| HOME | HELP | FEEDBACK | SUBSCRIPTIONS | ARCHIVE | SEARCH | TABLE OF CONTENTS |
Genetics, Vol. 168, 2383-2394, December 2004, Copyright © 2004
doi:10.1534/genetics.104.034447
| ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||
* Department of Statistics, University of Florida, Gainesville, Florida 32611
Department of Anatomy and Neurobiology, Washington University School of Medicine, St. Louis, Missouri 63110
1 Corresponding author: Department of Statistics, 533 McCarty Hall C, University of Florida, Gainesville, FL 32611.
E-mail: rwu{at}stat.ufl.edu
 |
ABSTRACT |
|---|
 TOP ABSTRACT A GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
Growth, from an evolutionary standpoint, represents the capability of an organism to sense differences in the internal or external environment, to redirect the developmental trajectory to better suit current environmental conditions, and to thereby increase fitness (VINICIUS and LAHR 2003). A general view derived from fundamental physical and physiological principles is that growth is more expeditious during the early exponential phase since growth rate is self-accelerating or autocatalytic and continues to proceed albeit at a lower rate later when growth rate becomes self-inhibiting (WEST et al. 2001). However, there is a ubiquitous phenomenon that some organisms produce distinctly different growth through several phases during development, while others display more subtle developmental differences. These differences in growth form or trajectory among organisms are thought to provide fuel for adaptation and evolution through alterations of the timing and rate of developmental events (RICE 1997).
The genetic control of growth has been studied using various statistical models. MEYER (1998) proposed random-regression models to understand the genetic control of growth trajectories for animal breeding, whereas KIRKPATRICK et al. (1990) derived a series of genetic models for infinite-dimensional traits in the evolutionary context. These models are particularly effective to study the genetic architecture of growth by modeling the (co)variance matrix for growth traits measured at different time points, although none of them can unravel the genetic mechanisms underlying growth differentiation at the molecular level.
The differentiation of growth trajectories can be attributed to a complex interaction between deterministic and opportunistic factors. The deterministic factors predispose an organism toward a specific growth shape (prototype), whereas the opportunistic factors modify it in response to the unique environment the organism experiences. The deterministic factors represent a suite of inherited loci formed during embryogenesis, but the opportunistic factors that can be genetic or environmental occur postembryonically through continuous growth. Genetic opportunistic factors are an array of new genes that are activated by the regulatory system of the organism in response to changing environments. Environmental opportunistic factors include various predictable environmental changes and unpredictable stochastic errors.
One of the most important tasks in growth analysis is to discern the genetic mechanisms underlying growth from environmental influences. The effects of deterministic factors or deterministic quantitative trait loci (dQTL) on growth are triggered by allelic sensitivity, which is represented as the differential expression of alleles at these QTL across development. As opposed to dQTL, opportunistic (or indeterministic) quantitative trait loci (oQTL) regulate growth trajectories by responding to specific environmental cues to turn on or adjust the expression of structural genes that directly influence the growth. Thus, deterministic effects due to dQTL occur at the level of the whole growth curve, basically affecting the whole growth process, while opportunistic effects due to oQTL cause deviations from the growth model by adjusting growth rate and acceleration curves in response to developmental signals. Although they are not mutually exclusive, the deterministic and opportunistic mechanisms contribute differently to differentiation in growth trajectories. WU (1998), for the first time, used QTL-mapping approaches to test the relative importance of these two mechanisms in shaping phenotypic plasticity. The "allelic sensitivity" hypothesis proposes that the QTL affecting the sensitivity of a trait to variation in an environment variable should map to the same regions as those that explain the variation in the across-environment mean trait value among genotypes. The "gene regulation" hypothesis predicts that the QTL regions that explain the variation among genotypes in the sensitivity of a trait to an environmental variable will be distinct from QTL that contribute to the variation in the trait among the genotypes within a given environment. Currently, QTL-mapping approaches have been used to study the phenotypic plasticity of life span in Drosophila (LEIPS and MACKAY 2000) and glucosinolate accumulation under methyl jasmonate treatment in Arabidopsis (KLIEBENSTEIN et al. 2003).
