Using genetic markers to estimate the pollen dispersal curve

Pollen dispersal is a critical process that shapes genetic diversity in natural populations of plants. Estimating the pollen dispersal curve can provide insight into the evolutionary dynamics of populations and is essential background for making predictions about changes induced by perturbations. Specifically, we would like to know whether the dispersal curve is exponential, thin-tailed (decreasing faster than exponential), or fat-tailed (decreasing slower than the exponential). In the latter case, rare events of long-distance dispersal will be much more likely. Here we generalize the previously developed TWOGENER method, assuming that the pollen dispersal curve belongs to particular one- or two-parameter families of dispersal curves and estimating simultaneously the parameters of the dispersal curve and the effective density of reproducing individuals in the population. We tested this method on simulated data, using an exponential power distribution, under thin-tailed, exponential and fat-tailed conditions. We find that even if our estimates show some bias and large mean squared error (MSE), we are able to estimate correctly the general trend of the curve - thin-tailed or fat-tailed - and the effective density. Moreover, the mean distance of dispersal can be correctly estimated with low bias and MSE, even if another family of dispersal curve is used for the estimation. Finally, we consider three case studies based on forest tree species. We find that dispersal is fat-tailed in all cases, and that the effective density estimated by our model is below the measured density in two of the cases. This latter result may reflect the difficulty of estimating two parameters, or it may be a biological consequence of variance in reproductive success of males in the population. Both the simulated and empirical findings demonstrate the strong potential of TWOGENER for evaluating the shape of the dispersal curve and the effective density of the population (d(e)).     

Evolutionary patterns from mass originations and mass extinctions

The Fossil Record 2 database gives a stratigraphic range of most known animal and plant families. We have used it to plot the number of families extant through time and argue for an exponential fit, rather than a logistic one, on the basis of power spectra of the residuals from the exponential. The times of origins and extinctions, when plotted for all families of marine and terrestrial organisms over the last 600 Myr, reveal different origination and extinction peaks. This suggests that patterns of biological evolution are driven by its own internal dynamics as well as responding to upsets from external causes. Spectral analysis shows that the residuals from the exponential model of the marine system are more consistent with 1/f noise suggesting that self-organized criticality phenomena may be involved.     

Scale-free neurodegeneration: cellular heterogeneity and the stretched exponential kinetics of cell death

Neurodegenerative disorders are an insidious group of diseases characterized by severe physical and cognitive effects that often have devastating consequences for the lives of affected individuals and their families. One feature common to a significant proportion of these diseases is that affected neurons commit to undergoing an active form of degeneration known as programmed cell death, or apoptosis. Although intense effort over the past several years has resulted is a remarkable increase in our understanding of the molecular events involved in neurodegeneration, our knowledge regarding the cellular and tissue properties that determine the temporal patterns of neuronal attrition is limited. We recently demonstrated that neurodegenerative kinetics in various diseases fit well to exponential decay functions, and proposed a universal one-hit switch mechanism in which mutant and injured neurons exist in a viable state characterized by an increased but constant risk of initiating apoptosis (Nature, 406, p. 195). Here we show that a heavy-tailed stretched exponential function is better able to account for neurodegenerative kinetic data. Moreover, normalization of all available data according to their corresponding best-fit stretched exponential parameters suggest that the generalized model is consistent with a universal mechanism of neuronal cell death that is greatly improved over the constant risk model. In contrast to the original model in which all cells exhibit an identical risk of initiating apoptosis, the stretched exponential model is consistent with each neuron experiencing a constant risk that is different from that experienced by other cells in the degenerating population, perhaps due to spatial differences in the cellular microenvironment. Intriguingly, the predicted distribution of risk across the cell population can be fit by a power-law function, further suggesting that scale-free properties of degenerating neuronal tissues might act as potent regulators of the kinetics of cell death in neural tissue.     

