Theory

We summarize the approach described by ter Kuile and Westerhoff (2001) and our application of it to metabolic scaling in orchid bees as follows:

The hierarchical regulation coefficient ρ_h, which is a measure of the contribution of enzyme concentration [E], and the metabolic regulation coefficient ρ_m, which is a measure of the contribution of mechanisms that result in the modulation of the activity of a constant concentration of enzyme, to variation in flux, are related as: (1)

At any enzyme-catalyzed step i, ρ_h is a function of the relative change in rate, ∂v_i, divided by the relative change in enzyme concentration, ∂e_i, times the ratio of change in e_i to the change in flux J. Assuming that∂ lnv_i/∂lne_i equals 1 (ter Kuile and Westerhoff, 2001), (2)

Similarly, at any step i, metabolic regulation results from a change in an enzyme-catalyzed rate divided by the change in concentration of its substrate, product or allosteric modulator (X), times the ratio of change in X to change in J, such that: (3)

Orthologous enzymes from organisms of similar body temperature display similar catalytic efficiencies (k_cat values) (Hochachka and Somero, 2002). Because (4)

it follows that interspecific variation in V_max measures variation in e_i. From Eq. 1, the slope of lnV_max plotted against lnJ yields the hierarchical regulation coefficient ρ_h. It follows that 1-ρ_h yields the metabolic regulation coefficientρ _m.

We note here that this approach was not devised for interspecific comparisons; ter Kuile and Westerhoff (2001) applied it to the analysis of variation in glycolytic rates in protozoa resulting from experimental manipulation. However, the mechanistic bases for and the quantitative relationships that describe intraspecific variation in flux among protozoa apply equally well to interspecific variation in flux among orchid bees. In a given steady-state situation (in this case, high rates of carbohydrate oxidation during hovering flight), the interspecific variation in flux can be accounted for either by interspecific variation in [E] or by interspecific variation in substrate, product or modulator concentrations. However, the interspecific variation in flux may also be due to variation in other morphological, physiological and biochemical traits besides these. Thus, in the work described herein, the value of slope of lnV_max plotted against lnJ is potentially influenced by other traits that covary with the biochemical parameters that we measured.

We address the above problem with the use of Felsenstein's method of phylogenetically independent contrasts (PIC analysis; Felsenstein, 1985; Garland et al., 1992). Briefly, in conventional statistical analyses of interspecific data, it is assumed that species are equally related to each other. In reality, they are related to each other by their phylogenetic relationships. Because closely related species tend to display greater similarity than more distantly related ones, estimates of ρ_h from slopes of linear regressions may be biased by phylogenetic relationships and correlated traits other than those hypothesized to account for interspecific variation in, as in the present study, flux rates. PIC analysis provides a statistical approach that allows for phylogenetic non-independence and makes possible the application of the approach described by ter Kuile and Westerhoff (2001) to the analysis of interspecific data. In the present work, PIC analysis was conducted using the PDAP (Midford et al., 2003) module in Mesquite (Maddison and Maddison, 2004). The analyses were performed using both gradual and speciational models of character evolution, with branch-lengths obtained from genetic distances inferred from cytochrome b partial sequences (Darveau et al., 2005a) or with branch lengths set to 1.

Sources of data

Glycolytic flux rates were estimated from V̇_CO2 during flight (Darveau et al., 2005a). Enzyme V_max values are from Darveau et al. (2005b). Hypothetical phylogenies used in PIC analysis were generated on the bases of partial sequences for the mitochondrial cytochrome b gene (Darveau et al., 2005a).

Calculations

Mathematical modeling of the phosphoglucoisomerase reaction was performed using Mathcad Professional, version 8 (MathSoft, Inc., Cambridge, MA, USA). Model assumptions, equations and parameters are provided in the text.

Metabolic regulation at the PGI step

In previous work using honeybees Apis mellifera, we combined empirical and modeling approaches to try to understand the relations between flux capacities and flux rates at the PGI step, a near-equilibrium, reversible, glycolytic reaction at which glucose 6-phosphate, G6P, is converted to fructose 6-phosphate, F6P (Staples and Suarez, 1997). At such a near-equilibrium step, the enzyme catalyzes reactions in both directions, and net glycolytic flux equals the forward rate minus the reverse rate. Using Haldane's equations (Haldane, 1930) in a similar manner as Veech et al. (1969), we showed that the high V_max at this step could be rationalized on the bases of the enzyme's dual roles, i.e. to maintain a net forward flux, equal to the overall rate of glycolysis, and to maintain the reaction close to equilibrium. We now consider how substrate and product concentrations in vivo would have to change to account for the range of net flux rates observed interspecifically among orchid bees. The Haldane (1930) equation applied to PGI is: (5)

