Ecological Archives E092-131-A2

Wilco C. E. P. Verberk, David T. Bilton, Piero Calosi, and John I. Spicer. 2011. Oxygen supply in aquatic ectotherms: Partial pressure and solubility together explain biodiversity and size patterns. Ecology 92:1555–1572.

Appendix B. Derivation of the OSI from first principles on gas diffusion.

Calculation of Oxygen Supply Index (OSI) and relative Oxygen Supply Index

Our integrated measure of environmental oxygen supply incorporates the three key features that drive oxygen diffusion from the environment into an organism; (1) oxygen diffusion rates, (2) oxygen partial pressures (pO2) and (3) oxygen solubility (αO2):

Oxygen Supply Index (OSI) ∝ αO2DO2pO2(B.1)

where αO2 = the solubility of oxygen in water (mol·m-3·Pa-1), DO2 = the diffusivity or oxygen diffusion coefficient in water (m2·s-1) and pO2 being the partial pressure of oxygen in water. αO2 is derived by taking the inverse of the Henry coefficient (k), which is calculated from (Benson and Krause 1984):

ln k = 3.71814 + 5596.17·T -1 – 1049668·T -2 + S*(0.0225 - 13.608·T -1 + 2565.68·T -2)(B.2)

where T is the absolute temperature (K) and S the salinity (following the practical salinity scale). The diffusivity of oxygen at a given temperature and dynamic viscosity of water, µ (kg·m-1·s-1), is calculated by use of the following equation (St-Denis & Fell 1971):

DO2 = 6.92·10-12·T·µ-1(B.3)

The dynamic viscosity, µ, is calculated in fresh water from the simplified correlation (Pátek et al. 2009) of the international formulation (Huber et al. 2009), which provides accurate values for the thermal and pressure ranges employed in this study (Pátek et al. 2009). Formulae given in Sündermann (1986) were used to adjust the calculated dynamic viscosity for fresh water to waters of different salinities. This involved calculating the density of water for different temperatures and salinities; these densities were used to convert dynamic viscosity to kinetic viscosity (m2·s-1).

To derive values for pO2 in water, we compared measured oxygen concentrations with predicted concentrations at equilibrium for a known αO2 and atmospheric pO2. Given Henry’s law that the amount of dissolved oxygen is directly proportional to atmospheric pO2 at equilibrium, a difference between predicted and measured oxygen concentrations reflects a departure from equilibrium; either super-saturation (measured concentrations higher than predicted) or under-saturation (measured concentrations lower than predicted). The oxygen concentration (moles·l-1) predicted at equilibrium is obtained by solving the equation (Benson et al. 1979):

pO2 = k · nO2/(nH2O + nO2)(B.4)

which describes the molar fraction of oxygen in solution at a given partial pressure, with nO2 and nH2O being the moles of oxygen and water, respectively. The moles of water (nH2O; moles·l-1) can be calculated from its molar weight and density, which varies with temperature as:

nH2O = (-0.0061·T2 + 3.3578·T + 535.93) / 18.01528(B.5)

Thus, the solution of equation 4 to calculate O2 at equilibrium (moles·l-1) is given by:

O2 (moles·l-1) = (pO2 / k · (-0.0061·T2 + 3.3578·T + 535.93) / 18.01528) / (1 - pO2 / k).(B.6)

αO2 is expressed in mol·m-3·Pa-1 (Piiper et al. 1971). Together with oxygen diffusivity (m2·s-1) and pO2 (Pa), the index of oxygen supply takes the dimensions of mol·m-1·s-1 (DO2·pO2·αO2 = [m2·s-1]·[Pa]·[mol·m-3·Pa-1]), which can be seen as the amount of oxygen, in moles, transferred across a given distance, in a given time.

Oxygen supply relative to demand was obtained by dividing the index of oxygen supply (OSI) by a metabolic correction factor which expresses changes in oxygen demand with temperature:

Relative Oxygen Supply Index (rOSI) = OSI / Q10ΔT/10(B.7)

This index of relative oxygen supply thus takes into account the combined effects of oxygen supplied from the environment and an organism’s internal oxygen demand. We used an overall Q10 value of 2.0, falling well with the range usually reported (Rostgaard and Jacobsen 2005). Since species will each respond differently to temperature, this relative oxygen supply index (rOSI) is expected to be useful only when comparing the species with a similar thermal physiology or assemblages with a similar diversity in thermal physiologies (Makarieva et al. 2005).

TABLE B1. Pearson R values for the relationships between species richness and the oxygen supply index (OSI) and relative oxygen index (rOSI). Results are shown separately for indices unadjusted and adjusted for the thickness of the laminar sub-layer. Significant relationships are in bold and significant differences due to such adjustment in the performance of the indices are indicated with asterisks (tested with pair-wise likelihood ratio tests). Note that unadjusted values are identical to those reported in Table 1, but are repeated here to facilitate comparison.

Unadjusted for laminar sub-layer Adjusted for laminar sub-layer
Oxygen supply
index1 (OSI)
Relative oxygen
supply index2 (rOSI)
Oxygen supply
index1 (OSI)
Relative oxygen
supply index2 (rOSI)
Higher species
richness at high
(low altitude)
(Jacobsen 2008) Rarefied
0.595 0.291 0.606 0.402
Ecuadorean Andes
(N = 30)
Richness 0.169 -0.161 0.214 -0.022
(Perry and
Schaeffer 1987)
0.565 -0.407 0.559 -0.363
Colorado mountains
(N = 12)
Richness 0.754* -0.735 0.852* -0.706
Lower species
richness at high
(high pollution)
(Jacobsen and
Marín 2008)
0.637 0.708 0.656 0.654
Bolivian Altiplano
(N = 12)
Richness 0.669 0.706 0.687 0.672
et al. 2008)
0.924* 0.899 -0.133* 0.905
Patagonian streams
(N = 6)
Richness 0.964* 0.927 0.034* 0.914

1 Calculations based on Tmin.
2 Calculations based on Tmax.


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