Heterogeneity in IBD allele sharing among covariate-defined subgroups: issues and findings for affected relatives

OBJECTIVES: Modelling of variation in identical-by-descent (IBD) allele sharing using covariates can increase power to detect linkage, identify covariate-defined subgroups linked to particular marker regions, and improve the design of subsequent studies to localize genes and characterize their effects. In this report, we highlight issues that arise in studies of families with affected relatives. METHODS: Mirea et al. [Genet Epidemiol 2003, in press] extended linear and exponential linkage likelihood models [Kong and Cox, Am J Hum Genet 1997;61: 1179-1188] to model variation in NPL scores among covariate-defined groups of families, and proposed likelihood ratio (LR) and t statistics to detect differences in allele sharing between groups defined by a binary covariate. Here we evaluate factors affecting the power of these tests analytically and by example, as well as effects of constraints, nuisance parameters, and incomplete data on test validity by simulation of locus heterogeneity in families with affected siblings or affected cousins. RESULTS: Provided constraints on the parameters are avoided, these tests are particularly useful when one subgroup has less than expected IBD sharing. The distribution of the LR statistic depends on the extent of linkage, particularly in the presence of constraints. The t statistic may be biased by group differences in information content. CONCLUSIONS: We recommend that constraints be applied cautiously, and covariate effects in IBD allele sharing models interpreted with care.     

Process‐based functions for seed retention on animals: a test of improved descriptions of dispersal using multiple data sets

Studies of external seed transport on animals usually assume that the probability of detachment is constant, so that seed retention should show a simple exponential relationship with time. This assumption has not been tested explicitly, and may lead to inaccurate representation of long distance seed dispersal by animals. We test the assumption by comparing the fit to empirical data of simple, two‐parameter functions. Fifty‐two data sets were obtained from five published studies, describing seed retention of 32 plant species on sheep, cattle, deer, goats and mice. Model selection suggested a simple exponential function was adequate for data sets in which seed retention was followed for short periods ( <48 h). The data gathered over longer periods (49–219 days) were best described by the power exponential function, a form of the stretched exponential which allows a changing dropping rate. In these cases the power exponential showed that seed dropping rate decreased with time, suggesting that seeds vary in attachment, with some seeds becoming deeply buried or wound up in the animal's coat. Comparison of fitted parameters across all the data sets also confirmed that seeds with adhesive structures have lower dropping rates than those without. We conclude that the seed dropping rate often changes with time during external transport on animals and that the power exponential is an effective function to describe this change. We advise that, to analyse seed dropping rates adequately, retention should be measured over reasonable time periods – until most seeds are dropped – and both the simple and power exponential functions should be fitted to the resulting data. To increase its utility, we provide functions describing the seed dropping rate and the dispersal kernel resulting from the power exponential relationship.     

Application of pattern-mixture models to outcomes that are potentially missing not at random using pseudo maximum likelihood estimation

In this work, we fit pattern-mixture models to data sets with responses that are potentially missing not at random (MNAR, Little and Rubin, 1987). In estimating the regression parameters that are identifiable, we use the pseudo maximum likelihood method based on exponential families. This procedure provides consistent estimators when the mean structure is correctly specified for each pattern, with further information on the variance structure giving an efficient estimator. The proposed method can be used to handle a variety of continuous and discrete outcomes. A test built on this approach is also developed for model simplification in order to improve efficiency. Simulations are carried out to compare the proposed estimation procedure with other methods. In combination with sensitivity analysis, our approach can be used to fit parsimonious semi-parametric pattern-mixture models to outcomes that are potentially MNAR. We apply the proposed method to an epidemiologic cohort study to examine cognition decline among elderly.     

Dispersal kernel of green rice leafhopper estimated from truncated data

Characterizing dispersal kernels from truncated data is important for managing and predicting population dynamics. We used mark-recapture data from 10 previously published replicated experiments at three host plant development stages (seedling, tillering, and heading) to estimate parameters of the normal and exponential dispersal kernels for green rice leafhopper, Nephotettix cincticeps (Uhler). We compared classic statistical methods for estimating untruncated distribution parameters from truncated data with maximum likelihood (MLE) and the method of statistical moments for simulated and empirical data. Simulations showed that both methods provided accurate parameter estimates with similar precision. The method of moments is algebraically complex, but simple to calculate, while the MLE methods require numerical solutions of nonlinear equations. Simulations also showed that accurate, precise estimates of the parameters of the untruncated distributions could be attained even under severe truncation with sufficient numbers of recaptures. Both diffusivity and the exponential mean were higher with later plant growth stage, showing that insects moved farther and faster at the heading stage. Precision of the estimates was not strongly related to percent capture, size of the experimental field, or the number of leafhoppers captured. The leptokurtic exponential kernel fit the data better than the normal kernel for all the experiments. These results support an alternative explanation for the strong density-dependent population regulation of this species at the heading stage. Instead of leafhopper density per se, the increase in movement at this stage could integrate the populations in the separate fields, leveling densities throughout the landscape.     

