Efficient biased estimation of evolutionary distances when substitution rates vary across sites

This paper deals with phylogenetic inference when the variability of substitution rates across sites (VRAS) is modeled by a gamma distribution. We show that underestimating VRAS, which results in underestimates for the evolutionary distances between sequences, usually improves the topological accuracy of phylogenetic tree inference by distance-based methods, especially when the molecular clock holds. We propose a method to estimate the gamma shape parameter value which is most suited for tree topology inference, given the sequences at hand. This method is based on the pairwise evolutionary distances between sequences and allows one to reconstruct the phylogeny of a high number of taxa (>1,000). Simulation results show that the topological accuracy is highly improved when using the gamma shape parameter value given by our method, compared with the true (unknown) value which was used to generate the data. Furthermore, when VRAS is high, the topological accuracy of our distance-based method is better than that of a maximum likelihood approach. Finally, a data set of Maoricicada species sequences is analyzed, which confirms the advantage of our method.     

Surface forces in lungs. II. Microstructural mechanics and lung stability

Recent lung microstructural models describing interactions between alveolar surface tension (gamma) and forces in structural elements of the alveolar duct predict that the component of lung recoil pressure due to gamma (P gamma) is proportional to gamma/V1/3, where V is the total lung volume. This relation is tested against experimental data obtained from pressure-volume measurements of excised rabbit lungs with different constant values of gamma. It is found that for values of gamma less than approximately 18 dyn/cm the data generally agree with the model predictions. With higher values of gamma, a mismatch between the data and predictions first occurs at low and high volumes and then spreads over the entire volume range. The mismatch at the lower volumes coincides with the appearance of nonuniformities of lung expansion. The nonuniformities are characterized by a coexistence of under- and overexpanded regions of the parenchyma referred to as a mixture of phases. These nonuniformities, as well as a pressure-volume curve with a shape similar to the shape of measured curves, are predicted from an analysis of lung stability. Results of this work indicate that if the lung expands uniformly, P gamma proportional to gamma/V1/3 is a good approximation over a wide range of volumes. The stability analysis indicates that the equilibrium configurations of the lung parenchyma when gamma is independent of interfacial area and elevated above normal values are nonuniform states of expansion, characterizable as a mixture of phases. This result confirms that a dependence of gamma on surface area is normally required to achieve stable, uniform states of lung expansion.     

A basic limitation on inferring phylogenies by pairwise sequence comparisons

Distance-based approaches in phylogenetics such as Neighbor-Joining are a fast and popular approach for building trees. These methods take pairs of sequences, and from them construct a value that, in expectation, is additive under a stochastic model of site substitution. Most models assume a distribution of rates across sites, often based on a gamma distribution. Provided the (shape) parameter of this distribution is known, the method can correctly reconstruct the tree. However, if the shape parameter is not known then we show that topologically different trees, with different shape parameters and associated positive branch lengths, can lead to exactly matching distributions on pairwise site patterns between all pairs of taxa. Thus, one could not distinguish between the two trees using pairs of sequences without some prior knowledge of the shape parameter. More surprisingly, this can happen for any choice of distinct shape parameters on the two trees, and thus the result is not peculiar to a particular or contrived selection of the shape parameters. On a positive note, we point out known conditions where identifiability can be restored (namely, when the branch lengths are clocklike, or if methods such as maximum likelihood are used).     

Continuous and tractable models for the variation of evolutionary rates

We propose a continuous model for variation in the evolutionary rate across sites and over the phylogenetic tree. We derive exact transition probabilities of substitutions under this model. Changes in rate are modelled using the CIR process, a diffusion widely used in financial applications. The model directly extends the standard gamma distributed rates across site model, with one additional parameter governing changes in rate down the tree. The parameters of the model can be estimated directly from two well-known statistics: the index of dispersion and the gamma shape parameter of the rates across sites model. The CIR model can be readily incorporated into probabilistic models for sequence evolution. We provide here an exact formula for the likelihood of a three-taxon tree. The likelihoods of larger trees can be evaluated using Monte-Carlo methods.     

Statistical Analysis of the Comet Assay Using a Mixture of Gamma Distributions

Tail moments in the single cell gel electrophoresis (comet) assay usually do not follow a normal distribution, making the statistical analysis complicated. Researchers have used a wide variety of statistical techniques in an attempt to overcome this problem. In many cases, the tail moments follow a bimodal distribution that can be modeled with a mixture of gamma distributions. This bimodality may be due to cells being in two different stages of the cell cycle at the time of treatment. Maximum likelihood, modified to accommodate censored data, can be used to estimate the five parameters of the gamma mixture distribution for each slide. A weighted analysis of variance on the parameter estimates for the gamma mixtures can be performed to determine differences in DNA damage between treatments. These methods were applied to an experiment on the effect of thymidine kinase in DNA damage and repair. Analysis based on the mixture of gamma distributions was found to be more statistically valid, more powerful, and more informative than analysis based on log-transformed tail moments.     

