Functional Mapping of Quantitative Trait Loci Underlying the Character Process: A Theoretical Framework



Functional Mapping of Quantitative Trait Loci Underlying the Character Process:

A Theoretical Framework

Chang-Xing Ma,*

,†

_{George Casella* and Rongling Wu*}

,1

*Department of Statistics, University of Florida, Gainesville, Florida 32611 and†_{Department of Statistics, Nankai University,} Tianjin 300071, China

Manuscript received September 10, 2001 Accepted for publication May 6, 2002

ABSTRACT

Unlike a character measured at a finite set of landmark points, function-valued traits are those that change as a function of some independent and continuous variable. These traits, also called infinite-dimensional characters, can be described as the character process and include a number of biologically, economically, or biomedically important features, such as growth trajectories, allometric scalings, and norms of reaction. Here we present a new statistical infrastructure for mapping quantitative trait loci (QTL) underlying the character process. This strategy, termedfunctional mapping, integrates mathematical relationships of different traits or variables within the genetic mapping framework. Logistic mapping proposed in this article can be viewed as an example of functional mapping. Logistic mapping is based on a universal biological law that for each and every living organism growth over time follows an exponen-tial growth curve (e.g., logistic or S-shaped). A maximum-likelihood approach based on a logistic-mixture model, implemented with the EM algorithm, is developed to provide the estimates of QTL positions, QTL effects, and other model parameters responsible for growth trajectories. Logistic mapping displays a tremendous potential to increase the power of QTL detection, the precision of parameter estimation, and the resolution of QTL localization due to the small number of parameters to be estimated, the pleiotropic effect of a QTL on growth, and/or residual correlations of growth at different ages. More importantly, logistic mapping allows for testing numerous biologically important hypotheses concerning the genetic basis of quantitative variation, thus gaining an insight into the critical role of development in shaping plant and animal evolution and domestication. The power of logistic mapping is demonstrated by an example of a forest tree, in which one QTL affecting stem growth processes is detected on a linkage group using our method, whereas it cannot be detected using current methods. The advantages of functional mapping are also discussed.

T

HE theoretical principle for analyzing quantitative simultaneously to map multiple QTL of epistatic interac-trait loci (QTL) dates back toSax(1923), who first tions throughout a linkage map. Currently, an upsurge associated pattern and pigment markers with seed size of QTL mapping methodologies has been developed to in beans. However, statistical methodologies for map- consider various situations regarding different marker ping QTL on a high-density linkage map of molecular types (dominant or codominant), different marker markers had not been well established until Lander spaces (sparse or dense), different experimental designs

and Botstein’s (1989) pioneering work. These au- (F2/backcross or full-sib family), or different mapping

thors employed an expectation-maximization (EM)- populations (autogamous or allogamous). The statisti-implemented maximum-likelihood approach, pro- cal methods used for QTL mapping in the literature posed by Dempster et al. (1977), to map QTL on a _{include regression analyses (Haleyand Knott 1992;} particular chromosomal interval bracketed by two _{Xu1995), maximum likelihood (LanderandBotstein} flanking markers. This so-called interval mapping _{1989; Zeng 1994; Kao et al. 1999), and the Bayesian} method was later improved by including markers from _{approach (Satagopanet al.1996;SillanpaaandArjas} other intervals as covariates to control the overall ge- _{1999;XuandYi2000). Many of these mapping methods} netic background (JansenandStam1994;Zeng1994). _{have been instrumental in the identification of QTL} The improved method, called composite interval map- _{responsible for variation in various complex traits} im-ping byZeng(1994), displays increased power in QTL _{portant to agriculture, forestry, biomedicine, or} biologi-detection because of reduced residual variance. Kao _{cal research (Tanksley1993;Wuet al.2000;Mackay}

et al.(1999) proposed using multiple marker intervals _{2001;Mauricio2001).}

It should be noted, however, that many quantitative traits, such as body size and body shape, are inherently 1_{Corresponding author:Department of Statistics, 533 McCarty Hall C,}

too complex to be described by a single value, because

University of Florida, Gainesville, FL 32611.

E-mail: [email protected] their phenotypes change with age, metabolic rate, or

vironmental stimulus. These traits, which can be ex- mapping. A maximum-likelihood-based method, imple-mented with the EM algorithm, is used to estimate QTL pressed as a function (or stochastic process) of some

independent and continuous variable, were thought of locations and effects on various biological processes. The newly developed method is applied in an example

as infinite-dimensional characters by Kirkpatrick and

Heckman (1989) orfunction-valued traitsbyPletcher to map the growth of a forest tree. Compared with

current mapping methods, our method incorporating

and Geyer(1999). The genetic determination of the

character process has long intrigued students in differ- growth trajectories tends to be more powerful and more precise in QTL detection and also has greater potential ent disciplines of biology, genetics, and breeding (e.g.,

Cheverud et al. 1983; Atchley1984; Wuet al. 1992; to increase mapping power, precision, and resolution by

reducing residual variance and the number of unknown

AtchleyandZhu1997;Rice1997). A simple approach

for mapping infinite-dimensional characters is to associ- parameters to be estimated. In practice, our method is economically more feasible than previous methods ate markers with phenotypes separately for different

ages, traits, or environments and compare the differ- because it needs a smaller sample size to obtain ade-quate power for QTL detection as a result of the use of ences of QTL expression across ages, traits, or

environ-ments (Cheverudet al.1996;Nuzhdinet al.1997;Ver- multiple measurements for each individual. It can be anticipated that the method proposed in this article will

haegenet al.1997;Emebiriet al.1998;Wuet al.1999).

