Quantile-specific heritability of sibling leptin concentrations and its implications for gene-environment interactions

“Quantile-dependent expressivity” occurs when the effect size of a genetic variant depends upon whether the phenotype (e.g., leptin) is high or low relative to its distribution. Leptin concentrations are strongly related to adiposity, whose heritability is quantile dependent. Whether inheritance of leptin concentrations is quantile dependent, and whether this explains the greater heritability in women than men in accordance with their greater adiposity, and explains other gene-environment interactions, remains to be determined. Therefore, leptin and leptin receptor concentrations from 3068 siblings in 1133 sibships from the Framingham Heart Study Third Generation Cohort were analyzed. Free leptin index (FLI) was calculated as the ratio of leptin to soluble leptin receptor concentrations. Full-sib (βFS) regression slopes were robustly estimated by quantile regression with nonparametric significance assigned from 1000 bootstrap samples. The analyses showed βFS increased significantly with increasing percentiles of the offspring’s age- and sex-adjusted leptin distribution (Plinear = 0.0001), which was accelerated at the higher concentrations (Pquadratic = 0.0003). βFS at the 90th percentile (0.418 ± 0.066) was 4.7-fold greater than at the 10th percentile (0.089 ± 0.032, Pdifference = 3.6 × 10−6). Consistent with quantile-dependent expressivity, the βFS was greater in female sibs, which was attributable to their higher leptin concentrations. Reported gene-environment interactions involving adiposity and LEP, LEPR, MnSOD, PPARγ, PPARγ2, and IRS-1 polymorphisms were consistent with quantile-dependent expressivity of leptin concentrations. βFS for leptin receptor concentrations and free leptin index also increased significantly with increasing percentiles of their distributions (Plinear = 0.04 and Plinear = 8.5 × 10−6, respectively). In conclusion, inherited genetic and shared environmental effects on leptin concentrations were quantile dependent, which likely explains male–female differences in heritability and some gene-environment interactions.

www.nature.com/scientificreports/ exhibit a strong relationship to body fat and adipocyte cell size 34 , consistent with its function as a quantitative endocrine signal of stored fat in adipose tissue. This might predict that genetic influences on leptin concentrations are also quantile-dependent, except that: (1) residual leptin heritability persists when adjusted for adiposity 9,12 ; (2) none of the non-FTO genetic loci previously associated with BMI attained < 10 −6 significance with leptin concentrations 22 , (3) the four non-FTO loci in or near LEP, SLC32A1, GCKR, CCNL1 that attained genome-wide association with leptin (P < 5 × 10 −8 ) persisted when adjusted for BMI 22 . An important consequence of quantile-dependent expressivity is that the selection of subjects for characteristics that distinguish high versus low phenotypes can yield different genetic effects 33 . Women secrete more leptin than men due to their larger percentage of body fat 35 , greater subcutaneous fat storage 35,36 , and low testosterone 37 . Being overweight or obese accentuates the sexual dimorphism in leptin secretion 38 . Genetic influences on leptin concentrations are also greater in women. For example, Martin et al. 39 reported that leptin heritability was greater in women than men (h 2 = 0.57 vs. h 2 = 0.31) and that women's had higher average leptin concentrations (29.34 ± 0.94 vs. 10.80 ± 0.56 ng/ml), as did Hasselbalch et al. 13 (i.e., h 2 female = 0.59 vs. h 2 male = 0.38), and Rotimi et al. 14 . Kaprio et al. 11 reported that additive genetic effects on leptin concentrations were five-fold larger in women than men in accordance with their higher average leptin concentration (16.8 ± 9.5 vs. 6.4 ± 3.5 ng/ml, P < 0.0001). Moreover, the effect of PPARγ2 (peroxisome proliferator-activated receptor γ2) rs1801282 genotypes on leptin concentrations is significantly greater in women than men 40 . In addition to sex, adiposity [41][42][43][44][45][46][47] , diet 48 , and smoking 49 are reported to modify the effects of genes on leptin concentrations.
We therefore sought to test whether shared environmental and inherited factors affecting leptin concentrations in sibs were quantile-dependent in a large population cohort (Framingham Heart Study 50 ). Untransformed concentrations were analyzed because quantile regression does not require normality 51,52 , and no biological justification has yet been given for its logarithmic transformation. We also re-analyze published studies of leptin that measured genetic variants directly from the perspective of quantile-dependent expressivity. The results suggest that quantile-dependent expressivity: (1) provides a simple explanation for the greater leptin heritability in women than men 11,13,14,39,40 and (2) is consistent with the genotype-specific effects of weight [41][42][43][44][45][46][47] , diet 48 , and smoking 49 on leptin concentrations.

