The development of any organism is a complex dynamic process that is controlled by a network of genes as well as by environmental factors. Traditional mapping approaches for analysing phenotypic data measured at a single time point are too simple to reveal the genetic control of developmental processes. A general statistical mapping framework, called functional mapping, has been proposed to characterize, in a single step, the quantitative trait loci (QTLs) or nucleotides (QTNs) that underlie a complex dynamic trait. Functional mapping estimates mathematical parameters that describe the developmental mechanisms of trait formation and expression for each QTL or QTN. The approach provides a useful quantitative and testable framework for assessing the interplay between gene actions or interactions and developmental changes.
Access optionsAccess options
Subscribe to Journal
Get full journal access for 1 year
only $22.08 per issue
All prices are NET prices.
VAT will be added later in the checkout.
Rent or Buy article
Get time limited or full article access on ReadCube.
All prices are NET prices.
Lynch, M. & Walsh, B. Genetics and Analysis of Quantitative Traits (Sinauer, Sunderland, Massachusetts, 1998).
Hallauer, A. R. & Miranda, F. J. B. Quantitative Genetics in Maize Breeding 2nd edn (Iowa State Univ. Press, Ames, Iowa, 1988).
Atchley, W. R. Ontogeny, timing of development, and genetic variance–covariance structure. Am. Nat. 123, 519–540 (1984).
Wolf, J. B., Frankino, W. A., Agrawal, A. F., Brodie, E. D. 3rd & Moore, A. J. Developmental interactions and the constituents of quantitative variation. Evolution 55, 232–245 (2001).
Drayne, D. et al. Genetic mapping of the human X-chromosome by using restriction fragment length polymorphisms. Proc. Natl Acad. Sci. USA 81, 2836–2839 (1984).
Dempster, A. P., Laird, N. M. & Rubin, D. B. Maximum likelihood from incomplete data via the EM algorithm. J. R. Stat. Soc. Ser. B 39, 1–38 (1977).
Lander, E. S. & Botstein, D. Mapping Mendelian factors underlying quantitative traits using RFLP linkage maps. Genetics 121, 185–199 (1989).
Zeng, Z. -B. Precision mapping of quantitative trait loci. Genetics 136, 1457–1468 (1994).
Jansen, R. C. & Stam, P. High resolution mapping of quantitative traits into multiple loci via interval mapping. Genetics 136, 1447–1455 (1994).
Hoeschele, I. in Handbook of Statistical Genetics (eds Balding, D. J., Bishop, M. & Cannings, C.) 599–644 (Wiley, New York, 2001).
Wu, R. L., Ma, C. -X. & Casella, G. Joint linkage and linkage disequilibrium mapping of quantitative trait loci in natural populations. Genetics 160, 779–792 (2002).
Wang, H. et al. Bayesian shrinkage estimation of QTL parameters. Genetics 170, 465–480 (2005).
Cheverud, J. M. et al. Quantitative trait loci for murine growth. Genetics 142, 1305–1319 (1996).
Mackay, T. F. C. Quantitative trait loci in Drosophila. Nature Rev. Genet. 2, 11–20 (2001).
Mauricio, R. Mapping quantitative trait loci in plants: uses and caveats for evolutionary biology. Nature Rev. Genet. 2, 370–381 (2001).
Peltonen, L. & McKusick, V. A. Dissecting human disease in the postgenomic era. Science 291, 1224–1229 (2001).
Andersson, L. & Georges, M. Domestic-animal genomics; Deciphering the genetics of complex traits. Nature Rev. Genet. 5, 202–212 (2004).
Mauricio, R. Ontogenetics of QTL: the genetic architecture of trichome density over time in Arabidopsis thaliana. Genetica 123, 75–85 (2004).
Jiang, C. & Zeng, Z. -B. Multiple trait analysis of genetic mapping for quantitative trait loci. Genetics 140, 1111–1127 (1995).
Diggle, P. J., Liang, K. Y. & Zeger, S. L. Analysis of Longitudinal Data (Oxford Univ. Press, Oxford, 1994).
Ma, C. X., Casella, G. & Wu, R. L. Functional mapping of quantitative trait loci underlying the character process: A theoretical framework. Genetics 161, 1751–1762 (2002).
