4. The size of a cell may be limited by the amount of nutrient that it can transport across its membrane. Though Escherichia coli is a rod shaped bacterium, for this problem we will assume that its cell is a perfect sphere. (Recall that the surface area of a sphere is A = 4pr², while the volume is given by V = 4pr³/3.) Estimates from the experiments of Tempest and Neijssel [2] indicate that for a particular set of experiments, the E. coli are capable of absorbing 1000 pmol of glucose/hr/mm² through their cell surface (pmol = picomoles). These cells are consuming glucose at a rate of 3200 pmol of glucose/hr/mm³ inside the cell for cellular functions with any net increase going to cell growth.

a. Let r be the radius of the cell. If the cell obtains all of its nutrient by absorbing it through its surface and uses the nutrient uniformly throughout the cell (its volume), then a function describing the net rate of glucose entering the cell is given by

F(r) = 1000 A(r) - 3200 V(r),

where A(r) and V(r) are the area and volume of the spherical cell depending on the radius r in mm. Graph this function for all r > 0 where F(r) is positive. Determine the radius of the cell that results in the maximum net increase of glucose entering the cell, which would allow for maximal growth. Give a brief biological explanation of how this model might relate to cell growth.

b. The cell stops growing when the net rate of glucose entering the cell is zero. Find the radius, volume and surface area of this largest possible cell (under these growing conditions).

c. Suppose that the cell could increase the transport of glucose across the cell surface to 1200 pmol of glucose/hr/mm² (a 20% increase). Repeat the calculations and graphs from Parts a and b, giving the percent changes from your earlier answers.

[2] D. W. Tempest and O. M. Neijssel (1987), "Growth yield and energy distribution", Chapter 52 in Escherichia coli and Salmonella typhimurium: Cellular and Molecular Biology by F. C. Neidhardt, American Society of Microbiologists.