Methods
The methods have been described previously [25][26][27][28][29][30][31][32][33] , but are repeated here for completeness. The data were obtained from the National Institutes of Health FRAMCOHORT, GEN3, FRAMOFFSPRING Research Materials obtained from the National Heart Lung and Blood Institute (NHLBI) Biologic Specimen and Data Repository Information Coordinating Center. Subjects were at least 16 years of age and not self-identified as nonwhite or Hispanic. Leptin and soluble leptin receptor concentrations were measured on stored EDTA plasma samples frozen at -80 °C from the first examination of the Framingham Third Generation Cohort 50 by ELISA (R&D Systems Inc., Minneapolis, MN) with an average interassay coefficients of variation < 5% 53 . Free leptin index, a purported measure of bioavailable leptin not bound to its soluble receptor, was calculated as the ratio of leptin to leptin-receptor concentrations. Subjects used in the current analyses were at least 16 years of age, were not taking medications for diabetes, and were self-identified as non-Hispanic white. These analyses were approved by Lawrence Berkeley National Laboratory Human Subjects Committee (HSC) for protocol "Gene-environment interaction vs quantile-dependent penetrance of established SNPs (107H021)" Approval number: 107H021-13MR20. LBNL holds the Office of Human Research Protections Federal wide Assurance number FWA 00006253. All surveys were conducted under the direction of the Framingham Heart Study human use committee guidelines, with signed informed consent from all participants or parent and/or legal guardian if < 18 years of age.
Statistics. Age and sex adjustment was performed using standard least-squares regression with the following independent variables: female (0,1), age, age 2 , female x age, and female x age 2 . Full-sibling correlations and regression coefficients (β FS ) were obtained by constructing all possible pairs using double entry 54 , with an adjusted Σ(k i − 1) degrees of freedom, where k i is the number of offspring in family i and the summation is taken over all i, i = 1,…, N nuclear families.
Simultaneous quantile regression was performed using the "sqreg" command of Stata (version. 11, StataCorp, College Station, TX) with one thousand bootstrap samples drawn to estimate the variance-covariance matrix for the 91 quantile regression coefficients (β FS ) between the 5th and 95th percentiles, and the post-estimation procedures (test and lincom) to test linear combinations of the β FS slopes after estimation with Σ(k i -1) degrees of freedom. Quantile-specific sib-sib concordance was assessed by: (1) estimating quantile-specific β FS -coefficient for the 5th, 6th, …, 95th percentiles of the sample distribution using simultaneous quantile regression (Fig. 1, the < 5th and > 95th percentiles ignored because they were thought to be less stable); (2) plotting the quantilespecific β FS coefficients versus the percentile of the trait distribution; and (3) testing whether the resulting graph is constant, or changes as a linear, quadratic, or cubic function of the percentile of the trait distribution using orthogonal polynomials 55 . Female β FS slopes refer to all sib-pairs where a female sib is the dependent variable and male or female sibs are the independent variable, male β FS slopes refer to all sib-pairs where a male sib is the dependent variable and male or female sibs are the independent variable. Unadjusted regression slope refer to an unadjusted sib value as the dependent variable versus the adjusted remaining sib values as the independent variables. Slopes are presented ± SE.
When β FS for male and female sib are compared on the same graph, their quantile-specific functions compare their slopes at the corresponding percentiles of their separate distribution (e.g., the slope at the 50th percentile of the females' distribution versus the slope at the 50th percentile of the males' distribution). However, the leptin concentration at the 50th percentile of the females' distribution will be greater then the 50th percentile of the males' distribution. Quantile-specific expressivity postulates that the genetic effects depend upon the leptin www.nature.com/scientificreports/ concentration. Therefore, additional displays were created based on probability-probability plots (P-P plots, Fig. 2) 56 that re-plot the males' and females β FS at the same leptin concentration. For example, Fig. 2 shows that the 50th percentile of the leptin distribution for male and female offspring combined was 7.07 ng/ml (horizontal axis). This corresponds to the 29.3 rd percentile of the female distribution and 73 rd percentile of the male distribution (vertical axis). Thus plotting the β FS at the females' 29.3th percentile and males' 73 rd percentile at the 50th percentile of their combined distribution results in their β FS 's being compared at the same leptin concentration. This process was repeated for each percentile of their combined distribution (interpolated where required) to compare male and female β FS when matched by leptin concentrations.