However, these separate analyses cannot provide effec- have great implications for the design of an efficient early selection program and the interface of genetics, tive estimates of genetic control over

infinite-dimen-sional characters, because they fail to capture the infor- development, and evolution. mation about the covariances of different traits or the

same trait measured at different ages or environments.

MODELING THE CHARACTER PROCESS Although multitrait mapping approaches can take into

account simultaneously different traits or the same trait Many biological processes in real life are expected to arise as curves, such as growth curves, allometric scal-measured at different ages or environments (Jiangand

Zeng1995;Korolet al.1995;Roninet al.1995;Eaves ings, hormone profiles, and norms of reaction. A growth curve ortrajectoryrepresents an individual as a function

et al.1996;KnottandHaley2000), their applications

are actually limited to bivariate, or at most trivariate, that relates the age of an individual to some measure of its size. Since there are an infinite number of ages, analyses. As the number of traits increases these

multitrait analysis approaches will have a reduced ability growth trajectories can be thought of as function-valued traits (PletcherandGeyer1999;JaffrezicandPletcher

to produce precise estimates of genetic parameters in

quantitative genetic studies (Shaw1987). 2000) or infinite-dimensional characters (Kirkpatrick

and Heckman 1989). Other examples of

function-val-To circumvent the difficulty in manipulating a large

number of correlated traits, new attempts were made ued traits include the continuous change of a morpho-logical or physiomorpho-logical variable with body size (allometric

byManginet al.(1998) and Korolet al.(2001), who

transformed the initial trait space into a space of a lower scaling,Niklas1994;Westet al.1997, 1999) and respon-sive phenotypes of a given genotype to a changing envi-dimension on the basis of principal component analysis

or interval-specific calculation of eigenvalues and eigen- ronment (reaction norm,Viaet al.1995). The common property of these function-valued traits is that they can vectors of the residual covariance matrix. These new

attempts have, to some extent, made the genetic map- be described as a function (or stochastic process) of some independent and continuous variable consisting ping of a large number of traits more tractable, but

they still treat infinite-dimensional characters as discrete of an infinite number of points, such as age, tempera-ture, light intensity, or biological size.

traits or eigenvalues and do not place the physiological

mechanisms predisposing for the phenotypic variation The model for QTL mapping we developed relies on concepts from functional analysis and stochastic pro-of infinite-dimensional characters in a mapping

frame-work. In real life, an infinite-dimensional character of- cesses. Throughout, we use growth trajectories as a con-crete example to illustrate the ideas, but allometric scal-ten changes its phenotype through particular

physiolog-ical regulations or developmental signals in the same ings, hormone profiles, and reaction norms can be treated in the same framework with appropriate modi-way that an organism tends to maximize its metabolic

capacity and internal efficiency as a consequence of natu- fications.

Growth trajectory: It is well known that there are ral selection. Therefore, the incorporation of the

under-pinning physiological or developmental mechanisms of biological laws underlying growth trajectories (Gould

1977;Alberchet al.1979). A growth law can be visual-trait variation into a QTL-mapping strategy may likely

produce more accurate results in terms of biological ized as the “force field” propelling a point through a phenotype space, tracing out the ontogenetic path. If reality.

The objective of this study is to propose a general the size of an organism is denoted by y, itsontogenetic

trajectory y(t) can be generated through the differential

theoretical framework for embedding biological

functions have been established to describe the growth gether with the initial growth and asymptotic growth, determines exclusively the difference of two growth trajectory. Basically, they are sorted into three

catego-ries: (1) exponential, (2) saturating, and (3) sigmoidal curves. Any two curves will not be distinguishable if they have the same values for these three variables.

(von Bertalanffy1957;Niklas1994). Each of these

growth models has a common feature that the develop-ment of ontogenetic trajectory is regulated by a set of

STATISTICAL MODELS “control parameters” such as onset age of growth, offset

signal for growth, growth rate during the period of _{Genetic design:The purpose of this article is to} intro-growth, and initial size at the commencement of the _{duce a novel idea to QTL mapping. Hence, we suppose} growth period. Also, each of these growth models exhib- _{a simplest backcross design derived from two contrasted} its an initial phase of exponential growth due simply _{homozygous inbred lines. Other more complex designs,} to the geometrically multiplying population of newly

such as an F2 or full-sib family, can also be used. In a differentiated cells. This initial growth phase has the

backcross population, there are two groups of genotypes property that small perturbations in growth rate or onset

at a locus, in which a marker-based genetic linkage map age are amplified enormously during ontogeny. Thus,

is constructed, aimed at the identification of QTL affect-it is easy to find examples of how a small “mutation” in

ing an age-dependent trait, such as body size or body a growth parameter causes a series of developmental

weight. In practice, the data are observed only at a finite alterations that produce a phenotype qualitatively

differ-set of times, 1, . . . ,m, rather than a continuum, so we ent from the normal one.

have only a finite set of data on each individuali, which In this article, we further limit our analysis to

sigmoi-can be considered as a multivariate trait vector,yi(1),

dal, or logistic, function (Pearl 1925). The logistic

. . . ,yi(m). This finite set of data can be modeled by a

curve is regarded as among the most important ones to

growth curve. Assume that a pleiotropic QTL of allele capture the age-specific change in growth (Niklas1994;

