Group Title: Molecular Pain 2008, 4:13
Title: A computational model for sex-specific genetic architecture of complex traits in humans: Implications for mapping pain sensitivity
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Title: A computational model for sex-specific genetic architecture of complex traits in humans: Implications for mapping pain sensitivity
Series Title: Molecular Pain 2008, 4:13
Physical Description: Archival
Creator: Wang C
Cheng Y
Liu T
Li Q
Fillingim RB
Wallace MR
Staud R
Kaplan L
Wu R
Publication Date: 39554
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Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Molecular Pain

BioM.ed Central


A computational model for sex-specific genetic architecture of
complex traits in humans: Implications for mapping pain sensitivity
Chenguang Wang', Yun Cheng', Tian Liu', Qin Li', Roger B Fillingim2,
Margaret R Wallace3, Roland Staud3, Lee Kaplan3 and Rongling Wu*1

Address: 'Department of Statistics, University of Florida, Gainesville, FL 32611 USA, 2Department of Community Dentistry and Behavioral
Science, University of Florida, Gainesville, FL 32611, USA and 3Department of Molecular Genetics and Microbiology, University of Florida,
Gainesville, FL 32611, USA
Email: Chenguang Wang; Yun Cheng; Tian Liu; Qin Li;
Roger B Fillingim; Margaret R Wallace; Roland Staud;
Lee Kaplan; Rongling Wu*
* Corresponding author

Published: 16 April 2008
Molecular Pain 2008, 4:13 doi:10.1 186/1744-8069-4-13

Received: 29 November 2007
Accepted: 16 April 2008

This article is available from:
2008 Wang et al; licensee BioMed Central Ltd.
This is an Open Access article distributed under the terms of the Creative Commons Attribution License (,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Understanding differences in the genetic architecture of complex traits between the two sexes has
significant implications for evolutionary studies and clinical diagnosis. However, our knowledge
about sex-specific genetic architecture is limited largely because of a lack of analytical models that
can detect and quantify the effects of sex on the complexity of quantitative genetic variation. Here,
we derived a statistical model for mapping DNA sequence variants that contribute to sex-specific
differences in allele frequencies, linkage disequilibria, and additive and dominance genetic effects
due to haplotype diversity. This model allows a genome-wide search for functional haplotypes and
the estimation and test of haplotype by sex interactions and sex-specific heritability. The model,
validated by simulation studies, was used to detect sex-specific functional haplotypes that encode
a pain sensitivity trait in humans. The model could have important implications for mapping
complex trait genes and studying the detailed genetic architecture of sex-specific differences.

Differences in males and females (sexual dimorphism) is
ubiquitous in many biological aspects [1-3]. In humans,
sexually dimorphic traits include those from morpholog-
ical shapes and body size to brain development to disease
susceptibility [4,5]. Substantial differences are also
observed in sensitivities to pain and pain-killing drugs,
and susceptibility to developing chronic pain between
men and women [6-8]. All these sex-specific differences
are due to varying expression of genes on the X/Y chromo-
some and autosomes, thought to result from differences
in cellular and hormonal environments between the two
sexes [9]. A growing body of research has been conducted

to elucidate the genetic control of sexual dimorphism in
various complex phenotypes by gene mapping
approaches [4,5,10,11]. Despite these efforts, however,
little is known about the genetic architecture underlying
sex-related variation in a quantitative trait.

Since sex is easily determined, the effects of sex on mor-
phological, developmental and pathological traits can be
directly observed. However, characterizing the impacts of
sex on the genetic architecture of these traits has been
challenged by a lack of powerful statistical approaches.
The motivation of this article is to develop a statistical and
computational model that can systematically search for

Page 1 of 10
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sex-specific genes contributing to quantitative variation
and formulate testable hypotheses regarding the interplay
between sexes and gene expression. Our model is princi-
pally different from those used in many previous studies
that are aimed to detect sex-specific quantitative trait loci
(QTLs) based on linkage or linkage disequilibrium analy-
sis [2,4,5,12,13]. Our model will be founded on the statis-
tical framework constructed by Liu et al. [14] to detect the
effects and diversity of haplotypes constructed by single
nucleotide polymorphisms (SNPs) that are genotyped at
candidate genes or genome-wide [15]. Our model has
been generalized to allow the test of sex differences in
haplotype frequencies, allele frequencies and linkage dis-
equilibria between different SNPs as well as additive and
dominant effects of haplotypes on complex traits. It has
power to identify sex-specific DNA sequence variants that
encode complex phenotypes in men and women.