where J is the net flux, [G6P] and [F6P] are glucose 6-phosphate and fructose 6-phosphate concentrations, respectively, K_f and K_r are Michaelis constants for the forward (G6P is substrate) and reverse (F6P is substrate) reactions, respectively. We used K_f=1.07 mmol l^-1 and K_r=0.117 mmol l^-1, obtained under simulated intracellular conditions using purified honeybee PGI (Staples and Suarez, 1997). It is assumed that both K_f and K_r are mass-independent and conserved across orchid bee species. K_eq is the equilibrium constant obtained in vitro, equal to 0.3 (Staples and Suarez, 1997). V_r represents the V_max value in the direction of F6P conversion to G6P. These scale isometrically in orchid bees and average 345.6 μmol g^-1 min^-1 (Darveau et al., 2005b). V_f is the V_max in the glycolytic direction, estimated from the Haldane (1930) equation for K_eq, (6)

as 948.2 μmol g^-1 min^-1. This is close to the V_f value measured empirically in honeybee flight muscles (Staples and Suarez, 1997). Eq. 5 is used to solve for the mass action ratios, i.e. the ratios of [F6P]/[G6P] in the flight muscles, required to account for the range of estimated flux rates. The decline in mass action ratio (Fig. 2A) is extremely small, a result that can also be demonstrated by holding [F6P] constant to 0.1 mmol l^-1, a value within the range measured in vivo (Staples and Suarez, 1997), and solving for the range in [G6P] required to obtain the range of observed net forward flux rates (Fig. 2B). The model predicts that a change of about 0.02 mmol l^-1 in [G6P] is enough to account for the entire range of variation in flux across orchid bee species. A highly amplified response to a small change in substrate concentration, as seen here, is a characteristic of near-equilibrium reactions (Newsholme and Crabtree, 1976). Given this result, it seems unlikely that the expected range of variation in [F6P]/[G6P] in orchid bee flight muscles would be detectable in a comparative, interspecific study. At many other steps in metabolism, this problem is exacerbated by even lower steady-state concentrations of substrate and product (e.g. Kashiwaya et al., 1994). Thus, a modeling approach such as the one described here may be the only way to address the issue of metabolic regulation at such steps.

A broader view of metabolic regulation

Metabolic control analysis using skinned muscle fibers from mammals has revealed that as much as half of the control of oxidative metabolism in muscles is exerted by actomyosin ATPase (Wisniewski et al., 1995). In the context of metabolic scaling in orchid bees, a possible interpretation of our results is that, as muscle operating frequencies decline with increasing body mass (Darveau et al., 2005a), muscle ATP hydrolysis rates decline (Darveau et al., 2005b) and this drives the allometric scaling of metabolism. Such an interpretation follows from the similarity in allometric exponents of wingbeat frequency and hovering metabolic rate and is consistent with the predominant role played by metabolic regulation seen in the results reported here.

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Fig. 2.

(A) Mass action ratio vs glycolytic flux rate (i.e. net forward flux at the phosphoglucoisomerase reaction). The Haldane relationship predicts that extremely small variation in the mass action ratio is enough to account for the range of glycolytic flux rates of 4.64-13.0 μmol g^-1 min^-1 observed across species. (B) Glycolytic flux rate (i.e. net forward flux at the PGI reaction) vs glucose 6-phosphate concentration. Holding fructose 6-phosphate concentration at 0.1 mmol l^-1, the Haldane relationship predicts that the range of glycolytic flux rates of 4.64-13.0 μmol g^-1 min^-1 observed across species is achieved by a change in glucose 6-phosphate concentration of about 0.02 mmol l^-1.

Further insights into the role played by metabolic regulation can be derived from work concerning bioenergetic scaling in mammalian hearts. Top-down control analysis has revealed that work rate (therefore ATP hydrolysis rate) dominates the control of oxidative metabolism in the heart (Diolez et al., 2002). Using ³¹P-NMR spectroscopy, Dobson and Headrick (1995) and Dobson and Himmelreich (2002) found that cytosolic free [ADP] declines as body mass increases in hearts, such that 1/[ADP] and the cytosolic phosphorylation potential [ATP]/[ADP][P_i] both scale allometrically with exponents close to -0.25. Dobson and colleagues propose that these result in higher `kinetic gain' such that small changes in [ADP] result in greater fractional changes in metabolic rate in the hearts of small mammals compared with larger ones. This scheme provides a plausible control mechanism by which allometry in work rate drives allometry in metabolic rates. That this applies to orchid bee flight muscles is a testable hypothesis.

Conclusion

The relative importance of the roles played by hierarchical regulation, resulting from variation in [E], and metabolic regulation in accounting for interspecific variation in metabolic rates is not well understood. Among orchid bees, the allometry in metabolic rates during hovering flight involves the hierarchical regulation of flux at the HK step and metabolic regulation at all other enzyme-catalyzed reactions that we examined. Thus, although techniques for the study of gene expression can be brought to bear on the problem of metabolic scaling, the predominance of metabolic regulation in accounting for interspecific variation in flux rates also warrants the application of a systems approach to the control of flux (Fell, 1997) and the manner in which it scales (Darveau et al., 2002; Suarez et al., 2004).