Mathematics, Statistics, and Computation Inspired by the Fluctuation Test: In Celebration of the 80th Anniversary of the Luria-Delbrück Experiment

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (31 October 2023) | Viewed by 16092

Special Issue Editor


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Guest Editor
School of Public Health, Texas A & M University, College Station, TX 77843, USA
Interests: biostatistics; infernce of microbial mutation rates; statistical education

Special Issue Information

Dear Colleagues,

The fluctuation test, invented by Luria and Delbrück in February 1943, is widely regarded as one of the most important microbiological experiments of the 20th century. Even in the era of high throughput DNA sequencing, the fluctuation test remains the most widely used method for determining microbial mutation rates in the laboratory. For example, the fluctuation test has been an important tool in research on microbial drug resistance, an active research topic closely interwoven with everyday human life. The fluctuation test protocol has also found numerous uses in cancer research and evolutionary studies. All of these fruitful applications of the classic experimental protocol hinge on innovations in mathematical modelling, statistical inference, and computer simulation. Today, the body of literature concerning various mathematical, statistical, and computational aspects of the classic protocol is enormous, and new research continues to emerge at an impressive pace. This is an exciting time to seek methodological innovations to expand the scope of application of the fluctuation test. I cordially invite you to contribute new practical research results, thought-provoking reviews, or innovative applications of the fluctuation test to this Special Issue in celebration of the approaching 80th anniversary of the Luria–Delbrück experiment.

Prof. Dr. Qi Zheng
Guest Editor

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Keywords

  • mutation rate
  • Luria–Delbrück protocol
  • microbial evolution
  • statistical inference
  • stochastic process
  • Monte Carlo simulation
  • mathematical modeling
  • mutant distribution
  • microbial drug resistance
  • fluctuation data
  • probability generating function
  • master equation
  • likelihood function

Published Papers (9 papers)

