Question

The bacteria E. Coli doubles in a little over 20 minutes in good lab conditions. (For purposes of this exercise, we will assume it takes exactly 20 minutes.)

a) Write a difference equation that models the growth of an E. Coli colony in a large nutrient dish, with an initial population of y₀=10000 bacteria, and time measured in increments of 20 minutes. What is the solution to this equation and initial condition?

b) How would the equation change if the time between measurements was one hour instead of 20 minutes? (Consider what happens to the size of the colony in one hour.)

c) Why would these models be inappropriate for a colony beginning with a single bacterium, or only a few?

d) Why would these models be inapropriate for tracking the growth of the colony over a period of months?

e) Suggest and alternate (nonlinear) model that might be more appropriate in tracking the growth of the colony from its inception until well after it fills the petri dish. Explain in biologicsl terms any numbers or parameters.

Answer 1

a) Differential Equation will be as follows:

'y' is the number of cells at any time t.

'r' is the growth rate.

upon solving this differential equation we get =

     Where c is the constant of integration.

at t = 0, y = y₀ = 10⁴    (Given)

now r and the doubling time are related to each other as follows:

Where is the doubling time, and equal to 20 mins (Given).

                t is in mins.

b) if t = 60 mins then,

the size of the colony increases but, this increase is restricted due to contact inhibittion among E.coli cells.

c) This model is the logarithmic growth of cells. If the number of cells are very less then the increase will be y₀*2ⁿ

Where n is the number of cycles and is calculated accordingly

This is not the logarthmic model we had proposed earlier, but it is a geometric progression model. Hence the logarithmic model will be inappropriate. At this moment the cells will be in the Lag phase of growth curve.

d) There are major 4 phases in cell growth :

1. Lag
2. Log
3. Stationary
4. Death

When we want to study the culture growth over a long period of time, the logarithmic model is not suitable. The logarithmic model is helpful only in the case when the cells are in between the Lag and the Stationary phase. Cells exhibit contact inhibition and ultimately we reach the point where cell duplicaation rate is equal to cell death rate. Thus new models are needed for examine cell growth over a long period of time.

e) The Baranyi model is extensively used for tracking the growth. he introduced the factor of representing the physiological state of the cells.The model focused on "work to be done" and product of the lag time () and specific growth rate (). This model consists of two simultaneous equations and is most often used by microbiologists to understand growth pattern.