The program FALCOR is a web tool designed for use with Luria-Delbruck fluctuation analysis to calculate the mutation rate from various mutation assays in bacteria and yeast (e.g. resistance to canavanine or erythromycin, reversion to
Trp+, etc.). Three calculation methods are available through this program:
Ma-Sandri-Sarkar Maximum Likelihood Estimator (MSS-MLE) Method,
Lea-Coulson Method of the Median (LC) and
Users of FALCOR should be familiar with fluctuation
analysis and how to conduct experiments properly [Rosche and Foster (2000); Foster (2006)]. Information regarding the
Implementation of the three methods,
Contact Information can be found below.
If you find FALCOR useful, please cite our paper.
(Fluctuation AnaLysis CalculatOR)
Originally described by Luria and Delbruck (1943), fluctuation analysis has
become the standard method in the field for calculating mutation rates. Briefly,
a small number of cells are used to inoculate parallel cultures in a
non-selective medium. The cultures are then grown to saturation to obtain equal
cell densities. Cells are then plated onto selective media to obtain the number
of mutants, r, and dilutions are plated onto rich medium to calculate the total
number of viable cells, Nt. Frequency is not a sufficiently accurate measure of
mutation; mutation rate should always be calculated (Rosche and Foster, 2000;
Schmidt et al., 2006).
A number of statistical methods have been developed to estimate the number of
mutations, m, from the observed values of mutants, r, across parallel cultures.
While the Lea-Coulson method of the median (LC), introduced in 1949, is the
classic model for the estimation of mutation rates, statistical analyses have
evolved to more accurately estimate m. However, the complex calculations
required place these more accurate methods beyond easy reach of bench
scientists. The Ma-Sandri-Sarkar Maximum Likelihood Estimator (MSS-MLE) is the
best method available to date; it is the most accurate and, unlike the LC
method, is valid over all values of r and m. Furthermore, the MSS-MLE method
calculates the mutation rate from the entire data set (not just the median),
providing more statistical power. A comprehensive evaluation of these methods
was conducted with experimental data by Rosche and Foster (2000). To facilitate
the use of these complicated methods by bench scientists, we developed a web
interface to implement these three most popular
It should be
noted that no methods currently exist to compare the values of m from cultures
with different values of Nt. When comparing data across strains, conditions, and
experiments, the values of Nt must statistically be equal. The comparison of
mutation rates, M, with a different final number of cells has not been
Values of 'r' represent the total
number of mutants (or revertants) on selective media plates. Values of 'N'
represent the total number of cells in the culture vial (Nt). To calculate 'N',
diluted cultures are plated onto rich medium. If the entire culture volume is
not plated, corrections can be applied when using the MSS-MLE method as in Eq.
(4). It is recommended that the user input data from Excel into the '2 column
entry' input box, with column 1 = r, column 2 = Nt. However, values can be
entered directly into the r and N input boxes. A sample data set is provided
along with corresponding output for the various methods of analysis as an Excel file, or as a txt file of the data. Additional data sets from Rosche and Foster (2000) and Qi Zheng (2002) are also provided.
Multiple data can be entered into the program at the same
time, and grouped together (as specified by the user) for various output. For
example, an experiment with 10 cultures is repeated 3 times. Grouping by 30
gives the median and confidence interval across all data points. Grouping by 10
gives the medians and confidence intervals for each of the 3 experiments.
The magnitude of the output can be controlled by the user by entering which log
value of 10 to express the rate (default value is
2) Combine rate output is designed to make it easier for creation of excel graphs. The rate and confidence interval range and difference about the median are
combined into one text field for easier output
The confidence interval difference should be used to make error bars in
MSS-Maximum Likelihood Estimator Method (MSS-MLE):
While the Lea-Coulson Method of the Median is the most commonly used in the
literature, the MSS-Maximum Likelihood Method is currently the best method to
estimate m. The MSS-MLE method uses an initial estimate of m to generate the
probability of observing r mutants on selective medium, pr (Eq. 1). The
likelihood function is the product of the pr's for each observed value of r (Eq.
2). The value of m is then adjusted until the likelihood function reaches a
maximum (Sarkar et al., 1992; Ma et al., 1992). The mutation rate, M, is then
defined as m/Nt, where Nt represents the average of the cell counts across the
cultures. The confidence intervals are calculated according to the method of
Stewart (1994) who discovered that the natural log of m is normally distributed.
The confidence intervals calculated by FALCOR are derived from an approximation
of this distribution (Eq. 3), as described by Rosche and Foster (2000).
