How to calculate mutation rates

Methods for Determining Spontaneous Mutation Rates
Patricia L. Foster
Department of Biology, Indiana University, 1001 E. Third Street, Bloomington, IN 47405, USA, Email
ude.anaidni|retsoflp#ude.anaidni|retsoflp, Tel. (+1) 812 855 4084, Fax (+1) 812 855 6705
Spontaneous mutations arise as a result of cellular processes that act upon or damage DNA. Accurate
determination of spontaneous mutation rates can contribute to our understanding of these processes
and the enzymatic pathways that deal with them. The methods that are used to calculate mutation
rates are based on the model for the expansion of mutant clones originally described by Luria and
Delbrück and extended by Lea and Coulson. The accurate determination of mutation rates depends
on understanding the strengths and limitations of these methods and how to optimize a fluctuation
assay for a given method. This chapter describes the proper design of a fluctuation assay, several of
the methods used to calculate mutation rates, and ways to evaluate the results statistically.
Spontaneous mutations are mutations that occur in the absence of exogenous agents. They may
be due to errors made by DNA polymerases during replication or repair, errors made during
recombination, the movement of genetic elements, or spontaneously occurring DNA damage.
The rate at which spontaneous mutations occur can yield useful information about cellular
processes. For example, the occurrence of specific classes of mutations in different mutant
backgrounds has been used to deduce the importance of various DNA repair pathways (Miller,
The mutation rate is the expected number of mutations that a cell will sustain during its lifetime.
The mutant fraction or frequency is the proportion of cells in a population that are mutant.1
Although mutant frequencies can be adequate indicators of the rate at which mutations are
induced by DNA damaging agents, they are inadequate indicators of spontaneous mutation
rates. This is because the population of mutants is composed of clones, each of which arose
from a cell that sustained a mutation. The size of a given mutant clone will depend on when
during the growth of the population the mutation occurred. This is the fundamental property
of spontaneous mutation that was exploited in the famous Luria and Delbrück fluctuation test
(Luria and Delbrück, 1943). Among replicate cultures, the distribution of the numbers of
mutations that were sustained is Poisson, but the distribution of the numbers of mutants that
result is far from Poisson, and is usually referred to as the Luria-Delbrück distribution.
There are two basic methods to determine mutation rates: mutant accumulation and fluctuation
analysis. These two methods are described below, with emphasis on fluctuation analysis.
The definitions of the terms used in this chapter are given in Table I. It is important to
distinguish between m, the mean number of mutations that occur during the growth of a culture,
and μ = the mutation rate, which is the mean number of mutations that occur during the lifetime
of a cell. Almost all methods to calculate mutation rates start by determining m and then obtain
1A mutation is a heritable change in the genetic material; a mutant is an individual that carries a mutation.
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μ by dividing m by some measure of the number of cell-lifetimes at risk for mutation, usually
Nt. It is also important to distinguish between the number of mutations per culture, m, and the
number of mutants per culture, r. r divided by Nt is the mutant fraction or mutant frequency,
The Lea-Coulson Model
The methods for calculating mutation rates discussed below are dependent on the model of
expansion of mutant clones originally described by Luria and Delbrück (Luria and Delbrück,
1943) and extended by Lea and Coulson (Lea and Coulson, 1949). It is usually called the Lea-
Coulson model and has the following assumptions.
1) the cells are growing exponentially
2) the probability of mutation is not influenced by previous mutational events
3) the probability of mutation is constant per cell-lifetime
4) the growth rates of mutants and nonmutants are the same
5) the proportion of mutants is always small
6) reverse mutations are negligible
7) cell death is negligible
8) all mutants are detected
9) no mutants arise after selection is imposed
In addition, for assays using batch cultures, the following assumptions apply
10) the initial number of cells is negligible compared to the final number of cells
11) the probability of mutation per cell-lifetime does not vary during the growth of the
Departures from the Lea-Coulson model affect the distribution of mutant numbers and impact
the mutation rate calculation. In general, most of the model’s requirements can be met with
proper experimental protocols, but some departures reflect real biological phenomenon.
