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# Biological adaptations in response to physical constraints: the case of Atlantic salmon

## Problem

Fick's first law of diffusion states that the flux $J$ of a substance (units: $\text{mol}×{\text{m}}^{-2}×{\text{s}}^{-1}$) is proportional to the gradient for the substance:
$J=-D×\frac{\mathrm{\Delta }C}{\mathrm{\Delta }x}$
where $\mathrm{\Delta }C$ is the difference in initial and final concentration of the substance (units: $\text{mol}×{\text{m}}^{-3}$), $\mathrm{\Delta }x$ is the distance of diffusion (in meters), and $D$ is the diffusion constant, which depends on environmental conditions and increases with temperature. Fick's second law of diffusion states that the time it takes a particle to diffuse a given distance is proportional to the square of that distance:
$t=\frac{{x}^{2}}{2D}$
Fick's laws can be used to model the rate of oxygen diffusion across biological membranes ($M{\text{O}}_{2}$) using the following equation
$M{\text{O}}_{2}=K×\frac{A}{L}×\mathrm{\Delta }P{O}_{2}$
where $K$ is Krogh's diffusion coefficient, $A$ is the surface area of the membrane, $L$ is the thickness of the membrane, and $\mathrm{\Delta }P{\text{O}}_{2}$ is the oxygen partial pressure difference across the membrane.
The physical restrictions modeled by these equations have lead to interesting evolutionary adaptations in the structures that animals use to absorb oxygen. For instance, Atlantic salmon hatchlings absorb the oxygen needed for cellular respiration cutaneously (through the skin). However, over the course of juvenile development, these animals experience a rapid increase in body size, and absorption of oxygen shifts almost totally to the branchial tissue of the gills. Researchers studied the measurable consequences of this developmental process by measuring oxygen absorption ($M{\text{O}}_{2}$) in branchial and cutaneous tissue and compared the results to measurements of surface area and anatomical diffusion factor (ADF), two different measures of the oxygen-absorbing capacity of tissues (figure 2). ADF is defined as the ratio of mass-specific surface area to harmonic mean diffusion distance. The harmonic mean of a data set is a measure of the average value of the data set. Data for measurements of harmonic mean diffusion distance are given in figure 2.
Figure 1. Relation between body mass and change in cutaneous and branchial oxygen absorption ($M{\text{O}}_{2}$), surface area (area), and anatomical diffusion factor (ADF) as a proportion of total body $M{\text{O}}_{2}$, area, and ADF over the course of juvenile development of atlantic salmon hatchlings.
Figure 2. Measurement of harmonic mean diffusion distance for filaments, skin, and lamellae for atlantic salmon (filaments and lamellae are types of branchial tissue).