Q1andQ2affecting growth curves or trajectories is

segre-Westet al.2001). The logistic growth curve as a

biologi-gating in the backcross population. This QTL is brack-cal law can be mathematibrack-cally described by

eted by two flanking markers␩ and␩ ⫹ 1, each with two genotypesM␩m␩,m␩m␩, andM␩⫹1m␩⫹1,m␩⫹1m␩⫹1,

g(t)⫽ a

1⫹ be⫺rt, (1) _{respectively. For a particular genotypej(j}⫽_{1 forQ}

1Q2 or 2 forQ2Q2) of this QTL, the parameters describing whereais the asymptotic or limit value of gwhent → _{its logistic curve are denoted bya}

j,bj, andrj. The

compari-∞,a/(1⫹b) is the initial value ofgwhent⫽ 0, andr _{sons of these parameters between two different}

geno-is the relative rate of growth (von Bertalanffy1957). _{types can determine whether and how this putative QTL} The logistic growth curve consists of two phases, an _{affects growth trajectories.}

exponential and an asymptotic. The overall form of _{Suppose that there are a total of}

N progeny in the the curve is determined by different combinations of _{backcross measured at each of}

mtimes. The trait pheno-parametersa,b, andr.If different genotypes at a putative _{type of progeny}

i measured at timet due to the QTL QTL have different combinations of these parameters,

located on an interval flanked by markers␩and␩ ⫹1 this implies that this QTL plays a role in governing the

can be expressed by a linear statistical model (

Kirkpat-difference of growth trajectories.

rickandHeckman1989;LanderandBotstein1989;

The logistic growth curve described in Equation 2 can

PletcherandGeyer1999),

be used to determine the coordinates of a biologically

important point in the entire growth trajectory—the in- yi(t)⫽ ␰i1g1(t)⫹(1⫺ ␰i1)g2(t)⫹ei(t) , (3)

flection point—where the exponential phase ends and

where␰ij is an indicator variable for the possible

geno-the asymptotic phase begins (Niklas1994). The time

types of the QTL for progenyi and defined as 1 if a at the inflection point corresponds to the time point at

particular QTL genotype is indicated and 0 otherwise; which a maximum growth rate occurs. The time (tI)

gj(t) is the genotypic value of the QTL for the trait at

and growth [g(tI)] at the inflection point for a QTL

timet; andei(t) is the residual effect of progenyi,

includ-genotype can be derived as

ing the aggregate effect of polygenes and error effect, and distributed as N(0,␴2

e(t)). The probability with

tI⫽

logb

r _{which␰ijtakes 1 or 0 depends on the two-locus genotype}

of the flanking markers␩and␩ ⫹1 and the position

g(tI)⫽

a

2. (2) of the QTL on the marker interval. The probability of a QTL genotype (Q1Q2orQ2Q2) conditional upon the four genotypes of the flanking markers (M␩m␩M␩⫹1m␩⫹1, The difference in the coordinates between different

M␩m␩m␩⫹1m␩⫹1, m␩m␩M␩⫹1m␩⫹1, and m␩m␩m␩⫹1m␩⫹1) genotypes provides important information about the

for progenyiin the backcross population was expressed genetics and evolution of growth trajectories (Niklas

pij⫽ Prob(␰ij ⫽j|␩,␩ ⫹1,␪), j⫽1 or 2 ,

where␪ is the ratio of the recombination fractions be- ⳵

⳵⍀␭

logL(⍀)⫽

兺

N

i⫽1

兺

2

j⫽1

pij

⳵ ⳵⍀␭fj(yi)

R2

j⫽1pijfj(yi)

tween marker ␩ and the QTL to the recombination fraction between the two markers.

⫽

_兺

N

i⫽1

兺

2

j⫽1

pijfj(yi)

R2

j⫽1pijfj(yi)

⳵

⳵⍀␭logfj(yi)

Statistical methods:The phenotypes of the trait at all time points 1, . . . , m for each QTL genotype group

follow a multivariate normal density, _⫽

_兺

N

i⫽1

兺

2

j⫽1

Pij

⳵

⳵⍀␭logfj(yi),

fj(y)⫽

1

(2␲)m/2|_R|1/2exp[⫺(y⫺gj)

T_R⫺1_(y⫺_g

j)/2] ,

where we define

wheregjis the vector of the expected genotypic values _P

ij⫽

pijfj(yi)

Rk

j⫽1pijfj(yi)

, (8)

of the trait for QTL genotype j measured for t times

andRis the residual variance-covariance matrix of the _{which could be thought of as a posterior probability} phenotypes measured at different ages. Indeed, gjcan _{that progeny}

ihave QTL genotypej.We then implement be modeled by the logistic curve of Equation 2 as _{the EM algorithm with the expanded parameter set {⍀,}

P}, whereP⫽{Pij,j⫽1, . . . ,k;i⫽1, . . . ,N}. Conditional

gj⫽[gj(t)]1⫻m⫽

冤

aj

1⫹bje⫺rjt

冥

1⫻m

, (4) _{onP, we solve for}

⳵

⳵⍀␭log L(⍀)⫽0 (9)

andRcan be assumed identical among different geno-types and modeled using AR(1) repeated

measure-ment errors (Davidian and Giltinan 1995;Verbeke to get our estimates of ⍀ (the M step; Equation 9).

andMolenberghs2000) as The estimates are then used to update P (the E step;

Equation 8), and the process is repeated until conver-gence. The values at convergence are the MLEs of⍀. The iterative expressions of estimating⍀from the

previ-R⫽ ␴2 e     

1 ␳ . . . ␳m⫺1

␳ 1 . . . ␳m⫺2 . . . .

