Efficient estimation of SNP heritability using Gaussian predictive process in large scale cohort studies

With the advent of high throughput genetic data, there have been attempts to estimate heritability from genome-wide SNP data on a cohort of distantly related individuals using linear mixed model (LMM). Fitting such an LMM in a large scale cohort study, however, is tremendously challenging due to its high dimensional linear algebraic operations. In this paper, we propose a new method named PredLMM approximating the aforementioned LMM motivated by the concepts of genetic coalescence and Gaussian predictive process. PredLMM has substantially better computational complexity than most of the existing LMM based methods and thus, provides a fast alternative for estimating heritability in large scale cohort studies. Theoretically, we show that under a model of genetic coalescence, the limiting form of our approximation is the celebrated predictive process approximation of large Gaussian process likelihoods that has well-established accuracy standards. We illustrate our approach with extensive simulation studies and use it to estimate the heritability of multiple quantitative traits from the UK Biobank cohort.     

Some Statistical Improvements for Estimating Population Size and Mutation Rate from Segregating Sites in DNA Sequences

In population genetics, under a neutral Wright-Fisher model, the scaling parameter straight theta=4Nmu represents twice the average number of new mutants per generation. The effective population size is N and mu is the mutation rate per sequence per generation. Watterson proposed a consistent estimator of this parameter based on the number of segregating sites in a sample of nucleotide sequences. We study the distribution of the Watterson estimator. Enlarging the size of the sample, we asymptotically set a Central Limit Theorem for the Watterson estimator. This exhibits asymptotic normality with a slow rate of convergence. We then prove the asymptotic efficiency of this estimator. In the second part, we illustrate the slow rate of convergence found in the Central Limit Theorem. To this end, by studying the confidence intervals, we show that the asymptotic Gaussian distribution is not a good approximation for the Watterson estimator.     

p-Value simulation for affected sib pair multiple testing

A standard approach to calculation of critical values for affected sib pair multiple testing is based on: (a) fully informative markers, (b) Haldane map function assumptions leading to a Markov chain model for inheritance vectors, (c) central limit approximation to averages of sampled inheritance vectors leading to an Ornstein-Uhlenbeck process approximation, and (d) simple approximations to the maximum of such a process. Under these assumptions, assuming equispaced or close to equispaced markers, if the sample size is large, an approximation is available that is easy to calculate and performs well. However, for small sample sizes, a large number of markers, and for small p-values, there is good reason to be cautious about the use of the Gaussian approximation. We develop an algorithm for calculation of multiple testing p-values based on the standard Markov chain model, avoiding the use of Gaussian (large sample) approximation. We illustrate the use of this algorithm by demonstrating some inadequacies of the Gaussian approximation.     

A Short Note on Short Dispersal Events

We study how the speed of spread for an integrodifference equation depends on the dispersal pattern of individuals. When the dispersal kernel has finite variance, the central limit theorem states that convolutions of the kernel with itself will approach a suitably chosen Gaussian distribution. Despite this fact, the speed of spread cannot be obtained from the Gaussian approximation. We give several examples and explanations for this fact. We then use the kurtosis of the kernel to derive an improved approximation that shows a very good fit to all the kernels tested. We apply the theory to one well-studied data set of dispersal of Drosophila pseudoobscura and to two one-parameter families of theoretical dispersal kernels. In particular, we find kernels that, despite having compact support, have a faster speed of spread than the Gaussian kernel.     

A consistent occupancy–climate relationship across birds and mammals of the Americas

At broad spatial scales, species richness is strongly related to climate. Yet, few ecological studies attempt to identify regularities in the individual species distributions that make up this pattern. Models used to describe species distributions typically model very complex responses to climate. Here, we test whether the variability in the distributions of birds and mammals of the Americas relates to mean annual temperature and precipitation in a simple, consistent way. Specifically, we test if simple mathematical models can predict, as a first approximation, the geographical variation in individual species’ probability of occupancy for 3277 non‐migratory bird and 1659 mammal species. We find a Gaussian model, where the probability of occupancy of a 10⁴ km² quadrat decreases symmetrically and gradually around a species ‘optimal’ temperature and precipitation, was generally the best model, explaining an average of 35% of the deviance in probability of occupancy. The inclusion of additional terms had very small and idiosyncratic effects across species. The Gaussian occupancy–climate relationship appears general among species and taxa and explains nearly as much deviance as complex models including many more parameters. Therefore, we propose that hypotheses aiming to explain the broad‐scale distribution of species or species richness must also predict generally Gaussian occupancy–climate relationships. Synthesis Science aims to identify regularities in a complex natural world. General patterns should be identified before one searches for potential mechanisms and contingencies. However, species geographic distributions are often modelled as complex (sometimes black box), species‐specific, functions of their environment. We asked whether a simple model could account for as much of the geographic variation in a species' probability of occupancy, and be widely applicable across thousands of species. As a first approximation, we found that a simple Gaussian occupancy‐climate relationship is very common in Nature, whether it be causal or not.     

