Paradoxical fixation of deleterious alleles in two-locus systems with epistasis.

The dynamics of four deterministic models of interaction between two Mendelian loci are explored numerically. At one locus there are two hypostatic alleles: h, the wild type and H, a deleterious mutant that in either heterozygous or homozygous state (depending on the specifics of the model) produces an abnormal phenotype. At another arbitrarily linked epistatic locus there are two alleles: e, the wild type with no effect on the expression of the H locus, and E, an epistatic mutant that in heterozygous or homozygous state (again, depending on the specifics of the model) blocks the expression of H. The parameters are the initial gamete frequencies, the recombination fraction, the genotypic viabilities, and the forward and back mutation rates at each locus. The He haplotype is eventually eliminated (unless back mutation occurs) from the population. If mutation is ignored, the evolutionary outcome is determined by the initial gamete frequencies and is either (1) an edge equilibrium comprising one pair of haplotypes only (he and He, or he and hE, or hE and HE, or He and HE, or, if there is no recombination, he and HE or hE and He); or (2) a corner equilibrium consisting of a single gametic type. Given that the forward mutation rates at both loci are greater than the back mutation rates, then the outcome is always the corner equilibrium in which HE is universal (apart from transient perturbations by mutation). In the process of fixation, the phenotypic impact of the deleterious allele H becomes neutralized by the epistatic allele E. The rate at which the initially harmful gene replaces the wild type gene depends on the recombination fraction, the genotypic viabilities, and the mutation rates.

The dynamics of quantifiable homeostasis. I. The individual.

In this field of enquiry, "homeostasis" is understood in the sense of Bernard, Cannon, and Wiener, that is, with the system of continuous adjustments, in a trait not inherently stable, to meet the challenges of the environment. It is shown that the clinical fitness of a trait may depend on factors other than the mean or the variance. In particular the broad pattern of variation of the phenotype may be of importance and hence the characteristics of the homeostatic mechanisms for the control of traits subject to variation: notably the homing value (the "setting" of the homeostat), and the strength with which the organism responds to departures from this value. These parameters are related, but perhaps only remotely, to the traditional notion of the value of a phenotype. In general, where the environmental value is variable, there exist circumstances in which the optimal control would be neither extremely tight nor extremely loose. For instance, anticipatory action on the part of the body may be vitiated by too tight a control. Some illustrations are given of genetic disorders in which there is a fault in the strength of control. While neither the pattern of inheritance nor the impact on the species is explored in detail, the broad implications are indicated. Compromise adjustments are explored where two or more traits are regulated through the same homeostatic device.

Recurrence risks for autosomal, epistatic two-locus systems: the effects of linkage disequilibrium.

The properties of the recurrence risk functions for Mendelian models involving two loci with epistatic interaction are examined. The two loci, H (hypostatic) and E (epistatic), are arbitrarily linked and diallelic. The general parameters are the genotypic frequencies and recombination fraction; the particular arguments are the number of affected and unaffected offspring. Limiting properties of the models are first described. Then, the more detailed properties of the models such as the sensitivity of the recurrence risk to the number of affected offspring are examined. If linkage disequilibrium is present, ie, the determinant of the gametic matrix is not zero, the predicted recurrence risk may resemble that empirically found in certain heritable diseases not evidently Mendelian.

The bingo model of survivorship. III. Genetic principles.

The broad relationships are explored between the genetic and the phenotypic structures of the bingo-gamma model (ie, the shortest waiting time among competing, independent, multiple-hit systems). Finite algorithms are derived to compute in closed form the joint and marginal distributions; the distribution, density, and hazard functions of time to failure; the respective total probabilities of dying from failure of each competing system; and the raw and central moments. The algorithm is the computer counterpart of a generating function. The number of competing systems and their individual orders and transition parameters may be chosen at will. Classical Galton-Fisher theory does not apply: neither means nor variances are additive nor are their effects homogeneous; rather, those systems with shorter mean survival more or less mask the impact of those with longer means. Thus even huge differences among means for alleles of any one component may be almost totally concealed phenotypically; even the maximal genetic covariation may in practice remain totally unrecognized and the heritability estimated close to zero. The proportional specific mortality is a less capricious index and is naturally additive, but, though a monotonic function of the underlying parameters, it is neither linear nor homogeneous.

