A statistical model for dissecting genomic imprinting through genetic mapping

Abstract As a result of nonequivalent genetic con-

tribution of maternal and paternal genomes to off-

springs, genomic imprinting or called parent-of-origin

effect, has been broadly identified in plants, animals

and humans. Its role in shaping organism’s develop-

ment has been unanimously recognized. However,

statistical methods for identifying imprinted quantita-

tive trait loci (iQTL) and estimating the imprinted ef-

fect have not been well developed. In this article, we

propose an efficient statistical procedure for genome-

wide estimating and testing the effects of significant

iQTL underlying the quantitative variation of inter-

ested traits. The developed model can be applied to

two different genetic cross designs, backcross and F2

families derived from inbred lines. The proposed pro-

cedure is built within the maximum likelihood frame-

work and implemented with the EM algorithm.

Extensive simulation studies show that the proposed

model is well performed in a variety of situations. To

demonstrate the usefulness of the proposed approach,

we apply the model to a published data in an F2 family

derived from LG/S and SM/S mouse stains. Two par-

tially maternal imprinting iQTL are identified which

regulate the growth of body weight. Our approach

provides a testable framework for identifying and

estimating iQTL involved in the genetic control of

complex traits.

Keywords EM algorithm Æ Genomic Imprinting ÆInbred Lines Æ Maximum likelihood Æ Quantitative trait

loci

Introduction

Gene imprinting refers to the phenomenon that the

expression of certain gene is determined by the non-

equivalent genetic contribution of maternal and

paternal genomes to offsprings (Pfeifer 2000). As a

result of genomic imprinting, monoallelic expression of

a gene is expected depending on the parental origin of

the alleles. Genomic imprinting is thought of as a

particular sub-type of dominant modification (Sapienza

1990). Therefore, a dosage difference would be

expected when one or more of these modifiers are sex-

linked. For example, the reciprocal heterozygotes may

have different average phenotypes if the underlying

gene is imprinted.

Representing a totally different genetic inheritance

pattern compared to the traditional Mendelian’s

inheritance, imprinting-like phenomena have been

ubiquitously observed in a wide range of phyla span-

ning from plants, animals to humans. Since the first

imprinted gene (Igf2) was identified in mouse

(DeChiara et al. 1991), there are over 80 imprinted

genes being identified in mammals (Morison et al.

2005). Imprinted genes were also identified in other

organisms such as plants (Alleman and Doctor 2000)

Y. Cui (&)Department of Statistics and Probability, Michigan StateUniversity, A-411 Wells Hall, East Lansing, MI 48824, USAe-mail: [email protected]

J. M. CheverudDepartment of Anatomy and Neurobiology, WashingtonUniversity Medical School, St. Louis, MO 63110, USA

R. WuDepartment of Statistics, University of Florida, Gainesville,FL 32611, USA

Genetica (2007) 130:227–239

DOI 10.1007/s10709-006-9101-x

123

ORIGINAL PAPER

A statistical model for dissecting genomic imprintingthrough genetic mapping

Yuehua Cui Æ James M. Cheverud Æ Rongling Wu

Received: 3 March 2006 / Accepted: 24 July 2006 / Published online: 6 September 2006� Springer Science+Business Media B.V. 2006

and animals (Jeon et al. 1999; Nezer et al. 1999; Van

Laere et al. 2003; Tuiskula-Haavisto et al. 2004). Luedi

et al. (2005) predicted that there are 600 potentially

imprinted genes in mouse genome and 64% of which

were predicted to exhibit maternal expression. Mean-

while, studies showed that many human diseases are

related to imprinted genes such as type I diabetes

(Paterson et al 1999), polycystic ovarian sybdrome

(Bennett et al. 1997), Prader-Willlis syndrome and

Angelman syndrome (Fall et al. 1999) and bipolar

disorder (McInnis et al. 2003).

Genomic imprinting can be considered as a highly

complex process involving a number of growth axes

operating coordinately at different development stage

(Bartolomei and Tilghman 1997). Incorporating such

information in genetic mapping should provide more

information for detecting genomic imprinting. While

genetic mapping approaches for hunting Mendelian

inheritance genes have been very successful, the search

for imprinted genes underlying complex traits has been

difficult and not been well established (de Koning et al.

2002). More recently, studies showed that genomic

imprinting displayed a diploid-type expression pattern

instead of being considered as a simple haploid

expression, and hence could be considered as a partial

effect, i.e., alleles derived from maternal and paternal

genome had different levels of expression (Naumova

and Croteau 2004; Sandovici et al. 2003, 2005).

Therefore, one can treat imprinted phenotype as

quantitative trait and quantitative genetic models can

be applied.

In this article, we develop an efficient statistical

model aimed to dissect genomic imprinting through

genome-wide scan for significant iQTL. The developed

model is based on experimental crosses of inbred lines

and can be implemented through two different exper-

imental cross designs, backcross or F2 family, as the

testing population. There are few studies focused on

mapping iQTL using inbred lines. Clapcott et al.

(2000) first proposed to use reciprocal backcross to

map iQTL underlying susceptibility to trypanosomiasis

in mice. However, the statistical procedures and

properties for their approach is not clearly demon-

strated. Cui et al. (2006) proposed an additive model

for dissecting genomic imprinting in an F2 design. The

model partitions the total additive effect as two sepa-

rate additive effects contributed from maternal and

paternal alleles, and considers the sex-specific differ-

ence in recombination fraction between male and fe-

male chromosomes. By integrating the backcross and

F2 designs into the same mapping framework, we in-

tend to propose a unified approach and to further

compare the performance of different methods.

