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**FALCOR**

*Fluctuation AnaLysis CalculatOR*

The program FALCOR is a web tool designed for use with Luria-Delbrück fluctuation analysis to calculate the mutation frequency and 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: (1) Ma-Sandri-Sarkar Maximum Likelihood Estimator (MSS-MLE) Method, (2) Lea-Coulson Method of the Median (LC) and (3) Frequency. It is assumed that any potential user of FALCOR has an understanding of fluctuation analysis, and how to conduct experiments properly. Foster (2006) provides an excellent review of fluctuation analysis, including the underlying assumptions that shape the current mathematical models. Information regarding the user interface and statistical calculations implemented by FALCOR can be found below.

The program FALCOR is a web tool designed for use with Luria-Delbrück fluctuation analysis to calculate the mutation frequency and 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: (1) Ma-Sandri-Sarkar Maximum Likelihood Estimator (MSS-MLE) Method, (2) Lea-Coulson Method of the Median (LC) and (3) Frequency. It is assumed that any potential user of FALCOR has an understanding of fluctuation analysis, and how to conduct experiments properly. Foster (2006) provides an excellent review of fluctuation analysis, including the underlying assumptions that shape the current mathematical models. Information regarding the user interface and statistical calculations implemented by FALCOR can be found below.

__BACKGROUND:__

Originally described by Luria & Delbrück (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, *N*_{t}. 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.

__INTERFACE:__

*Input*

Values
of '*r*' represent the number of
mutants (or revertants) on selective media plates.
Values of '*N*' represent the total
number of cells plated onto the selective media. To calculate '*N*', diluted cultures are plated onto
rich medium. Values of '*r*' and '*N*' have to be normalized to the same
volume (such as 1 mL). As such, you obtain '*r*' mutants when 1 mL
of an overnight culture is plated on selective media, with 1 mL of media containing '*N*'
number of cells. It is recommended that the user input data from Excel into the
'2 column entry' input box, with column 1 = r, column 2 = N. 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.
Click here for data set

*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*

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). "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. 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, *p*_{r} (Eq.
1). The likelihood function is the product of the *p*_{r}’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*/*Ñ*_{t}, where *Ñ*_{t} represents
the average of the cell counts across the cultures. As such, FALCOR first
normalizes all values of *r* to *Ñ*_{t}, as
determined by how the data is grouped. 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:

Eq.
2:

Eq.
3: ; where

*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. 4).
The mutation rate, *M*, is then
determined for each data point: *m*/*N*_{t}.
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. 5.

Eq. 4:

Eq. 5:

*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 *J. Genet. ***49**: 264-285.

Luria,S.E. and Delbrück,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-Delbrück
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.

__CITATION: __

Papers citing this tool can be found here

__CURRENT CONTACT INFORMATION: __

Ping Liang, PhD, Professor

Department of Biological Sciences

Brock University

St. Catharines, Ontario

Canada L2S 3A1

EMail: pliang at brocku.ca

__
CREDITS:__

Last modified:Tuesday, 30-Aug-2022 10:19:58 EDT .