Effect of body size and temperature on standard metabolic rate

Metabolism is proportional to a constant power of the body weight or length as described by the allometric equation y = ax^b where y is the metabolic rate (measured as oxygen consumption), x is the body weight or length, b is the allometric coefficient and a corresponds to the metabolic rate of an animal per unit of weight (VO₂ or k ˙ also called reaction rate). Accordingly, reaction rate (oxygen consumption) curves were constructed using the general model below accounting for the effect of temperature (Arrhenius relationship) and body size scaling:

where T is the absolute temperature (in Kelvin: °C + 273.15 °K), T₁ is a chosen reference temperature (here 20 °C, 293.15 °K), T_A is the Arrhenius temperature, k ˙ is the reaction rate and k ˙1 its value at the reference temperature T₁, X is a body size dimension (WW, DW, L3, TL or NSEG), and b is the scaling exponent of the allometric effect.

Curve fitting was achieved using the downhill simplex method of the Nelder–Mead model and standard deviations were estimated using an asymptotic method with MATLAB R2010b. All fits were tested by analysis of variance (p < 0.001) with residuals being tested for normality and homogeneity of variance as well as parameter significance testing using Student’s t-test (p < 0.05).

The Van’t Hoff coefficient (Q₁₀) is the factor that should be applied to reaction rates for every 10 °C increase (Kooijman, 2010) and it was determined according to:

where k ˙ is reaction rate, T the absolute temperature (in Kelvin) and T₁ a chosen temperature. The relationship between Q₁₀ and T_A is the following (Kooijman, 2010):