To test the existence of imprinting effect and further

quantify the degree of imprinting effect, two imprinting

index parameters are considered. They are statistically

testable and their estimates provide a quantitative

measure of imprinting effect. A genome-wide scan for

iQTL is performed based on the interval mapping

theory and parameters are estimated under the maxi-

mum likelihood framework implemented with the EM

algorithm (Dempster et al. 1977). Extensive computer

simulation studies are performed to investigate the

accuracy and precision of parameter estimation under

various study designs such as different sample sizes,

different heritability levels, and different gene action

modes. Finally, the model is applied to genome-wide

scan for iQTL that affect body weight in an F2 family

derived from two inbred mouse strains. Comparisons

of model performance between the current approach

and Cui et al. (2006) are discussed.

Model

Consider an autosomal locus having two segregating

alleles A and a. To distinguish the parental origin of

these two types of allele for an individual, two different

notations are used. We let Am and am be two alleles

which are derived from the maternal parent, and Af and

af be two alleles which are derived from the paternal

parent. Different allelic combinations of these four

types of alleles end up with four possible imprinted

genotypes AmAf, Amaf, amAf, amaf. Based on traditional

Mendelian inheritance, the genetic values of individuals

who carrying genotype Amaf or amAf are not distin-

guishable since these two contrasting genotypes only

show one observable heterozygotic genotype.

However, genetic imprinting theory suggests that

alleles inherited from one parent are not completely

expressed. Assuming that the two reciprocal hetero-

zygotes, Amaf and amAf, are distinguishable, the genetic

values of the four genotypes can be expressed as

l1 ¼ lþ a; for AmAf

l2 ¼ lþ ac1; for Amaf

l3 ¼ lþ ac2; for amAf

l4 ¼ l� a; for amaf

8>><

>>:

ð1Þ

where l is the overall mean; a is the additive genetic

effect; c1 and c2 are two imprinting index parameters,

with c1 = 1 and c2 = – 1 corresponding to complete

paternal imprinting and c1 = – 1 and c2 = 1 corre-

sponding to complete maternal imprinting. When

c1 = c2 = c, this model reduces to general Mendelian’s

genetic model with no imprinting effect. With the

228 Genetica (2007) 130:227–239

123

above parameter settings, this model allows us to

test both iQTL and Mendelian’s QTL. When the

inequality c1 „ c2 is satisfied, c1 > c2 implies that

the gene is partially maternal imprinting and c1 < c2

implies partially paternal imprinting. Therefore, the

imprinting index parameters c1 and c2 measure the

magnitude of the imprinting effect. The proposed

model considers the effect of allelic parental origin,

and hence provide a complete dissection of the allelic

parent-of-origin effect. We define this model as mul-

tiplicative model to distinguish with our previously

proposed model (Cui et al. 2006) since the genetic

effects for heterozygotes are modelled by a multipli-

cative imprinting effect.

To have a complete dissection of the imprinting

effect as shown in model (1), we need a cross design

that can provide enough information to distinguish the

genetic effect of two reciprocal heterozygotes, Amaf

and amAf. Here we propose two different genetic

experiment designs.

Genetic design I

This design is defined as backcross design. Consider

an F1 heterozygote, Aa, derived from two homolo-

gous inbred lines, P1 (AA) and P2 (aa). This F1 is

crossed with one of the two parents to generate the

backcross progeny. To detect the imprinting effect,

two backcross families should be considered. The first

one considers F1 (Amam) as mother to backcross

with one of the parents P1 (AfAf) and is denoted as

F1 · P1. With this mating design, two possible

genotypes (AmAf and amAf) in the backcross progeny

are expected. The second one is to use F1 (Amam) as

mother to backcross with another parent P2 (afaf) and

is denoted as F1 · P2. Similarly, with this mating

design two different genotypes (Amaf and amaf) in the

backcross progenies can be obtained. The two back-

cross families contain four possible genotypes illus-

trated in model (1), and more importantly, the two

reciprocal heterozygotes, Amaf (from F1 · P1 family)

and amAf (from F1 · P2 family), can be well iden-

tified. Thus, this design provides a testable population

allowing us to assess gene imprinting effect.

One can also use F1 as paternal parents to get the

backcross population. Table 1 tabulates the detailed

designs as well as the genetic value for corresponding

progeny genotype. Both backcross designs end up with

four distinguishable genotypes through which the four

genetic parameters (l,a,c1,c2) can be estimated. Two

sets of phenotypes (y1 and y2) and two sets of marker

data (M1 and M2) can be obtained corresponding to

two different backcross types.

Suppose there is a putative quantitative trait locus

(QTL) that segregates with two alleles A and a in this

pedigree. Based on this genetic design, our goal is to

map iQTL using codominant molecular markers, and

further quantify the imprinted effects. To illustrate the

idea, the detailed parameter estimation and hypothesis

testings procedures are demonstrated based on design

B1. The same procedure can be applied to design B2

with little modification.

Genetic design II

This design is defined as F2 design (denoted as F2 in

Table 1). Consider an F1 family derived from two

homologous inbred lines, P1 (AA) and P2 (aa). By

crossing the F1 individuals, we get the F2 family. Sim-

ilarly, we assume there is a putative quantitative trait

locus (QTL) that segregates with two alleles A and a in

this pedigree. Three distinguishable QTL genotypes,

AA, Aa and aa, are expected based on the traditional

Mendelian inheritance theory, irrespective of the

parental origin of alleles. To distinguish the two

reciprocal heterozygotes, Amaf and amAf, the sex

difference in recombination fraction between two

homologous chromosomes is considered, through

which the genetic mean of the two reciprocal hetero-

zygotes can be separated. With this design, the phe-

notype data (y) and marker data (M) can be generated.