Suppose there is a diversity of haplotypes constructed by
two SNPs each with two alleles designated as 1 and 0. Let
p and q be the 1-allele frequencies for the first and second
SNP, respectively. Thus, the 0-allele frequencies at these
two different SNPs will be 1 p and 1 q. The two SNPs
that are segregating in a natural human population form
four haplotypes, [11], [10], [01], and [00], whose frequen-
cies are constructed by allele frequencies and linkage dis-
equilibrium (D) between the two SNPs, i.e.,

pll = pq+D
Po = p(1-q)- D
Po0 = (1-p)q- D
Poo = (1-p)(1-q)+D

The parameters contained in equation (1) can be used to
describe some important aspects of the genetic structure
and diversity of a natural population. Thus, differences in
genetic architecture between the two different sexes can be
characterized by these sex-specific parameters. Because it
is easy to derive the closed forms for estimating haplotype
frequencies [14], we will estimate the linkage disequilib-
rium from the estimated haplotype frequencies.

Let OMp = (PM11 PM10' PM0'PMo0) and

Fp = (PF11' PFo' PF0ol PFoo) be the vectors of haplotype fre-
quencies among males and females, respectively. All the
genotypes for the two SNPs are consistent with diplo-
types, except for the double heterozygote, 10/10, that
belongs to a diplotype of either [11] [00] or [10] [01]
(Table 1). Assuming that the population is at Hardy
Weinberg equilibrium, the frequency of a diplotype is

expressed as the product of the frequencies of the two hap-
lotypes that construct it. Table 1 characterizes the differ-
ences in diplotype frequencies between the males and

If haplotypes triggers an effect on a quantitative trait, this
means that at least one haplotype performs differently
from the rest of the haplotypes. Without loss of generality,
let haplotype [11] be such a distinct haplotype, called risk
haplotype, designated as A. All the other non-risk haplo-
types, [10], [01] and [00], are collectively expressed as A.
The risk and non-risk haplotypes form three composite
diplotypes AA (symbolized as 2), AA (symbolized as 1)
and AA (symbolized as 0). The genotypic values of the
three composite diplotypes may be different between the
two sexes, arrayed in (M2, PM1, PM0) for the males and

(PF2pFi', Fo0) for the females, respectively. Let (aM, dM)
and (aF, dF) be the additive and dominance genetic effects
due to the risk and non-risk haplotypes in males and
females, respectively.

Assume that a total of n subjects (including nM males and
nF females) sampled from the population are phenotyped
for a quantitative trait. In each sex, there are nine possible
genotypes for the two SNPs, each genotype with an
observed number generally expressed as n for the

males and n F for the females (r, > r', r2 r'2, r3 r'3
rm r'f / r2r2

Many physiological traits scale with body weight (W)
according to a power function with a certain allometric
exponent. Thus, we implement this allometric scaling law
to describe the phenotypic value of a trait for subject i
within male or female subpopulations in terms of the
haplotypes considered as

yM, = aMWM +xaM+zMdM+eM,

YF = FW F + xFaF + ZFdF + eF,

where (a ai) or (ca, A) are body weight-related allom-
etric coefficients, (xM z ) or (XF, F, ) are the indicator
variables associated with the additive and dominance
effects, respectively, and eM, or eF is the residual error,

normally distributed as N(0, o1) or N(0, o) The geno-

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Molecular Pain 2008, 4:13

Table I: Diplotypes and their frequencies for each of nine genotypes at two SNPs, and composite diplotypes for one assumed sex-
specific risk haplotype chosen from four possible haplotypes.

Risk Haplotype in Males

Risk Haplotype in Females

Genotype Configuration Male Freq.

[i1][i1] p
e 1V,1

[I1] [10] 2pM11PMIo

[10] [10] 2

[11] [0] 2pM11PMo,

Female Freq. [I I] [10] [01] [00] [II] [10] [01]




2pF PFoI
2PFP eF01

88 BB BB B


AA 88




[10] [00] 2pMloPMoo

[01] [01] 2
1][00] PMo
[OI] [00] 2PmOlPmOO




201Fo PFoo

[00] [00] P200





Two alleles for each of the two SNPs are denoted as I and 0, respectively. Genotypes at different SNPs are separated by a slash. Diplotypes are the
combination of two bracketed maternally and paternally derived haplotypes. By assuming different haplotypes as a risk haplotype (denoted as A for
males and B for females), composite diplotypes are accordingly defined and their genotypic values are given.

typic values of composite diplotypes and variance are
arrayed by a quantitative genetic parameter vector
OM = (aM,lM,aM,dM, 2) for the males and

Fq = (aF,PF,aF,dF, 02) for the females, respectively.

