The Site-Frequency Spectrum of Linked Sites

The site-frequency spectrum, representing the distribution of allele frequencies at a set of polymorphic sites, is a commonly used summary statistic in population genetics. Explicit forms of the spectrum are known for both models with and without selection if independence among sites is assumed. The availability of these explicit forms has allowed for maximum likelihood estimation of selection, developed first in the Poisson random field model of Sawyer and Hartl, which is now the primary method for estimating selection directly from DNA sequence data. The independence assumption, which amounts to assume free recombination between sites, is, however, a limiting case for many population genetics models. Here, we extend the site-frequency spectrum theory to consider the case where the sites are completely linked. We use diffusion approximation to calculate the joint distribution of the allele frequencies of linked sites for models without selection and for models with equal coefficient selection. The joint distribution is derived by first constructing Green’s functions corresponding to multiallele diffusion equations. We show that the site-frequency spectrum is highly correlated between frequencies that are complementary (i.e., sum to 1), and the correlation is significantly elevated by positive selection. The results presented here can be used to extend the Poisson random field to allow for estimating selection for correlated sites. More generally, the Green’s function construction should be able to aid in studying the genetic drift of multiple alleles in other cases.     

On the Overdispersed Molecular Clock

Rates of molecular evolution at some loci are more irregular than described by simple Poisson processes. Three situations under which molecular evolution would not follow simple Poisson processes are reevaluated from the viewpoint of the neutrality hypothesis: concomitant or multiple substitutions in a gene, fluctuating substitution rates in time caused by coupled effects of deleterious mutations and bottlenecks, and changes in the degree of selective constraints against a gene (neutral space) caused by successive substitutions. The common underlying assumption that these causes are lineage nonspecific excludes the case where mutation rates themselves change systematically among lineages or taxonomic groups, and severely limits the extent of variation in the number of substitutions among lineages. Even under this stringent condition, however, the third hypothesis, the fluctuating neutral space model, can generate fairly large variation. This is described by a time-dependent renewal process, which does not exhibit any episodic nature of molecular evolution. It is argued that the observed elevated variances in the number of nucleotide or amino acid substitutions do not immediately call for positive Darwinian selection in molecular evolution.     

Non-poisson analysis of endonucleases with substrate sequence specificities

Endonucleases with substrate-sequence specificities, such as restriction enzymes, usually cleave small, defined nucleic acid molecules used in enzyme assays at one or only a few sites. The methods in common use for analysis of endonucleases are based on the Poisson distribution. A critical, but usually unstated, assumption of this distribution, however, is that there is a large number of possible reactive sites on each substrate molecule. Thus use of the Poisson distribution may introduce large errors into analysis of such assays. Here we develop a series of appropriate expressions for use in analyzing endonucleases with substrate-sequence specificities.     

Statistical methods for estimating doubling time in in vitro cell growth

Summary Doubling time has been widely used to represent the growth pattern of cells. A traditional method for finding the doubling time is to apply gray-scaled cells, where the logarithmic transformed scale is used. As an alternative statistical method, the log-linear model was recently proposed, for which actual cell numbers are used instead of the transformed gray-scaled cells. In this paper, I extend the log-linear model and propose the extended log-linear model. This model is designed for extra-Poisson variation, where the log-linear model produces the less appropriate estimate of the doubling time. Moreover, I compare statistical properties of the gray-scaled method, the log-linear model, and the extended log-linear model. For this purpose, I perform a Monte Carlo simulation study with three data-generating models: the additive error model, the multiplicative error model, and the overdispersed Poisson model. From the simulation study, I found that the gray-scaled method highly depends on the normality assumption of the gray-scaled cells; hence, this method is appropriate when the error model is multiplicative with the log-normally distributed errors. However, it is less efficient for other types of error distributions, especially when the error model is additive or the errors follow the Poisson distribution. The estimated standard error for the doubling time is not accurate in this case. The log-linear model was found to be efficient when the errors follow the Poisson distribution or nearly Poisson distribution. The efficiency of the log-linear model was decreased accordingly as the overdispersion increased, compared to the extended log-linear model. When the error model is additive or multiplicative with Gamma-distributed errors, the log-linear model is more efficient than the gray-scaled method. The extended log-linear model performs well overall for all three data-generating models. The loss of efficiency of the extended log-linear model is observed only when the error model is multiplicative with log-normally distributed errors, where the gray-scaled method is appropriate. However, the extended log-linear model is more efficient than the log-linear model in this case.     

