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

Abstract

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.

Introduction

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,

1996).

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.

Terminology

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,

f.

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

culture

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

−

r1

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

Nλ

(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,

1964).

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

r

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 ~

obs

z )− 0.693

ln(r ~

obs

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

Nt

), 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

Nt

, Drake’s formula can be rearranged into

the same form as Lea and Coulson’s formula, Eq. (6):

r∼

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 ×

p0

2 ; p2 = m2

× (p0

3 +

p1

2 );

p3 = m3

× (p0

4 +

p1

3 +

p2

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

c

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

i=Nt

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) ≈

m

Nt

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

2Nt

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)

Nt

0.6932m

Nt

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

freedom:

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

derived.

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.,

1990).

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.

Acknowledgements

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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