The log-likelihood of haplotype frequencies, genotypic
values of composite diplotypes and residual variances
given sex-specific phenotypic (YM, Yp) and SNP data (SM,
Sp) is factorized into two parts, expressed as

logL(O,,OM ,;OFp,OF I yMSM;YF,SF)
= logL(OMp,OF I SM,SF)+logL(OM ,OF I YM,SM,OMp;Y,SF,OFp)

= logL(O8H SM)+logL(8 q SF)+logL(OM y ,SM,MO,)+ ogL(OF. YFpSF,,Oq)
where equation (4) is derived from equation (3) because
the males and females are assumed to be independent,

" S i ( SJ)= constant

+2n'11 logPk,
+n ,... _( P )

+2n1/oo logphko
+ 10/111 -I-I Ph )

+ ..... -I Pkoo + )

+ ...... i _- i Ph o)

+2n0 /11 logp No

+ .. ... _(- Pkoo
i. I

1 ..- |y ,,S,,O,)=

1 logfk2(k)
0 log fhk (Y, )

+C log y,(yk.)

+ i log[4af,l (y,) + (1 0h)fh, (Y,)1

+ '1'/oolog fkf (h, )

+jc / logfk (Y,,)
+1 no/l log[,(Yk,)
+ o log fko )

n o/loo
+ -Iolg )

where ft (y,) is a normal distribution density function

of composite diplotypej (j = 2, 1, 0) for sex k, and

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00/1 1



Molecular Pain 2008, 4:13

= PklPk00
Pkoo00Pk0 +PkPk01
is the relative proportion of diplotype [11] [00] within the
double heterozygote for sex k.

It can be seen from equation (3) or (4) that maximizing
L(OM ,0 M q; F, IF M,SM,YF,SF) is equivalent to

maximizing log (0kp IS) and log L(Ok YPI,Sk,Okp)

individually in equation (5).

The EM algorithm
A closed-form solution for the EM algorithm [14] has
been derived to estimate the unknown parameters that
maximize the two sex-specific likelihood of (5). The esti-
mates of sex-specific haplotype frequencies are based on
the log-likelihood function (0)p Sk), whereas the esti-

mates of sex-specific genotypic values of composite diplo-
types and the residual variance are based on the log-
likelihood function L(O, y S ) These two dif-

ferent types of parameters can be estimated using a two-
stage hierarchical EM algorithm (see ref. [14] for a
detailed implementation).

Model selection
According to equation (5), the summed likelihood across
the sexes, L(OM,O M q I y,,SM)+ L(Op, OFq yF,SF), is
formulated by assuming that haplotype [11] is a risk hap-
lotype. However, a real risk haplotype is unknown from
raw data (yk, SJ). An additional step for choosing the most
likely risk haplotype should be implemented. The sim-
plest way to do so is to calculate the likelihood values by
assuming that any one of the four haplotypes can be a risk
haplotype (Table 1). Thus, we obtain four possible likeli-
hood values as follows:

No. Haplotype
1 [11]

L(6mM., IM ySM) + LI(e, F. I YF, SF)

2 [10] L,(OM, 62M, YM SM) + L2(OF, '2F I YF, SF)
3 [01] L3(Mp, 6,3Mq y, SM)+L3(Fp,'3Fq YF,SF)
4 [00] L4(OM, 4M YM, SM)+L4(OF '04F F,SF)
The largest likelihood value calculated is thought to corre-
spond to the most likely risk haplotype. Under an optimal

risk haplotype, we estimate sex-specific quantitative
genetic parameters Mq and 0F .

Hypothesis tests
The genetic architecture of a quantitative trait is character-
ized by population (including haplotype frequencies,
allele frequencies, and linkage disequilibria) and quanti-
tative genetic parameters (including haplotype effects and
mode of inheritance for haplotypes). The model proposed
provides a meaningful way for estimating the genetic
architecture of a trait and further testing sex-specific differ-
ences in genetic control.