Mutant Accumulation
When a population is growing exponentially the appearance of new mutants plus the
proliferation of preexisting mutants results in a constant increase in the mutant fraction each
generation. The mutation rate is this increase, and can be determined by measuring the change
in the mutant fraction over time (Figure 1). However, an important caveat is that the population
must be of sufficient size so that the probability that mutations occur each generation is
essentially unity; otherwise, the chance occurrence of mutations will dominate the population
(Luria, 1951);. For batch cultures, the population must be large enough so that the average
number of mutations per culture, m, is much greater than one. For a continuously dividing cell
population, μ can be calculated by Eq. (1) (Drake, 1970):
μ = (r2
N 2

N 1 )÷ ln(N 2N 1) = (f 2 − f 1) ÷ (lnN 2 − lnN 1) Eq. (1)
Although conceptually simple, there are important technical difficulties that limit the use of
this method. In general, by the time the population reaches a sufficient size, some mutations
have already occurred, and these produce clones of mutants that make it impossible to
accurately measure the accumulation of new mutants. Thus, to measure mutant accumulation
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a large population with few mutants must be generated. One way to do this is by generating
and testing a number of populations and using the ones with a low mutant fraction. This is
convenient if the population can be stabilized and used repeatedly for experiments, for example
stocks of bacteriophage or viruses. Alternatively, a population can be purged of pre-existing
mutants if the mutational target gives a phenotype that can be selected both for and against
(i.e. a “counterselectable marker”). Selection against the mutants can be used to purge the
population, and then selection for the mutants can be used to measure mutation rates. For
example, Lac+ bacteria with a temperature sensitive galE mutation are selected against by
lactose at the nonpermissive temperature, but selected for by lactose at the permissive
temperature (Reddy and Gowrishankar, 1997). In mammalian cells, mutations in the
hypoxanthine-guanine phosphoribosyl transferase (hprt) locus make cells sensitive to HAT
medium (which contains hypoxantine, aminopterin, and thymidine) but resistant to 6-
thioguanine (Glaab and Tindall, 1997). A few other counterselectable markers are available
(Reyrat et al., 1998). Another possibility is to use a mutational target that allows mutants to be
eliminated by cell sorting. For example, mutants that allow green fluorescent protein (GFP) to
be produced can be eliminated by fluorescence-activated cell sorting (FACS); new mutants
can then be detected by flow cytometry (Bachl et al., 1999)
Mutant accumulation has been extensively used to measure mutation rates in chemostats
(Kubitschek and Bendigkeit, 1964;Novick and Szilard, 1950) Cell number, N, is constant in a
chemostat, so mutant accumulation is a function of the growth rate, λ, and the mutation rate
per cell per generation μ. Thus:
μ = 1

(r2 − r1)
(t2 − t1) Eq. (2)
Where t1 and t2 are the times at which the numbers of mutants, r1 and r2, are measured.
Fluctuation Analysis
Experimental Design
A normal fluctuation test begins by inoculating a small number of cells into a large number of
parallel cultures. The cultures are allowed to grow, usually to saturation, and then each culture
is plated on a selective medium that allows the mutants to produce colonies. The total number
of cells is determined by plating appropriate dilutions of a few cultures on nonselective
medium. The distribution of the numbers of mutants among the parallel cultures is used to
calculate the mutation rate. This basic design, invented by Luria and Delbrück (Luria and
Delbrück, 1943), can be used for single-celled microorganisms, cultured cells, bacteriophage,
and viruses. So that individual mutants can be counted, a solid medium is usually used for
selection, but the p0 method (see below) also can be used with liquid medium.
The goal of designing a fluctuation assay is to maximize the precision with which the mutation
rate is estimated. Precision is a measure of reproducibility, not accuracy (accuracy is how well
the resulting estimate reflects the actual mutation rate, and that will depend on how well the
underlying assumptions reflect reality). The important design parameters are: m, the number
of mutations per culture; r, the number of mutants per culture; N0, the initial number of cells;
Nt, the final number of cells; V, the culture volume; and, C, the number of parallel cultures.
