Biophysics Problem 37

A cell's oxygen requirement is proportional to its mass, but its oxygen intake is proportional to its surface area. Show that a cell cannot grow indefinitely and survive.

Since the oxygen required is proportional to mass and mass itself is proportional to volume \((M = \rho V),\) the oxygen required scales as \(L^3.\)
Thus the oxygen required will equal some constant multiplied by \(L^3,\) i.e.,
\(O_{required} = K_1 L^3,\)


where \(K_1\) is this constant.
Since the oxygen supplied is proportional to \emph{surface area}, it similarly scales as some constant times \(L^2,\) i.e.,
\(O_{supplied} = K_2 L^2,\)
where \(K_2\) is a different constant than \(K_1.\)

In the limiting case, the required and supplied oxygen will be equal.  Therefore,
\(K_1 L^3 = K_2 L^2\\ L = \frac{K_1}{K_2}\)
When \((K_1/K_2) < L,\) the above equation will not be balanced and the cell's oxygen requirements cannot be satisfied.