Functional mapping of growth and development

Biol. Rev. (2010), 85, pp. 207–216. 207doi:10.1111/j.1469-185X.2009.00096.x

Functional mapping of growthand development

Yao Li1,2 and Rongling Wu2,3,4∗1Department of Statistics, University of Florida, Gainesville, FL 32611 USA2Department of Public Health Sciences, Pennsylvania State College of Medicine, Hershey, PA 17033 USA3Department of Statistics, Pennsylvania State University, University Park, PA 16802 USA4College of Forest Resources and Environments, Nanjing Forestry University, Nanjing 210037, China

(Received 5 February 2009; revised 27 July 2009; accepted 30 July 2009)

ABSTRACT

Understanding how an organism develops into a fully functioning adult from a mass of undifferentiated cellsmay reveal different strategies that allow the organism to survive under limiting conditions. Here, we review ananalytical model for characterizing quantitative trait loci (QTLs) that underlie variation in growth trajectoriesand developmental timing. This model, called functional mapping, incorporates fundamental principles behindbiological processes or networks that are bridged with mathematical functions into a statistical mapping framework.Functional mapping estimates parameters that determine the shape and function of a particular biological process,thus providing a flexible platform to test biologically meaningful hypotheses regarding the complex relationshipsbetween gene action and development.

Key words: growth, development, quantitative trait loci, functional mapping, complex trait.

CONTENTS

I. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208II. Functional Mapping: A Basic Framework . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

(1) Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208(2) Modeling the mean vector . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209(3) Modeling the covariance matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209(4) Computational algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209(5) Application . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210(6) Statistical considerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210

III. Ontogenetics of QTL . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212(1) Global tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212(2) Regional tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212(3) Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212

IV. QTLS For Ontogenetic Allometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213V. Implications for Phylogeny . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

VI. Future Prospects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214VII. Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214

VIII. Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215IX. References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215

∗ Address for correspondence: Tel: (717)531-0003 ext 287520; Fax: (717)531-5779; E-mail: [email protected]

Biological Reviews 85 (2010) 207–216 © 2009 The Authors. Journal compilation © 2009 Cambridge Philosophical Society

208 Yao Li and Rongling Wu

I. INTRODUCTION

Growth is an intrinsic property of life, requiring the functionalintegration of multiple genetic, hormonal, metabolic andenvironmental factors into a hierarchical, sequential andinteractive movement through space and time. The formand process of growth can be described by mathematicalfunctions obtained on the basis of the goodness of fitto repeated measurements on a time scale (Brody, 1945;von Bertalanffy, 1957). Mathematical functions have provenuseful to discern various developmental features of growth(Richards, 1959). In particular, based on fundamentalprinciples for the allocation of metabolic energy betweenthe maintenance of existing tissue and the production ofnew biomass, West, Brown & Enquist (2001) derived ageneral quantitative model for growth and explained whysuch a model exists for a spectrum of biological organizationsincluding tumors in animals (Guiot et al., 2003, 2006).

Most earlier studies of growth focused on its physiologicalunderpinnings at different organizational levels frommolecular to cellular to tissue to whole organism (Scanes,2003). Interest in exploring the genetic control of growthwas initiated by quantitative genetic analyses (Atchley,1984) and recently intensified with the advent of powerfulmolecular technologies (Cheverud et al., 1996). Traditionalquantitative genetic principles are integrated with growthmodels to estimate more precisely the genetic variation ofgrowth processes. Meyer (2000) proposed random regressionmodels to study the ontogenetic control of growth for animalbreeding, whereas others utilized orthogonal polynomials orother mathematical functions to derive a series of geneticmodels for growth trajectories in an evolutionary context(Kirkpatrick, Hill & Thomapson, 1994; Pletcher & Geyer,1999; Jaffrezix & Pletcher, 2000). All these mentionedmodels have been instrumental in understanding the geneticarchitecture of growth by modeling the covariance matrixfor growth traits measured at different time points, althoughnone incorporates molecular markers to identify individualgenes for growth differentiation.