The first step is to determine a preliminary r and m by plating aliquots from a few parallel
cultures on the selective medium. A preliminary m is calculated from the mutant numbers
obtained (eg., using Method 2 or 3) and then the other parameters adjusted so that the final
m is within a useful range. The value of m will determine which methods can be used to calculate
the mutation rate. None of the methods are reliable if m is less than 0.3 unless a prohibitive
number of cultures are used. However, if m is above 15 some of the methods are not valid
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(Rosche and Foster, 2000). Obviously, the final m also has to be small enough so that the
number of colonies on the selective plates is countable. However, if there are a few outlier high
counts, they can be truncated at 150 with little loss of precision (Asteris and Sarkar, 1996;Jones
et al., 1999).
The desired m is achieved by adjusting Nt, the final number of cells, either by manipulating
the cell density or the culture volume. When using defined medium, cell density can be adjusted
by limiting the carbon source. However, using other required growth factors, such as vitamins
or amino acids, is not recommended because non-requiring mutants will be selected and cell
physiology may change. For example, in tryptophan-limited chemostats, bacterial mutation
rates become time-dependent instead of generation-dependent (Kubitschek and Bendigkeit,
The desired Nt can be achieved by adjusting the culture volume. It is usually considered
necessary to plate all the cells from each culture on the selective medium because sampling,
or low plating efficiency (which is the same thing), increases the proportion of cultures with
small numbers of mutants and narrows the distribution (Crane et al., 1996;Stewart et al.,
1990). But this requirement restricts the volume of culture that can be used without
concentrating the cells (which can be tedious with many cultures). However, if several mutant
phenotypes are to be assayed in the same cultures, sampling is unavoidable. In addition, because
a large culture contains more “information” than a small culture, it is better to plate a small
aliquot from a large culture than all of a small culture if a proper correction can be applied
(Jones et al., 1999). Some, but not all, of the methods for calculate mutation rates discussed
below are amenable to such corrections.
The validity of the mutation rate calculation requires that Nt be the same in each culture.
Usually, but not always, this can be accomplished by growing cells to saturation. If achieving
an uniform Nt is a problem, the cell number in each culture can be monitored before mutant
selection by measuring the optical density or by counting cells microscopically (e.g. using a
Petroff-Hausser chamber). Because there is currently no valid method to correct for different
Nt’s, deviant cultures must be eliminated from the analysis.
The initial inoculum, N0, must contain no preexisting mutants and must be small relative to
Nt. Most of the methods to calculate the mutation rate are valid if N0 is at least 1/1000 of Nt
(Sarkar et al., 1992), but this may not insure that N0 contains no mutants. A reasonable rule of
thumb is that N0 roughly equals (Nt/m) × 10−5. the best way to insure uniformity is to grow a
starter culture in the same medium that will be used for the fluctuation assay, dilute these cells
to the appropriate density in a large volume of fresh medium, and then distribute aliquots into
individual cultures tubes for nonselective growth.
The precision of the estimate of m depends on C, the number of parallel cultures. Most
experiments have 10 to 100 parallel cultures, with about 40 being most common. There is little
gain in precision if C is larger unless m is less than about 0.3 (Jones et al., 1999;Rosche and
Foster, 2000).
The fluctuation test was originally designed to test whether mutations occur before or after
selection is imposed (Luria and Delbrück, 1943). If the selection is lethal, then the only mutants
that appear must have pre-existed. However, if the selection is not lethal, for example reversion
of an auxotrophy or utilization of a carbon source, mutants can arise both before and after
selection has been applied. Post-plating mutants can arise because the cells are proliferating
on the selective medium (i.e. the selection is not stringent), or they can result from mutations
that occur in non-growing cells (adaptive mutations). In either case, the distribution of mutant
numbers will be a combination of the Luria-Delbrück and Poisson, and the m estimated for
pre-plating mutations will be inflated by the post-plating mutations (Cairns et al., 1988). If
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non-mutant cells grow on the selective medium because of contaminating nutrients, one
solution is to add an excess of scavenger cells that cannot mutate (because, for example, they
have a deletion of the relevant gene) to consume the contaminants (Cairns and Foster, 1991).
