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Discontinuous gas exchange in ticks is thought to be one of the ways in which these animals maintain the very low metabolic rates required by their sit-and-wait strategy, which includes long periods of fasting (Lighton and Fielden 1995). Scorpions are also thought to have uncharacteristically low metabolic rates, and this has prompted considerable speculation regarding the benefits of low metabolic rates in both groups (Lighton et al. 2001). In turn, this speculation has raised the question of what a 'characteristic' metabolic rate is for arthropods, including insects, of a given size. In other words, what values should the coefficient (c) and exponent (z) assume in the scaling relationship

where B is metabolic rate (usually expressed in mW) and M is body mass (usually expressed in g). This question has long occupied physiologists and ecologists, and can indeed be considered one of the most contentious, yet basic issues in environmental physiology. The controversy concerns both the empirical value of z (but also of c, see Heusner 1991), and the theoretical reasons why a particular value of z should be expected. In addition, theoretical investigations have focused mostly on endotherms, while empirical work often includes unicells, ectotherms (vertebrate and invertebrate) and vertebrate endotherms (Robinson et al. 1983; West et al. 2002).

From a theoretical perspective, for endotherms, it was originally suggested (as far back as 1883 by Rubner) on the basis of a simple dimensional analysis that z = 0.67, the same as the scaling relationship for surface area. Heusner (1991) subsequently argued that this exponent reflects an underlying dimensional relationship between mass and power and is, therefore, of less interest than the coefficient (c) of the relationship. The early surface area arguments appeared to be at odds with the empirically derived data (see Dodds et al. 2001 for discussion), spawning many more theoretical investigations. These have been based, inter alia, on

(1) the temperature dependence of metabolic rate (Gray 1981);

(2) considerations of organisms in terms of heterogeneous catalysis of enzyme reactors, fractal geometry, and nonlinear relationships for transport processes (Sernetz et al. 1985);

(3) the contribution of non-metabolizing and metabolizing tissues to total mass (Spaargaren 1992);

(4) nutrient supply networks (West et al. 1997, 1999);

(5) dimensional analyses and four-dimensional biology (Dodds et al. 2001); and

(6) multiple control sites in metabolic pathways (Darveau et al. 2002).

Of these analyses, those based on nutrient supply networks have gained the most attention recently. This is predominantly because it has been argued that the nutrient supply network model provides a mechanistic basis for understanding the primary role of body size in all aspects of biology, and that the quarter power scaling (i.e. z = 0.75) is the 'single most pervasive theme underlying all biological diversity' (West et al. 1997, 1999). These analyses assume that the rate at which energy is dissipated is minimized, transport systems have a space-filling fractal-like branching pattern, and the final branch of the network is size invariant. Based on these assumptions, and several others, such as branching patterns that preserve cross-sectional area, a scaling exponent of 3/4 is derived, which is apparently in keeping with previous empirical estimates of this value (see also Gillooly et al. 2001). Subsequently, Dodds et al. (2001) have argued that there are several inconsistencies in West et al.'s (1997, 1999) mathematical arguments, suggesting that the exponent need not be 0.75, as the latter have claimed. For example, Dodds et al. (2001) argued that under a pulsatile flow system, and using the arguments of West et al. (1997), the scaling exponent should be 6/7, rather than 3/4. Since then, several, often critical, discussions have appeared of the assumptions of the nutrient supply network models, the empirical data that are used to support them, and their implications (see Functional Ecology 2004, Volume 18:2).

By contrast, Darveau et al. (2002) have argued (for endothermic vertebrates) that searching for a single rate-limiting step, which enforces scaling of metabolism, is simplistic, even though many previous models are based on such an assumption. Metabolic control is vested in both energy demand and in energy supply pathways and at multiple sites. Thus, there must be multiple causes of metabolic rate allometry. They argue that this relationship is best expressed as

where M is body mass, a is the intercept, b is the scaling exponent of process i, and ci the control coefficient of that process. At rest, all steps in the O2 delivery system display large excess capacities (arguably for any organisms that have factorial aerobic scopes above 1), therefore it is energy demand that is likely to set the scaling exponent. The two most likely energy sinks are protein turnover (25 per cent of total ATP demand) and the Na^K+-ATPase (25 per cent), and using the scaling exponents and maximum and minimum values for the coefficients of these processes Darveau et al. (2002) find that the scaling exponent for basal metabolic rate lies between 0.76 and 0.79. On the other hand, maximal metabolic rate is likely to be limited by O2 supply because it is known that, at maximum metabolic rate, supply shows almost no reserve capacity, whereas acto-myosin and the Ca2 + pump (in muscles which are using 90 per cent of the energy during activity) have considerable reserve capacity. In consequence, the scaling exponent during maximal metabolic demand lies between 0.82 and 0.92. Thus, in this multi-site control model, bi and ci for all major control sites in ATP-turnover pathways determine the overall scaling of whole-organism bioenergetics, depending on the relative importance of limitations in supply versus demand. Any environmental influences on scaling (e.g. Lovegrove 2000) must consequently have an effect through these proximal causes.