␳m⫺1 ␳m⫺2 _{. . . 1}

     . (5)

ous step are given inappendix a.The standard errors of the MLEs are estimated using the inverse of the Fisher information matrix.

For simplicity, the matrixRof Equation 5 assumes

vari-In practical computations, the QTL position parame-ance stationarity,i.e., there is the same residual variance

ter ␪ can be viewed as a fixed parameter because a (␴2_{) for the trait at each time, and covariance}

stationar-putative QTL can be searched at every 1 or 2 cM on a ity;i.e., the covariance between different measurements

map interval bracketed by two markers throughout the decreases proportionally (in␳) with increased time

in-entire linkage map. The amount of support for a QTL at terval (see alsoPletcherandGeyer1999). These two

a particular map position is often displayed graphically assumptions, although providing a reasonable

approxi-through the use of likelihood maps or profiles, which mation in some situations, can be readily relaxed. In

plot the likelihood-ratio test statistic as a function of

thediscussion, we propose a few different approaches

map position of the putative QTL. to the relaxation of these two assumptions.

The likelihood of the backcross progeny withm

-dimen-sional measurements can be represented by a multivari- HYPOTHESIS TESTS ate mixture model

After the MLEs of the parameters of interest are ob-tained, a number of biologically meaningful hypotheses

L(⍀)⫽

兿

N

i⫽1

冤

兺

2

j⫽1

pijfj(yi)

冥

, (6)

can be tested on the basis of the logistic-based genetic model. First, the hypothesis about the existence of a where the vector ⍀⫽ (aj, bj,rj,␪, ␳,␴2)T contains

un-QTL affecting an overall growth curve can be formu-known parameters to be estimated for the QTL effect,

lated as QTL position, and residual (co)variances. The

maxi-mum-likelihood estimates (MLEs) of the unknown pa- H0: a1⫽a2,b1⫽b2,r1⫽r2 rameters for a pleiotropic QTL can be computed by

H1: at least one of the equalities above does not hold, implementing the EM algorithm (Dempsteret al.1977;

(10)

LanderandBotstein1989;Zeng1994). The

log-likeli-hood is given by where H0corresponds to the reduced model, in which

the data can be fit by a single logistic curve, and H1 log L(⍀)⫽

兺

N

i⫽1 log

冤

兺

2

j⫽1

pijfj(yi)

冥

, (7) corresponds to the full model, in which there exist two

different logistic curves to fit the data.

time at which the detected QTL starts to exert or ceases the way in which QTL trigger an effect on growth and development.

an effect on growth trajectories, by comparing the

differ-ence of the expected means between different geno- The test statistics for testing the hypotheses (10–15) are calculated as the log-likelihood ratio (LR) of the types at various time points. At a given time t*, the

hypothesis is full over reduced model:

H0: g1(t*)⫽ g2(t*)

LR⫽ ⫺2 log

冤

L(⍀˜)

L(⍀ˆ)

冥

, H1: g1(t*)⬆ g2(t*) . (11)

where ⍀˜ and ⍀ˆ denote the ML estimates of the un-If H0is rejected, this means that the QTL has a

signifi-known parameters under H0 and H1, respectively. But cant effect on variation in growth at timet*. Testing the

the determination of the distribution of the LR is a hypotheses (11) is equivalent to testing the difference of

difficult statistical issue. For a two-normal mixture the model with no restriction and the model with the

model, like ours in this study,Loiselet al.(1994) proved restriction:

that the limiting distribution of LR under H0 for (10) is a 50:50 mixture of␹2

1and␹22if␴is unknown. The test

a1 1⫹b1e⫺r1t*

⫽ a2

1⫹b2e⫺r2t* .

statistics for the other hypotheses (12–15) can be viewed as being asymptotically␹2_{distributed with 1 d.f.} Becauset* is given, one of the six logistic parameters

can be expressed as a function of the other five and,

thus, there is one fewer parameter to be estimated for _EXAMPLE the model with the above restriction (the reduced

The Populus map:We use an example of a forest tree model) than the model with no restriction (the full

to demonstrate the power of our statistical model for model). By scanning time points from 1 tom, one can

mapping QTL affecting growth trajectories. The study find the time point at which the QTL starts or ceases

material used was derived from the triple hybridization to exert an effect on growth.

of Populus (poplar). A Populus deltoides clone (desig-Third, the genotypic differences in time (tI) and

nated I-69) was used as a female parent to mate with growth [g(tI)] at the inflection point of maximum

an interspecificP. deltoides⫻P. nigraclone (designated growth rate (Equation 2) can be tested. The test for the

I-45) as a male parent (Wu et al. 1992). The hybrids genotypic difference is based on the restriction

betweenP. deltoidesandP. nigraare called Euramerica poplar (P. euramericana). Both P. deltoides I-69 and P.

logb1

r1

⫽log b2

r2

(12)

euramericanaI-45 were selected at the Research Institute

for Poplars in Italy in the 1950s and were introduced

fortI, and to China in 1972. In the spring of 1988, a total of 450

1-year-old rooted three-way hybrid seedlings were

a1 2 ⫽

a2

2 (13) planted at a spacing of 4_{Xuchou City, Jiangsu Province, China. The total stem}⫻ 5 m at a forest farm near

heights and diameters measured at the end of each of forg(tI).

