On the distribution of developmental errors: comparing the normal, gamma, and log-normal distribution

The amount of between‐individual variation in the unobservable developmental instability (DI) has been the subject of intense recent debates. The unexpectedly high estimates of between‐individual variation in DI based on distributional characteristics of observable asymmetry values (of on average bilaterally symmetric traits) rely on statistical models that assume an underlying normal distribution of developmental errors. This prompted doubts on the assumption of the Gaussian nature of developmental errors. However, when applying other candidate distributions [log‐normal and gamma (γ)], recent analyses of empirical datasets have indicated that estimates remain generally high. Yet, all estimates were based on bilaterally symmetric traits, which did not allow for a formal comparison of the alternative distributions. In the present study, we extend a recent statistical model to allow statistical comparison of the different distributions based on traits that developed repeatedly under the same conditions, such as flower traits and regrown feathers. We analyse simulated and empirical data and show that: (1) it is statistically difficult to differentiate among the three alternatives when variances are small relative to the mean, as is often the case with DI; (2) the normal distribution fits the log‐normal or γ relatively well under those circumstances; (3) the deviance information criterion (DIC) is able to pick up differences in model fit among the three alternative distributions, yet more strongly so when levels of DI were high; (4) empirical datasets show a better fit of the normal over the log‐normal and γ‐distributions as judged by the DIC; and (5) estimates of between‐individual variation in DI in the three empirical datasets were relatively high (> 50%) under each distributional assumption. In conclusion, and based on our three datasets, the normal approximation appears to be a reasonable choice for statistical models of DI and the remarkably high estimates of variation in DI cannot be attributed to non‐normal developmental noise. Nevertheless, our method should be applied to a broad range of traits and organisms to evaluate the generality of this result. We argue that there is an urgent need for studies that reveal the underlying mechanisms of developmental noise and stability, as well as the role of developmental selection, in order to be able to determine the biological importance of the highly skewed distributions of developmental instability often observed. © 2007 The Linnean Society of London, Biological Journal of the Linnean Society, 2007, 92 , 197–210.     

The distribution of the intervals between neural impulses in the maintained discharges of retinal ganglion cells

Simulated neural impulse trains were generated by a digital realization of the integrate-and-fire model. The variability in these impulse trains had as its origin a random noise of specified distribution. Three different distributions were used: the normal (Gaussian) distribution (no skew, normokurtic), a first-order gamma distribution (positive skew, leptokurtic), and a uniform distribution (no skew, platykurtic). Despite these differences in the distribution of the variability, the distributions of the intervals between impulses were nearly indistinguishable. These inter-impulse distributions were better fit with a hyperbolic gamma distribution than a hyperbolic normal distribution, although one might expect a better approximation for normally distributed inverse intervals. Consideration of why the inter-impulse distribution is independent of the distribution of the causative noise suggests two putative interval distributions that do not depend on the assumed noise distribution: the log normal distribution, which is predicated on the assumption that long intervals occur with the joint probability of small input values, and the random walk equation, which is the diffusion equation applied to a random walk model of the impulse generating process. Either of these equations provides a more satisfactory fit to the simulated impulse trains than the hyperbolic normal or hyperbolic gamma distributions. These equations also provide better fits to impulse trains derived from the maintained discharges of ganglion cells in the retinae of cats or goldfish. It is noted that both equations are free from the constraint that the coefficient of variation (CV) have a maximum of unity. The concluding discussion argues against the random walk equation because it embodies a constraint that is not valid, and because it implies specific parameters that may be spurious.     

Effect of Selection on a Quantitative Characteristic

In this paper we study the effect of selection procedures on certain parameters of the distribution function (d.f.) (such as mean, percentile etc.) of a quantitative characteristic X, in successive generations when the d.f. is governed by a single locus. We study the changes in gene frequencies under the truncation and genotype selection procedures by obtaining approximations to the gene frequencies, since exact expressions are not available. Using these approximations for the gene frequencies, we compute the selection differentials of X for different values of n, the number of generations. We also obtain the limiting distributions as n → ∞ and compute the number of generations required for the above parameter(s) of the d.f. to reach a value close enough to the limiting value.     

