Stochastic spread of Wolbachia

Wolbachia are very common, maternally transmitted endosymbionts of insects. They often spread by a mechanism termed cytoplasmic incompatibility (CI) that involves reduced egg hatch when Wolbachia-free ova are fertilized by sperm from Wolbachia-infected males. Because the progeny of Wolbachia-infected females generally do not suffer CI-induced mortality, infected females are often at a reproductive advantage in polymorphic populations. Deterministic models show that Wolbachia that impose no costs on their hosts and have perfect maternal transmission will spread from arbitrarily low frequencies (though initially very slowly); otherwise, there will be a threshold frequency below which Wolbachia frequencies decline to extinction and above which they increase to fixation or a high stable equilibrium. Stochastic theory was used to calculate the probability of fixation in populations of different size for arbitrary current frequencies of Wolbachia, with special attention paid to the case of spread after the arrival of a single infected female. Exact results are given based on a Moran process that assumes a specific demographic model, and approximate results are obtained using the more general Wright-Fisher theory. A new analytical approximation for the probability of fixation is derived, which performs well for small population sizes. The significance of stochastic effects in the natural spread of Wolbachia and their relevance to the use of Wolbachia as a drive mechanism in vector and pest management are discussed.     

Dynamics of neutral and selected alleles when the offspring distribution is skewed

We analyze the dynamics of two alternative alleles in a simple model of a population that allows for large family sizes in the distribution of offspring number. This population model was first introduced by Eldon and Wakeley, who described the backward-time genealogical relationships among sampled individuals, assuming neutrality. We study the corresponding forward-time dynamics of allele frequencies, with or without selection. We derive a continuum approximation, analogous to Kimura's diffusion approximation, and we describe three distinct regimes of behavior that correspond to distinct regimes in the coalescent processes of Eldon and Wakeley. We demonstrate that the effect of selection is strongly amplified in the Eldon-Wakeley model, compared to the Wright-Fisher model with the same variance effective population size. Remarkably, an advantageous allele can even be guaranteed to fix in the Eldon-Wakeley model, despite the presence of genetic drift. We compute the selection coefficient required for such behavior in populations of Pacific oysters, based on estimates of their family sizes. Our analysis underscores that populations with the same effective population size may nevertheless experience radically different forms of genetic drift, depending on the reproductive mechanism, with significant consequences for the resulting allele dynamics.     

Spatial Stochastic Models for Cancer Initiation and Progression

The multistage carcinogenesis hypothesis has been formulated by a number of authors as a stochastic process. However, most previous models assumed “perfect mixing” in the population of cells, and included no information about spatial locations. In this work, we studied the role of spatial dynamics in carcinogenesis. We formulated a 1D spatial generalization of a constant population (Moran) birth–death process, and described the dynamics analytically. We found that in the spatial model, the probability of fixation of advantageous and disadvantageous mutants is lower, and the rate of generation of double-hit mutants (the so-called tunneling rate) is higher, compared to those for the space-free model. This means that the results previously obtained for space-free models give an underestimation for rates of cancer initiation in the case where the first event is the generation of a double-hit mutant, e.g. the inactivation of a tumor-suppressor gene.     

Evolution of resistance and progression to disease during clonal expansion of cancer

Inspired by previous work of Iwasa et al. (2006) and Haeno et al. (2007), we consider an exponentially growing population of cancerous cells that will evolve resistance to treatment after one mutation or display a disease phenotype after two or more mutations. We prove results about the distribution of the first time when k mutations have accumulated in some cell, and about the growth of the number of type-k cells. We show that our results can be used to derive the previous results about a tumor grown to a fixed size.     

On the Stochastic Theory of Compartments: The Two-Compartment and the Mammillary Compartment Systems

A two-compartment and a mammillary compartment system with arbitrary sojourn time distributions for each of the compartments is analysed in this paper, both with and without input. Several results found earlier by other authors for these systems are generalized and some new results are also added.     

