On an unbiased and consistent estimator for mutation rates

Determination of Mutation Rates with Two Symmetric and Asymmetric Mutation Types

We revisit our earlier paper, with two of the coauthors, in which we proposed an unbiased and consistent estimator μ^n for an unknown mutation rate μ of microorganisms. Previously, we proved that the associated sequence of estimators μ^n converges to μ almost surely pointwise on a nonextinct set Ω0. Here, we show that this sequence converges also in the mean square with respect to conditional probability measure P0·=P·∩Ω0/PΩ0 and that, with respect to P0, the estimator is asymptotically unbiased. We further assume that a microorganism can mutate or turn to a different variant of one of the two types. In particular, it can mean that bacteria under attack by a virus or chemical agent are either perishing or surviving, turning them to stronger variant. We propose estimators for their respective types and show that they are a.s. pointwise and L2-consistent and asymptotically unbiased with respect to measure P0.

Toward a Unique Definition of the Mutation Rate

In around 1943, while writing a classic paper with Luria, Delbrück envisioned two kinds of mutation rates: One was expressed as mutations per bacterium per unit time, the other as mutations per bacterium per division cycle. Due to minor mathematical errors, the precise connection between the two concepts remained elusive for Delbrück. As a result, researchers and educators alike are still grappling with the definition of the mutation rate. Within the context of microbial mutation, the current author proposes an idealized model to bring new clarity to the distinction between the two forms of the mutation rate that Delbrück once attempted to define and characterize. The paper also critiques two incorrect estimators of mutation rates and brings to light two important yet unexplored "invariance" hypotheses about mutation rates.

A second look at the final number of cells in a fluctuation experiment

In a fluctuation experiment, the number of cells existing in a culture immediately before plating (commonly known as Nt) varies across the parallel cultures. However, most existing mathematical models for fluctuation assay data do not recognize the variation in Nt. Despite repeated attempts in the past to integrate this source of variability in the estimation of microbial mutation rates, several questions of practical importance remain unanswered. The present investigation finds that the variation needs accounting for only when the coefficient of variation for Nt is large, and experimental data suggest that the coefficient of variation is often moderate or small. Moreover, an increase in the inoculum size can reduce the coefficient of variation. Through extensive simulation, several existing methods that accommodate the variation in Nt are compared. It was found that a newly devised likelihood method based on the existing gamma mixture model outperforms other existing methods. The investigation focuses on the estimation of mutation rates using the Lea-Coulson model, under which mutation is selectively neutral; however, the paper also explores the major findings' implications for the comparison of mutation rates using the likelihood ratio test, and for the estimation of mutation rates using the Mandelbrot-Koch model that allows for non-neutral mutations.

The Luria-Delbrück protocol is still the most practical

It is theoretically appealing to determine microbial mutation rates by counting mutant cells in two successive generations. However, major experimental difficulties have been largely ignored, causing unreliable estimates of mutation rates to be unwittingly accepted. Counting mutant cells twice in the same liquid culture incurs appreciable errors that are difficult to quantify; maintaining synchronous cell growth for 25 or more generations is an equally daunting laboratory feat.

Fast maximum likelihood estimation of mutation rates using a birth-death process

Since fluctuation analysis was first introduced by Luria and Delbrück in 1943, it has been widely used to make inference about spontaneous mutation rates in cultured cells. Under certain model assumptions, the probability distribution of the number of mutants that appear in a fluctuation experiment can be derived explicitly, which provides the basis of mutation rate estimation. It has been shown that, among various existing estimators, the maximum likelihood estimator usually demonstrates some desirable properties such as consistency and lower mean squared error. However, its application in real experimental data is often hindered by slow computation of likelihood due to the recursive form of the mutant-count distribution. We propose a fast maximum likelihood estimator of mutation rates, MLE-BD, based on a birth-death process model with non-differential growth assumption. Simulation studies demonstrate that, compared with the conventional maximum likelihood estimator derived from the Luria-Delbrück distribution, MLE-BD achieves substantial improvement on computational speed and is applicable to arbitrarily large number of mutants. In addition, it still retains good accuracy on point estimation.

