On the proportion of cancer stem cells in a tumour

It is now generally accepted that cancers contain a sub-population, the cancer stem cells (CSCs), which initiate and drive a tumour’s growth. At least until recently it has been widely assumed that only a small proportion of the cells in a tumour are CSCs. Here we use a mathematical model, supported by experimental evidence, to show that such an assumption is unwarranted. We show that CSCs may comprise any possible proportion of the tumour, and that the higher the proportion the more aggressive the tumour is likely to be.     

The distribution of the proportion of the genome which is homozygous by descent in inbred individuals.

The expected proportion of loci which are homozygous by descent in an inbred individual is the inbreeding coefficient of that individual. Confidence limits on this proportion depend on the number of loci controlling the trait under investigation and the linkage relationships among the loci. Such information is generally not available. If it is assumed that the number of loci is large, the variance of the proportion of homozygous loci is approximated by the variance of the total map length which is homozygous by descent; this is calculated for several regular mating systems, including selfing, sib mating, and parent-off-spring mating. In particular, the confidence interval on the proportion of homozygous loci in Drosophila is likely to be large, and this has a number of implications to the interpretation and analysis of experimental data.     

On the two-locus sampling distribution

Two methods are discussed for evaluating the distribution of the configuration of unlabeled gametic types in a random sample of size n from the two-locus infinitely-many-neutral-alleles diffusion model at stationarity. Both involve finding systems of linear equations satisfied by the desired probabilities. The first approach, which is due to Golding, is to include additional probabilities in the system that allow some members of the sample to be specified at only one locus. The second approach, which is new, considers the joint distribution of the sample configuration and the number of recombination events since the time of the most recent common ancestor. The first approach is used for numerical computation, whereas the second approach is used to derive a two-locus version of Hoppe's urn model. The latter permits efficient simulation of the two-locus sampling distribution, provided the recombination parameter is not too large.Supported in part by NSF grants DMS-8704369 and DMS-8902991     

On the geographical distribution of the Juglandaceae

The present paper aims to discuss the geog raphical distribution of the Juglandaceae on the basis of unity of the phylogeny and the process of dispersal in the plants. The paper is divided into the following three parts: 1. The systematic positions and the distribution patterns of nine living genera in the family Juglandaceae (namely, Engelhardia, Oreomunnea, Alfaroa, Pterocarya, Cyclocarya, Juglans, Carya, Annamocarya and Platycarya) are briefly discussed. The evolutional relationships between the different genera of the Juglandaceae are elucidated. The fossil distribution and the geological date of the plant groups are reviewed. Through the analysis for the geographical distribution of the Juglandaceous genera, the distribution patterns may be divided as follows: A. The tropical distribution pattern a. The genera of tropical Asia distribution: Engelhardia, Annamocarya. b. The genera of tropical Central America distribution: Oreomunnea, Alfaroa. B. The temperate distribution pattern c. The genus of disjunct distribution between Western Asia and Eastern Asia: Pterocarya. d. The genus of disjunct distribution between Eurasia and America: Juglans. e. The genus of disjunct distribution between Eastern Asia and North America: Carya. f. The genera whose distribution is confined to Eastern Asia: Cyclocarya, Platycarya. 2. The distribution of species According to Takhtajan’s view point of phytochoria, the number of species in every region are counted. It has shown clearily that the Eastern Asian Region and the Cotinental South-east Asian Region are most abundant in number of genera and species. Of the 71 living species, 53 are regional endemic elements, namely 74.6% of the total species. The author is of the opinion that most endemic species in Eurasia are of old endemic nature and in America of new endimic nature. There are now 7 genera and 28 species in China, whose south-western and central parts are most abundant in species, with Province Yunnan being richest in genera and species. 3. Discussions of the distribution patterns of the Juglandaceae A. The centre of floristic region B. The centre of floristic regions is determined by the following two principles: a. A large number of species concentrate in a district, namely the centre of the majority; b. Species of a district can reflect the main stages of the systematic evolution of the Juglandaceae, namely the centre of diversity. It has shown clearly that the southern part of Eastern Asian region and the northern part of Continental South-east Asian Region (i.c. Southern China and Northern Indo-China) are the main distribution centre of the Juglandaceae, while the southern part of Sonora Region and Caribbean Region (i.c. South-western U.S.A., Mexico and Central America) are the secondary distribution centre. As far as fossil records goes, it has shown that in Tertiary period the Juglandaceae were widely distributed in northern Eurasia and North America, growing not only in Europe and the Caucasus but also as far as in Greenland and Alaska. It may be considered that the Juglandaceae might be originated from Laurasia. According to the analysis of distribution pattern for living primitive genus, for example, Engelhardia, South-western China and Northern Indo-China may be the birthplace of the most primitive Juglandaceous plants. It also can be seen that the primitive genera and the primitive sections of every genus in the Juglandaceae have mostly distributed in the tropics or subtropics. At the same time, according to the analysis of morphological characters, such as naked buds in the primitive taxa of this family, it is considered that this character has relationship with the living conditions of their ancestors. All the evidence seems to show that the Juglandaceae are of forest origin in the tropical mountains having seasonal drying period. B. The time of the origin The geological times of fossil records are analyzed. It is concluded that the origin of the Juglandaceae dates back at least as early as the Cretaceous period. C. The routes of despersal After the emergence of the Juglandaceous plant on earth, it had first developed and dispersed in Southern China and Indo-China. Under conditions of the stable temperature and humidity in North Hemisphere during the period of its origin and development, the Juglandaceous plants had rapidly developed and distributed in Eurasia and dispersed to North America by two routes: Europe-Greenland-North America route and Asia-Bering Land-bridge-North America route. From Central America it later reached South America. D. The formaation of the modern distribution pattern and reasons for this formation. According to the fossil records, the formation of two disjunct areas was not due to the origin of synchronous development, nor to the parallel evolution in the two continents of Eurasia and America, nor can it be interpreted as due to result of transmissive function. The modern distribution pattern has developed as a result of the tectonic movement and of the climatic change after the Tertiary period. Because of the continental drift, the Eurasian Continent was separated from the North American Continent, it had formed a disjunction between Eurasia and North America. Especially, under the glaciation during the Late Tertiary and Quaternary Periods, the continents in Eurasia and North America were covered by ice sheet with the exception of “plant refuges”, most plants in the area were destroyed, but the southern part of Eastern Asia remained practically intact and most of the plants including the Juglandaceae were preserved from destruction by ice and thence became a main centre of survival in the North Hemisphere, likewise, there is another centre of survival in the same latitude in North America and Central America. E. Finally, the probable evolutionary relationships of the genera of the Juglanda-ceae is presented by the dendrogram in the text.     

