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Transitions at CpG dinucleotides, referred to as "CpG substitutions", are a major mutational input into vertebrate genomes and a leading cause of human genetic disease. The prevalence of CpG substitutions is due to their mutational origin, which is dependent on DNA methylation. In comparison, other single nucleotide substitutions (for example those occurring at GpC dinucleotides) mainly arise from errors during DNA replication. Here we analyzed high quality BAC-based data from human, chimpanzee, and baboon to investigate regional variation of CpG substitution rates. We show that CpG substitutions occur approximately 15 times more frequently than other single nucleotide substitutions in primate genomes, and that they exhibit substantial regional variation. Patterns of CpG rate variation are consistent with differences in methylation level and susceptibility to subsequent deamination. In particular, we propose a "distance-decaying" hypothesis, positing that due to the molecular mechanism of a CpG substitution, rates are correlated with the stability of double-stranded DNA surrounding each CpG dinucleotide, and the effect of local DNA stability may decrease with distance from the CpG dinucleotide.Consistent with our "distance-decaying" hypothesis, rates of CpG substitution are strongly (negatively) correlated with regional G+C content. The influence of G+C content decays as the distance from the target CpG site increases. We estimate that the influence of local G+C content extends up to 1,500 approximately 2,000 bps centered on each CpG site. We also show that the distance-decaying relationship persisted when we controlled for the effect of long-range homogeneity of nucleotide composition. GpC sites, in contrast, do not exhibit such "distance-decaying" relationship. Our results highlight an example of the distinctive properties of methylation-dependent substitutions versus substitutions mostly arising from errors during DNA replication. Furthermore, the negative relationship between G+C content and CpG rates may provide an explanation for the observation that GC-rich SINEs show lower CpG rates than other repetitive elements.

We address phylogenetic reconstruction when the data is generated from a mixture distribution. Such topics have gained considerable attention in the biological community with the clear evidence of heterogeneity of mutation rates. In our work we consider data coming from a mixture of trees which share a common topology, but differ in their edge weights (i.e., branch lengths). We first show the pitfalls of popular methods, including maximum likelihood and Markov chain Monte Carlo algorithms. We then determine in which evolutionary models, reconstructing the tree topology, under a mixture distribution, is (im)possible. We prove that every model whose transition matrices can be parameterized by an open set of multilinear polynomials, either has non-identifiable mixture distributions, in which case reconstruction is impossible in general, or there exist linear tests which identify the topology. This duality theorem, relies on our notion of linear tests and uses ideas from convex programming duality. Linear tests are closely related to linear invariants, which were first introduced by Lake, and are natural from an algebraic geometry perspective.

Different genes often have different phylogenetic histories. Even within regions having the same phylogenetic history, the mutation rates often vary. We investigate the prospects of phylogenetic reconstruction when all the characters are generated from the same tree topology, but the branch lengths vary (with possibly different tree shapes). Furthering work of Kolaczkowski and Thornton (2004, Nature 431: 980-984) and Chang (1996, Math. Biosci. 134: 189-216), we show examples where maximum likelihood (under a homogeneous model) is an inconsistent estimator of the tree. We then explore the prospects of phylogenetic inference under a heterogeneous model. In some models, there are examples where phylogenetic inference under any method is impossible - despite the fact that there is a common tree topology. In particular, there are nonidentifiable mixture distributions, i.e., multiple topologies generate identical mixture distributions. We address which evolutionary models have nonidentifiable mixture distributions and prove that the following duality theorem holds for most DNA substitution models. The model has either: (i) nonidentifiability - two different tree topologies can produce identical mixture distributions, and hence distinguishing between the two topologies is impossible; or (ii) linear tests - there exist linear tests which identify the common tree topology for character data generated by a mixture distribution. The theorem holds for models whose transition matrices can be parameterized by open sets, which includes most of the popular models, such as Tamura-Nei and Kimura's 2-parameter model. The duality theorem relies on our notion of linear tests, which are related to Lake's linear invariants.

Using data from primates, we show that molecular clocks in sites that have been part of a CpG dinucleotide in recent past (CpG sites) and non-CpG sites are of markedly different nature, reflecting differences in their molecular origins. Notably, single nucleotide substitutions at non-CpG sites show clear generation-time dependency, indicating that most of these substitutions occur by errors during DNA replication. On the other hand, substitutions at CpG sites occur relatively constantly over time, as expected from their primary origin due to methylation. Therefore, molecular clocks are heterogeneous even within a genome. Furthermore, we propose that varying frequencies of CpG dinucleotides in different genomic regions may have contributed significantly to conflicting earlier results on rate constancy of mammalian molecular clock. Our conclusion that different regions of genomes follow different molecular clocks should be considered when inferring divergence times using molecular data and in phylogenetic analysis., Competing Interests: Competing interests. The authors have declared that no competing interests exist.

Markov chain Monte Carlo (MCMC) algorithms play a critical role in the Bayesian approach to phylogenetic inference. We present a theoretical analysis of the rate of convergence of many of the widely used Markov chains. For N characters generated from a uniform mixture of two trees, we prove that the Markov chains take an exponentially long (in N) number of iterations to converge to the posterior distribution. Nevertheless, the likelihood plots for sample runs of the Markov chains deceivingly suggest that the chains converge rapidly to a unique tree. Our results rely on novel mathematical understanding of the log-likelihood function on the space of phylogenetic trees. The practical implications of our work are that Bayesian MCMC methods can be misleading when the data are generated from a mixture of trees. Thus, in cases of data containing potentially conflicting phylogenetic signals, phylogenetic reconstruction should be performed separately on each signal.