# Fluctuations and noise I – Mutations

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```					   Fluctuations and noise I –
Mutations

l Luria SE, Delbruck M. 1943. Mutations of bacteria from virus
sensitivity to virus resistance. Genetics 28: 491-511.
l Fidler IJ, Kripke ML. 1977. Metastasis results from
preexisting variant cells within a malignant tumor. Science
197: 893-5.
Binomial distribution
The binomial distribution is the discrete probability of the number
of successes in a sequence of n independent yes/no experiments
(Bernoulli trial), each of which yields success with probability p.

Binomial distribution approaches to Poisson distribution when
the probability P is small （when p<<1, mean = variance）,
while approaches to Gaussian distribution when n is large enough.
Poisson process and Poisson distribution:
A Poisson process can be simply viewed as the chemical reaction:

l is the probability per unit time the reaction occurs, with a jump: n ® n+1

The Markov chain can be described by the Master equation (i.e., CKE):

A sample path of Poisson process:
Consider that a ligand binds a receptor, and it has
the probability per unit time k to dissociate from
the receptor. What is the probability P(t) the
ligand remains bound after the time t ?
Lytic vs Lysogenic infection

Lysogenic bacterium is usually immune to the virus,
and provide an example of acquired immunity.
The Luria–Delbrück experiment (also called the Fluctuation
Test)

The two possibilities tested
by the Luria–Delbrück
experiment. (A) If mutations
are induced by the media,
roughly the same number of
mutants are expected to
appear on each plate. (B) If
mutations arise
spontaneously during cell
divisions prior to plating,
each plate will have a highly
variable number of mutants.
LD’s theory:

Nt: the number of bacteria at t:

a, mutation rate:

m, the number of mutation:

The probability of no mutation:    For calculating the rate a

the number of resistant
bacteria at time t:
The number of mutation dm from

These number of mutations dm follows a Poisson distribution. At
the observation time t, the mean resistant bacteria stemming from
these mutations is expanded exp(tau) folds, and accordingly the
variance is exp(2tau) times greater. (Is there any approximation in
this derivation?) Thus,
The authors guessed a long-tailed distribution with very high
variance. The difficulty when testing a very flat distribution
with a small sample is: The average number of resistant
bacteria derived from a limited number of experimental
cultures will probably be considerably smaller than theoretical
predictions.
The likely average r of the number of the resistant bacteria :

t_0 is chosen as the time up to which just one mutation
occurred on average in a group of C similar cultures.
LD obtained the relation for a :

The final relation
to be tested by
experiments:
Modern theory
The very fast growing in the variance indicates a fast
collapsing distribution .
Experimental results are always larger than theory predicted
Luria–Delbrück experiment:

1 First grew cultures of B in nutrient broth or in
synthetic medium.
2 Inoculated a small number of bacteria into separate
culture tubes. (cultures to be tested, 10 cc or 0.2 cc)
3 After a period of growth, they plated equal volumes
of these separate cultures onto agar plates
containing virus. (samples of the cultures to be
tested, 0.05 cc or 0.08 cc)
4 Counting the number of resistant clones.

plating
nutrient   Inoculation
culture tubes             agar plates
broth
Nutrient Broth                 Agar plate

Colonized agar plate, divided into four. A
nutrient agar plate is a petri dish containing a
layer of agar gel that also contains some
proteins, minerals, sugar and vitamins
The Luria–Delbrück experiment (1943) (also called
the Fluctuation Test (彷徨实验)) demonstrates that
in bacteria, genetic mutations arise in the absence
of selection, rather than being a response to
selection. Therefore, Darwin‘s theory of natural
selection acting on random mutations applies to
bacteria as well as to more complex organisms. Max
Delbrück and Salvador Luria won the 1969 Nobel
Prize in Physiology or Medicine in part for this work.
（wikipedia）
Discuss:

The number of metastatic foci in the lungs of the mice
receiving the cloned sublines was similar to the number of
foci seen in mice receiving the parent line.

This would indicate that the parent population was
homogeneous and that the metastatic foci probably
resulted from adaptation during the process of
metastasis.
Alternatively, the cloned sublines gave rise to widely
different numbers of lung colonies.

This would suggest that the parent tumor was
heterogeneous and that cells of both high and low
metastatic potential preexisted in the parent population.
Scheme of the Fluctuation test

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