Statistical analysis of in situ hybridization data: derivation and use of the zmax test.

There are many situations in which grain distributions resulting from in situ hybridization of radioactively labeled probes to unique genes should be subjected to a statistical analysis. However, the problems posed by analysis of in situ hybridization data are not straightforward, and no completely satisfying method is currently available. We have developed a procedure in which the major and any number of minor site(s) of hybridization may be specifically located and the significance of each tested. This zmax procedure first tests the overall distribution for departure from randomness and then identifies significantly overlabeled whole chromosomes (or chromosome arms or other large segments), a process that may be repeated to pinpoint significantly overlabeled regions within these chromosomes. We describe in detail the derivation of the zmax statistic, present tables of significant zmax levels, and show with examples how zmax is used in tests of significance of in situ hybridization data.     

Effect of postnatal development on calcium currents and slow charge movement in mammalian skeletal muscle

Single- (whole-cell patch) and two-electrode voltage-clamp techniques were used to measure transient (Ifast) and sustained (Islow) calcium currents, linear capacitance, and slow, voltage-dependent charge movements in freshly dissociated fibers of the flexor digitorum brevis (FDB) muscle of rats of various postnatal ages. Peak Ifast was largest in FDB fibers of neonatal (1-5 d) rats, having a magnitude in 10 mM external Ca of 1.4 +/- 0.9 pA/pF (mean +/- SD; current normalized by linear fiber capacitance). Peak Ifast was smaller in FDB fibers of older animals, and by approximately 3 wk postnatal, it was so small as to be unmeasurable. By contrast, the magnitudes of Islow and charge movement increased substantially during postnatal development. Peak Islow was 3.6 +/- 2.5 pA/pF in FDB fibers of 1-5-d rats and increased to 16.4 +/- 6.5 pA/pF in 45-50-d-old rats; for these same two age groups, Qmax, the total mobile charge measurable as charge movement, was 6.0 +/- 1.7 and 23.8 +/- 4.0 nC/microF, respectively. As both Islow and charge movement are thought to arise in the transverse-tubular system, linear capacitance normalized by the area of fiber surface was determined as an indirect measure of the membrane area of the t-system relative to that of the fiber surface. This parameter increased from 1.5 +/- 0.2 microF/cm2 in 2-d fibers to 2.9 +/- 0.4 microF/cm2 in 44-d fibers. The increases in peak Islow, Qmax, and normalized linear capacitance all had similar time courses. Although the function of Islow is unknown, the substantial postnatal increase in its magnitude suggests that it plays an important role in the physiology of skeletal muscle.     

Statistical analysis of in situ hybridization data: Derivation and use of the Zmax test

There are many situations in which grain distributions resulting from in situ hybridization of radioactively labeled probes to unique genes should be subjected to a statistical analysis. However, the problems posed by analysis of in situ hybridization data are not straightforward, and no completely satisfying method is currently available. We have developed a procedure in which the major and any number of minor site(s) of hybridization may be specifically located and the significance of each tested. This Z_max procedure first tests the overall distribution for departure from randomness and then identifies significantly overlabeled whole chromosomes (or chromosome arms or other large segments), a process that may be repeated to pinpoint significantly overlabeled regions within these chromosomes. We describe in detail the derivation of the Z_max statistic, present tables of significant Z_max levels, and show with examples how Z_max is used in tests of significance of in situ hybridization data.     

A resolution of the ascertainment sampling problem. II. Generalizations and numerical results

The ascertainment problem arises when families are sampled by a nonrandom process and some assumption about this sampling process must be made in order to estimate genetic parameters. Under classical ascertainment assumptions, estimation of genetic parameters cannot be separated from estimation of the parameters of the ascertainment process, so that any misspecification of the ascertainment process causes biases in estimation of the genetic parameters. Ewens and Shute proposed a resolution to this problem, involving conditioning the likelihood of the sample on the part of the data which is "relevant to ascertainment." The usefulness of this approach can only be assessed by examining the properties (in particular, bias and standard error) of the estimates which arise by using it for a wide range of parameter values and family size distributions and then comparing these biases and standard errors with those arising under classical ascertainment procedures. These comparisons are carried out in the present paper, and we also compare the proposed method with procedures which condition on, or ignore, parts of the data.     

Multifactorial analysis of family data ascertained through truncation: a comparative evaluation of two methods of statistical inference.