If the non-mutant cells grow because the selection is not stringent, the time it takes for a mutant
to form a colony on the selective medium can be determined and then mutant colonies can be
counted at the earliest possible time after plating.
Analyzing the Results of Fluctuation Assays
Fluctuation assays give the distribution of the numbers of mutants per culture, r, which is used
to calculated m, the mean or most likely number of mutations per culture. m is not itself a
particularly interesting parameter since it depends on the cell density and the volume of the
culture. However, it is mathematically tractable and yields the mutation rate when divided by
some measure of the number of cells. Although the distribution of mutants is not Poisson, the
distribution of mutations is, so m is a Poisson parameter. There are many methods to calculate
m (often called estimators) but they are all based on the theoretical distribution of mutant clonesizes
described by Luria and Delbrück (Luria and Delbrück, 1943) and Lea and Coulson (Lea
and Coulson, 1949). Each method has its advantages and disadvantages, and the choice of
method depends on the particular conditions of the experiment and the mathematical
sophistication and persistence of the user. The MSS maximum likelihood method (Method 5)
is the gold standard because it utilizes all of the results of an experiment and is valid over the
entire range of mutation rates. Of the less complicated methods, the Lea-Coulson method of
the median (Method 2) and the Jones median estimator (Method 3) are reliable when mutation
rates are low to moderate, and the p0 method (Method 1) can be used when mutation rates are
very low (m ≤1). Drake’s Formula (Method 4) is particularly useful when comparing data
reported as mutant frequencies instead of mutation rates. Methods 6 and 7 can be useful when
not all the requirements of the clone-size distribution are met. No method using the mean
number of mutants is valid, and none are given here. To see how these various methods behave
with real data, see Rosche and Foster, 2000 (Rosche and Foster, 2000).
Method 1: The p0 method The distribution of the number of mutations that occur during
the growth of parallel cultures has a Poisson distribution. If there are no mutants, there were
no mutations, and so the mean number of mutations can be calculated from p0, the proportion
of cultures with no mutants (Luria and Delbrück, 1943). The zeroth term of the Poisson
distribution is:
p0 =e −m Eq. (3)
So m is:
m = − ln p0 Eq. (4)
Although simple, the p0 method is limited. Its range of usefulness is 0.7 ≥p0≥0.1 (0.3 ≤ m ≤
2.3) and it performs best when p0 is about 0.3. The p0 method is inefficient (i.e., requires more
cultures for the same precision) compared to other methods (Koziol, 1991;Rosche and Foster,
2000). In addition, p0 is sensitive to several biologically relevant factors that complicate
fluctuation analysis. Phenotypic lag (the delay in expression of a phenotype), poor plating
efficiency, and selection against mutants all inflate p0 and result in an erroneously low m.
However, if all cells (not just mutants) have a plating efficiency of less than one, a correction
factor can be applied to m. The same correction can be applied if only a fraction, of each culture
is plated (Jones, 1993;Stewart et al., 1990). The actual m is calculated from the observed m
using Eq. (5) where z is either the plating efficiency or the fraction plated2:
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mact =mobs
z − 1
z ln(z ) Eq. (5)
Method 2: Lea-Coulson median estimator This method is based on the observation that
for 4 ≤ m ≤ 15, a plot of the probability that a culture contains r or fewer mutants, versus
m − ln(m) gives a skewed curve about a median of 1.24 (Lea and Coulson, 1949).
Rearranging gives the following transcendental equation relating m to the median, r̃:
r ~
m − ln(m) − 1.24 = 0 Eq. (6)
that can be solved easily by iteration (an example of how to use a spread sheet to solve this
equation is given in Figure 2). The Lea and Coulson method of the median is easy to apply and
remarkably accurate with in computer simulations (Asteris and Sarkar, 1996;Stewart, 1994)
and with real data (Rosche and Foster, 2000). It performs well over the range
2.5 ≤ r ~ ≤ 60(1.5 ≤m ≤ 15) (Rosche and Foster, 2000). Because it uses the median, it is
relatively insensitive to deviations that affect either end of the distribution, especially if r̃ is
relatively large. However, because little of the information obtained from the fluctuation test
is used, the method is relatively inefficient (Rosche and Foster, 2000).