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Research

37 pages, 3336 KiB  
Article
Estimating the Rate of Mutation to a Mutator Phenotype
by Isaac Vázquez-Mendoza, Erika E. Rodríguez-Torres, Mojgan Ezadian, Lindi M. Wahl and Philip J. Gerrish
Axioms 2024, 13(2), 117; https://doi.org/10.3390/axioms13020117 - 11 Feb 2024
Viewed by 1201
Abstract
A mutator is a variant in a population of organisms whose mutation rate is higher than the average mutation rate in the population. For genetic and population dynamics reasons, mutators are produced and survive with much greater frequency than anti-mutators (variants with a [...] Read more.
A mutator is a variant in a population of organisms whose mutation rate is higher than the average mutation rate in the population. For genetic and population dynamics reasons, mutators are produced and survive with much greater frequency than anti-mutators (variants with a lower-than-average mutation rate). This strong asymmetry is a consequence of both fundamental genetics and natural selection; it can lead to a ratchet-like increase in the mutation rate. The rate at which mutators appear is, therefore, a parameter that should be of great interest to evolutionary biologists generally; for example, it can influence: (1) the survival duration of a species, especially asexual species (which are known to be short-lived), (2) the evolution of recombination, a process that can ameliorate the deleterious effects of mutator abundance, (3) the rate at which cancer appears, (4) the ability of pathogens to escape immune surveillance in their hosts, (5) the long-term fate of mitochondria, etc. In spite of its great relevance to basic and applied science, the rate of mutation to a mutator phenotype continues to be essentially unknown. The reasons for this gap in our knowledge are largely methodological; in general, a mutator phenotype cannot be observed directly, but must instead be inferred from the numbers of some neutral “marker” mutation that can be observed directly: different mutation-rate variants will produce this marker mutation at different rates. Here, we derive the expected distribution of the numbers of the marker mutants observed, accounting for the fact that some of the mutants will have been produced by a mutator phenotype that itself arose by mutation during the growth of the culture. These developments, together with previous enhancements of the Luria–Delbrück assay (by one of us, dubbed the “Jones protocol”), make possible a novel experimental protocol for estimating the rate of mutation to a mutator phenotype. Simulated experiments using biologically reasonable parameters that employ this protocol show that such experiments in the lab can give us fairly accurate estimates of the rate of mutation to a mutator phenotype. Although our ability to estimate mutation-to-mutator rates from simulated experiments is promising, we view this study as a proof-of-concept study and an important first step towards practical empirical estimation. Full article
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15 pages, 888 KiB  
Article
Trends in the Use of Proper Methods for Estimating Mutation Rates in Fluctuation Experiments
by Guillem A. Devin and Alejandro Couce
Axioms 2023, 12(12), 1100; https://doi.org/10.3390/axioms12121100 - 1 Dec 2023
Viewed by 1521
Abstract
The accurate quantification of mutation rates holds significance across diverse fields, including evolution, cancer research, and antimicrobial resistance. Eighty years ago, Luria and Delbrück demonstrated that the proper quantification of mutation rates requires one to account for the non-linear relationship between the number [...] Read more.
The accurate quantification of mutation rates holds significance across diverse fields, including evolution, cancer research, and antimicrobial resistance. Eighty years ago, Luria and Delbrück demonstrated that the proper quantification of mutation rates requires one to account for the non-linear relationship between the number of mutations and the final number of mutants in a cell population. An extensive body of literature has since emerged, offering increasingly efficient methods to account for this phenomenon, with different alternatives balancing accuracy and user-friendliness for experimentalists. Nevertheless, statistically inappropriate approaches, such as using arithmetic averages of mutant frequencies as a proxy for the mutation rate, continue to be commonplace. Here, we conducted a comprehensive re-analysis of 140 publications from the last two decades, revealing general trends in the adoption of proper mutation rate estimation methods. Our findings demonstrate an upward trajectory in the utilization of best statistical practices, likely due to the wider availability of off-the-shelf computational tools. However, the usage of inappropriate statistical approaches varies substantially across specific research areas, and it is still present even in journals with the highest impact factors. These findings aim to inspire both experimentalists and theoreticians to find ways to further promote the adoption of best statistical practices for the reliable estimation of mutation rates in all fields. Full article
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31 pages, 1611 KiB  
Article
Fascination with Fluctuation: Luria and Delbrück’s Legacy
by Raina S. Robeva and John R. Jungck
Axioms 2023, 12(3), 280; https://doi.org/10.3390/axioms12030280 - 7 Mar 2023
Viewed by 3680
Abstract
