Mutation Rates

"The stars might lie but the numbers never do." Mary Chapin Carpenter

The Fluctuation Test

Since the beginning of this century it was widely known that when a population of bacteria is exposed to a toxic environment, some rare cells may acquire the ability to grow much better than most of the other cells in the population. There are three possible ways bacteria might acquire such new properties:

  1. some particularly robust cells in the population might adjust to the harsh environment (i.e., adaption);
  2. exposure to the toxic conditions might produce rare, resistant mutations in the population of bacteria (i.e., induced mutations); or
  3. spontaneous mutants might have ocurred in the bacterial population prior to exposure to the toxic conditions, yielding resistant progeny cells (i.e., spontaneous mutations).

It is easy to distinguish cells that have simply adapted to the toxic conditions from resistant mutants. Adaption is a transient property that depends upon continued exposure to the toxic conditions. Thus, if the cells are transferred from the toxic conditions and grown for many generations under favorable growth conditions they would de-adapt, so most of the cells would fail to thrive if subsequently transferred back into the toxic growth conditions.

In contrast, distinguishing between induced vs spontaneous mutations is much more difficult. For many years, most microbiologists believed that mutations in bacteria were induced by exposure to a particular environment. (At the time Salvador Luria said that "bacteriology is the last stronghold of Lamarckism".)

The first rigorous evidence that mutations in bacteria followed the same Darwinian principles as in eukaryotic cells came from a clever study by Luria and Delbruck [Luria and Delbruck. 1943. Genetics 28: 491-511]. They studied mutations that made E. coli resistant to phage T1. Phage T1 interacts with specific receptors on the surface of E. coli, enters the cell, and subsequently kills the cell. Thus, when E. coli is spread on a plate with 1010 phage T1, most of the cells are killed. However, rare T1 resistant (TonR) colonies can arise due to mutations in E. coli that alter the T1 receptor in the cell wall. Luria noted that the two theories of mutation made different statistical predictions. If the TonR mutations were induced by exposure to phage T1, then every population of cells would be expected to have an equal probability of developing resistance and hence a nearly equal number of TonR colonies would be produced from different cultures. For example, if there was a 10-8 probability that exposure to phage T1 would induce a TonR mutant, then approximately 10 colonies would arise on each plate spread with 109 bacteria. In contrast, if TonR mutations were due to random, spontaneous mutations that occured sometime during the growth of the culture prior to exposure to phage T1, then the number of TonR colonies would vary widely between each different culture. For example, although there is an equal probability (say 10-8) that a TonR mutant would arise per cell division, the number of resistant bacteria in each culture would depend upon whether the mutation occured during one of the first cell divisions or one of the last cell divisions. The figure below shows a cartoon of the alternative predictions. (TonS cells are indicated in white and TonR cells are indicated in black. The shaded area indicates when the cells were exposed to phage T1.

The fluctuation test

In either of these two cases, if multiple samples from a single culture of bacteria were plated on phage T1, each of the resulting plates should yield approximately the same number of colonies. However, the two possibilities can be distinguished mathematically by comparing the mean and variance of the number of the number of mutants in each culture:

Mean and variance

where m = Number of mutants per culture and n = number of cultures. If approximately the same number of resistant mutants are obtained on each plate as with multiple samples from a single culture or as predicted by the directed-mutagenesis hypothesis, the mean should be approximately equal to the variance. In contrast, if there is large variation in the number of mutants per plate, the mean will be much less than the variance. The results obtained (see below) indicated that mutation to TonR is a random event.

The Poisson distribution

The Poisson equation describes the probability distribution of random events -- for example, the probability that random mutations in a cell will affect ("hit") a particular gene ("target").

Poisson equation

Px = probability that a target will have exactly x hits
h = average number of hits per target

(For a non-mathematical analogy, see the cartoon drawn to illustrate the poisson distribution by one of the participants in the Cold Spring Harbor Laboratory Phage course over 50 years ago.)

The simplest way of determining Px is to determine the frequency of zero events and plug this value into the Poisson equation:

Solving the Poisson equation for P<SUB>0</SUB>

Thus, for example a fluctuation test where 11 of 20 tubes had no mutants:

An example

Once the value for h has been determined from the number of tubes with 0 mutants, this value can then be used to calculate Px for the other number of mutants. The theoretical values calculated from this prediction of randomness can then be compared with the observed values.

Note that this simple solution is not accurate if less than 10% of the tubes have no mutants or more than 70% of the tubes have no mutants. (A more complex equation is necessary to accurately estimate higher mutation frequencies -- for a thorough mathematical explanation see [Lea and Coulson. 1949. J. Genetics 49: 264-285].)

