Descent measures for two loci with some applications

For any four genes, two at each of two loci, in a population, a 15 component descent measure has been introduced. These components are the probabilities of the 15 possible arrangements on a set of initial gametes of those genes of which the four of interest are copies. Since identity by descent of genes is equivalent to their being copies of a single gene on an initial gamete, descent measures have inbreeding coefficients as special cases. The individual descent measure, defined for four genes on two uniting gametes can be evaluated for any pedigree by means of an algorithm developed here. If initial gametic frequencies are specified, descent measures allow genotypic frequencies and disequilibria functions at one and two loci to be found. The procedures are illustrated for selfing and for sib mating. Several applications of the descent measures are discussed.     

Effect of Mating Structure on Variation in Linkage Disequilibrium

Measurement of linkage disequilibrium involves two sampling processes. First, there is the sampling of gametes in the population to form successive generations, and this generates disequilibrium dependent on the effective population size (N_e) and the mating structure. Second, there is sampling of a finite number (n) of individuals to estimate the population disequilibrium.——Two-locus descent measures are used to describe the mating system and are transformed to disequilibrium moments at the final sampling. Approximate eigenvectors for the transition matrix of descent measures are used to obtain formulae for the variance of the observed disequilibria as a function of N_e, mating structure, n, and linkage or recombination parameter.——The variance of disequilibrium is the same for monoecious populations with or without random selfing and for dioecious populations with random pairing for each progeny. With monogamy, the variance is slightly higher, the proportional difference being greater for unlinked loci.     

Prediction of multi-locus inbreeding coefficients and relation to linkage disequilibrium in random mating populations

An algorithm to predict the level of identity by descent simultaneously at multiple loci is presented, which can in principle be extended to any number of loci. The model assumes a random mating population, with random association of haplotypes. The relationship is shown between coefficients of multi-locus identity or non-identity by descent and moments of multi-locus linkage disequilibrium. Thus, these moments can be computed from the multilocus identity or, using algorithms derived previously to predict the disequilibria moments, vice-versa. The results can be applied to predict multi-locus identity in, for example, gene mapping.     

The demands of stair descent relative to maximum capacities in elderly and young adults.

In this study, we aimed to establish the joint moment and joint range of motion requirements of stair descent and the demands relative to maximal capacities in elderly and young adults. Participants descended a custom-built standard dimension four-step staircase, at their self-selected speed in a step-over manner. Kinetic data were acquired from force platforms embedded into each of the steps and into the floor at the base of the stairs. A motion analysis system was used to acquire kinematic data and joint moments were calculated using the kinematic and kinetic data. Maximum capacities (joint moment and joint range of motion) were assessed using a dynamometer. During stair descent the elderly generated lower absolute ankle joint moments than the young, which enabled them to operate at a similar relative proportion of their maximal capacity compared to young adults (75%). The knee joint moments during stair descent were similar between groups, but the elderly operated at a higher proportion of their maximal capacity (elderly: 42%; young: 30%). Ankle plantarflexion-dorsiflexion angle changes were similar between groups, which meant that the elderly operated at a higher proportion of their maximal assisted dorsiflexion angle. These results indicate that the elderly redistribute the joint moments in order to maintain the task demands within 'safe' limits.     

Extinction and descent

The probability of lineal extinction is sensitive to all the moments of the reproductive success probability distribution. In particular, high variance in reproductive success is associated with high probability of lineal extinction. Where male variance in reproductive success exceeds female variance, strictly patrilineal lines of descent will become extinct more rapidly than strictly matrilineal lines of descent. Patrilineal genealogies will be expected to be shallower and broader than matrilineal genealogies under such conditions. Potential implications of this genealogical asymmetry for human descent systems include the greater information content of patrilineal kinship reckoning compared with any other unilineal system and the greater effectiveness of patrilineal kinship as a vehicle for corporate action.     