The identification of specific quantitative trait loci(QTLs) that regulate growth trajectories through coordinatedinteractions with internal and external environments hasbeen a thorny issue in genetic research. An approximateapproach to mapping growth traits has been to associateDNA-based linkage maps with phenotypes (Lander &Botstein, 1989) for different ages or stages of developmentand to compare the differences across these stages, or to usemultiple-trait mapping where the same character is measuredat different times although this approach is limited to two orat most three dimensions of measurements. In either case,the approximation does not capture the dynamics of theprocess, which greatly limits the scope of inference aboutits genetic architecture. Functional mapping (Ma, Casella &Wu, 2002; Wu et al., 2003, 2004a,b,c; Wu & Lin, 2006), i.e.the integration of the mathematical aspects of the biologicalmechanisms and processes of the trait within the statisticalgenetic mapping framework, is the natural way to approachthis kind of complexity.

Instead of estimating individual genetic effects at allpossible points, functional mapping only needs to estimatea few parameters that determine the shape and functionof a particular biological process. This approach not onlyfacilitates the estimation process of genetic parameters, butalso provides results that are closer to biological reality dueto the incorporation of biological principles behind growthtrajectories. For example, Ma et al. (2002) used functionalmapping to detect a QTL that affects growth trajectories ofstem diameter in forest trees. This QTL was found to remainsilent until tree canopies close in the stand, which is consistentwith the ecological prediction from asymmetric competitionof growing trees in a closed site. Herein, we describe the basicprinciple of functional mapping that has shown promise insynthetic approaches to the genetic analysis of complex traits.Through the combination of statistical modeling, genetics,and growth analysis, functional mapping can address manyquestions, such as the patterns of genetic control over growth,the duration of QTL effects, as well as the mechanistic causesof change or no change in growth form. Functional mappingmakes it methodologically feasible to test various biologicallymeaningful hypotheses at the interplay between genetics,developmental biology and evolution.

II. FUNCTIONAL MAPPING: A BASICFRAMEWORK

(1) Theory

The statistical foundation of QTL mapping is based on thefinite mixture model (Lander & Botstein, 1989), in which eachobservation y is assumed to have arisen from one of a knownor unknown number of components, each component beingmodeled by a density from a parametric family f . Assumingthat there are J QTL genotypes contributing to the variationin a quantitative trait, the distribution (p) of observation y isexpressed as a mixture of J densities weighted by mixtureproportions, i.e.,

p(y|ω, ϕ, ν) = ω1f1(y; ϕ1, ν) + · · · + ωJ fJ (y; ϕJ , ν) (1)

where ω = (ω1, . . . , ωJ ) are the mixture proportions (i.e.QTL genotype frequencies) which are constrained to benon-negative and sum to unity; ϕ = (ϕ1, . . . , ϕJ ) are thecomponent- (i.e. QTL genotype) specific parameters, with ϕ j

being specific to component j; and ν are parameters whichare common to all components.

To describe growth comprehensively and precisely, weneed to measure trait values, arrayed by y, at a series ofdiscrete time points (say T ). If all samples are assumed tohave independent responses in time, the likelihood function(L) of such longitudinal data measured for a random sampleof size n is constructed as

L(ω, ϕ, ν|y) =n∏

i=1

[ω1f1(yi; ϕ1, ν) + · · · + ωJ fJ (yi; ϕ1, ν)]

(2)


Functional mapping of growth and development 209

where the multivariate normal distribution of individual i

measured at T time points is expressed as

f (yi; ϕ j , ν) = 1(2π )T /2|�|1/2

exp[−1

2(yi − uj )�−1(yi − uj )′

],

(3)

where yi = [yi (1), . . . ,yi (T )] is a vector of observationmeasured at T time points, uj = [μj (1), . . . , μj (T )] is a vectorof expected values (μj ) for QTL genotype j at different points,and � is the residual covariance matrix. Assuming that thereis no epistasis between the QTLs considered, the relationshipbetween the observation and expected genotypic mean ata particular time t can be described by a linear regressionmodel,

yi (t) = xib +J∑

j=1

ξ ij uj (t) + εi (t) + ei (t), (4)

where xi is a (1 × m) design matrix of individual i for m

covariates (including sex, nutritional level, or temperature),b is the vector for m covariate effects, ξ ij is the indicatorvariable denoted as 1 if a QTL genotype j is consideredfor individual i and as 0 otherwise, and εi (t) and ei (t) area permanent error and random error, respectively, whichtogether follow a normal distribution with mean 0 andvariance (1–η) σ 2