On estimating the proportion of variance in a phenotypic trait attributable to a measured locus

The measured genotype approach can be used to estimate the variance contributions of specific candidate loci to quantitative traits of interest. We show here that both the naive estimate of measured-locus heritability, obtained by invoking infinite-sample theory, and an estimate obtained from a bias-corrected variance estimate based on finite-sample theory, produce biased estimates of heritability. We identify the sources of bias, and quantify their effects. The two sources of bias are: (1) the estimation of heritability from population samples as the ratio of two variances, and (2) the existence of sampling error. We show that neither heritability estimator is less biased (in absolute value) than the other in all situations, and the choice of an ideal estimator is therefore a function of the sample size and magnitude of the locus-specific contribution to the overall phenotypic variance. In most cases the bias is small, so that the practical implications of using either estimator are expected to be minimal.     

On a property of the exponential distribution

If t is an independent exponentially distributed random variable, the distribution p = [t - x] is a modified geometric distribution, similar to the result of HAWKINS and KOTZ (1976), x is uniform.     

On the size distribution of live genera

This article deals with the theoretical size (number of species) distribution of live genera, arising from a simple model of macroevolution in which speciations and extinctions are assumed to occur independently and at random, and in which new genera are formed by the random splitting of existing genera. Mathematically, the distribution is that of the state of a homogeneous birth-and-death process after an exponentially distributed time. An ordinary differential equation for the generating function of the distribution is derived and solved and a recurrence relation for computing the probabilities in the distribution presented. Some properties of the distribution, including asymptotic behaviour, are examined and the distribution of the time since establishment of a genus of a given size derived. Fitting the distribution to empirical taxon size distributions by maximum likelihood is discussed and two examples are presented.     

On the statistical distribution of mutant bacteria

A problem in probability is stated with included the problem of the distribution of bacterial mutants as a special case. This problem is solved exactly but since the resulting expressions are too complicated for practical use, various approximate expressions for the distribution are considered, especially for the bacterial mutation case. Research sponsored by the Office of Naval Research while the author was at the California Institute of Technology.     

On the galactosyl distribution of commercial galactomannans

A simple method was developed that enabled the enzymatic determination of the galactose distribution in galactomannans. endo-Mannanase of Aspergillus niger was used to degrade the galactomannan polymers and the degradation products were determined with high-performance anion-exchange chromatography. A whole range of commercial high-to-low substituted galactomannans was analyzed in this way. It was found that differences in the anion-exchange chromatograms reflected dissimilarities in the distribution of galactose and could be used directly to discern these dissimilarities. The differences among the various elution profiles were used to construct a similarity distance tree. In addition to this approach, the absolute amount of non-substituted mannose released by the enzyme was found to be a good discriminating factor. In this way, galactomannans with regular, blockwise, and randomly distributed galactose could be discerned. All guars and the highly substituted gum of Prosopis juliflora were found to have a blockwise distribution of galactose. For different batches of tara gum both random and blockwise distributions were found. Among batches of locust bean gum the greatest variation was observed: both random, blockwise, and ordered galactose distributions were present. Cassia gum was found to have a highly regular distribution of galactose.     

On the distribution of plant abundance data

Plant abundance data are often analysed using standard statistical procedures without considering their distributional features and the underlying ecological processes. However, plant abundance data, e.g. when measured in biodiversity monitoring programs, are often sampled using a hierarchical sampling procedure, and since plant abundance data in a hierarchical sampling procedure are typically both zero-inflated and over-dispersed, the use of a standard statistical procedure is sub-optimal and not the best possible practice in the modelling of plant abundance data. Two distributions (the zero-inflated generalised binomial distribution and the zero-inflated bounded beta distribution) are suggested as possible distributions for analysing either discrete, continuous, or ordinal hierarchically sampled plant cover data.