Fixation probabilities of random mutants under frequency dependent selection

Evolutionary game dynamics describes frequency dependent selection in asexual, haploid populations. It typically considers predefined strategies and fixed payoff matrices. Mutations occur between these known types only. Here, we consider a situation in which a mutation has produced an entirely new type which is characterized by a random payoff matrix that does not change during the fixation or extinction of the mutant. Based on the probability distribution underlying the payoff values, we address the fixation probability of the new mutant. It turns out that for weak selection, only the first moments of the distribution matter. For strong selection, the probability that a new payoff entry is larger than the wild type's payoff against itself is the crucial quantity.     

Fixation probabilities in a spatially heterogeneous environment

 We consider a simple model of a one-locus, two-allele population inhibiting a two-patch system and experiencing spatially heterogeneous viability selection. The populaton size is finite. We use a diffusion approximation and singular perturbation techniques to find the probability of fixation of a mutant allele. We focus on situations in which each allele is advantageous in one patch and deleterious in the other patch. Our theoretical results support the previous conclusions that, under certain conditions, small populations respond faster to selection than do large populations. We emphasize that knowledge of the dependence of migration rates on population size is crucial in evaluating the effects of population size on the rate of evolution.     

Waiting times to appearance and dominance of advantageous mutants: estimation based on the likelihood

The germinal center reaction (GCR) of vertebrate immunity provides a remarkable example of evolutionary succession, in which an advantageous phenotype arises as a spontaneous mutation from the parental type and eventually displaces the parental type altogether. In the case of the immune response to the hapten (4-hydroxy-3-nitrophenyl)acetyl (NP), as with several other designed immunogens, the process is dominated by a single key mutation, which greatly simplifies the modeling of and analysis of data. We developed a two-stage model of this process in which the primary stage represents the appearance and establishment of the mutant population as a stochastic process while the second stage represents the growth and dominance of the clone as a deterministic process, conditional on its time of establishment from stage one. We applied this model to the analysis of population samples from several germinal center (GC) reactions and used maximum-likelihood methods to estimate the waiting times to arrival and to dominance of the mutant clone. We determined the sampling properties of the maximum-likelihood estimates using Monte Carlo methods and compared them to their asymptotic distributions. The methods we present here are well-suited for use in the analysis of other systems, such as tumor growth and the experimental evolution of bacteria.     

A note on a method to aid sampling populations for characteristics

A method is described for determining the minimum sample size for examining characters in animal or microbial populations.     

Fixation times and probabilities for an independent loci model in genetics

Assuming that two genetic loci evolve independently and that there is no mutation and selection, formulae are obtained for mean first and final fixation times and first fixation probabilities. These results, together with known results for the case of complete linkage, give some indication of the effect of linkage on these quantities.     

Fixation probabilities depend on life history: fecundity, generation time and survival in a burst-death model

The burst-death model has been developed to describe the life history of organisms with variable generation times and a burst of a fixed number of offspring. The model also includes an optional constant clearance rate, such as washout from a chemostat, and the possibility of sustained periods of population growth followed by severe bottlenecks, as in serial passaging. In this model, a beneficial mutation can either increase the burst rate or the burst size, or reduce the clearance rate, thus increasing survival. In this article we examine the effects of these three possible mechanisms on both the Malthusian fitness and the fixation probability of the lineage. We find that equivalent relative increases in the burst rate or burst size confer equivalent increases in the Malthusian fitness of a lineage, whereas increasing survival typically has a more moderate effect on Malthusian fitness. In contrast, for beneficial mutations that confer the same increase in fitness, mutations that increase survival are the most likely to fix, followed by mutations that increase the burst rate. Mutations that increase the burst size are the least likely to fix. These results imply that mutant lineages with the highest Malthusian fitness are not, in many cases, the most likely to escape extinction.     