Statistical methods

Finite mixture model

Statistical finite mixture model has been proven to be a

powerful tool for mapping QTL (Lander and Botstein

Table 1 Genotype and genotypic values under genomicimprinting in backcross and F2 designs

DesignType

CrossType

Mother Father Offspring Offspringgenetic value

B1 F1 · P1 Amam AfAf AmAf l1 = l + aamAf l2

¢ = l + ac2

F1 · P2 Amam afaf Amaf l3¢ = l + ac1

amaf l4 = l – aB2 P1 · F1 AmAm Afaf AmAf l1 = l + a

Amaf l2 = l + ac1

P2 · F1 amam Afaf amAf l3 = l + ac2

amaf l4 = l – aF2 F1 · F1 Amam Afaf AmAf l1 = l + a

Amaf l2 = l + ac1

amAf l3 = l + ac2

amaf l4 = l – a

Genetica (2007) 130:227–239 229

123

1989). In the mixture model, each observation y is as-

sumed to have arisen from one of J (J possibly un-

known but finite) components, each component being

modelled by a density from the parametric family f. So

the distribution of y can be modelled as a linear com-

bination of density functions which are specific to dif-

ferent QTL genotypes

y � f ðyjM;u; gÞ ¼ p1f1ðyjM;u1; gÞ þ � � �þ pJfJðyjM;uJ; gÞ

ð2Þ

where y refers to the response (phenotype); M

refers to the marker information; p = (p1,...,pJ)¢ refers

to the mixture proportions (i.e., QTL genotype

frequencies) which are constrained to be nonnegative

andP

j=1J pj = 1; / = (/1,...,/J)¢ are the component

(or QTL genotype) specific parameters, with /j

being specific to component j; and g is a parameter

(i.e., residual variance) which is common to all

components. Corresponding to our genetic

designs, the mixture model for individual i can be

expressed as

f ðy1ijM1;u; gÞ ¼ p1jif1ðy1ijM1;u1; gÞþp2jif2ðy1ijM1;u2; gÞ

f ðy2i0 jM2;u; gÞ ¼ p3ji0f3ðy2i0 jM2;u3; gÞþp4ji0f4ðy2i0 jM2;u4; gÞ

8>><

>>:

ð3Þ

for genetic design B1 and

f ðyijM;u; gÞ ¼ p01jif1ðyijM;u1; gÞ þ � � �þ p04jif4ðyijM;u4; gÞ

ð4Þ

for genetic design F2.

Using Haldane mapping function (Haldane 1922),

the mixture proportion (pj|i, pjji0 and pj|i¢ ) can be

expressed as a function of the QTL position through

the recombination fractions between the QTL and

the two flanking markers that bracket it. Therefore,

with known marker genotypes, these frequencies in

the mixture model (3) and (4) should be replaced by

the conditional probabilities of the QTL genotype

given on the two flanking marker genotypes. General

genetic literature has provided the conditional prob-

abilities (pj|i and pjji0) of QTL genotypes for back-

cross family as function of the recombination

fractions among the QTL and the two flanking

markers (Cui et al. 2004). The conditional probability

(pj|i¢ ) in mixture model (4) considering sex difference

in recombination fraction has also been derived (see

Cui et al. 2006).

Likelihood function

Assume there are n1 and n2 backcross progenies

genotyped with co-dominant molecular markers from

two backcross families (F1 · P1 and F1 · P2).

These markers are used to construct a genetic link-

age map. Phenotype data are also observed for a

quantitative trait. Let n = n1 + n2 denote the total

number of observations. In such an imprinted family,

the same allele A or a of a QTL is expressed dif-

ferently, depending on their parental origin. There-

fore, we assumes four types of genotypes as

illustrated in Table 1 after considering the allelic

parental origin.

With the assumption of independence, and based on

the mixture model (3), the likelihood function of the

marker data (M = [M1; M2]) and trait phenotypes

(y = [y1;y 2]) controlled by the putative QTL that is

located between two flanking markers in the imprinted

backcross family can be formulated as

LðHjM; y1; y2Þ ¼Yn1

i¼1

p1jif1ðy1iÞ þ p2jif2ðy1iÞ� �

Yn2

i0¼1

p3ji0f3ðy2i0 Þ þ p4ji0f4ðy2i0 Þ� �

ð5Þ

where the unknown vector H contains the QTL posi-

tion, QTL effects and residual variance, and the den-

sity function fj (j = 1,...,4) is assumed to be normal with

mean lj and common variance r2. The location of the

putative QTL is modelled through the conditional

probability pj|i and pjji0 .Similarly, the likelihood function for the F2 family

with n individuals can be formulated based on the

mixture model (4) as

LðHjM;yÞ¼Yn

i¼1

p01jif1ðyiÞþp02jif2ðyiÞþp03jif3ðyiÞþp04jif4ðyiÞh i

ð6Þ

Computational algorithm

Setting up the likelihood function is the first step

toward a complete dissection of iQTL effect. There are

a number of ways to estimate the unknown parameters

H ¼ fHp;Hqg such as EM, Newton Raphson and

Fisher Scoring, where Hp contains the QTL location

parameter and Hq contains all the genetic parameters.

Among the algorithms being used, EM algorithm

(Dempster et al. 1977) has been broadly applied in

230 Genetica (2007) 130:227–239

123

QTL mapping study. In this study, parameters are

estimated by a standard maximum likelihood approach

implemented with the EM algorithm. This includes

differentiating the log-likelihood function (Eqs. (5) and

(6)) with respect to each unknown parameter, setting

the derivatives equal zero and solving the derived log-

likelihood equations. The detailed derivations of the

parameter estimation are given in Appendix A.

In practice, we usually do not estimate the location

of QTL. Instead, the QTL position parameter can be

viewed as a fixed parameter by searching for a putative

QTL at every 1 or 2 cM on a map interval bracketed by

two markers throughout the entire linkage map. The

log-likelihood ratio test statistic for a QTL at a map

position is displayed graphically to generate a log-

likelihood ratio profile plot called LR profile plot. The

genomic position that corresponds to a peak of the

profile is the maximum likelihood estimate (MLE) of

the QTL location. The searching approach for design I

is based on a regular linkage map. For F2 design, we

consider the sex difference of genetic distance between

male and female chromosome and reestimate the

recombination fraction between marker and a putative

QTL for male (rf) and female (rm) chromosome

respectively. These newly estimated recombination

fractions are used to get the conditional probability

pj|i¢ in mixture model (4).