Method 3: The Jones median estimator This estimator is based on the theoretical
dilution of the experimental cultures that would be necessary to produce a distribution with a
median of 0.5 (Jones et al., 1994). The basic equation is:
m = r ~ − ln(2)
ln( r ~ ) − ln ln(2)

r ~ − 0.693

ln( r ~ ) + 0.367
Eq. (7)
Two advantages of the Jones estimator are that it is explicit and that it accommodates dilutions.
If z = the fraction of the culture that is plated or the plating efficiency, and r ~
obs is the observed
median, then (Jones et al., 1994):
m = (r ~
z )− 0.693
ln(r ~
z )+ 0.367
Eq. (8)
In computer simulations over the range 3 ≤ r ~ ≤ 40(1.5 ≤m ≤ 10) the Jones estimator proved
to be as reliable and more efficient than the Lea and Coulson median estimator (Jones et al.,
1994). The Jones estimator also performs well with real data (Rosche and Foster, 2000).
Method 4: Drake’s formula Drake’s formula (Drake, 1991) is an easy way to calculate
mutation rates from mutant frequencies, and is especially useful in comparing data from
different sources. Because it uses frequencies, Drake’s formula gives the mutation rate, μ,
instead of m (with μ =m
), Starting from Eq. (1) above, Drake sets N1 to be 1/μ, the
population size at which the probability of mutation approaches unity. Assuming that no
2When z = 1, z − 1
z ln(z ) = 1 from I’Hôpital’s rule, and mact = mobs
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mutations occur before the population reached this size, f1 is zero; f2 is the final mutant
frequency, f; and, N2 is the final population size, Nt. This gives:
u = f ÷ ln(uNt ) Eq. (9)
that can be solved for μ by iteration. Drakes’ formula is based on the same assumption discussed
above for Eq. (1), i.e. that mutations occur only during the deterministic period of mutant
accumulation. Using the median frequency (if available) instead of the mean minimizes the
influence of jackpots(Drake, 1991). Since μ =m
, Drake’s formula can be rearranged into
the same form as Lea and Coulson’s formula, Eq. (6):
m − ln(m) = 0 Eq. (10)
When m < 4, estimates of m obtained with Eq. (10) are significantly higher than those obtained
with Eq. (6), but asymptotically approach those obtained with Eq. (6) as m becomes larger. If
m ≥30, the differences are trivial (Rosche and Foster, 2000).
Method 5: The MSS-maximum likelihood method (Sarkar et al., 1992) described a
recursive algorithm based on the Lea-Coulson generating function (Lea and Coulson, 1949)
that efficiently computes the Luria-Delbrück distribution for a given value of m3. Known as
the MSS algorithm, it is:
p0 =e −m
; pr = mr
Σ i =0
r −1 pi
(r − i + 1) Eq. (11)
Note that the equation to calculate p0 is the same as Eq. (3) and the proportion of cultures with
each of the other possible values of r is given by the equation on the right. The algorithm is
recursive, meaning that at a given m, the proportions of cultures with 0, 1, 2, 3, etc. mutants
are p0=e −m; p1 =m ×
2 ; p2 = m2
× (p0
3 +
2 );
p3 = m3
× (p0
4 +
3 +
2 ); etc. for all possible values or r.
