model can also be used to predict the scaling of water mass with total or dry mass. The water mass of a given plant is equal to the sum of the water mass of each non-vascular cell plus the water mass of the network supplying those cells. The WBE model predicts that the fluid volume, and thus mass (assuming constant fluid density) of the internal vascular network should scale proportionally to *M*^{25/24} (West *et al*. 1999a). However, within succulents most cells are living and function in part to store water. Many cacti stems are 90–94% water (Gibson & Nobel 1986). Thus the cellular water mass of the plant should be equal to the sum of the water mass of its cells, *M*_{w} = Σ*M*_{wc}. Alternatively, the cellular water mass of a given succulent is equal to the number of cells multiplied by the water mass of the average cell: *M*_{w} = *N*_{c} × *<M>*_{wc}*.* If the average water mass of cells does not change with whole-plant mass *<M>*_{wc}∝*M*^{0}, this leads to an isometric scaling relationship between whole-plant cellular water mass and mass (dry) (*M*_{w}∝*M*^{1}). Thus for all intents and purposes water mass should scale nearly isometrically with whole-plant mass (*M*_{w}∝*M*^{1}) (Table 2; Fig. 3). Given the above derivations, by substitution the model also predicts the scaling of diameter, height and spread with water mass (Table 2; Fig. 3).

Our combined theoretical and empirical analyses specifically address the following objectives: (1) to contrast the different assumptions and expectations for both WBE and minimal branching models; (2) to derive expected exponents for bivariate scaling relationships (*H*, *D*, *S*, *A*_{E}, *M*, *M*_{w}) for plants exhibiting minimal branching (Table 2); (3) to test the minimal model's predictions with both intraspecific and interspecific data from a large biometric database on Sonoran Desert succulent plants; and (4) to evaluate the relative merits of these general models.