ε + ησ 2e . Of the residual variance, σ 2

ε (t) isthe variance due to the permanent error, σ 2

e (t) is the variancedue to the random error, and η is the proportion of therandom variance due to the total residual variance. Sincethe random errors can be assumed to be independent atdifferent time points, the covariance between residual errorsat any two time points, t1 and t2, is expressed as (1–η) σε(t1,t2). All these time-dependent variances and covariances formthe residual covariance matrix � of multivariate normaldistribution function (3).

The mixture proportions, (ω1, . . . , ωJ ), in equation (1)can be viewed as the prior probabilities of QTL genotypesin a mapping population. When known markers that areco-segregating with the putative QTLs are incorporatedinto the mixture model, these mixture proportions aresubstituted by the conditional probabilities of QTL genotypesgiven the marker genotypes. These conditional probabilitiescan be derived, with their expression dependent on thetype of population. For a structured pedigree, they areexpressed in terms of the recombination fraction, whereas,for a natural population, they are expressed in terms oflinkage disequilibrium. The derivations of the conditionalprobabilities of QTL genotypes are given in general QTLmapping literature (Wu, Ma & Casella, 2007; Wu & Lin,2008). The parameters that describe QTL locations arearrayed in vector �.

To solve the likelihood function implemented with growthdata measured at multiple time points, a general statisticalframework is formulated for functional mapping of growthcurves. This framework includes two tasks: modeling themean vector and modeling the covariance matrix; these aredescribed below.

(2) Modeling the mean vector

The time-dependent expected values of QTL genotype j canbe approximated by using a growth equation,

uj (t) = g(t; j ), (5)

where g is a mathematical function describing growthtrajectories specified by a set of curve parameters arrayed invector �j . A number of growth equations that characterizethe developmental nature of an organism have been proposed(Brody, 1945; von Bertalanffy, 1957; Richards, 1959; Bock &Thissen, 1976; Jolicoeur, Pontier & Abidi, 1992; West et al.,2001; Sumiya et al., 2001; Aggrey, 2009).

The overall form of the growth curve of QTL genotypej is determined by the set of curve parameters containedin �j . Different combinations of these parameters amonggenotypes at a putative QTL imply that this QTL plays arole in governing the differentiation of growth trajectories.Thus, by testing for the difference in �j among differentgenotypes, we can determine whether a specific QTL existsthat confers an effect on growth processes and how such aQTL alters growth trajectories.

(3) Modeling the covariance matrix

The structure of longitudinal residual covariance matrix∑

can be modeled by a proper statistical approach. The first-order autoregressive [AR(1)] model (Diggle, Liang & Zeger,2002) is the simplest choice, expressed as

σ 2ε (1) = · · · = σ 2

ε (T ) = σ 2εandσ 2

e (1) = · · · = σ 2e (T ) = σ 2

e

(6)

for permanent- and random-error variances, and

cov(t1, t2) = σ 2ερ

|t2−t1| (7)

for the covariance between any two time points t1 and t2,where ρ(0 < ρ < 1) is the proportion parameter with whichthe correlation decays with time lag. The parameters thatmodel the covariance structure are arrayed in vector �. Forthe AR(1) model, we have � = (η, ρ, σ 2).

