A logistic mixture model for characterizing genetic determinants causing differentiation in growth trajectories

Abstract

The logistic or S-shaped curve of growth is one of the few universal laws in biology. It is certain that there exist specific genes affecting growth curves, but, due to a lack of statistical models, it is unclear how these genes cause phenotypic differentiation in growth and developmental trajectories. In this paper we present a statistical model for detecting major genes responsible for growth trajectories. This model is incorporated with pervasive logistic growth curves under the maximum likelihood framework and, thus, is expected to improve over previous models in both parameter estimation and inference. The power of this model is demonstrated by an example using forest tree data, in which evidence of major genes affecting stem growth processes is successfully detected. The implications for this model and its extensions are discussed.

QTL methodology for response curves on the basis of non-linear mixed models, with an illustration to senescence in potato

“…As a competitor to the use of non-linear mixed models for the QTL modeling of growth trajectories, mixture model approaches using EM estimation procedures have been proposed (Ma et al 2002;Wu et al 2002aWu et al , 2003a. The difference between the mixture model approach and our mixed model approach is that we approximate the mixture density for the phenotype in relation to the possible QTL genotypes for a particular place at the genome (Jansen 1992(Jansen , 1993Zeng 1993Zeng , 1994) with a normal density following the regression approach of Haley and Knott (1992) and Martínez and Curnow (1992).…”

“…They combine logistic growth curves and QTL mapping within a mixture model approach, modelling growth curves parameters as a function of molecular marker information. The approach is implemented within an expectation-maximization (EM) algorithm and proved to be powerful and to produce accurate estimates of QTL effects and positions (Wu et al 2002a(Wu et al , 2003a. The methodology was further generalized to allow changing growth rates during development (Wu et al 2003b).…”

Functional mapping for genetic control of programmed cell death

“…Although this presents one of the most difficult tasks in genetic studies, some of the key issues have been overcome by Wu and colleagues (27,(37)(38)(39)(40)(41), who proposed a so-called "functional mapping" to map and identify specific QTL that underlie the developmental changes of complex traits. The rationale of functional mapping is to express the genotypic values of QTL at a series of time points in terms of a continuous growth function with respect to time t. Under this formulation, the parameters describing longitudinal trajectories, rather than time-dependent genotypic values as carried out in traditional mapping strategies, are estimated within a maximum likelihood framework.…”

A statistical model for functional mapping of quantitative trait loci regulating drug response

“…Using the genetic variance due to the QTL for the response at the last measurement point, we calculated the residual variances under different heritability levels (H 2 ¼ 0.1 and 0.4). These residual variances, plus a given residual correlation (r ¼ 0.7), form a residual (co)-variance matrix according to equation (5). The phenotypic values of drug effect for 400 random patients are simulated by the summations of genotypic values predicted by the curves and residual errors following multivariate normal distributions, with MV N(0, R e ).…”

“…The genetic architecture of function-valued traits can be studied using the marker-based functional mapping model, developed by Wu and colleagues. [5][6][7][8][9][10] Different from traditional methods for mapping a complex trait described by a single value, functional mapping has power to map dynamic QTL responsible for a biological process that need be measured at a finite number of time points. In modeling functional mapping, fundamental principles behind biological or biochemical networks described by mathematical functions are incorporated into a QTL mapping framework.…”