While Luria and Delbrück’s seminal work has found its way to some college biology textbooks, it is now largely absent from those in mathematics. This is a significant omission, and we consider it a missed opportunity to present a celebrated conceptual model that [...] Read more.
While Luria and Delbrück’s seminal work has found its way to some college biology textbooks, it is now largely absent from those in mathematics. This is a significant omission, and we consider it a missed opportunity to present a celebrated conceptual model that provides an authentic and, in many ways, intuitive example of the quantifiable nature of stochasticity. We argue that it is an important topic that could enrich the educational literature in mathematics, from the introductory to advanced levels, opening many doors to undergraduate research. The paper has two main parts. First, we present in detail the mathematical theory behind the Luria–Delbrück model and make suggestions for further readings from the literature. We also give ideas for inclusion in various mathematics courses and for projects that can be used in regular courses, independent projects, or as starting points for student research. Second, we briefly review available hands-on activities as pedagogical ways to facilitate problem posing, problem-based learning, and investigative case-based learning and to expose students to experiments leading to Poisson distributions. These help students with even limited mathematics backgrounds understand the significance of Luria–Delbrück’s work for determining mutation rates and its impact on many fields, including cancer chemotherapy, antibiotic resistance, radiation, and environmental screening for mutagens and teratogens. Full article
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20 pages, 1256 KiB  
Article
The Kinetic Theory of Mutation Rates
by Lorenzo Pareschi and Giuseppe Toscani
Axioms 2023, 12(3), 265; https://doi.org/10.3390/axioms12030265 - 3 Mar 2023
Viewed by 1219
Abstract
The Luria–Delbrück mutation model is a cornerstone of evolution theory and has been mathematically formulated in a number of ways. In this paper, we illustrate how this model of mutation rates can be derived by means of classical statistical mechanics tools—in particular, by [...] Read more.
The Luria–Delbrück mutation model is a cornerstone of evolution theory and has been mathematically formulated in a number of ways. In this paper, we illustrate how this model of mutation rates can be derived by means of classical statistical mechanics tools—in particular, by modeling the phenomenon resorting to methodologies borrowed from classical kinetic theory of rarefied gases. The aim is to construct a linear kinetic model that can reproduce the Luria–Delbrück distribution starting from the elementary interactions that qualitatively and quantitatively describe the variations in mutated cells. The kinetic description is easily adaptable to different situations and makes it possible to clearly identify the differences between the elementary variations, leading to the Luria–Delbrück, Lea–Coulson, and Kendall formulations, respectively. The kinetic approach additionally emphasizes basic principles which not only help to unify existing results but also allow for useful extensions. Full article
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20 pages, 721 KiB  
Article
Reversible Transitions in a Fluctuation Assay Modify the Tail of Luria–Delbrück Distribution
by Pavol Bokes, Anna Hlubinová and Abhyudai Singh
Axioms 2023, 12(3), 249; https://doi.org/10.3390/axioms12030249 - 1 Mar 2023
Viewed by 1290
Abstract
We consider a fluctuation test experiment in which cell colonies were grown from a single cell until they reach a given population size and were then exposed to treatment. While they grow, the cells may, with a low probability, acquire resistance to treatment [...] Read more.
We consider a fluctuation test experiment in which cell colonies were grown from a single cell until they reach a given population size and were then exposed to treatment. While they grow, the cells may, with a low probability, acquire resistance to treatment and pass it on to their offspring. Unlike the classical Luria–Delbrück fluctuation test, and motivated by recent work on drug-resistance acquisition in cancer/microbial cells, we allowed the resistant cell state to switch back to a drug-sensitive state. This modification did not affect the central part of the Luria–Delbrück distribution of the number of resistant survivors: the previously developed approximation by the Landau probability density function applied. However, the right tail of the modified distribution deviated from the power law decay of the Landau distribution. Here, we demonstrate that the correction factor was equal to the Landau cumulative distribution function. We interpreted the appearance of the Landau laws from the standpoint of singular perturbation theory and used the asymptotic matching principle to construct uniformly valid approximations. Additionally, we describe the corrections to the distribution tails in populations initially consisting of multiple sensitive cells, a mixture of sensitive and resistant cells, and a cell with a randomly drawn state. Full article
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18 pages, 2436 KiB  
Article
SimuBP: A Simulator of Population Dynamics and Mutations Based on Branching Processes
by Xiaowei Wu
Axioms 2023, 12(2), 101; https://doi.org/10.3390/axioms12020101 - 18 Jan 2023
Viewed by 1418
Abstract