Density gradient proportionality constants,density distributions,and pressure effects for three salt solutions

Earlier determinations of density gradient proportionality constants β⁰, density distributions ρ(r), and the effect of pressure on density gradients in the analytical ultracentrifuge have been of limited precision and usefulness in the study of proteins and polypeptides. Reasons for these difficulties are that numerous intermediate relationships were required in the calculations, and the density ranges studied were generally above 1.2 g/ml. Relations are derived in the present paper to directly compute β⁰(ρ) values and β⁰′(ρ) values from the original data without any intermediate expansions or approximations. Data are presented for CsCl, CsBr, and Cs₂SO₄ and compared with literature values. Density distributions are computed for all three salts under a wide variety of experimental conditions of density, column length, and angular velocity. These values of ρ(r) and r_e are obtained by a numerical iterative technique. Values obtained by this new method are compared with values obtained using closed-form expressions. The effects of pressure on the composition density gradient for the three salts given above are calculated and found to be significant for Cs₂SO₄ solutions.     

Recent Selection on Synonymous Codon Usage in Drosophila

Evidence from a variety of sources indicates that selection has influenced synonymous codon usage in Drosophila. It has generally been difficult, however, to distinguish selection that acted in the distant past from ongoing selection. However, under a neutral model, polymorphisms usually reflect more recent mutations than fixed differences between species and may, therefore, be useful for inferring recent selection. If the ancestral state is preferred, selection should shift the frequency distribution of derived states/site toward lower values; if the ancestral is unpreferred, selection should increase the number of derived states/site. Polymorphisms were classified as ancestrally preferred or unpreferred for several genes of D. simulans and D. melanogaster. A computer simulation of coalescence was employed to derive the expected frequency distributions of derived states/site under various modifications of the Wright–Fisher neutral model, and distributions of test statistics (t and Mann–Whitney U) were derived by appropriate sampling. One-tailed tests were applied to transformed frequency data to assess whether the two frequency distributions deviated from neutral expectations in the direction predicted by selection on codon usage. Several genes from D. simulans appear to be subject to recent selection on synonymous codons, including one gene with low codon bias, esterase-6. Selection may also be acting in D. melanogaster. Received: 15 April 1998 / Accepted: 13 May 1999     

A diversity of beta diversities: straightening up a concept gone awry. Part 1. Defining beta diversity as a function of alpha and gamma diversity

The term beta diversity has been used to refer to a wide variety of phenomena. Although all of these encompass some kind of compositional heterogeneity between places, many are not related to each other in any predictable way. The present two‐part review aims to put the different phenomena that have been called a beta component of diversity into a common conceptual framework, and to explain what each of them measures. In this first part, the focus is on defining beta diversity. This involves deciding what diversity is and how the observed total or gamma diversity (γ) is partitioned into alpha (α) and beta (β) components. Several different definitions of “beta diversity” that result from these decisions have been used in the ecological literature. True beta diversity is obtained when the total effective number of species in a dataset (true gamma diversityγ) is multiplicatively partitioned into the effective number of species per compositionally distinct virtual sampling unit (true alpha diversityα_d) and the effective number of such compositional units (β_Md=γ/α_d). All true diversities quantify the effective number of types of entities. Because the other variants of “beta diversity” that have been used by ecologists quantify other phenomena, an alternative nomenclature is proposed here for the seven most popular beta components: regional‐to‐local diversity ratio, two‐way diversity ratio, absolute effective species turnover (=regional diversity excess), Whittaker's effective species turnover, proportional effective species turnover, regional entropy excess and regional variance excess. In the second part of the review, the focus will be on how to quantify these phenomena in practice. This involves deciding how the sampling units that contribute to total diversity are selected, and whether the entity that is quantified is all of “beta diversity”, a specific part of “beta diversity”, the rate of change in “beta diversity” in relation to a given external factor, or something else.     

Evaluation of serum protein electrophoresis in greater rhea (<Emphasis Type="Italic">Rhea americana</Emphasis> Linnaeus, 1758)

Serum proteins from 47 healthy greater rheas (Rhea americana; male and female) were separated by electrophoresis in order to characterize normal reference ranges. Determination of total protein concentration was performed through biuret reaction. The mean value of total serum protein was 4.4 g/dL. Absolute concentrations of serum proteins were determined by agarose gel electrophoretic fractioning. Five fractions were analyzed: albumin, alpha, beta1, beta2, and gamma globulins, and the average values in grams per deciliter were 2.39, 0.32, 0.45, 0.30, and 0.79, respectively.     

Model Construction and Model Examination on the Basis of Simulated Selection

By reason nonlinear relations founded between selection differential and realised selection response we have been made investigations about variants of the genetic-statistical model, which include this nonlinearity. The variations of the model would not only referred to the postulate pattern of the connection between phenotype, genotype and environment but also enclosed the postulate assumption about the distribution of the variates. In an investigated special case the linear model equation P = G ± e was held, however the distributions of P and G were defined over a limited range in one direction. For P we have defined a modified normal distribution and the distribution of the random vector (G, e) non normal regarded with cov (G, e) ≠ 0, By means of a solution set of an integral equation a density function of the random vector (P, G) has been received, in which the expectation of the selection response of the usual genetic-statistical model approximate is included as a special case. The genetical parameters has been derived, which result from changed model. However their representation was only possible partially as an integral function. A subsequent paper informs of the examination this mode! variants, which depend on a parameter of the nonlinearity c.     

