title: A mathematical framework for functional mapping of complex phenotypes using delay differential equations.
abstract: All biological phenomena occurring at different levels of organization from cells to organisms can be modeled as a dynamic system, in which the underlying components interact dynamically to comprehend its biological function. Such a systems modeling approach facilitates the use of biochemically and biophysically detailed mathematical models to describe and quantify "living cells," leading to an in-depth and precise understanding of the behavior, development and function of a biological system. Here, we illustrate how this approach can be used to map genes or quantitative trait loci (QTLs) that control a complex trait using the example of the circadian rhythm system which has been at the forefront of analytical mathematical modeling for many years. We integrate a system of biologically meaningful delay differential equations (DDEs) into functional mapping, a statistical model designed to map dynamic QTLs involved in biological processes. The DDEs model the ability of circadian rhythm to generate autonomously sustained oscillations with a period close to 24h, in terms of time-varying mRNA and protein abundances. By incorporating the Runge-Kutta fourth order algorithm within the likelihood-based context of functional mapping, we estimated the genetic parameters that define the periodic pattern of QTL effects on time-varying mRNA and protein abundances and their dynamic association as well as the linkage disequilibrium of the QTL and a marker. We prove theorems about how to choose appropriate parameters to guarantee periodic oscillations. We further used simulation studies to investigate how a QTL influences the period and the amplitude of circadian oscillations through changing model parameters. The model provides a quantitative framework for assessing the interplay between genetic effects of QTLs and rhythmic responses.
authors.affiliation: Department of Statistics, The Pennsylvania State University, University Park, PA 16802, USA.
authors.forename: Guifang
authors.initials: G
authors.lastname: Fu
authors.forename: Zhong
authors.initials: Z
authors.lastname: Wang
authors.forename: Jiahan
authors.initials: J
authors.lastname: Li
authors.forename: Rongling
authors.initials: R
authors.lastname: Wu
mesh_terms: D000465:Algorithms; D001683:Biological Clocks / Q000235:genetics; D002874:Chromosome Mapping / Q000379:methods*; D002940:Circadian Rhythm / Q000235:genetics* / Q000502:physiology; D006801:Humans; D015810:Linkage Disequilibrium; D008957:Models, Genetic; D010641:Phenotype; D040641:Quantitative Trait Loci / Q000235:genetics*; D012333:RNA, Messenger / Q000235:genetics; D049490:Systems Biology / Q000379:methods
chemical_list: D012333:RNA, Messenger