Results
Women and men were of similar age {female vs. male mean (SD): 39  Quantile-dependent expressivity. Figure 1A presents the full-sib regression slopes (β FS ) at the 10th, 25th, 50th, 75th, and 90th percentiles of the sibs' age-and sex-adjusted leptin distribution. The slopes get progressively steeper with increasing percentiles of the distribution. β FS at the 90th percentile was 4.7-fold greater than at the 10th percentile (P difference = 3.6 × 10 −6 ). These slopes, along with those of the other percentiles between the 5th and 95th percentiles, are presented in the quantile-specific β FS plot in Fig. 1B. They show β FS increased with increasing percentiles of the offspring's distribution (i.e., slope ± SE increased 0.0034 ± 0.0009 per percentile, P linear = 0.0001) and that the increase accelerated at higher concentrations (P quadratic = 0.0003). Quantile-specific β FS was significant (P ≤ 0.005) for all individual percentiles between the 10th and 94th percentiles of the sibs' leptin distribution. If β FS was the same over all quantiles as traditionally assumed, then the line segments in Fig. 1A would be parallel, and the graph in Fig. 1B would show a flat line having zero slope. Figure 3 show significant quantile-dependent increases in the slopes for the soluble leptin receptor concentrations and the free leptin index, i.e., each one-percent increase in the phenotype distribution increased β FS by 0.0014 ± 0.0007 (P linear = 0.04) for leptin receptor concentrations, and by 0.0043 ± 0.0010 (P linear = 8.5 × 10 −6 ) for the free leptin index. The increases were nonlinear for both the leptin receptor and the free leptin index (i.e., significant convexity for leptin index and significant concavity for the leptin receptor, with some cubic effects). Figure 1B showed that leptin heritability increased significantly with increasing quantiles of the offspring's leptin distribution when male and female age-and sex-adjusted sibling data were combined. However, Fig. 4 shows the leptin distribution in females is shifted towards to the right of the males' distribution, ergo the females' β FS should be greater than that of the males. In fact, as traditionally estimated by least squares regression, leptin's β FS was higher in females than males (0.26 ± 0.03 vs. 0.07 ± 0.01 for the total sample, P < 10 −15 ). Moreover, Fig. 5A shows that the quantile-specific β FS was higher in females than males at each percentile of their respective distribution.

Male-female differences in heritability.
The problem with Fig. 5A is that comparing male and female β FS at their 10th percentiles means comparing the male β FS at an unadjusted leptin concentration of 1.38 ng/ml with the female β FS at an unadjusted concentration of 3.38 ng/ml. At the 50th percentile, the males' β FS at 4.20 ng/ml is being compared to the female β FS at 11.97 ng/ml, and at the 90th percentile, the males' β FS at 12.52 ng/ml is being compared to the females' β FS at 41.10 ng/ml. Specifically, quantile-dependent expressivity predicts an increase in genetic effects with increasing leptin concentrations. Therefore the male and female β FS graphs were re-plotted to correspond to the same leptin concentrations in Fig. 5B using a probability-probability (P-P) plot (Fig. 2, see methods). The significant www.nature.com/scientificreports/ differences between the male and female β FS plots were eliminated when matched by leptin concentrations. In fact, the relationship of β FS to the percentiles of the leptin distribution was more easily described by quantile regression of the leptin concentrations unadjusted for sex in Fig. 5C. The bump below the 40th percentile of the age and sex-adjusted leptin distribution in Fig. 1B was eliminated for the unadjusted concentrations, along with the significant cubic effect (adjusted: P = 0.02; unadjusted P = 0.25). The preceding analyses of β FS in Framingham Study sibships lack the specificity of directly measured genotypes. This limitation may be partly addressed by reinterpreting published studies that measured genetic variants directly from the perspective of quantile-dependent expressivity (Figs. 6 and 7). Specifically, in each case, the difference in genetic effect size by environmental condition (adiposity, diet, smoking) or disease status (multiple sclerosis, systemic lupus erythematosus, psoriasis) corresponds to a larger genetic effect for the higher average leptin concentration, i.e., quantile-dependent expressivity.
