### Calculate mutation rates from fluctuation assays

$$bz$$-$$rates$$ is a web-tool that allows to compute mutation rates from fluctuation assays. $$bz$$-$$rates$$ estimator is the Generating Function from Hamon & Ycart 2012. This estimator calculates the parameters m (mean number of mutations) and b (mutant relative fitness), for a given fluctuation assay dataset and therfore allows to calculate a mutation rate taking into account differential growth rate.

### News

2015/05/06: several updates in the input form. Generating Function kept as the only estimator

2015/03/10: usage of scientific notation in the 'N0' field is now possible

2015/02/28: increased the authorized maximum number of plated cells

 Mutation rate calculator
 b : the relative fitness of mutant versus WT cells is known Algorithms N0 : initial number of cells per culture b (0 < b < ∞) z : plating efficiency or fraction of a culture plated (0 < z ≤ 1) Copy / paste your fluctuation data here Nmutants [tab/space] Ncells Nmutants : number of mutants on the plate Ncells : number of plated cells
 No Yes Generating Function (Hamon & Ycart 2012) Maximum Likelihood (Hamon & Ycart 2012)
 Results

The 'Results' box outputs the following information:
\begin{array}{ll} \hline \small{m} & \small{\text{mean number of mutations per culture not corrected by the plating efficiency (z)}} \\ \small{\mu} & \small{\text{mutation rate per cell per division not corrected by the plating efficiency (z)}} \\ \small{m_{corr}} & \small{\text{number of mutations per culture corrected by the plating efficiency (z)}} \\ \small{\boldsymbol{\mu_{corr}}} & \small{\textbf{mutation rate per cell per division corrected by the plating efficiency (z)}} \\ \small{CL_{lower}} & \small{\text{lower 95% confidence limit for } m_{corr}} \\ \small{CL_{upper}} & \small{\text{upper 95% confidence limit for } m_{corr}} \\ \small{b} & \small{\text{mutant cells relative fitness predicted by the Generating Function (only output if b is left empty in the input field)}} \\ \small{b_{lower}} & \small{\text{lower 95% confidence limit for b (only output if b is left empty in the input field)}} \\ \small{b_{upper}} & \small{\text{upper 95% confidence limit for b (only output if b is left empty in the input field)}} \\ \small{meanNc} & \small{\text{average number of plated cells per culture } (\overline{Nc})} \\ \small{sdNc} & \small{\text{standard deviation of the number of plated cells } (\sigma_{Nc})} \\ \small{\chi^{2}} & \small{\text{Pearson's chi-square value} } \\ \small{\chi^{2}-p} & \small{\text{Pearson's chi-square p-value} } \\ \hline \end{array}

### Motivation

A classical approach to calculate mutation rates ($$\mu$$) in microorganisms consists in performing fluctuation analyses through multiple cultures grown in parallel under identical conditions (Luria & Delbrück 1943). Each individual culture is started with a small inoculum ($$N_{0}$$) and mutational events occur independently in each culture. At the end of the experiment, a $$\mu$$ can be estimated from the proportions of mutant cells in the different cultures. Since the seminal work of Lea & Coulson in 1949, numerous methods were proposed to calculate $$\mu$$ (for review see Foster 2006). They all rely on the estimation of $$m$$, the mean number of mutations per culture.

The estimation of $$m$$ can be affected by the fraction of cells that is plated on selective media for each culture. This criteria is defined as the plating efficiency ($$z$$) which accounts for the fact that if the entire culture is not plated, then not all the mutants will be experimentally detected. A correction proposed by Stewart and colleagues to account for this potential bias (Stewart & al. 1990) is also implemented in $$bz$$-$$rates$$.

### Estimating $$m$$

When the mutant relative fitness is not provided by the user, $$bz$$-$$rates$$ uses the GF from Hamon & Ycart 2012 to estimate $$m$$ and $$b$$. The GF uses the probability generating function to estimate the compound Poisson distribution. See Hamon & Ycart 2012 for further details. When $$b$$ is known, only $$m$$ is estimated by the generating function.

### Correcting for the plating efficiency $$z$$

$$m_{corr}$$ is calculated using the equation 41 from Stewart et al. 1990: $$m_{corr} = m \frac{z-1}{z\cdot ln(z)} \nonumber$$

### Calculating the mutation rates $$\mu$$ and $$\mu_{corr}$$

$$\mu$$ and $$\mu_{corr}$$ are calculated with the following formulas: \begin{align} \mu &= \frac{m} { \overline{Nc} } \nonumber \\ \nonumber \\ \mu_{corr} &= \frac{m_{corr}} { \overline{Nt} } \nonumber \end{align}

### References

Foster (2006). Methods for determining spontaneous mutation rates. Methods in Enzymology, 409(05), 195–213. doi:10.1016/S0076-6879(05)09012-9
Hamon & Ycart (2012). Statistics for the Luria-Delbrück distribution. Electronic Journal of Statistics, 6, 1251–1272. doi:10.1214/12-EJS711
Jaeger & Sarkar (1995). On the distribution of bacterial mutants: the effects of differential fitness of mutants and non-mutants. Genetica, 217–223.
Luria & Delbrück (1943). Mutations of bacteria from virus sensitivity to virus resistance. Genetics, 28(6), 491.
Sarkar & Sandri (1992). On fluctuation analysis: a new, simple and efficient method for computing the expected number of mutants. Genetica, 173–179.
Stewart & al. (1990). Fluctuation Analysis: The Probability Distribution of the Number Mutants Under Different Conditions. Genetics, 124(1), 175–185.

### FAQ

Although bz-rates is built as a simplified tool to compute mutation rates, users should have in mind the goods practices for fluctuation tests analysis: Determining mutation rates in bacterial populations, Rosche & Foster, 2000, DOI:10.1006/meth.1999.0901

Is it possible to measure mutation rates when the number of cells per culture differs: No. In practice, bz-rate averages the number of plated cells per culture. Therefore, bz-rates will always output a result for a given dataset. However, they will only be valid if the total number of cells per culture and the number of plated cells per culture is identical (or at least quite close).