After haplotype frequencies are estimated, allele frequen-
cies and linkage disequilibrium between the two SNPs
with each sex can be calculated as

Male Female
AllelefrequencyforSNP1 p = phM +pM pF = PF +PFp
Allele frequency for SNP 2 qM = p/m + Pho qF = pFp +PF,,
Linkage disequilibrium DM = pmIiPMo PMi0pm DF = PF,,PFo PFjoPF,
The influence of haplotypes on a quantitative trait is
quantified in terms of the additive (a) and dominant
genetic effects (d), and the mode of inheritance (p), which
are estimated for each sex. Each of these population and
quantitative genetic parameters can be tested when appro-
priate hypotheses are formulated.

Overall genetic control
Haplotype effects on the trait, i.e., the existence of func-
tional haplotypes, in both male and female populations
can be tested using the following hypotheses expressed as

Ho: /'M I /- M and /F -- forj =2,1,0
H1 : At least one equality in Ho does not hold
The log-likelihood ratio test statistic (LR) under these two
hypotheses can be similarly calculated,

LR= -_1I... I,,. .. y yM,;yF)- .I l'. ;, OF yM, SM, OM ;yFSF,O F)],

where the L0 and L1 are the plug-in likelihood values
under the null and alternative hypotheses of (8), respec-
tively. Although the critical threshold for determining the
existence of a functional haplotype can be based on
empirical permutation tests, the LR may asymptotically
follow a X2 distribution with four degrees of freedom, so
that the threshold can be obtained from the
distribution table.

Sex-specific population genetic architecture
The male and female populations may be different in
terms of population genetic parameters. Such sex-specific

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Molecular Pain 2008, 4:13

differences can be tested by formulating the following

Ho : PM = PF
H1 : PM PF

for allele frequency at SNP 1,

Ho: q= qF
H : qM# qF

for allele frequency at SNP 2, and

Ho: DM =DF

for the linkage disequilibrium between the two SNPs.

For each of the hypotheses (10)-(12), the LR values are
calculated, which are each thought to asymptotically fol-
low a X2-distribution with one degree of freedom. Sex-spe-
cific differences in overall population genetic architecture
can be tested with the null hypothesis H0: PM = PF, qM = qF,
and Dm = DF, with the corresponding LR value to be ; 2-
distributed with three degrees of freedom.

Sex-specific quantitative genetic architecture
Sex-specific differences in overall quantitative genetic
architecture can be tested by formulating the hypotheses

H : aM = aFanddM = dF
H1 : At least one equality in Ho does not hold

The LR value calculated under the null and alternative
hypotheses is suggested to follow a X2-distribution with
two degrees of freedom. The rejection of the null hypoth-
esis implies that the effects of the same haplotype are dif-
ferent between the two sexes. If there exists a sex-specific
difference, the next step is to test whether this difference is
due to the additive or dominant genetic effects, or both.

Sex-specific risk haplotypes
In the preceding sections, the same risk haplotype was
assumed between the male and female populations. It is
possible that the two sexes have different risk haplotypes.
Let pM1 Om,, = 2, 1, 0) and PF, (j]= 2, 1, 0) be the geno-

typic values of composite diplotypes for the males and
females constructed by a sex-specific rick haplotype. By
reformulating the likelihood log L(OkP I y k, S k, ,O ) of

equation (5) based on sex-specific composite diplotypes,
these genotypic values can be estimated with the EM algo-
rithm. A best combination of risk haplotypes between the
two sexes can be determined from the AIC values.

Multi-locus haplotyping
Three-SNP model
Consider three associated SNPs, Si, S2, and S3, each with
two alleles denoted by 1 and 0. Letp, q and r, and D12, D13,
D23 and D123 be the 1-allele frequencies for the three SNPs,
and the linkage disequilibria between SNPs 1 and 2, SNPs
1 and 3, SNPs 2 and 3 and among the three SNPs, respec-
tively. Eight haplotypes, [111], [110], [101], [100], [011],
[010], [001] and [000], formed by these three SNPs, have
sex-specific frequencies arrayed in
0hp = (Pke11,'Pke11'ok, ph'Pkl oo'Poh, 'PkOlo'Pk ooI',Pkooo) for
sex k. Each of these haplotype frequencies is constructed
by allele frequencies at different SNPs and their linkage
disequilibria of different orders, expressed as

Pk,,, = P k + PDkh, + qDh,, + rhDh + Dhq
Pkpo = pAq((1 r) phDh qDk3 + (1 )Dh Dh2
Pkh = p(1l- qk)rT- P-hDh +(1- q)D h -h rkDh Dk
Pk_ = p1(1- q4)(1 r) + phDh ( -(- qk)Dk( (1 r)Dhk + Dk
Pk0 =(1 pe)qh7h + (1 p)Dh qDk rhDhk Dh
P (,,, = (1 P)q)(l A)-(1- p )Dh + qkDk (1 r)Dhk + Dk2
Pk,,, = (1 Pe)(1 ql)- (1 p )D ( -(1- q)Dk3 r7-Dk + Dk12
Pk, = (1- ph)(1- q4)(1 r) + (1 p)Dk + (1 q)Dk3 +(1- r-)Dkh + Dh,
for sex k.

