A generalized Luria-Delbrück model

We develop extensions of the Luria-Delbrück model that explicitly consider non-exponential growth of normal cells and a birth-death process with mean exponential or Gompertz growth of mutants. Death of mutant cells can be important in clones arising during cancer progression. The use of a birth-death process for growth of mutant cells, as opposed to a pure birth process as in previous work on the Luria-Delbrück model, leads to a large increase in the extra Poisson variation in the size of the mutant cell populations, which needs to be addressed in statistical analyses. We also discuss connections with previous work on carcinogenesis models.     

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.     

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.     

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.     

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.     

Discussion on research methods of bacterial resistant mutation mechanisms under selective culture—uncertainty analysis of data from the Luria-Delbrück fluctuation experiment

Whether bacterial drug-resistance is drug-induced or results from rapid propagation of random spontaneous mutations in the flora prior to exposure, remains a long-term key issue concerned and debated in both genetics and medicinal fields. In a pioneering study, Luria and Delbrück exposed E. coli to T1 phage, to investigate whether the number of resistant colonies followed the Poisson distribution. They deduced that the development of resistant colonies is independent of phage presence. Similar results have since been obtained on solid medium containing antibacterial agents. Luria and Delbrück??s conclusions were long considered a gold standard for analyzing drug resistance mutations. More recently, the concept of adaptive mutation has triggered controversy over this approach. Microbiological observation shows that, following exposure to drugs of various concentrations, drug-resistant cells emerge and multiply depending on the time course, and show a process function, inconsistent with the definition of Poisson distribution (which assumes not only that resistance is independent of drug quantity but follows no specific time course). At the same time, since cells tend to aggregate after division rather than separating, colonies growing on drug plates arise from the multiplication of resistant bacteria cells of various initial population sizes. Thus, statistical analysis based on equivalence of initial populations will yield erroneous results. In this paper, 310 data from the Luria-Delbrück fluctuation experiment were reanalyzed from this perspective. In most cases, a high-end abnormal value, resulting from the non-synchronous variation of the two above-mentioned time variables, was observed. Therefore, the mean value cannot be regarded as an unbiased expectation estimate. The ratio between mean value and variance was similarly incomparable, because two different sampling methods were used. In fact, the Luria-Delbrück data appear to follow an aggregated, rather than Poisson distribution. In summary, the statistical analysis of Luria and Delbrück is insufficient to describe rules of resistant mutant development and multiplication. Correction of this historical misunderstanding will enable new insight into bacterial resistance mechanisms.     

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.     

An explicit representation of the Luria–Delbrück distribution

The probability distribution of the number of mutant cells in a growing single-cell population is presented in explicit form. We use a discrete model for mutation and population growth which in the limit of large cell numbers and small mutation rates reduces to certain classical models of the Luria-Delbrück distribution. Our results hold for arbitrarily large values of the mutation rate and for cell populations of arbitrary size. We discuss the influence of cell death on fluctuation experiments and investigate a version of our model that accounts for the possibility that both daughter cells of a non-mutant cell might be mutants. An algorithm is presented for the quick calculation of the distribution. Then, we focus on the derivation of two essentially different limit laws, the first of which applies if the population size tends to infinity while the mutation rate tends to zero such that the product of mutation rate times population size converges. The second limit law emerges after a suitable rescaling of the distribution of non-mutant cells in the population and applies if the product of mutation rate times population size tends to infinity. We discuss the distribution of mutation events for arbitrary values of the mutation rate and cell populations of arbitrary size, and, finally, consider limit laws for this distribution with respect to the behavior of the product of mutation rate times population size. Thus, the present paper substantially extends results due to Lea and Coulson (1949), Bartlett (1955), Stewart et al. (1990), and others.     

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.     

Exploring the common molecular basis for the universal DNA mutation bias: revival of Löwdin mutation model

Recently, numerous genome analyses revealed the existence of a universal G:C→A:T mutation bias in bacteria, fungi, plants and animals. To explore the molecular basis for this mutation bias, we examined the three well-known DNA mutation models, i.e., oxidative damage model, UV-radiation damage model and CpG hypermutation model. It was revealed that these models cannot provide a sufficient explanation to the universal mutation bias. Therefore, we resorted to a DNA mutation model proposed by L?wdin 40 years ago, which was based on inter-base double proton transfers (DPT). Since DPT is a fundamental and spontaneous chemical process and occurs much more frequently within GC pairs than AT pairs, L?wdin model offers a common explanation for the observed universal mutation bias and thus has broad biological implications.     

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.     

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.     

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.     

On the appropriateness of use of a continuous formulation for the modelling of discrete multireactant systems following Michaëlis–Menten kinetics

The possibility of solving the mass balances to a multiplicity of substrates within a CSTR in the presence of a chemical reaction following Michaelis-Menten kinetics using the assumption that the discrete distribution of said substrates is well approximated by an equivalent continuous distribution on the molecular weight is explored. The applicability of such reasoning is tested with a convenient numerical example. In addition to providing the limiting behavior of the discrete formulation as the number of homologous substrates increases, the continuous formulation yields in general simpler functional forms for the final distribution of substrates than the discrete counterpart due to the recursive nature of the solution in the latter case.List of Symbols C{N. M} mol/m³ concentration of substrate containing N monomer residues each with molecular weight M - {N, M} normalized value of C{N. M} - C {M} mol/m³ da concentration of substrate of molecular weight M - _in normalized value of C {M} at the i-th iteration of a finite difference method - {M} normalized value of C {M} - C ₀{N.M} mol/m³ inlet concentration of substrate containing N monomer residues each with molecular weight M - {N ·M} normalized value of C₀{N. M} - _{0 i} normalized value of C ₀ {M} at the i-th iteration of a finite difference method - C ₀ {M} mol/m³ da initial concentration of substrate of molecular weight M - C _tot mol/m³ (constant) overall concentration of substrates (discrete model) - C _tot mol/m³ (constant) overall concentration of substrates (continuous model) - D deviation of the continuous approach relative to the discrete approach - i dummy integer variable - I arbitrary integration constant - j dummy integer variable - k dummy integer variable - K _m mol/m³ Michaëlis-Menten constant for the substrates - l dummy integer variable - M da molecular weight of substrate - M normalized value of M - M da maximum molecular weight of a reacting substrate - N number of monomer residues of a reacting substrate - N maximum number of monomer residues of a reacting substrate - N total number of increments for the finite difference method - Q m³/s volumetric flow rate of liquid through the reactor - S inert product molecule - S _i substrate containing i monomer residues - V m³ volume of the reactor - v _max mol/m³ s reaction rate under saturating conditions of the enzyme active site with substrate - v _max{N. M} mol/m³ s reaction rate under saturating conditions of the enzyme active site with substrate containing N monomer residues with molecular weight M - _max{N · M} dimensionless value of v_max{N. M} (discrete model) - _max{M} dimensionless value of v _max {M} (continuous model) - mol/m³ s molecular weight-averaged value of v_max (discrete model) - mol.da/m³s molecular weight-averaged value of v_max (continuous model) - v _max {M} mol.da/m³s reaction rate under saturating conditions of the enzyme active site with substrate with molecular weight M - _max {M} dimensionless value of v_max{M} - _{max, (i)} dimensionless value of v_max{M} at the i-th iteration of a finite difference method - v _max mol/m³ s reference constant value of v _max Greek Symbols dimensionless operating parameter (discrete distribution) - dimensionless operating parameter (continuous distribution) - M da (average) molecular weight of a monomeric subunit - M selected increment for the finite difference method - auxiliary corrective factor (discrete model)