Question

a. Write down a differential equation for bacterial growth during the exponential phase

(batch culture), where k is the instantaneous growth rate.

b. Solve the equation. Your solution should be a continuous function that gives the number

of cells as a function of time.

c. Write an equation for the doubling time (generation time) in terms of k.

d. Parameterize your equation as follows: Assume that you start with 10 cells, and that it

takes each cell 30 minutes to divide. Then, using Matlab, plot your result on i) a linear

plot and ii) a log-linear plot.

Answer 1

a. Bacterial growth is the proliferation of fully grown mother bacterium into two daughter cells having half the cell size of its mother. There are four phases in bacterial growth namely lag phase, exponential (log) phase, stationary phase and death phase. Exponential phase is the period of exponential growth of the bacteria. Lets see about differential equation for bacterial growth during the exponential phase in batch culture (limited amount of nutrients-single input and single output).

dN/dt = kN where N is the population size, k is the instantaneous growth rate and t is the time.

N(0) = N₀ in initial condition. If k > 0, then above equation becomes

N(t) = N₀e^kt.

b. At the beginning t = 0, the initial concentration of bacterial cells is 1,000 CFU/mL. At time t = 6 h, the concentration of cells is 16,000 CFU/mL. What is the growth rate.

We know that, N(t) = N₀e^kt

Where: N₀ = initial concentration of cells = 1,000 CFU/mL
N = concentration of cells after time t = 16,000 CFU/mL
t = 6 h

kt = ln (N(t)/N₀)

k(6) = ln (16000/1000) = 2.77258872224

k = 2.8/6 = 0.462

Therefore growth rate is 0.462 or 46.2%.

c. For the bacteria, the doubling time (T) is readily computed from

2N₀ = N₀e^kT which becomes T = ln(2)/k

d. Given that we start with 10 cells, and that it takes each cell 30 minutes to divide. Here N₀ = 10 cells, t = 30 mins. But k value (growth rate) was not mentioned and they didn't say it is doubled or not. Here some more information is required in order to plot the graph using the above equation.