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PubMed1998

A closed-form description of tumour control with fractionated radiotherapy and repopulation.

PURPOSE: To derive a closed form expression of tumour control probability (TCP) following the geometric stochastic approach of Tucker and Taylor. METHODS: A model was constructed based upon a Galton-Watson branching process with cell killing represented by a Bernoulli random variable, and repopulation represented by a Yule Fury process. A closed-form expression of the probability-generating function was derived, which yielded an explicit expression for the mean number of surviving clonogens and the TCP. RESULTS: The mean number of surviving cells, after i clonogens have been treated with n fractions of irradiation, was [equation: see text], where s is the surviving fraction, lambda is the rate of cell division, and delta t is the interfraction time interval. The tumour control probability was [equation: see text]. CONCLUSIONS: Tucker and Taylor provided improvements upon the conventional Poisson model for TCP, mainly through numerical simulation. Here a model based upon their geometric stochastic approach has been derived in closed form. The resultant equations provide a simpler alternative to numerical simulation allowing the effects of fractionated radiotherapy on a replicating population of tumour cells to be more easily predicted.

Cell Division↗
PubMed1962

Electron microscopic observations of rat and mouse cerebellum in tissue culture.

Closely ordered stages of myelin formation in cultures of newborn rat and mouse cerebellum, selected by direct light microscopy, were studied with the electron microscope. Electron micrographs of these cultures reveal the presence of neurons, axons, neuroglia, microglia, and ependymal cells. The appearance of the neuron is identical to that previously described in vivo. The neuroglial cell has long, branching processes, and its cytoplasm is characterized by packets of long, narrow fibrils. During myelin formation, a glial cell process surrounds the axon. This process may form an internal mesaxon and may spiral for several turns around the axon. Other glial cell processes may interdigitate with or overlay the innermost process to contribute to the multilamellated structure. The glial processes flatten and the cytoplasmic surfaces of the cell membrane come into contact to form the lamellae of the myelin sheath. These adhesions may be temporarily incomplete as evidenced by sequestered islands of glial cytoplasm among the myelin lamellae. Ultimately, a compact, apparently spiral, myelin sheath is formed. These findings are discussed in relation to in vivo central myelin formation.

Animals↗
PubMed2004

From a phylogenetic tree to a reticulated network.

In many phylogenetic problems, assuming that species have evolved from a common ancestor by a simple branching process is unrealistic. Reticulate phylogenetic models, however, have been largely neglected because the concept of reticulate evolution have not been supported by using appropriate analytical tools and software. The reticulate model can adequately describe such complicated mechanisms as hybridization between species or lateral gene transfer in bacteria. In this paper, we describe a new algorithm for inferring reticulate phylogenies from evolutionary distances among species. The algorithm is capable of detecting contradictory signals encompassed in a phylogenetic tree and identifying possible reticulate events that may have occurred during evolution. The algorithm produces a reticulate phylogeny by gradually improving upon the initial solution provided by a phylogenetic tree model. The new algorithm is compared to the popular SplitsGraph method in a reanalysis of the evolution of photosynthetic organisms. A computer program to construct and visualize reticulate phylogenies, called T-Rex (Tree and Reticulogram Reconstruction), is available to researchers at the following URL: www.fas.umontreal.ca/biol/casgrain/en/labo/t-rex.

Algorithms↗
PubMed1992

Quantitative PCR with internal controls.

We examine the use of internal controls for estimating the expected initial copy number of the target in a polymerase chain reaction (PCR). We base our investigation on an extended branching-process model. In terms of that model, we delineate the necessary assumptions for this methodology to yield approximately unbiased answers, and we provide means for testing some of those assumptions. We show how to design a series of PCRs to attain optimal precision of the estimate. We provide an algorithm for conducting the statistical analysis of the data, including a formula for a confidence interval for the unknown expected initial copy number.

Algorithms↗
PubMed1987

Distribution and abundance of insertion sequences among natural isolates of Escherichia coli.

A reference collection of 71 natural isolates of Escherichia coli (the ECOR collection) has been studied with respect to the distribution and abundance of transposable insertion sequences using DNA hybridization. The data include 1173 occurrences of six unrelated insertion sequences (IS1, IS2, IS3, IS4, IS5 and IS30). The number of insertion elements per strain, and the sizes of DNA restriction fragments containing them, is highly variable and can be used to discriminate even among closely related strains. The occurrence and abundance of pairs of unrelated insertion sequences are apparently statistically independent, but significant correlations result from stratifications in the reference collection. However, there is a highly significant positive association among the insertion sequences considered in the aggregate. Nine branching process models, which differ in assumptions regarding the regulation of transposition and the effect of copy number on fitness, have been evaluated with regard to their fit of the observed distributions. No single model fits all copy number distributions. The best models incorporate no regulation of transposition and a moderate to strong decrease in fitness with increasing copy number for IS1 and IS5, strong regulation of transposition and a negligible to weak decrease in fitness with increasing copy number for IS3, and less than strong regulation of transposition for IS2, IS4 and IS30.

