Sequential Markov coalescent algorithms for population models with demographic structure

We analyse sequential Markov coalescent algorithms for populations with demographic structure: for a bottleneck model, a population-divergence model, and for a two-island model with migration. The sequential Markov coalescent method is an approximation to the coalescent suggested by McVean and Cardin, and by Marjoram and Wall. Within this algorithm we compute, for two individuals randomly sampled from the population, the correlation between times to the most recent common ancestor and the linkage probability corresponding to two different loci with recombination rate R between them. These quantities characterise the linkage between the two loci in question. We find that the sequential Markov coalescent method approximates the coalescent well in general in models with demographic structure. An exception is the case where individuals are sampled from populations separated by reduced gene flow. In this situation, the correlations may be significantly underestimated. We explain why this is the case.     

The joint allele frequency spectrum of multiple populations: a coalescent theory approach

The allele frequency spectrum is a series of statistics that describe genetic polymorphism, and is commonly used for inferring population genetic parameters and detecting natural selection. Population genetic theory on the allele frequency spectrum for a single population has been well studied using both coalescent theory and diffusion equations. Recently, the theory was extended to the joint allele frequency spectrum (JAFS) for three populations using diffusion equations and was shown to be very useful in inferring human demographic history. In this paper, I show that the JAFS can be analytically derived with coalescent theory for a basic model of two isolated populations and then extended to multiple populations and various complex scenarios, such as those involving population growth and bottleneck, migration, and positive selection. Simulation study is used to demonstrate the accuracy and applicability of the theoretical model. The coalescent theory-based approach for the JAFS can characterize the demographic history with comprehensive statistical models as the diffusion approach does, and in addition gains several novel advantages: the computational complexity of calculating the JAFS with coalescent theory is reduced, and thus it is feasible to analytically obtain the JAFS for multiple populations; the hitchhiking effect can be efficiently modeled in coalescent theory, enabling the development of methodologies for detecting selection via multi-population polymorphism data. As an alternative to the diffusion approximation approach, the coalescent theory for the JAFS also provides a foundation for population genetic inference with the advent of large-scale genomic polymorphism data.     

The coalescent and the genealogical process in geographically structured population

We shall extend Kingman's coalescent to the geographically structured population model with migration among colonies. It is described by a continuous-time Markov chain, which is proved to be a dual process of the diffusion process of stepping-stone model. We shall derive a system of equations for the spatial distribution of a common ancestor of sampled genes from colonies and the mean time to getting to one common ancestor. These equations are solved in three particular models; a two-population model, the island model and the one-dimensional stepping-stone model with symmetric nearest-neighbour migration.     

Proteins as micro viscosimeters: Brownian motion revisited

Translational and rotational diffusion coefficients of proteins in solution strongly deviate from the Stokes–Einstein laws when the ambient viscosity is induced by macromolecular co-solutes rather than by a solvent of negligible size as was assumed by A. Einstein one century ago for deriving the laws of Brownian motion and diffusion. Rotational and translational motions experience different micro viscosities and both become a function of the size ratio of protein and macromolecular co-solute. Possible consequences upon fluorescence spectroscopy observations of diffusing proteins within living cells are discussed.     

A vectorized method of importance sampling with applications to models of mutation and migration

An importance-sampling method is presented for computing the likelihood of the configuration of population genetic data under general assumptions about population history and transitions among states. The configuration of the data is the number of chromosomes sampled that are in each of a finite set of states. Transitions among states are governed by a Markov chain with transition probabilities dependent on one or more parameters. The method assumes that the joint distribution of coalescence times of the underlying gene genealogy is independent of the genetic state of each lineage. Given a set of coalescence times, the probability that a pair of lineages is chosen to coalesce in each replicate is proportional to the contribution that the coalescence event makes to the probability of the data. This method can be applied to gene genealogies generated by the neutral coalescent process and to genealogies generated by other processes, such as a linear birth-death process which provides a good approximation to the dynamics of low-frequency alleles. Two applications are described. In the first, the fit of allele frequencies at two microsatellite loci sampled in a Sardinian population to the one-step mutation model is tested. The one-step model is rejected for one locus but not for the other. The second application is to low-frequency alleles in a geographically subdivided population. The geographic location is the allelic state, and the alleles are assumed to be sufficiently rare that their dynamics can be approximated by a linear birth-death process in which the birth and death rates are independent of geographic location. The analysis of eight low-frequency allozyme alleles found in the glaucous-winged gull, Larus glaucescens, illustrates how geographically restricted dispersal can be detected.     

Stochastic growth and extinction in a spatial geometric Brownian population model with migration and correlated noise

A continuous spatial model for populations that are not density-regulated is analyzed. The model is a generalization of the geometric Brownian motion often used to describe populations at a single location. The locations are linked by migration and spatial correlation in the noise. At any point of time, the population size at a given location is log normally distributed so the log population size constitutes a Gaussian field. The model is homogeneous in space but not in time. In particular, we analyze the case when the stochastic growth rate is negative and the total population approaches extinction. The properties of the extinction process is studied by considering local quasi-extinctions. A major conclusion is that migration tends to increase the time to extinction provided that there is no cost of migration. However, as the area occupied by the species starts to decrease, the decrease is faster for populations with larger migration.     