Here, we develop a novel statistical model for understanding the genetic architecture of growth trajectories by integrating the two underlying genetic mechanisms, allelic sensitivity and gene regulation, and the physiological principles behind growth and development in the QTL-mapping context. Wu and coworkers have proposed a host of statistical models for functional mapping of QTL responsible for the shape of growth trajectories using the growth data measured at many discrete time points (WU et al. 2002, 2003a,b; MA et al. 2002). These models have been successfully used to identify QTL affecting stem size growth in a forest tree and extended to map QTL for cancer growth, drug response, and human immunodeficiency virus (HIV)-1 dynamics. Beyond these previous models, the new model proposed in this article can provide greater insights into the physiological control mechanisms of growth and thereby provide a more rapid way to alter growth patterns toward a predetermined direction. We used body weight data in an F2 hybrid progeny of mice (CHEVERUD et al. 1996) to investigate the usefulness and power of this new model.
|
A GENERAL FRAMEWORK FOR FUNCTIONAL MAPPING |
|---|
|
TOPABSTRACTA GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
 | (1) |
= (1, ... , L)T are the mixture proportions that are constrained to be nonnegative and sum to unity; 
= (1, ... , L)T are the component-specific parameters, with l being specific to component l; and 
is a parameter that is common to all components. T stands for the vector transpose.
The estimation of QTL genotype frequencies (specified by ), QTL effects (specified by ), and the residual variance within each QTL genotype (specified by ) involves three steps. First, derive the unknown QTL genotype frequencies in the given mapping population conditional upon known marker genotype according to Mendelian (for linkage analysis) or population genetic principles (for linkage disequilibrium analysis). Second, model the genetic effects and residual variance (including residual covariances for a multivariate analysis) specified in the distribution on the basis of quantitative genetic theory. Third, develop an efficient computational algorithm to solve all the unknown parameters in the mixture described by Equation 1.
Step 1—linkage analysis or linkage disequilibrium analysis:
The genotype frequencies of unknown QTL genotypes given marker genotypes can be derived, depending on the mapping population considered. In experimental crosses, as are used in mouse, rice, and forest trees, the derivation of these conditional QTL genotype frequencies is based on classic Mendelian genetic theories. The conditional probabilities of QTL genotypes, conditional upon marker genotypes, are expressed in terms of the recombination fractions between the markers and QTL. For random samples from a natural population, population genetic theory is used to postulate the association between the markers and QTL described by the coefficient of their linkage disequilibrium (LD). The LD-based mapping strategy is commonly used in humans or other organisms in which it is difficult or even impossible to generate inbred lines or make controlled crosses.
Step 2—modeling the mean vector and covariance structure:
Genetic effects (including the additive, dominant, and epistatic) of QTL on the phenotype are modeled on the basis of quantitative genetic theories (MATHER and JINKS 1982). But for genetic mapping aimed to detect QTL affecting growth trajectories, simple quantitative genetic modeling is not fully adequate to reveal dynamic changes of genetic control over growth. A considerable body of evidence suggests that the kinetics of ontogenetic growth of biological entities obey a universal growth law that can be described by mathematical functions (BRODY 1945; NIKLAS 1994; GUIOT et al. 2003). WEST et al. (2001) derived a conceptual model from basic cellular mechanisms and physical principles to explain why such a growth law is followed by any tissue, organ, and organism. In cancer research, universal tumor (sigmoidal) growth curves are found to result from chromosomal instability (CIN) in tumor initiation (RASNICK and DUESBERG 1999; FABARIUS et al. 2002). The occurrence of CIN, due to gene mutation, results in whole chromosomes or large fractions of chromosomes being gained or lost during cell division. This results in an imbalance in the number of chromosomes per cell (aneuploidy) and an enhanced rate of loss of heterozygosity (STOCK and BIALY 2003).
