Theories of optimization, form and function in branching architecture in plants

Optimality models are used as a tool for studying design in the context of natural selection. It has often been pointed out (e.g. Parker & Maynard Smith 1990) that it is not necessary to assume that organisms are optimal for optimality theory to be relevant: it can be valuable in providing a null hypothesis in investigations of adaptation and design. Despite this parsimonious view point, optimality has grown up as a major paradigm in biology closely associated with evolution theory and is the source of a major debate within the study of evolution (e.g. see Dupre 1987). One of the difficulties with the hypothesis of optimality is the availability of observations to test it. It seems logical to suppose that the most likely candidates for nearoptimal design as a result of evolutionary processes are those which share the following features: evidence of evolutionary convergence, a relatively simple form-function relationship, evidence of steep gradients in the 'adaptive landscape' and of increasing fitness over evolutionary time. One example that may qualify and where optimality arguments have frequently been used is the branching design of biological structures whose function is to fill space with a conservative use of materials (Zamir 1976; Honda & Fisher 1978; leFevre 1983; Niklas & Kerchner 1984; Niklas 1986; Morgan & Cannell 1988b). The proposition of evolutionary optimization in these forms is by no means certain. In this paper, we aim to show that some ideas growing up in the medical and engineering literatures, combined with ideas present in botany could substantially advance the study of form-function relations of branched structures and subject the optimization hypothesis to more critical assessment. In plants, above-ground branching pattern directly affects light capture, water transport, mechanical support, reproduction, wind resistance and ultimately, the competitive advantage of trees (KUppers 1989). Branch design can influence the relative growth rate of a tree via its effect on the light-extinction coefficient through determining the spatial distribution of foliage (Cannell 1989). A wide variety of botanical branched forms exist in nature, classified by Halle, Olderman & Tomlinson (1978). That these have persisted through evolutionary time to the present indicates that they are all functional forms and comply with the principle of adequate design (Rashevsky 1960). The term 'functional form' is used throughout this work to mean a form which performs a clear functional role with sufficient success that the organism to which it belongs reproduces enough for a viable species (i.e. one that persists over evolutionary time). Theoretical studies have shown that it is relatively simple to simulate branched forms with a basic branch generating rule, manipulating the values of relatively few parameters. These parameters, though, are not necessarily those that are genetically coded (Niklas 1994). It is likely that more than one function (or design objective) has exerted an influence over each genetically coded parameter of form. There is also likely to be a degree of pleiotropy further confusing relations between form and function. Frequently there will be multiple strategies leading to different forms for any given objective function and these could be combined so that trade-offs between them produce a set of different functional forms for the one design objective [see the example of Campbell, Grime & Mackey (1991) below]. The objective space, which plots all possible forms against one or more objective functions, can be seen as having contours of indifference resulting from the combination of strategies and of different objective functions which may not be orthogonal, indeed they may be highly correlated. Consequently there is not just a single optimal point nor several but rather some subspace (e.g. a surface) in adaptive space where different solutions represent 'adequate design'. This may help to explain the wide variety of form in nature: even when forms occupy what appear to be very similar niches (Tomlinson 1983). Despite this, there have been a number of demonstrations of apparently optimal form in branch design (reviewed by Fisher 1992) but these have all concerned optimization against only one design objective at a time. Our interest is in the effect of multiple objectives on the hypothesis of optimization in branched structures in the context of constraints, especially ontological constraints.