How to model a complex trait. 2. Analysis with two disease loci

Complex traits are often governed by more than one trait locus. The first step towards an adequate model for such diseases is a linkage analysis with two trait loci. Such an analysis can be expected to have higher power to detect linkage than a standard single-trait-locus linkage analysis. However, it is crucial to accurately specify the parameters of the two-locus model. Here, we recapitulate the general two-locus model with and without genomic imprinting. We relate heterogeneity, multiplicative, and additive two-locus models to biological or pathophysiological mechanisms, and give the corresponding averaged ("best-fitting") single-trait-locus models for each of the two loci. Furthermore, we derive the two-locus penetrances from the averaged single-locus models, under the assumption of one of the three model classes mentioned above. Using these formulae, if the best-fitting single-locus models are available, investigators may perform a two-trait-locus linkage analysis under a realistic model. This procedure will maximize the power to detect linkage for traits which are governed by two or more loci, and lead to more accurate estimates of the disease-locus positions.     

Asynchronous division in cell colonies. I. General considerations and a linear control model

Asynchronous cell division is examined generally for features which have mathematical analogs in the field of statistical physics. A simple linear model for the controlling chemical kinetics is proposed, which yields the essential features of the experimental division time distribution. These results suggest an experiment by which the concept of talandic temperature (Goodwin, 1963) might be studied quantitatively.     

The complex genetic and molecular basis of a model quantitative trait

Quantitative traits are often influenced by many loci with small effects. Identifying most of these loci and resolving them to specific genes or genetic variants is challenging. Yet, achieving such a detailed understanding of quantitative traits is important, as it can improve our knowledge of the genetic and molecular basis of heritable phenotypic variation. In this study, we use a genetic mapping strategy that involves recurrent backcrossing with phenotypic selection to obtain new insights into an ecologically, industrially, and medically relevant quantitative trait—tolerance of oxidative stress, as measured based on resistance to hydrogen peroxide. We examine the genetic basis of hydrogen peroxide resistance in three related yeast crosses and detect 64 distinct genomic loci that likely influence the trait. By precisely resolving or cloning a number of these loci, we demonstrate that a broad spectrum of cellular processes contribute to hydrogen peroxide resistance, including DNA repair, scavenging of reactive oxygen species, stress-induced MAPK signaling, translation, and water transport. Consistent with the complex genetic and molecular basis of hydrogen peroxide resistance, we show two examples where multiple distinct causal genetic variants underlie what appears to be a single locus. Our results improve understanding of the genetic and molecular basis of a highly complex, model quantitative trait.     

Reproductive biology of the Pteridophyta 1. General considerations and a study of Onoclea sensibilis L.

Many pteridophytes are capable of two levels of selling, intra and intergametophytic. Since intragametophytio selling (the origin of both gametes from one gametophyte) results in zygotic homozygosity, this introduces a factor in pteridophyte evolution seldom contemplated. Morphological and genetic techniques are presented which may give an estimate of the frequency of intragametophytic selfing in nature.
Morphological criteria such as the sequence of the development and the spatial arrangement of the gametangia may be indicative of the importance of intragametophytic selfing in the mating system. Estimates of the genetic load (sporophytio lethals) in a population are related to the frequency of the occurrence of intragametophytio selfing.
Onoclea sensibilis L. morphologically has a mating system which gives a low probability of intragametophytio selfing. A genetic load was found in this species.     

A random model approach to mapping quantitative trait loci for complex binary traits in outbred populations.

Mapping quantitative trait loci (QTL) for complex binary traits is more challenging than for normally distributed traits due to the nonlinear relationship between the observed phenotype and unobservable genetic effects, especially when the mapping population contains multiple outbred families. Because the number of alleles of a QTL depends on the number of founders in an outbred population, it is more appropriate to treat the effect of each allele as a random variable so that a single variance rather than individual allelic effects is estimated and tested. Such a method is called the random model approach. In this study, we develop the random model approach of QTL mapping for binary traits in outbred populations. An EM-algorithm with a Fisher-scoring algorithm embedded in each E-step is adopted here to estimate the genetic variances. A simple Monte Carlo integration technique is used here to calculate the likelihood-ratio test statistic. For the first time we show that QTL of complex binary traits in an outbred population can be scanned along a chromosome for their positions, estimated for their explained variances, and tested for their statistical significance. Application of the method is illustrated using a set of simulated data.     