Instead of estimating the genetic effects on the phenotypes measured at different time points, our function-mapping model estimates the parameters that describe growth curves (called the curve parameters). Also, different from traditional multivariate mapping approaches that estimate an entire unstructured (co)variance matrix (JIANG and ZENG 1995), functional mapping estimates only the parameters that characterize the structure of the (co)variance matrix (called the matrix parameters). The estimation of the curve parameters depends on practical biological problems; for example, cancer growth can be fit by a sigmoidal curve whereas HIV-1 dynamics are fit by a biexponential function. The best way in which the (co)variance matrix is structured for a data set should be chosen on the basis of modeling tests. For instance, stationary autoregressive models may be enough for some growth data sets, but for many other data sets nonstationary antedependence (AD) models may be more appropriate (NúñEZ-ANTóN and ZIMMERMAN 2000; ZIMMERMAN and NúñEZ-ANTóN 2001).
Step 3—computational algorithms:
A number of computational algorithms can be used to solve the mixture model of Equation 1 containing the unknown parameters 
= (, , ). These algorithms include least-squares regression analysis, a maximum-likelihood method implemented with the EM algorithm, and a Bayesian approach implemented with a Markov chain Monte Carlo algorithm. For our functional mapping, the simplex algorithm, widely used in operations research (NELDER and MEAD 1965), is found to provide a fast and precise maximum-likelihood estimate (MLE) of the curve and matrix parameters (ZHAO et al. 2004). Additional advantages of the simplex algorithm are that it is derivative free and easy to implement.
For linkage analysis, the genotypic frequencies of QTL genotypes are functions of the recombination fraction between the QTL and markers. The estimation of these recombination fractions can be obtained using a grid approach by viewing the QTL position as a known parameter between two flanking markers. The profile for the support of the likelihood for the existence of a QTL is drawn across a linkage group and the peak of the profile corresponds to the position of the QTL over the genome. For LD analysis, the QTL-marker genotypic frequencies are determined by a group of population genetic parameters, i.e., the QTL and marker allele frequencies and QTL-marker linkage disequilibria. Recently, a closed-form solution for the EM algorithm was proposed to estimate these population parameters through the estimation of haplotype frequencies (LOU et al. 2003).
|
A MECHANISTIC MODEL FOR MAPPING GROWTH |
|---|
|
TOPABSTRACTA GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
time points, so we have a finite set of data on each individual, which can be considered as a multivariate trait vector, y = [y(1), ... , y()]. As discussed in the Introduction, growth is controlled by two types of genetic machineries, one being the dQTL and the other being the oQTL. Cumulative growth for individual i measured at time t can be partitioned into two different components in terms of the underlying dQTL and residual effects. Statistically, this can be expressed as
 | (2) |
Growth rate for an individual during a time interval is the difference between the growth measured at two different time points, expressed as
 |
Incorporating the component partitioning of growth as described by Equation 2, we have
 | (3) |
The first term of Equation 3 describes the allelic sensitivity of the dQTL from time t to t + 1. We assume that the residual increment [ei(t + 1) – ei(t)] from time t to t + 1 includes the opportunistic genetic component stimulated by new environmental signals [
i(t, t + 1)] and random errors [
i(t, t + 1)]. Thus, the second term of Equation 3 describes the gene regulation control of the oQTL that are newly activated by environmental signals from time t to t + 1. As compared to the growth model (Equation 2) that can identify only the deterministic genetic control, the growth rate model (Equation 3) can identify both deterministic and opportunistic genetic control. Whereas the determinate gene affects growth through its allelic sensitivity, as indicated by gi(t + 1) – gi(t), the indeterminate gene may control growth through regulatory interactions.
Growth acceleration [
yi(t, t + 1)] is defined as the difference between growth rates at different time intervals, i.e.,
 |
On the basis of Equation 3, we have
 | (4) |
ei(t, t + 1)] contains opportunistic genetic control [i(t, t + 1)] and random error components [i(t, t + 1)].
