Q5
LILI Genetics 107
rev 1953 The Estimation of Mutation Rates Univ. Wis.
Adapted from Luria and Delbruck, Genetics 28:491; 1943, Armitage, Jour. Hygiene
51:162, 1953, and Lea and Coulson, Jour. Genetics 49:264, 1949.
1. Assumptions
a. Growth is synchronous and uniform for mutant and non mutant cells.
b. Mutations occur at celi division, yielding one mitant, one non mutant
offspring,
ec. All mutant ceils are effectively counted precisely at time of assay.
2. Definitions
N Cell mmber per culture.
t Generation number. Defined by the law cf geometric growth N « Qe or
t= Log, N,
r Number of mutant cells.
m Number of mutational events. d average mutant clone size = r/m.
a Probability that a division wilL yield one mutant offspring.
At each generation, N increases to 2N by means of N divisions. By summing the
total divisions, itis evident that N-N divisions are involved in the increase
from an inoculum No to a final value N.~ As Ng is usually negligible in canparison
te N, we may usually write that N divisions have occurred in the growth of a
culture of size N. Therefore:
(1) ms aN
Null fraction method
The probability that a mutation will not occur at a given division is l-a,.
Therefore the probability that no mutations will have occurred in a culture is
(2) Po
(22) pose eX or (2b) loge po = ~all, a = I/N logy V/po = 2:3 log,» 2
N Po
where py is that fraction of a series of cultures which contains no mtants.
(1-a)" which can be shown to be closely approximated ty
Average number of mutants
(Warning; Carefully distinguish r fromm }!) It will be shown that each genera-
tion contributes an equal number of mutants r to the final crop. At the i'th
generation, Ny cells are produced from N3/2 by means of Ny/2 = 21/2 divisions,
Cn the average, there will then be a.21/2 mutations at this generation, Each
of the cells of the i'th generation will increase by a factor N/Ns = 2°/21 by
the time of assay, The total erep of mutants fran the i'th generation will
therefore be a.21/2,2¢/2t or a.2/e2, Summing over all n generations we find
the total mutant crop
(3) 4
u
at2t/2 = atN/2 = a N logy N/2
2r/M log N = .602 r/W Log, N.
i
a
(4) and d = r/m = t/2,
Luria and Delbruck's likely average correction
L and D point out that (3) displays the average value of r, including the
contribution of a great many mutants from rare, very carly mutations, They set
up another expression which they consider will give the "likely average" in any
given experiment of € cultures, each size N.
It is assumed that no mutations are likely to have occurred prior tots i.
i is selected arbitrarily, as a function of mutation rate, so that in the entire
experiment with C cultures there will have been one premature mutation, that is,
so that a.C.Nj = 1, or 24 = 1/aC and i = -logeaC. Thus it is assumed that the
mutations in each culture have occurred throughout t-i, rather than all t,
generations. (3) then becomes
(5) rt = 1/2,(t-i) aN = (aN/2)(t+i) = (aN/2) (logoN + logs aC) = (aN/2) logs acN,
This cannot be solved explicitly for a, but may be handled numerically or with
the help of a chart provided by L and D, This treatment has been criticized by
Armitage; it has also been abused by workers who have pooled estimates of a
from different experiments, rather than summing the pooled data, At best this
technique does not mitigate the very high variance of r, which makes feasible
estimates of its mean very difficult.
Use oi the median and upper quartile
The (limited) solution of the theoretical distributicn of r by Lea and
Coulson allowed the development of two other measures, the medien ro and the
upper quartile rz (i.e. the values standing at the positions (n+1)/2 and 3(n+1)/4
in a series of n observations ranked by size),
They have shown that Po/tm ~log, m= 1.24 and r3/m ~ log, m= 4.09, respectively.
Tables to assist the calculation of m (=aN, and hénce of a) are given by these
authors. These methods do not make full use of all the numerical data, but provide
more stable estimates of r than methods based on the experimental mean,