Double site enzyme and squatting: where one regulatory ligand is also a substrate of the reaction

A regulatory model termed "squatting" has already been proposed (Mazat et al., 1977). It involves competition between at least two ligands binding to two regulatory sites. In this paper we study the case where one of the ligands is a substrate or a product of the reaction, one of the sites where competition occurs being the catalytic one. Complex regulatory patterns are evidenced and some experimental examples are analysed according to this model. It is worth noting that this model takes into account competition between chemically related ligands as occurs in vivo.     

Directional selection and the site-frequency spectrum.

In this article we explore statistical properties of the maximum-likelihood estimates (MLEs) of the selection and mutation parameters in a Poisson random field population genetics model of directional selection at DNA sites. We derive the asymptotic variances and covariance of the MLEs and explore the power of the likelihood ratio tests (LRT) of neutrality for varying levels of mutation and selection as well as the robustness of the LRT to deviations from the assumption of free recombination among sites. We also discuss the coverage of confidence intervals on the basis of two standard-likelihood methods. We find that the LRT has high power to detect deviations from neutrality and that the maximum-likelihood estimation performs very well when the ancestral states of all mutations in the sample are known. When the ancestral states are not known, the test has high power to detect deviations from neutrality for negative selection but not for positive selection. We also find that the LRT is not robust to deviations from the assumption of independence among sites.     

Single-salt behavior of a symmetrical 4-site channel with barriers at its middle and ends

The theory of a symmetrical 3-barrier, 4-site, single-filing ionic channel is developed. The model goes beyond earlier models by including additional sites, as well as barriers which need not be symmetrical in the applied field, and contains the earlier models as special cases. It is itself a special case of the most general 4-site model, which has 5 barriers. By considering the barriers at the mouth and middle of the channel to be sufficiently larger than the barriers separating the sites in each channel half, these barriers can be neglected; thus this case reduces to a 3-barrier model where the sites in each channel half can then be assumed to be in equilibrium with each other. The alternative 3-barrier, 4-site case, where the barrier between the sites is considered to be larger than that at the mouth of the channel, is considered elsewhere. Pure cation permeation is considered and only single-salt properties of the system are analyzed, namely occupancy, conductance, flux ratio exponent and current-voltage relation. The concentration dependences of these properties are computed and interrelated and, where possible, also given in analytical form. The mathematical relations are obtained for a channel which is symmetrical around its middle, which is the appropriate assumption for the gramicidin channel. However, the barriers themselves are allowed to be asymmetric with respect to the potential dependence, which has been found to be essential for gramicidin. Mathematically, a straight-forward matrix formulation is used; but a general theoretical method is presented for reducing a complex model (with more than 2 sites) to a simpler cases when equilibrium exists across one or several barriers, as is often the cases. This method is a prototype which makes analytical solutions of complex barrier models possible in many cases.     

Effect of nonindependent substitution on phylogenetic accuracy

All current phylogenetic methods assume that DNA substitutions are independent among sites. However, ample empirical evidence suggests that the process of substitution is not independent but is, in fact, temporally and spatially correlated. The robustness of several commonly used phylogenetic methods to the assumption of independent substitution is examined. A compound Poisson process is used to model DNA substitution. This model assumes that substitution events are Poisson-distributed in time and that the number of substitutions associated with each event is geometrically distributed. The asymptotic properties of phylogenetic methods do not appear to change under a compound Poisson process of DNA substitution. Moreover, the rank order of the performance of different methods does not change. However, all phylogenetic methods become less efficient when substitution follows a compound Poisson process.     

