ALL Metrics
-
Views
-
Downloads
Get PDF
Get XML
Cite
Export
Track
Brief Report
Revised

The number of neutral mutants in an expanding Luria-Delbrück population is approximately Fréchet

[version 2; peer review: 2 approved]
PUBLISHED 03 Mar 2023
Author details Author details
OPEN PEER REVIEW
REVIEWER STATUS

This article is included in the Genomics and Genetics gateway.

Abstract

Background: A growing population of cells accumulates mutations. A single mutation early in the growth process carries forward to all descendant cells, causing the final population to have a lot of mutant cells. When the first mutation happens later in growth, the final population typically has fewer mutants. The number of mutant cells in the final population follows the Luria-Delbrück distribution. The mathematical form of the distribution is known only from its probability generating function. For larger populations of cells, one typically uses computer simulations to estimate the distribution.
Methods: This article searches for a simple approximation of the Luria-Delbrück distribution, with an explicit mathematical form that can be used easily in calculations.
Results: The Fréchet distribution provides a good approximation for the Luria-Delbrück distribution for neutral mutations, which do not cause a growth rate change relative to the original cells.
Conclusions: The Fréchet distribution apparently provides a good match through its description of extreme value problems for multiplicative processes such as exponential growth.

Keywords

Population genetics, probability distributions, extreme value distributions

Revised Amendments from Version 1

In Equation 1, I replaced m < z with m ≤ z so that the new equation is F(z)=Prob(m≤z)=exp⁡(−(z−βs)−α)

See the author's detailed response to the review by Qi Zheng
See the author's detailed response to the review by Pavol Bokes

Suppose a single cell expands exponentially to a population of size N, with a mutation rate of u per cell division. The number of mutant cells, m, in the final population depends on the number of mutations that occur and when those mutations occur. For example, a single mutation in the final round of cell division is limited to one cell. By contrast, a single mutation transmitted to one of the daughters in the first cellular division may occur in approximately one-half of the final population.

The distribution of the number mutants, m, is known as the Luria–Delbrück distribution1. That distribution is widely used to estimate the mutation rate. The distribution also arises when studying the amount of mutational mosaicism within multicellular individuals24.

Currently, for experiments with a small number of mutational events, one typically calculates the distribution with a probability generating function5,6. However, that approach becomes numerically inaccurate for larger numbers of mutational events, in which case the distribution is calculated by computer simulation.

This article shows that the Fréchet distribution provides a good approximation for the number of neutral mutants. In particular, the probability that the number of mutants, m, is less than z is approximately

F(z)=Prob(mz)=exp((zβs)α),(1)

in which exp(z) = ez is the exponential function. The probability of being in the upper tail, m > z, is 1 − F(z). The three parameters set the shape, α, the scale, s, and the minimum value, β, such that z, m > β.

This form of the Fréchet distribution has three parameters. I found that the following parameterization matches closely the Luria–Delbrück process for neutral mutations

α=e/2s=eNuβ=Nulog(Nue(1+α))

in which e is the base of the natural logarithm. This parameterization depends on the single parameter, Nu, the final population size times the mutation rate.

Figure 1 shows the good fit. Two aspects of mismatch occur. First, the number of mutants is discrete, whereas the Fréchet is continuous. As Nu declines to one, significant amounts of probability mass concentrate at particular mutant number values, causing discrepancy between the distributions. Nonetheless, the Fréchet remains a good approximation.

c7235814-c1b4-458f-88b1-a9132aa4e117_figure1.gif

Figure 1. Cumulative distribution of the number of neutral mutants in an expanding population.

Each population begins with one cell and grows to N cells. Mutation occurs at rate u. Blue curves show the distribution from a computer simulation using the simu.cultures command of the R package rSalvador7. Orange curves show the Fréchet distribution in Equation 1. In rSalvador, I used sample sizes of 106 or 107, values of Nu varying as shown above the plots, and values of N ranging from 106 to 1010. The Julia software code to produce this figure is available from Zenodo8. The input data for calculating the empirical Luria-Delbrück CDF is also available from Zenodo9.

Second, the lower tail of the Luria–Delbrück process spreads to lower values than the Fréchet. One can see this mismatch most clearly in the figure for Nu ≥ 100.

This mismatch may occur because the Luria–Delbrück process transitions from a highly stochastic process in earlier cellular generations to a nearly deterministic accumulation of mutations in later cellular generations, when the larger population size reduces the coefficient of variation in the number of new mutations. The Fréchet applies most closely to the earlier generations for the following reasons.

In an expanding population, the earliest mutation strongly influences the final number of mutants. An early mutant carries forward to all descendant cells in an expanding mutant clone. If we start with the final cells and then look back through the cellular generations toward the original progenitor, the mutation with the most extreme time from the end toward the beginning tends to dominate the final mutant number.

The extreme value of a temporal extent often has a Gumbel distribution. In this case, once the mutation arises, it increases multiplicatively by cell division to affect the final mutation count. Substituting the extreme Gumbel time for its multiplicative consequence provides a common way to observe a Fréchet probability pattern.

Prior mathematical work also supports the Fréchet approximation. Kessler and Levine10 showed that the Luria–Delbrück distribution converges to a Landau distribution for large Nu, in which the Landau distribution is a special case of the Lévy α-stable distribution. However, the Landau distribution does not have a closed-form expression for its probability or cumulative distribution functions.

Separately, Simon11 showed the close match between the Lévy α-stable distribution and the Fréchet distribution. That match of a Lévy distribution to the Fréchet distribution had not previously been associated with the Luria–Delbrück distribution. The Fréchet parameterization in this article provides a simple expression that can be used to develop further theory and applications of the Luria–Delbrück process.

