FALCOR
(Fluctuation AnaLysis CalculatOR)

BACKGROUND:
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 methods.

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 statistically validated.


INTERFACE :
Input
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.

Grouping
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.

Rate Output
1) 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 10^-7).
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 handling.
3) The confidence interval difference should be used to make error bars in Excel.


IMPLEMENTATION :
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 likelihood function


Eq. (3): Confidence intervals for the MSS-MLE method
;where

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.

Eq.(4): Correcting for low plating efficiency (sampling) with the MSS-MLE method
; 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 C.

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 Mann-Whitney U test.

Eq. (6): Lea-Coulson Equation for estimating 'm' from 'r'.


Eq. (7): Binomial distribution function used to calculate 95% confidence intervals using the LC and frequency methods.


Frequency:
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).


REFERENCES :
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, 4-17.
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, 137: 1139-1146.
Zheng,Q. (2002) Statistical and algorithmic methods for fluctuation analysis with SALVADOR as an implementation. Math. Biosci., 176: 237-252.


CONTACT INFORMATION :
Brandon M. Hall
Roswell Park Cancer Institute
BLSC L3-123
Elm & Carlton Streets
Buffalo, NY 14263
brandon.hall@roswellpark.org