Growth equations:
A number of mathematical functions have been proposed to deal with time-dependent growth. Our analysis is based on a general RICHARDS (1959) function for growth, which is expressed as
 | (5) |
 | (6) |
 | (7) |
Figure 1 illustrates the shapes of growth curve, growth rate curve, and growth acceleration curves based on Equations 5, 6, and 7, respectively. Genetic differentiation in growth curve is due to interactions between the dQTL and oQTL, with the dQTL contributing to the mean growth trajectory, as specified by Equation 5, and the oQTL contributing to the deviation of the trajectory by adjusting growth rate (Equation 6) and acceleration curves (Equation 7) in response to developmental signals.
|
– 
, or a different pair of flanking markers, 1 – 1 bracketing the dQTL, 2 – 2 bracketing the rate oQTL, and 3 – 3 bracketing the acceleration oQTL. The QTL genotypes at a QTL are unknown, but can be predicted from observed genotypes of flanking markers on the basis of their linkage relationship with the QTL. The conditional probability of a QTL genotype, conditional upon known marker genotypes, is expressed in terms of the recombination fractions among the markers and QTL. Such conditional probabilities form the vector for the proportions of components () contained in the mixture model.
We use a set of curve parameters 
u = (au, bu, cu, ku) to determine the growth curve of genotype u at the dQTL (u = 0 for aa, 1 for Aa, and 2 for AA), 
v = (av, bv, cv, kv) to determine the growth rate curve of genotype v at the rate oQTL (v = 0 for bb, 1 for Bb, and 2 for BB), and 
w = (aw, bw, cw, kw) to determine the growth acceleration curve of genotype w at the acceleration QTL (w = 0 for cc, 1 for Cc, and 2 for CC). The genotypic values of the dQTL for growth at different time points are fit by Equation 5, whereas the genotypic values of the rate oQTL for growth rate and of the acceleration oQTL for growth acceleration at different time intervals are fit by Equations 6 and 7, respectively. As seen below, u and v are jointly estimated on the basis of Equation 3, and u and w are jointly estimated on the basis of Equation 4.
In our new functional mapping for growth rate or growth acceleration, we propose to simultaneously estimate the joint QTL genotypes at the dQTL and oQTL on the basis of the observed joint marker genotypes at two pairs of flanking markers. The conditional probabilities of joint QTL genotypes given joint marker genotypes can be derived separately for two different cases. First, if the two QTL are linked with different marker intervals, these conditional joint QTL genotype frequencies for individual i carrying a particular marker genotype (uv|i or uw|i) are calculated as the product of the conditional QTL genotype frequencies (u|i, v|i, w|i) as predicted by respective marker genotypes, i.e.,
 |
 |
Note that the conditional probability given a marker genotype is equivalent to the conditional probability given a particular individual i because we know the marker genotype of each individual. Second, if two QTL are linked with the same marker interval, the conditional joint QTL genotype frequencies at these two QTL given the corresponding marker pairs should be derived on the basis of classic Mendelian genetic principles (WU and CASELLA 2005).
A mechanism-based mixture model:
The traditional genetic mapping model of a longitudinal trait assumes a simple genetic control mode over growth. For example, the growth model based on Equation 2 can detect only dQTL that govern growth through the life of an organism. Here, we propose to use the growth rate (Equation 3) or acceleration model (Equation 4) to characterize more precise genetic control mechanisms underlying growth. Such mechanisms include both dQTL and oQTL control. Assume that the growth of an entity is measured at discrete time points for an F2 population of size n. Growth rates are calculated as the differences of growth between two adjacent time points. For these growth rates to reflect the scale of growth of an individual, we need to include the measurement at one of the two extreme points. Without loss of generality, the growth measured at the first time point is included in growth rate data for individual i, arrayed by yi = [yi(1), yi(1, 2), ... , yi( – 1, )]. The likelihood function of all the growth rate data (y) can be formulated by a mixture model,
 | (8) |
includes the QTL position in terms of the conditional QTL genotype probabilities given marker genotypes, i.e., mixture proportions = (1, ... , L)T, the QTL genotypic means and residual (co)variances contained in the multivariate normal distribution for QTL genotype l, fl(y), with the -dimensional mean vector m and the x (co)variance matrix 
. We have
 | (9) |
(fit by Equation 5) and the regulatory interaction, (t, t + 1), due to the oQTL (fit by Equation 6). Thus, to model the mean vector in Equation 9, we need to estimate only the curve parameters for the growth (u) and growth rate curves (v).