(4) Computational algorithms

The EM algorithm, originally advocated by Dempster, Laird& Rubin (1977), has been implemented to estimate threegroups of unknown parameters in the functional mappingmodel, that is, the QTL-location parameters (�) that specifythe co-segregation patterns of QTL and markers in thepopulation, the curve parameters (�j ) that model the meanvector, and the parameters (�) that model the longitudinalcovariance structure. These unknowns correspond to mixtureproportions (denoted by ωj ), component-specific parameters(denoted by ϕ j ) and common parameters (denoted by ν),respectively, in the mixture model [equation (1)]. A detaileddescription of the EM algorithm was given in Ma et al. (2002).

More recently, Liu & Wu (2009) constructed a Bayesianframework for functional mapping. Markov chain Monte



Carlo (MCMC) algorithms were implemented to estimatemodel parameters contained in �j (j = 1, . . . , J) and �. TheBayesian algorithm shows two major advantages, comparedwith the maximum likelihood approach. First, the formerallows the estimates of confidence intervals for the estimatesof each parameter. Second, it increases computationalefficiency, especially when functional mapping has a complexmean-covariance modeling structure.

(5) Application

An F2 population of 535 mice was founded by two strains,the Large (LG/J) and Small (SM/J) (Cheverud et al., 1996).The F2 hybrids were weighed in a growth chamber at 10weekly intervals starting at age seven days. The raw masseswere corrected for the effects of each covariate due to dam,litter size at birth, parity and sex. From this F2 population, 75microsatellite markers were genotyped, with which a geneticmap of the mouse genome composed of 19 autosomes wasconstructed.

Body mass of this F2 population during the first 10 weeks ofgrowth can be statistically fitted by a logistic growth equation(P < 0.0001; see Fig. 1A for sample mouse #44),

g(t) = a

1 + be−rt, (8)

where g(t) is the body weight of the mouse at age t, a isthe asymptotic value when t tends to be infinite, b is aparameter that describes the initial growth of the mouse,and r is the relative growth rate. Thus, the growth-law-incorporated functional mapping strategy can be used tomap the QTL responsible for genetic differences in growthtrajectories. Almost all of the 19 autosomes were found tocarry QTLs for body mass growth. Most of these QTLs werealso detected from the traditional interval mapping approachfor trait analysis at a single time point. However, functionalmapping displays greater power to detect a significant QTLand better resolution into QTL localization than intervalmapping because the former capitalizes on more phenotypicinformation than the latter (Ma et al., 2002; Wu & Lin,2006). In Fig. 1B, this is illustrated by using an examplefrom mouse chromosome 1, on which a significant QTLis observed at α = 0.001 that causes the differentiation ingrowth trajectories. This is shown by a quite steep peak(bracketed by D1Mit20 and D1Mit7) of the profile of log-likelihood ratio (LR) test statistics for hypothesizing a QTLacross genomic positions (second curve in Fig. 1B). A flatterLR profile was drawn using results from interval mappingfor body mass at the age of maximum difference (10 weeks)(broken curve in Fig. 1B).

Functional mapping can also generate biologically moremeaningful results than current mapping approaches. Theage-specific expression of a QTL detected on mousechromosome 1 is examined by comparing the growth curvesof three corresponding genotypes in the F2 population(Fig. 1C). Statistical tests suggest that the QTL effect hastwo distinct phases during development. The homozygote

derived from the LG/J allele exhibits better growth afterbirth than homozygote derived from the SM/J allele andthe heterozygote, suggesting that the QTL is operationalin an additive manner. This additive gene action modeis maintained until about five weeks of age at which theQTL has an unobservable effect. After this age, the twohomozygotes tend to converge, but the heterozygote, due toits increased growth rate, becomes more and more divergentfrom the homozygotes, suggesting a gradually increased roleof an overdominance effect in shaping growth trajectories.Age five weeks can be viewed as a transition point for theQTL expression; before five weeks the additive effect ispredominant, whereas after this time the overdominanceeffect is of greater importance.

In a study of the same mapping population (Zhaoet al., 2004), functional mapping was modeled within theframework for genotype-sex interactions, to investigate howa growth QTL is expressed differently between the two sexes.While some QTLs detected display sex-invariant effects, thedynamic expression of other QTLs differs between the maleand female mice.