Sex-specific population genetic architecture can be tested
by comparing the differences in allele frequencies (Pk, 1 .
rk) and linkage disequilibria of different orders
(Dk2, Di3, ,Dtk D ) between the males and females.

In a natural population, there are 27 genotypes for the
three SNPs. The frequency of each genotype is expressed
in terms of haplotype frequencies. Some genotypes are
consistent with diplotypes, whereas the others that are
heterozygous at two or more SNPs are not. Each double
heterozygote contains two different diplotypes. One triple
heterozygote, i.e., 10/10/10, contains four different diplo-
types, [111] [000] (in a probability of 2Pk111,Pko ), [110]

[001] (in a probability of 2PkploPoo ), [101] [010] (in a
probability of 2Pk101Pk010 ) and [100] [011] (in a probabil-
ity of 2ploopol0). The relative frequencies of different
diplotypes for this double or triple heterozygote are a
function of haplotype frequencies (Supporting Informa-
tion Table 1). The integrative EM algorithm can be
employed to estimate the MLEs of haplotype frequencies.
A general formula for estimating haplotype frequencies
can be derived.

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Molecular Pain 2008, 4:13

By assuming [111] as a risk haplotype (labeled by A) and
all the others as non-risk haplotypes (labelled by A ), the
formulation of genotypic values for three composite
diplotypes, /p for AA, p, for AA and p0 for AA can be
derived. Similar procedures described for the two-SNP
model can be obtained to estimate and test sex-specific
additive and dominance genetic effects when a haplotype
contains three SNPs.

L-SNP model
It is possible that the two- and three-SNP models are too
simple to characterize genetic variants for quantitative var-
iation. We can develop a model that includes an arbitrary
number of SNPs whose sequences are associated with the
phenotypic variation. A key issue for the multi-SNP
sequencing model is how to distinguish among 2 -1 differ-
ent diplotypes for the same genotype heterozygous at
loci. The relative frequencies of these diplotypes can be
expressed in terms of haplotype frequencies.

Consider a a functional haplotype that contains L SNPs
among which there exist linkage disequilibria of different
orders. The two alleles, 1 and 0, at each of these SNPs are
symbolized by r, ..., rL, respectively. Let p ,...,p be the
allele frequencies for these different SNPs within sex k. A
haplotype frequency, denoted as pr1-q r, is decomposed
into the following components:

P kr
= pp No LD
+(-1)' --- p DL .)L +(-1) 2 -pD Digenic LD
+(-1) L P D ++(-1 ...pD Trigenic LD

+(-1)L(- _).+ +.Di L L-genic LD
where Dk's are the linkage disequilibria of different orders
among particular SNPs for sex k.

Sex-specific difference in terms of allele frequencies and
linkage disequilibria between different SNPs as well as
haplotype additive and dominance effects can be tested by
formulating the corresponding hypotheses.

Pain genetics study
The model proposed was used to detect differences in the
genetic architecture of pain sensitivity between men and
women. Genetic and phenotypic data were from a pain
genetics project in which 237 subjects (including 143 men
and 94 women) from five different races were sampled for
six SNPs at three candidate genes. As a demonstration of
the utilization of the model, we will focus on two SNPs,

OPRDT80G (with two alleles T and G) and OPRDT921C
(with two alleles T and C), at the delta opioid receptor.
Pain testing procedures followed Fillingim et al. [16]. The
phenotypic values of traits were subtracted by the means
for each race to remove the effect due to races.