The diffusion process approach to one-compartmental stochastic models: A mathematical note

We consider the one-dimension (one-compartment) exponential model using a diffusion process approach. In particular, we summarize the known results in the case where the stochastic component of the model is a Gaussian white noise process with mean zero and variance σ². Finally, we briefly illustrate a number of cases where similar forms of model arise.     

Couple stresses and discrete potentials in the vertex model of cellular monolayers

The vertex model is widely used to simulate the mechanical properties of confluent epithelia and other multicellular tissues. This inherently discrete framework allows a Cauchy stress to be attributed to each cell, and its symmetric component has been widely reported, at least for planar monolayers. Here, we consider the stress attributed to the neighbourhood of each tricellular junction, evaluating in particular its leading-order antisymmetric component and the associated couple stresses, which characterise the degree to which individual cells experience (and resist) in-plane bending deformations. We develop discrete potential theory for localised monolayers having disordered internal structure and use this to derive the analogues of Airy and Mindlin stress functions. These scalar potentials typically have broad-banded spectra, highlighting the contributions of small-scale defects and boundary layers to global stress patterns. An affine approximation attributes couple stresses to pressure differences between cells sharing a trijunction, but simulations indicate an additional role for non-affine deformations.

     

A minimally parametrized branching process explaining plateau phase of qPCR amplification

Quantitative polymerase chain reaction (qPCR) is a commonly used molecular biology technique for measuring the concentration of a target nucleic acid sequence in a sample. The whole qPCR amplification process usually consists of an exponential, a linear and a plateau phase. In qPCR experiments, amplification curves of samples with different template concentrations often, even though not always, have the same plateau height. The biological theory for this phenomenon is that the plateau height is determined by reaction kinetics. Does it mean that the target concentration has no effect on the final plateau height? We proposed a branching process based on Michaelis–Menten kinetics. Our model can describe all phases of qPCR amplification despite its simplicity (it depends on only one parameter). We theoretically showed, through almost sure convergence, that amplification curves will eventually plateau at finite values in any experiment, under any condition. We conclude that the plateau height is largely determined by reaction kinetics but could also be affected by the template concentration. This is in accordance with the current biological theory.

     

Multi-locus Analysis of Genomic Time Series Data from Experimental Evolution

Genomic time series data generated by evolve-and-resequence (E&R) experiments offer a powerful window into the mechanisms that drive evolution. However, standard population genetic inference procedures do not account for sampling serially over time, and new methods are needed to make full use of modern experimental evolution data. To address this problem, we develop a Gaussian process approximation to the multi-locus Wright-Fisher process with selection over a time course of tens of generations. The mean and covariance structure of the Gaussian process are obtained by computing the corresponding moments in discrete-time Wright-Fisher models conditioned on the presence of a linked selected site. This enables our method to account for the effects of linkage and selection, both along the genome and across sampled time points, in an approximate but principled manner. We first use simulated data to demonstrate the power of our method to correctly detect, locate and estimate the fitness of a selected allele from among several linked sites. We study how this power changes for different values of selection strength, initial haplotypic diversity, population size, sampling frequency, experimental duration, number of replicates, and sequencing coverage depth. In addition to providing quantitative estimates of selection parameters from experimental evolution data, our model can be used by practitioners to design E&R experiments with requisite power. We also explore how our likelihood-based approach can be used to infer other model parameters, including effective population size and recombination rate. Then, we apply our method to analyze genome-wide data from a real E&R experiment designed to study the adaptation of D. melanogaster to a new laboratory environment with alternating cold and hot temperatures.     