After the point estimates of parameters are obtained

by the EM algorithm, the asymptotic sampling errors

of the estimates contained in bH can be obtained. The

techniques for doing so involve calculation of the

incomplete-data information matrix which is the neg-

ative second-order derivative of the incomplete-data

log-likelihood. The incomplete-data information can

be calculated by extracting the information for the

missing data from the information for the complete

data (Louis 1982).

Hypothesis tests and threshold determination

As a commonly used test approach for testing the

existence of a QTL, likelihood ratio (LR) test is

applied. Here we propose two different hypothesis

tests which are detailed as follows.

Overall test

The existence of a QTL on the effect of a quantitative

trait is called the overall test. The LR test statistic is

computed between the full (there is a QTL) and re-

duced model (there is no QTL). These two models

correspond to two alternative hypotheses, expressed as

H0 : a ¼ c1 ¼ c2 ¼ 0;H1 : at least one parameter is not zero.

�

Letting bH and bH be the MLEs of the unknown

parameters under H0 and H1, respectively, the log-

likelihood ratio can be calculated as

LR ¼ �2½log Lð bHjM; yÞ � log Lð bHjM; yÞ�; ð7Þ

The LR is asymptotically v2-distributed with the

degrees of freedom that are equal to the number of

unknown parameters estimated. For a mixture model

like ours here, this may be violated due to some reg-

ularity problem, for example, the reduced model is not

completely nested within the full model. The critical

threshold value for declaring the existence of the

testing QTL is empirically calculated on the basis of

permutation tests (Churchill and Doerge 1994). By

repeatedly shuffling the relationships between marker

genotypes and phenotypes many times (1000 times

say), a series of the log-likelihood ratios are calculated,

from the distribution of which the critical threshold is

determined.

Imprinting test

To have a full dissection of the imprinting effect of the

detected QTL, we propose the following three

imprinting hypothesis tests (iTest) to examine the

imprinting property of the detected QTL.

iTest 1: Test the existence of the imprinted

effect—Due to the imprinting effect, two parent-of-

origin dependent heterozygote, Amaf and amAf, may

have different genetic values. If the condition c1 = c2 is

satisfied, then the model is reduced to the traditional

Mendelian’s genetic model and no imprinting effect

exists. The corresponding hypothesis can be formu-

lated as

H0 : c1 ¼ c2;H1 : c1 6¼ c2:

�

where rejecting H0 provides evidence of genomic

imprinting.

iTest 2: Test if a QTL detected is completely

maternal imprinting – Complete maternal imprinting

corresponds to complete inactivation of alleles derived

from maternal parent. Under the current genetic

model, we are able to test if an iQTL detected expe-

riences monoallelic expression. This is equivalent to

test if the two equations c1 = – 1 and c2 = 1 are true.

Correspondingly, the hypothesis can be formulated as

Genetica (2007) 130:227–239 231

123

H0 : c1 ¼ �1; c2 ¼ 1H1 : at least one of the equalities does not hold:

�

iTest 3: Test if a QTL detected is completely

paternal imprinting – Complete paternal imprinting

corresponds to complete inactivation of alleles derived

from paternal parent. Correspondingly, the hypothesis

can be formulated as

H0 : c1 ¼ 1; c2 ¼ �1H1 : at least one of the equalities does not hold:

�

The sequence of the above proposed test is given as

follows. We first conduct the overall test. If a significant

QTL is detected, then iTest 1 should be performed. We

stop to do further test if iTest 1 is not significant.

Otherwise, we continue iTests 2 and 3. If both iTests 2

and 3 are not significant, the estimates of c1 and c2

provides a measure of the direction and degree of the

imprinting effect where c1 > c2 implies partially

paternal imprinting and c1 < c2 implies partially

maternal imprinting. Since the threshold values ob-

tained using permutation test are limited specifically to

the data set on which the permutations are performed,

the threshold might be biased when sample size is

small. Therefore, the critical thresholds for iTests 1 – 3

are determined by using parametric bootstrap (detailed

in Appendix B).

Results

Simulation study

We perform simulation studies to investigate the the-

oretical properties of our model. We assume the two

backcross families in design B1 share the same linkage

map. A genome composed of three small linkage

groups each with 6 evenly spaced markers (M1 �M6)

is simulated. The total length for each group is 100 cM.

The size of backcross family for F1 · P1 and F1 · P2

is denoted as n1 and n2 respectively. Without loss of

generality, we assume n1 = n2 and n = n1 + n2 is the

total sample size. For a given marker, there are two

genotypes, one heterozygous and the other homozy-

gous. Markers are simulated according to the Mende-

lian inheritance laws. The distribution of markers

simulated depends on the cross type. Three different

putative QTL that display different imprinting features

are simulated on different linkage groups, with the

locations and effect values given in Tables 2–4. The

position of each QTL is assumed to be located at

48 cM away from the first marker (M1) at each linkage

group. Similarly, marker information and the putative

QTL information can be simulated for design F2.

Based on the simulated QTL genotype, the corre-

sponding phenotype data can be simulated. For

example, if the simulated QTL genotype is Amaf for

individual i, then the phenotype yi is simulated from a

normal distribution with mean l + ac1 and variance r2.

We performed a series of simulation study with

various sample size (n = 200 vs n = 400), and different

imprinting mechanism (no imprinting, completely

maternal imprinting and partially maternal imprint-

ing). To examine the impacts of parameter spaces on

parameter estimation, we simulated the data under

different heritability levels (H2 = 0.1, 0.25, 0.4). These

simulation designs are aimed to have a better under-

standing of the model performance under different

situations. We would like to know which factor affects

the estimation and further provide some empirical

evidences for scientists to design their experiment.