The estimation of all elements in is not efficient. An approach for structuring the matrix for repeated measurements should be used. One of the simplest approaches is the first-order autoregressive [AR(1)] model in which we need to estimate only two parameters, residual variance (
2) and residual correlation per unit time interval (
). The main advantages of the AR(1) model include easy mathematical manipulation and increased power. However, this structuring approach needs two strong assumptions, i.e., constant residual variance over time and residual correlation decaying in a proportion () with time interval. WU et al. (2004b) implemented a transform-both-sides (TBS) model to make the first assumption more realistic. Many other parametric approaches have been proposed to structure the matrix and some of them may need the estimation of more parameters. The choice of the most parsimonious approach is discussed in NúñEZ-ANTóN and ZIMMERMAN (2000).
Similarly, the opportunistic QTL for growth acceleration (y) can be mapped by formulating a likelihood function expressed as
 | (10) |
The mean vector in this case is given by
 | (11) |
, can be structured using an appropriate approach.
Computational algorithms:
The likelihood functions described by Equations 8 and 10 contain the unknown vector composed of the curve (u, v, w) and matrix parameters [, 2 if the AR(1) model is used] and unknown QTL positions. In QTL mapping, QTL positions can be viewed as a fixed parameter (WU and CASELLA 2005). When the QTL are assumed to be at particular positions, conditional joint QTL genotype frequencies () given marker genotypes are known. With known QTL positions, we need to estimate only curve and matrix parameters. By moving QTL positions every 4 cM along a genome, a landscape for the likelihood-ratio test statistics determined by two genomic positions each for a different QTL can be drawn.
The estimation of the curve and matrix parameters can be based on the EM algorithm. But this may be extremely difficult because the nonlinear equations contained in our model are very complicated to differentiate. We have implemented the simplex algorithm (NELDER and MEAD 1965) to solve the likelihood equations (ZHAO et al. 2004).
|
HYPOTHESIS TESTS |
|---|
|
TOPABSTRACTA GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
The existence of the growth QTL:
Testing whether specific QTL exist to affect the presence of QTL for growth is a first step toward the understanding of the detailed genetic architecture of complex phenotypes. The genetic control over entire growth trajectories can be tested by formulating the following hypotheses, H0: 0 = 1 = 2; 0 = 1 = 2 H1: at least one of the equalities above does not hold, (12) when growth rate data are analyzed, or H0: 0 = 1 = 2; 0 = 1 = 2 H1: at least one of the equalities above does not hold, (13) when growth acceleration data are analyzed. The H0 states that there are no QTL affecting growth trajectories (the reduced model), whereas the H1 proposes that such QTL do exist (the full model). The test statistic for testing the hypotheses is calculated as the log-likelihood ratio of the reduced to the full model,
 |
The existence of dQTL:
Our model can test the existence of dQTL for growth. This test relies on linkage mapping in which particular QTL can map to testable genomic positions.
The null hypothesis for this test is given by
 |
The H1 is the same as Equation 12 for the growth rate model and Equation 13 for the growth acceleration model.
The existence of oQTL:
The null hypothesis for testing the existence of oQTL can be formulated by
 |
 |
In practice, it is interesting to test whether the dQTL and oQTL are identical. This test can be formulated by assuming that there is the same genomic position for the two QTL and that the curve parameters affected by dQTL and oQTL are identical among the corresponding QTL genotypes. Under the growth rate and acceleration model, these two assumptions are expressed, respectively, as
 |
 |
WU et al. (2004a) proposed a functional mapping model for testing epistatic interaction effects on growth trajectories. This model can be extended in this study to test if the dQTL and oQTL interact to affect growth curves, growth rate curves, or growth acceleration curves.
|
RESULTS |
|---|
|
TOPABSTRACTA GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
|
) for the growth rate data is derived on the basis of the AR(1) model for the growth data. The covariance between the growth at time point 1 and the growth rate between two adjacent time points (t, t + 1) is derived as
 | (14) |
Assuming two different time points t2 
t1, the covariance between these two growth rates is derived as
 | (15) |
The covariances in Equations 14 and 15 compose the matrix that has the closed forms for its determinant,
 |
 |
and 
denote the elements of the inverse. The existence of the closed forms of the determinant and inverse largely facilitates the extension of our model to any number of time points measured and can also significantly increase the computational efficiency.