(6) Statistical considerations

The central theme of functional mapping is to model thedynamic pattern of genetic control of growth trajectories intime. Growth, an increase of size or mass with age, obeys somebiological rules. Rigorous mathematical equations have beenestablished to model growth trajectories based on a statisticalfit to observational data (Brody, 1945; von Bertalanffy,1957; Richards, 1959; Bock & Thissen, 1976; Jolicoeuret al., 1992; Sumiya et al., 2001; Aggrey, 2008). Fromfundamental principles of biophysics and biochemistry, Westet al. (2001) showed that growth is inherently characterizedby an exponential phase. To describe the growth curves ofdifferent organisms, many types of growth equations havebeen derived, which mostly include the monomolecular,Gompertz, and logistic curves. A general model that coversthese growth types was given in Richards (1959). Unlike thoseof microorganisms, plants and animals, human growth curvestypically include three different phases, early childhood, mid-childhood, and adolescence. Mathematical equations thatspecify these three phases have been derived and used tomodel the growth pattern of human body size (Bock &Thissen, 1976; Jolicoeur et al., 1992; Sumiya et al., 2001).

It should be noted that the practical use of functionalmapping needs robust statistical tests for growth equationsbecause growth curves may vary among different tissues,organs and organisms, and change between environments(Niklas, 1994). In many situations, there may be no explicitgrowth curves that fit longitudinal data satisfactorily. Aseries of nonparametric models have been incorporated intofunctional mapping, aimed at characterizing the temporalpattern of genetic effects on an arbitrary form of growthcurves (Cui et al., 2008; Yang, Wu & Casella, 2009). Amore sophisticated semiparametric model that integratesthe parametric modeling of one stage of growth and thenon-parametric modeling of another stage has been derived



A

B

C

Fig. 1. Example of functional mapping (Lin et al., 2005). (A) A typical example of growth trajectory fitted by the logistic equationfrom an F2 population of two mouse inbred strains, LG/J and SM/J. (B) The profile of the log-likelihood ratios (LR) between thefull model (there is a QTL) and the reduced (no QTL) model for body mass growth trajectories across chromosome 1 for the mousemap. The solid curve corresponds to the result from our new model and the broken curve, from traditional interval mapping.The genomic position corresponding to the peak of the curve is the maximum likelihood estimate of the QTL localization. Thevertical broken lines indicate the positions of markers on this chromosome shown beneath. The map distances (in centiMorgan)between two markers were calculated using the Haldane mapping function. The thresholds for claiming the genome-wide existenceof a QTL were obtained from permutation tests. The distribution of each of the LR values over 1000 simulation replicates can beapproximated by a χ2 distribution. The 99.9th percentile of the distribution of the maximum is used as an empirical critical value todeclare genome-wide existence of a QTL at α = 0.001. These percentiles are indicated by two horizontal lines (the upper one forour new model and the lower one for traditional interval mapping). (C) Three growth curves each presenting a group of genotypes,the homozygote derived from the LG/J allele (solid), the heterozygote (dotted) and the homozygote derived from the SM/J allele(dashed), at the QTL detected on chromosome 1 in the F2 population of mice. The times at the inflection point are indicated by thevertical lines each corresponding to a QTL genotype. Data from Cheverud et al. (1996).



(Cui, Zhu & Wu 2006). The semiparametric model providesa favorable balance between statistical fitting and biologicalinterpretation, as validated in its application to the geneticmapping of plant yield traits.

Functional mapping manipulates longitudinal data mea-sured across time. A parsimonious model is needed to modelthe longitudinal covariance structure. Although the AR(1)model has elegant mathematical properties that facilitatecomputational implementation, its (co)variance stationarityassumption is likely to be oversimplified in practice. Toremove the heteroscedastic problem of the residual variancein the AR(1) model, a transform-both-sides (TBS) modelhas been embedded into the growth-incorporated finite mix-ture model. Both empirical analyses with real examples andcomputer simulations suggest that the TBS-based model canincrease the precision of parameter estimation and com-putational efficiency (Wu et al., 2004b). Furthermore, theTBS model preserves the original biological means of thecurve parameters although statistical analyses are based ontransformed data.

