Integration of functional mapping and delay differential equations to map the genes that regulate circadian rhythms

Most living organisms develop the capacity of generating autonomously sustained oscillations with a period close to 24h. Genes have been thought to play a central role in the regulation of this process, but the detection of these genes (or quantitative trait loci, QTLs) has been made possible with a newly developed functional mapping model. Functional mapping is the integration between genetic mapping and biological principles through mathematical equations. In this article, we equip functional mapping with a biologically meaningful delay differential equations (DDEs) to model circadian rhythms measured in terms of mRNA and protein abundance in time course. The integrative model provides a quantitative framework for assessing the interplay between genetic effects of QTLs and rhythmic responses. By embedding Runge-Kutta forth order algorithm within the likelihood-based context, we obtained estimates of genetic parameters including QTL allele frequency, QTL effects, and the linkage disequilibrium of the QTL and a marker. The potential implications of the new model are discussed.