Two-locus inbreeding measures for recurrent selection

Summary For a population undergoing recurrent selection, a method is presented for determining the average inbreeding coefficients at the end of each breeding cycle. The coefficients are derived in terms of probability measures that genes are identical by descent. For the one-locus case, two digametic measures are defined and employed in the derivation of a recurrence formula for the inbreeding coefficient. Two further classes of measures, trigametic and quadrigametic, are required for transition from one cycle to the previous one to allow the calculation of the inbreeding function for the two-locus case. Numerical values of the average probability of double identity by descent for populations with various imposed assumptions are listed to illustrate the effects of linkage and population size on the accrual of inbreeding and hence of homozygosity.Paper number 5018 of the Journal Series of the North Carolina Agricultural Experiment Station, Raleigh, North Carolina. This investigation was supported in part by NIH research grant number GM 11546 from the National Institute of General Medical Sciences.     

Estimation of low back moments from video analysis: a validation study

This study aimed to develop, compare and validate two versions of a video analysis method for assessment of low back moments during occupational lifting tasks since for epidemiological studies and ergonomic practice relatively cheap and easily applicable methods to assess low back loads are needed. Ten healthy subjects participated in a protocol comprising 12 lifting conditions. Low back moments were assessed using two variants of a video analysis method and a lab-based reference method. Repeated measures ANOVAs showed no overall differences in peak moments between the two versions of the video analysis method and the reference method. However, two conditions showed a minor overestimation of one of the video analysis method moments. Standard deviations were considerable suggesting that errors in the video analysis were random. Furthermore, there was a small underestimation of dynamic components and overestimation of the static components of the moments. Intraclass correlations coefficients for peak moments showed high correspondence (>0.85) of the video analyses with the reference method. It is concluded that, when a sufficient number of measurements can be taken, the video analysis method for assessment of low back loads during lifting tasks provides valid estimates of low back moments in ergonomic practice and epidemiological studies for lifts up to a moderate level of asymmetry.     

Analysis of linkage disequilibrium in an island model

Linkage disequilibria for two loci in a finite island model were parameterized. The total linkage disequilibrium was decomposed into three components, gametic, demic, and population, for which corresponding unbiased estimators were established. Other statistics encountered provided measures of differentiation corresponding to the hierarchical structure of the ecological model. Under the assumption of linkage equilibrium, the variances and covariances of these estimators and statistics were formulated in terms of descent measures, functions of gene frequencies, and the numbers of individuals, demes, and populations sampled. The functions of gene frequencies fall into two classes, one representing the differentiation of genes at each locus, and the other representing the association of genes between the loci. For a neutral model with extinction, migration, and linkage, transition equations were derived for the descent measures which also take into account deme size and numbers of dems within the population. With the addition of unequal mutation rates for a finite number of alleles at each locus, the transition equations were solved for the descent measures in the equilibrium state. This permitted the exact numerical evaluation of the effects of the sampling and ecological dimensions and of extinction, migration, and mutation rates in any parameter range. Some numerical results were presented for the effects of linkage, extinction, migration, and sampling on the variances of various measures of linkage disequilibrium and genetic differentiation. Also, some results were compared with the approximate numerical results of Ohta which agreed fairly well in the parameter ranges she considered, but not so well in other ranges.     

Joint moments’ contributions to vertically accelerate the center of mass during stair ambulation in the elderly: An induced acceleration approach

