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.

Predicting recurrence risks under epistatic models

We present the expected recurrence risks to a sib of an affected proband under 4 simple 2-locus epistatic models for various allele frequencies at both the disease locus and the epistatic locus. Four obvious epistatic models are considered: an autosomal recessive disease with both 1) dominant and 2) recessive masking by the epistatic locus, and an autosomal dominant disease again with both 3) dominant and 4) recessive masking. Expected recurrence risks to a sib of an affected proband and to a sib of an affected proband with another normal sib are presented in the absence of information on parental status. Similar risks are presented for the case where both parents are known to be phenotypically normal. These recurrence risks were calculated using a convenient matrix notation which allows sequential calculation of genotypic probabilities. In general, 2-locus epistatic models can give surprisingly low recurrence risks, and often these risks, especially for models of recessive diseases, fall into the range associated with a more general multifactorial model for liability.

Genetic ascertainment with heterogeneous risk

Analysis of the bias of ascertainment is reformulated to deal with more general patterns commonly encountered in practice. The goal is to provide a unifying theory that will both replace the traditional, rather piecemeal, treatment of the problem and free it from certain restrictive assumptions. A compact algebraic method is furnished for analyzing the properties of the distributions by means of the probability generating function (PGF). The scope of the generalization is illustrated by applying it to the various classical patterns of bias of ascertainment. It is extended to other patterns in which the conditions of ascertainment, though more plausible, are also logically more complicated. It also accommodates cases hitherto inadequately dealt with, such as where the segregation ratios are heterogeneous (for example because of age-dependence); and cases where the ascertainment function is of arbitrary form and denies us such valuable, but demanding, assumptions as independence. Not only is the result unifying, but it leads to usable results in specific application such as diseases that depend on age or birth order. While the commonest applications are in human genetics, there are many other issues (such as the use of batteries of tests) in which it is equally important.

Pulmonary function in the Marfan syndrome

To explore the causes of reported abnormalities in pulmonary function in persons with the Marfan syndrome, we analyzed retrospective anthropometric and pulmonary function tests in 79 patients. For ten subjects, there were matched, related control subjects who did not have a connective tissue disorder. Patients had lower values of FVC and TLC than the values expected for their standing heights (83 and 91 percent). However, when sitting height was used to calculate expected spirometric values, patients free of severe deformity of the thoracic cage did not have significant spirometric abnormalities (FVC 105 percent, FEV1 92 percent). However, patients with moderate-to-severe pectus excavatum or scoliosis (common features of the Marfan syndrome) had marked reductions in total lung capacity as well as in FVC and FEV1, suggesting a restrictive ventilatory defect. The abnormalities of pulmonary function often found in clinical laboratories can be explained in most cases by inappropriate use of standing height to calculate expected values, by thoracic cage deformity, or by both. We have found no evidence from standard tests of pulmonary function for a connective tissue defect of lung parenchyma that is of clinical importance in Marfan patients who lack chest wall deformity.

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.

The dynamics of angular homeostasis: I. General principles

A general model is proposed casting aspects of ontogeny in quantitative terms amenable to genetic analysis. Its primordial construct is a chain of cells (termed a "pursuer") growing under the influence of a signal towards a fixed structure termed a "target." There is provision for graduated correction of the direction of growth of the pursuer. The determinants of scale include the size of the cells and the distance from the target. The minimum number of parameters is two: the initial angle of growth; and the force of the correction of errors of direction. Both are potentially of genetic interest. The impact of variation in these factors on the path of growth is studied. These findings are readily translated into biological terms, notably in congenital defects of the heart. Besides the primordial purposes, there are other objectives to the process. Some membranes require free edges, or large curvatures, or circular arrays. These secondary qualities require that the cells never reach the target. The target then becomes simply a construction point: that is, while remaining a center of attraction, it is no longer a true goal. If, because of undercorrection, the cell line misses the target at the first pass, it assumes a permanent orbit about it. The orbit rapidly comes to lie on a circle, with a radius independent of the initial angle of growth but related to the cell size and the restoration constant. From this property, several kinds of structures other than a simple bridge may result, especially when a series of lines of growth together form a tissue: a cusp, a free-floating membrane, or a circular membrane to fill a gap.