Table 2 The MLEs of the QTL position and effect parameters with no imprinting based on 200 simulation replicates

Design H2 n Position (48 cM) l = 5 a = 0.5 c1 = 0.4 c2 = 0.4 r2

B1 0.1 200 50.60 (16.16) 5.02 (0.12) 0.50 (0.12) 0.42 (0.58) 0.37 (0.60) 1.30 (0.19)400 49.64 (11.42) 5.00 (0.09) 0.50 (0.08) 0.43 (0.37) 0.42 (0.32) 1.29 (0.16)

0.25 200 49.23 (9.92) 5.02 (0.07) 0.48 (0.07) 0.41 (0.27) 0.35 (0.25) 0.49 (0.12)400 48.59 (6.72) 5.01 (0.05) 0.49 (0.05) 0.42 (0.20) 0.37 (0.17) 0.48 (0.11)

0.4 200 48.56 (8.13) 5.02 (0.05) 0.48 (0.05) 0.42 (0.19) 0.33 (0.19) 0.28 (0.09)400 48.45 (5.83) 5.01 (0.04) 0.48 (0.04) 0.43 (0.14) 0.35 (0.13) 0.27 (0.08)

F2 0.1 200 47.67 (9.69) 5.01 (0.19) 0.50 (0.13) 0.23 (1.12) 0.55 (1.27) 1.12 (0.19)400 47.46 (5.72) 5.01 (0.14) 0.50 (0.10) 0.30 (0.85) 0.45 (0.84) 1.14 (0.13)

0.25 200 48.13 (4.72) 5.00 (0.10) 0.50 (0.07) 0.38 (0.50) 0.43 (0.48) 0.38 (0.05)400 48.01 (2.73) 5.00 (0.07) 0.50 (0.05) 0.33 (0.39) 0.46 (0.37) 0.38 (0.04)

0.4 200 48.07 (2.80) 5.00 (0.07) 0.51 (0.05) 0.33 (0.35) 0.45 (0.35) 0.19 (0.03)400 48.08 (1.77) 4.99 (0.05) 0.51 (0.04) 0.32 (0.27) 0.44 (0.26) 0.19 (0.02)

The square roots of the mean squared errors of the MLEs are given in parentheses

The locations of the the QTL is described by the map distances (in cM) from the first marker of the linkage group (100 cM long). Thehypothesized r2 value is 1.22 for H2 = 0.1, 0.42 for H2 = 0.25 and 0.20 for H2 = 0.4

232 Genetica (2007) 130:227–239

123

In general, our iQTL interval mapping model can

provide reasonable estimates of QTL locations, QTL

effects and residual variances for various imprinting

mechanism, no imprinting (Table 2), completely

maternal imprinting (Table 3), and partially maternal

imprinting (Table 4). Several interesting findings are

summaries as follows. First of all, the QTL location can

be reasonably estimated with the precision depending

on the sample size and heritability level and also the

imprinting mechanism. As we expected, better esti-

mation can be obtained with large sample size and

higher heritability level. The square root of mean

square root (SMSE) decreases as sample size increases

from 200 to 400, and similar pattern can be observed

for increased heritability level. Comparing the two

designs, design B1 provides more precise estimation

than design F2 when the QTL is completely maternal

imprinted. However, poor estimation is observed for

design B1 when the QTL is partially imprinted.

Second, most genetic parameters can be quite reli-

ably estimated using our model. As expected, the

additive genetic effect (a) can be more reliably esti-

mated than the imprinting effects (c1 and c2). Increased

H2 and sample sizes can always improve the precision

of the parameter estimation (Tables 2–4). But relative

to the effect of increasing sample sizes, the increase of

H2 from 0.1 to 0.4 can lead to more significant

improvement for the estimation precision than the

increase of sample size n from 200 to 400. For example,

the SMSE of the MLE of the maternal imprinting

parameter c1 for design B1 in Table 3 would decrease

by more than 2-fold when increasing H2 from 0.1 to 0.4

than increasing n from 200 to 400. This suggests that in

practice well-managed experiments, through which

Table 3 The MLEs of the QTL position and effect parameters with completely maternal imprinting based on 200 simulation replicates

Design H2 n Position (48 cM) l = 5 a = 0.5 c1 = – 1 c2 = 1 r2

B1 0.1 200 49.61 (10.81) 5.03 (0.16) 0.49 (0.17) – 1.22 (1.03) 1.12 (1.08) 2.44 (0.38)400 49.52 (6.76) 5.01 (0.13) 0.50 (0.11) – 1.03 (0.58) 1.06 (0.54) 2.42 (0.33)

0.25 200 49.21 (6.28) 5.00 (0.10) 0.48 (0.10) – 1.03 (0.42) 1.03 (0.41) 0.95 (0.25)400 48.96 (4.12) 5.01 (0.07) 0.48 (0.07) – 1.03 (0.29) 1.00 (0.29) 0.91 (0.22)

0.4 200 49.19 (5.48) 5.00 (0.07) 0.46 (0.07) – 1.05 (0.31) 1.04 (0.28) 0.49 (0.16)400 49.10 (3.92) 5.00 (0.04) 0.47 (0.05) – 1.00 (0.20) 1.01 (0.21) 0.47 (0.13)

F2 0.1 200 46.54 (11.39) 4.98 (0.26) 0.51 (0.19) – 0.76 (2.43) 1.25 (4.01) 2.10 (0.34)400 47.06 (9.82) 4.99 (0.17) 0.51 (0.19) – 0.86 (1.13) 0.87 (1.08) 2.16 (0.23)

0.25 200 47.48 (7.79) 5.00 (0.14) 0.51 (0.10) – 0.96 (0.76) 0.95 (0.82) 0.72 (0.11)400 47.89 (3.65) 4.97 (0.10) 0.52 (0.07) – 0.92 (0.50) 0.95 (0.49) 0.74 (0.08)

0.4 200 47.78 (5.23) 4.99 (0.10) 0.51 (0.07) – 1.04 (0.13) 0.98 (0.40) 0.36 (0.06)400 48.25 (2.60) 5.00 (0.07) 0.51 (0.05) – 1.00 (0.27) 1.00 (0.29) 0.37 (0.04)