This algorithm can be used to estimate m from experimental results using a maximum
likelihood function, the formula for which is (Ma et al., 1992):
f (r ∣m) = Π
i =1
f (ri ∣m) Eq. (12)
wheref(r∣m) = prfrom Eq. (11) and C is the number of cultures. The procedure is to start with
a trial m (obtained from Eq. (6), for example) and use Eq. (11) to calculate the probability,
pr, of obtaining each possible r from 0 to the maximum value obtained (even if a given r was
not obtained in the experiment it has to be included in the recursive equation). As mentioned
above, for most experiments, values of r greater than 150 can be lumped into one category
(Asteris and Sarkar, 1996). The likelihood function, Eq. (12), is the product of these calculated
pr’s for r obtained in the experiment. The easiest way to do this calculation is to arrange the
3The algorithm itself was described earlier (Gurland, 1958;Gurland, 1963) but was independently derived by Sarkar and coworkers and
applied to fluctuation analysis (Ma et al., 1992;Sarkar et al., 1992)
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mutant counts in order and count the number of cultures that had each r mutants, cr. Then the
product of the pr’s is:
(p0)c0 × (p1)c1 × (p2)c2 × (p3)c3 × (p4)c4 … Eq. (13)
where each pr is from Eq. (11) (alternatively cr ln(pr) for each r can be added). Note that values
of r that were not obtained in the experiment give a value of 1 in Eq. (13) and so do not have
to be included. The procedure is repeated with different m’s over a small range until a m that
maximizes Eq. (13) is found.
As mentioned above, the MSS-maximum likelihood method is the best method currently
available to estimate m. It uses all the results from a fluctuation experiment and is valid over
the entire range of values of m. In addition, computer simulations have shown it to behave in
a manner that allows statistical evaluation (Stewart, 1994). A comparison with other methods
using real data can be found in Rosche and Foster (Rosche and Foster, 2000)
Method 6: Accumulation of clones Luria (Luria, 1951) pointed out that Pr, the proportion
of cultures with r or more mutants, approaches 2m/r during the deterministic portion of growth
(i.e. when m is 1 or greater). Formally:
Pr = Σ i =r
pi = 2m
r Eq. (14)
Taking logarithms gives
ln(Pr ) = − ln(r ) + ln(2m) Eq. (15)
A plot of ln(Pr) versus ln(r) will yield a straight line with a slope of -1 and an intercept (where
ln(r) = 0) equal to ln(2m). Dividing Pr at the intercept by 2 gives m.
Method 7: The quartiles method The median is the central (50%) quartile of a distribution.
More of fluctuation assay can be incorporated in the calculation of m if the upper (75%) and
lower (25%) quartiles are also used. By regressing m versus the theoretical values of r at the
quartiles, Koch (Koch, 1982) derived the following empirical equations:
m1 = 1.7335 + 0.4474 Q1 − 0.002755(Q1)2 Eq. (16)
m2 = 1.1580 + 0.2730 Q2 − 0.000761(Q2)2 Eq. (17)
m3 = 0.6658 + 0.1497 Q3 − 0.0001387(Q3)2 Eq. (18)
where Q1, Q2 and Q3 are the values of r at 25%, 50%, and 75% of the ranked series of observed
r’s. For a perfect Luria-Delbrück distribution, the three m’s should be equal. These equations
are valid over the range 2≤m≤14; Koch also gives graphs that can be used up to values of m =
120 (Koch, 1982).
Calculating the Mutation Rate
The mutation rate, μ, is the mean number of mutations, m, normalized to some measure of the
number of cells at risk for mutation. Three such measures are used, each of which is based on
different assumptions about the underlying mutational process. If the probability of mutation
is constant over the cell cycle, then m should be divided by the number of cell divisions that
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have taken place. Since the final number of cells in a culture, Nt, arose from Nt-1 divisions,
the mutation rate is (Luria and Delbrück, 1943):
u =m
(Nt − 1) ≈
Eq. (18)
The same calculation applies if mutations are assumed to occur at or during division, (Armitage,
1952). If mutations are assumed to occur at the beginning of the cell cycle (i.e. shortly after
division), then m should be divided by the total number of cells that ever existed in the culture,
which is 2 Nt (because Nt cells had Nt/2 parents, Nt/4 grandparents, etc., and the sum of the
series is 2 Nt) . Thus, the mutation rate is (Armitage, 1952):
u =m
Eq. (19)
However, cells usually are not growing synchronously, and in an asynchronous population
there are an average of N/ln(2) cells during one generation period. Thus, the total number of
divisions during the growth of a culture is Nt/ln(2) and the mutation rate is (Armitage, 1952):
u =mln(2)


Eq. (21)
For the same m, these three equations will give mutation rates that differ by 1: 0.5: 0.693. It is
best to use one consistently, and to describe which one was used so that readers can compare
results obtained with different methods.