Mitochondrial mismatch analysis is insensitive to the mutational process

Mismatch distributions are histograms showing the pattern of nucleotide (or restriction) site differences between pairs of individuals in a sample. They can be used to test hypotheses about the history of population size and subdivision (if selective neutrality is assumed) or about selection (if a constant population size is assumed). Previous work has assumed that mutations never strike the same site twice, an assumption that is called the model of infinite sites. Fortunately, the results are surprisingly robust even when this assumption is violated. We show here that (1) confidence regions inferred using the infinite- sites model differ little from those inferred using a model of finite sites with uniform site-specific mutation rates, and (2) even when site- specific mutation rates follow a gamma distribution, confidence regions are little changed until the gamma shape parameter falls well below its plausible range, to roughly 0.01. In addition, we evaluate and reject the proposition that mismatch waves are produced by pooling data from several subdivisions of a structured population.      

Stage duration distributions in matrix population models

Matrix population models are a standard tool for studying stage‐structured populations, but they are not flexible in describing stage duration distributions. This study describes a method for modeling various such distributions in matrix models. The method uses a mixture of two negative binomial distributions (parametrized using a maximum likelihood method) to approximate a target (true) distribution. To examine the performance of the method, populations consisting of two life stages (juvenile and adult) were considered. The juvenile duration distribution followed a gamma distribution, lognormal distribution, or zero‐truncated (over‐dispersed) Poisson distribution, each of which represents a target distribution to be approximated by a mixture distribution. The true population growth rate based on a target distribution was obtained using an individual‐based model, and the extent to which matrix models can approximate the target dynamics was examined. The results show that the method generally works well for the examined target distributions, but is prone to biased predictions under some conditions. In addition, the method works uniformly better than an existing method whose performance was also examined for comparison. Other details regarding parameter estimation and model development are also discussed.     

Joint inference of the distribution of fitness effects of deleterious mutations and population demography based on nucleotide polymorphism frequencies

The distribution of fitness effects of new mutations (DFE) is important for addressing several questions in genetics, including the nature of quantitative variation and the evolutionary fate of small populations. Properties of the DFE can be inferred by comparing the distributions of the frequencies of segregating nucleotide polymorphisms at selected and neutral sites in a population sample, but demographic changes alter the spectrum of allele frequencies at both neutral and selected sites, so can bias estimates of the DFE if not accounted for. We have developed a maximum-likelihood approach, based on the expected allele-frequency distribution generated by transition matrix methods, to estimate parameters of the DFE while simultaneously estimating parameters of a demographic model that allows a population size change at some time in the past. We tested the method using simulations and found that it accurately recovers simulated parameter values, even if the simulated demography differs substantially from that assumed in our analysis. We use our method to estimate parameters of the DFE for amino acid-changing mutations in humans and Drosophila melanogaster. For a model of unconditionally deleterious mutations, with effects sampled from a gamma distribution, the mean estimate for the distribution shape parameter is approximately 0.2 for human populations, which implies that the DFE is strongly leptokurtic. For Drosophila populations, we estimate that the shape parameter is approximately 0.35. Differences in the shape of the distribution and the mean selection coefficient between humans and Drosophila result in significantly more strongly deleterious mutations in Drosophila than in humans, and, conversely, nearly neutral mutations are significantly less frequent.     

A third form of the G protein beta subunit. 2. Purification and biochemical properties.

Visual excitation in cones is thought to involve a cone-specific G protein (cone transducin) that transduces the light signal detected by the cone visual pigment into an increase in the enzymatic activity of a cGMP phosphodiesterase. In the preceding paper, we have shown that the G beta 3 isoform of G proteins is specifically localized in bovine cone photoreceptors and proposed that it might be a component of cone transducin. We reported here the purification from bovine retinal extract of a cone-specific T beta 3 gamma complex (where T is transducin), which is composed of a G beta 3 subunit and an immunochemically distinct G gamma subunit. Our purification of this complex is based on a two-stage procedure; the first stage consists of a series of column chromatographies that yield a mixture of purified T beta gamma substantially enriched in T beta 3 gamma, and the second stage involves the removal of all of the rod-specific T beta 1 gamma from the mixture using an affinity column of immobilized monoclonal antibodies directed against the rod T gamma subunit of transducin. Using this procedure, we were able to obtain sufficient amounts of T beta 1 gamma and T beta 3 gamma to begin a comparative study of their properties. We showed that T beta 3 gamma is distinguishable from T beta 1 gamma by isoelectric focusing under nondenaturing conditions. The G beta 3 polypeptide of T beta 3 gamma also migrates slightly slower than the G beta 1 polypeptide of T beta 1 gamma on denaturing polyacrylamide gels. Analysis of the interactions of T beta 3 gamma with other retinal proteins indicated that it has a lower affinity for the T alpha subunit of rod transducin but appears to complex with a phosducin-like protein. The differences in the intrinsic biochemical properties of T beta 3 gamma as compared to T beta 1 gamma may partially account for the lower light sensitivity of cones.     