Group coancestry-controlled selection in a Pinus sylvestris L. breeding program

 Integer Linear Programming was used to maximize genetic gain from selection at a given level of relatedness. Variances and breeding values for total height were available for 296 plus-trees of Pinus sylvestris which had been evaluated by open-pollinated progeny testing at a single test site in northern Sweden. Second-generation breeding and selection scenarios for this breeding population were evaluated using simulated data derived deterministically from normal distributions of estimated breeding values of progeny around mid-parent family means. The study considered two mating designs, assortative and non-assortative single-pair mating, and two selection criteria, individual phenotype and performance of half-sib progeny. Relatedness (group coancestry) was restricted to a level equivalent to that given by within-family selection of 2 trees per family from each of 25 families (the current standard in Sweden). Selection that allows the best-performing families to contribute a greater number of progeny was superior, both when the breeding population size was limited to 50 individuals and when it was allowed to be larger. The selected set giving the greatest average breeding value under restricted group coancestry included the best individual from families that would have been rejected under application of standard within-family selection. We made a comparison of the present value on retrieved gain between phenotypic selection and evaluation by progeny testing. Received: 24 November 1998 / Accepted: 14 December 1998     

Reversal time distribution in the perception of visual ambiguous stimuli

Reversal of perspective for ambiguous optical stimuli (Necker cube, Schröder staircase, honeycomb) has been studied, determining the statistical distribution of time intervals spent on each percept. The experimental distributions can be fitted with the gamma function, characterized by two parameters n, b. The two parameters are not independent, showing a correlatiomn = 0.74.Subsequent intervals appear to be largely independent; from the beta distribution for the fraction of time spent on a given percept, one can show that the subjects differ only in regard to the variance of this variable.     

Probability distributions of ancestries and genealogical distances on stochastically generated rooted binary trees

The stationary birth-only, or Yule-Furry, process for rooted binary trees has been analysed with a view to developing explicit expressions for two fundamental statistical distributions: the probability that a randomly selected leaf is preceded by N nodes, or “ancestors”, and the probability that two randomly selected leaves are separated by N nodes. For continuous-time Yule processes, the first of these distributions is presented in closed analytical form as a function of time, with time being measured with respect to the moment of “birth” of the common ancestor (which is essentially inaccessible to phylogenetic analysis), or with respect to the instant at which the first bifurcation occurred.The second distribution is shown to follow in an iterative manner from a hierarchy of second-order ordinary differential equations.For Yule trees of a given number n of tips, expressions have been derived for the mean and variance for each of these distributions as functions of n, as well as for the distributions themselves.In addition, it is shown how the methods developed to obtain these distributions can be employed to find, with minor effort, expressions for the expectation values of two statistics on Yule trees, the Sackin index (sum over all root-to-leaf distances), and the sum over all leaf-to-leaf distances.     

The mouse salivary androgen-binding protein (ABP) gene cluster on Chromosomes 7: characterization and evolutionary relationships

Mouse salivary androgen-binding protein (ABP) is a pair of dimers, composed of an alpha subunit disulfide bridged to either a beta or a gamma subunit. It has been proposed that each subunit is encoded by a distinct gene: Abpa, Abpb, and Abpg for the alpha, beta, and gamma subunits, respectively. We report here the structures and sequences of the genes that encode these three subunits. Each gene has three exons separated by two introns. Mouse salivary ABP is a member of the secretoglobin family, and we compare the structure of the three ABP subunit genes to those of 18 other mammalian secretoglobins. We map the three genes as a gene cluster located 10 cM from the centromere of Chromosome (Chr) 7 and show that Abpa is the closest of the three to the gene for glucose phosphate isomerase (GPI) and that Abpg is the closest to the centromere, with Abpb mapping between them. Abpa is oriented in the opposite direction to Abpb and Abpg, with its 5 end directed toward their 5 ends. We compare the location of these genes with other secretoglobin genes in the mouse genome and with the known locations of secretoglobin genes in the human genome and present evidence that strong positive selection has driven the divergence of the coding regions of Abpb and Abpg since the putative duplication event that created them. The nucleotide sequence data reported in this paper have been submitted to GenBank and have been assigned the accession numbers Abpa: AF144714; Abpb: AY325897; Abpg: 325898. *Current address: (Christina M. Laukaitis) Department of Internal Medicine, St. Vincent Hospital, 2100 W. 86th St., Indianapolis, IN 46260, USA     