Wu, R. L., Ma, C. -X., Zhao, W. & Casella, G. Functional mapping of quantitative trait loci underlying growth rates: A parametric model. Physiol. Genomics 14, 241–249 (2003).
Wu, R. L., Ma, C. -X., Lou, Y. -X. & Casella, G. Molecular dissection of allometry, ontogeny and plasticity: A genomic view of developmental biology. BioScience 53, 1041–1047 (2003).
Wu, R. L., Ma, C. -X., Lin, M. & Casella, G. A general framework for analyzing the genetic architecture of developmental characteristics. Genetics 166, 1541–1551 (2004).
Wu, R. L., Ma, C. X., Lin, M., Wang, Z. H. & Casella, G. Functional mapping of quantitative trait loci underlying growth trajectories using a transform-both-sides logistic model. Biometrics 60, 729–738 (2004).
Wu, R. L., Ma, C. X., Littell, R. C. & Casella, G. A statistical model for the genetic origin of allometric scaling laws in biology. J. Theor. Biol. 217, 275–287 (2002).
Wu, R. L., Wang, Z. H., Zhao, W. & Cheverud, J. M. A mechanistic model for genetic machinery of ontogenetic growth. Genetics 168, 2383–2394 (2004).
Brody, S. Bioenergetics and Growth (Reinhold, New York, 1945).
von Bertalanffy, L. Quantitative laws for metabolism and growth. Quart. Rev. Biol. 32, 217–231 (1957).
Richards, F. J. A flexible growth function for empirical use. J. Exp. Bot. 10, 290–300 (1959).
Rice, S. H. The analysis of ontogenetic trajectories: When a change in size or shape is not heterochrony. Proc. Natl Acad. Sci. USA 94, 907–912 (1997).
West, G. B., Brown, J. H. & Enquist, B. J. A general model for ontogenetic growth. Nature 413, 628–631 (2001).
Anholt, R. R. & Mackay, T. F. C. Quantitative genetic analyses of complex behaviours in Drosophila. Nature Rev. Genet. 5, 838–849 (2004).
Whitlock, M. C., Phillips, P. C., Moore, F. B. & Tonsor, S. J. Multiple fitness peaks and epistasis. Ann. Rev. Ecol. Syst. 26, 601–629 (1995).
Wolf, J. B. Gene interactions from maternal effects. Evolution 54, 1882–1898 (2000).
Wolf, J. B., Brodie, E. D. 3rd & Wade, M. J. Epistasis and the Evolutionary Process (Oxford Univ. Press, Oxford, 2000).
Carlborg O & Haley, C. S. Epsitasis: too often neglected in complex trait studies? Nature Rev. Genet. 5, 618–625 (2004).
Moore, J. H. The ubiquitous nature of epistasis in determining susceptibility to common human diseases. Hum. Hered. 56, 73–82 (2003).
Wu, R. L., Ma, C. -X., Hou, W., Corva, P. & Medrano, J. F. Functional mapping of quantitative trait loci that interact with the hg gene to regulate growth trajectories in mice. Genetics 171, 239–249 (2005).
Scheiner, S. M. Genetics and evolution of phenotypic plasticity. Ann. Rev. Ecol. Sys. 24, 35–68 (1993).
Schlichting, C. D. & Pigliucci, M. Phenotypic Evolution: A Reaction Norm Perspective (Sinauer, Sunderland, Massachusetts, 1998).
Via, S. et al. Adaptive phenotypic plasticity: Consensus and controversy. Trends Ecol. Evol. 5, 212–217 (1995).
Wu, R. L. The detection of plasticity genes in heterogeneous environments. Evolution 52, 967–977 (1998).
Leips, J. & Mackay, T. F. C. Quantitative trait loci for life span in Drosophila melanogaster: Interactions with genetic background and larval density. Genetics 155, 1773–1788.
Kingsolver, J. G. & Woods, H. A. Thermal sensitivity of growth and feeding in Manduca sexta caterpillars. Physiol. Zool. 70, 631–638 (1997).
Chapman, T., Arnqvist, G., Bangham, J. & Rowe, L. Sexual conflict. Trends Ecol. Evol. 18, 41–47 (2003).
Zhao, W., Ma, C. -X., Cheverud, J. M. & Wu, R. L. A unifying statistical model for QTL mapping of genotype × sex interaction for developmental trajectories. Physiol. Genomics 19: 218–227 (2004).