Effects of correlated and synchronized stochastic inputs to leaky integrator neuronal model

We present an analysis of neuronal model behaviour with correlated synaptic inputs including the cases that correlated inputs are equivalent to exactly synchronized inputs and correlated inputs are not equivalent to exactly synchronized inputs. For the former case, it is found that the fully (synaptically) correlated inputs assumption (see Section 1 for definition), which is used in most, if not all, theoretical and experimental work in the past few years, results in a waste of resources and might be an unrealistic assumption; with an exactly balanced excitatory and inhibitory, and synaptically correlated input, the integrate-and-fire model simply behaves as a synchrony detector in certain parameter regions; the well-known diffusion model, upon which most theoretical work is based, fails to approximate the model with synaptically correlated Poisson inputs. A novel way to approximate synaptically correlated Poisson inputs is then presented;an optimization principle on neuronal models with partially (synaptically) correlated inputs is proposed, which enables us to predict microscopic structures in neuronal systems. For the latter case,with tightly synchronized inputs (see Section 1 for definition), the model behaviour depends on its integration time of input signals and could exhibit bursting discharge.for loosely synchronized inputs, we found that correlated inputs are equivalent to the post-spike voltage reset mechanism proposed in the literature.     

“Addition” and “multiplication” theorems for the inputs of two neurons converging on a third

The output of a neuron innervated by two other neurons which, in turn, are subjected to two independent Poisson showers of stimuli, is derived as a function of the frequencies of the Poisson showers under two distinct assumptions, 1) where either of the two neurons can fire the third, and 2) where the stimuli from both neurons must impinge within a certain time interval to fire the third. For very small frequencies, the output of the third neuron is very nearly the sum of the input frequencies in the first case and proportional to the product of the input frequencies in the second case. Hence the designation “addition” and “multiplication” theorems. This treatment is a generalization of a previous treatment where the Poisson shower was assumed identical for the two outer neurons.     

Electroosmotic Flow in Nanoscale Parallel-plate Channels: Molecular Simulation Study and Comparison with Classical Poisson–Boltzmann Theory

Molecular dynamics simulations have been carried out for simple electrolyte systems to study the electrokinetically driven osmotic flow in parallel-plate channels of widths ～10–120?nm. The results are compared with the classical theory predictions based on the solution to the Poisson–Boltzmann equation. We find that despite some of the limitations in the Poisson–Boltzmann equation, such as assumption of the Boltzmann distribution for the ions, the classical theory captures the general trend of the variations of the osmotic flow with channel width, as characterized by the mobility of the fluid in channels between ～10 and 120?nm at moderate to low ion concentration. At moderate concentration (corresponding to relatively low surface potential), the classical theory is almost quantitative. The theory and simulation show more disagreement at low concentration, primarily caused by the high surface potential where the assumption of Boltzmann distribution becomes inaccurate. We discuss the limitations of the Poisson–Boltzmann equation as applied to the nanoscale channels.     

Analysis of Competition Binding Assays: Assessment of the Range of Validity of a Commonly Invoked Assumption

Abstract

A common assumption invoked in the analysis of competition binding assays is that the fractional saturation of sites with the unlabeled ligand is given by 1 - (the concentration of bound labeled ligand in the presence of unlabeled ligand)/(the concentration of bound labeled ligand in the absence of unlabeled ligand). This assumption is critically evaluated in the context of several binding models: (a) binding of univalent ligands to multiple classes of equivalent and independent sites, with and without nonspecific binding; (b) cooperative binding of univalent ligands; and (c) binding of multivalent ligands to a single class of univalent acceptors. We show that the conventional assumption is only valid when the labeled ligand is mainly in the free form, occupies a small fraction of the total sites and binds univalently to all sites in an equivalent and independent manner, and when the unlabeled ligand forms l : l complexes with the acceptor sites. When these conditions are satisfied, the conventional assumption is valid even if the unlabeled ligand binds to nonequivalent sites or exhibits cooperativity. Finally, we apply the theory derived for case (a) above to the binding of fluoresceinated epidermal growth factor to A431 cells and demonstrate that the analysis of data obtained from both conventional and competition assays provides information which is difficult, if not impossible, to obtain from either assay alone.     

On the generalized poisson regression mixture model for mapping quantitative trait loci with count data

Statistical methods for mapping quantitative trait loci (QTL) have been extensively studied. While most existing methods assume normal distribution of the phenotype, the normality assumption could be easily violated when phenotypes are measured in counts. One natural choice to deal with count traits is to apply the classical Poisson regression model. However, conditional on covariates, the Poisson assumption of mean-variance equality may not be valid when data are potentially under- or overdispersed. In this article, we propose an interval-mapping approach for phenotypes measured in counts. We model the effects of QTL through a generalized Poisson regression model and develop efficient likelihood-based inference procedures. This approach, implemented with the EM algorithm, allows for a genomewide scan for the existence of QTL throughout the entire genome. The performance of the proposed method is evaluated through extensive simulation studies along with comparisons with existing approaches such as the Poisson regression and the generalized estimating equation approach. An application to a rice tiller number data set is given. Our approach provides a standard procedure for mapping QTL involved in the genetic control of complex traits measured in counts.     

