Toric ideals of homogeneous phylogenetic models

We consider the phylogenetic tree model in which every node of the tree is observed and binary and the transitions are given by the same matrix on each edge of the tree. We are able to compute the Grobner basis and Markov basis of the toric ideal of invariants for trees with up to 11 nodes. These are perhaps the first non-trivial Grobner bases calculations in 2^11 indeterminates. We conjecture that there is a quadratic Grobner basis for binary trees. Finally, we give a explicit description of the polytope associated to this toric ideal for an infinite family of binary trees and conjecture that there is a universal bound on the number of vertices of this polytope for binary trees.Comment: 6 pages, 17 figure

Conjunctive Bayesian networks

Conjunctive Bayesian networks (CBNs) are graphical models that describe the accumulation of events which are constrained in the order of their occurrence. A CBN is given by a partial order on a (finite) set of events. CBNs generalize the oncogenetic tree models of Desper et al. by allowing the occurrence of an event to depend on more than one predecessor event. The present paper studies the statistical and algebraic properties of CBNs. We determine the maximum likelihood parameters and present a combinatorial solution to the model selection problem. Our method performs well on two datasets where the events are HIV mutations associated with drug resistance. Concluding with a study of the algebraic properties of CBNs, we show that CBNs are toric varieties after a coordinate transformation and that their ideals possess a quadratic Gr\"{o}bner basis.Comment: Published in at http://dx.doi.org/10.3150/07-BEJ6133 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Phylogenetic Algebraic Geometry

Phylogenetic algebraic geometry is concerned with certain complex projective algebraic varieties derived from finite trees. Real positive points on these varieties represent probabilistic models of evolution. For small trees, we recover classical geometric objects, such as toric and determinantal varieties and their secant varieties, but larger trees lead to new and largely unexplored territory. This paper gives a self-contained introduction to this subject and offers numerous open problems for algebraic geometers.Comment: 15 pages, 7 figure

Genome-wide analysis points to roles for extracellular matrix remodeling, the visual cycle, and neuronal development in myopia

Myopia, or nearsightedness, is the most common eye disorder, resulting primarily from excess elongation of the eye. The etiology of myopia, although known to be complex, is poorly understood. Here we report the largest ever genome-wide association study (43,360 participants) on myopia in Europeans. We performed a survival analysis on age of myopia onset and identified 19 significant associations (p < 5e-8), two of which are replications of earlier associations with refractive error. These 19 associations in total explain 2.7% of the variance in myopia age of onset, and point towards a number of different mechanisms behind the development of myopia. One association is in the gene PRSS56, which has previously been linked to abnormally small eyes; one is in a gene that forms part of the extracellular matrix (LAMA2); two are in or near genes involved in the regeneration of 11-cis-retinal (RGR and RDH5); two are near genes known to be involved in the growth and guidance of retinal ganglion cells (ZIC2, SFRP1); and five are in or near genes involved in neuronal signaling or development. These novel findings point towards multiple genetic factors involved in the development of myopia and suggest that complex interactions between extracellular matrix remodeling, neuronal development, and visual signals from the retina may underlie the development of myopia in humans

Efficient Replication of Over 180 Genetic Associations with Self-Reported Medical Data

While the cost and speed of generating genomic data have come down dramatically in recent years, the slow pace of collecting medical data for large cohorts continues to hamper genetic research. Here we evaluate a novel online framework for amassing large amounts of medical information in a recontactable cohort by assessing our ability to replicate genetic associations using these data. Using web-based questionnaires, we gathered self-reported data on 50 medical phenotypes from a generally unselected cohort of over 20,000 genotyped individuals. Of a list of genetic associations curated by NHGRI, we successfully replicated about 75% of the associations that we expected to (based on the number of cases in our cohort and reported odds ratios, and excluding a set of associations with contradictory published evidence). Altogether we replicated over 180 previously reported associations, including many for type 2 diabetes, prostate cancer, cholesterol levels, and multiple sclerosis. We found significant variation across categories of conditions in the percentage of expected associations that we were able to replicate, which may reflect systematic inflation of the effects in some initial reports, or differences across diseases in the likelihood of misdiagnosis or misreport. We also demonstrated that we could improve replication success by taking advantage of our recontactable cohort, offering more in-depth questions to refine self-reported diagnoses. Our data suggests that online collection of self-reported data in a recontactable cohort may be a viable method for both broad and deep phenotyping in large populations