Optimisation of NMR dynamic models II. A new methodology for the dual optimisation of the model-free parameters and the Brownian rotational diffusion tensor

Finding the dynamics of an entire macromolecule is a complex problem as the model-free parameter values are intricately linked to the Brownian rotational diffusion of the molecule, mathematically through the autocorrelation function of the motion and statistically through model selection. The solution to this problem was formulated using set theory as an element of the universal set —the union of all model-free spaces (d’Auvergne EJ and Gooley PR (2007) Mol BioSyst 3(7), 483–494). The current procedure commonly used to find the universal solution is to initially estimate the diffusion tensor parameters, to optimise the model-free parameters of numerous models, and then to choose the best model via model selection. The global model is then optimised and the procedure repeated until convergence. In this paper a new methodology is presented which takes a different approach to this diffusion seeded model-free paradigm. Rather than starting with the diffusion tensor this iterative protocol begins by optimising the model-free parameters in the absence of any global model parameters, selecting between all the model-free models, and finally optimising the diffusion tensor. The new model-free optimisation protocol will be validated using synthetic data from Schurr JM et al. (1994) J Magn Reson B 105(3), 211–224 and the relaxation data of the bacteriorhodopsin (1–36)BR fragment from Orekhov VY (1999) J Biomol NMR 14(4), 345–356. To demonstrate the importance of this new procedure the NMR relaxation data of the Olfactory Marker Protein (OMP) of Gitti R et al. (2005) Biochem 44(28), 9673–9679 is reanalysed. The result is that the dynamics for certain secondary structural elements is very different from those originally reported. Electronic supplementary material The online version of this article (doi:) contains supplementary material, which is available to authorized users.     

Genetic diversity and dispersal of Phragmites australis in a small river system

Even though the reed, Phragmites australis, is an extensively studied wetland species, little is known about reproduction and dispersal modes within and among reed populations at the scale of small river systems. Using microsatellite analysis of 189 individuals from three adjacent river catchments in the Czech Republic, we elucidated the role of the river corridors in the dispersal of P. australis. Using Bayesian clustering of individuals, we found that 19% of clusters were distributed only along one river, which implied dispersal by water (or by wind) along river corridors, whereas 38% of clusters were widely distributed and were likely the product of wind long-distance dispersal among rivers. Intensive exchange of propagules among river systems is further demonstrated by only 6% of total variance being attributed to the variance among rivers in the AMOVA-analysis. Spatial autocorrelation analysis revealed a decreasing pattern up to 5–10 km and no clear pattern over longer distances. This gives an evidence for pollen and seed dispersal at short distances (up to 1 km), whereas most likely only seed dispersal at longer distances up to 10 km. We found five multilocus genotypes distributed in two different populations. The distances between populations with the same genotype ranged from 0.5 to 10.8 km. This can be interpreted as long-distance vegetative dispersal.     

Dynamic Monte Carlo simulation of non-equilibrium Brownian diffusion of single-chain macromolecules

We performed dynamic Monte Carlo simulations of biased diffusion of 3D phantom single lattice polymer. We observed spontaneous deformation of polymer coil when the external driving forces exceed a critical strength. In addition, longer chains require lower critical strengths, at which their activated velocities deviate from Newtonian-fluid behaviours and merge into a master curve exhibiting shear-thinning followed with shear thickening. We attributed the cause of deformation to the random updating of monomers. The latter represents the dynamic heterogeneity along the real polymer chain, and raises a nonlinear asymmetric accumulation of local acceleration and then an internal tension between chain middle and chain end, as evidenced by our previous Brownian Dynamics simulations. Our results unravel a single-molecular-level source of nonlinear dynamics, which has been overlooked in current theoretical considerations on the basis of Rouse ideal-chain model.     

Spatially explicit neutral models for population genetics and community ecology: Extensions of the Neyman-Scott clustering process

Spatially explicit models relating to plant populations have developed little since Felsenstein (1975) pointed out that if limited seed dispersal causes clustering of individuals, such models cannot reach an equilibrium. This paper aims to resolve this issue by modifying the Neyman-Scott cluster point process. The new point processes are dynamic models with random immigration, and the continuous increase in the clustering of individuals stops at some level. Hence, an equilibrium state is achieved, and new individual-based spatially explicit neutral coalescent models are established. By fitting the spatial structure at equilibrium to individual spatial distribution data, we can indirectly estimate seed dispersal and effective population density. These estimates are improved when genetic data are available, and become even more sophisticated if spatial distribution and genetic data pertaining to the offspring are also available.     

Island models and the coalescent process

Using a coalescent approach, we derive several classical results and extend them to more general models. We find that the classic result for constant population size and constant migration rates holds in models with varying population size and varying migration rates with the obvious substitution of effective population size and mean migration fraction. In addition, the relationship of a 'local' F _ST to local gene flow is derived. This result may be useful for analysing gene flow in a regional subset of a large global population, using only data from the regional subset.     

Modelling pairwise dependence of maxima in space

We model pairwise dependence of temporal maxima, such as annualmaxima of precipitation, that have been recorded in space, eitheron a regular grid or at irregularly spaced locations. The constructionof our estimators stems from the variogram concept. The asymptoticproperties of our pairwise dependence estimators are establishedthrough properties of empirical processes. The performance ofour approach is illustrated by simulations and by the treatmentof a real dataset. In addition to bringing new results aboutthe asymptotic behaviour of copula estimators, the latter beinglinked to first-order variograms, one main advantage of ourapproach is to propose a simple connection between extreme valuetheory and geostatistics.     

Clades, clans, and reciprocal monophyly under neutral evolutionary models

The Yule model and the coalescent model are two neutral stochastic models for generating trees in phylogenetics and population genetics, respectively. Although these models are quite different, they lead to identical distributions concerning the probability that pre-specified groups of taxa form monophyletic groups (clades) in the tree. We extend earlier work to derive exact formulae for the probability of finding one or more groups of taxa as clades in a rooted tree, or as ‘clans’ in an unrooted tree. Our findings are relevant for calculating the statistical significance of observed monophyly and reciprocal monophyly in phylogenetics.