These two SNPs construct four haplotypes, [TC], [TT],
[GC], and [GT], which yield 10 diplotypes, [TC] [TC], [TC]
[TT], [TT] [TT], [TC] [GC], [TC] [GT], [TT] [GC], [TT] [GT],
[GC] [GC], [GC] [GT], and [GT] [GT] and nine genotypes,
GG/CT and GG/TT. Based on the observed numbers of
each genotype in the male and female populations, we
estimated sex-specific haplotype frequencies (Table 1).
The pattern of haplotype distribution is consistent
between the two sexes, with haplotypes [TC] and [TT]
jointly occupying a majority proportion in the popula-
tions. Haplotype [GT] is very rare, with the frequency
close to zero. SNP OPRDT80G has a low heterozygosity
because the frequency of its commoner allele (T) is closer
to 0.90, whereas there is a high heterozygosity for SNP
OPRDT921C in terms of its averaged allele frequencies.
The two SNPs are highly significantly associated at p =
3.41 x 10-5 for males and p = 9.63 x 10-5 for females, with
a normalized linkage disequilibrium of D' = 1.00, because
alleles T from OPRDT80G and T from OPRDT921C as
well alleles G from OPRDT80G and C from OPRDT921C
tend to form the same haplotypes more frequently than at
random. There is no sex-specific difference in allele fre-
quencies at the two SNPs and their linkage disequilib-

By assuming that one of the haplotypes is a risk haplo-
type, we estimated the effects of each haplotype on a pain
sensitivity trait, assessed with a baseline pressure pain
threshold measured at the ulna, in the pooled male and
female population. The likelihood of haplotype [TC],
[TT] and [GC] as a risk haplotype are -594.8, -594.5, and -
596.1, and thus the most likely risk haplotype is [TT]. The
genotypic values of composite diplotypes constructed by
this risk haplotype and its non-risk haplotype counterpart
were estimated and compared between different sexes. In
both males and female, the three composite diplotypes do
not display significant genetic differences in the pain trait
studied, but the directions of the additive and dominance
effects are different between the two sexes (Table 2). In
males, the non-risk haplotype tends to increase pressure
pain thresholds, and it is overdominant to the risk haplo-
type, leading to increased pressure pain thresholds at a
marginal significance level (p = 0.058) (Fig. 1). By con-
trast, in females, the non-risk haplotype tends to reduce
pressure pain thresholds, and it also tends to be overdom-
inant to the risk haplotype by reducing pressure pain
thresholds. These discrepancies in both effect size and
direction (Fig. 2) make the overall quantitative genetic

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Molecular Pain 2008, 4:13

Table 2: The estimates and tests of population genetic structure for two SNPs, OPRDT80G (with alleles T and G) and OPRDT92 IC
(with two alleles C and T), and quantitative genetic effects of haplotypes constructed by these two SNPs on baseline pressure pain
thresholds measured at the ulna in males and females.


Genetic Parameter




p-value Sex-specific LR Sex-specific p-value

Haplotype Frequency








Allele Frequency and Linkage Disequilibrium

PT (OPRDT80G) 0.869

qC (OPRDT921C) 0.495

-0.066 3.41 x 10-5







-0.066 9.63 x 10-s

Additive and Dominant Effects and Inheritance Mode for Risk Haplotype [TT]

a and d




architecture of the pain sensitivity trait significantly differ-
ent between the two sexes (p = 1.49 x 10-7) (Table 2).
Although the additive genetic effect displays a gene by sex
interaction at the p = 0.03 significance level, a gene by sex
interaction for the dominance effect is highly significant
at p = 6.16 x 10-7. No significant difference was observed
in inheritance mode between males and females.

Monte carlo simulation
Simulation studies were performed to test the statistical
properties of the model proposed. Given a certain sample
size (n), we simulated two SNPs by assuming different
allele frequencies and linkage disequilibria between two
sexes. The hypothesized allele frequencies at the two SNPs
are PM = 0.5, q,= 0.6 and DM = 0.1 for males and F,= 0.8,
q, = 0.9 and DF = 0.06 for females. By postulating one of
the four haplotypes constructed by the two SNPs as a risk
haplotype, we calculated the genetic variance among three
composite diplotypes using the additive effect a = 0.6 and
dominance effect d = 0.8, from which the residual vari-

ance was calculated when the heritability (H2) of a trait is
given. The phenotypic values of the trait were simulated
by assuming that they follow a normal distribution under
four different simulation designs, (1) n = 100 and H2 =
0.1, (2) n = 400 and H2 = 0.1, (3) n = 100 and H2 = 0.4,
and (4) n = 400 and H2 = 0.4.

The new model was used to analyze the simulated SNP
and phenotypic data with the results tabulated in Table 3.
Population genetic parameters including allele frequen-
cies (p and q) and linkage disequilibrium (D) can well be
estimated, with increasing precision when the sample size
increases from 100 to 400. The power to detect the given
sex-specific differences in allele frequencies and linkage
disequilibrium is as high as 0.95 even with a modest sam-
ple size (100). Although the traditional model that does
not implement sex-specific differences can provide precise
estimates of these parameters, the estimates are generally
between the true values of males and females.