Subsequently, it has been shown that equation (6) is technically flawed because it violates dimensional homogeneity, and the utility of this approach for understanding scaling has been questioned (Banavar et al. 2003; West et al. 2003). Darveau et al. (2003) claim that the technical problems can be overcome by modifying equation (6) to

where MR0 is the metabolic rate of an organism of characteristic body mass M0. However, this begs the question of what a characteristic mass might be of, for example, mammals that show eight orders of magnitude variation in mass, and indeed the choice of M0 must be arbitrary (Banavar et al. 2003). Notwithstanding these difficulties, the analysis by Darveau et al. (2002) draws attention to the importance of multiple control pathways in metabolism.

What these theoretical analyses imply for the scaling of insect metabolic rate is not clear. West et al. (1997) argue that their models apply as much to insects as to other organisms. However, they also assume that gas exchange in insects takes place solely by diffusion, and that tracheal cross-sectional area remains constant during branching of the system. Clearly there are problems with both of these assumptions (Sections 3.3, 3.4.1). Moreover, empirical studies suggest that in insects SMR scales neither as M0.75, nor as M0.67. Although several early studies provided estimates of the scaling relationship for SMRs in insects (e.g. Bartholomew and Casey 1977; Lighton and Wehner 1993), the first consensus scaling relationship across several taxa, that took into account the likely effects of activity (Section 3.1), was the one provided by Lighton and Fielden (1995), and subsequently modified by Lighton et al. (2001). They argued that all non-tick, non-scorpion arthropods share a single allometric relation:

where mass is in g and metabolic rate in mW, at 25°C. The scaling exponent in this case lies midway between earlier assessments suggesting that the exponent is approximately 0.75 and others suggesting that it is closer to 1 (e.g. von Bertalanffy 1957). It is also statistically indistinguishable from the 6/7 exponent predicted by Dodds et al. (2001). Although tracheae are clearly capable of some flexibility (Herford 1938; Westneat et al. 2003), thus tempting speculation regarding the similarity of the empirical and theoretically derived values, pulsatile flow of the kind seen in mammals seems unlikely in insects. Moreover, it seems even more unlikely in scorpions, which have a similar exponent (Lighton et al. 2001), but lack tracheae.

Other empirical investigations have revealed a wide range of interspecific scaling exponents for metabolic rate in insects (0.5-1.0), although in many cases only a small number of species was examined (Hack 1997; Davis et al. 1999; Duncan et al. 2002a), or data obtained using a wide variety of methods were included in the analysis (Addo-Bediako et al. 2002). These results have led several authors to suggest that carefully collected data from a wider variety of species are required before a consensus scaling relationship for insect SMRs is adopted, especially because Lighton and Fielden's (1995) analysis was based on a limited range of taxa (mostly tenebrionid beetles and ants) (Duncan et al. 2002a). There can be little doubt that additional investigations of insect metabolic rate, which control carefully for both activity and feeding status (see Section 3.5.3), would be useful for elucidating the consensus scaling equation for insect metabolic rate. They might reveal that there is no consensus relationship, but that the relationship varies with size, taxonomic group, and geography, as seems to be the case for endothermic vertebrates (Heusner 1991; Lovegrove 2000; Dodds et al. 2001). Moreover, such a comparative analysis, including estimations based on phylogenetic independent contrasts (Harvey and Pagel 1991) (something that is rarely done for insects), might also reveal a very different scaling relationship to those that have been found to date, although even here analyses are likely to be problematic (see Symonds and Elgar 2002). Nonetheless, determining what form the scaling relationship for metabolic rate takes in insects, and whether it is consistent at the intra- and interspecific levels is of considerable importance. The available data and analyses suggest that in insects the intraspecific and interspecific scaling relationships are quite different, with intraspecific relationships being indistinguishable from the 0.67 value predicted solely on geometric considerations (Bartholomew et al. 1988; Lighton 1989), and the interspecific slope being much higher. These results contradict West et al.'s (1997, 2002) models which suggest that the scaling relationship at the intra- and interspecific levels should be identical, and support previous suggestions that there is no reason why interspecific and intraspecific scaling relationships should be the same (see Chown and Gaston 1999 for review). From an ecological perspective, the scaling of metabolic rate provides insight into the likely energy use of insects of different sizes, which has a host of implications. These include those for the evolution of body size frequency distributions (Kozlowski and Gawelczyk 2002), for understanding energy use and abundance in local communities (Blackburn and Gaston 1999), and for understanding patterns in the change of insect body sizes across latitude (Chown and Gaston 1999).

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