Methods for studying recombination on chromosomes that undergo nondisjunction

A lod score method is provided for mapping genes relative to the centromere using family data from autosomal trisomies. Such gene-centromere mapping can be performed whenever two or more members of a meiotic tetrad can be recovered. The critical mapping parameter is not the recombination value theta or the map distance omega, but the probability of nonreduction in a heterozygous host, the probability of heterozygosity (nonreduction) is 1-gamma/2 for a meiosis I error and gamma for a meiosis II error. Under various assumptions regarding chiasma interference, gamma can be related to theta and omega. We provide specific methods for estimating gamma and theta from trisomy data using maximum likelihood, so that recombination may be studied on chromosomes that underwent nondisjunction.     

A simple method for estimating the parameter of substitution rate variation among sites

When the rate variation among sites is described by a gamma distribution, an important problem is how to estimate the shape parameter alpha, which is an index of the degree of among-site rate variation. The parsimony-based methods for estimating alpha are simple but biased, i.e., alpha tends to be overestimated. On the other hand, the likelihood-based methods are asymptotically unbiased but take a huge amount of computational time. In this paper, we have developed a new method to solve this problem: we first estimate the expected number of substitutions at each site, which is corrected for multiple hits, and then estimate the parameter alpha. Our method is computationally as fast as the parsimony method, and the estimation accuracy is much higher than that of parsimony and similar to that of the likelihood method.      

Log‐gamma linear‐mixed effects models for multiple outcomes with application to a longitudinal glaucoma study

Glaucoma is a progressive disease due to damage in the optic nerve with associated functional losses. Although the relationship between structural and functional progression in glaucoma is well established, there is disagreement on how this association evolves over time. In addressing this issue, we propose a new class of non‐Gaussian linear‐mixed models to estimate the correlations among subject‐specific effects in multivariate longitudinal studies with a skewed distribution of random effects, to be used in a study of glaucoma. This class provides an efficient estimation of subject‐specific effects by modeling the skewed random effects through the log‐gamma distribution. It also provides more reliable estimates of the correlations between the random effects. To validate the log‐gamma assumption against the usual normality assumption of the random effects, we propose a lack‐of‐fit test using the profile likelihood function of the shape parameter. We apply this method to data from a prospective observation study, the Diagnostic Innovations in Glaucoma Study, to present a statistically significant association between structural and functional change rates that leads to a better understanding of the progression of glaucoma over time.     

Use of mixture distributions to deconvolute the behavior of "hits" and controls in high-throughput screening data

The stochastic nature of high-throughput screening (HTS) data indicates that information may be gleaned by applying statistical methods to HTS data. A foundation of parametric statistics is the study and elucidation of population distributions, which can be modeled using modern spreadsheet software. The methods and results described here use fundamental concepts of statistical population distributions analyzed using a spreadsheet to provide tools in a developing armamentarium for extracting information from HTS data. Specific examples using two HTS kinase assays are analyzed. The analyses use normal and gamma distributions, which combine to form mixture distributions. HTS data were found to be described well using such mixture distributions, and deconvolution of the mixtures to the constituent gamma and normal parts provided insight into how the assays performed. In particular, the proportion of hits confirmed was predicted from the original HTS data and used to assess screening assay performance. The analyses also provide a method for determining how hit thresholds--values used to separate active from inactive compounds--affect the proportion of compounds verified as active and how the threshold can be chosen to optimize the selection process.     