Distribution of Polymorphus minutus among its intermediate hosts

Hirsch R. P. 1979. Distribution of Polymorphus minutus among its intermediate hosts. International journal for Parasitology10: 243–248. In 1971, Crofton investigated patterns of distribution of Polymorphus minutus in the intermediate host, Gammarus pulex. Among his conclusions were: (1) P. minutus populations occur in patterns similar to negative binomial distributions, and (2) parasite-induced host mortality results in patterns similar to truncated (high end) negative binomial distributions. Those conclusions, however, were not tested by statistical analyses. To test Crofton's observations, Chi-square goodness of fit tests were applied to data used by Crofton and an additional two stations sampled by Hynes & Nicholas in 1963. Analyses were expanded to include five theoretical distributions, four patterns of host mortality and various rates of host mortality. Truncated forms of negative binomial, positive binomial and Poisson distributions were also investigated where nontruncated distributions failed to fit observed distributions. It was found that negative binomial distributions most frequently describe patterns of P. minutus distribution with the exception of one population described by Poisson and another by positive binomial distributions. Crofton's assumption that truncated distributions result from parasite-induced host mortality seems unlikely in light of those analyses.     

Dual synaptic plasticity in the hippocampus: Hebbian and spatiotemporal learning dynamics

We assume that Hebbian learning dynamics (HLD) and spatiotemporal learning dynamics (SLD) are involved in the mechanism of synaptic plasticity in the hippocampal neurons. While HLD is driven by pre- and postsynaptic spike timings through the backpropagating action potential, SLD is evoked by presynaptic spike timings alone. Since the backpropagation attenuates as it nears the distal dendrites, we assume an extreme case as a neuron model where HLD exists only at proximal dendrites and SLD exists only at the distal dendrites. We examined how the synaptic weights change in response to three types of synaptic inputs in computer simulations. First, in response to a Poisson train having a constant mean frequency, the synaptic weights in HLD and SLD are qualitatively similar. Second, SLD responds more rapidly than HLD to synchronous input patterns, while each responds to them. Third, HLD responds more rapidly to more frequent inputs, while SLD shows fluctuating synaptic weights. These results suggest an encoding hypothesis in that a transient synchronous structure in spatiotemporal input patterns will be encoded into distal dendrites through SLD and that persistent synchrony or firing rate information will be encoded into proximal dendrites through HLD.     

An age-stratified poisson model for comparing trends in cancer rates across overlapping regions

The annual percent change (APC) has been used as a measure to describe the trend in the age-adjusted cancer incidence or mortality rate over relatively short time intervals. The yearly data on these age-adjusted rates are available from the Surveillance, Epidemiology, and End Results (SEER) Program of the National Cancer Institute. The traditional methods to estimate the APC is to fit a linear regression of logarithm of age-adjusted rates on time using the least squares method or the weighted least squares method, and use the estimate of the slope parameter to define the APC as the percent change in the rates between two consecutive years. For comparing the APC for two regions, one uses a t-test which assumes that the two datasets on the logarithm of the age-adjusted rates are independent and normally distributed with a common variance. Two modifications of this test, when there is an overlap between the two regions or between the time intervals for the two datasets have been recently developed. The first modification relaxes the assumption of the independence of the two datasets but still assumes the common variance. The second modification relaxes the assumption of the common variance also, but assumes that the variances of the age-adjusted rates are obtained using Poisson distributions for the mortality or incidence counts. In this paper, a unified approach to the problem of estimating the APC is undertaken by modeling the counts to follow an age-stratified Poisson regression model, and by deriving a corrected Z -test for testing the equality of two APCs. A simulation study is carried out to assess the performance of the test and an application of the test to compare the trends, for a selected number of cancer sites, for two overlapping regions and with varied degree of overlapping time intervals is presented.     