A bivariate quantitative genetic model for a threshold trait and a survival trait

Many of the functional traits considered in animal breeding can be analyzed as threshold traits or survival traits with examples including disease traits, conformation scores, calving difficulty and longevity. In this paper we derive and implement a bivariate quantitative genetic model for a threshold character and a survival trait that are genetically and environmentally correlated. For the survival trait, we considered the Weibull log-normal animal frailty model. A Bayesian approach using Gibbs sampling was adopted in which model parameters were augmented with unobserved liabilities associated with the threshold trait. The fully conditional posterior distributions associated with parameters of the threshold trait reduced to well known distributions. For the survival trait the two baseline Weibull parameters were updated jointly by a Metropolis-Hastings step. The remaining model parameters with non-normalized fully conditional distributions were updated univariately using adaptive rejection sampling. The Gibbs sampler was tested in a simulation study and illustrated in a joint analysis of calving difficulty and longevity of dairy cattle. The simulation study showed that the estimated marginal posterior distributions covered well and placed high density to the true values used in the simulation of data. The data analysis of calving difficulty and longevity showed that genetic variation exists for both traits. The additive genetic correlation was moderately favorable with marginal posterior mean equal to 0.37 and 95% central posterior credibility interval ranging between 0.11 and 0.61. Therefore, this study suggests that selection for improving one of the two traits will be beneficial for the other trait as well.     

How the Atg1 complex assembles to initiate autophagy

The Atg1 complex, comprising Atg1, Atg13, Atg17, Atg29, and Atg31, is a key initiator of autophagy. The Atg17-Atg31-Atg29 subcomplex is constitutively present at the phagophore assembly site (PAS), while Atg1 and Atg13 join the complex when autophagy is triggered by starvation or other signals. We sought to understand the energetics and dynamics of assembly using isothermal titration calorimetry (ITC), sedimentation velocity analytical ultracentrifugation, and hydrogen-deuterium exchange (HDX). We showed that the membrane and Atg13-binding domain of Atg1, Atg1_EAT, is dynamic on its own, but is rigidified in its high-affinity (∼100 nM) complex with Atg13. Atg1_EAT and Atg13 form a 2:2 dimeric assembly and together associate with lower affinity (∼10 μM) with the 2:2:2 Atg17-Atg31-Atg29 complex. These results lead to an overall model for the assembly pathway of the Atg1 complex. The model highlights the Atg13-Atg17 binding event as the weakest link in the assembly process and thus as a natural regulatory checkpoint.     

How the Atg1 complex assembles to initiate autophagy

The Atg1 complex, comprising Atg1, Atg13, Atg17, Atg29, and Atg31, is a key initiator of autophagy. The Atg17-Atg31-Atg29 subcomplex is constitutively present at the phagophore assembly site (PAS), while Atg1 and Atg13 join the complex when autophagy is triggered by starvation or other signals. We sought to understand the energetics and dynamics of assembly using isothermal titration calorimetry (ITC), sedimentation velocity analytical ultracentrifugation, and hydrogen-deuterium exchange (HDX). We showed that the membrane and Atg13-binding domain of Atg1, Atg1_EAT, is dynamic on its own, but is rigidified in its high-affinity (～100 nM) complex with Atg13. Atg1_EAT and Atg13 form a 2:2 dimeric assembly and together associate with lower affinity (～10 μM) with the 2:2:2 Atg17-Atg31-Atg29 complex. These results lead to an overall model for the assembly pathway of the Atg1 complex. The model highlights the Atg13-Atg17 binding event as the weakest link in the assembly process and thus as a natural regulatory checkpoint.     

General thermodynamic considerations of receptor interactions.

Assuming that the biological response is directly proportional to the fractional degree of receptor occupancy, the mathematical relationships between free ligand concentration, receptor occupation and biological activity are developed for a number of equilibrium models. The models considered include simple 1:1 binding with and without conformational changes in the receptor, the coupled binding of two distinct effectors to a single macromolecule, and a system involving indirect coupling between two effectors that bind to two distinct components of the receptor system. This latter model is elaborated into the concept of a domain of receptor-enzyme pairs such that occupation of a single receptor may activate the entire domain of enzymes. This model can explain discrepancies between activation and binding isotherms as has been found with some beta-adrenergic agonist-sensitive adenylate cyclase systems.     

The mouse as a model for human biology: a resource guide for complex trait analysis

The mouse has been a powerful force in elucidating the genetic basis of human physiology and pathophysiology. From its beginnings as the model organism for cancer research and transplantation biology to the present, when dissection of the genetic basis of complex disease is at the forefront of genomics research, an enormous and remarkable mouse resource infrastructure has accumulated. This review summarizes those resources and provides practical guidelines for their use, particularly in the analysis of quantitative traits.     

