Improved inference of mutation rates: I. An integral representation for the Luria-Delbrück distribution

The estimation of mutation rates is ordinarily performed using results based on the Luria-Delbrück distribution. There are certain difficulties associated with the use of this distribution in practice, some of which we address in this paper (others in the companion paper, Oprea and Kepler, Theor. Popul. Biol., 2001). The distribution is difficult to compute exactly, especially for large values of the random variable. To overcome this problem, we derive an integral representation of the Luria-Delbrück distribution that can be computed easily for large culture sizes. In addition, we introduce the usual assumption of very small probability of having a large proportion of mutants only after the generating function has been computed. Thus, we obtain information on the moments for the more general case. We examine the asymptotic behavior of this system. We find a scaling or "standardization" technique that reduces the family of distributions parameterized by three parameters (mutation rate, initial cell number, and final cell number) to a single distribution with no parameters, valid so long as the product of the mutation rate and the final culture is sufficiently large. We provide a pair of techniques for computing confidence intervals for the mutation rate. In the second paper of this series, we use the distribution derived here to find approximate distributions for the case where the cell cycle time is not well-described as an exponential random variable as is implicitly assumed by Luria-Delbrück distribution.     

On Haldane's formulation of Luria and Delbrück's mutation model

Two formulations of Luria and Delbrück's mutation model have been in common use since the 1940s. While mathematicians focused their attention on the formulation of Lea and Coulson that assumes asynchronous cell growth, biologists found more appealing the formulation of Haldane that assumes synchronous cell growth. This article attempts to solve several outstanding issues for the latter formulation. First, it provides an exact, closed-form expression for the mutant distribution by correcting a minor error in the literature. Second, it presents a novel algorithm for computing the mutant distribution, which leads to novel methods for computing point and interval estimates of mutation rates based on the maximum likelihood principle. Third, it critically examines existing methods based on the mean number of mutants. Finally, it compares the two formulations to underline their strengths and shortcomings.     

Progress of a half century in the study of the Luria-Delbrück distribution

The Luria-Delbrück mutation model has been mathematically formulated in a number of ways. This review article examines four most important formulations, focusing on important practical issues closely linked with the distribution of the number of mutants. These issues include the probability generating functions, moments (cumulants), computational methods and asymptotics. This review emphasizes basic principles which not only help to unify existing results but also allow for a few useful extensions. In addition, the review offers a historical perspective and some new explanations of divergent moments.     

Conceptual models and analytical tools: The biology of physicist Max Delbrück

Journal of the History of Biology -     

An atypical fauna in the Lower Devonian Hunsrück Slate of Germany

The Hunsrück Slate is a world renowned conservation lagerstätte, stretching SW-NE in a narrow band between the villages of Bundenbach and Gemünden, Hunsrück region, Germany. A great variety of complete Lower Devonian fossils is preserved with their soft parts pyritised due to rapid burial by sediment. The fossils of the Hunsrück Slate outside the narrow band are scarcely known. They represent the normal Situation: quiet low energy Sedimentation. This paper describes this overall “Rhenish” neritic fauna under the heading “atypical”, in contrast to the famous fossils of the conservation lagerstätte. The fauna described here lacks starfishes, mitrates and chelicerates. The paper begins with an overview of the recent relevant literature (stratigraphic position of the Hunsrück Slate, palaeoecology). The slate fossils were collected in two quarries WNW and NE of Bundenbach (Lingenbach and Karschheck, respectively) and most closely resemble that of the Wisper Valley in the Taunus region. Their geologic age is Lower Devonian, early Emsian, Ulmen Substage. Special interest is given to Community structures, rugose corals, bivalves, gastropods, trilobites, conulariids, brachiopods and crinoids. In the systematic part, the crinoidOrthocrinus simplex is redescribed. Also, two new species are introduced: the rugose coralVolgerophyllum karschheckensis n. gen., n. sp., and the crinoidAcanthocrinus spinosus n. sp.     

Bringing physics to bear on the phenomenon of life: the divergent positions of Bohr, Delbrück, and Schrödinger

The received view on the contributions of the physics community to the birth of molecular biology tends to present the physics community as sharing a basic level consensus on how physics should be brought to bear on biology. I argue, however, that a close examination of the views of three leading physicists involved in the birth of molecular biology, Bohr, Delbrück, and Schr?dinger, suggests that there existed fundamental disagreements on how physics should be employed to solve problems in biology even within the physics community. In particular, I focus on how these three figures differed sharply in their assessment of the relevance of complementarity, the potential of chemical methods, and the relative importance of classical physics. In addition, I assess and develop Roll-Hansen's attempt to conceptualize this history in terms of models of scientific change advanced by Kuhn and Lakatos. Though neither model is fully successful in explaining the divergence of views among these three physicists, I argue that the extent and quality of difference in their views help elucidate and extend some themes that are left opaque in Kuhn's model.     