Calculation of Mutation Rates

Instead of the mutation rate, often the "mutant frequency" is reported. The mutant frequency is simply the ratio of mutants divided by the total number of bacteria in the population. The mutant frequency is easy to measure, but is less reliable than the mutation rate because it may show large fluctuations depending upon when the first mutation appeared in the population.

The mutation rate is the number of mutations per cell division. Because the cell population is so large, the number of cell divisions is approximately equal to the number of cells in the population (N).

Mutation rate

In the example shown above, if h was determined from a fluctuation test with 107 cells per tube, then the mutation rate would be:

An example

[Some mathematical problems associated with calculating mutation rates by this approach, and alternative approachs are described in the listed references by Foster (2000) and Rosche and Foster (2000).]

What is the typical rate of spontaneous mutations?

Rates of spontaneous mutation seem to be determined by evolutionary balances between the deleterious consequences of many mutations and the additional energy and time required to further reduce mutation rates. Bacteria, Archae, and Eukaryotic microbes produce about one mutation per 300 chromosome replications. For E. coli this works out to be between 10-6 and 10-7 mutations per gene per generation, however it is important to note that there are certain "hot spots" or "cold spots" for spontaneous mutations. (A "hot spot" is a site that has a higher rate of mutations than predicted from a normal distribution, and a "cold spot" is a site with a lower rate of mutations than predicted from a normal distribution.) Higher eukaryotes have the same rate of spontaneous mutation, so that rates per sexual generation are about one mutation per gamete (close to the maximum compatible with life). RNA viruses have much higher mutation rates -- about one mutation per genome per chromosome replication -- and even small increases in their mutation rates are lethal.

Are all mutations random?

In 1988 Cairns and colleagues renewed the controversy about whether some mutations are induced by the selective conditions [Cairns, J., J. Overbaugh, and S. Miller. 1988. Nature 335: 142-145]. In contrast to the experiments done by Luria and Delbruck where any T1s mutants rapidly died on the selection plates, Cairns and colleagues used a "nonlethal selection" -- E. coli Lac- mutants were plated on medium with lactose as a carbon source and the number of Lac+ revertants were determined over a period of several weeks. Under these conditions the Lac- cells cannot form visible colonies, but the cells do not die. They observed that although some Lac+ revertants appear within a few days as expected for pre-existing revertants, Lac+ revertants continued to appear over time suggesting that starvation of the Lac- mutants stimulated mutation to Lac+. Hence, they dubbed this process "adaptative mutagenesis". This Lamarkian interpretation stimulated a decade of research on this question. After the dust settled from the heated controversy, the take-home point seems to be that starvation (and possibly other stressful conditions) may stimulate increased mutagenesis (possibly by several different mechanisms), and only those mutations that allow the cells to respond to the environmental stress accumulate in the resulting population [for a recent review see Foster, 1999].

Sample problems

To determine the frequency of StrR mutants a fluctuation test was done with 50 tubes each containing 108 cells and 42 of the tubes contained no mutants. Use the Luria-Delbruck calculation to determine the mutation rate to StrR.


First calculate the average number of hits per cell
h = -ln (42/50) = -ln(0.84) = 0.17

Then divide the average number of hits per cell by the number of cells in the population
a = h / N = 0.17 / 108 = 1.7 x 10-9

To determine the frequency of putP mutants a fluctuation test was done using 20 tubes with a final concentration of 107 bacteria each. From each tube 0.1 ml of culture was plated on medium that selects for putP mutants. Seventeen of the tubes yielded putP mutants but 3 of the tubes yielded no mutants. Based upon these results, use the Luria-Delbruck calculation to determine the mutation rate to putP-.


First calculate the average number of hits per cell
h = -ln (3/20) = 1.9

Then divide the average number of hits per cell by the number of cells in the population
a = h / N = 1.9 / 107 = 1.9 x 10-7

Suggest two reasons why the rate of mutation to StrR is so much less than the rate of mutation to Pro-.

ANSWER: One reasonable explanation is that any mutation that disrupts any of the proline biosynthetic genes would result in a Pro- phenotype, but only very specific base substitutions in ribosomal genes result in streptomycin resistance (i.e., Str resistance is a smaller target size for mutations) -- this is the actual reason. A second potential reason could be that there are redundant genes that encode the wild-type Str sensitive phenotype and the Str resistant mutant phenotype is recessive to the wild-type.

To get a feeling for the probability of random events, try the simple Poisson Distribution program at


Return to Microbial Genetics supplement.

Please send comments, suggestions, or questions to
Last modified July 10, 2004