Originating from the Luria–Delbrück experiment in 1943, fluctuation analysis (FA) has been well developed to demonstrate random mutagenesis in microbial cell populations and infer mutation rates. Despite the remarkable progress in its theory and applications, FA often faces difficulties in the computation perspective, [...] Read more.
Originating from the Luria–Delbrück experiment in 1943, fluctuation analysis (FA) has been well developed to demonstrate random mutagenesis in microbial cell populations and infer mutation rates. Despite the remarkable progress in its theory and applications, FA often faces difficulties in the computation perspective, due to the lack of appropriate simulators. Existing simulation algorithms are usually designed specifically for particular scenarios, thus their applications may be largely restricted. There is a pressing need for more flexible simulators that rely on minimum model assumptions and are highly adaptable to produce data for a wide range of scenarios. In this study, we propose SimuBP, a simulator of population dynamics and mutations based on branching processes. SimuBP generates data based on a general two-type branching process, which is able to mimic the real cell proliferation and mutation process. Through simulations under traditional FA assumptions, we demonstrate that the data generated by SimuBP follow expected distributions, and exhibit high consistency with those generated by two alternative simulators. The most impressive feature of SimuBP lies in its flexibility, which enables the simulation of data analogous to real fluctuation experiments. We demonstrate the application of SimuBP through examples of estimating mutation rates. Full article
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10 pages, 378 KiB  
Article
Numbers of Mutations within Multicellular Bodies: Why It Matters
by Steven A. Frank
Axioms 2023, 12(1), 12; https://doi.org/10.3390/axioms12010012 - 23 Dec 2022
Cited by 3 | Viewed by 1527
Abstract
Multicellular organisms often start life as a single cell. Subsequent cell division builds the body. Each mutational event during those developmental cell divisions carries forward to all descendant cells. The overall number of mutant cells in the body follows the Luria–Delbrück process. This [...] Read more.
Multicellular organisms often start life as a single cell. Subsequent cell division builds the body. Each mutational event during those developmental cell divisions carries forward to all descendant cells. The overall number of mutant cells in the body follows the Luria–Delbrück process. This article first reviews the basic quantitative principles by which one can understand the likely number of mutant cells and the variation in mutational burden between individuals. A recent Fréchet distribution approximation simplifies calculation of likelihoods and intuitive understanding of process. The second part of the article highlights consequences of somatic mutational mosaicism for understanding diseases such as cancer, neurodegeneration, and atherosclerosis. Full article
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18 pages, 510 KiB  
Article
A Fresh Approach to a Special Type of the Luria–Delbrück Distribution
by Qi Zheng
Axioms 2022, 11(12), 730; https://doi.org/10.3390/axioms11120730 - 14 Dec 2022
Cited by 1 | Viewed by 1045
Abstract
The mutant distribution that accommodates both fitness and plating efficiency is an important class of the Luria–Delbrück distribution. Practical algorithms for computing this distribution do not coincide with the theoretically most elegant ones, as existing generic methods often either produce unreliable results or [...] Read more.
The mutant distribution that accommodates both fitness and plating efficiency is an important class of the Luria–Delbrück distribution. Practical algorithms for computing this distribution do not coincide with the theoretically most elegant ones, as existing generic methods often either produce unreliable results or freeze the computational process altogether when employed to solve real-world research problems. Exploiting properties of the hypergeometric function, this paper offers an algorithm that considerably expands the scope of application of this important class of the Luria–Delbrück distribution. An integration method is also devised to complement the novel algorithm. Asymptotic properties of the mutant probability are derived to help gauge the new algorithm. An illustrative example and simulation results provide further guidelines on the use of the new algorithm. Full article
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29 pages, 396 KiB  
Article
Mutant Number Laws and Infinite Divisibility
by Anthony G. Pakes
Axioms 2022, 11(11), 584; https://doi.org/10.3390/axioms11110584 - 24 Oct 2022
Cited by 1 | Viewed by 1056
Abstract
Concepts of infinitely divisible distributions are reviewed and applied to mutant number distributions derived from the Lea-Coulson and other models which describe the Luria-Delbrück fluctuation test. A key finding is that mutant number distributions arising from a generalised Lea-Coulson model for which normal [...] Read more.
Concepts of infinitely divisible distributions are reviewed and applied to mutant number distributions derived from the Lea-Coulson and other models which describe the Luria-Delbrück fluctuation test. A key finding is that mutant number distributions arising from a generalised Lea-Coulson model for which normal cell growth is non-decreasing are unimodal. An integral criterion is given which separates the cases of a mode at the origin, or not. Full article
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