The square roots of the mean squared errors of the MLEs are given in parentheses

The locations of the the QTL is described by the map distances (in cM) from the first marker of the linkage group (100 cM long). Thehypothesized r2 value is 2.25 for H2 = 0.1, 0.75 for H2 = 0.25 and 0.38 for H2 = 0.4

Table 4 The MLEs of the QTL position and effect parameters with partially maternal imprinting based on 200 simulation replicates

Design H2 n Position (48 cM) l = 5 a = 0.5 c1 = 0.5 c2 = – 0.4 r2

B1 0.1 200 51.75 (21.81) 5.03 (0.20) 0.49 (0.15) 0.63 (1.05) – 0.63 (1.20) 1.36 (0.18)400 50.69 (13.73) 4.99 (0.14) 0.51 (0.09) 0.55 (0.48) – 0.38 (0.47) 1.37 (0.12)

0.25 200 50.79 (14.01) 5.02 (0.11) 0.50 (0.08) 0.51 (0.38) – 0.46 (0.37) 0.47 (0.07)400 49.74 (9.26) 5.00 (0.08) 0.50 (0.05) 0.51 (0.22) – 0.40 (0.22) 0.47 (0.05)

0.4 200 49.75 (9.91) 5.01 (0.07) 0.50 (0.05) 0.50 (0.21) – 0.43 (0.18) 0.24 (0.04)400 49.24 (6.91) 5.00 (0.05) 0.50 (0.03) 0.51 (0.15) – 0.41 (0.14) 0.25 (0.03)

F2 0.1 200 46.89 (11.48) 4.98 (0.13) 0.51 (0.13) 0.41 (1.04) – 0.45 (1.34) 1.25 (0.21)400 47.5 (7.25) 4.99 (0.13) 0.51 (0.09) 0.41 (0.82) – 0.32 (0.81) 1.30 (0.14)

0.25 200 47.11 (7.01) 5.00 (0.12) 0.50 (0.08) 0.38 (0.61) – 0.30 (0.55) 0.43 (0.07)400 47.61 (3.29) 5.00 (0.08) 0.50 (0.06) 0.40 (0.49) – 0.32 (0.44) 0.44 (0.04)

0.4 200 47.42 (2.83) 5.00 (0.08) 0.50 (0.05) 0.41 (0.44) – 0.33 (0.41) 0.21 (0.03)400 47.75 (2.15) 5.00 (0.05) 0.50 (0.04) 0.42 (0.39) – 0.35 (0.33) 0.22 (0.02)

The square roots of the mean squared errors of the MLEs are given in parentheses

The locations of the the QTL is described by the map distances (in cM) from the first marker of the linkage group (100 cM long). Thehypothesized r2 value is 1.35 for H2 = 0.1, 0.45 for H2 = 0.25 and 0.23 for H2 = 0.4

Genetica (2007) 130:227–239 233

123

residual errors are reduced and therefore H2 is

increased, are more important than simply increasing

sample sizes. The similar findings are obtained in

design F2. Overall, design B1 outperforms design F2

in terms of the estimation precision for most genetic

parameters (indicated by SMSE).

A case study

Currently, we do not have real data for the backcross

design to test our model. We applied our model to a

published data set (Vaughn et al. 1999) based on de-

sign F2 aimed to map iQTL that contribute to varia-

tion in body growth in an animal model system –

mouse. The F2 progeny was measured for their body

mass at 10 weekly intervals starting at age 7 days. A

regular genetic linkage map was constructed. The

sex-specific genetic distance is reconstructed based on

the marker data and the fact of 1.25:1 genetic distance

ratio between female and male chromosome (Dietrich

et al. 1996).

The similar LR profile plot is obtained by using the

model proposed in this study as the one shown in Cui

et al. (2006). To save space, we only give the likelihood

profile plot for data observed at Week 4 (Fig. 1). In

Fig. 1, the solid curve corresponds to the LR profile

and the dashed line corresponds to the 5% significant

threshold out of 1000 permutations. Clearly, there are

one QTL detected on chromosome 6 (Q6) and one

detected at chromosome 15 (Q15). Table 5 tabulates

the maximum likelihood estimates (MLEs) of the QTL

position and effect parameters, their asymptotic sam-

pling error and the P-values for iTest 1–3.

Even though the current model and our previous

model (Cui et al. 2006) agree on the number and

location of QTL detected, the imprinting property for

the identified QTL is different. It can be seen from

the testing result that Q6 shows evidence of imprinting

0

10

20

30

40D1 D2 D3 D4 D5

10

20

30

40D6

LR

D7 D8 D9 D10 D11

0

10

20

30

40D12 D13 D14 D15 D16 D17 D18 D19

10 cM

Test Position

Fig. 1 The profiles of the log-likelihood ratios (LR) between thefull (there is a QTL) and reduced model (there is no QTL) forbody mass growth observed at Week 4 across the 19 mousechromosomes using the linkage map constructed from microsat-ellite markers (Vaughn et al. 1999). The genomic positionscorresponding to the peak of the curve are the MLEs of the QTL

localizations (indicated by the arrows). The genome-widethreshold values for claiming the existence of QTL is given asthe horizonal dashed lines. Tick marks on the x-axis representthe positions of markers on the linkage group, whose names aregiven in Vaughn et al. (1999)

234 Genetica (2007) 130:227–239

123

(P-value = 0.025 in iTest 1), and this QTL is a partially

maternal iQTL (P-value = 0.03 in iTest 2 and 0.039 in

iTest 3, c1\c2). However, different from previous

findings, Q6 shows a time-related (age-dependent)

effect which shows partial imprinting only during week

4 and 5. After week 6, the imprinting effect switches off

and this QTL functions as a regular Mendelian QTL

until the end of the observation period, week 10. An-

other inconsistency is that QTL in chromosome 15

(Q15) does not show evidence of imprinting (P-va-

lue = 0.72 in iTest 1).