Estimation of a monotone percentile residual life function under random censorship

In this paper, we introduce a new estimator of a percentile residual life function with censored data under a monotonicity constraint. Specifically, it is assumed that the percentile residual life is a decreasing function. This assumption is useful when estimating the percentile residual life of units, which degenerate with age. We establish a law of the iterated logarithm for the proposed estimator, and its ‐equivalence to the unrestricted estimator. The asymptotic normal distribution of the estimator and its strong approximation to a Gaussian process are also established. We investigate the finite sample performance of the monotone estimator in an extensive simulation study. Finally, data from a clinical trial in primary biliary cirrhosis of the liver are analyzed with the proposed methods. One of the conclusions of our work is that the restricted estimator may be much more efficient than the unrestricted one.     

A probabilistic view on the deterministic mutation–selection equation: dynamics,equilibria, and ancestry via individual lines of descent

We reconsider the deterministic haploid mutation–selection equation with two types. This is an ordinary differential equation that describes the type distribution (forward in time) in a population of infinite size. This paper establishes ancestral (random) structures inherent in this deterministic model. In a first step, we obtain a representation of the deterministic equation’s solution (and, in particular, of its equilibria) in terms of an ancestral process called the killed ancestral selection graph. This representation allows one to understand the bifurcations related to the error threshold phenomenon from a genealogical point of view. Next, we characterise the ancestral type distribution by means of the pruned lookdown ancestral selection graph and study its properties at equilibrium. We also provide an alternative characterisation in terms of a piecewise-deterministic Markov process. Throughout, emphasis is on the underlying dualities as well as on explicit results.

     

Evolutionary games under incompetence

The adaptation process of a species to a new environment is a significant area of study in biology. As part of natural selection, adaptation is a mutation process which improves survival skills and reproductive functions of species. Here, we investigate this process by combining the idea of incompetence with evolutionary game theory. In the sense of evolution, incompetence and training can be interpreted as a special learning process. With focus on the social side of the problem, we analyze the influence of incompetence on behavior of species. We introduce an incompetence parameter into a learning function in a single-population game and analyze its effect on the outcome of the replicator dynamics. Incompetence can change the outcome of the game and its dynamics, indicating its significance within what are inherently imperfect natural systems.

     

The impact of exclusion processes on angiogenesis models

Angiogenesis is the process by which new blood vessels form from existing vessels. During angiogenesis, tip cells migrate via diffusion and chemotaxis, new tip cells are introduced through branching, loops form via tip-to-tip and tip-to-sprout anastomosis, and a vessel network forms as endothelial cells, known as stalk cells, follow the paths of tip cells (a process known as the snail-trail). Using a mean-field approximation, we systematically derive one-dimensional non-linear continuum models from a lattice-based cellular automaton model of angiogenesis in the corneal assay, explicitly accounting for cell volume. We compare our continuum models and a well-known phenomenological snail-trail model that is linear in the diffusive, chemotactic and branching terms, with averaged cellular automaton simulation results to distinguish macroscale volume exclusion effects and determine whether linear models can capture them. We conclude that, in general, both linear and non-linear models can be used at low cell densities when single or multi-species exclusion effects are negligible at the macroscale. When cell densities increase, our non-linear model should be used to capture non-linear tip cell behavior that occurs when single-species exclusion effects are pronounced, and alternative models should be derived for non-negligible multi-species exclusion effects.

     

Resolution enhancement for lung 4D-CT based on transversal structures by using multiple Gaussian process regression learning

PurposeFour-dimensional computed tomography (4D-CT) plays a useful role in many clinical situations. However, due to the hardware limitation of system, dense sampling along superior–inferior direction is often not practical. In this paper, we develop a novel multiple Gaussian process regression model to enhance the superior-inferior resolution for lung 4D-CT based on transversal structures.MethodsThe proposed strategy is based on the observation that high resolution transversal images can recover missing pixels in the superior-inferior direction. Based on this observation and motived by random forest algorithm, we employ multiple Gaussian process regression model learned from transversal images to improve superior–inferior resolution. Specifically, we first randomly sample 3 × 3 patches from original transversal images. The central pixel of these patches and the eight-neighbour pixels of their corresponding degraded versions form the label and input of training data, respectively. Multiple Gaussian process regression model is then built on the basis of multiple training subsets obtained by random sampling. Finally, the central pixel of the patch is estimated based on the proposed model, with the eight-neighbour pixels of each 3 × 3 patch from interpolated superior-inferior direction images as inputs.ResultsThe performance of our method is extensively evaluated using simulated and publicly available datasets. Our experiments show the remarkable performance of the proposed method.ConclusionsIn this paper, we propose a new approach to improve the 4D-CT resolution, which does not require any external data and hardware support, and can produce clear coronal/sagittal images for easy viewing.     