Adiposity. Becer Fig. 6A-C shows that the leptin difference between obese and non-obese patients was greater in LEPR R-allele carriers than QQ homozygotes (18.2 ± 1.8 vs. 12.3 ± 2.0 ng/ml, P = 0.03), greater in MnSOD AlaVal heterozygotes (16.8 ± 1.6 ng/ml) and ValVal homozygotes (19.5 ± 1.3 ng/ml) than AlaAla homozygotes (11.8 ± 1.5 ng/ ml, P = 0.02 and 0.0001, respectively), and greater in PPARγ2 AlaAla homozygotes than carriers of the Pro-allele (16.8 ± 2.2 vs. 11.7 ± 0.8 ng/ml, P = 0.03). However, mean leptin concentrations were nearly three-fold higher in the obese. Consistent with quantile dependent expressivity, the line graphs of Fig. 6A-C show the difference between genotypes was greater at the higher mean leptin concentrations of the obese subjects than at the lower mean leptin concentrations of the non-obese subjects. www.nature.com/scientificreports/ The histogram in Fig. 6D shows that the leptin difference between obese and non-obese diabetics reported by Simon et al. 40 was greater in the Pro12Ala genotype than Pro12Pro genotype of PPARγ2 (40.3 ± 10.7 vs. 7.2 ± 2.2 ng/ml, P interaction = 0.002). The line graph shows this may be attributable to the larger genotype difference (28.4 ± 10.7 ng/ml) in obese subjects because of their higher average leptin concentrations (23.8 ± 1.9 ng/ml) vis-à-vis the smaller genotype difference (-4.7 ± 2.5 ng/ml) in non-obese subjects because of their lower average concentrations (13.9 ± 1.1 ng/ml).
Meirhaeghe et al. 44 reported there was a significant gene-adiposity interaction (P < 0.03) involving the silent C → T substitution in exon 6 of the PPARγ gene. The histogram in Fig. 6E shows a greater leptin difference between obese and non-obese subjects in carriers of the T-allele than CC homozygotes (23.2 ± 3.1 vs. 15.4 ± 1.7 ng/ml). However, average leptin concentrations were greater in the obese than nonobese subjects (30.3 ± 1.4 vs. 12.6 ± 0.4 ng/ml) and, as shown in the accompanying line graph, the difference between genotypes was substantially greater in the obese (6.7 ± 3.4 vs. −1.1 ± 0.8 ng/ml).
Data reported by Krempler et al. 45 showed that the leptin difference between obese and nonobese subjects was greater in wild type homozygotes than heterozygotes of the IRS-1 codon 972 variant (P interaction = 0.0004, Fig. 6F). However, average leptin concentrations were greater in the obese than non-obese subjects (36.7 ± 1.5 vs. 8.7 ± 0.5 ng/ml) and, as shown in the accompanying line graph, the difference between genotypes was substantially greater in the obese (11.1 ± 2.9 vs. −0.8 ± 1.7 ng/ml).
Another study, by Le Stunff et al. 47 , reported a greater effect of fat mass on serum leptin concentrations in obese girls who were + / + homozygotes (regression equation: leptin = 6 + 0.7kg fat ) than −/− homozygotes (leptin = -8.3 + 1.9kg fat ) of the LEP -2,549 polymorphism. The result was replicated in two separate cohorts. However, average leptin concentrations increased with increasing fat mass. There was no genotype difference for the less fat girls who had lower leptin concentration, and diverging leptin concentrations between genotypes as average leptin concentrations increased with increasing fat mass (their Figs. 2 and 3).
Scientific Reports | (2020) 10:22152 | https://doi.org/10.1038/s41598-020-79116-1 www.nature.com/scientificreports/ average leptin concentrations were, however, higher in the MS than matched control patients (15.70 ± 0.28 vs. 8.39 ± 0.27 ng/ml, P < 0.0001), and that this corresponded to greater differences between genotypes (GG minus AA: 7.81 ± 0.66 in MS vs. 5.00 ± 0.69 ng/ml in matched controls). The Fig. 7B histogram shows that MS's and control's leptin differences also significantly differed by the LEPR 223A/G polymorphism, P = 0.005 for AA versus AG genotypes, P = 3.7 × 10 −5 for AA versus GG genotypes, and P = 0.05 for AG versus GG genotypes. The associated line graph shows that the effect can again be attributed to the genotype differences being greater in the MS than healthy patients in accordance with the greater mean concentrations in MS than healthy patients.
Systemic lupus erythematosus. Afroze et al. 59 reported that leptin concentrations were significantly higher in G-allele carriers than AA homozygotes of the LEPR 223A/G polymorphism (25.6 ± 1.2 vs. 16.4 ± 2.2 ng/ ml, P < 0.001) for patients with systemic lupus erythematosus and apparently not in matched controls in accordance with the higher average leptin concentrations of the patients (23.9 ± 1.95 vs. 14.8 ± 1.04 ng/ml, P < 0.001).

