Estimating the Mutation Rate of Bacteria

1
The Luria-Delbrück Model

• Estimating the Mutation Rate of Bacteria
• Poisson Processes
• Probability Generating Functions

2
The Problem
Question Did the mutation to resistance
happen BECAUSE of the presence of a virus, or
even BEFORE adding the virus to the culture?
3
The Problem
The mutation happens
BECAUSE of the virus (acquired immunity)
BEFORE adding the virus (mutation to immunity)
VIRUS
VIRUS
4
The Problem
Parameters in both cases
µ the mutation rate (unknown) ß1 growth rate
for normal cells (known) ß2 growth rate for
mutant cells (known)
After growing several of those cultures, we can
measure

• - the number of cultures with no mutation
• the average number of mutant cells
• the sample variance of the number of mutant cells

5
The Proof
In 1943, S.E. Luria and M. Delbrück use the
interplay of mean and variance of both
distributions to show that the number of
mutations follows the Luria-Delbrück
distribution, and not the Poisson distribution.
Mutations happen independent of the presence of
viruses.
The data can then be used to estimate the
mutation rate µ.
6
The Luria-Delbrück Model
Assumptions
1. We start at time t0 with a single
non-resistent cell. 2. Normal cells grow with
rate ß1, that means . 3. A
mutation is the division of a normal cell into
one normal and one resistant cell. 4.
Mutations occur randomly with rate µN(t), that
means 5. The offspring of every mutant cell
grows with rate ß2.
7
Poisson Processes

• The numbers of mutations in disjoint time
  intervals
• are independent.

Note
8
The Luria-Delbrück Model
Let
Then
9
The Luria-Delbrück Model
Let p0(t) be the probability for no mutation
until time t.
10
Estimating the Mutation Rate
To estimate the mutation rate µ, solve for µ,
where
(P0 method)
11
Probability Generating Functions
Goal Calculate the complete distribution of X(t).
If X(t) were natural, we could set and calculate
the
12
Probability Generating Functions
Why is this useful?
- Can get back distribution - -
etc. - There are
techniques to find G(z,t).
13
The Lea-Coulson Formulation
To use these techniques, we need to change
the model such that X(t) takes only values
0,1,2,3
This could be done by simply setting
(discretized Luria- Delbrück model)
This leads to a closed form of the p.g.f. that
allows us to calculate the in finite
time.
14
The Lea-Coulson Formulation
Lea and Coulson used a different approach
They defined
,
where the are independent Yule
processes with birth rate ß2, that is a Poisson
process with rate function .
Thus X(t) is a filtered Poisson process.
15
The Lea-Coulson Distribution
We will now show how to calculate the p.g.f. of
X(t), using a standard technique We start with
an equation for the probability that the number
of resistant cells grows by one in the
time interval t,t?t, and then derive a partial
differential equation for the p.g.f.
16
The Lea-Coulson Distribution
17
The Lea-Coulson Distribution
The End