18. MODELLING NON-LINEAR GENETIC CORRELATIONS BETWEEN BMI AND DEPRESSION

Genetic correlations estimate the proportion of variance shared by two traits as a result of genetic causes. They collapse the bivariate relationship between two outcomes into a single parameter. While correlations do not strictly assume linearity of effects, they may miss dependence of relation between two variables that are non-linearly associated. An example is observed with Body Mass Index (BMI), which has a U- or J-shaped phenotypic association with depression, such that both high and low BMI are associated with increased depression risk or depressive symptoms. However, this non-linear association is not reflected in estimates of genetic correlations between both phenotypes. Rather, this association is represented by point estimates which average out these effects and treat them as though they are linear. We set out to model the genetic relationship between BMI and depression, while imposing minimal prior constraints on the shape of this association. Assuming BMI and depression to be the sum of additive genetic (a_bmi and a_dep) and environmental effects (e_bmi and e_dep), we were interested in estimating the functional relationship that describes the expected value of a_dep given a value of a_bmi. We estimated the function f(a_bmi) from multiple estimates of the derivative f'(a_bmi), each based on pairs of sections of BMI, stratified from low to high. To do this, we obtained BMI information from 454,441 UK Biobank participants with genotype information available. Participants were each stratified into one of 30 quantile-based “bins” according to their BMI. From low to high BMI, participants were stratified into 2 half percentile bins (e.g 0%-0.5%), 4 one percentile bins (e.g. 2%-3%), 18 five percentile bins (e.g. 10%-15%), 4 one percentile bins (e.g. 95-96%), and 2 half percentile bins (99%-99.5%), depending on the position of their BMI in the total sample distribution. Bin sizes were chosen to emphasize the tails of the BMI distribution as we hypothesise that this is where specific genetic effects may be detected. We subsequently performed 435 case-control linear mixed model GWAS of all pairwise combinations of the BMI bins and estimated the genetic correlation between depression and each BMI GWAS using LD score regression. As both the distance between bins (either in terms of BMI or bin number) and the genetic correlation with depression are known, we translated each correlation estimate to an angle that estimates the distance between the BMI bins as a function of depression liability. By combining the angles from all genetic correlation estimates, we end up with a line that represents the shape of the genetic relationship between BMI and depression. We additionally compared results modelling the genetic relationship between BMI and depression, to results modelling the genetic relationship between BMI and anorexia nervosa. We estimated a non-linear function that explains the genetic relationship between BMI and depression. Specifically, we found that the shape of the genetic relationship replicates the same U- or J-shape observed in phenotypic associations between both traits i.e. both low and high BMI are genetically correlated with higher risk of depression. Biologically, this could suggest that different genetic variants are involved at the BMI extremes, or that genes are differentially expressed across the BMI range. These are important avenues for future research to explore.