Angular homeostasis. V: Some issues in genetics, ontogeny, and evolution.

We discuss a provisional model that deals with aspects of normal control of the direction in which cells grow; hence, the organization of structure. The interacting issues of genetics, ontogeny, and phylogeny in relationship to normal development and its defects are addressed. Our main goal is to define a model with a small and sufficient set of conditions that would make it possible to undertake statistically efficient genetic studies of certain congenital defects. But to do so in plausible and heuristic terms, one must address two classical questions: 1) How is the genome translated (or sometimes mistranslated) into a phenotype that is clinically conspicuous and that can be the object of genetic selection, and hence of evolution? 2) Granted that evolution of species occurs in small, rare steps, what is the path that calls for the smallest number of mutations through which ontogenesis could have developed, the intermediate stages being conserved because of actual phenotypic advantage?

The bingo model of survivorship: 1. probabilistic aspects.

A "bingo" model is one in which the pattern of survival of a system is determined by whichever of several components, each with its own particular distribution for survival, fails first. The model is motivated by the study of lifespan in animals. A number of properties of such systems are discussed in general. They include the use of a special criterion of skewness that probably corresponds more closely than traditional measures to what the eye observes in casually inspecting data. This criterion is the ratio, r(h), of the probability density at a point an arbitrary distance, h, above the mode to that an equal distance below the mode. If this ratio is positive for all positive arguments, the distribution is considered positively asymmetrical and conversely. Details of the bingo model are worked out for several types of base distributions: the rectangular, the triangular, the logistic, and by numerical methods, the normal, lognormal, and gamma.

Angular homeostasis II: Pursuit of a moving target in a plane and some implications for cardiac teratology.

A model of cardiac ontogenesis is analyzed. It is cast in terms of the geometry of the pursuit of a linearly moving target by the growth of a chain of cells in the same plane, the pursuer, which at each step adjusts its direction of growth towards the current position of the target. The endpoint is the fusion between them, which can occur in 2 modes: either by the leading cell of the pursuer catching up with the target (pursuer-mediated fusion, or PMF) or by the target running into the preformed side of the pursuer (target-mediated fusion, or TMF). The causal specifications are the step size, the speed of the pursuer, the speed of the target, the restoration constant, and the initial direction of the pursuer; the outcome variables are the number of steps to fusion and the mode of fusion. The pattern of behavior is complicated, being more-or-less regular over large tracts of values, interspersed with abrupt, threshold-like changes that may generate a dichotomous pattern of inheritance despite a continuous gradation of genetic or other causes. The temporary abolition of the correction process (a change introduced to simulate the pattern of the effect of a teratogen) may delay fusion and suggest how a septum may fail to fuse, the ductus arteriosus to close, or an endocardial cushion to form. But the model also predicts that under certain plausible conditions, the "teratogen" would speed up fusion and hence perhaps offset a genetic predisposition to a congenital defect.

The bingo model of survivorship. V. The problems of conformation to the empirical evidence.

We discuss the statistical and biological problems of adapting the theoretical bingo model to the analysis of empirical data. A distinction is made between an idealized pathogenetic model, which aims to represent the disease in as much authentic detail as the present state of knowledge allows and in components that have literal interpretation, and an empirical model, which deals with those effects of the pathogenetic model that one may hope to observe clinically. We review a variety of empirical models distinguishable by the amount of data available on intermediate degrees of damage short of total destruction. The relationship of damage to time is explored, and we consider the criteria and usefulness of linearization of this relationship where the diachronic ("longitudinal") data are few and extend over a comparatively short time. Every time a patient is examined, the degree of cumulative damage is assessed in each of the body systems of interest. Thus the examination will furnish a set of measurements, which is obtained on each of several examinations, taken over a period that for preference is long relative to the survival of the system. Specific disorders discussed include dentition and enlargement of the aorta with age in the Marfan syndrome.

The bingo model. IV. The statistics of survivorship in the bingo-gamma model.