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1.49 x 10-7


6.16 x 10-7


Molecular Pain 2008, 4:13



Figure I
Different genotypic values of baseline pressure pain thresh-
olds measured at the ulna for composite diplotypes, AA, AA,
and AA, constructed by risk haplotype [TT] and non-risk
haplotype in males and females. The origin indicates the
mean of the genotypic values between the two homozygotes.

The estimation of quantitative genetic parameters includ-
ing the additive (a) and dominance effects (d) needs the
determination of an optimal risk haplotype. When all
possible risk haplotypes were assumed for the simulated
data, we found that the true risk haplotype gave the largest
likelihood among the four possible cases. In general,
quantitative genetic parameters can well be estimated, but
the estimation precision increases dramatically with sam-
ple size and heritability. Although the additive effect can
be obtained with reasonable precision at a modest herita-
bility (0.1) with a modest sample size (100), the precise
estimation of the dominance effect relies upon a larger
heritability and sample size. Also, the given difference in
the additive effect between two sexes can be detected with
great power, even when both the sample size and herita-
bility are small. But the same size of sex-specific difference
in the dominance effect can be detected with the same
power only when the sample size is 400 and heritability is
0.4 (Table 3). The traditional model gave biased estimates
of the sex-specific additive and dominance effects regard-
less of increasing sample size and heritability.

We conducted an additional simulation study, in which
the data simulated under the assumption of no sex-spe-
cific differences in all genetic parameters were analyzed by
the new and traditional model. As expected, both the
models provide reasonable estimates of population and
quantitative genetic parameters, with estimation precision
increasing with increasing sample size and heritability
(data not shown). This, in conjunction with the results in

Figure 2
Different additive (a) and dominant genetic effects (d) of hap-
lotypes on baseline pressure pain thresholds measured at the
ulna in males and females.

Table 3, suggests that the new model provides a general
tool for study the genetic architecture of a complex trait,
regardless of whether the genetic control of the trait is sex-

The genetic architecture of a quantitative trait is complex
in terms of interactions between its underlying genetic fac-
tors and various environments including sex [1-3]. How-
ever, in many current studies, gene by sex interactions are
often ignored simply because existing analytical models
are not incorporated by environmental factors. While dif-
ferent phenotypic expressions of a trait between the two
sexes can be easily measured [4,5], sex-specific discrep-
ancy in the genetic control of the trait can be discerned
only when a sophisticated model is used. There is strong
evidence for sex-specific genetic influence [17,18] even for
traits that display no sexual dimorphism [19]. Although
quantitative genetic models are available to estimate sex-
specific heritabilities due to aggregative effects of many
genes [5,19-22] or map sex-specific QTLs for phenotypic
variation [4,5,11-13,23], the model proposed in this arti-
cle can dissect sex-specific genetic control at the DNA
sequence levels.

Our model is founded on the conceptual framework for
haplotyping a trait with single nucleotide polymorphisms
(SNPs) formulated by Liu et al. [14]. Since haplotypes
constructed by physically associated SNPs are thought to
affect the expressivity of a complex trait [23,24], it is more
precise to characterize such haplotype effects by incorpo-
rating gene by gene and gene by sex interactions. Lin and
Wu [25] extended Liu et al.'s [14] model to estimate hap-

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Molecular Pain 2008, 4:13

Table 3: The MLEs of population and quantitative genetic parameters and the standard errors of the estimates obtained by the new
model and the power to detect sex-specific differences under different simulation designs. Parameter estimates by a conventional
model are also given.