Base Sequence↗
PubMed1988

Why do unrelated insertion sequences occur together in the genome of Escherichia coli?

Natural isolates of Escherichia coli are polymorphic for the presence or absence of insertion sequences. Among the ECOR reference collection of 71 natural isolates studied for the number of copies of the insertion sequences IS1, IS2, IS3, IS4, IS5 and IS30, the number of strains containing no copies of the insertion sequences were 11, 28, 23, 43, 46 and 36, respectively. Significant correlations occur in the ECOR strains in the presence or absence of unrelated insertion sequences in the chromosome and plasmid complements. Strains containing any insertion sequence are more likely to contain additional, unrelated insertion sequences than would be expected by chance. We suggest that the positive correlations result from horizontal transfer mediated by plasmids. A branching-process model for the plasmid-mediated transmission of insertion sequences among hosts yields such a correlation, even in the absence of interactions affecting transposition or fitness. The predictions of the model are quantitatively in agreement with the observed correlations among insertion sequences.

DNA Transposable Elements↗
PubMed1997

The probability of fixation in populations of changing size.

The rate of adaptive evolution of a population ultimately depends on the rate of incorporation of beneficial mutations. Even beneficial mutations may, however, be lost from a population since mutant individuals may, by chance, fail to reproduce. In this paper, we calculate the probability of fixation of beneficial mutations that occur in populations of changing size. We examine a number of demographic models, including a population whose size changes once, a population experiencing exponential growth or decline, one that is experiencing logistic growth or decline, and a population that fluctuates in size. The results are based on a branching process model but are shown to be approximate solutions to the diffusion equation describing changes in the probability of fixation over time. Using the diffusion equation, the probability of fixation of deleterious alleles can also be determined for populations that are changing in size. The results developed in this paper can be used to estimate the fixation flux, defined as the rate at which beneficial alleles fix within a population. The fixation flux measures the rate of adaptive evolution of a population and, as we shall see, depends strongly on changes that occur in population size.

Alleles↗
PubMed2003

Probability of fixation of an advantageous mutant in a viral quasispecies.

The probability that an advantageous mutant rises to fixation in a viral quasispecies is investigated in the framework of multitype branching processes. Whether fixation is possible depends on the overall growth rate of the quasispecies that will form if invasion is successful rather than on the individual fitness of the invading mutant. The exact fixation probability can be calculated only if the fitnesses of all potential members of the invading quasispecies are known. Quasispecies fixation has two important characteristics: First, a sequence with negative selection coefficient has a positive fixation probability as long as it has the potential to grow into a quasispecies with an overall growth rate that exceeds that of the established quasispecies. Second, the fixation probabilities of sequences with identical fitnesses can nevertheless vary over many orders of magnitudes. Two approximations for the probability of fixation are introduced. Both approximations require only partial knowledge about the potential members of the invading quasispecies. The performance of these two approximations is compared to the exact fixation probability on a network of RNA sequences with identical secondary structure.

Animals↗
PubMed1974

The survival of mutants at very low frequencies in tribolium populations.

Forty population cages, each with 499 adult T. castaneum of the wild-type UPF strain, received a bb female newly mated with UPF males. Half of the immigrants had a Chicago Black genetic background, the other half a UPF background. These conditions simulate, respectively, the fate of a rare, genetically differing immigrant or the fate of a mutation in populations of considerable size. Adults were censused for 11 discrete generations. The semi-dominant autosomal black gene survived in 26 out of 40 cultures by the end of the experiment, demonstrating its selective advantage at these very low frequencies. The gene increased from an initial frequency of 0.002 to 0.055 (at generation 11) in at least one replicate. Although frequency-dependent fitness has been shown for black at higher frequencies, no such dependence could be demonstrated at the low frequencies of this study. The cultures simulating mutations (immigrants with native backgrounds) had a higher average gene frequency, different distribution of gene frequencies across replicates, and a lower extinction rate of black than did the cultures with alien background immigrants. The observations only partially fitted expectation based on a branching process model. The data show a tendency for the persistence of a few heterozygotes in cultures and for a deficiency of cultures that lost the mutant or those with many heterozygotes. The increase in frequency of black cannot be attributed to increased reproductive success of heterozygotes. The advantage of heterozygotes appears due to delayed developmental period as a result of tactile stimulation and probable differential cannibalism among pupae.

Animals↗
PubMed2000

Functional analysis of ARHGAP6, a novel GTPase-activating protein for RhoA.