Our model identified two significant QTL located on mouse chromosome 1 responsible for the growth process of body weight. Figure 3 gives the landscape of the log-likelihood ratio (LR) test statistics for claiming the existence of a dQTL () and an oQTL (
) on chromosome 1. The peak of the landscape that corresponds to the MLEs of the locations of these two QTL has the LR value of 391, well beyond the critical threshold (202) obtained from 100 permutation tests. As indicated in Figure 3, dQTL and oQTL are located at 30.3 and 93.6 cM, respectively, from one end of the chromosome. The hypothesis test about whether the dQTL and oQTL are the same QTL is rejected, suggesting that mass growth in mice is genetically controlled with two different mechanisms, such as allelic sensitivity and gene regulation.
|
|
|
|
|
DISCUSSION |
|---|
|
TOPABSTRACTA GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
Two distinct mechanisms underlie ontogenetic growth: allelic sensitivity, that is, the differential allelic expression of the same dQTL across ages, and regulatory gene control, in which new oQTL are activated as the organism grows. Our model proposed in this article can identify these two mechanisms underlying ontogenetic growth and development. It allows for the test of a number of biologically important variables in organismic development, for example, the timing of maximal growth rate and the duration of linear growth phase (MA et al. 2002). The knowledge about the genetic control of the timing of developmental features can enhance our understanding of the genetic mechanisms underlying growth trajectories.
Our model provides information about the dynamic changes of allelic expression over time. Its application to a case study of mouse growth basically supports the previous detection of QTL by traditional mapping approaches (CHEVERUD et al. 1996). However, our model has greater power to detect significant QTL and provides more precise estimates of QTL parameters than traditional interval mapping does for single-time growth because our model simultaneously analyzes growth data measured at all possible time points. Furthermore, with the different growth curves, each corresponding to a QTL genotype, our model can be extended to investigate possible pleiotropic effects of the growth QTL on many different developmental events, such as the timing of sexual maturity and reproductive fitness, or biomedically important traits, such as metabolic rate and fatness. We can therefore integrate growth and development, which are historically regarded as two different biological problems, into a comprehensive framework under which their common or unique underlying genetic machineries are identified.
The maximum-likelihood-based statistical theory has proven to be effective for deriving our QTL-mapping methodology incorporated by growth laws (MA et al. 2002; WU et al. 2002, 2004a,b). However, in this study, attempting to embed more sophisticated developmental principles (and therefore more unknown parameters) in the mapping model, we encounter a considerable computational load although it is not a prohibitive factor. Also, it is likely that sample size and the number of time points used in practice are not sufficiently large, which affects the statistical inference of our model based on traditional maximum-likelihood theory. These considerations make it worthwhile to derive our functional mapping model in the Bayesian context (ROBERT and CASELLA 1999).
Our model will be particularly useful in biomedical research. For example, in cancer clinics, the characterization of the timing at which the exponential growth phase begins and the linear growth phase ends in tumor growth enables us to determine the time from initiation to clinical symptoms and from first symptoms to serious clinical problems. Furthermore, it will determine how long after treatment the tumor will take to recur if the treatment is unsuccessful. It may also determine the outcome of a treatment that takes several weeks or months to complete, such as in many radiotherapy or chemotherapy schedules. Knowledge of a tumor's growth rate and growth potential can therefore be important for both prognosis and treatment planning (IZUMI et al. 2003).
With the availability of the complete genome sequence of humans, it seems feasible to map and clone the genes that are responsible for the observed variations in growth parameters. Finding the genes and functions that underlie variation in growth characteristics is going to be an exciting challenge for future research. Such work holds great promise for development of gene therapies to prevent and control cancers.
|
ACKNOWLEDGEMENTS |
|---|
|
TOPABSTRACTA GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
|
LITERATURE CITED |
|---|
|
TOPABSTRACTA GENERAL FRAMEWORK FOR...A MECHANISTIC MODEL FOR...HYPOTHESIS TESTSRESULTSDISCUSSIONACKNOWLEDGEMENTSLITERATURE CITED |
|---|
BRODY, S., 1945 Bioenergetics and Growth. Reinhold Publishing, New York.