Falls are a serious problem faced by the elderly. Older adults report mostly to fall while performing locomotor activities, especially the ones requiring stair negotiation. During these tasks, older adults, when compared with young adults, seem to redistribute their lower limb joint moments. This may indicate that older adults use a different strategy to accelerate the body upward during these tasks. The purposes of this study were to quantify the contributions of each lower limb joint moment to vertically accelerate the center of mass during stair ascent and descent, in a sample of community-dwelling older adults, and to verify if those contributions were correlated with age and functional fitness level. A joint moment induced acceleration analysis was performed in 29 older adults while ascending and descending stairs at their preferred speed. Agreeing with previous studies, during both tasks, the ankle plantarflexor and the knee extensor joint moments were the main contributors to support the body. Although having a smaller contribution to vertically accelerate the body, during stair descent, the hip joint moment contribution was related with the balance score. Further, older adults, when compared with the results reported previously for young adults, seem to use more their knee extensor moment than the ankle plantarflexor moment to support the body when the COM downward velocity is increasing. By contributing for a better understanding of stair negotiation in community dwelling older adults, this study may help to support the design of interventions aiming at fall prevention and/or mobility enhancement within this population.     

Wide step width reduces knee abduction moment of obese adults during stair negotiation

PurposeAn increased likelihood of developing obesity-related knee osteoarthritis may be associated with increased peak internal knee abduction moments (KAbM). Increases in step width (SW) may act to reduce this moment. The purpose of this study was to determine the effects of increased SW on knee biomechanics during stair negotiation of healthy-weight and obese participants.MethodsParticipants (24: 10 obese and 14 healthy-weight) used stairs and walked over level ground while walking at their preferred speed in two different SW conditions – preferred and wide (200% preferred). A 2 × 2 (group × condition) mixed model analysis of variance was performed to analyze differences between groups and conditions (p < 0.05).ResultsIncreased SW increased the loading-response peak knee extension moment during descent and level gait, decreased loading-response KAbMs, knee extension and abduction range of motion (ROM) during ascent, and knee adduction ROM during descent. Increased SW increased loading-response peak mediolateral ground reaction force (GRF), increased peak knee abduction angle during ascent, and decreased peak knee adduction angle during descent and level gait. Obese participants experienced disproportionate changes in loading-response mediolateral GRF, KAbM and peak adduction angle during level walking, and peak knee abduction angle and ROM during ascent.ConclusionIncreased SW successfully decreased loading-response peak KAbM. Implications of this finding are that increased SW may decrease medial compartment knee joint loading, decreasing pain and reducing joint deterioration. Increased SW influenced obese and healthy-weight participants differently and should be investigated further.     

Application of an image-based weighted measure of skeletal bending stiffness to great ape mandibles

Traditional measures of structural stiffness in the primate skeleton do not consider the heterogeneous material stiffness distribution of bone. This assumption of homogeneity introduces an unknown degree of error in estimating stiffness in skeletal elements. Measures of weighted stiffness can be developed by including heterogeneous grayscale variations evident in computed tomographic (CT) images. Since gray scale correlates with material stiffness, the distribution of bone quality and quantity can be simultaneously considered. We developed weighted measures of bending resistance and applied these to CT images at three locations along the mandibular corpus in the hominoids Gorilla, Pongo, and Pan. We calculated the traditional (unweighted) moment of inertia for comparison to our weighted measure, which weighs each pixel by its gray-scale value. This weighing results in assignment of reduced moment of inertia values to sections of reduced density. Our weighted and unweighted moments differ by up to 22%. These differences are not consistent among sections, however, such that they cannot be calculated by simple correction of unweighted moments. The effect of this result is that the rank ordering of individual sections within species changes if weighted moments are considered. These results suggest that the use of weighted moments may spur different interpretations of comparative data sets that rely on stiffness measures as estimates of biomechanical competence.     

Effect of mating structure on variation in inbreeding

An analysis is made of the variation among individuals in finite populations of the proportion of their genes which are identical by descent. There are two causes of this variability: variation in pedigree among individuals, and linkage which causes whole blocks of genes to be identical or nonidentical by descent. The variation between and within populations is analyzed in detail for several mating systems: monoecious populations with and without random selfing, and dioecious populations with and without a hierarchical mating structure. Transition matrices for two-locus descent measures are given for each system. Total variability is obtained by integrating these measures over the distribution of map distances over whole chromosomes. Approximate methods are also developed for unlinked loci. Unless populations have a very small effective size (N_e) there is little variation in inbreeding between populations. For unlinked loci, the coefficient of variation in nonidentity within populations approaches about 1/√3N_e for random selfing, 1/√6N_e for monogamous matings and 1/√12N_e for monoecy with selfing excluded or dioecy with random pairing. If there is no association between map distance and initial heterozygosity or effect on quantitative traits, the coefficient of variation in mean heterozygosity over the genome is related to that in nonidentity, and the additional variation in a quantitative trait due to dominant genes equals the product of the square of the initial inbreeding depression and the squared coefficient of variation of nonidentity.     

