Thursday, January 21, 2010
a limit on the size of a bacterium
01.21.10. The way a bacterium breathes limits how large it can grow. Unlike a human, who actively takes in oxygen and expels carbon dioxide, a bacterium passively takes in oxygen and other nutrients, absorbing through its membrane only those molecules that strike its surface while diffusing throughout the surrounding medium. The maximum metabolic intake M (molecules per time per mass) of such a passively "breathing" organism is readily calculated (see, e.g., Philip Nelson's Biological Physics, Sec. 4.6.2, p. 138) as M = 3Dc/(nR^2), where c and D are the concentration and diffusion constant, respectively, of the external molecule (e.g. oxygen), and n and R are the density and radius, respectively, of the bacterium. Solving this equation for R yields an upper bound for the size of the bacterium, i.e. R < sqrt[3Dc/(nM)]. Students, armed with values of D, c, n, and M, solved for and graphed the result R < 5 microns (which is actually a quite reasonable limit for bacterial size). As a "challenge" question, students were also asked to figure out how algebraically to get from the M equation to the R equation by understanding the steps in an analogous problem.