# study the Curtis's metabolic model

## Demand vs Supply of O2

Both are dependent on temperature (T) and biomass (B), and the supply of O2 has one more constrain: the ambient pO2.

• Demand: f(T, B)

• Supply: f(T, B, pO2)

## Definition of temperature sensitivity (Arrhenius function)

$$\gamma_T(E) = exp((\frac{1}{T} - \frac{1}{T_{ref}}) \frac{-E}{k_B})$$

where kB is the Boltzmann constant. E is the parameter varied in Demand and Supply function.

## Demand function

$$f(T, B) = \alpha_D \cdot B^{\delta} \cdot \gamma_T(E_d)$$

αD is the rate coefficient has units of O2 per unit body mass per time (μmol O2 g−3/4 h−1)

## Supply function

$$f(T, B, \ce{pO2}) = \alpha_S \cdot \gamma_T(E_s) \cdot B^{\sigma} \cdot \ce{pO2}$$

The function 𝛼̂ s(𝑇) represents the efficacy of the O2 supply. It is a rate coefficient (in μmol O2 $g^{−3/4} h^{−1} atm^{−1}$)

## Metabolic index

$$\Phi = \frac{f(T,B)}{f(T, B, \ce{pO2})}\ \Phi = \frac{\alpha_S}{\alpha_D} \cdot B^{\delta - \sigma} \cdot \ce{pO2} \cdot\gamma_{T}(E_d - E_s)$$

Unknown parameters:

• $E_d, E_s$
• $\alpha_D, \alpha_S$
• $\sigma, \delta$ (for S and D respectively)

When $\Phi = 1$, the organism has the minimum O2 balance to survive.

## Derived Physilogical Traits:

$\frac{\alpha_S}{\alpha_D}$: the vulnerability to hypoxia, which is measurable as the lowest O2 pressure (Pcrit) that can sustain resting metabolic demand (Φ = 1)

$E_0 = E_d - E_s$: the sensitivity of hypoxia tolerance to temperature

$\epsilon = \sigma - \delta$: the allometric scaling (i.e., related to body size or biomass) of the supply-to-demand ratio

Pcrit: lab-measured hypoxic thresholds

Deutsch, C. et al. Climate change tightens a metabolic constraint on marine habitats. Science (2015)

Deutsch, C. et al. Metabolic trait diversity shapes marine biogeography. Nature (2020)

Justin P. and Deutsch C. Avoiding ocean mass extinction from climate warming. Science (2022)

Deutsch, C. et al. Impact of warming on aquatic body sizes explained by metabolic scaling from microbes to macrofauna. PNAS (2022).