Eq. (1): Probability function of observing 'r' mutants
Eq. (2): Maximum
(3): Confidence intervals for the MSS-MLE method
Low plating efficiency, or sampling, can be used to increase the accuracy in
measuring high mutation rates (by allowing larger culture volumes), or as a way
to determine the rates of mutations at multiple loci from the same culture
(plating the culture volume across various selective plates). Of the methods
offered by FALCOR, errors in low plating efficiency can be corrected
when using the MSS-MLE method. If the entire culture is not plated, Eq.
(4) can be used to correct the observed number of mutations in the culture,
adjusting it to what would be observed if the entire culture was plated (plating
Nt, the total number of cells). For example, 200ul of 1ml cultures are plated,
with an observed m of 10. The correction factor for plating 1/5 or 20% of the
culture is not simply 5, where mact for the culture would be 50. By using
Eq.(4), we can calculate that the correction factor is (0.2-1)/(0.2 ln (0.2)),
which equals 2.485. The actual value of m is then 24.85. To calculate M, mact/Nt
is used, where Nt is the number of cells in the entire 1ml culture. As such, the
correction factor of Eq. (4) can be applied directly to the calculated value of
M given by FALCOR.
Correcting for low plating efficiency (sampling) with the MSS-MLE
; where z is the
fraction of culture plated.
Eq. (5): Student's t-test for significance
testing using the MSS-MLE method
Since the m calculated from the MSS-MLE method is normally distributed,
statistical analysis of the MSS-MLE method can be carried out with a
simple students t-test
(Rosche and Foster, 2000). When comparing two values
of m, the value of t can be calculated from Eq. (5). The value of t can then be
used to determine the level of significance (p-value) in a standard two-tailed t
table with the degrees of freedom of C1 + C2 - 2. The values of m can be
obtained from the first output box of FALCOR, and the standard deviation is the
same as described in Eq.(3), which is dependent upon m and
Lea-Coulson Method of the Median:
FALCOR implements a slightly modified version of the LC method (Schmidt et al.
2006). Briefly, the value of m is calculated from r via the Lea-Coulson equation
(Eq. 6). The mutation rate, M, is then determined for each data point: m/Nt. The
program then sorts the values of M and determines the median. This method
performs well over the range 2.5 < r < 60 (1.5 < m < 15). Confidence
intervals are derived from the cumulative binomial distribution of the
rank-values (Rosche and Foster, 2000). The probability mass function of a
binomial distribution used by FALCOR is given in Eq. (7). Sorted values of M are
listed by FALCOR for significance testing via the
Lea-Coulson Equation for estimating 'm'
Eq. (7): Binomial distribution function used to calculate
95% confidence intervals using the LC and frequency methods.
This method determines the frequency of mutation (i.e., r/N). However, frequency
is highly inaccurate, and in cases of measuring spontaneous mutations, rates
should be calculated to obtain a accurate representation of the data.
Frequencies are useful for determining the level of induced mutations in an
population. FALCOR calculates frequency, providing statistical interpretation of
the data with confidence intervals about the median, as with the LC method
(based on Eq. 5).
Foster,P.L. (2006) Methods for Determining Spontaneous Mutation Rates. Methods
Enzymol., 409, 195-213.
Lea,D.E. and Coulson,C.A. (1949) The distribution of the numbers of
mutants in bacterial populations. J. Genet. 49: 264-285.
Luria,S.E. and Delbruck,M. (1943) Mutations of bacteria from virus sensitivity to virus
resistance. Genetics, 28: 491-511.
Ma,W.T., Sandri,G.v.H. and Sarkar,S. (1992). Analysis of the Luria-Delbruck distribution using discrete convolution
powers. J Appl Prob, 29: 255-267.
Rosche,W.A. and Foster,P.L. (2000) Determining Mutation Rates in Bacterial Populations. Methods, 20,
Sarkar,S., Ma,W.T. and Sandri,G.v.H. (1992) On fluctuation analysis: a new, simple and efficient method for computing the expected number of mutants.
Genetica, 85: 173-179.
Schmidt,K.H., Pennaneach,V., Putnam,C.D. and Kolodner,R.D. (2006) Chapter 27: Analysis of Gross-Chromosomal Rearrangements in
Saccharomyces cerevisiae. Methods Enzymol., 409: 462-476.
Stewart,F.M. (1994) Fluctuation tests: how reliable are the estimates of mutation rates? Genetics,
Zheng,Q. (2002) Statistical and algorithmic methods for fluctuation analysis with SALVADOR as an implementation. Math. Biosci., 176: 237-252.
Brandon M. Hall
Park Cancer Institute
Elm & Carlton Streets
Hall, B.M., Ma, C., Liang, P. & Singh, K.K. (2009) Fluctuation AnaLysis CalculatOR (FALCOR): a web tool for the determination of mutation rate using Luria-Delbruck fluctuation analysis. Bioinformatics, 25(12): 1564-1565.