3. The shape of a cell affects its surface area to volume ratio. This can be significant in the cell's ability to absorb nutrients or survive toxins. You are given that the volume of a sphere and cylinder are (4/3)pr3 and pr2h, respectively, where r is the radius and h is the height. The surface area for a sphere and a cylinder are 4pr2 and (2prh + 2pr2), respectively.

a. Complete the following table (from Biology 350), which examines cellular geometry.


Organism

Shape

Diam
mm

Height
mm

Volume
mm 3

Surface
mm 2

S.A.:Vol.
Ratio

Mycoplasm

Sphere

0.3

--




Coccus

Sphere

1.5

--




E. coli

Cylinder

0.75

4.0




Yeast

Cylinder

5.0

8.0




Diatom

Cylinder

20

60





Note that the diameter and not the radius is given.

b. Suppose that the Coccus bacteria and E. coli satisfy the discrete Malthusian growth equation

Pn+1 = (1+k)Pn, P0 = 1000,

where P0 is the initial population and the doubling time for the population is 25 min. Find the value of k and write the general solution, then determine how long it takes for the total surface area of each of these growing populations to reach 1 m2. (Recall that 1 mm = 10-6 m.)

c. Assume the same population dynamics as given above. Determine how long it takes for each of the populations to grow to where their volumes occupy 1 cm3. (Recall that 1 cm = 10-2 m.)

d. Michael Crichton in the Andromeda Strain (1969) states that

"A single cell of the bacterium E. coli would, under ideal circumstances, divide every twenty minutes... [I]t can be shown that in a single day, one cell of E. coli could produce a super-colony equal in size and weight to the entire planet Earth."
The diameter of the Earth is 12,756 km, so assuming it is a perfect sphere, determine how long it takes for an ideally growing colony of E. coli (doubling every 20 min with the volume you computed above) to equal the volume of the Earth. (Don't forget that 1 km = 1000 m. Also, you have to find a new value of k and start with P0 = 1.) How does your answer compare to the statement of Michael Crichton?