04.22.09. Bacteria swim to find nutrients and escape other organisms. Evolutionary pressures have optimized bacteria's size and structure for, among other survival requirements, the ability to swim fast. An undersized bacterium may lack flagellar strength, while an oversized bacterium may incur too much frictional drag. We quantified this intuition by devising a mathematical model from the following (somewhat contrived) considerations: (1) a bacterium's flagella are 4 micrometers longer than its radius, (2) drag is proportional to cross-sectional area, and (3) flagellar length is 6 times more important for survival than overcoming drag. With guidance, students turned these criteria into an equation for a bacterium's velocity v as a function of its radius r, v = 6*(r+4) - 3*r^2, and simplified the equation to a standard form. Students were asked "What value of r gives the largest v?", which led naturally into generating test data and plotting the equation. As shown by the plot, swimming velocity is maximal with a radius of 1 micrometer under this model, which is a biologically realistic value.