Zhao, W., Zhu, J., Gallo-Meagher, M. & Wu, R. L. A unified statistical model for functional mapping of genotype × environment interactions for ontogenetic development. Genetics 168, 1751–1762 (2004).
Gillooly, J. F., Brown, J. H., West, G. B., Savage, V. M. & Charnov, E. L. Effects of size and temperature on metabolic rate. Science 293, 2248–2251 (2001).
West, G. B., Brown, J. H. & Enquist, B. J. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126 (1997).
West, G. B., Brown, J. H. & Enquist, B. J. The fourth dimension of life: Fractal geometry and allometric scaling of organisms. Science 284, 1677–1679 (1999).
Guiot, C. P., Degiorgis, G., Delsanto, P. P., Gabriele, P. & Seisboeck, T. S. Does tumor growth follow a 'universal law'? J. Theor. Biol. 225, 147–151 (2003).
Ambros, V. Control of developmental timing in Caenorhabditis elegans. Curr. Opin. Genet. Dev. 10, 428–33 (2000).
Rougvie, A. E. Control of developmental timing in animals. Nature Rev. Genet. 2, 690–701 (2001).
Niklas, K. J. Plant Allometry: the scaling of form and process (Univ. Chicago Press, Chicago, 1994).
Heath, S. C. Markov chain Monte Carlo segregation and linkage analysis for oligogenic models. Am. J. Hum. Genet. 61, 748–760 (1997).
Meyer, K. Random regression to model phenotypic variation in monthly weights of Australian beef cows. Livestock Prod. Sci. 65, 19–38 (2000).
Macgregor, S., Knott, S. A., White, I. & Visscher, P. M. Quantitative trait locus analysis of longitudinal quantitative trait data in complex pedigrees. Genetics 171, 1365–1376 (2005).
Lou, X. -Y. et al. A haplotype-based algorithm for multilocus linkage disequilibrium mapping of quantitative trait loci with epistasis in natural populations. Genetics 163, 1533–1548 (2003).
Wall, J. D. & Pritchard, J. K. Haplotype blocks and linkage disequilibrium in the human genome. Nature Rev. Genet. 4, 587–597 (2003).
Wang, Z. H. & Wu, R. L. A statistical model for high-resolution mapping of quantitative trait loci determining human HIV-1 dynamics. Stat. Med. 23, 3033–3051 (2004).
Perelson, A. S., Neumann, A. U., Markowitz, M., Leonard, J. M. & Ho, D. D. HIV-1 dynamics in vivo: Virion clearance rate, infected cell life-span, and viral generation time. Science 271, 1582–1586 (1996).
Nowak, M. A. & May, R. M. Virus Dynamics (Oxford Univ. Press, New York, 2000).
Gong, Y. et al. A statistical model for high-resolution mapping of quantitative trait loci affecting pharmacodynamic processes. Pharmacogenomics J. 4, 315–321 (2004).
Wu, R. L. & Zeng, Z. -B. Joint linkage and linkage disequilibrium mapping in natural populations. Genetics 157, 899–909 (2001).
Frary, A. et al. fw2.2: a quantitative trait locus key to the evolution of tomato fruit size. Science 289, 85–88 (2000).
Cooper, R. S. & Psaty, B. M. Genomics and medicine: Distraction, incremental progress, or the dawn of a new age? Ann. Int. Med. 138, 576–680 (2003).
Liu, T., Johnson, J. A., Casella, G. & Wu, R. L. Sequencing complex diseases with HapMap. Genetics 168, 503–511 (2004).
Yalcin, B., Flint, J. & Mott, R. Using progenitor strain information to identify quantitative trait nucleotides in outbred mice. Genetics 171, 673–681 (2005).
Lin, M., Aquilante, C., Johnson, J. A. & Wu, R. L. Sequencing drug response with HapMap. Pharmacogenomics J. 5, 149–156 (2005).
Lin, M. & Wu, R. L. Theoretical basis for the identification of allelic variants that encode drug efficacy and toxicity. Genetics 170, 919–928 (2005).
Pletcher, S. D. & Geyer, C. J. The genetic analysis of age-dependent traits: Modeling the character process. Genetics 153, 825–835 (1999).