Prediction of function divergence in protein families using the substitution rate variation parameter alpha

Protein families typically embody a range of related functions and may thus be decomposed into subfamilies with, for example, distinct substrate specificities. Detection of functionally divergent subfamilies is possible by methods for recognizing branches of adaptive evolution in a gene tree. As the number of genome sequences is growing rapidly, it is highly desirable to automatically detect subfamily function divergence. To this end, we here introduce a method for large-scale prediction of function divergence within protein families. It is called the alpha shift measure (ASM) as it is based on detecting a shift in the shape parameter (alpha [alpha]) of the substitution rate gamma distribution. Four different methods for estimating alpha were investigated. We benchmarked the accuracy of ASM using function annotation from Enzyme Commission numbers within Pfam protein families divided into subfamilies by the automatic tree-based method BETE. In a test using 563 subfamily pairs in 162 families, ASM outperformed functional site-based methods using rate or conservation shifting (rate shift measure [RSM] and conservation shift measure [CSM]). The best results were obtained using the "GZ-Gamma" method for estimating alpha. By combining ASM with RSM and CSM using linear discriminant analysis, the prediction accuracy was further improved.     

Shape analysis of gamma rhythm supports a superlinear inhibitory regime in an inhibition-stabilized network

Visual inspection of stimulus-induced gamma oscillations (30–70 Hz) often reveals a non-sinusoidal shape. Such distortions are a hallmark of non-linear systems and are also observed in mean-field models of gamma oscillations. A thorough characterization of the shape of the gamma cycle can therefore provide additional constraints on the operating regime of such models. However, the gamma waveform has not been quantitatively characterized, partially because the first harmonic of gamma, which arises because of the non-sinusoidal nature of the signal, is typically weak and gets masked due to a broadband increase in power related to spiking. To address this, we recorded local field potential (LFP) from the primary visual cortex (V1) of two awake female macaques while presenting full-field gratings or iso-luminant chromatic hues that produced huge gamma oscillations with prominent peaks at harmonic frequencies in the power spectra. We found that gamma and its first harmonic always maintained a specific phase relationship, resulting in a distinctive shape with a sharp trough and a shallow peak. Interestingly, a Wilson-Cowan (WC) model operating in an inhibition stabilized mode could replicate this shape, but only when the inhibitory population operated in the super-linear regime, as predicted recently. However, another recently developed model of gamma that operates in a linear regime driven by stochastic noise failed to produce salient harmonics or the observed shape. Our results impose additional constraints on models that generate gamma oscillations and their operating regimes.     

One interferon gamma receptor binds one interferon gamma dimer

We investigated the stoichiometry of the interferon gamma and interferon gamma receptor interaction, using recombinant interferon gamma and recombinant soluble interferon gamma receptor, applying chemical cross-linking and chromatographic techniques, and analyzing the resulting products in denaturing polyacrylamide gels. Interferon gamma cross-linked to itself produced a major band of an apparent molecular mass of 34 kDa, which suggests that it exists as a dimer in physiological buffer and which agrees with published data. Soluble interferon gamma receptor cross-linked to itself produced mainly a 28-kDa band, suggesting that the interferon gamma receptor exists as a monomer. Interferon gamma cross-linked to the soluble interferon gamma receptor resulted in the formation of two main products of apparent molecular masses of 60 and 44 kDa. The predominant 60-kDa band resulted from the cross-linking of one interferon gamma dimer (34 kDa) to one interferon gamma receptor molecule (27 kDa). The 44-kDa band was formed by the cross-linking of one interferon gamma molecule to one interferon gamma receptor. Kinetic studies showed that the cross-linking of interferon gamma dimer to the soluble receptor proceeds through the intermediate formed by cross-linking one molecule of the interferon gamma dimer to the receptor. Reducing and dissociating agents inhibited complex formation. When chromatographed on Sephadex G-100, interferon gamma was eluted as a protein of 34-kDa molecular mass, the soluble interferon gamma receptor as a protein of 40 kDa, and their mixture was eluted in one peak corresponding to an apparent molecular mass of 73 kDa. Sodium dodecyl sulfate-polyacrylamide gel analysis of the eluted mixture showed the presence of both interferon gamma and interferon gamma receptor at a ratio of 2:1. The found results suggest that the interferon gamma receptor binds interferon gamma as a dimer.     

Position of gamma-chain carboxy-terminal regions in fibrinogen/fibrin cross-linking mixtures