There is a wealth of statistical literature for nonstationarycovariance models (Diggle et al., 2002). One representativemodel is the structured antedependence (SAD) approachthat can specify the age-specific change of correlation in theanalysis of longitudinal traits (Zimmerman & Nunez-Anton,2001). The SAD model of different orders can be readilyemployed in the model framework of functional mapping forgrowth trajectories (Zhao et al., 2005a,b). It is also essentialto extend functional mapping to include a situation in whichindividuals are measured at unequally spaced time intervals,and where different subjects have different measurementpatterns.

In practice, longitudinal growth data may be measured atirregular intervals and also may be highly sparse, with theextent and distribution depending on different individuals.Aggrey (2008) showed that growth parameters derived fromintensely measured data are different from those of sparselymeasured data. Also, special statistical models are neededto approximate the structure of the covariance matrixwith highly sparse longitudinal data (Fan et al. 2007). Theresolution of all these statistical issues is crucial to broadenthe application of functional mapping to a wide spectrum ofbiological and experimental problems.

III. ONTOGENETICS OF QTL

Functional mapping allows the identification of ontogeneticchanges in QTL expression over time. The influences ofa QTL on growth form and pattern can be understoodfrom the mathematical properties of growth curves. Almostall nonhuman organisms share the same basic S-shaped orsigmoidal growth curve: the growth of chickens, rats, cattle,crops and trees conforms to this sigmoid curve; this can bea case at different organizational levels from cells, tissues,to organs, and whole organisms. The sigmoid curve has awell-studied mathematical form shown in Fig. 2A. By taking

the first and second derivatives of the growth curve, thecurves for the velocity or rate of growth and acceleration canbe obtained (see Fig. 2B and C).

(1) Global tests

Functional mapping can test how a QTL affects the shape ofthe growth curve, growth rate curve and growth accelerationcurve and whether these three types of curves share thesame QTL or are controlled by different QTLs. QTLsthat affect the entire process of growth are called long orpermanent QTLs. The additive, dominant and epistaticeffects of QTLs on growth processes can be tested in greatdetail.

(2) Regional tests

It is likely that an important developmental event occursin a time interval rather than simply at a particulartime point. How a QTL exerts its effects on a stage ofdevelopment can be tested. The inflection point of thegrowth curve, Pi is that at which the growth rate reachesa maximum (Fig. 2A, B). The growth curve can thereforebe divided into two phases, an exponential growth phase(from time t = 0 to Pi) and an asymptotic growth phase(from Pi to infinity). Thus, determination of the timing ofthe inflection point can help to understand better the shapeand process of growth. Functional mapping can test how aQTL affects the exponential and asymptotic growth phasesand whether these two periods of growth share a commonQTL.

Two points P1 and P2 partition the growth curve intothree phases, the exponential growth (from time t = 0 toP1), linear growth (from P1 to P2) and the ageing phases(from P2 to infinity). The coordinates of the three points, P1,Pi and P2, can be obtained by calculating the second andthird derivatives of the growth curve with respect to time;the timing (abscissa) of each point is then derived (Liu et al.,2004). The genetic control of these three periods of growthcan then be tested.

In practice, it is interesting to use the time duration ofthe linear growth phase (or grand period of growth) incomparisons because this is a key determinant of final size.Functional mapping can detect and test whether a specificQTL determines the grand period of growth.