Alpha status and agonistic alliance in wild chimpanzees (Pan troglodytes schweinfurthii)

Intermale competition was studied in a habituated group of wild chimpanzees (Pan troglodytes schweinfurthii) in the Mahale Mountains of Tanzania. The group of 27 individuals included 3 adult males and 12 adult females. The beta-ranked male was observed to attack the alpha male and usurp his status. The beta male was supported by an ally, the gamma male. Earlier in the competition the gamma male had supported the alpha male. When the alpha male was displaced, he completely lost what had been an exclusive copulatory right. It was clear that alliance with the most subordinate male was a critically effective strategy for the major competitors. The alliance strategy of the gamma male invested him with important power. The strategy of changing alliance is referred to as “allegiance fickleness.” It is suggested that allegiance fickleness could provide a basis for individual selection of social intelligence.     

Distribution of basement membrane anchoring molecules in normal and transformed endometrium: Altered expression of laminin γ2 chain and collagen type XVII in endometrial adenocarcinomas

Basement membranes (BMs) play an important role in anchoring epithelial cells and separating them from the adjacent stroma. Altered composition and assembly of BMs may influence carcinoma cell growth and invasion. Using immunohistochemistry and in situ hybridization, we investigated the expressions of the BMs components laminin-5 (Ln-5) subunits and collagen types IV, VII and XVII in normal endometrium and compared them to the expression pattern in hyperplastic and neoplastic endometrium. Chains of Ln-5 (332) and types IV, VII and XVII collagens were observed in normal endometrium. In hyperplastic endometrium, laminin 2 chain and type XVII collagen showed intensified expression in foci of dispersed epithelial cells. Individual carcinoma cells in adenocarcinomas of low differentiation grade displayed increased laminin 2 chain and type XVII collagen immunoreactivity and mRNA synthesis, whereas type VII collagen usually showed decreased expression. Laminin and type IV collagen showed BM disruptions, especially in tumors with low differentiation. Our results indicate that all the BM anchoring molecules investigated are expressed in normal endometrium, but the expression of laminin 2 chain and collagen type XVII is altered in endometrial adenocarcinomas, which support their role in malignant growth.     

New roles for the gamma rhythm: population tuning and preprocessing for the Beta rhythm

Gamma (30–80 Hz) and beta (12–30 Hz) oscillations such as those displayed by in vitro hippocampal (CA1) slice preparations and by in vivo neocortical EEGs often occur successively, with a spontaneous transition between them. In the gamma rhythm, pyramidal cells fire together with the interneurons, while in the beta rhythm, pyramidal cells fire on a subset of cycles of the interneurons. It is shown that gamma and beta rhythms have different properties with respect to creation of cell assemblies. In the presence of heterogeneous inputs to the pyramidal cells, the gamma rhythm creates an assembly of firing pyramidal cells from cells whose drive exceeds a threshold. During the gamma to beta transition, a slow outward potassium current is activated, and as a result the cell assembly vanishes. The slow currents make each of the pyramidal cells fire with a beta rhythm, but the field potential of the network still displays a gamma rhythm. Hebbian changes of connections among the pyramidal cells give rise to a beta rhythm, and the cell assemblies are recovered with a temporal separation between cells firing in different cycles. We present experimental evidence showing that such a separation can occur in hippocampal slices.     

Organ-specific crystalline structures of ferritin cores inβ-thalassemia/hemoglobin E

Summary The cores of ferritins isolated from different organs of human subjects with-thalassemia/hemoglobin E (-thal/HbE) disease have different size distributions and crystallinities depending on the source organ. These patients have not been treated by hypertransfusion regimen or iron chelation therapy.-Thal/HbE spleens and livers yield ferritin cores which are less crystalline than those isolated from normal spleens and livers, reflecting the more rapid deposition of iron in the diseased state. Ferritins isolated from the hearts and pancreases of-thal/HbE subjects were found to have larger, more crystalline cores than those from the-thal/HbE livers and spleens, possibly as a consequence of the role of the heart and pancreas as long-term iron deposition sites in this iron overload pathology.     