Discussion
Quantile-regression does not require normality 51,52 , and provides the opportunity to assess quantile-specific genetic effects as originally measured. This approach led to the novel finding that genetic inheritance and shared environmental factors affecting leptin concentrations were over four-fold greater at the 90th than the 10th percentiles of the leptin distribution (Fig. 1). Traditionally, the decision to log transform data is driven solely by the statistical requirements of parametric testing. With respect to analyzing genotype-phenotype associations, the logarithmic and other normalizing transformations of right-skewed data accentuates the contribution of lower phenotypic values while diminishing the contribution of higher values. The logarithmic transformation eliminated the increase in β FS with increasing percentiles of the leptin distribution and the greater β FS in women than men, consistent with the conclusion that the genetic effects are concentration dependent. However, we are not aware of any biological rationale for analyzing normally distributed blood proteins as plasma concentrations and asymmetrically distributed blood proteins as log-concentrations. In fact, the majority of studies reporting gene-environment interactions involve untransformed leptin concentrations 42,43,[45][46][47]49,[58][59][60] . Women have higher leptin concentrations than men due to their female body fat distribution and/or low testosterone [35][36][37] . The goal of sex-adjustment is to eliminate the male-female difference, usually through a translational adjustment of their respective distributions, to ideally attain comparability at each percentile of their respective distributions. Figure 1 suggests that the higher leptin concentrations in women than men should result in stronger female inherited or shared environmental effects on their leptin concentrations, as observed in Fig. 5. This resulted in a significant sex-difference between male and female β FS when their age-and sex-adjusted data were matched at their corresponding percentiles (Fig. 5A), but not when matched by leptin concentrations (Fig. 5B), or when their leptin concentrations were analyzed without adjusting for sex (Fig. 5C). Rather than postulating sex-specific genetic effects 39 , we propose that the greater female than male leptin heritability reported by others 11,13,14,39,40 may be entirely attributable to the women's higher leptin concentrations. Caveats and limitations. An important limitation of our analysis is that β FS does not only measure heritability (i.e. the proportion of the phenotype variance due to additive genetic effects). Falconer's formula equate β FS to (0.5V A + 0.25V D + V Ec )/V P where V A is the additive genetic variance, V D the dominance variance, V Ec the common environment variance, and V P the phenotype variance 57 . Although there is no way to separate V A , V D , and V Ec or to correct for assortative mating in our analyses, 2β FS (i.e., 0.34 ± 0.04) is smaller than 75% of the heritability estimates published by others [8][9][10][11][12][13][14][15][16][17][18][19] , suggesting that V D , V Ec , and assortative mating effects are modest, and the observed quantile-effects are largely genetic. Other studies, in fact suggest spouse concordance and shared environmental effects are modest. For example, Hasselbach et al. 13 identified no significant shared environmental effect. The lack of common familial environmental influence and spousal effects were also reported by Rotimi et al. 14 . Liu et al. 12 did not identify any significant spousal effect, and attributed only 12% of leptin variance to the shared sibling environment. www.nature.com/scientificreports/ None of the SNPs identified to date explain any more than a few percent of leptin or soluble leptin receptor heritability 22,23 , which means that the effects of any particular SNP is not necessarily constrained by results of Fig. 1. Not all studies show an increase in genetic effect size with increasing leptin concentrations, e.g., the C/T exon 6 PPARγ polymorphism had the same effect in obese women and men despite the women's two-fold greater leptin concentrations 44 . Our analyses were derived from an exclusively White population which may not apply to other racial groups, e.g., Luke et al. 's 61 report that that lower leptin concentrations of Nigerians (6.4 ± 0.3 ng/ml) than Jamaicans (15.0 ± 0.7 ng/ml) or African Americans (18.8 ± 0.4 ng/ml) did not correspond to lower leptin heritability (h 2 : 0.38, 0.25, and 0.43, respectively).

Conclusion.
Our principle finding is that the full-sib regression slope increases with increasing percentiles of the sibs' leptin concentrations, and that this increase accelerates dramatically at higher portions of its distribution. Included in the regression slope are genetic effects, which on the basis of other heritability studies, we presume to be substantial. This suggests the expressivity of leptin concentrations is quantile-dependent, that quantile-dependent expressivity likely explains the larger genetic effects on women's than men's leptin concentrations, and may contribute to many purported gene-environment interactions affecting leptin. In seeking genetic variants affecting leptin and other traits, it may not make sense to accentuate the weaker genetic effects at the lower phenotype values while de-emphsizing the stronger genetic effects at the higher phenotype values.