When family data are ascertained through single selection based on truncation, a prevailing method of analysis is to condition the likelihood function on the proband's actual phenotypic value. An alternative method conditions the likelihood function on the event that the proband's measurement lies in the truncation region. Both methods are contrasted here by using Monte Carlo simulations; identical sets of data were analyzed using both methods. The results suggest that, under either method, (1) parameter estimates are nearly unbiased and (2) likelihood-ratio tests of null hypotheses are approximately distributed as chi 2. However, conditioning on the proband's actual phenotypic value yields considerably less efficient estimates and reduced power for hypothesis tests. A corresponding result also holds under complete ascertainment. It is argued, therefore, that whenever sufficient information is available on the nature of truncation, the alternative approach should be used.     

The effects of a known family-size distribution on the estimation of genetic parameters.

We consider the question: In a segregation analysis, can knowledge of the family-size distribution (FSD) in the population from which a sample is drawn improve the estimators of genetic parameters? In other words, should one incorporate the population FSD into a segregation analysis if one knows it? If so, then under what circumstances? And how much improvement may result? We examine the variance and bias of the maximum likelihood estimators both asymptotically and in finite samples. We consider Poisson and geometric FSDs, as well as a simple two-valued FSD in which all families in the population have either one or two children. We limit our study to a simple genetic model with truncate selection. We find that if the FSD is completely specified, then the asymptotic variance of the estimator may be reduced by as much as 5%-10%, especially when the FSD is heavily skewed toward small families. Results in small samples are less clear-cut. For some of the simple two-valued FSDs, the variance of the estimator in small samples of one- and two-child families may actually be increased slightly when the FSD is included in the analysis. If one knows only the statistical form of the FSD, but not its parameter, then the estimator is improved only minutely. Our study also underlines the fact that results derived from asymptotic maximum likelihood theory do not necessarily hold in small samples. We conclude that in most practical applications it is not worth incorporating the FSD into a segregation analysis. However, this practice may be justified under special circumstances where the FSD is completely specified, without error, and the population consists overwhelmingly of small families.     

A note on the sampling theory for infinite alleles and infinite sites models

This note considers sampling theory for a selectively neutral locus where it is supposed that the data provide nucleotide sequences for the genes sampled. It thus anticipates that technical advances will soon provide data of this form in volume approaching that currently obtained from electrophoresis. The assumption made on the nature of the data will require us to use, in the terminology ofKimura (Theor. Pop. Biol.2, 174–208 (1971)), the “infinite sites” model of Karlin and McGregor (Proc. Fifth Berkeley Symp. Math. Statist. Prob.4, 415–438 (1967)) rather that the “infinite alleles” model of Kimura and Crow (Genetics49, 174–738 (1964)). We emphasize that these two models refer not to two different real-world circumstances, but rather to two different assumptions concerning our capacity to investigate the real world. We compare our results where appropriate with corresponding sampling theory of Ewens (Theor. Pop. Biol.3, 87–112 (1972)) for the “infinite alleles” model. Note finally that some of our results depend on an assumption of independence of behavior at individual sites; a parallel paper byWatterson (submitted for publication (1974)) assumes no recombination between sites. Real-world behavior will lie between these two assumptions, closer to the situation assumed by Watterson than in this note. Our analysis provides upper bounds for increased efficiency in using complete nucleotide sequences.     

Molecular evolution of voltage-sensitive ion channel genes: on the origins of electrical excitability

We have analyzed nucleic acid and amino acid sequence alignments of a variety of voltage-sensitive ion channels, using several methods for phylogenetic tree reconstruction. Ancient duplications within this family gave rise to three distantly related groups, one consisting of the Na+ and Ca++ channels, another the K+ channels, and a third including the cyclic nucleotide-binding channels. A series of gene duplications produced at least seven mammalian homologues of the Drosophila Shaker K+ channel; clones of only three of these genes are available from all three mammalian species examined (mouse, rat, and human), pointing to specific genes that have yet to be recovered in one or another of these species. The Shaw-related K+ channels and the Na+ channel family have also undergone considerable expansion in mammals, relative to flies. These expansions presumably reflect the needs of the high degree of physiological and neuronal complexity of mammals. Analysis of the separate domains of the four-domain channels (Ca++ and Na+) supports their having evolved by two sequential gene duplications and implies the historical existence of a functional two-domain channel.      

On the concept of the effective population size

The concept of the effective population size is discussed. It is shown that the “eigenvalue” and the “inbreeding” effective population sizes are in principle different, even though they have been sometimes identified in the literature. On the other hand the “eigenvalue” and “variance” effective sizes are usually both close when the latter exists. Since, however, there are many models for which a variance effective size cannot in principle exist, it seems useful to introduce the eigenvalue effective size and to examine some of its properties.