On Bartlett's formulation of the Luria-Delbrück mutation model

Current knowledge of microbial mutation rates was accumulated largely by means of fluctuation experiments. A mathematical model describing the cell dynamics in a fluctuation experiment is indispensable to the estimation of mutation rates through fluctuation experiments. In almost six decades the model formulated by Lea and Coulson dominated in research and application, although the model formulated by Bartlett is generally believed to describe the cell dynamics more faithfully. The neglect of the Bartlett formulation was mainly due to mathematical difficulties. The present investigation overcomes some of these difficulties, thereby paving the way for the application of the Bartlett formulation in estimating mutation rates. Specifically, the article offers an algorithm for computing the distribution function of the number of mutants under the Bartlett formulation. The article also provides algorithms for computing point and interval estimates of mutation rates that are based on the maximum-likelihood principle. In addition, the article examines and compares the asymptotic behavior of the distributions of the number of mutants under the two formulations.     

Improved inference of mutation rates: II. Generalization of the Luria-Delbrück distribution for realistic cell-cycle time distributions

In the first paper of this series (Kepler and Oprea, Theor. Popul. Biol. 2001) we found a continuum approximation of the Luria-Delbrück distribution in terms of a scaled variable related to the proportion of mutants in the culture. Here we show that the Luria-Delbrück distribution is inaccurate when realistic division processes are being considered due to the non-Markovian character of the cell cycle. We derive the expectation of the proportion of mutants in the culture for arbitrary cell-cycle time distributions. We then introduce a two-parameter generalization of the continuum Luria-Delbrück distribution for two of the more commonly used cell-cycle time distributions: gamma and shifted exponential. We obtain the generalized distribution by defining a map from the actual parameters to "effective" parameters. The effective mutation rate is obtained analytically, while the effective population size is obtained by fitting simulation data. Our simulations show that the second parameter depend mostly on the coefficient of variation of the cell-cycle time distribution.     

Comments of R. Glück

Abstract

This paper discusses several liposomal approaches: A T-independent response against hapten or lipopolysaccharide by the addition of a T-epitope (HA2 molecule of influenza); the same response against a synthetic peptide-based vaccine; finally the first commercial liposomal hepatitis A vaccine is discussed.     

Pyritisation of plant microfossils from the Devonian Hunsrück Slate of Germany

The extraordinary pyritised fossils of the Hunsrück Slate of Germany provide critical information on marine life in the Devonian. Evidence for terrestrial life is confined mainly to rare incomplete plant macrofossils, often of uncertain affinity. The discovery of plant microfossils preserved as pyrite in-fills of cells with decay-resistant organic walls provides additional evidence of life on land around the margins of the Hunsrück Slate sea. The microfossils include groups of elongate water-conducting cells: some show annular or helical indentations which represent thickenings of the original cell wall, others show casts of bordered pits. Spores, possible sporangia, and marine acritarchs are also present. In most cases pyritisation was rapid enough to prevent collapse of the cells, and they retain their original 3-dimensionality. Wall structures subsequently decayed except where they were resistant enough to survive as an organic residue.     

Use of the Burrows–Wheeler similarity distribution to the comparison of the proteins

In this paper, we present an approach based on Burrows–Wheeler transform to compare the protein sequences. The strings representing amino acid sequences do not reflect the chemical physical properties better, and it is very hard to extract any key features by reading these long character strings directly. The use of the Burrows–Wheeler similarity distribution needs a suitable representation which can reflect some interesting properties of the proteins. For the comparison of the primary protein sequences we convert the protein sequences into digital codes by the Ponnuswamy hydrophobicity index, and for the comparison of the structure of the proteins we adjust the topology of protein structure strings, which are simple but useful representation of the secondary structure of proteins to match the Burrows–Wheeler similarity distribution. At last, some experiments show that the approach proposed in this paper is a powerful and useful tool for the comparison of proteins.     

The arthropodCheloniellon from the devonian hunsrück shale

Cheloniellon calmani Broili 1932, is restudied with the aid of the radio-graphic technique applied to the two well known specimens (CI and CII), and also two other specimens (CIII and B).Bundenbachiellus giganteus (B) (Broili 1929) is probably a synonym, but the incomplete and distorted condition makes it appear expedient to consider this name as a nomen nudum. New radiographs of CI, CII and CIII allow a better reconstruction of C.calmani than the one based on the distorted specimen CI. The new picture shows a head with two tergite plates on the dorsal side and one pair of antennae, a second preoral pair of appendages and four pairs of probably uniramous legs with gnathobases on the ventral side. Only the last pair is situated beneath the second tergite. The body has eight tergites with wide pleura corresponding to eight pairs of biramous appendages. The outer branch carries very stout lamellar spines. The ninth tergite is small and cylindrical and corresponds to long furcal rami. The posterior end is probably formed by a conical piece, possibly a telson, without appendages.Cheloniellon was a benthic carnivore. The second preoral appendages and postoral gnathobases make it unexpectedly similar to eurypterids and xiphosurids, and it may be a late representative of a group of trilobitomorphs that gave rise to the Chelicerata.     