11 growing seasons are used in this example. Fourth, when there is no double “crossover” between

A genetic linkage map has been constructed using 90 the growth curves of the two QTL genotypes, the effect

genotypes randomly selected from the 450 hybrids with of QTL⫻age interaction on the overall growth curve

random amplified polymorphic DNAs (RAPDs), ampli-can be tested by comparing the genotypic differences

fied fragment length polymorphisms (AFLPs), and in-at time t ⫽ 0 and t ⫽ ∞, which is expressed by the

tersimple sequence repeats (ISSRs;Yinet al.2002). This restriction

map comprises the 19 largest linkage groups for each parental map, which represent roughly 19 pairs of

chro-a1 1⫹b1

⫺ a2

1⫹b2

⫽a1⫺a2. (14)

mosomes. We chose linkage group 10 from theP.

del-toides parent map to detect QTL affecting diameter

Similarly, the effect of QTL⫻ age interaction on the _{growth using our newly developed method.}

growth at any two different time pointst1andt2can be _{Logistic curves:By plotting total growth against year,} tested with the restriction _{it is observed that each of the 90 mapped genotypes}

follows the S-shaped (logistic) growth curve. Figure 1

a1 1⫹b1e⫺r1t1

⫺ a2 1⫹b2e⫺r2t1

⫽ a1 1⫹b1e⫺r1t2

⫺ a2

1⫹b2e⫺r2t2

. _{illustrates S-shaped growth curves for individual stem}

diameters over 11 years. A least-squares approach was (15)

used to fit diameter growth with the logistic curve (Equa-tion 1) for each genotype. On the basis of statistical Testing QTL⫻age interactions on the basis of

Figure1.—Plots of stem diameter growthvs.ages for each of the 90 genotypes used to construct linkage maps in poplar hybrids (Yinet al.2002). The growth of these genotypes can be well fit by a particular logistic curve. Thex- andy-axes of the plots denote age (in years) and stem diameter (in centimeters).

(r2⬎_{0.95). Also, different curve shapes of these geno- the empirical critical threshold at the significance level}

P⫽ 0.01. types imply possible genetic control over growth

trajec-tories. The statistical model built upon the logistic To compare the power of our method with previous methods, the same material is subjected to interval map-growth curve model is used to map QTL responsible

for growth trajectories in diameters. ping (LanderandBotstein1989) and composite inter-val mapping (Zeng1994) on the basis of the most

differ-QTL detection: Using our logistic mapping model,

one QTL is detected on linkage group 10 for the growth entiated phenotypes measured at year 11 (Figure 1). Neither of these two mapping methods can declare the trajectory of stem diameter in the interspecific hybrids

of poplar (Figure 2). The critical value for claiming the existence of a significant QTL given their lower LR values. Figure 2 illustrates the result from interval map-existence of QTL can be determined on the basis of the

Bonferroni argument for the sparse-map case (Lander ping for diameters. No LR value from interval mapping is larger than the threshold (7.68) obtained from

per-andBotstein1989) or by permutation tests proposed

by Doerge and Churchill (1996). In this example, mutation tests at the significance levelP ⫽0.05.

Similar conclusions about the difference of QTL de-the chromosome-wide empirical estimate of de-the critical

value is obtained from 1000 permutation tests. It is tection between our method and current methods are obtained for many other linkage groups (results not found that the critical values for declaring the existence

of a QTL on the linkage group under consideration are shown). These suggest that our method incorporating logistic growth curves has greater power to detect a 34.69 and 45.56 at the significance levelsP⫽0.05 and

0.01, respectively. The profile of the log-likelihood ratios significant QTL than the current methods.

The dynamic pattern of QTL expression:Our method of the fullvs.reduced model across the length of linkage

Figure 2.—The profile of the log-likelihood ratios be-tween the full and reduced (no QTL) model for diameter growth trajectories across link-age group 10 in thePopulus del-toidesparent map. The genomic positions corresponding to the peak of the curve are the MLEs of the QTL localization. The solid and broken curves/lines indicate the results from our method and interval mapping, respectively. The threshold val-ues for both methods are given as horizontal lines. The vertical dotted lines indicate the posi-tions of markers on the linkage group (Yin et al.2002), whose names are given beneath.

curves of diameter are drawn using the estimates of the genetic basis of quantitative traits can be better unraveled (Mackay2001) and, ultimately, genetic im-logistic parameters for two genotypes at the QTL

de-tected on linkage group 10 (Figure 3). On the basis of provement for these complex traits can be made more efficient (Tanksley1993;Wuet al.2000). Analyses and the hypothesis test (11), this QTL is detected to be

inactive until trees grew toⵑ6 years in the field. And interpretations of genomic data, however, are strongly dependent upon the study material, data structure, ge-its effect on diameter growth increased with age. At 11

years old, genotype Q1Q2 exhibited diameter growth netic model, and statistical method used. As a result, considerable attention has been paid to the develop-4.5 cm more than its alternativeQ2Q2. This difference

appears to increase after age 11 years, as predicted from ment of powerful experimental designs and analytical methodologies that can increase the power, precision, the logistic curves estimated (Figure 3). Apparently, this

QTL interacts significantly with age to affect stem diame- and resolution of QTL mapping. Currently, there have been many strategies proposed to increase QTL map-ter growth.

