Ángel López-Urrutia and Xosé Anxelu G. Morán. 2007. Resource limitation of bacterial production distorts the temperature dependence of oceanic carbon cycling. Ecology 88:817–822.

Appendix A. Modeling heterotrophic bacterial metabolism.

In order to provide a coherent formulation for the effects of resource availablity and temperature on bacterial metabolism we derive a general model for bacterial metabolism. Here, we formulate a set of equations that are used in estimating bacterial growth efficiencies (BGE) and in the prediction of bacterial cell specific respiration (BR_i) and cell specific biomass production (BP_i) from environmental variables like temperature (T, in kelvins) and chlorophyll concentration (chl in mg/m³). According to our conceptual model the total carbon assimilated by a bacteria cell (i.e., the bacterial carbon demand BCD) depends both on temperature and on resource availability following the equation:

BCD = b0e-E/kT

(A.1)

where b₀ is a normalization constant independent of temperature and resource availability; e^-E/kT is Boltzmann's factor where E is the average activation energy for bacterial metabolism and k is Boltzmann's constant; and chl /(chl + K_m) is the Michaelis-Menten functional response of bacterial metabolism to resource availability, where chl serves as a proxy for resource concentration.

Part of this assimilated carbon is devoted to cell maintenance and respired as CO₂. We have shown that this bacterial respiration is independent of resource concentration but depends on temperature following Boltzmann's factor

BRi = r0e-E/kT

(A.2)

where r₀ is a normalization constant independent of temperature. We estimated r₀ and E from our data by regressing the natural logarithm of BR_i against 1/kT (Fig 1; Table 1).

The remaining assimilated carbon not used for respiration is used for biomass increase, that is for cell-specific bacterial production

BPi = BCD - BRi = b0e-E/kT - r0e-E/kT

(A.3)

To estimate the parameters in Eq. A.3 we have to be aware of several limitations of our data set and of the Leucine incorporation technique used to measure BP_i. First, although the equation above allows for the fact that sometimes net bacterial growth can be zero or negative when available resource concentrations are very low or zero (in which case we would expect a sort period of degrowth before bacteria die), this is not allowed by the Leucine incorporation technique. To measure BP_i we add a radiolabelled resource (3H-leucine) to the media and hence it is methodologically impossible to measure growth at zero resource concentration. It is therefore not surprising that measured BP_i is always greater than zero. Furthermore, we should keep in mind that chl is just a proxy for resource availability and even if chl was zero a small amount of substrate would still be available for bacterial growth. This lead us to reformulate Eq. A.3 using b₀ and r₀ that differ from the parameters in Eqs. A.1 and A.2 .

BPi' = b0'e-E/kT - r0'e-E/kT = e-E/kT[b0' - r0']

(A.4)

Second, the correlation between temperature and chlorophyll concentration and the lack of high chlorophyll concentration data at high temperatures cautions against any attempts to fit a multiple nonlinear regression to our data. We decided to fit Eq. A.4 in a two step approach. First, we show that the activation energy for BP_i is not significantly different to the activation energy for BR_i (see Fig. 2), so we divided BP_i by e^-E/kT (where E is the value obtained in the fit of Eq. A.2; Table 1) and used nonlinear regression following equation

BPi'*eE/kT = b0' - r0'

(A.5)

to estimate parameters b₀^', r₀^' and K_m. With this estimated parameters it is then straightforward to formulate Eq. A.4 (see Table 1).

The formula that describes the dependence of BGE on resource availability can be easily derived as

BGE = 1 - = 1 -

(A.6)

and symplifying we arrive to

BGE = 1 -

(A.7)

if we substitute the parameters in Eq. A.7 with those fitted using Eqs. A.2 and A.5 we arrive to the equation presented in Table 1 and represented in Fig 3C.