Statistical methods to evaluate mutation rates
The estimates of m or μ obtained from fluctuation tests are neither normally distributed nor
unbiased; therefore, no matter how many times a fluctuation experiment is repeated, it is not
valid to take the mean and standard deviation of the results (Asteris and Sarkar, 1996;Jones et
al., 1994;Stewart, 1994). There are two approaches that allow reasonable confidence limits to
be placed around estimates of mutation rates. The first approach is to put confidence limits
around the parameter used to calculate m and calculate new m’s using these values; the new
m’s will be estimates for the confidence limits of m (Wierdl et al., 1996). This approach is valid
only for parameters that have defined distributions, such as p0 and the median. The second
approach is to find a transforming function that gives m a normal distribution; this has been
successful only for the MSS maximum likelihood method (Method 5). Once confidence limits
are obtained for m, these can be divided by Nt (or 2 Nt or Nt/ln2) to estimate the confidence
limits for μ, the mutation rate. Of course, this procedure ignores the variance of the
determination of Nt, (which is approximately Nt). Although nontrivial, it is probably justifiable
to ignore this variance as long as Nt is determined accurately.4
Confidence Limits for p0—Because a culture either has mutants or it does not, p0 can be
considered a binomial parameter with p0 = p and (1− p0)=q (Lea and Coulson, 1949). For a
sample population of n, the standard deviation of p is
σp = pq
n − 1 Eq. (22)
4It is a little-appreciated fact that the expected value of the ratio of two variables is not the ratio of their expected values (i.e. not the ratio
of their means). Furthermore, the calculation of the variance of a ratio is fairly complicated (e.g. see (Rice, 1995). However, if the
denominator is larger than the numerator, the variance of the ratio will be smaller than the variance of the numerator, and thus no great
harm should be done by ignoring the variance of the denominator.
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But for most fluctuation assays, using the binomial is inappropriate and gives meaningless
intervals. There are several more wildly applicable methods to calculate the CLs for a
proportion; the following one uses the F statistic (Zar, 1984):
CL Upper =
(np + 1)Fdf 1
nq + (np + 1)Fdf 1
Eq. (23)
CL Lower = np
np + (nq + 1)Fdf 2
Eq. (24)
where F is evaluated at the desired α (α = level of significance) and the following degrees of
df 1 : v1 = 2(nq + 1); v2 = 2np Eq. (25)
df 2 : v1 = 2(np + 1); v2 = 2nq Eq. (26)
Confidence limits for the median—The median is, by definition, the value at which the
cumulative binomial probability is equal to 0.5 (i.e. 50% of the values are above and 50% of
the values are below the median). Therefore, the 95% CLs for the median are the values above
and below which less than 5% of the values are expected to fall, given n trials and a probability
of 0.5. These values can be found by calculating the binomial probabilities for each possible
rank-value of a given sample population and then finding the upper and lower rank-values that
symmetrically include 95% probability (or any other desired probability). The CLs for the
median are the actual sample values that correspond to these rank-values. Conveniently, the
binomial probabilities have already been calculated in tables found in many statistic books
(e.g., (Zar, 1984). Thus, to calculate CLs for the median
1. Use a table or calculate the binomial probability for each possible rank-value
(including 0) for the given population size, n = C, and p = q = 0.5.
2. Pick i, the highest rank-value that has a probability equal to or less than α/2 .
3. Pick the j = n-(i+1) rank-value.
4. Order the samples by increasing value
5. Pick the (i+1)th and the jth values. These are the CLs for the median value.
Confidence limits for m obtained from the MSS maximum likelihood method—
Using simulated fluctuation tests, Stewart (Stewart, 1994) evaluated the distributions of m’s
obtained using several of the common methods. He found that the natural logarithms of m’s
obtained using the MSS maximum likelihood method (Method 5) are approximately normally
distributed. From this, Stewart calculated the standard deviation of ln(m):
σln(m) ≈ 1.225m −0.315
C Eq. (27)
where C is the number of cultures. Since ln(m) is normally distributed, the 95% confidence
limits for ln(m) should be
ln(m) ± 1.96σln(m) Eq. (28)
While this is a reasonable approximation, the true confidence limits must be calculated from
the actual m and σ of the population, not the experimentally determined m and σ of the sample.