The survivorship (time to death or failure) of a bingo-gamma (BG) model is defined as the minimum among the waiting times for completion among k independent gamma processes. The ith process is of order ni, with a mean rate for the occurrence of hits of ai. In this paper we address the case where, for all competing processes, the order and the rate at which hits occur are the same but both they and k are unknown. We denote by k the multiplicity, by n the order or the number of hits to failure, and by a the transition parameter. The joint maximum likelihood estimator (MLE) of the three parameters of this BG process is developed. An algorithm for calculating it has been devised and a computer program in BASIC has been written. The properties of the MLE have been explored systematically, mainly by Monte Carlo simulation. The distributions, means, variances, covariances, and correlation coefficients of the three parameters are explored for samples of size 25 and samples of size 100. Also, the simple average of the observed survival times (which gives a method of moments estimator of the mean survival) is compared with the MLE of the mean survival; the two estimators seem to be unbiased and about equally efficient.

Median nerve compression in Weill-Marchesani syndrome.

Weill-Marchesani syndrome is a rare, generalized disorder of connective tissue manifested by short stature, brachymorphia, and spherophakia. Inheritance is autosomal recessive. In the less than 50 reported cases, joint stiffness in the hands and thenar atrophy have been noted in adults. A kindred is reported here in which release of multiple trigger fingers and bilateral carpal tunnel syndrome in childhood has improved hand function in a brother and sister.

Identification of female carriers of adrenoleukodystrophy.

The concentrations of very long chain fatty acids in plasma and cultured skin fibroblasts were studied in 96 women who were obligate heterozygotes for X-linked adrenoleukodystrophy, in 34 women who were mothers of single probands with ALD, and in 32 normal women of comparable age. Discriminant analysis was used to develop a classification function for the plasma values. With this function, plasma values in 88% of the women who were obligate heterozygotes for ALD and 77% of the mothers of single probands were identified as abnormal. With subsequent inclusion of the fibroblast assay data, abnormal plasma values were found in 93% of the obligate heterozygotes.

Angular homeostasis: IV. Polygonal orbits.

Some properties are discussed of regular polygons that may result from angular homeostatic processes in stable orbit. To characterize these "homeostatic polygons" we need to discuss the winding number, the sidedness (integer, fractional and irrational), multiplicity, envelopes, and density. A regular (i.e., equilateral, equiangular) polygon may be closed in one revolution about its unique center, in multiple revolutions, or not at all. A homeostatic polygon can be generated only if all vertices are included in a single polygon, which occurs if and only if the number of vertices and the number of revolutions required to complete the polygon are relatively prime. For the homeostatic polygon to have a finite number of sides (without repeating itself) the angle subtended by any two successive vertices at the center must be a rational multiple of 2 pi. Biological implications of these properties are illustrated.

Respiratory complications of achondroplasia.

Nine patients with achondroplasia who were seen over a three-year period developed significant respiratory complications. Eight had sleep-disordered breathing, including obstructive sleep apnea in five, for which two required tracheostomy. Of the seven patients with significant hypoxemia, five had clinical evidence of cor pulmonale and recurrent pulmonary infiltrates. Two patients died, one with autopsy findings of compression of the medulla at the level of the foramen magnum and one with respiratory and cardiac failure. Appropriate therapy for our patients depended on recognition of the mechanisms that led to the respiratory complications, including (1) chest deformity, (2) upper airway obstruction and sleep-disordered breathing, (3) neurologic complications, and (4) coincidental chronic pulmonary conditions such as asthma.

Immunologic studies of cartilage-hair hypoplasia in the Amish.

The immunologic status of 18 Old Order Amish persons with cartilage-hair hypoplasia and 9 unaffected sibs was studied. Although none of the subjects had a history suggestive of persistent immune dysfunction, the subjects with cartilage-hair hypoplasia had significantly lower lymphocyte mitogenic and allogeneic cell stimulation responses when compared to unaffected sibs and unrelated control subjects. The abnormalities of cellular immune function found in the 18 affected subjects were similar to those reported in Finnish subjects with cartilage-hair hypoplasia.