Jaffrezix, F. & Pletcher, S. D. Statistical models for estimating the genetic basis of repeated measures and other function-valued traits. Genetics 156, 913–922 (2000).
Kirkpatrick, M. & Heckman, N. A quantitative genetic model for growth, shape, reaction norms, and other infinite-dimensional characters. J. Math. Biol. 27, 429–450 (1989).
Kirkpatrick, M., Hill, W. G. & Thompson, R. Estimating the covariance structure of traits during growth and aging, illustrated with lactation in dairy cattle. Genet. Res. 64, 57–69 (1994).
Norton, L. A Gompertzian model of human breast cancer growth. Cancer Res. 48, 7067–7071 (1988).
Gatenby, R. A. & Maini, P. K. Mathematical oncology: Cancer summed up. Nature 421, 321 (2003).
Michor, F., Iwasa, Y. & Nowak, M. A. Dynamics of cancer progression. Nature Rev. Cancer 4, 197–205 (2004).
Izumi, Y. et al. Responses to antiangiogenesis treatment of spontaneous autochthonous tumors and their isografts. Cancer Res. 63, 747–751 (2003).
Raff, R. A. Evo-devo: the evolution of a new discipline. Nature Rev. Genet. 1, 74–79 (2000).
Arthur, W. The emerging conceptual framework of evolutionary developmental biology. Nature 415, 757–764 (2002).
Vinicius, L. & Lahr, M. M. Morphometric heterochrony and the evolution of growth. Evolution 57, 2459–2468 (2003).
Dusheck, J. It's the ecology, stupid! Nature 418, 578–579 (2002).
Zhao, W., Chen, Y. Q., Casella, G., Cheverud, J. M. & Wu, R. L. A nonstationary model for functional mapping of complex traits. Bioinformatics 21, 2469–2477 (2005).
Lin, M. & Wu, R. L. A unifying model for nonparametric functional mapping of longitudinal trajectories and time-to-events. BMC Bioinformatics (in the press).
Vaughn, T. T. et al. Mapping quantitative trait loci for murine growth — A closer look at genetic architecture. Genet. Res. 74, 313–322 (1999).
The authors thank the three anonymous referees for their constructive comments that have improved the presentation of this manuscript. This work was supported by an Outstanding Young Investigator Award of the National Natural Science Foundation of China, a University of Florida Research Opportunity Fund, a University of South Florida Biodefense grant and the National Institutes of Health.
The authors declare no competing financial interests.
The change in proportion of various parts of an organism as a consequence of growth.
- Allometric scaling law
Metabolic rates or other biological variables that scale as multiples of one-quarter of body mass.
- Biexponential equation
An equation that describes two subsequent processes in which the responses change exponentially with a variable in each process.
- Dynamic biological thermal function
A function that describes the change of growth rate or other variables of an organism with different temperatures.
- Exercise stress test
A general screening tool to test the effect of exercise on the heart.
- Finite mixture model
A type of density model that comprises several component functions, usually Gaussian functions, which are combined to provide a multimodal density.
- Fourier series equation
An expansion of a periodic function in terms of an infinite sum of sines and cosines.
- Linkage disequilibrium
The non-random co-segregation of alleles at different loci in a population.
- Log-likelihood ratio
A test statistic that is expressed as the log ratio of the maximum value of the likelihood function under the constraint of the null hypothesis to the maximum value without that constraint.
- Logistic equation
Also called an S-shaped curve. It models a process of growth in which the initial stage of growth is approximately exponential. As competition arises, the growth slows, and at maturity, growth stops.
- Model selection
A process in which the best model is selected from many competing models that fit the data.
Functions that have the form f(x) = anxn + a−1xn−1 + ... + a1x + a0, where n is a non-negative integer.
- Shrinkage estimation
An estimating procedure by which all candidate variables are taken into account in the model, but their estimated effects are forced to shrink towards zero.
- Wavelet transform approach
An approach that compresses high-order dimensional data to a low-order representation without losing the original information.
About this article
A Gaussian process model and Bayesian variable selection for mapping function-valued quantitative traits with incomplete phenotypic data
Accuracy assessment of plant height using an unmanned aerial vehicle for quantitative genomic analysis in bread wheat
Plant Methods (2019)
Modelling strategies for assessing and increasing the effectiveness of new phenotyping techniques in plant breeding
Plant Science (2019)