The decay of linkage disequilibrium under random union of gametes: how to calculate Bennett's principal components

How rapidly does an arbitrary pattern of statistical association among a set of loci decay under meiosis and random union of gametes? This problem is non-trivial, even in the case of an infinitely large population where selection and other forces are absent. J. H. Bennett (1954, Ann. Hum. Genet. 18, 311-317) found that, for an arbitrary number of loci with an arbitrary linkage map, it is possible to define measures of linkage disequilibrium that decay geometrically with time. He found a recursive method for deriving expressions for these variables in terms of "allelic moments" (the factorial moments about the origin of the "allelic indicators"), and expressions for the allelic moments in terms of his new variables. However, Bennett no where stated his recursive algorithm explicitly, nor did he give a general formula for his measures of linkage disequilibrium, for an arbitrary number of loci. Recursive definitions of Bennett's variables were obtained by Lyubich. However, the expressions generated by these recursions are not the same as those found by Bennett. (They do not express Bennett's variables as functions of the allelic moments.) Lyubich's derivations employ genetic algebras. Here, I present a method for obtaining explicit expressions for Bennett's variables in terms of the allelic moments. I show that the transformation from the allelic moments to Bennett's variables and the inverse transformation always have the form that Bennett claimed. (This transformation and its inverse have essentially the same form.) I present general recursions for calculating the coefficients in the forward transformation and the coefficients in the inverse transformation. My derivations involve combinatorial arguments and ordinary algebra only. The special case of unlinked loci is briefly discussed.     

A stochastic analysis of first-order reaction networks

A stochastic model for a general system of first-order reactions in which each reaction may be either a conversion reaction or a catalytic reaction is derived. The governing master equation is formulated in a manner that explicitly separates the effects of network topology from other aspects, and the evolution equations for the first two moments are derived. We find the surprising, and apparently unknown, result that the time evolution of the second moments can be represented explicitly in terms of the eigenvalues and projections of the matrix that governs the evolution of the means. The model is used to analyze the effects of network topology and the reaction type on the moments of the probability distribution. In particular, it is shown that for an open system of first-order conversion reactions, the distribution of all the system components is a Poisson distribution at steady state. Two different measures of the noise have been used previously, and it is shown that different qualitative and quantitative conclusions can result, depending on which measure is used. The effect of catalytic reactions on the variance of the system components is also analyzed, and the master equation for a coupled system of first-order reactions and diffusion is derived. All authors contributed equally to this work.     

Covariances of relatives stemming from a population undergoing mixed self and random mating

We consider covariances of all parent and first-generation relatives from outcrossing or self-fertilization in a parent population that is in equilibrium with respect to these processes. The results, which are for any number of alleles and loci with additive and dominance effects, are phrased in terms of six quadratic genetic components whose coefficients are given by descent measures for equilibrium populations. Because of the variation in the inbreeding coefficients for this system of mating, the expressions include joint contributions of loci to the variances and covariances of relatives. By inclusion of the full complement of relatives, all quadratic components can be estimated. The findings of Ghai (1982, Biometrics 38, 87-92) for compound functions of the covariances with two alleles at a single locus are analyzed in terms of the more general model.     

