Mathematical issues arising from the directed mutation controversy

What are we missing in teaching the Luria-Delbrück experiment?

The importance of teaching the Luria-Delbrück experiment to biology students is increasingly recognized by educators, and improved pedagogical methods for teaching the classic experiment have been proposed and tested in the classroom. However, there are still obstacles that impede the proper teaching of the classic experiment. This note proposes two strategies to further improve the teaching of the classic experiment. The first strategy is to be frank with an inherent limitation of the classic experiment, and instructors should explain from a logical point of view why the classic experiment cannot be used to refute the possibility of directed mutation. The second strategy is to emphasize the pioneering work of Delbrück on developing the mutant distribution that enables researchers to estimate microbial mutation rates using data generated by fluctuation experiments, and instructors should shift their attention to the overlooked essential role of the mutant distribution.

Estimation of Rates of Non-neutral Mutations When Bacteria are Exposed to Subinhibitory Levels of Antibiotics

The increasingly widespread use of antibiotics has raised an urgent public health question: how does nonlethal exposure of antibiotics increase rates of bacterial mutations that confer antibiotic resistance? Quantitative studies of this sort are rare, and interested investigators tend to neglect scenarios where the mutation inducer and the selective agent are the same. One possible reason for this puzzling neglect is that such an investigation may lead to results that are prone to a Lamarckian interpretation, but a more tangible reason is that successful estimation of mutation rates in such scenarios requires new computational methods. This study presents computational methods tailored for a mutation model in which some wide-type cells may be killed by nonlethal exposure to an antibiotic, but in which mutants proliferate unimpeded. Methods for computing maximum likelihood estimates of mutation rates and their corresponding confidence intervals are devised and evaluated by simulations.

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.

Combinatorial Genetic Modeling of pfcrt-Mediated Drug Resistance Evolution in Plasmodium falciparum

The emergence of drug resistance continuously threatens global control of infectious diseases, including malaria caused by the protozoan parasite Plasmodium falciparum. A critical parasite determinant is the P. falciparum chloroquine resistance transporter (PfCRT), the primary mediator of chloroquine (CQ) resistance (CQR), and a pleiotropic modulator of susceptibility to several first-line artemisinin-based combination therapy partner drugs. Aside from the validated CQR molecular marker K76T, P. falciparum parasites have acquired at least three additional pfcrt mutations, whose contributions to resistance and fitness have been heretofore unclear. Focusing on the quadruple-mutant Ecuadorian PfCRT haplotype Ecu1110 (K76T/A220S/N326D/I356L), we genetically modified the pfcrt locus of isogenic, asexual blood stage P. falciparum parasites using zinc-finger nucleases, producing all possible combinations of intermediate pfcrt alleles. Our analysis included the related quintuple-mutant PfCRT haplotype 7G8 (Ecu1110 + C72S) that is widespread throughout South America and the Western Pacific. Drug susceptibilities and in vitro growth profiles of our combinatorial pfcrt-modified parasites were used to simulate the mutational trajectories accessible to parasites as they evolved CQR. Our results uncover unique contributions to parasite drug resistance and growth for mutations beyond K76T and predict critical roles for the CQ metabolite monodesethyl-CQ and the related quinoline-type drug amodiaquine in driving mutant pfcrt evolution. Modeling outputs further highlight the influence of parasite proliferation rates alongside gains in drug resistance in dictating successful trajectories. Our findings suggest that P. falciparum parasites have navigated constrained pfcrt adaptive landscapes by means of probabilistically rare mutational bursts that led to the infrequent emergence of pfcrt alleles in the field.

Spontaneous emergence of multiple drug resistance in tuberculosis before and during therapy

The emergence of drug resistance in M. tuberculosis undermines the efficacy of tuberculosis (TB) treatment in individuals and of TB control programs in populations. Multiple drug resistance is often attributed to sequential functional monotherapy, and standard initial treatment regimens have therefore been designed to include simultaneous use of four different antibiotics. Despite the widespread use of combination therapy, highly resistant M. tb strains have emerged in many settings. Here we use a stochastic birth-death model to estimate the probability of the emergence of multidrug resistance during the growth of a population of initially drug sensitive TB bacilli within an infected host. We find that the probability of the emergence of resistance to the two principal anti-TB drugs before initiation of therapy ranges from 10⁻⁵ to 10⁻⁴; while rare, this is several orders of magnitude higher than previous estimates. This finding suggests that multidrug resistant M. tb may not be an entirely “man-made” phenomenon and may help explain how highly drug resistant forms of TB have independently emerged in many settings.

On a logical difficulty in the directed mutation debate

This paper calls attention to an overlooked logical difficulty that has impeded the directed mutation debate for over half a century. It further suggests that the random mutation hypothesis be regarded at present as a null hypothesis in evolutionary biology.

New algorithms for Luria-Delbrück fluctuation analysis

Fluctuation analysis is the most widely used approach in estimating microbial mutation rates. Development of methods for point and interval estimation of mutation rates has long been hampered by lack of closed form expressions for the probability mass function of the number of mutants in a parallel culture. This paper uses sequence convolution to derive exact algorithms for computing the score function and observed Fisher information, leading to efficient computation of maximum likelihood estimates and profile likelihood based confidence intervals for the expected number of mutations occurring in a test tube. These algorithms and their implementation in SALVADOR 2.0 facilitate routine use of modern statistical techniques in fluctuation analysis by biologists engaged in mutation research.