The effect of travel loss on evolutionarily stable distributions of populations in space

A key assumption of the ideal free distribution (IFD) is that there are no costs in moving between habitat patches. However, because many populations exhibit more or less continuous population movement between patches and traveling cost is a frequent factor, it is important to determine the effects of costs on expected population movement patterns and spatial distributions. We consider a food chain (tritrophic or bitrophic) in which one species moves between patches, with energy cost or mortality risk in movement. In the two-patch case, assuming forced movement in one direction, an evolutionarily stable strategy requires bidirectional movement, even if costs during movement are high. In the N-patch case, assuming that at least one patch is linked bidirectionally to all other patches, optimal movement rates can lead to source-sink dynamics where patches with negative growth rates are maintained by other patches with positive growth rates. As well, dispersal between patches is not balanced (even in the two-patch case), leading to a deviation from the IFD. Our results indicate that cost-associated forced movement can have important consequences for spatial metapopulation dynamics. Relevance to marine reserve design and the study of stream communities subject to drift is discussed.     

Co‐occurrence models fail to infer underlying patterns of avoidance and aggregation when closure is violated

Advances in multi‐species monitoring have prompted an increase in the use of multi‐species occupancy analyses to assess patterns of co‐occurrence among species, even when data were collected at scales likely violating the assumption that sites were closed to changes in the occupancy state for the target species. Violating the closure assumption may lead to erroneous conclusions related to patterns of co‐occurrence among species. Occurrence for two hypothetical species was simulated under patterns of avoidance, aggregation, or independence, when the closure assumption was either met or not. Simulated populations were sampled at two levels (N = 250 or 100 sites) and two scales of temporal resolution for surveys. Sample data were analyzed with conditional two‐species occupancy models, and performance was assessed based on the proportion of simulations recovering the true pattern of co‐occurrence. Estimates of occupancy were unbiased when closure was met, but biased when closure violations occurred; bias increased when sample size was small and encounter histories were collapsed to a large‐scale temporal resolution. When closure was met and patterns of avoidance and aggregation were simulated, conditional two‐species models tended to correctly find support for non‐independence, and estimated species interaction factors (SIF) aligned with predicted values. By contrast, when closure was violated, models tended to incorrectly infer a pattern of independence and power to detect simulated patterns of avoidance or aggregation that decreased with smaller sample size. Results suggest that when the closure assumption is violated, co‐occurrence models often fail to detect underlying patterns of avoidance or aggregation, and incorrectly identify a pattern of independence among species, which could have negative consequences for our understanding of species interactions and conservation efforts. Thus, when closure is violated, inferred patterns of independence from multi‐species occupancy should be interpreted cautiously, and evidence of avoidance or aggregation is likely a conservative estimate of true pattern or interaction.     

Maximum-likelihood estimation of phylogeny from DNA sequences when substitution rates differ over sites

Felsenstein's maximum-likelihood approach for inferring phylogeny from DNA sequences assumes that the rate of nucleotide substitution is constant over different nucleotide sites. This assumption is sometimes unrealistic, as has been revealed by analysis of real sequence data. In the present paper Felsenstein's method is extended to the case where substitution rates over sites are described by the gamma distribution. A numerical example is presented to show that the method fits the data better than do previous models.      

Ligand-dependent aggregation and cooperativity: a critique

Ligand-dependent site-site (or subunit-subunit) interactions provide the basis for explaining cooperativity in chemical reactions. Even in the simplest possible nonaggregating system, interpretation of the interactions in terms of structural details requires an explicit assumption (or model) for the binding of the ligand to the sites when there are no interactions. This paper develops in detail the processes by which aggregation will yield ligand-dependent cooperativity. Two conceptually distinct free energy differences may contribute to cooperativity in an aggregation reaction. One is the free energy difference in ligand binding between the monomer and the aggregate. The other is derived from ligand-dependent interactions between the sites of the aggregate. In this analysis an explicit distinction is made between the experimentally accessible constants and those derived from assumed models. Experimental measurements of an aggregation cycle in which all of the species in equilibrium are defined do not allow for an evaluation of the energies of interaction without some model (or assumption). In the analysis presented, an explicit assumption is employed relating the constant for binding of the ligand to the isolated monomer and the constant for the binding of the ligand to aggregate under conditions where there are no ligand-dependent interactions.