CHEVERUD, J. M., E. J. ROUTMAN, F. A. M. DUARTE, B. VAN SWINDEREN, K. COTHRAN et al., 1996 Quantitative trait loci for murine growth. Genetics 142: 1305–1319.[Abstract]
CHURCHILL, G. A., and R. W. DOERGE, 1994 Empirical threshold values for quantitative trait mapping. Genetics 138: 963–971.[Abstract]
FABARIUS, A., A. WILLER, G. YERGANIAN, R. HEHLMANN and P. DUESBERG, 2002 Specific aneusomies in Chinese hamster cells at different stages of neoplastic transformation, initiated by nitrosomethylurea. Proc. Natl. Acad. Sci. USA 99: 6778–6783.
GREGORCZYK, A., 1998 Richards plant growth model. J. Agron. Crop Sci. 181: 243–247.
GUIOT, C. P., G. DEGIORGIS, P. P. DELSANTO, P. GABRIELE and T. S. D. SEISBOECK, 2003 Does tumor growth follow a "universal law"? J. Theor. Biol. 225: 147–151.[CrossRef][Medline]
HART, D., E. SHOCHAT and Z. AGUR, 1998 The growth law of primary breast cancer as inferred from mammography screening trials data. Br. J. Cancer 78: 382–387.[Medline]
IZUMI, Y., E. D. TOMASO, A. HOOPER, P. HUANG, J. HUBER et al., 2003 Responses to antiangiogenesis treatment of spontaneous autochthonous tumors and their isografts. Cancer Res. 63: 747–751.
KIRKPATRICK, M., D. LOFSVOLD and M. BULMER, 1990 Analysis of the inheritance, selection and evolution of growth trajectories. Genetics 124: 979–993.[Abstract]
KLIEBENSTEIN, D. J., A. FIGUTHA and T. MITCHELL-OLDS, 2003 Genetic architecture of plastic methyl jasmonate responses in Arabidopsis thaliana. Genetics 161: 1685–1696.
JIANG, C., and Z-B. ZENG, 1995 Multiple trait analysis of genetic mapping for quantitative trait loci. Genetics 140: 1111–1127.[Abstract]
LEIPS, J., and T. F. C. MACKAY, 2000 Quantitative trait loci for lifespan in Drosophila melanogaster: interactions with genetic background and larval density. Genetics 155: 1773–1788.
LOU, X.-Y., G. CASELLA, R. C. LITTELL, M. K. C. YANG and R. L. WU, 2003 A haplotype-based algorithm for multilocus linkage disequilibrium mapping of quantitative trait loci with epistasis in natural populations. Genetics 163: 1533–1548.
MA, C.-X., G. CASELLA and R. L. WU, 2002 Functional mapping of quantitative trait loci underlying the character process: a theoretical framework. Genetics 161: 1751–1762.
MATHER, K., and J. L. JINKS, 1982 Biometrical Genetics, Ed. 3. Chapman & Hall, London.
MEYER, K., 1998 Estimating covariance functions for longitudinal data using a random regression model. Genet. Sel. Evol. 30: 221–240.
NELDER, J. A., and R. MEAD, 1965 A simplex method for function minimization. Comput. J. 7: 308–313.
NIKLAS, K. L., 1994 Plant Allometry: The Scaling of Form and Process. University of Chicago Press, Chicago.
NORTON, L., 1988 A Gompertzian model of human breast cancer growth. Cancer Res. 48: 7067–7071.[Medline]
NúñEZ-ANTóN, V., and D. L. ZIMMERMAN, 2000 Modeling nonstationary longitudinal data. Biometrics 56: 699–705.[CrossRef][Medline]
RASNICK, D., and P. H. DUESBERG, 1999 How aneuploidy affects metabolic control and causes cancer. Biochem. J. 340: 621–630.