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Methods to calculate or estimate the correct confidence limits are given in Stewart (Stewart,
1994). A close approximation can be obtained from the following equations (Rosche and
Foster, 2000)
CL +95% = ln(m) + 1.96σ (e 1.96σ )−0.315 Eq. (29)
CL −95% = ln(m) − 1.96σ (e 1.96σ )+0.315 Eq. (30)
Once the upper and lower limits for ln(m) are obtained, the upper and lower limits for m are
simply the antilogs.
Departures from the Lea-Coulson Model
All the methods for calculating mutation rates discussed above depend on the Lea-Coulson
model of expansion of mutant clones (Lea and Coulson, 1949); therefore, calculated mutation
rates will be wrong If the assumptions of the model are violated. However, with some care,
several biological meaningful departures can be accommodated, and meaningful mutation rates
Sampling or low plating efficiency—If only a sample of a culture is plated, or if the cells
(not just the mutants) have a plating efficiency less than 100%, all clones will be reduced in
size by the same relative amount and m will be too small. The observed m can be corrected
using Eq. (5) if Method 1 is used, or Eq. (8) if Method 3 is used.
Phenotype lag—If the expression of a mutant phenotype is delayed for several generations,
mutants that arise in the last few generations of growth will result in few mutant progeny,
whereas mutants that arise early will contribute a normal number. Thus, the lower end of the
distribution will be affected, but, depending on the length of the lag, the upper end will not,
resulting in an inflated m. The actual m can be estimated graphically with Method 6 by using
only the upper part of the curve and eliminating any obvious jackpots (Rosche and Foster,
2000). If the length of the phenotypic lag is known, Koch (Koch, 1982) gives a method for
estimating m from the quartiles (Method 7).
Selection against mutants—If during non-selective growth mutants grow more slowly
that nonmutants, the result is the opposite of what happens if there is phenotypic lag: mutants
that arise in the last few generations of growth will contribute a normal number of mutant
progeny, but mutants that arise early will contribute few. This shifts the distribution of mutant
numbers from the Luria-Delbrück toward the Poisson (Koch, 1982;Stewart et al., 1990). Koch
(Koch, 1982) gives graphs that can be used to estimate m from the quartiles when the growth
rate of mutants ranges from 60 to 90% that of nonmutants. If there is more than one type of
mutant and each type has a different growth rate, the distribution can be approximated by
assuming there is only one type whose growth rate is the average of the two (Stewart et al.,
Adaptive mutation—If mutations occur after the cells are plated on selective medium, the
distribution of mutant numbers will have a Poisson component and a plot of ln(Pr) versus ln
(r) (Eq. (15)) will give a curve that is a combination of the Luria-Delbrück and Poisson (Cairns
et al., 1988). The two components can be estimated by fitting the experimental values to the
combined distributions (Cairns and Foster, 1991).
Other curve fitting—Using simulated data, Steward et al. (Stewart et al., 1990) have
determined the effects of several deviations from the Lea-Coulson model on the shape of the
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ln(Pr) versus ln(r) curve. Experimental data can be fit to these curves to test whether a given
factor is operative. However, all of the deviant curves are rather similar, so any conclusion that
a given factor is distorting the distribution would have to be tested experimentally.
Research in the author’s laboratory is supported by USPHS grant NIH-NIGMS G65175.
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Fig 1.
An illustration of the constant increase in the mutant fraction after a population reaches a size
sufficiently large so that the accumulation of mutants is simply a function of population size.
Luria’s conventions are followed {Luria, 1951 632 /id}): k = the generation numbered
backwards from 0; N = the number of cells present at each generation; Nt = the final number
of cells in the population; μ = the mutation rate per cell (assuming a synchronous population).
At each generation there are Nt/2k individuals that produce μNt/2k new mutations, which will
produce a total of μNt mutant progeny by the last generation.
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