(3) Example

From Fig. 1, we further analyzed the difference in growth rateamong the three genotypes at the QTL detected on mousechromosome 1. The inflection point of the growth curveoccurs about three days earlier for the homozygote derivedfrom the LG/J allele than the homozygote derived fromthe SM/J allele and the heterozygote. It is possible that thisdifference in development causes LG/J homozygote to reachasymptotic growth earlier than the other genotypes (Fig. 1C).As will be seen below, with the three different growth curves,each corresponding to a QTL genotype, we can investigate



Fig. 2. The growth curve (A), growth rate curve (B) and growth acceleration curve (C). The point Pi is the inflection point, at whichgrowth rate reaches its maximum. The points P1 and P2 indicate the timing of maximum acceleration and maximum decelerationof growth; these being the first and second inflection points of the growth rate curve, respectively.

possible pleiotropic effects of this growth QTL on manydevelopmental events, such as the timing of sexual maturity,reproductive fitness, or biomedically important traits such asmetabolic rate and fatness.

IV. QTLS FOR ONTOGENETIC ALLOMETRY

Growth is not an isolated phenomenon, rather it should beviewed as a dynamic system in which multiple biologicalaspects are coordinated to control growth processes.Functional mapping has been extended to model the growthtrajectories of multivariate traits by increasing the dimensionsof the multivariate normal distribution. Zhao et al. (2005b)provided a detailed procedure for formulating the likelihoodof multivariate longitudinal traits and demonstrated thestatistical behaviour of the multivariate model by using anexample of functional mapping in forest trees. The extendedfunctional mapping model allows the estimation of specificQTLs that affect the body, brain and reproductive growth,and investigation of the genetic mechanisms, pleiotropy orlinkage, responsible for the correlations among these threetypes of growth processes.

Trait correlations obey certain rules that may be specifiedby allometry. Allometry describes the relationship of thegrowth of one part of an organism to the growth of thewhole organism (Niklas et al., 1994). Isometric growth isregarded as the growth of various parts of an organism

in a one-to-one ratio; for most living organisms, growthis non-siometric. Allometric patterns of growth have beenused as a criterion to distinguish among different species,populations, and individuals. Although both rats and humansshow increased brain growth relative to body growth, thesetwo species differ dramatically in the relative rates of growthof the brain, reproductive system and body (Bogin, 1999).In rats, reproductive maturation occurs before the brain orbody reaches final adult size (Fig. 3A), whereas humans delaybody growth and reproductive development but do not delaybrain growth (Fig. 3B).

Functional mapping can be integrated with ontogeneticchanges in allometry through the power equation (Liet al., 2007), thus allowing the identification of QTLs thatdetermine the degree and pattern of the response of abody part to body size during development. This integrativefunctional mapping model can be readily extended topredict how the structure and functioning of a biologicalsystem is affected by genetic interactions derived fromdifferent regions of the genome. Functional mapping hasbeen shown to be powerful for testing whether QTLs thataffect body growth processes are also responsible for time-to-event phenotypes, such as age-at-onset traits (Lin & Wu,2006). Age-at-onset traits can be related to reproductivebehaviours including the time to reproductive maturation.After a QTL that determines the differentiation of a growthprocess is estimated, the next step is to test whether this QTLpleiotropically exerts an effect on the time to reproductivematuration.



Fig. 3. Growth of different types of tissue in rats (A) and humans(B). The rat data are from Timiras (1972) and the human datafrom Cabana, Jolicoeur & Michael (1993) and Scammon (1930).The figures were adapted from Bogin (1999).

V. IMPLICATIONS FOR PHYLOGENY

The evolution of complex organisms, such as animals andplants, does not result simply from the direct transformationof adult ancestors into adult descendants, but rather involvesa cascade of developmental processes that produce thenew features of each generation. There is much interest inevolutionary studies identifying the genetic or developmentalchanges in the rate or timing of developmental processes thatmust take place to derive a particular phenotype from itsancestor (Rice, 1997; Vinicius & Lahr, 2003; Raff, 2000).

A difference in the rate or timing of growth anddevelopmental events or patterns between a descendantand its ancestor is called heterochrony (McNamara, 1996).It has been argued that heterochrony plays a key role inthe evolution, complexity, and diversity of the biospherethroughout Earth’s history. In his seminal book, McNamara(1996) shows how changes in size, shape, and behaviourduring animal ontogeny have resulted in the speciation of,and general trends in, life forms from trilobites to dogs.Functional mapping can be integrated with phylogeny toprovide a method for detecting the genes responsible forheterochrony and analyzing patterns of genetic variation

that reveal evolutionary changes in growth trajectory(Rice, 2008).