Population Genetics and Molecular Evolution of DNA Sequences in Transposable Elements. I. A Simulation Framework

A population genetic simulation framework is developed to understand the behavior and molecular evolution of DNA sequences of transposable elements. Our model incorporates random transposition and excision of transposable element (TE) copies, two modes of selection against TEs, and degeneration of transpositional activity by point mutations. We first investigated the relationships between the behavior of the copy number of TEs and these parameters. Our results show that when selection is weak, the genome can maintain a relatively large number of TEs, but most of them are less active. In contrast, with strong selection, the genome can maintain only a limited number of TEs but the proportion of active copies is large. In such a case, there could be substantial fluctuations of the copy number over generations. We also explored how DNA sequences of TEs evolve through the simulations. In general, active copies form clusters around the original sequence, while less active copies have long branches specific to themselves, exhibiting a star-shaped phylogeny. It is demonstrated that the phylogeny of TE sequences could be informative to understand the dynamics of TE evolution.     

Evolution and biodiversity of Antarctic organisms: a molecular perspective

The Antarctic biota is highly endemic, and the diversity and abundance of taxonomic groups differ from elsewhere in the world. Such characteristics have resulted from evolution in isolation in an increasingly extreme environment over the last 100 Myr. Studies on Antarctic species represent some of the best examples of natural selection at the molecular, structural and physiological levels. Analyses of molecular genetics data are consistent with the diversity and distribution of marine and terrestrial taxa having been strongly influenced by geological and climatic cooling events over the last 70 Myr. Such events have resulted in vicariance driven by continental drift and thermal isolation of the Antarctic, and in pulses of species range contraction into refugia and subsequent expansion and secondary contact of genetically distinct populations or sister species during cycles of glaciation. Limited habitat availability has played a major role in structuring populations of species both in the past and in the present day. For these reasons, despite the apparent simplicity or homogeneity of Antarctic terrestrial and marine environments, populations of species are often geographically structured into genetically distinct lineages. In some cases, genetic studies have revealed that species defined by morphological characters are complexes of cryptic or sibling species. Climate change will cause changes in the distribution of many Antarctic and sub-Antarctic species through affecting population-level processes such as life history and dispersal.     

Evolution of Relapse-Proficient Subclones Constrained by Collateral Sensitivity to Oncogene Overdose in Wnt-Driven Mammary Cancer

1. Download high-res image (186KB)
2. Download full-size image

     

Evolution of population structure in a highly social top predator, the killer whale

Intraspecific resource partitioning and social affiliations both have the potential to structure populations, though it is rarely possible to directly assess the impact of these mechanisms on genetic diversity and population divergence. Here, we address this for killer whales (Orcinus orca), which specialize on prey species and hunting strategy and have long-term social affiliations involving both males and females. We used genetic markers to assess the structure and demographic history of regional populations and test the hypothesis that known foraging specializations and matrifocal sociality contributed significantly to the evolution of population structure. We find genetic structure in sympatry between populations of foraging specialists (ecotypes) and evidence for isolation by distance within an ecotype. Fitting of an isolation with migration model suggested ongoing, low-level migration between regional populations (within and between ecotypes) and small effective sizes for extant local populations. The founding of local populations by matrifocal social groups was indicated by the pattern of fixed mtDNA haplotypes in regional populations. Simulations indicate that this occurred within the last 20,000 years (after the last glacial maximum). Our data indicate a key role for social and foraging behavior in the evolution of genetic structure among conspecific populations of the killer whale.     

普通群体基因频率逐代演变的一个不变量与平衡区域

建立普通群体基因型频率逐代演变规律的数学模型,发现在各代中RR与rr的频率的差为不变量,它与平衡状态的Rr的频率一一对应。在育种工作中这个不变量是可观测的,作物选种即是人为改变群体基因频率,因此它有一定的参考意义。获得由第一代基因型频率计算平衡状态基因型频率的公式,推导出平衡状态基因型频率与异交率的换算方法。     

种子与花粉的随机迁移对植物群体遗传结构分化的影响

将Ｗｒｉｇｈｔ的经典岛屿模型拓广到植物群体上,同时考虑了含有花粉和种子随机迁移的影响。并给出了３种不同遗传方式的基因（双亲遗传,父本和母本遗传）频率的期望均值和方差。理论结果证明花粉或种子的随机迁移可增加基因频率方差,其幅度取决于迁移率和迁移基因频率的方差。同绝对迁移率一样,花粉和种子的迁移率方差及迁移基因频率的方差对群体遗传结构的分化有着同样的重要。一个重要结论就是花粉或种子的随机迁移率和随机迁     