If two growth curves predicted by a QTL have differ- ping. These include: (1) selecting two highly differenti-ated inbred lines to make a segregating generation, such ent ages and/or growth at the inflection point, this

indicates that the inflection point is under genetic deter- as the F2or backcross; (2) increasing the sample size of a mapping population by genotyping more progeny; (3) mination. It is found that the QTL detected on linkage

group 10 exerts strong control over the inflection point saturating the map density using informative markers, especially in genomic regions carrying QTL; (4) using (Figure 3). The genetic control of the inflection point

suggests that the growth trajectory can be genetically composite interval mapping and multiple interval map-ping (Jansen and Stam 1994; Zeng 1994; Kao et al.

modified to increase a tree’s capacity to effectively

ac-quire spatial resources. 1999); and (5) developing more powerful computa-tional technologies, such as the Bayesian approach im-plemented with the Markov chain Monte Carlo (MCMC) DISCUSSION

algorithm (Satagopanet al.1996;XuandYi2000). In this article, we propose that a simultaneous analysis of Beyond the traditional models and tools used for

quantitative genetic studies, current genome technolo- repeated measurements for a quantitative trait based on biological mechanisms can be used as an alternative gies permit us to dissect quantitative traits into

approaches, our approach estimates a highly reduced number of model parameters, which can make an ini-tially high-dimensional mapping model more tractable and the estimates of QTL parameters more precise.

Composite interval mapping can improve mapping precision to some extent when multiple QTL are located on the same linkage group, but their use frequently depends upon many other factors,e.g., marker spacing, the choice of markers as cofactors, and genotyped sam-ple size (Broman2001). Multiple interval mapping, pro-posed by Zeng and co-workers, can simultaneously model multiple marker intervals so that multiple QTL and their epistatic interactions can be estimated (Kao

et al.1999). Yet, a serious difficulty may be encountered

when multiple interval mapping is extended to simulta-neously map multiple quantitative traits or repeated measurements at different ages, because a high number

Figure3.—Two growth curves each presenting two groups _{of QTL effects should be modeled in these cases. Our}

of genotypes at the QTL detected on linkage group 10 in the _{logistic-mixture model, when built upon composite}

in-Populus deltoidesparent map. The times at the inflection point

terval mapping or multiple interval mapping, can make (tI1andtI2) are indicated for the two QTL genotypesQ1Q2and

these two approaches more tractable by reducing the

Q2Q2, respectively. The differentiation pattern of growth curves

beyond the maximum observed age (11), affected by the QTL, number of model parameters to be estimated. In fact, is represented by extended broken curves. _{evidence for more than one QTL observed on some} linkage groups from our approach in the poplar exam-ple (results not shown) prompts us to build the logistic It is well demonstrated that increased sample sizes and _{mixture model upon composite interval mapping or} marker densities can almost always improve precision in _{multiple interval mapping and provide better resolution} QTL mapping, but they could be economically expen- _{of multiple linked QTL for growth processes.}

sive in practice. Our mapping approach for repeated _{We have used the method of maximum likelihood to} measurements based on growth curves can extract maxi- _{estimate the unknown parameters with their MLEs. The} mum information about QTL effects and positions con- _{MLEs are attractive in terms of their properties of} invari-tained in an arbitrary segregating family and, thus, con- _{ance, consistency, and asymptotic efficiency. Our} ap-fers an advantage for QTL detection in the situation _{proach, built upon the traditional maximum-likelihood} where a limited size of genotyped samples or a limited _{method, is readily accessible to the general genetics} level of marker density is used. In an example with a _{community. Using prior information on parameters,} small sample size (N ⫽90) using forest tree data, our _{however, we can incorporate the logistic-mixture model} logistic mixture model offers improved power to detect _{in the Bayesian paradigm (Satagopanet al. 1996;Xu} a number of QTL underlying stem growth, in contrast _{andYi2000). By specifying the prior density of} parame-to traditional approaches based on a single trait, which _{ters, MCMC can be used to evaluate the posterior density} do not detect any QTL. Such differences are not surpris- _{and provide posterior distributions of QTL effects and} ing because a single-trait analysis approach typically can- _{positions (RobertandCasella1999).}

not detect the QTL of small effect (Beaviset al.1994). _{Although the results of our approach are quantified} The increased detection power of our approach re- _{by differences in the parameters controlling the overall} sults from the simultaneous use of multiple measure- _{shapes of different logistic curves, they can also be} inter-ments that are correlated due to either the effect of _{preted as regular genetic parameters,}

i.e., the additive pleiotropic QTL or residual covariances or both. This,

or dominant effect of a QTL on growth at an arbitrary in principle, is similar to the result from multitrait

map-time point and the percentage of the total phenotypic ping, as shown inJiangandZeng(1995),Roninet al.

variance explained by this QTL. According to classical (1995),Eaveset al.(1996),Manginet al.(1998),Knott

quantitative genetics theory, the expected genetic values

andHaley(2000), andKorolet al.(1995, 2001).

