Reducing degeneracy in maximum entropy models of networks.

Based on Jaynes's maximum entropy principle, exponential random graphs provide a family of principled models that allow the prediction of network properties as constrained by empirical data (observables). However, their use is often hindered by the degeneracy problem characterized by spontaneous symmetry breaking, where predictions fail. Here we show that degeneracy appears when the corresponding density of states function is not log-concave, which is typically the consequence of nonlinear relationships between the constraining observables. Exploiting these nonlinear relationships here we propose a solution to the degeneracy problem for a large class of systems via transformations that render the density of states function log-concave. The effectiveness of the method is demonstrated on examples.

The inverse problem for certain tree parameters.

Let p be a graph parameter that assigns a positive integer value to every graph. The inverse problem for p asks for a graph within a prescribed class (here, we will only be concerned with trees), given the value of p. In this context, it is of interest to know whether such a graph can be found for all or at least almost all integer values of p. We will provide a very general setting for this type of problem over the set of all trees, describe some simple examples and finally consider the interesting parameter "number of subtrees", where the problem can be reduced to some number-theoretic considerations. Specifically, we will prove that every positive integer, with only 34 exceptions, is the number of subtrees of some tree.

The allele frequency spectrum in genome-wide human variation data reveals signals of differential demographic history in three large world populations.

We have studied a genome-wide set of single-nucleotide polymorphism (SNP) allele frequency measures for African-American, East Asian, and European-American samples. For this analysis we derived a simple, closed mathematical formulation for the spectrum of expected allele frequencies when the sampled populations have experienced nonstationary demographic histories. The direct calculation generates the spectrum orders of magnitude faster than coalescent simulations do and allows us to generate spectra for a large number of alternative histories on a multidimensional parameter grid. Model-fitting experiments using this grid reveal significant population-specific differences among the demographic histories that best describe the observed allele frequency spectra. European and Asian spectra show a bottleneck-shaped history: a reduction of effective population size in the past followed by a recent phase of size recovery. In contrast, the African-American spectrum shows a history of moderate but uninterrupted population expansion. These differences are expected to have profound consequences for the design of medical association studies. The analytical methods developed for this study, i.e., a closed mathematical formulation for the allele frequency spectrum, correcting the ascertainment bias introduced by shallow SNP sampling, and dealing with variable sample sizes provide a general framework for the analysis of public variation data.

Sequence variations in the public human genome data reflect a bottlenecked population history.

Single-nucleotide polymorphisms (SNPs) constitute the great majority of variations in the human genome, and as heritable variable landmarks they are useful markers for disease mapping and resolving population structure. Redundant coverage in overlaps of large-insert genomic clones, sequenced as part of the Human Genome Project, comprises a quarter of the genome, and it is representative in terms of base compositional and functional sequence features. We mined these regions to produce 500,000 high-confidence SNP candidates as a uniform resource for describing nucleotide diversity and its regional variation within the genome. Distributions of marker density observed at different overlap length scales under a model of recombination and population size change show that the history of the population represented by the public genome sequence is one of collapse followed by a recent phase of mild size recovery. The inferred times of collapse and recovery are Upper Paleolithic, in agreement with archaeological evidence of the initial modern human colonization of Europe.

Connected gene neighborhoods in prokaryotic genomes.

A computational method was developed for delineating connected gene neighborhoods in bacterial and archaeal genomes. These gene neighborhoods are not typically present, in their entirety, in any single genome, but are held together by overlapping, partially conserved gene arrays. The procedure was applied to comparing the orders of orthologous genes, which were extracted from the database of Clusters of Orthologous Groups of proteins (COGs), in 31 prokaryotic genomes and resulted in the identification of 188 clusters of gene arrays, which included 1001 of 2890 COGs. These clusters were projected onto actual genomes to produce extended neighborhoods including additional genes, which are adjacent to the genes from the clusters and are transcribed in the same direction, which resulted in a total of 2387 COGs being included in the neighborhoods. Most of the neighborhoods consist predominantly of genes united by a coherent functional theme, but also include a minority of genes without an obvious functional connection to the main theme. We hypothesize that although some of the latter genes might have unsuspected roles, others are maintained within gene arrays because of the advantage of expression at a level that is typical of the given neighborhood. We designate this phenomenon 'genomic hitchhiking'. The largest neighborhood includes 79 genes (COGs) and consists of overlapping, rearranged ribosomal protein superoperons; apparent genome hitchhiking is particularly typical of this neighborhood and other neighborhoods that consist of genes coding for translation machinery components. Several neighborhoods involve previously undetected connections between genes, allowing new functional predictions. Gene neighborhoods appear to evolve via complex rearrangement, with different combinations of genes from a neighborhood fixed in different lineages.