Microphthalmia with linear skin defects (MLS) is an X-linked dominant, male-lethal syndrome characterized by microphthalmia, aplastic skin and agenesis of the corpus callosum, and is caused by the deletion of a 500 kb critical region in Xp22.3. Our laboratory isolated a novel rho GTPase-activating protein (rhoGAP) gene named ARHGAP6 from the MLS region. ARHGAP6 contains 14 exons encoding a 974 amino acid protein with three putative SH3-binding domains. Because exons 2-14 are deleted in all MLS patients, we hypothesized that ARHGAP6 may be responsible for some of the phenotypic features of MLS. We pursued two approaches to study the function of ARHGAP6 and its role in the pathogenesis of MLS: gene targeting of the rhoGAP domain in mouse embryonic stem cells and in vitro expression studies. Surprisingly, loss of the rhoGAP function of Arhgap6 does not cause any detectable phenotypic or behavioral abnormalities in the mutant mice. Transfected mammalian cells expressing ARHGAP6 lose their actin stress fibers, retract from the growth surface and extend thin, branching processes resembling filopodia. The ARHGAP6 protein co-localizes with actin filaments through an N-terminal domain and recruits F-actin into the growing processes. Mutation of a conserved arginine residue in the rhoGAP domain prevents the loss of stress fibers but has little effect on process outgrowth. These results suggest that ARHGAP6 has two independent functions: one as a GAP with specificity for RhoA and the other as a cytoskeletal protein that promotes actin remodeling.

Actins↗
PubMed1985

The history of a lethal X-linked mutant.

Much attention has recently been focused upon the role of linkage analysis in genetic counselling, and enthusiasm has led to some misleading claims. While linkage analysis will undoubtedly play an important role for traits where there are extensive pedigrees of cases, it can achieve nothing where each case is the sole appearance of a mutation. The situation for a lethal recessive X-linked trait lies, in probabilistic terms, close to this latter extreme, particularly in the current world of small families. In view of current practice and research, it seems important to have a precise quantitative assessment of the patterns of carriers and cases generated by such an X-linked lethal mutation, with particular reference to the problem of Duchenne muscular dystrophy. In this paper a branching-process analysis is used to provide such an assessment. This approach has the advantage of requiring neither possibly unrealistic equilibrium assumptions nor assumptions about mutation rates. It also allows the effects of a variety of family size distributions to be analysed. It is found that an X-linked recessive lethal mutation produces few cases (if any), and will do so within a very few generations of its occurrence. Within the usual range of modern family size distributions the mean time to appearance is remarkably constant. The mean number of carriers at the time of the first case is small, as is the expected number of future cases. Except where the family size distribution has large variance, a high proportion of cases are the first in their family.

Biometry↗
PubMed1990

Epidemics in heterogeneous populations: II. Nonexponential incubation periods and variable infectiousness.

Two stochastic models for the spread of an infection through a heterogeneous population are considered. First, we consider a model where the incubation period has an increasing hazard rate but constant infectiousness; in the second model, the incubation period is the sum of p exponentially distributed stages, each with its own mean and level of infectiousness. By using multitype birth-death and branching processes as approximations to each epidemic model, it is shown that the epidemics initially have underlying exponential growth. Furthermore, the growth rate theta is an increasing function of the Frobenius root of the matrix of reproductive ratios R0. The results have applications in long-term sensitivity analyses, model fitting, and the determination of optimal vaccination strategies.

Epidemiologic Methods↗
PubMed2001

Activation of NG2-positive oligodendrocyte progenitor cells during post-ischemic reperfusion in the rat brain.

This study examines the alteration of oligodendrocyte progenitor cells which express membrane NG2 chondroitin sulfate proteoglycan after focal ischemia in the rat brain. Adult male Sprague-Dawley rats were subjected to 90 min occlusion of the middle cerebral artery, followed by reperfusion time of up to 2 weeks. The distribution and morphological changes in NG2-positive oligodendrocyte progenitor cells were immunohistochemically examined. Stellate-shaped NG2-positive cells with multiple branched processes were detected in both the gray and white matter of normal brain. After 2 weeks of reperfusion, NG2-positive cells in the area surrounding the infarction site (peri-infarct area) clearly showed enlarged cell bodies with hypertrophied processes. These stained strongly for NG2. Although the number of NG2-positive cells was increased significantly in the peri-infarct area, it decreased markedly in the infarct core compared to controls. Double immunostaining revealed that these NG2-positive cells were neither astrocytes nor microglia, but NG2-positive oligodendrocyte progenitor cells. These progenitor cells are known to differentiate into oligodendrocytes. As such, this upregulation of NG2 expression may be an adaptive mechanism attempting to remyelinate rat brain tissue after ischemic insult. Only further study will elucidate this hypothesis.