Mitotic evolution of Plasmodium falciparum shows a stable core genome but recombination in antigen families

Malaria parasites elude eradication attempts both within the human host and across nations. At the individual level, parasites evade the host immune responses through antigenic variation. At the global level, parasites escape drug pressure through single nucleotide variants and gene copy amplification events conferring drug resistance. Despite their importance to global health, the rates at which these genomic alterations emerge have not been determined. We studied the complete genomes of different Plasmodium falciparum clones that had been propagated asexually over one year in the presence and absence of drug pressure. A combination of whole-genome microarray analysis and next-generation deep resequencing (totaling 14 terabases) revealed a stable core genome with only 38 novel single nucleotide variants appearing in seventeen evolved clones (avg. 5.4 per clone). In clones exposed to atovaquone, we found cytochrome b mutations as well as an amplification event encompassing the P. falciparum multidrug resistance associated protein (mrp1) on chromosome 1. We observed 18 large-scale (>1 kb on average) deletions of telomere-proximal regions encoding multigene families, involved in immune evasion (9.5×10⁻⁶ structural variants per base pair per generation). Six of these deletions were associated with chromosomal crossovers generated during mitosis. We found only minor differences in rates between genetically distinct strains and between parasites cultured in the presence or absence of drug. Using these derived mutation rates for P. falciparum (1.0–9.7×10⁻⁹ mutations per base pair per generation), we can now model the frequency at which drug or immune resistance alleles will emerge under a well-defined set of assumptions. Further, the detection of mitotic recombination events in var gene families illustrates how multigene families can arise and change over time in P. falciparum. These results will help improve our understanding of how P. falciparum evolves to evade control efforts within both the individual hosts and large populations.

Malaria is one of the six diseases that together are responsible for 90% of all infectious disease deaths throughout the world. The five species of Plasmodium that cause human malaria take over 655,000 lives each year. Parasites evade the immune response through antigenic variation and develop resistance to anti-malarial drugs through genetic changes in either the drug target or genes conferring resistance. We used whole-genome sequencing and microarray techniques to study evolution in P. falciparum parasites propagated in vitro for up to 180 generations. We determined the mutation rate and found that the core genome of a single clone is stable, while the subtelomeric regions are prone to acquire structural variants. These changes occur mainly in multigene families involved in immune evasion. Our findings indicate that the parasite specifically increases the sequence variability in multigene families through mitotic recombination. This high plasticity of the parasite genome suggests that multiple haplotypes will be present in a natural infection initiated by a single parasite.

Accumulation of neutral mutations in growing cell colonies with competition

Neutral mutations play an important role in many biological processes including cancer initiation and progression, the generation of drug resistance in bacterial and viral diseases as well as cancers, and the development of organs in multicellular organisms. In this paper we study how neutral mutants are accumulated in nonlinearly growing colonies of cells subject to growth constraints such as crowding or lack of resources. We investigate different types of growth control which range from "division-controlled" to "death-controlled" growth (and various mixtures of both). In division-controlled growth, the burden of handling overcrowding lies with the process of cell-divisions, the divisions slow down as the carrying capacity is approached. In death-controlled growth, it is death rate that increases to slow down expansion. We show that division-controlled growth minimizes the number of accumulated mutations, and death-controlled growth corresponds to the maximum number of mutants. We check that these results hold in both deterministic and stochastic settings. We further develop a general (deterministic) theory of neutral mutations and achieve an analytical understanding of the mutant accumulation in colonies of a given size in the absence of back-mutations. The long-term dynamics of mutants in the presence of back-mutations is also addressed. In particular, with equal forward- and back-mutation rates, if division-controlled and a death-controlled types are competing for space and nutrients, cells obeying division-controlled growth will dominate the population.