Another iQTL detected is located on chromosome

10, and it shows partial maternal imprinting effect

starting from week 7. One Mendelian QTL detected is

on chromosome 7. These two QTL are consistent with

our previous findings (Cui et al. 2006). Totally, we

identified four major QTL and two of them are Men-

delian QTL and two of them are iQTL.

Discussion

The discovery of genomic imprinting leads to a totally

different area of epigenetic study, epigenetics, which is

different from the traditional Mendelian landscape. As

a normal mechanism of genetic regulation, genomic

imprinting has provided remarkable insights into pre-

viously puzzling human diseases such as Prader-Willi,

Angelman, and Beckwith-Wiedemann syndromes

(Falls et al. 1999). The identification of imprinted

genes in plants, animals and humans also greatly

changed the way scientists think of traditional Men-

delian genetics. Advanced biotechnology such as

positional cloning and candidate gene testing has sig-

nificantly facilitated the identification of imprinted

genes with monoallelic expression (Pfeifer 2000).

While considering that these techniques are still

expensive to implement in lab, genetic mapping pro-

vide an alternative way for identifying potentially im-

printed genes responsible for quantitative variation

(de Koning et al. 2000, 2002; Hanson et al. 2001;

Knapp et al. 2004; Shete et al. 2003; Strauch et al.

2000; Tuiskula-Haavisto et al. 2004). These methods

are focused on family based study or outbred species.

However, family based data is not available for plants

and using outbred lines might introduce confounding

genetic effects for imprinting due to high heterozy-

gosity at a given QTL (Lin et al. 2003).

To overcome these problems for mapping iQTL,

inbred lines serve as a good alternative since one can

easily select the lines with interests of specific pheno-

type. Here we propose a statistical mapping approach

to detect genome-wide significant iQTL in experi-

mental crosses of inbred species. A quantitative genetic

model is presented through which two different genetic

designs can be applied to map iQTL. This model

considers two imprinting parameters which measure

the degree of imprinting, and hence provide a complete

dissection of the imprinting effect. Several hypothesis

tests can be conducted focusing on these two parame-

ters to provide a complete dissection of imprinting

property of any detected significant QTL. As such, the

developed model is both statistically and biologically

interpretable. The partial imprinting effect shown by

the iQTL detected in this study is consistent with cur-

rent findings (Sandovici et al. 2003, 2005; Naumova

and Croteau 2004; Spencer 2002).

Simulation study demonstrates the usefulness of the

developed model based on backcross and F2 design in

which robust parameter estimation can be obtained

with modest heritability level and sample size. Low

heritability (H2 = 0.1) and small sample size (n = 200)

result in poor parameter estimation. This information

is useful in practice as we need to be cautious about the

interpretation of imprinting effect detected with real

data when the proportion of variance explained by the

QTL is low. Although backcross design outperforms F2

design in most cases in which backcross design pro-

vides more precise genetic parameter estimation than

the F2 design, the trade-off is poor variance and QTL

Table 5 The MLEs of the QTL position and effect parameters in the F2 progeny derived from the LG/J and SM/J strains estimated bythe imprinting model at week 4

Chrom QTL position l a c1 c2 r2 iTest 1 iTest 2 iTest 3 H2

15 (Q15) D15Mit5 – D15Mit2 MLE 18.2051 0.8785 0.6111 – 0.0842 4.7351 8.61%(D15Mit5 + 6.2 cM) A.S.E.* (0.2198) (0.1594) (0.9443) (1.0000) (0.4123)

P-value 0.726 (Q6) D6Nds5 – D6Mit58 MLE 18.6372 0.7892 – 2.1856 0.8213 3.9928 21.41%

(D6Nds5 + 14 cM) A.S.E.* (0.2300) (0.1562) (0.7124) (0.4809) (0.4248)P-value 0.025 0.03 0.039

The asymptotic standard errors are given in the parenthesis

*A.S.E. refers to the asymptotic standard error computed using Louis method (Louis 1982)

Genetica (2007) 130:227–239 235

123

location estimation in backcross design. However, the

bias could be greatly reduced if the two design popu-

lations ( B1 and B2) are combined together. Statisti-

cally, design B2 can be viewed as a replicate of design

B1. With increased total sample size, we would expect

increased mapping power and reduced bias for

parameter estimation. A number of studies has shown

the advantage of combing different related crosses in

QTL mapping study (Xu 1998; Zou et al. 2001).

Therefore, backcross design is more preferable to the

F2 design in real experiment.

As shown in the paper, the genetic difference be-

tween male and female chromosomes provides the

theoretical basis for distinguishing the effects of two

heterozygotes Amaf and amAf. However, the difference

is just a genomewide averaged difference and real

distance may vary by intervals. Moreover, many plants

and animals do not show evidence of differences for

two homologous chromosomes. These greatly restrict

the usefulness of design F2. For these concerns, we

would recommend researchers to use the backcross

instead of the F2 design when they design their

experiments. For existing F2 data, it is still worth to try

our F2 model to explore some potential imprinting

information.

In our previous paper (Cui et al. 2006), the genetic

model is derived by partition the additive effect into

two parts and hence it can be defined as an additive

model. The genetic model proposed in this study only

put restrictions on the two reciprocal heterozygotes

and can be defined as a multiplicative model. Even

though the results obtained by both models are broadly

consistent in which both models agree on the same

number of QTL detected spanning across the entire

genome, disagreement arises for the imprinting prop-

erty of detected QTL Q6 and Q15 as indicated by the

case study. These discrepancies may be due to the

difference in the threshold criteria used to claim the

existence of an iQTL between the two approaches or

could be due to the nature of the genes. For example,

Q6 shows some evidences of time-related (or age-

dependent) effect. One of the possible explanations for

this phenomenon is to use the loss of imprinting (LOI)

theory where the imprinting effect could be removed

by epigenetic modification (Feinberg 2001). Evidence

of LOI has been reported on chromosome 6 in mouse

genome for a maternally imprinted gene DLX5 (Hor-

ike et al. 2005). This gene is located on a well identified

imprinted gene cluster (Ono et al. 2003). Since im-

printed genes tend to cluster together on the genome

(Pferfer 2000), it is possible that Q6 is also located on

this region and experiences LOI. Therefore, before

seeking for further lab verification of the imprinting

property of any detected iQTL, it is safe to try both

models.