Animals↗
PubMed1997

Survival probability of drug resistant mutants in malaria parasites.

This study predicts the ultimate probability of survival of a newly arisen drug resistant mutant in a population of malaria parasites, with a view to understanding what conditions favour the evolution of drug resistance. Using branching process theory and a population genetics transmission model, the probabilities of survival of one- and two-locus new mutants are calculated as functions of the degree of drug pressure, the mean and variation in transmission rate, and the degree of natural selection against the mutant. Probability of survival increases approximately linearly with drug pressure, the slope of the line increasing with mean transmission rate. Thus increased drug pressure, especially in combination with high transmission rates, strongly favours the evolution of drug resistance. These conclusions also hold for the case of multiple drug resistance where it is coded for by two unlinked loci: the greater effective recombination breakdown in high transmission areas is counteracted by greater effective selection so that the net effect of higher transmission rates is to favour the evolution of multiple drug resistance. High variability in transmission rate and natural selection against the mutants are unfavourable to mutant survival, though these are relatively weak forces.

Animals↗
PubMed2003

Evolutionary dynamics of escape from biomedical intervention.

Viruses, bacteria, eukaryotic parasites, cancer cells, agricultural pests and other inconvenient animates have an unfortunate tendency to escape from selection pressures that are meant to control them. Chemotherapy, anti-viral drugs or antibiotics fail because their targets do not hold still, but evolve resistance. A major problem in developing vaccines is that microbes evolve and escape from immune responses. The fundamental question is the following: if a genetically diverse population of replicating organisms is challenged with a selection pressure that has the potential to eradicate it, what is the probability that this population will produce escape mutants? Here, we use multi-type branching processes to describe the accumulation of mutants in independent lineages. We calculate escape dynamics for arbitrary mutation networks and fitness landscapes. Our theory shows how to estimate the probability of success or failure of biomedical intervention, such as drug treatment and vaccination, against rapidly evolving organisms.

Biological Evolution↗
PubMed1992

The insertion sequence IS200 fingerprints chromosomal genotypes and epidemiological relationships in Salmonella heidelberg.

In Salmonella heidelberg the copy number of the Salmonella-specific insertion element IS200 was found to vary from four to six. All strains tested contained at least one common insertion site which was serovar specific, and most strains contained three common sites. Concurrent analysis of plasmids indicated that all insertion sequence copies were chromosomally located, and also supported the equivalence of an IS200 fingerprint and clonality. Seven intra-serovar clonal lines were thereby identified. One of these was associated with human infections, including septicaemias. Another was associated with chicken as a host: all these strains also carried a unique plasmid of 23 MDa, which was typed as a member of the IncX group. The chromosomal fingerprint of a third clone showed it to be a descendant of the chicken line marked by a single IS200 transposition. One or two representatives of four other clonal lines were identified. These lines of S. heidelberg could be related by divergent evolution, and the most recent relatives conformed to a continuous branching process model of IS200 transposition. This insertion sequence provided a highly discriminatory molecular marker of the S. heidelberg chromosome, and two of the seven clonal lines so identified were associated with distinct clinical/epidemiological contexts.

Animals↗
PubMed1999

Dynamical real space renormalization group applied to sandpile models.

A general framework for the renormalization group analysis of self-organized critical sandpile models is formulated. The usual real space renormalization scheme for lattice models when applied to nonequilibrium dynamical models must be supplemented by feedback relations coming from the stationarity conditions. On the basis of these ideas the dynamically driven renormalization group is applied to describe the boundary and bulk critical behavior of sandpile models. A detailed description of the branching nature of sandpile avalanches is given in terms of the generating functions of the underlying branching process.

Journal Article↗
PubMed2000

Boundary spatiotemporal correlations in a self-organized critical model of punctuated equilibrium

In a semi-infinite geometry, a one-dimensional, M-component model of biological evolution realizes microscopically an inhomogeneous branching process for M-->infinity. This implies a size distribution exponent tau(')=7/4 for avalanches starting at a free, "dissipative" end of the evolutionary chain. A bulklike behavior with tau(')=3/2 is restored by "conservative" boundary conditions. These are such as to strictly fix to its critical, bulk value the average number of species directly involved in an evolutionary avalanche by the mutating species located at the chain end. A two-site correlation function exponent tau(')(R)=4 is also calculated exactly in the "dissipative" case, when one of the points is at the border. Together with accurate numerical determinations of the time recurrence exponent tau(')(first), these results show also that, no matter whether dissipation is present or not, boundary avalanches have the same space and time fractal dimensions as those in the bulk, and their distribution exponents obey the basic scaling laws holding there.

Journal Article↗