VI. FUTURE PROSPECTS

Growth is characteristic of every living entity when it is inits early stage of development. Growth analysis by variousmathematical, biophysical or physiological approaches hasfascinated students of biology for many decades. Mechanisticmodeling of growth processes and patterns has beeninstrumental in predicting meat production in agricultureand stimulating the integration of developmental eventsinto evolutionary studies. In medicine, mathematical modelsconstructed in terms of the difference between anabolismand catabolism have been used to predict dynamic changesin tumours and viral loads over time and these models haveimportant clinical consequences by deciphering the naturalhistory of the diseases (Michor, Iwasa & Nowak, 2004).

Growth, from an evolutionary standpoint, represents thecapability of an organism to sense differences in the internalor external environment, to redirect their developmentaltrajectory to better suit current environmental conditions,and thereby to increase fitness (Vinicius, 2003). A generalview derived from fundamental physical and physiologicalprinciples is that growth is more expeditious during theearly exponential phase since growth rate is self-acceleratingor autocatalytic and continues to proceed albeit at alower rate later when growth rate becomes self-inhibiting.However, some organisms produce distinctly different growthrates through several phases during development, whileothers display more subtle developmental differences. Thesedifferences in growth form or trajectory among organismsprobably allow adaptation and evolution through alterationsin the timing and rate of developmental events (Rice, 2000).

We have described methods of computing complexphenotypes by incorporating growth equations for knowndevelopmental and physiological processes and examininghow these processes behave under genetic control. Withthe availability of complete genome sequences, it seemsfeasible to map and clone the genes that are responsible forthe observed variations in growth parameters (Weedon et al.,2007, 2008; Sanna et al., 2008; Lettre et al., 2008). Finding thegenes and functions that underlie variation in developmentalcharacteristics based on the strategy of functional mappingwill be an exciting challenge for future research.

VII. CONCLUSIONS

(1) Growth and development are controlled by specific genesthrough growth hormones. Some genes act in a lifelongprocess where other genes are only turned on for a shortperiod of time and then shut off. The identification ofdifferent patterns of gene expression can gain insights intothe mechanistic and developmental regulation of growth,



ultimately holding a promise to alter the process and form ofgrowth traits.

(2) Functional mapping has proven to be powerful forlocating genes that control growth processes and identifyingthe similarities and differences of genes for vegetative growthand reproductive growth. The merits of functional mappingoriginate from the integration of biological principlesgoverning growth and development into a genetic mappingframework by parsimonious modeling of temporal trendsand longitudinal covariance structures.

(3) There are notable differences in the pattern of growthand development between humans and non-humans. Thegrowth of non-humans generally follows a simple sigmoidcurve, whereas the growth of humans is very complex,spanning several distinct stages of development. Functionalmapping was originally derived for mapping growth curvesof a forest tree and has been used to map growth genesin other non-human organisms including the mouse, rice,and soybean. There is a underscoring need to extendfunctional mapping to map genes for human growthcurves.

(4) Traditional evolutionary genetics studies phylogeneticdifferences in phenotypic traits, usually measured at adultages. A growing interest is devoted to characterize species-specific differences in the growth and developmental processof traits, leading to the birth of a new study paradigm ofevolutionary developmental biology, known as ‘‘evo-devo.’’The marriage of functional mapping with evo-devo willprovide an unprecedented opportunity to identify so-called‘‘heterochrony genes’’ and facilitate the study of the evolutionof development.

VIII. ACKNOWLEDGEMENTS

We thank Dr Changxing Ma for drawing Fig. 1 and hisstimulating discussion regarding functional mapping, andtwo reviewers for their constructive comments on themanuscript. This work was partially supported by JointNSF/NIH grant DMS/NIGMS-0540745.

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