There are conflicting ideas regarding the location of the carboxyl-terminal regions of cross-linked gamma-chain dimers in double-stranded fibrin fibrils. Some investigators believe that the chains are always oriented longitudinally along each fibril strand and traverse the contacting ends of abutting fibrin D domains ("DD-long" cross-linking). Other investigations have indicated instead that the chains are situated transversely between adjacent D domains in opposing fibril strands (transverse cross-linking). To distinguish between these two possibilities, the gamma dimer composition of factor XIIIa-cross-linked fibrin/fibrinogen complexes that had been formed through noncovalent D/E interactions between fibrinogen D domains and fibrin E domains was examined. Two factor XIIIa-mediated cross-linking conditions were employed. In the first, fibrin/fibrinogen complexes were formed between (125)I-labeled fibrinogen 2 ("peak 2" fibrinogen), each heterodimeric molecule containing one gamma(A) and one larger gamma' chain, and nonlabeled fibrin 1 molecules ("peak 1" fibrin), each containing two gamma(A) chains. If DD-long cross-linking occurred, (125)I-labeled gamma(A)-gamma(A), gamma(A)-gamma', and gamma'-gamma'dimers in a 1:2:1 ratio would result. Transverse cross-linking would yield a 1:1 mixture of (125)I-labeled gamma(A)-gamma(A) and gamma(A)-gamma' dimers, without any gamma'-gamma' dimers. Autoradiographic analyses of reduced SDS-PAGE gels from protocol 1 revealed (125)I-labeled gamma(A)-gamma(A) and gamma(A)-gamma' dimers at a ratio of approximately 1:1. No labeled gamma'-gamma' dimers were detected. Protocol 2 used a converse mixture, (125)I-fibrin 2 and nonlabeled fibrinogen 1. DD-long cross-linking of this mixture would yield only nonradioactive gamma(A)-gamma(A) dimers, whereas transverse cross-linking would yield a 1:1 mixture of (125)I-labeled gamma(A)-gamma(A) and gamma(A)-gamma' dimers. Autoradiographic analyses of this mixture yielded (125)I-labeled gamma(A)-gamma(A) and gamma(A)-gamma' dimers in a 1:1 ratio. These findings provide no evidence that longitudinal (DD-long) gamma chain positioning occurs in cross-linked fibrin and indicate instead that most, if not all, gamma-chain positioning in an assembled fibrin polymer is transverse.     

Distribution of epidermal ridge minutiae

The distribution of epidermal ridge minutiae on the distal portion of male human thumbprints has been characterized. For each of 412 thumbprints, a centrally located focal minutia was chosen and neighboring minutiae were sampled. Minutiae were considered to be neighbors if there were no other minutiae appearing in the intervening region defined by the two minutia events and the ridge system. For each neighbor minutia, the total ridge distance between the focal and neighbor minutiae was measured. This distance is the total length of ridges appearing in the intervening region. The number of neighbor minutiae occurring about the focal minutia was found to be normally distributed with a mean of 6.42 (n = 412). The distribution of total ridge distances was not adequately described by the negative exponential distribution, but was well described by a gamma distribution with a shape parameter of 0.193 and a scale parameter of 5.91 mm. This gamma distribution reflects a local overdispersion of minutiae. This study is noteworthy as the first to describe the distribution of epidermal ridge minutiae within the ridge structure. The results support prior work based on quadrat sampling and eliminate two possible sources of error. A possible explanation for the overdispersed distribution lies in the growth stress model for minutia formation. Minutia formation may alleviate local growth stress, thereby removing the impetus for formation of additional minutiae in the immediately surrounding region.     

Nonideal mixing of phosphatidylserine and phosphatidylcholine in the fluid lamellar phase.

The mixing of phosphatidylserine (PS) and phosphatidylcholine (PC) in fluid bilayer model membranes was studied by measuring binding of aqueous Ca2+ ions. The measured [Ca2+]aq was used to derive the activity coefficient for PS, gamma PS, in the lipid mixture. For (16:0, 18:1) PS in binary mixtures with either (16:0, 18:1)PC, (14:1, 14:1)PC, or (18:1, 18:1)PC, gamma PS > 1; i.e., mixing is nonideal, with PS and PC clustered rather than randomly distributed, despite the electrostatic repulsion between PS headgroups. To understand better this mixing behavior, Monte Carlo simulations of the PS/PC distributions were performed, using Kawasaki relaxation. The excess energy was divided into an electrostatic term Uel and one adjustable term including all other nonideal energy contributions, delta Em. Uel was calculated using a discrete charge theory. Kirkwood's coupling parameter method was used to calculate the excess free energy of mixing, delta GEmix, hence In gamma PS,calc. The values of In gamma PS,calc were equalized by adjusting delta Em in order to find the simulated PS/PC distribution that corresponded to the experimental results. We were thus able to compare the smeared charge calculation of [Ca2+]surf with a calculation ("masked evaluation method") that recognized clustering of the negatively charged PS: clustering was found to have a modest effect on [Ca2+]surf, relative to the smeared charge model. Even though both PS and PC tend to cluster, the long-range nature of the electrostatic repulsion reduces the extent of PS clustering at low PS mole fraction compared to PC clustering at an equivalent low PC mole fraction.