Daily Newsletter March 7, 2012
Today's Topic: Bacterial Growth Formula
Today we are going to do a little math. Don't get scare, it is simple, but powerful. A critical concept that all microbiologists, and even cell biologists, must deal with is the rate at which cells grow. Even if you never use these equations again, going through them will build a mental picture of rate at which bacterial cells can multiply, and how large populations can become. On a practical side, it is critical as it helps you to understand when your population reaches a point that would be equivalent to a late log stage population; a working culture as it were.
The core bacterial growth formula is N
t = N
o x 2
n
N
o is the original culture, or the 0 generation.
t is time.
n is the number of generations that bacteria goes through
N
t is the final population size (the population at a specific time interval).
This formula is based on the binary fission form of bacterial growth, and describes the
log phase of growth.
Thus, it describes the doubling of a bacterial population at every time interval. This time interval is dependent upon the organism, and the environmental conditions.
Using this formula, you can solve more meaningful problems than just the doubling of bacteria.
Question: What would be a more meaningful question to ask about bacterial growth?
n = the number of generations a population has undergone.
n = (log N
t – log N
o) / log 2 = (log N
t – log N
o) / 0.301
So if you know the original population size, and the final population size, you can discover the number of generations a bacteria has gone through. Through mathematical manipulation, we can solve for other problems. In the next formula set we look at the generations per hour of a given bacterial culture.
k = the mean growth rate of a bacterial population (generations per hour).
k = n/t = (log N
t – log N
o) / (0.301 x t)
If we know the original and final population size, we can extrapolate n, the number of generations that have occured in the population. If we divide n by the time it took, you will get k, the number of generations that occur per hour.
Why would you want to know the number of generations per hour?
An inverse of k allows us to learn how many hours it takes to have one generation. This is also known as a generation time. If you ever read that an organism has a generation time of x minutes, you are looking at the solution to the problem below. As you can see, this provides a powerful tool in understanding an organism.
g = the mean doubling time of a bacterial population (hours per generation).
g = 1/k
Will an organism have the same g in all environmental conditions? Why?
Daily Challenge: Problems
Here are some problems to help you go through these equations. Show your work and answer the associated questions.
Problem: You have been asked to growth, in batch culture, a population of
Rhodococcus spp. (the designation ssp represents a generic species; either the species is unknown, unnamed, or in this case unimportant). You are using lactose as a carbon source, and providing a complete array of other essential nutrients by adding Yeast Extract to the nutrient broth.
Rhodococcus has a doubling time of 20 minutes in this environment. You start with 100 ml of a 10
4 culture. Your fermentation vessel is 10L (so it contains 10L of broth, including the starting culture). How long will it take to reach a population of 10
9,?
HINT: What are you looking for? number of generations? generation time? Something else? Which formula will give you the answer?
Question: Using the same system, an assistant attempts to replicate your system, but instead of lactose, they use sucrose as the carbon source. After 72 hours, they have a population of 10
7. What was the growth rate of this particular species of
Rhodococcus in this environment? Was this a more effective environment?
Question: Using the original fermentation equipment, you are asked to grow a population of
Rhodococcus spp. to 10
8,/sup> in 24 hours. Is this possible? Explain.
Learning Objectives:
What would you describe as the learning objectives today? Are you being asked to memorize these equations? Are you being asked to understand these equations? Are you being asked to use these equations? How could you use these equations?