Allometric analysis using the multivariate shifted exponential normal distribution

In allometric studies, the joint distribution of the log-transformed morphometric variables is typically elliptical and with heavy tails. To account for these peculiarities, we introduce the multivariate shifted exponential normal (MSEN) distribution , an elliptical heavy-tailed generalization of the multivariate normal (MN). The MSEN belongs to the family of MN scale mixtures (MNSMs) by choosing a convenient shifted exponential as mixing distribution. The probability density function of the MSEN has a simple closed-form characterized by only one additional parameter, with respect to the nested MN, governing the tail weight. The first four moments exist and the excess kurtosis can assume any positive value. The membership to the family of MNSMs allows us a simple computation of the maximum likelihood (ML) estimates of the parameters via the expectation-maximization (EM) algorithm; advantageously, the M-step is computationally simplified by closed-form updates of all the parameters. We also evaluate the existence of the ML estimates. Since the parameter governing the tail weight is estimated from the data, robust estimates of the mean vector of the nested MN distribution are automatically obtained by downweighting; we show this aspect theoretically but also by means of a simulation study. We fit the MSEN distribution to multivariate allometric data where we show its usefulness also in comparison with other well-established multivariate elliptical distributions.     

A resolution of the ascertainment sampling problem I. Theory

We consider the "ascertainment problem" arising when families are sampled by a nonrandom sampling process and, for the purpose of estimating genetic parameters, some assumption must be made about the process by which families enter the sample. A resolution of this problem, involving conditioning the likelihood of the sample on that part of the data relevant to ascertainment, is put forward. Numerical examples illustrating the properties of the procedure are provided.     

A method for the estimation of parameters for natural stage-structured populations

A relatively simple method is proposed for the estimation of parameters of stage-structured populations from sample data for situation where (a) unit time survival rates may vary with time, and (b) the distribution of entry times to stage 1 is too complicated to be fitted with a simple parametric model such as a normal or gamma distribution. The key aspects of this model are that the entry time distribution is approximated by an exponential function withp parameters, the unit time survival rates in stages are approximated by anr parameter exponential polynomial in the stage number, and the durations of stages are assumed to be the same for all individuals. The new method is applied to four Zooplankton data sets, with parametric bootstrapping used to assess the bias and variation in estimates. It is concluded that good estimates of demographic parameters from stagefrequency data from natural populations will usually only be possible if extra information such as the durations of stages is known.     

Complex segregation analysis of obsessive-compulsive disorder in families with pediatric probands

OBJECTIVE: The purpose of this study was to assess the mode of inheritance for obsessive-compulsive disorder (OCD) in families ascertained through pediatric probands. METHODS: We ascertained 52 families (35 case and 17 control families) through probands between the ages of 10 and 17 years. Direct interviews were completed with 215 individuals. Family informant data were collected on another 450 individuals without direct interviews, forming two data sets with one contained within the other. Complex segregation analyses were performed using regressive models as programmed in REGTL in the S.A.G.E. package.All models used in the analyses included sex-specific age and type parameters. RESULTS: All models that excluded a residual effect of an affected parent were rejected. With that parameter included, the environmental and sporadic models were rejected in comparisons with the most general model in both data sets (all p < 0.005). With the direct interview data, the general codominant Mendelian model was not rejected when compared with the most general model (p = 0.140). We could not distinguish between any of the simple Mendelian models using either data set. However, the dominant Mendelian model provided a somewhat better fit than the other Mendelian models to the direct interview data. CONCLUSIONS: The results provide evidence for a major susceptibility locus in families with OCD when age at onset is incorporated into the model. Mendelian factors at most partially explained the familial aggregation of the phenotype, and residual familial effects were necessary to fit the data adequately. The results support the importance of linkage efforts by suggesting that a major locus is segregating within a proportion of families with OCD ascertained through pediatric probands.     

