Bulletin of Mathematical Biology

, Volume 69, Issue 8, pp 2723–2735 | Cite as

Toward the Human Genotope

  • Peter Huggins
  • Lior PachterEmail author
  • Bernd Sturmfels
Origianal Article


The human genotope is the convex hull of all allele frequency vectors that can be obtained from the genotypes present in the human population. In this paper, we take a few initial steps toward a description of this object, which may be fundamental for future population based genetics studies. Here we use data from the HapMap Project, restricted to two ENCODE regions, to study a subpolytope of the human genotope. We study three different approaches for obtaining informative low-dimensional projections of this subpolytope. The projections are specified by projection onto few tag SNPs, principal component analysis, and archetypal analysis. We describe the application of our geometric approach to identifying structure in populations based on single nucleotide polymorphisms.


ENCODE project Genotope Human variation Polytope Single nucleotide polymorphism 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. Beerenwinkel, N., Pachter, L., Sturmfels, B., 2006. Epistasis and shapes of fitness landscapes, Statistica Sinica, to appear. ArXiv:q-bio.PE/0603034. Google Scholar
  2. Beerenwinkel, N., Pachter, L., Sturmfels, B., Elena, S., Lenski, R., 2007. Analysis of epistatic interactions and fitness landscapes using a new geometric approach. BMC Evol. Biol. 7, 60. CrossRefGoogle Scholar
  3. Cavalli-Sforza, L.L., Menozzi, P., Piazza, A., 1994. The History and Geography of Human Genes. Princeton University Press, Princeton. Google Scholar
  4. Chesler, E.J., Lu, L., Shou, S., Qu, Y., Gu, J., Wang, J., Hsu, H.C., Mountz, J.D., Baldwin, N.E., Langston, M.A., et al., 2005. Complex trait analysis of gene expression uncovers polygenic and pleiotropic networks that modulate nervous system function. Nat. Genet. 37, 233–242. CrossRefGoogle Scholar
  5. Christiansen, F.B., 2000. Population Genetics of Multiple Loci. Wiley, New York. Google Scholar
  6. Cutler, A., Breiman, L., 1994. Archetypal analysis. Technometrics 36, 338–347. zbMATHCrossRefMathSciNetGoogle Scholar
  7. Dewey, C., Huggins, P., Woods, K., Sturmfels, B., Pachter, L., 2006. Parametric alignment of Drosophila genomes. PLoS Comput. Biol. 2(6), e73. CrossRefGoogle Scholar
  8. ENCODE project consortium, 2004. The ENCODE (ENCylopedia Of DNA Elements) project. Science 306(5696), 636–640. CrossRefGoogle Scholar
  9. Gawrilow, E., Joswig, M., 2005. Geometric reasoning with polymake. ArXiv:math/0507273. Google Scholar
  10. Grosse, I., Bernaola-Galván, P., Carpena, P., Román-Roldán, R., Oliver, J., Stanley, H.E., 2002. Analysis of symbolic sequences using the Jensen–Shannon divergence. Phys. Rev. E 65(041904-1), 1063–1065. Google Scholar
  11. Hallgrímsdóttir, I., Yuster, D., 2007. A complete classification of two-locus disease models. BMC Genet., in press. Google Scholar
  12. International HapMap Consortium, 2005. A haplotype map of the human genome. Nature 437(7063), 1299–1320. CrossRefGoogle Scholar
  13. Kimmel, G., Shamir, R., 2005. A block-free hidden Markov model for genotypes and its application to disease association. J. Comput. Biol. 12(10), 1243–1260. CrossRefGoogle Scholar
  14. Ott, J., 1999. Analysis of Human Genetic Linkage, 3rd edn. Johns Hopkins University Press, Baltimore. Google Scholar
  15. Price, A.L., Patterson, N.J., Plenge, R.M., Weinblatt, M.E., Shadlick, N.A., Reich, D., 2006. Principal components analysis corrects for stratification in genome-wide association studies. Nat. Genet. 38, 904–909. CrossRefGoogle Scholar
  16. Pritchard, J.K., Stephens, M., Donnelly, P., 2006. Inference of population structure using multilocus genotype data. Genetics 55, 945–959. Google Scholar
  17. Rinaldo, A., Bacanu, S.A., Devlin, B., Sonpar, V., Wasserman, L., Roeder, K., 2005. Characterization of multilocus linkage disequilibrium. Genet. Eipdemiol. 28(3), 193–206. CrossRefGoogle Scholar
  18. Sturm, J.F., 1999. Using SeDuMi, a Matlab toolbox for optimization over symmetric cones. Optim. Methods Softw. 11–12, 625–653. CrossRefMathSciNetGoogle Scholar

Copyright information

© Society for Mathematical Biology 2007

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at BerkeleyBerkeleyUSA

Personalised recommendations