The statistics of quantifiable homeostasis. I. Simple linear homeostasis

The statistical properties are explored of the least-squares estimators of the parameters of a deterministic linear model for homeostasis with random, normally distributed technical errors. The parameters are: the amplitude of perturbation (A), the homing value (H), the restoration constant (B), and the lag time (L). The former two, although important in their own right, tell us nothing about the processes of biologic restitution and are thus considered "nuisance" parameters; the latter two are the "business" parameters, which tell us about the processes of biologic adjustment. The properties studied include bias and precision and their relationships to the number of data points and to the size of the technical error. Efforts have been concentrated on data points equally spaced over six lag times (which corresponds roughly to the period of observation used in loading studies, such as glucose tolerance tests). The distributions of these four estimators are studied in sets of 100 Monte Carlo simulations, each with 15, 30, and 100 data points, respectively. The estimators are virtually unbiased and all satisfactorily close to Gaussian. The residual mean-square errors (the divisor being four less than the number of data points, since four parameters are estimated) seem to be unbiased estimators of the variances of the errors. Moreover, when suitably scaled, the residual sums of squares have distributions close to the chi-square distribution for the appropriate degrees of freedom and with the corresponding means and variances. The correlations among the estimators are modest and, except where A is involved, small.

Homeostasis. VII. A conspectus

The characteristics and properties of physiological homeostasis are broadly reviewed in terms that should pave the way for definitive genetic analysis. The ideas are illustrated by common physiological and pathophysiological examples with special reference to type II diabetes mellitus.

The genetics of quantifiable homeostasis: I. The general issues

Some fundamental issues in developing the quantitative genetics of heritable traits are addressed. Such a trait must be capable of being represented as a quantity. Choice of the metric can be satisfactorily made only in the light of some insight into the nature of the trait and what its revealing characteristics may be. This insight is likely to change as more is known about the process, so the form and detail of the analysis is not static but perhaps continually evolving. Much has been made out of sample means and, at a more sophisticated level, Galton-Fisher theory (the analysis of the first two moments of multivariate Gaussian variates). However, at least some processes seriously violate the assumptions underlying that theory. One such group comprises traits in which the means are unvarying and the genetic message lies in other aspects of the distribution; in these circumstances the standard summary statistics such as heritability, regression, and correlation may be totally uninformative, distracting, or even misleading. This paradox is illustrated by reference to homeostasis, where the Mendelian component lies, not in the mean, but in the degree of laxity of control. Since estimating higher moments is notoriously imprecise, the suggested remedy is the familial and commonsense one of exploring the genetics of the intermediate states and underlying mechanisms and not only that of the resultant ("bottom-line") phenotype. In our chosen problem, the task of relating the outcome to the basic mechanism of inheritance calls for a detailed understanding of the homeostasis involved.

Familial aggregation in Alzheimer dementia--I. A model for the age-dependent expression of an autosomal dominant gene

An autosomal dominant genetic etiology has been proposed for Alzheimer Dementia (AD), but many cases appear to be sporadic. Evaluation of the possible genetic transmission of AD from its familial aggregation requires consideration of (1) the proportion of index cases with genetic disease, and (2) the consequences of typically very late onset. To investigate these factors, a provisional biomathematical genetic model was developed from the empirical age-specific incidence of AD in relatives. Based upon the premise of an autosomal dominant AD gene in proband families, the modeling technique provides estimates of the proportion of genetic index cases (as opposed to phenocopies) and the parameters of age-dependent gene expression. With appropriate parameters the model accurately reflects the age-specific familial risk of AD, suggesting the appropriateness of its underlying assumptions. The estimated proportions of genetic index cases suggest that heritable disease constitutes a majority of AD. In cases ascertained by the presence of aphasia or apraxia the estimated proportion of genetic cases is 100%. The greatest likelihood of gene expression is in the ninth decade, however, suggesting that most genetically predisposed relatives will die from other causes before developing AD.

Familial aggregation in Alzheimer dementia--II. Clinical genetic implications of age-dependent onset

A biomathematical genetic model for the age-specific risk of Alzheimer Dementia (AD) was applied to two problems in the clinical genetics of this disorder. In a test of the ability of a clinical marker specifically to identify genetic AD, cases grouped by the phenotype of amnesia with aphasia or apraxia (aaa) were shown to have familial risk that suggested a pure genetic illness, and differed significantly (p = 0.006) from cases without this phenotype. The model was also used in a Bayesian paradigm to assess the probability that individual cases had hereditary disease, given their family history. Here the results were surprisingly ambiguous: Even with no affected relatives, there is a substantial likelihood that many AD cases may have a genetic illness. Hence, one cannot reliably classify individual cases as "familial" or "sporadic" from family history alone. The phenotype of aaa (or other suitable marker) appears to be more reliable than the degree of manifest familial aggregation as an indicator of genetic AD.