A note on a method for determining advantageous properties of an anti-arrhythmic drug based on a mathematical model of cardiac cells

Regional hyperkalemia during acute ischemia may provoke cardiac arrhythmias such as ventricular fibrillation. Despite intense research efforts over the last decades, the problem of finding an efficient anti-arrhythmic drug without dangerous side effects is still open. One approach to analyze the effect of anti-arrhythmic drugs is to do simulations based on mathematical models of collections of cardiomyocytes. Such simulations have recently illuminated the pro-arrhythmic capability of well-established anti-arrhythmic drugs.The purpose of the present note is to introduce a method intended for computing advantageous properties of an anti-arrhythmic drug. For a given model of a normal and an ischemic cell, we introduce a drug as a vector of non-negative real numbers whose components are multiplied by individual terms representing specific ionic currents. The drug vector is computed such that the action potentials of the resulting drugged cells are as close as possible to the action potential of a normal (not drugged) cell. Numerical simulations based on the Luo-Rudy I model and the Hund-Rudy model show that the classical shortened action potential obtained due to hyperkalemia is prolonged by using the drug computed by this method. Furthermore, for both models a 2D collection of spatially coupled ischemic cells give arrhythmogenic solutions before the drug is applied, and stable solutions after the drug is applied. It is emphasized that we do not address the possibility of realizing a drug with the properties computed in this note.     

Extinction probabilities in branching processes: A note on holgate and Lakhani's paper

Within the class of offspring distributions with given meanm>1 and probability of no offspringp _o, the probabilityq of ultimate extinction in a Galton-Watson branching process starting from one individual satisfiesp ₀<q(m,p₀)≤q<1. A short table illustrates the lower boundq(m,p ₀). Work done at the Mathematics Department, University of Washington, Seattle, U.S.A.     

Fixation probabilities for the Moran process with three or more strategies: general and coupling results

We study fixation probabilities for the Moran stochastic process for the evolution of a population with three or more types of individuals and frequency-dependent fitnesses. Contrary to the case of populations with two types of individuals, in which fixation probabilities may be calculated by an exact formula, here we must solve a large system of linear equations. We first show that this system always has a unique solution. Other results are upper and lower bounds for the fixation probabilities obtained by coupling the Moran process with three strategies with birth–death processes with only two strategies. We also apply our bounds to the problem of evolution of cooperation in a population with three types of individuals already studied in a deterministic setting by Núñez Rodríguez and Neves (J Math Biol 73:1665–1690, 2016). We argue that cooperators will be fixated in the population with probability arbitrarily close to 1 for a large region of initial conditions and large enough population sizes.

     

A note on 'stomaching' for the quantitative sampling of epiphyton

SUMMARY. 1. Methods for the quantitative removal of epiphyton from Cladophora glomerata were compared.
2. Stomaching for 3–5 min effected maximum epiphyte release. Agitation for up to 40 min in water or hydroiysing solutions of hydrochloric acid or F.A.A. removed significantly fewer epiphytes.
3. Stomaching is recommended as a new, simple, quick and effective method for quantitative epiphyton sampling.     

A note on semiparametric efficient inference for two-stage outcome-dependent sampling with a continuous outcome

Outcome-dependent sampling designs have been shown to be a cost-effectiveway to enhance study efficiency. We show that the outcome-dependentsampling design with a continuous outcome can be viewed as anextension of the two-stage case-control designs to the continuous-outcomecase. We further show that the two-stage outcome-dependent samplinghas a natural link with the missing-data and biased-samplingframeworks. Through the use of semiparametric inference andmissing-data techniques, we show that a certain semiparametricmaximum-likelihood estimator is computationally convenient andachieves the semiparametric efficient information bound. Wedemonstrate this both theoretically and through simulation.     

Fixation probabilities when generation times are variable: the burst death model

Estimating the fixation probability of a beneficial mutation has a rich history in theoretical population genetics. Typically, to attain mathematical tractability, we assume that generation times are fixed, while the number of offspring per individual is stochastic. However, fixation probabilities are extremely sensitive to these assumptions regarding life history. In this article, we compute the fixation probability for a "burst-death" life-history model. The model assumes that generation times are exponentially distributed, but the number of offspring per individual is constant. We estimate the fixation probability for populations of constant size and for populations that grow exponentially between periodic population bottlenecks. We find that the fixation probability is, in general, substantially lower in the burst-death model than in classical models. We also note striking qualitative differences between the fates of beneficial mutations that increase burst size and mutations that increase the burst rate. In particular, once the burst size is sufficiently large relative to the wild type, the burst-death model predicts that fixation probability depends only on burst rate.     