Age-structured population growth rates in constant and variable environments: a near equilibrium approach

General measures summarizing the shapes of mortality and fecundity schedules are proposed. These measures are derived from moments of probability distributions related to mortality and fecundity schedules. Like moments, these measures form infinite sequences, but the first terms of these sequences are of particular value in approximating the long-term growth rate of an age- structured population that is growing slowly. Higher order terms are needed for approximating faster growing populations. These approximations offer a general nonparametric approach to the study of life-history evolution in both constant and variable environments. These techniques provide simple quantitative representations of the classical findings that, with fixed expected lifetime and net reproductive rate, type I mortality and early peak reproduction increase the absolute magnitude of the population growth rate, while type III mortality and delayed peak reproduction reduce this absolute magnitude.     

Introducing a new staircase design to quantify healthy knee function during stair ascent and descent

The design, manufacture and validation of a new free standing staircase for motion analysis measurements are described in this paper. The errors in vertical force measurements introduced when the stairs interface with a force plate (FP) are less than 0.6%. The centre of pressure error introduced is less than 0.7 mm compared to the error from the FP. The challenges of introducing stair gait into a clinical trial with a limited number of FPs and time limitations for assessment sessions are addressed by introducing this cost effective solution.

The staircase was used in a study to measure non-pathological knee function of 10 subjects performing stair ascent and descent. The resulting knee kinematics and knee joint moments are in agreement with previous studies. The kinematic and joint moment profiles provide a normative range, which will be useful in future studies for identifying alterations in joint function associated with pathology and intervention.     

Mechanics of toe and heel landing in stepping down in ongoing gait

When stepping down from a height difference in ongoing gait, subjects are known to use a heel landing at small height differences and switch to toe landing for larger height differences. We hypothesized that in toe landing, the leading leg can perform more negative work, to control the momentum gained during the descent, than in heel landing. Ten young male participants walked over a 10-m walkway at 5km/h to step down a height difference of 10cm halfway, using a heel or toe landing in five trials each. Kinematic data and ground reaction forces under the leading and trailing legs were recorded. Inverse dynamical analysis of both strategies showed that the leading leg performed more negative work in toe landing, while the vertical velocity at ground contact was lower. In addition, the impact forces were lower in toe landing than in heel landing. Toe landing was found to reduce gait velocity in the first step on the lower level and required higher moments and negative power around the ankle joint than heel landing. Our results indicate that heel landing may be preferred when stepping down small height differences, because it is less demanding especially for the plantar flexor muscles, while toe landing may be preferred for stepping down larger height differences, because it improves control over the momentum gained during the descent.     

Effect of gene conversion on variances of digenic identity measures

The variances and covariances of digenic descent measures are studied for a two-locus model incorporating mutation, gene conversion, recombination, drift, and finite sampling. Gene conversion can occur between allelic pairs of genes or between non-allelic pairs on the same or different gametes within individuals. Most interest therefore centers on pairs of genes, and five digenic identity measures are required. The behavior over time of these measures is studied, with an emphasis on the effects of gene conversion. Because of the stochastic nature of the forces of drift, recombination, mutation, and conversion, the actual identity status of gene pairs can vary from expectation among replicate populations. To study this variation we compute the expected variances and covariances of the measures, and show that this requires the introduction of trigenic and quadrigenic measures. Allowing for conversion between genes on different gametes requires a large number of these higher-order measures.     

Divorce among the Yoruba

Divorce rates among the northern Yoruba are not only higher than one might expect in patrilineal societies, but little different from those of the southern Yoruba, with cognatic descent groups. This initial observation challenges the hypothesis that divorce rates are directly correlated with the line of descent. Data on over 300 divorces were collected in four Yoruba towns—two with agnatic and two with cognatic descent groups. The divorces were classed by the length of the terminated marriage and its fertility, and the distribution of the types of divorce were then compared in the four towns. The results suggest that the rate of divorce is primarily correlated with the degree to which a woman is alienated by marriage from her own descent group. But a number of secondary factors seem to be important, and these may cumulatively raise or reduce the divorce rate to a marked degree.