Hydration from hydrodynamics. General considerations and applications of bead modelling to globular proteins.

The effect of hydration on hydrodynamic properties of globular proteins can be expressed in terms of two quantities: the delta (g/g) parameter and the thickness of the hydration layer. The two paradigms on hydration that originate these alternative measures are described and compared. For the numerical calculation of hydrodynamic properties, from which estimates of hydration can be made, we employ the bead modelling with atomic resolution implemented in programs HYDROPRO and HYDRONMR. As typical, average values, we find 0.3 g/g and a thickness of only approximately 1.2 A. However, noticeable differences in this parameter are found from one protein to another. We have made a numerical analysis, which leaves apart marginal influences of modelling imperfections by simulating properties of a spherical protein. This analysis confirms that the errors that one can attribute to the experimental quantities suffice to explain the observed fluctuations in the hydration parameters. However, for the main purpose of predicting protein solution properties, the above mentioned typical values may be safely used. Particularly for atomic bead modelling, a hydrodynamic radius of approximately 3.2 A yields predictions in very good agreement with experiments.     

Hereditary spastic paraplegia: LOD-score considerations for confirmation of linkage in a heterogeneous trait.

Hereditary spastic paraplegia (HSP) is a degenerative disorder of the motor system, defined by progressive weakness and spasticity of the lower limbs. HSP may be inherited as an autosomal dominant (AD), autosomal recessive, or an X-linked trait. AD HSP is genetically heterogeneous, and three loci have been identified so far: SPG3 maps to chromosome 14q, SPG4 to 2p, and SPG4a to 15q. We have undertaken linkage analysis with 21 uncomplicated AD families to the three AD HSP loci. We report significant linkage for three of our families to the SPG4 locus and exclude several families by multipoint linkage. We used linkage information from several different research teams to evaluate the statistical probability of linkage to the SPG4 locus for uncomplicated AD HSP families and established the critical LOD-score value necessary for confirmation of linkage to the SPG4 locus from Bayesian statistics. In addition, we calculated the empirical P-values for the LOD scores obtained with all families with computer simulation methods. Power to detect significant linkage, as well as type I error probabilities, were evaluated. This combined analytical approach permitted conclusive linkage analyses on small to medium-size families, under the restrictions of genetic heterogeneity.     

Cultural and biological evolutionary processes, selection for a trait under complex transmission.

We consider the evolution of a trait, which is under both genetic and phenotypic transmission. An individual is always born in one state but can be converted to the other before reaching adulthood. If the conversion takes place by a learning process, the native state is called “unskilled,” and that acquired by learning is called “skilled.” If phenotypic conversion takes place by way of infection, the native state is uninfected, and can be converted to infected. Native and converted phenotypes may be subject to selection; acquiring a skill may lead to selective advantage of skilled versus unskilled, while contracting a disease may involve a selective disadvantage. Conversion probability is a function of the parental phenotypes. In some of our models we assume that only one parent has teaching ability (or transmits the disease) and in others we consider more general situations. The probability of learning (or of taking the disease) may be determined by the individual's genotype. A diallelic locus is considered. The evolution of the genotypes and the phenotypes is studied in a variety of situations. Equilibria, and in a few simple cases the dynamics of the phenotypes and genotypes in the population are given. The usual equilibrium for heterozygote advantage is found to depend, in the present case, on the parameters of the learning process. Oscillatory equilibria and more than one stable equilibrium can exist in certain circumstances. Even in the absence of genotypic differences for the conversion probability gene frequencies may change.     

A mixture model approach to the mapping of quantitative trait loci in complex populations with an application to multiple cattle families.

A mixture model approach is presented for the mapping of one or more quantitative trait loci (QTLs) in complex populations. In order to exploit the full power of complete linkage maps the simultaneous likelihood of phenotype and a multilocus (all markers and putative QTLs) genotype is computed. Maximum likelihood estimation in our mixture models is implemented via an Expectation-Maximization algorithm: exact, stochastic or Monte Carlo EM by using a simple and flexible Gibbs sampler. Parameters include allele frequencies of markers and QTLs, discrete or normal effects of biallelic or multiallelic QTLs, and homogeneous or heterogeneous residual variances. As an illustration a dairy cattle data set consisting of twenty half-sib families has been reanalyzed. We discuss the potential which our and other approaches have for realistic multiple-QTL analyses in complex populations.