The gamma-subunit of ENaC is more important for channel surface expression than the beta-subunit

The amiloride-sensitiveepithelial sodium channel (ENaC) plays a critical role in fluid andelectrolyte homeostasis and is composed of three homologous subunits:, , and . Only heteromultimeric channels made of ENaCare efficiently expressed at the cell surface, resulting in maximallyamiloride-sensitive currents. To study the relative importance ofvarious regions of the - and -subunits for the expression offunctional ENaC channels at the cell surface, we constructedhemagglutinin (HA)-tagged --chimeric subunits composed of -and -subunit regions and coexpressed them with HA-tagged - and-subunits in Xenopus laevis oocytes. The whole cellamiloride-sensitive sodium current (I_ami) andsurface expression of channels were assessed in parallel using thetwo-electrode voltage-clamp technique and a chemiluminescence assay.Because coexpression of ENaC resulted in largerI_ami and surface expression compared withcoexpression of ENaC, we hypothesized that the -subunit ismore important for ENaC trafficking than the -subunit. Usingchimeras, we demonstrated that channel activity is largely preservedwhen the highly conserved second cysteine rich domains (CRD2) of the- and -subunits are exchanged. In contrast, exchanging the wholeextracellular loops of the - and the -subunits largely reducedENaC currents and ENaC expression in the membrane. This indicates thatthere is limited interchangeability between molecular regions of thetwo subunits. Interestingly, our chimera studies demonstrated that theintracellular termini and the two transmembrane domains of ENaC aremore important for the expression of functional channels at the cellsurface than the corresponding regions of ENaC.

     

Simultaneous Comparisons of Multiple Treatments to Two (or More) Control

DUNNETT (1955) developed a procedure simultaneously comparing k treatments to one control with an exact overall type I error of α when all sampling distributions are normal. Sometimes it is desirable to compare k treatments to m≧2 controls, in particular to two controls. For instance, several new therapies (e.g., pain relievers) could be compared to two standard therapies (e.g., Aspirin and Tylenol). Alternatively, a standard therapy could be very expensive, difficult to apply and/or have bad side effects, making it useful to compare each new therapy to both standard therapy and no therapy (Placebo). Dunnett's method is expanded here to give comparisons of mean values for k treatments to mean values for m≧2 controls at an exact overall type I error of α when all sampling distributions are normal. Tabled values needed to make exact simultaneous comparisons at α = .05 are given for m = 2. An application is made to an example from the literature.     

Patterns of nucleotide sequence variation in <Emphasis Type="Italic">ICAM1</Emphasis> and <Emphasis Type="Italic">TNF</Emphasis> genes in twelve ethnic groups of India: roles of demographic history and natural selection

We have studied DNA sequence variation in and around the genes ICAM1 and TNF, which play functional and correlated roles in inflammatory processes and immune cell responses, in 12 diverse ethnic groups of India, with a view to investigating the relative roles of demographic history and natural selection in shaping the observed patterns of variation. The total numbers of single nucleotide polymorphisms (SNPs) detected at the ICAM1 and TNF loci were 29 and 12, respectively. Haplotype and allele frequencies differed significantly across populations. The site frequency spectra at these loci were significantly different from those expected under neutrality, and showed an excess of intermediate-frequency variants consistent with balancing selection. However, as expected under balancing selection, there was no significant reduction of F _ST values compared to neutral autosomal loci. Mismatch distributions were consistent with population expansion for both loci. On the other hand, the phylogenetic network among haplotypes for the TNF locus was similar to expectations under population expansion, while that for the ICAM1 was as expected under balancing selection. Nucleotide diversity at the ICAM1 locus was an order of magnitude lower in the promoter region, compared to the introns or exons, but no such difference was noted for the TNF gene. Thus, we conclude that the pattern of nucleotide variation in these genes has been modulated by both demographic history and selection. This is not surprising in view of the known allelic associations of several polymorphisms in these genes with various diseases, both infectious and noninfectious.     

A Report on a Monte Carlo Simulation of the Sampling Distribution of the Incidence Test and its Two Related Tests

A large-scale (5000 throws) Monte Carlo simulation experiment was carried out to study the nature of the sampling distributions of the incidence test and its two related tests, the FRIEDMAN test and the dual test, and to evaluate the goodness of the proposed gamma approximation relative to the conventional chi-square approximation. For all three tests, on the basis of the experimental results obtained for k = 3, 4, 5, and N = k(1) (120/k), the gamma approximation should be preferred over the chi-square. Exact companion tabulations show that use of the chi-square approximation entails an appreciable conservative bias. Extensive listings are provided of the moments of the experimental distributions of the three test statistics, and of the observed distributions of the corresponding gamma probabilities. The simulation experiment was run on a UNIVAC 1108 large-scale computer system.