RICE, S. H., 1997 The analysis of ontogenetic trajectories: when a change in size or shape is not heterochrony. Proc. Natl. Acad. Sci. USA 94: 907–912.
RICHARDS, F. J., 1959 A flexible growth function for empirical use. J. Exp. Bot. 10: 290–300.
ROBERT, C. P., and G. CASELLA, 1999 Monte Carlo Statistical Methods. Springer, New York.
SCANES, C. G., 2003 Biology of Growth of Domestic Animals. Culinary and Hospitality Industry Publications Services, Weimar, TX.
STOCK, R. P., and H. BIALY, 2003 The sigmoidal curve of cancer. Nat. Biotech. 21: 13–14.[CrossRef][Medline]
VINICIUS, L., and M. M. LAHR, 2003 Morphometric heterochrony and the evolution of growth. Evolution 57: 2459–2468.[CrossRef][Medline]
VON BERTALANFFY, L., 1957 Quantitative laws for metabolism and growth. Q. Rev. Biol. 32: 217–231.[CrossRef][Medline]
WEST, G. B., J. H. BROWN and B. J. ENQUIST, 2001 A general model for ontogenetic growth. Nature 413: 628–631.
WU, R. L., 1998 The detection of plasticity genes in heterogeneous environments. Evolution 54: 967–977.[CrossRef]
WU, R. L., C.-X. MA, M. CHANG, R. C. LITTELL, S. S. WU et al., 2002 A logistic mixture model for characterizing genetic determinants causing differentiation in growth trajectories. Genet. Res. 19: 235–245.
WU, R. L., C. X. MA, M. C. K. YANG, M. CHANG, U. SANTRA et al., 2003a Quantitative trait loci for growth in Populus. Genet. Res. 81: 51–64.[CrossRef][Medline]
WU, R. L., C.-X. MA, W. ZHAO and G. CASELLA, 2003b Functional mapping of quantitative trait loci underlying growth rates: a parametric model. Physiol. Genomics 14: 241–249.
WU, R. L., C.-X. MA, M. LIN and G. CASELLA, 2004a A general framework for analyzing the genetic architecture of developmental characteristics. Genetics 166: 1541–1551.
WU, R. L., C. X. MA, M. LIN, Z. H. WANG and G. CASELLA, 2004b Functional mapping of quantitative trait loci underlying growth trajectories using a transform-both-sides logistic model. Biometrics 60: 729–738.[CrossRef][Medline]
ZHAO, W., R. L. WU, C.-X. MA and G. CASELLA, 2004 A fast algorithm for functional mapping of complex traits. Genetics 167: 2133–2137.
ZIMMERMAN, D. L., and V. NúñEZ-ANTóN, 2001 Parametric modeling of growth curve data: an overview (with discussions). Test 10: 1–73.[CrossRef]
This article has been cited by other articles:
 |
|
 |
|
  R. Yang, H. Gao, X. Wang, J. Zhang, Z.-B. Zeng, and R. Wu A Semiparametric Approach for Composite Functional Mapping of Dynamic Quantitative Traits Genetics, November 1, 2007; 177(3): 1859 - 1870. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
W. Zhao, H. Li, W. Hou, and R. Wu Wavelet-Based Parametric Functional Mapping of Developmental Trajectories With High-Dimensional Data Genetics, July 1, 2007; 176(3): 1879 - 1892. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 M. Lin, H. Li, W. Hou, J. A. Johnson, and R. Wu Modeling sequence sequence interactions for drug response Bioinformatics, May 15, 2007; 23(10): 1251 - 1257. [Abstract] [Full Text] [PDF]
|
|
|
|
 |
|
 Y. Cui, J. Zhu, and R. Wu Functional mapping for genetic control of programmed cell death Physiol Genomics, May 16, 2006; 25(3): 458 - 469. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
R. Wu and W. Hou A Hyperspace Model to Decipher the Genetic Architecture of Developmental Processes: Allometry Meets Ontogeny Genetics, January 1, 2006; 172(1): 627 - 637. [Abstract] [Full Text] [PDF]
|
|
|
|
|
|
|