Evolution of genes and taxa: a primer

The rapidly growing fields of molecular evolution and systematics have much to offer to molecular biology, but like any field have their own repertoire of terms and concepts. Homology, for example, is a central theme in evolutionary biology whose definition is complex and often controversial. Homology extends to multigene families, where the distinction between orthology and paralogy is key. Nucleotide sequence alignment is also a homology issue, and is a key stage in any evolutionary analysis of sequence data. Models based on our understanding of the processes of nucleotide substitution are used both in the estimation of the number of evolutionary changes between aligned sequences and in phylogeny reconstruction from sequence data. The three common methods of phylogeny reconstruction – parsimony, distance and maximum likelihood – differ in their use of these models. All three face similar problems in finding optimal – and reliable – solutions among the vast number of possible trees. Moreover, even optimal trees for a given gene may not reflect the relationships of the organisms from which the gene was sampled. Knowledge of how genes evolve and at what rate is critical for understanding gene function across species or within gene families. The Neutral Theory of Molecular Evolution serves as the null model of molecular evolution and plays a central role in data analysis. Three areas in which the Neutral Theory plays a vital role are: interpreting ratios of nonsynonymous to synonymous nucleotide substitutions, assessing the reliability of molecular clocks, and providing a foundation for molecular population genetics.     

Biological applications of the theory of birth-and-death processes

In this review, we discuss applications of the theory of birth-and-death processes to problems in biology, primarily, those of evolutionary genomics. The mathematical principles of the theory of these processes are briefly described. Birth-and-death processes, with some straightforward additions such as innovation, are a simple, natural and formal framework for modeling a vast variety of biological processes such as population dynamics, speciation, genome evolution, including growth of paralogous gene families and horizontal gene transfer and somatic evolution of cancers. We further describe how empirical data, e.g. distributions of paralogous gene family size, can be used to choose the model that best reflects the actual course of evolution among different versions of birth-death-and-innovation models. We conclude that birth-and-death processes, thanks to their mathematical transparency, flexibility and relevance to fundamental biological processes, are going to be an indispensable mathematical tool for the burgeoning field of systems biology.     

Conservation genetics of Amorpha georgiana (Fabaceae), an endangered legume of the Southeastern United States

Amorpha georgiana (Fabaceae) is an endangered legume species found in longleaf pine savannas in the Southeastern United States. Approximately 900 individuals and 14 populations remain, most of which are concentrated in North Carolina. Eleven microsatellite loci were used to explore genetic diversity, population structure and recent population bottlenecks using genotypic data from 132 individuals collected at ten different localities. Although A. georgiana is quite rare, it exhibited high levels of genetic diversity (17.7 alleles/locus; H _o = 0.65, H _E = 0.75). Most of the genetic variation was found within rather than between populations of this species. The single remaining Georgia population was well differentiated from populations of the Carolinas ( F _ST > 0.1), which had weaker structure among them ( F _ST < 0.1). Only a geographically disjunct population showed strong evidence of a recent population bottleneck, perhaps due to a recent founder event. Hybridization with A. herbacea was also detected. For conservation management plans, A. georgiana populations in each geographic region (North Carolina, South Carolina and Georgia) plus a disjunct population in North Carolina (Holly Shelter) should be treated as separate management units for which in situ conservation, including habitat restoration and use of prescribed burns, should ensure persistence of this species and preservation of its evolutionary potential.     

Demographic and genetic interrelationships among the Gavlis of Dharwad

Lot of work has been done in recent years on the genetics of isolated and small population groups. But J. Sutter (1963) notes that these studies have not yielded satisfactory results, because these investigators have applied the formulae and models constructed by the mathematicians which are based on the assumption of panmixia, whereas panmixia cannot occur in human populations especially if the population is very small. Sometimes we speak of genetic drift and selection without taking into account the fact that the population at the same time is controlled by two most important demographic parameters of fertility and mortality which can alter genetic drift and selection. The geneticists are primarily interested in fertility. They want to determine, for any given couple, the number of offspring reaching the age of reproduction. One might therefore assume that the measurement of fertility should play a major role in population genetics. Thus, there is an urgent need for the establishment of meaningful relationship between demography and population genetics. In view of the above facts, an attempt is made in the present study to analyse the “Demographic and Genetic Interrelationships among the Gavlis of Dharwad” so as to throw light on some of the complex genetic issues like endogamy, inbreeding and selection potential.     