Our presentation of this paper is focused on the

genetic model and its implementations on two genetic

designs. The model can be applied to plant or animal

systems. Despite its usefulness of identifying and

estimating the effect of genomic imprinting, the

model developed in this study can be considered as

simple. Several studies have shown that maternal ef-

fect plays a very important role in many species

(Wade 1998; Wolf 2000, 2002; Agrawal et al. 2001).

Its effect in shaping the development of the offspring

is beyond the direct inheritance of alleles and can be

thought of as an indirect contribution of maternal to

its offspring. Both genomic imprinting and maternal

effect may have similar contribution to the phenotype

of offsprings from the same mother. Having a clear

dissection of these two effect could result in a better

understanding of the genetic architecture of any bio-

logical traits caused by epigenetic modification. An-

other limitation of the current model is that gene

epistasis is not considered where complicated inter-

actions could occur among the same or different

generations. Evidence of interaction between im-

printed genes in mouse has been reported (Cattanach

et al. 2004). Our follow-up papers are concerned with

developing models to dissect the contribution of

maternal effect and genomic imprinting involving

complicated genetic interactions.

Acknowledgements We thank the anonymous referees and theeditor for their valuable comments on the manuscript. Thisresearch was supported by a start-up fund from Michigan StateUniversity.

Appendix A: EM algorithm for estimating the genetic

parameters

The following algorithm is applied to Design I. Similar

algorithm can be obtained for Design II with little

modification and hence is omitted here.

The log-likelihood function of Eq. (5) can be written

as

log LðHjM; y1; y2Þ ¼Xn1

i¼1

log p1jif1ðy1iÞ þ p2jif2ðy1iÞ� �

þXn2

i0¼1

log p3ji0f3ðy2i0 Þ þ p4ji0f4ðy2i0 Þ� �

with a derivative for a particular element Q‘,

236 Genetica (2007) 130:227–239

123

@

@H‘log LðHÞ

¼Xn1

i¼1

X2

j¼1

pjji@@H‘

fjðy1iÞP2

j¼1 pjjifjðy1iÞ

þXn2

i0¼1

X4

j0¼3

pj0 ji0@@H‘

fj0 ðy2i0 ÞP4

j0¼3 pj0 ji0fj0 ðy2i0 Þ

¼Xn1

i¼1

X2

j¼1

pjjifjðy1iÞP2

j¼1 pjjifjðy1iÞ@

@H‘log fjðy1iÞ

þXn2

i0¼1

X4

j0¼3

pj0 ji0 fj0 ðy2i0 ÞP4

j0¼3 pj0 ji0fj0 ðy2i0 Þ@

@H‘log fj0 ðy2i0 Þ

¼Xn1

i¼1

X2

j¼1

Pjji@

@H‘log fjðy1iÞ

þXn2

i0¼1

X4

j0¼3

Pj0ji0@

@H‘log fj0 ðy2i0 Þ

ðA1Þ

where we define

Pjji ¼pjjifjðy1iÞ

P2j¼1 pjjifjðy1iÞ

for j ¼ 1; 2 ðA2Þ

and

Pj0ji0 ¼pj0 ji0fj0 ðy2i0 Þ

P4j0¼3 pj0ji0 fj0 ðy2i0 Þ

for j0 ¼ 3; 4 ðA3Þ

which can be thought of as posterior probabilities of

QTL genotypes given on the marker genotypes derived

from the two backcross families. Given the initial val-

ues for the unknown parameters H, we can update

Pj ¼ Pjji and Pj0 ¼ Pj0 ji0 (E-step). The estimated pos-

terior probabilities are used to obtain the new MLEs of

H (M step) based on the log-likelihood equations

l ¼ y1 � a½10P1 þ 10P2c1� þ y2 � a½10P3c2 þ 10P4�n

ðA4Þ

a ¼ ½P01 þP02c1�ðy1 � lÞ þ ½P03ð1þ c2Þ � 1�ðy2 � lÞ

10ðP1 þP2c1 þP3 þP3c2ÞðA5Þ

c1 ¼P02ðy1 � lÞ

a10P2ðA6Þ

c2 ¼P03ðy2 � lÞ

a10P3ðA7Þ

which are derived by letting the derivative in Eq. (A1)

equal to zero. This iterative process is repeated be-

tween Eqs. (A2) and (A8) until the specified conver-

gence criteria is satisfied. The values at convergence

are regarded as the MLEs.

Appendix B: Assessing statistical significance

by parametric bootstrap

To assess the statistical significance of the iTest 1–3, we

use parametric bootstrap method. The detailed

approach is described as follows:

Step 1: Obtain the MLEs of parameters under the H0,

denoted as H0.

Step 2: Simulate full phenotype information based on

H0. Note at this step, we only simulate phenotype data

and keep the original marker information.

Step 3: With the simulated phenotype and the original

marker data, we calculate the LR statistic at the test

position (specific chromosome location) and save the

LR value denoted as LR*.

Step 4: Repeat Step 2–3 for B times and obtain the

bootstrap LR test statistics LR*1,...,LR*

B.

Step 5: Calculate the bootstrap empirical P-value as

P-value ¼ 1

B

XB

i¼1

I½LR�i[LRobs�

where I is the indicator function with value 1 if LR*i >

LRobs and 0 otherwise, and LRobs is the observed LR

test statistic calculated based on real data.

This empirical P-value is then compared with a

significant level a. A general recommendation is to

generate B = 1000 bootstrap samples through which

the empirical P-value is calculated.

r2 ¼ P01ðy1 � l� aÞ þP02ðy1 � l� ac1Þ þP03ðy2 � l� ac2Þ þP04ðy2 � lþ aÞn

ðA8Þ

Genetica (2007) 130:227–239 237

123

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