A Stochastic Differential Equation Model For Functional Mapping Of Complex Traits

The genetic study of complex traits is an important topic in a wide spectrum of disciplines including biology,biochemistry,agriculture and forestry.Complex traits are controlled by genes,the environment and their interactions.Because of this,studying complex traits is very challenging,representing one of the most difficult tasks in modern biology.Genetic mapping and association studies have been widely used as a powerful tool to dissect the genetic architecture of complex traits.Conventional mapping approaches strive to illustrate the genetic effects of genes expressed in deterministic environments,but how genes trigger their effect is not only determined by deterministic environments,but also by unpredictable,stochastic environments.It has been recognized at the cell level that the response of the organism to stochastic environment can shape its genetic variation and evolution.However,whether and how the efficiency of genetic mapping can be improved by implementing stochastic environments has remained unclear,largely limiting the progress of genetic mapping to study complex traits.In this thesis,a stochastic differential equation(SDE)model is developed for functional mapping of dynamic traits to systematically disentangle the influence of environmental stochasticity on genetic mapping.The SED-based functional mapping shows remarkable advantages in statistical power and biological relevance.I have obtained the following results and conclusions:(1)The formation of complex traits include the time dimension.Because of this property,functional mapping incorporates growth equations into genetic mapping,leading to increasing mapping efficiency.Here,I derive a SDE model describing growth laws and integrates it into functional mapping,further enhancing the efficiency of functional mapping by capturing the influence of stochastic environments on complex traits.According to the constructed stochastic differential equation,combined with the related properties of stochastic differential equation,the positivity,existence,uniqueness,boundedness,and asymptotic stability of the equation are proved.the existence of its unique and asymptotically stable positive solution is guaranteed.(2)The new model was used to analyze growth traits of seedlings from a full-sib F1 progeny of Euphrates poplar(Populus euphratica Oliver),characterizing important growth quantitative trait loci(QTLs)that are not detected by traditional deterministic models,such as ordinary differential equations(ODE).Even in a well-controlled experimental condition,SDE provides a better fitness of stem height and root length(R~2=0.98–1.00)than ODE(R~2=0.83–0.99).SDE-based functional mapping identifies 20and 22 QTLs for stem height and root length,respectively.Gene enrichment analysis suggests that some of these QTLs reside in candidate genes encoding protease synthesis,toxin catabolism and cellular polysaccharide catabolism.Of these QTLs,only 11 and 10 were detected for stem height and root length,respectively,by ODE-based functional mapping.(3)SDE and evolutionary game theory were integrated to study trait correlations expressed in stochastic environments.Conventional correlation analysis is based on a linear model,failing to consider the nonlinearity of trait interdependence due to stochastic environments.I incorporate SDE into evolutionary game theory to derive a generalized Lotka-Volterra SDE(GLVSDE)model.GLVSDE can not only estimate the strength of trait correlation,but also characterize the causality of the correlation and the direction of the causality.By analyzing stem height and root length traits in Euphrates poplar,GLVSDE identifies the antagonistic relationship between these two traits.When the organism is encountered with an adverse environment,its traits may grow reciprocally at the cost of each other.This model provides a means of quantifying and predicting the dynamic relationship between the organism and its environment..(4)Computer simulation studies were performed to investigate the statistical properties of the new model.In an ideal condition,such as large sample size and homogeneous environment,SDE-and ODE-based mapping models have a similar performance in terms of parameter estimation precision and detection power.However,for a modest sample size and heterogenous environment,SDE performs better than ODE in all aspects.This result suggests that SDE-based functional mapping is a more general approach for QTL mapping.In this thesis,SDE-based functional mapping model was proposed for the first time.This model can remarkably increase the efficiency of genetic mapping for complex traits that are controlled by both deterministic and stochastic environments.Also,the model produces results that are more biologically relevant composed to those by a traditional deterministic model.This model holds a great promise to serve as a powerful tool for dissecting complex traits in agriculture,forestry,and biomedicine.