A note on Cannings and Thompson''s sequential sampling scheme for pedigrees.

We consider sequential sampling of pedigrees for genetic analysis. Cannings and Thompson (1977) gave simple, general guidelines for valid sequential sampling schemes. We show that their formulation of the likelihood contains an error, which is, however, easily corrected so as to maintain the validity of the sequential scheme. We also point out that although sequential and fixed-structure pedigree sampling do have the same likelihoods (as Cannings and Thompson showed), and therefore yield the same maximum likelihood point estimates of genetic parameters, they do not necessarily yield the same significance tests or confidence intervals.     

Protein probabilities in shotgun proteomics: evaluating different estimation methods using a semi-random sampling model

The calculation of protein probabilities is one of the most intractable problems in large-scale proteomic research. Current available estimating methods, for example, ProteinProphet, PROT_PROBE, Poisson model and two-peptide hits, employ different models trying to resolve this problem. Until now, no efficient method is used for comparative evaluation of the above methods in large-scale datasets. In order to evaluate these various methods, we developed a semi-random sampling model to simulate large-scale proteomic data. In this model, the identified peptides were sampled from the designed proteins and their cross-correlation scores were simulated according to the results from reverse database searching. The simulated result of 18 control proteins was consistent with the experimental one, demonstrating the efficiency of our model. According to the simulated results of human liver sample, ProteinProphet returned slightly higher probabilities and lower specificity than real cases. PROT_PROBE was a more efficient method with higher specificity. Predicted results from a Poisson model roughly coincide with real datasets, and the method of two-peptide hits seems solid but imprecise. However, the probabilities of identified proteins are strongly correlated with several experimental factors including spectra number, database size and protein abundance distribution.     

A note on auxotrophic mutants of cowpea rhizobia

We have examined a variety of common mutagens in producing auxotrophic mutants in cowpea rhizobia strains JRC23 and IRC256. While NTG (N-methyl-N-nitro-N-nitrosoguanidine), EMS (ethylmethane sulfonate), NA (nitrous acid), and UV (ultraviolet) irradiation were mutagenic with the strain JRC23, these mutagenic agents did not mutate strain IRC256. On the contrary, transposon mutagenesis with Tn5 yielded auxotrophs in strain IRC256 but not in strain JRC23, while only methionine (Met^–) auxotrophs from strain JRC23, histidine (His^–), and adenine plus thiamine (Ade^–+Thi^–) auxotrophs from strain IRC256 were isolated.     

Photographic sampling: a photographic sampling method for mites on plants

A photographic sampling method for mites on plants was evaluated using Tetranychus urticae and Phytoseiulus persimilis on pepper plants. It was found to be 92% accurate for T. urticae eggs and 98% accurate for P. persimilis eggs at densities up to 45 eggs per cm² for T. urticae, and up to 3 eggs per cm² for P. persimilis. The motiles of the two species were not confused, nor were they confused with exuviae or other matter.     

A note on the sampling theory for infinite alleles and infinite sites models

This note considers sampling theory for a selectively neutral locus where it is supposed that the data provide nucleotide sequences for the genes sampled. It thus anticipates that technical advances will soon provide data of this form in volume approaching that currently obtained from electrophoresis. The assumption made on the nature of the data will require us to use, in the terminology ofKimura (Theor. Pop. Biol.2, 174–208 (1971)), the “infinite sites” model of Karlin and McGregor (Proc. Fifth Berkeley Symp. Math. Statist. Prob.4, 415–438 (1967)) rather that the “infinite alleles” model of Kimura and Crow (Genetics49, 174–738 (1964)). We emphasize that these two models refer not to two different real-world circumstances, but rather to two different assumptions concerning our capacity to investigate the real world. We compare our results where appropriate with corresponding sampling theory of Ewens (Theor. Pop. Biol.3, 87–112 (1972)) for the “infinite alleles” model. Note finally that some of our results depend on an assumption of independence of behavior at individual sites; a parallel paper byWatterson (submitted for publication (1974)) assumes no recombination between sites. Real-world behavior will lie between these two assumptions, closer to the situation assumed by Watterson than in this note. Our analysis provides upper bounds for increased efficiency in using complete nucleotide sequences.