How-for QTL genotypesQ1Q2andQ2Q2at timetcan be ex-ever, beyond these multitrait mapping approaches, our

pressed, respectively, as growth-based approach treats phenotypic values as a

function of age, thus having the ability to analyze a

g1(t)⫽ ␮(t)⫹

1 2␣(t) quantitative trait measured at an unlimited number of

time points by modeling the full, continuous growth

trajectory. Moreover, instead of estimating a large _g

where ␣(t) is the additive genetic effect of the QTL of the ubiquitous phenomena in biology, holding for every cell, organ, tissue, organism, or population, in a detected on growth at timet, which can be solved from

the above equations. The additive genetic variance of range from microbes (10⫺13_{g) to blue whales (10}8_g), no matter what species it is derived from (Westet al.

growth at timet contributed by this QTL is expressed

as 2001). The pattern of the logistic growth curve can be

different among species, populations, and genotypes

(Hofet al.1999;Robertet al.1999). However, it is also

␴2

g(t)⫽ 1 4␣

2_(t).

worthwhile to incorporate other biologically meaning-ful models (reviewed inNiklas1994) into our analysis, Thus, the percentage of the total phenotypic variance

as long as they fit well a dataset for particular species, accounted for by this QTL is

environments, or developmental stages.

To incorporate a general biological process, we

R(t)⫽ ␴

2 g(t)

␴2

g(t)⫹ ␴2e(t) .

should first have a descriptive mathematical function that is expressed as

These parameters described above can also be used to investigate the contribution of a QTL to growth at a

point. However, it is important in practice to know how _y⫽

   

f(x) for allometric laws

g(t) for growth models

h(z) for reaction norms

, (16)

much a QTL contributes to the differentiation of overall growth curves or the differentiation of growth at a time

whereyis the biological trait of interest,xis the body interval. This can be formulated by calculating the

inte-size,tis the age, andzis an environmental variable like gral of the difference of two logistic curves on a

particu-temperature, nutrition, or light intensity. The forms of lar time interval. Inappendix b, the formula for

calculat-mathematical functions,f(x),g(t), andh(z), which can ing the integral of a logistic curve is given. With the

be linear or nonlinear, are generally different, de-genetic contributions of a QTL to growth, our approach

pending on specific questions of interest. Generally, the can increase the power of discriminating various

impor-establishment of appropriate mathematical functions is tant hypotheses that concern the genetic architecture

based on the goodness of fit to observational data of developmental features (Vaughnet al.1999). Using

(Niklas1994). Alternatively, these mathematical

func-the logistic mixture model, func-the pattern of gene

expres-tions are derived from an optimality perspective. For sion for each different QTL can be explicitly described,

example, Westet al.(1997, 1999) proposed a fractal-thus leading to insights on fundamentally important

like network system for the absorption and internal dis-biological questions; e.g., when and how does a QTL

tribution of metabolites to explain quarter-power scal-affect the phenotype of a quantitative trait during the

entire growth trajectory? How does a QTL interact with ing laws pervasive in the living world. In addition,West

et al.(2001) explained why the growth of an organism

age to affect the growth and development of an

organ-ism? These questions will also have implications for ap- follows a sigmoid curve based on fundamental princi-ples for the allocation of metabolic energy between plied breeding programs. If a tree breeder intends to

select superior genotypes with high fiber yield at harvest- maintenance of existing tissue and the production of new biomass.

ing ages (say 15 years) on the basis of their early

perfor-mance, a marker-assisted selection strategy incorporat- The method proposed in this study can be extended to other situations, such as partially informative markers ing a QTL like one detected on linkage group 10 from

our method (Figure 3) can be expected to increase or dominant markers, to deal with linked QTL of epista-sis or to combine it with selective genotyping. In this the efficiency of early selection because such a QTL

predisposes for productive final fiber yield at age 15 study, it is assumed that residual variances and covari-ances among different ages are stationary. This assump-years. With no information about the developmental

change of QTL expression, however, this breeder is tion simplifies the mathematical manipulation of the residual variance-covariance matrix (inversion, factor-unable to identify and, therefore, to make use of this

QTL in his early selection. ization, etc.), but may be deviate from reality. The exten-sion of our analysis to nonstationary variance-covariance Our method can be extended to incorporate a

gen-eral biological process of an organism into a QTL map- structures is possible, as proposed by Nunez-Anton

(1997) and Nunez-Anton andZimmerman (2000) in ping framework. Such a process can be allometric

scal-ings (Westet al. 1997, 1999), growth models (Gould their structured antedependent models. Also, Kirkpat-rick and co-workers proposed Legendre polynomials 1977;Alberchet al.1979), or continuous responses to

the environment in which an organism is reared (Via to model the dynamic changes of genetic or residual variance and covariance with age (Kirkpatrick and

et al.1995). However, for clarity of description, we based

our analysis on growth curves only. For growth models, Heckman 1989;Kirkpatricket al.1990, 1994). These parametric models for covariance function were im-we further limited our analysis to sigmoidal or logistic

tative genetics of development—genetic correlations among

age-positive definite property of the functions. These

differ-specific trait values and the evolution of ontogeny. Evolution37:

ent models for covariance function with some modifica- _895–905.

Cheverud, J. M., E. J. Routman, F. A. M. Duarte, B. van Swinderen, tions can be incorporated into our mapping strategy.