Somatic mosaicism and cancer: inference based on a conditional Luria-Delbrück distribution

Somatic mosaicism for mutations in disease-causing genes has been reported in several recent studies. Examples include hemophilia A, many skin disorders, and several cancers such as retinoblastoma and familial adenomatous polyposis. Many of these disorders require multiple mutations in order to express the disease phenotype. For example, two recessive mutations to the retinoblastoma locus are required to initiate retinoblastomal tumors. I develop a mathematical framework for somatic mosaicism in which two recessive mutations cause disease. With my framework, I analyse the following question: Given an observed frequency of cells with two mutations and an easily scored aberrant phenotype, what is the conditional frequency distribution of cells carrying one mutation and therefore susceptible to transformation by a second mutation? This question is important because a high frequency of carrier cells can cause genetic counselors to misdiagnose a mosaic as an inherited heterozygote carrier and because widespread mosaicism can lead to some germline transmission. As more data accumulate, the observed distribution of mosaics can be compared against my predicted distribution. These sorts of studies will contribute to a broader understanding of the distribution of somatic mutations, a central topic in the study of cancer.     

Measuring and Overcoming Limits of the Saffman-Delbrück Model for Soap Film Viscosities

We observe tracer particles diffusing in soap films to measure the two-dimensional (2D) viscous properties of the films. Saffman-Delbrück type models relate the single-particle diffusivity to parameters of the film (such as thickness h) for thin films, but the relation breaks down for thicker films. Notably, the diffusivity is faster than expected for thicker films, with the crossover at h/d = 5.2 ± 0.9 using the tracer particle diameter d. This indicates a crossover from purely 2D diffusion to diffusion that is more three-dimensional. We demonstrate that measuring the correlations of particle pairs as a function of their separation overcomes the limitations of the Saffman-Delbrück model and allows one to measure the viscosity of a soap film for any thickness.     

Morphology and distribution of Müller cells in the retina of the toad Bufo marinus

We have previously shown that an antibody against neuron-specific enolase (NSE) selectively labels Müller cells (MCs) in the anuran retina (Wilhelm et al. 1992). In the present study the light- and electron-microscopic morphology of MCs and their distribution were described in the retina of the toad, Bufo marinus, using the above antibody. The somata of MCs were located in the proximal part of the inner nuclear layer and were interconnected with each other by their processes. The MCs were uniformly distributed across the retina with an average density of 1500 cells/mm². Processes of MCs encircled the somata of photoreceptor cells isolating them from each other by glial sheath, except for those of the double cones. Some of the photoreceptor pedicles remained free of glial sheath. Electron-microscopic observations confirmed that MC processes provide an extensive scaffolding across the neural retina. At the outer border of the ganglion cell layer these processes formed a non-continuous sheath. The MC processes traversed through the ganglion cell layer and spread beneath it between the neuronal somata and the underlying optic axons. These processes formed a continuous inner limiting membrane separating the optic fibre layer from the vitreous tissue. Neither astrocytic nor oligodendrocytic elements were found in the optic fibre layer. The significance of the uniform MC distribution and the functional implications of the observed pattern of MC scaffolding are discussed.     

An angiosperm-wide analysis of the gynodioecy–dioecy pathway

Background and Aims

About 6 % of an estimated total of 240 000 species of angiosperms are dioecious. The main precursors of this sexual system are thought to be monoecy and gynodioecy. A previous angiosperm-wide study revealed that many dioecious species have evolved through the monoecy pathway; some case studies and a large body of theoretical research also provide evidence in support of the gynodioecy pathway. If plants have evolved through the gynodioecy pathway, gynodioecious and dioecious species should co-occur in the same genera. However, to date, no large-scale analysis has been conducted to determine the prevalence of the gynodioecy pathway in angiosperms. In this study, this gap in knowledge was addressed by performing an angiosperm-wide survey in order to test for co-occurrence as evidence of the gynodioecy pathway.

Methods

Data from different sources were compiled to obtain (to our knowledge) the largest dataset on gynodioecy available, with 275 genera that include at least one gynodioecious species. This dataset was combined with a dioecy dataset from the literature, and a study was made of how often dioecious and gynodioecious species could be found in the same genera using a contingency table framework.

Key Results

It was found that, overall, angiosperm genera with both gynodioecious and dioecious species occur more frequently than expected, in agreement with the gynodioecy pathway. Importantly, this trend holds when studying different classes separately (or sub-classes, orders and families), suggesting that the gynodioecy pathway is not restricted to a few taxa but may instead be widespread in angiosperms.

Conclusions

This work complements that previously carried out on the monoecy pathway and suggests that gynodioecy is also a common pathway in angiosperms. The results also identify angiosperm families where some (or all) dioecious species may have evolved from gynodioecious precursors. These families could be the targets of future small-scale studies on transitions to dioecy taking phylogeny explicitly into account.