The Luria & Delbruck Fluctuation Test.

(1943; Genetics 28:491-511)

Dean W. Gabriel

When a human acquires immunity to disease, he starts out in a state in which he is susceptible, but winds up resistant.  Is the resistant state heretible?  A bacterial population may be used to model the situation.  The starting bacterial culture is susceptible, and yet it winds up resistant, after most are killed off.  The question is, does exposure to the disease cause the immunity, or does exposure to the disease select immune mutants?

Luria & Delbruck used E. coli bacteria and a virus (phage) that lyses the bacteria. The observation was that if you start with a pure (from single colony) culture, add the phage, it will clear the culture. After a few hours or days, the culture is back. Nothing was added. Where did the resistant bacteria come from? It wasn't an introduction, so it was either mutation or some kind of resistance response (say, similar to the way we might resist or die from, the plague).

Hypothesis 1: (Mutation hypothesis): There is a finite and small probability (a) that a mutation from susceptible to resistant will occur at a constant rate.

Hypothesis 2: (Physiological resistance hypothesis): there is a finite and small probability that each cell in the culture might survive the phage attack, and all subsequent progeny from survivors will be resistant.

Hypothesis 1 corollaries:

---will generate resistant clones, and their prevalence will be determined by how early in culture growth the mutation event occurred.

---the proportion of resistant bacteria will increase over time, prior to selection.

--if you study a large number of small cultures, the proportion of resistant bacteria in each culture will vary or fluctuate wildly, depending on how early the mutation occurred.

Hypothesis 2 corollaries:

--no clones. Some bacteria will survive and reproduce resistant daughter cells. Selection begins only after the addition of phage, not before.

--the proportion of potential survivors (resistant) bacteria will remain constant.

--if you study a large number of small cultures, the proportion of resistant bacteria will remain constant; size of the culture is irrelevant.

The experiment:

First, to demonstrate good sampling techniques, a large culture was used. A carefully measured sample was drawn, plated on an agar plate loaded with phage, and the total # of survivors was counted. A dilution series was also performed to get the total number of bacteria per ml of culture.

Experiment 1 Experiment 2 Experiment 3
Sample 1 14 46 4
Sample 2 15 56 2
Sample 3 13 52 2
Sample 4 21 48 1
mean 16.7 51.4 3.3
variance 15 27 3.8

Note that in each experiment, the sampling error is low. But between experiments, there is variation, indicating some support for the mutation theory.

They then grew very small cultures of E. coli, and sampled and plated as before. The results were typically:
Exp 1 Exp 3 Exp 4
Culture 1 10 30 1
Culture 2 18 10 0
Culture 3 225 40 0
Culture 4 10 45 7
Culture 5 14 153 303
mean 26.7 62 26.2
variance 1,217 3,498 2,178

With a large number of small cultures, the total number of resistant bacteria fluctuates wildly from culture to culture, within a given experiment (same medium, same day, same phase of the moon, etc.)    Note the variance.   This provides good evidence for the mutation hypothesis. A possible objection: maybe the probability of survival by the second hypothesis is highly variable in small cultures. This is possible, but remained a poor possibility in light of the authors' demonstration that despite the wild fluctuations in numbers among cultures, the mutation rate remained constant.

(The calculations provided below are for your benefit, and will not be required for exams: )

The calculation is based on two things:  the generation time--the time for doubling (a common calculation in bacteriology), and on a determination of the mutation rate (a difficult calculation that is not ordinary):.
First, the doubling time:

Nt= 2n (No), where Nt is total number (measured), No is starting number (measured) , and n is # of bacterial generations (calculated).

Then, ln Nt = n(ln 2) + ln No

n = ln Nt - ln No
        ln 2

If a = mutation rate or chance of mutation per individual per generation, then

a = m

where m = number of mutations (not number of mutants [which could be measured directly]!)

The problem is how to correlate the total number of mutants with the number of mutations. The total number of mutants depends on the growth rate of cells carrying "old" mutations ("mutants") plus cells with the new mutations. That is, dm (increase in number of mutants) = Mtdt (total number of mutants at time t) + a Nt dt (new mutants added).

But we cannot determine exactly how far back old mutations occurred, and their subsequent clonal replication began; that is, we cannot directly measure the number of mutations, m.

We can, however, apply a probability function to estimate the average number of mutations per milliliter. This is because there are two possible "states", mutant or wild type, resistant or susceptible. Let P = the probability of the mutant state; and Q = the probability of wild type. Since the number of mutations is exceedingly small, we cannot apply a binomial distribution here. Instead we use a Poisson distribution, where

Po = e-y, P1 = e-y y, P2 = e-y y2 etc.

in the above, y represents the average number of events per unit of space or time. In our case, it represents m, the average number of mutations per ml.

If we look at experiment 4, above, for example, we see that

Po = 2/5. Therefore, 2/5 = e-m. It is then a simple matter to solve for m.

In the small culture experiments, the mutation rates were:
Exp. 1 1.8 X 10-8
Exp 2 1.4 X 10-8
Exp 3 4.1 X 10-8
Exp. 4 2.1 X 10-8
Exp.5 1.1 X 10-8
Therefore the mutation rates are evidently constant, even while wide fluctuations occur.

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