K. Cothran et al., 1996 Quantitative trait loci for murine

Functional mapping: SinceLanderand Botstein’s

growth. Genetics142:1305–1319.

(1989) interval mapping, there has been a wealth of _{Davidian, M.,andD. M. Giltinan,1995 Nonlinear Models for Repeated}

Measurement Data.Chapman & Hall, London.

literature reporting on the development of statistical

Dempster, A. P., N. M. LairdandD. B. Rubin,1977 Maximum

methods for QTL mapping. The transition from a usual

likelihood from incomplete data via EM algorithm. J. R. Stat.

single- or two-trait analysis to treatment of multiple mea- _{Soc. Ser. B39:1–38.}

Doerge, R. W.,andG. A. Churchill,1996 Permutation tests for

surements from different traits significantly improves

multiple loci affecting a quantitative character. Genetics142:

all aspects of utilization of the mapping information

285–294.

contained in the data. In traditional mapping strategies, _{Eaves, L. J., M. C. NealeandH. Maes,1996 Multivariate multipoint} the combination of statistics and molecular genetics linkage analysis of quantitative trait loci. Behav. Genet.26:519–

525.

makes it possible to identify QTL that contribute to

Emebiri, L. C., M. E. Devey, A. C. MathesonandM. U. Slee,1998

complex traits. However, in this study we attempt to _{Age-related changes in the expression of QTLs for growth in} combine powerful statistics and molecular genetics with radiata pine seedlings. Theor. Appl. Genet.97:1053–1061.

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new strategy, which is calledfunctional mappingdue to

Hof, L., L. C. P. Keizer, I. A. M. ElberseandO. A. Dolstra,1999

the implementation of different mathematical functions _{A model describing the flowering of single plants, and the} herita-of biological means, herita-offers four significant advantages bility of flowering traits of Dimorphotheca pluvialis. Euphytica

110:35–44.

over previous strategies when applied to QTL mapping:

Jaffrezic, F.,andS. D. Pletcher,2000 Statistical models for

esti-(1) Results from functional mapping are closer to bio- _{mating the genetic basis of repeated measures and other} function-logical reality because the underlying biofunction-logical mecha- valued traits. Genetics156:913–922.

Jansen, R. C.,andP. Stam,1994 High resolution of quantitative

nisms are considered; (2) smaller sample sizes may be

traits into multiple loci via interval mapping. Genetics136:1447–

used to achieve adequate power and precision for QTL _1455.

detection because multiple measurements on the same Jiang, C., andZ-B. Zeng, 1995 Multiple trait analysis of genetic mapping for quantitative trait loci. Genetics140:1111–1127.

individuals increase precision for mapping; (3) a large

Kao, C. H., Z-B. ZengandR. D. Teasdale,1999 Multiple interval

number of variables can be analyzed simultaneously by

mapping for quantitative trait loci. Genetics152:1203–1216.

treating growth or a process as a smooth curve, and _{Kirkpatrick, M.,}and N. Heckman,1989 A quantitative genetic model for growth, shape, reaction norms, and other

infinite-also the estimates of a small number of parameters can

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Kirkpatrick, M., W. G. HillandR. Thompson,1994 Estimating the

and this has a direct impact on applied breeding and

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the developmental studies of genetics and evolution. _{with lactation in dairy cattle. Genet. Res.64:57–69.}

Knott, S. A.,and C. S. Haley,2000 Multitrait least squares for We thank Dr. Alan Agresti, Dr. Myron Chang, Dr. Ramon Littell

quantitative trait loci detection. Genetics156:899–911. and Dr. Sam Wu for their helpful discussions on this study and three

Korol, A. B., Y. I. RoninandV. M. Kirzhner,1995 Interval mapping anonymous referees for their constructive comments on the earlier _{of quantitative trait loci employing correlated trait complexes.} version of this manuscript. This work is partially supported by grants _{Genetics140:1137–1147.}

from the National Science Foundation to G.C. (DMS9971586) and _{Korol, A. B., Y. I. Ronin, A. M. Itskovich, J. Peng}andE. Nevo,

an Outstanding Young Investigator Award of the National Natural 2001 Enhanced efficiency of quantitative trait loci mapping analysis based on multivariate complexes of quantitative traits. Science Foundation of China to R.W. (30128017). The publication

Genetics157:1789–1803. of this manuscript is approved as journal series R-08640 by the Florida

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the above equations will be used to provide new estimators Lettingy⫽b⫹er t_{, we have}

of⍀in the next step.

dy

dt⫽ re

r t

and, thus, APPENDIX B

Below, we describe a mathematical procedure for calcu- dy

r ⫽e

r t_dt.

lating the integral of a logistic curve,

Also, whent⫽t1ort2, we have the limits ofyasb⫹ er t1

g(t)⫽ a

1⫹be⫺r t, _orb⫹_er t_{2, respectively. Therefore, we have}

on the interval [t1 t2]. The integral of the curve on this _G[t

1 t2]⫽

冮

t2

t1

a

rydy

interval is expressed as

G[t1t2]⫽

冮

t₂

t1

a

1⫹be⫺r tdt ⫽

a

rlny|

b⫹ert2

b⫹ert1

⫽

冮

t2

t1

aer t

er t⫹_bdt. ⫽

a

r[ln(b⫹e

r t₂₎⫺_ln(