Conditionally specified continuous distributions

A distribution is conditionally specified when its model constraintsare expressed conditionally. For example, Besag's (1974) spatialmodel was specified conditioned on the neighbouring states,and pseudolikelihood is intended to approximate the likelihoodusing conditional likelihoods. There are three issues of interest:existence, uniqueness and computation of a joint distribution.In the literature, most results and proofs are for discreteprobabilities; here we exclusively study distributions withcontinuous state space. We examine all three issues using thedependence functions derived from decomposition of the conditionaldensities. We show that certain dependence functions of thejoint density are shared with its conditional densities. Therefore,two conditional densities involving the same set of variablesare compatible if their overlapping dependence functions areidentical. We prove that the joint density is unique when theset of dependence functions is both compatible and complete.In addition, a joint density, apart from a constant, can becomputed from the dependence functions in closed form. Sinceall of the results are expressed in terms of dependence functions,we consider our approach to be dependence-based, whereas methodsin the literature are generally density-based. Applicationsof the dependence-based formulation are discussed.     

Calculation of Steel's cell loss factor and of the parameters of cellular kinetics for a population with an exponential growth of the cell number and cell death at G0 phase with a probability equal to 1

The method of calculation of three cell kinetics parameters (the Steel's cell loss factor phi, the proliferative pool Pc, and the mean number m of the proliferating cells after mitotic division of one cell) was shown to be the same for the exponential growth state of cell number with cell death at the G0-phase, and for the exponential growth state with cell death occurring immediately after mitosis. The value of the mean number delta of non-proliferating cells that appeared after mitotic division of one cell is different for these two models of the exponential growth state with the equal values of the other three parameters (phi, Pc, and m). A method is proposed for calculating the parameter delta on the data of the percentage of labeled cells obtained in the experiments with continuous cultivation of cells in the nutrient medium containing 3H-thymidine. The kinetics of cell line HL-60 (the experimental data of Foa et al., 1982) can be described at the first approximation, by a model of the exponential growth state with the cell death at the G0-phase, with Pc = 0.80, phi = 0.24, m = 1.61, delta = 0.39, and the life time of the non-proliferating cells tQ = 24 hours.     

Small-sample confidence regions in exponential families

This article presents an algorithm for small-sample conditional confidence regions for two or more parameters for any discrete regression model in the generalized linear interactive model family. Regions are constructed by careful inversion of conditional hypothesis tests. This method presupposes the use of approximate or exact techniques for enumerating the sample space for some components of the vector of sufficient statistics conditional on other components. Such enumeration may be performed exactly or by exact or approximate Monte Carlo, including the algorithms of Kolassa and Tanner (1994, Journal of the American Statistical Association 89, 697-702; 1999, Biometrics 55, 246-251). This method also assumes that one can compute certain conditional probabilities for a fixed value of the parameter vector. Because of a property of exponential families, one can use this set of conditional probabilities to directly compute the conditional probabilities associated with any other value of the vector of the parameters of interest. This observation dramatically reduces the computational effort required to invert the hypothesis test to obtain the confidence region. To construct a region with confidence level 1 - alpha, the algorithm begins with a grid of values for the parameters of interest. For each parameter vector on the grid (corresponding to the current null hypothesis), one transforms the initial set of conditional probabilities using exponential tilting and then calculates the p value for this current null hypothesis. The confidence region is the set of parameter values for which the p value is at least alpha.     