Simulation Design

Genetic Parameter


0.5025 0.0360
0.5964 0.0349
0.0961 0.0194

0.6055 0.2008
0.3061 0.3371

0.5013 0.0172
0.60 1 1 0.0177
0.0986 0.0098

0.6050 0.1 I 14
0.2980 0.1604

0.5008 0.0366
0.5945 0.0338
0.0965 0.0178

0.5989 0.0924
0.2934 0.1352

0.5014 0.0181
0.6000 0.0171
0.1001 0.0092

0.6003 0.0439
0.3008 0.0659


0.7829 0.0265
0.8732 0.0193
0.0181 0.0099

0.3045 0.1277
0.6093 0.1798

0.7980 0.0141
0.8958 0.0101
0.0084 0.0053

0.3056 0.0728
0.5986 0.0933

0.7835 0.0269
0.8731 I 0.0195
0.0161 0.0088

0.3012 0.0505
0.6021 0.0652

0.7974 0.0136
0.8968 0.0102
0.0077 0.0048

0.3010 0.0301
0.6006 0.0372


Traditional Model

0.6433 0.0224
0.7347 0.0201
0.0779 0.0120

0.4789 0. 192
0.5843 0.1875

0.6496 0.01 I I
0.7484 0.0100
0.0775 0.0061

0.67 0.4907 0.0616
0.42 0.5715 0.0966

0.6421 0.0221
0.7338 0.0194
0.0747 0.0 I 15

0.4582 0.0483
0.5355 0.0715

0.6494 0.01 13
0.7484 0.0099
0.0745 0.0058

0.4684 0.0251
0.5499 0.0360

lotype-haplotype interactions. The new model reported
here can not only estimate sex-specific genetic parameters,
but also provide a series of statistical procedures for test-
ing sex-specific differences in the genetic architecture of
quantitative variation. In a natural population, the struc-
ture and pattern of genetic variation can be studied by
population genetic parameters, such as haplotype fre-
quencies, allele frequencies and linkage disequilibria.
Thus, the understanding of differences in these parame-
ters between the two sexes help to infer the sex-specific
genetic structure of a natural population and its evolu-
tionary processes. As shown through simulation studies,
our model is alert to discern sex-specific differences in
basic population genetic parameters.

Liu et al. [14] assumed a so-called reference or risk haplo-
type that triggers an effect on complex traits in a different
way from the other haplotypes. Thus, the combinations
between the risk and non-risk haplotypes (composite
diplotypes) will perform differently, depending on the
type of combination, i.e., risk by risk, risk by non-risk and

non-risk by non-risk. Liu et al. [14] proposed the concepts
of the additive effect due to the substitution of the non-
risk (or risk) haplotype by the risk (or non-risk) haplotype
and the dominance effect due to the interaction between
the risk and non-risk haplotypes. These concepts have
been integrated into the current model that allows the test
of additive by sex and dominance by sex interaction
effects. Simulation studies suggest that the new model dis-
plays adequate power to detect differences in these quan-
titative genetic parameters between the sexes, although
the detection of sex-specific dominance effects needs a
much larger sample size and/or heritability level.

Our model was used to analyze a real data set for pain
genetics. Our analyses of a pain sensitivity trait-baseline
pressure pain threshold measured at the ulna-by estimat-
ing genetic parameters and testing their sex-specific differ-
ences in a combination of male and female samples
revealed that males and females have different population
structure at two SNPs genotyped from a candidate gene
(delta opioid receptor) for human pain and that haplo-

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Molecular Pain 2008, 4:13

types exert different genetic effects on the trait between the
two sexes. A further test indicates that the risk haplotype
[TT] detected by the model exemplifies sex-specific modes
of inheritance in affecting the pain trait. While there are
no differences among composite diplotypes in males, a
significant additive effect was detected in females. Both
additive and dominance effects due to the risk haplotype
identified are different between the two sexes. Anholt and
Mackay [2] described three major mechanisms that
explain sex-specific difference in trait control, i.e., sex-spe-
cific effects (a gene affects only one sex), sex-biased effects
(a gene affects both sexes but to different degrees), and
sex-antagonistic effects (a gene affects both sexes but in
opposite directions). In our example, the functional hap-
lotype detected affects the pain trait in a sex-antagonistic
effect manner, a mechanism thought to help the mainte-
nance of genetic variation in natural populations [26].

In practice, failure to model sex-specific architecture may
significantly hamper the ability to detect signals of func-
tional genetic variants in genomewide screens. Although
combining male and female data to increase sample size
are tempting approaches to increase power, the estimates
in this way will be biased from true sex-specific differ-
ences. Possible mechanisms that cause sex differences
include parent-of-origin effects [27], linkage to or interac-
tion with sex chromosomes, or differences arising from
sex-specific hormonal environments. Our gene by sex
interaction model that is incorporated by these mecha-
nisms can be modified to consider interactions between
genes and any other environments such as life style. Our
interaction models should provide a more powerful tool
to draw a detailed and precise picture of the genetic archi-
tecture of any complex traits that are important to human

Authors' contributions
CW, YC, TL and QL wrote the programs and performed
data analyses. RF, MW, and RL designed the experiment.
LK performed the experiment. RW conceived the idea and
wrote the paper. All authors read and approved the final

The preparation of this manuscript is partially supported by a joint NSF/
NIH grant (0540745) to RW and an NIH grant (RO I NS041670-05AI) to

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