The dynamics of quantifiable homeostasis. V: Homeostasis of higher order

The properties are explored of a cybernetic process with lag L and force of restoration equal to the size of displacement one lag unit earlier, raised to any arbitrary positive power (w greater than 1), multiplied by the restoration constant b, the sign being opposite to that of the displacement at that time. No closed-form solution is available, but a converging Taylor series expansion is presented that gives a solution of arbitrary precision. Unlike the linear system previously discussed, the amplitude of the displacement (A) is germane, and it is shown that the criterion that the effect of a perturbation be damped out is whether or not LbAw-1 exceeds a constant, kw, that depends only on w. The counterpart of this pattern of higher-order homeostasis is closer to the biological behavior of many forms of homeostasis than the scaled invariance characteristic of the linear process. kw is shown to be bounded between pi/2 and 2, and increases monotonically with w. Exactly computed values and large-sample approximations are given for the critical values below which no overshoot at all occurs. It is shown that where it does not, the minimum expected cost decreases with increasing w and reaches a minimum at infinite w. However, this evolutionary gain is vitiated where large displacements are involved, and may need to be redressed by a change in the restoration constant. Thus, the dynamic balance becomes much more subtle than with lower-order processes. Several brief clinical illustrations of these ideas are presented.

The dynamics of quantifiable homeostasis. III: a linear model of certain metrical diseases

A generalization of the linear, lagged, homeostatic process shows that whether a displacement of the trait function dies out with time, continues indefinitely, or shows a steadily amplifying (wild) oscillation depends on the value assumed by the product of the lag time and the restoration constant. Moreover, it is shown that if a steady displacing force is used rather than an instantaneous displacement, a new homing value results which is given by the ratio of the displacing force to the restoration coefficient. Combining these two developments furnishes grounds for determining whether or not an overshoot will occur when administration of a drug is stopped (for instance, the rebound thrombosis on discontinuing heparin). Further developments of these ideas show how the diabetes that begins in mature patients can be wholly accounted for by the well-known prolongation of the lag in insulin response that occurs in that disorder. If wild oscillation is to be avoided as the lag time increases, the restoration constant must be weakened (evidently by a systematic reduction in insulin receptors) and this weakening means that the homing value is displaced. Thus the hyperglycemia in this diabetes is to be seen as the price paid for avoiding wild oscillation. Provided that the therapeutic use of exogenous insulin is systematic and regular, rather than cybernetic, its success where endogenous (cybernetic) insulin secretion has failed is readily understood. The point is illustrated by a familiar analogy of a car driver with slow responses. The genetic and evolutionary implications of these ideas are outlined.

The dynamics of quantifiable homeostasis. IV: Zero-order homeostasis

A zero-order homeostatic process is one in which the corrective force a has sign opposite to that of what the displacement from the homing value was one lag (L) time earlier, but is unrelated to magnitude of displacement. The properties of a zero-order process are considered in some detail. Its stable state is an oscillation with a period of 4L and a maximum amplitude of aL. It is suggested that this property is exploited in evolution for intrinsically rhythmical processes, and several examples (respiration, menstruation, and heart beat) are discussed. The parameters may be modulated by an ancillary cybernetic (retreat) circuit and hence the properties of the oscillation controlled as need be. Zero-order oscillation has means of conserving information through phase shifts. Also, when it is combined with a linear cybernetic process (with restoration constant b), it prevents information in the latter from dying out, even if Lb is less than pi/2 (which in a pure linear process would ensure extinction of all signs of perturbation). A further elaboration is the Dilman process of zero order in which there is a threshold of displacement below which homeostatic responses are not evoked. The merits of the value of the threshold are discussed. The properties of the zero-order process with a first-order retreat function furnish a tentative explanation for why hormonal homeostasis does not operate directly through the pituitary but involves the hypothalamus and intermediate hormone-releasing factors.