Modeling the Evolution of Insect Phenology

Climate change is likely to disrupt the timing of developmental events (phenology) in insect populations in which development time is largely determined by temperature. Shifting phenology puts insects at risk of being exposed to seasonal weather extremes during sensitive life stages and losing synchrony with biotic resources. Additionally, warming may result in loss of developmental synchronization within a population making it difficult to find mates or mount mass attacks against well-defended resources at low population densities. It is unknown whether genetic evolution of development time can occur rapidly enough to moderate these effects. We present a novel approach to modeling the evolution of phenology by allowing the parameters of a phenology model to evolve in response to selection on emergence time and density. We use the Laplace method to find asymptotic approximations for the temporal variation in mean phenotype and phenotypic variance arising in the evolution model that are used to characterize invariant distributions of the model under periodic temperatures at leading order. At these steady distributions the mean phenotype allows for parents and offspring to be oviposited at the same time of year in consecutive years. Numerical simulations show that populations evolve to these steady distributions under periodic temperatures. We consider an example of how the evolution model predicts populations will evolve in response to warming temperatures and shifting resource phenology.     

Evolution of linkage disequilibrium of the founders in exponentially growing populations

Evolution of linkage disequilibrium of the founders in exponentially growing populations was studied using a time-inhomogeneous It? process model. The model is an extension of the diffusion approximation of the Wright-Fisher model. As a measure of linkage disequilibrium, the squared standard linkage deviation, which is defined by a ratio of the moments, was considered. A system of ordinary differential equations that these moments obey was obtained. This system can be solved numerically. By simulations, it was shown that the squared standard linkage deviation gives a good approximation of the expectation of the squared correlation coefficient of gamete frequencies. In addition, a perturbative solution was obtained when the growth rate is not large. By using the perturbation, an asymptotic formula for the squared standard linkage deviation after a large number of generations was obtained. According to the formula, the squared standard linkage deviation tends to be 1/(4Nc), where N is the current size of the population and c is the recombination fraction between two loci. It is dependent on neither the initial effective size, the growth rate, nor the mutation rate. In exponentially growing populations, linkage disequilibrium will be asymptotically the same as that in a constant size population, the effective size of which is the current effective size.     

Stochastic analysis of the relationships between saturated hydraulic conductivity variance and solute dispersion in heterogeneous soils

The effect of three‐dimensional heterogeneity of saturated hydraulic conductivity on the vertical transport of solutes in soils is examined by means of controlled numerical experiments. Saturated hydraulic conductivity, an important transport parameter that controls the dispersion of pollutants in heterogeneous soils, is assumed to be composed of a homogeneous mean value and a perturbation caused by the vertical variability of soil properties, producing a stochastic process in the mean flow direction. The spatial heterogeneity of porous soils is characterized by the variance and correlation scale of the saturated hydraulic conductivity in the transport domain. Numerical experiments are carried out to evaluate the extent of contaminant dispersion in Hawaiian Oxic soils when uncertainty exists as a result of the spatial heterogeneity of saturated hydraulic conductivity. Statistical analysis of the saturated hydraulic conductivity measurements on undisturbed soil cores from two locations in Hawaiian Oxic soils indicated two different soils with the same mean and different variances. The partial differential equations describing three‐dimensional transient flow and solute transport in soils with a random conductivity field were solved to evaluate the effect of these two variance levels on the transport of a contaminant plume originating from the surface. The significance of the variance on the spatial and temporal distribution of tracer concentrations is demonstrated using solute breakthrough curves at various depths in the soil profile. The longitudinal macrodispersivity resulting from tracer spreading in the heterogeneous soils with a finite local dispersivity is also analyzed. The analysis shows a similar solute dispersion behavior for the two variances. However, there is an overall reduction in the dispersion of solutes resulting from a uniform velocity field with the same mean. Macrodispersivity values in heterogeneous soils are proportional to the variance at smaller travel distances but converge to the same value at larger travel distances.