Comments on this article Comments (0)

Version 2
VERSION 2 PUBLISHED 04 Nov 2022
Comment
Author details Author details
Competing interests
Grant information
Copyright
Download
 
Export To
metrics
Views Downloads
F1000Research - -
PubMed Central
Data from PMC are received and updated monthly.
- -
Citations
CITE
how to cite this article
Frank SA. The number of neutral mutants in an expanding Luria-Delbrück population is approximately Fréchet [version 2; peer review: 2 approved]. F1000Research 2023, 11:1254 (https://doi.org/10.12688/f1000research.127469.2)
NOTE: If applicable, it is important to ensure the information in square brackets after the title is included in all citations of this article.
track
receive updates on this article
Track an article to receive email alerts on any updates to this article.

Open Peer Review

Current Reviewer Status: ?
Key to Reviewer Statuses VIEW
ApprovedThe paper is scientifically sound in its current form and only minor, if any, improvements are suggested
Approved with reservations A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.
Not approvedFundamental flaws in the paper seriously undermine the findings and conclusions
Version 1
VERSION 1
PUBLISHED 04 Nov 2022
Views
23
Cite
Reviewer Report 20 Feb 2023
Pavol Bokes, Department of Applied Mathematics and Statistics, Comenius University, Bratislava, Slovakia 
Approved
VIEWS 23
The paper compares the empirical (simulational) Luria-Delbruck mutant-number distribution to a Frechet distribution. The advantage of the Frechet distribution over other known approximations, e.g. the skewed alpha-stable distributions, is that it possesses a closed-form cumulative distribution function (cdf), see Equation ... Continue reading
CITE
CITE
HOW TO CITE THIS REPORT
Bokes P. Reviewer Report For: The number of neutral mutants in an expanding Luria-Delbrück population is approximately Fréchet [version 2; peer review: 2 approved]. F1000Research 2023, 11:1254 (https://doi.org/10.5256/f1000research.139979.r162863)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
  • Author Response (F1000Research Advisory Board Member) 21 Feb 2023
    Steven Frank, Department of Ecology and Evolutionary Biology, University of California, Irvine, 92697-2525, USA
    21 Feb 2023
    Author Response F1000Research Advisory Board Member
    Thank you, I appreciate these comments. I agree that looking closely at the upper tail in a log-log plot would provide additional insight about the frequency of rare but potentially ... Continue reading
COMMENTS ON THIS REPORT
  • Author Response (F1000Research Advisory Board Member) 21 Feb 2023
    Steven Frank, Department of Ecology and Evolutionary Biology, University of California, Irvine, 92697-2525, USA
    21 Feb 2023
    Author Response F1000Research Advisory Board Member
    Thank you, I appreciate these comments. I agree that looking closely at the upper tail in a log-log plot would provide additional insight about the frequency of rare but potentially ... Continue reading
Views
32
Cite
Reviewer Report 23 Nov 2022
Qi Zheng, Department of Epidemiology and Biostatistics, Texas A&M University School of Public Health, College Station, TX, USA 
Approved
VIEWS 32
In his brief report the author presents an interesting approximation of the Luria-Delbruck distribution, which microbiologists use to help determine microbial mutation rates in the laboratory. Specifically, equation (1) in the brief report is an approximation of the cumulative probability. ... Continue reading
CITE
CITE
HOW TO CITE THIS REPORT
Zheng Q. Reviewer Report For: The number of neutral mutants in an expanding Luria-Delbrück population is approximately Fréchet [version 2; peer review: 2 approved]. F1000Research 2023, 11:1254 (https://doi.org/10.5256/f1000research.139979.r155112)
NOTE: it is important to ensure the information in square brackets after the title is included in all citations of this article.
  • Author Response (F1000Research Advisory Board Member) 21 Feb 2023
    Steven Frank, Department of Ecology and Evolutionary Biology, University of California, Irvine, 92697-2525, USA
    21 Feb 2023
    Author Response F1000Research Advisory Board Member
    Thank you for the careful reading. With regard to the comment about m < z versus m <= z, the calculations to make figure 1 used m <= z for ... Continue reading
COMMENTS ON THIS REPORT
  • Author Response (F1000Research Advisory Board Member) 21 Feb 2023
    Steven Frank, Department of Ecology and Evolutionary Biology, University of California, Irvine, 92697-2525, USA
    21 Feb 2023
    Author Response F1000Research Advisory Board Member
    Thank you for the careful reading. With regard to the comment about m < z versus m <= z, the calculations to make figure 1 used m <= z for ... Continue reading

Comments on this article Comments (0)

Version 2
VERSION 2 PUBLISHED 04 Nov 2022
Comment
Alongside their report, reviewers assign a status to the article:
Approved - the paper is scientifically sound in its current form and only minor, if any, improvements are suggested
Approved with reservations - A number of small changes, sometimes more significant revisions are required to address specific details and improve the papers academic merit.
Not approved - fundamental flaws in the paper seriously undermine the findings and conclusions
Sign In
If you've forgotten your password, please enter your email address below and we'll send you instructions on how to reset your password.

The email address should be the one you originally registered with F1000.

Email address not valid, please try again

You registered with F1000 via Google, so we cannot reset your password.

To sign in, please click here.

If you still need help with your Google account password, please click here.

You registered with F1000 via Facebook, so we cannot reset your password.

To sign in, please click here.

If you still need help with your Facebook account password, please click here.

Code not correct, please try again
Email us for further assistance.
Server error, please try again.