LAMARC 2.0: maximum likelihood and Bayesian estimation of population parameters

We present a Markov chain Monte Carlo coalescent genealogy sampler, LAMARC 2.0, which estimates population genetic parameters from genetic data. LAMARC can co-estimate subpopulation Theta = 4N(e)mu, immigration rates, subpopulation exponential growth rates and overall recombination rate, or a user-specified subset of these parameters. It can perform either maximum-likelihood or Bayesian analysis, and accomodates nucleotide sequence, SNP, microsatellite or elecrophoretic data, with resolved or unresolved haplotypes. It is available as portable source code and executables for all three major platforms. AVAILABILITY: LAMARC 2.0 is freely available at http://evolution.gs.washington.edu/lamarc     

Comparison of different neuron models to conductance-based post-stimulus time histograms obtained in cortical pyramidal cells using dynamic-clamp in vitro

A wide diversity of models have been proposed to account for the spiking response of central neurons, from the integrate-and-fire (IF) model and its quadratic and exponential variants, to multiple-variable models such as the Izhikevich (IZ) model and the well-known Hodgkin–Huxley (HH) type models. Such models can capture different aspects of the spiking response of neurons, but there is few objective comparison of their performance. In this article, we provide such a comparison in the context of well-defined stimulation protocols, including, for each cell, DC stimulation, and a series of excitatory conductance injections, arising in the presence of synaptic background activity. We use the dynamic-clamp technique to characterize the response of regular-spiking neurons from guinea-pig visual cortex by computing families of post-stimulus time histograms (PSTH), for different stimulus intensities, and for two different background activities (low- and high-conductance states). The data obtained are then used to fit different classes of models such as the IF, IZ, or HH types, which are constrained by the whole data set. This analysis shows that HH models are generally more accurate to fit the series of experimental PSTH, but their performance is almost equaled by much simpler models, such as the exponential or pulse-based IF models. Similar conclusions were also reached by performing partial fitting of the data, and examining the ability of different models to predict responses that were not used for the fitting. Although such results must be qualified by using more sophisticated stimulation protocols, they suggest that nonlinear IF models can capture surprisingly well the response of cortical regular-spiking neurons and appear as useful candidates for network simulations with conductance-based synaptic interactions.     

Exponential Sum-Fitting of Dwell-Time Distributions without Specifying Starting Parameters

Fitting dwell-time distributions with sums of exponentials is widely used to characterize histograms of open- and closed-interval durations recorded from single ion channels, as well as for other physical phenomena. However, it can be difficult to identify the contributing exponential components. Here we extend previous methods of exponential sum-fitting to present a maximum-likelihood approach that consistently detects all significant exponentials without the need for user-specified starting parameters. Instead of searching for exponentials, the fitting starts with a very large number of initial exponentials with logarithmically spaced time constants, so that none are missed. Maximum-likelihood fitting then determines the areas of all the initial exponentials keeping the time constants fixed. In an iterative manner, with refitting after each step, the analysis then removes exponentials with negligible area and combines closely spaced adjacent exponentials, until only those exponentials that make significant contributions to the dwell-time distribution remain. There is no limit on the number of significant exponentials and no starting parameters need be specified. We demonstrate fully automated detection for both experimental and simulated data, as well as for classical exponential-sum-fitting problems.     

NEE观测误差分布类型对陆地生态系统机理模型参数估计的影响——以长白山温带阔叶红松林为例

基于观测数据的陆地生态系统模型参数估计有助于提高模型的模拟和预测能力,降低模拟不确定性.在已有参数估计研究中,涡度相关技术测定的净生态系统碳交换量(NEE)数据的随机误差通常被假设为服从零均值的正态分布.然而近年来已有研究表明NEE数据的随机误差更服从双指数分布.为探讨NEE观测误差分布类型的不同选择对陆地生态系统机理模型参数估计以及碳通量模拟结果造成的差异,以长白山温带阔叶红松林为研究区域,采用马尔可夫链-蒙特卡罗方法,利用2003～2005年测定的NEE数据对陆地生态系统机理模型CEVSA2的敏感参数进行估计,对比分析了两种误差分布类型(正态分布和双指数分布)的参数估计结果以及碳通量模拟的差异.结果表明,基于正态观测误差模拟的总初级生产力和生态系统呼吸的年总量分别比基于双指数观测误差的模拟结果高61～86 g C m-2 a-1和107～116 g C m-2 a-1,导致前者模拟的NEE年总量较后者低29～47 g C m-2 a-1,特别在生长旺季期间有明显低估.在参数估计研究中,不能忽略观测误差的分布类型以及相应的目标函数的选择,它们的不合理设置可能对参数估计以及模拟结果产生较大影响.