The bingo model of survivorship. II: statistical aspects of the bingo model of multiplicity 1 with application to hereditary polyposis of the colon

Some Mendelian disorders (Huntington chorea, hereditary polyposis coli) are not manifest at birth but show a distribution in the age of onset. Patients at risk fall into three groups. In type I, they are affected when first examined. In type II, they are not affected at one visit, but are at a later visit. Those of type III (who comprise an indistinguishable mixture of those who have, and those who have not, inherited the gene) are never found to be affected. This paper posits a model that the age of onset is logistic. (It is a degenerate bingo model in which competing causes of death may be ignored.) The statistical properties of maximum likelihood estimation (MLE) are explored by Monte Carlo simulation of this logistic function with known arbitrary parameters. Two schemes are used: point-prevalence (or synchronic) data of types I and III, and piecewise longitudinal (diachronic) data; this allows all three types to be included. Samples of various sizes between 25 and 100 are used. While estimates of the parameters are positively biased (especially with small samples), the estimate of the mean appears to be consistent, almost unbiased, and fairly precise, though somewhat larger than the estimates from the lower bound (a fact that calls for some caution in interpreting actual data). The MLE was applied to 109 patients with the Gardner syndrome (GS); measures of variability found by applying MLE to four random subsets of 25 each were compared against the asymptotic estimates. The analysis was also applied to 36 persons with familial polyposis coli (FPC). The mean age of onset in GS and FPC was similar, and since they are rather earlier than is currently believed, it is recommended that regular supervision be started at not later than 10 years of age.

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. II. Characterization of linear processes

This paper deals with the representation of physiological homeostatic processes by mathematical models that lead to lag-differential equations. It deals exclusively with feedback processes that are linear. It is surmised that in the response to standard perturbations from equilibrium, in many systems the strength at which the restorative force works (denoted by the constant b) is a genetic characteristic, while the average or homing value may or may not be. The class of equations with linear feedback is solved. The solution provides predictions against which experimental data may be tested. Three types of displacement from the homing value are recognized: drifts, catastrophes, and saccades. Several methods of evaluating the mathematical functions are described. They are forward recursion, backward recursion, numerical integration, steady-state solutions, and inversion of the Heaviside expansion of the Laplace transform. Conditions for homeostatic processes that result in damped nonoscillating responses, damped oscillating responses, stable oscillations, and uncontrolled oscillations are derived. The concept of an associated penalty that depends on the degree of displacement from the homing value is discussed. Penalties are evaluated by numerical integration for linear, quadratic, and cubic costs. Costs for polynomial functions may be found by linear weighting. The optimum strength of response (ie, the one that produces the minimum cost) is calculated for each of these cost functions.

Genetic aspects of multiple polyposis coli

Questions on this topic that concern the clinician are addressed: whether it is inherited and, if so, whether Mendelian; what the segregation ratios will be; whether it is heterogeneous; how its high frequency can be accounted for; and what preventive measures are available. In several cases the question has to be reworded in order to make it sufficiently precise for it to be used in practice without misleading the patient. Framed in this fashion applicable answers are relatively easily available although many genetic puzzles remain. To date, no antenatal diagnosis is available.

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.

Inheritance of the mitral valve prolapse syndrome. Discussion of a three-dimensional penetrance model

The families of 12 probands with classic mitral valve prolapse were studied for evidence of mitral valve prolapse. Seventy parents, sibs, and progeny were included in the analysis. Forty-seven percent (16 of 34) of progeny were affected compared with 30 percent (3 of 10) of parents. Thirty-eight percent (10 of 26) of sibs were affected. A three-compartmental penetrance model was devised to account for the variation in expression with age. This includes a latent stage (time before onset of signs), an affected stage, and a stage in which the subjects are withdrawn (because of treatment, regression, or death). The implications of this model are discussed.

The analysis and interpretation of experiments: some philosophical issues

The epistemology and ontology of experimentation are discussed in depth with special reference to biology and medicine. Two types of experiments are distinguished: exploratory (or "blazing") and consolidating. They have objectives and canons that are strikingly different. A contrast is drawn between the literalism of the most pragmatic scientists and the formalism of most statisticians. The terms and notions of the one may have imperfect correspondence with those of the other, or perhaps none at all. The dangers are pointed out of both the conclusion that the gap between the two is unbridgeable and the contrary view that the gap is trivial and merits no serious attention. Substantive discussions address the choice and meaning of a metric, the idea of interaction and time series analysis, the ontology of variation, and the epistemology of randomization, hypothesis-seeking, and multiple comparisons.