Novel moment closure approximations in stochastic epidemics

Moment closure approximations are used to provide analytic approximations to non-linear stochastic population models. They often provide insights into model behaviour and help validate simulation results. However, existing closure schemes typically fail in situations where the population distribution is highly skewed or extinctions occur. In this study we address these problems by introducing novel second-and third-order moment closure approximations which we apply to the stochastic SI and SIS epidemic models. In the case of the SI model, which has a highly skewed distribution of infection, we develop a second-order approximation based on the beta-binomial distribution. In addition, a closure approximation based on mixture distribution is developed in order to capture the behaviour of the stochastic SIS model around the threshold between persistence and extinction. This mixture approximation comprises a probability distribution designed to capture the quasi-equilibrium probabilities of the system and a probability mass at 0 which represents the probability of extinction. Two third-order versions of this mixture approximation are considered in which the log-normal and the beta-binomial are used to model the quasi-equilibrium distribution. Comparison with simulation results shows: (1) the beta-binomial approximation is flexible in shape and matches the skewness predicted by simulation as shown by the stochastic SI model and (2) mixture approximations are able to predict transient and extinction behaviour as shown by the stochastic SIS model, in marked contrast with existing approaches. We also apply our mixture approximation to approximate a likehood function and carry out point and interval parameter estimation.     

Critical test of bead–spring model to resolve the scaling laws of polymer melts: a molecular dynamics study

Abstract

To examine the intrinsic nature of the bead–spring Kremer–Grest (KG) model, long-time molecular dynamics simulations are performed. Certain scaling laws for representative polymer properties are compared with theoretical predictions. The results for static properties satisfy the expected static Gaussian nature, irrespective of the chain length. In contrast, the results for the dynamic properties of short chains show a clear discrepancy from theoretical predictions that assume ideal chain motion. This is clear evidence that the Gaussian nature of the dynamics of short chains is not necessarily established for the actual KG model, despite it being designed to have Gaussian characteristics by virtue of its stochastic equations of motion. This intrinsic nature of the KG model should be considered carefully when using this model for applications that involve relatively short chains.     

Transient dynamics in altered disturbance regimes: recovery may start quickly, then slow

When a pattern of spatial or temporal environmental variation changes, it takes time for populations to reach their new stationary distributions, and during this time, the competitive landscape is also in flux. As a first step toward understanding community responses to altered variational regimes, I investigate the convergence of an annual–perennial plant system to its stationary spatiotemporal distribution following a change in environmental variation. I find that, to good approximation, convergence is the sum of two separate processes: global convergence, which governs changes in the total population, and local convergence, which governs population redistribution. While the slower process (global or local) eventually governs convergence, the faster process may initially dominate if it starts further from its stationary distribution, so that the populations converge quickly at first, then slow down. That is, when disturbances are spatially heterogeneous, a system may be initially more resilient under some initial conditions than others.

Geographical Variation in a Quantitative Character

A model for the evolution of the local averages of a quantitative character under migration, selection, and random genetic drift in a subdivided population is formulated and investigated. Generations are discrete and nonoverlapping; the monoecious, diploid population mates at random in each deme. All three evolutionary forces are weak, but the migration pattern and the local population numbers are otherwise arbitrary. The character is determined by purely additive gene action and a stochastically independent environment; its distribution is Gaussian with a constant variance; and it is under Gaussian stabilizing selection with the same parameters in every deme. Linkage disequilibrium is neglected. Most of the results concern the covariances of the local averages. For a finite number of demes, explicit formulas are derived for (i) the asymptotic rate and pattern of convergence to equilibrium, (ii) the variance of a suitably weighted average of the local averages, and (iii) the equilibrium covariances when selection and random drift are much weaker than migration. Essentially complete analyses of equilibrium and convergence are presented for random outbreeding and site homing, the Levene and island models, the circular habitat and the unbounded linear stepping-stone model in the diffusion approximation, and the exact unbounded stepping-stone model in one and two dimensions.