On the distribution of interspecies correlation for Markov models of character evolution on Yule trees

Probability Distribution of Tree Age for the Simple Birth-Death Process, with Applications to Distributions of Number of Ancestral Lineages and Divergence Times for Pairs of Taxa in a Yule Tree

In this contribution, a general expression is derived for the probability density of the time to the most recent common ancestor (TMRCA) of a simple birth-death tree, a widely used stochastic null-model of biological speciation and extinction, conditioned on the constant birth and death rates and number of extant lineages. This density is contrasted with a previous result which was obtained using a uniform prior for the time of origin. The new distribution is applied to two problems of phylogenetic interest. First, that of the probability of the number of taxa existing at any time in the past in a tree of a known number of extant species, and given birth and death rates, and second, that of determining the TMRCA of two randomly selected taxa in an unobserved tree that is produced by a simple birth-only, or Yule, process. In the latter case, it is assumed that only the rate of bifurcation (speciation) and the size, or number of tips, are known. This is shown to lead to a closed-form analytical expression for the probability distribution of this parameter, which is arrived at based on the known mathematical form of the age distribution of Yule trees of a given size and branching rate, which is derived here de novo, and a similar distribution which additionally is conditioned on tree age. The new distribution is the exact Yule prior for divergence times of pairs of taxa under the stated conditions and is potentially useful in statistical (Bayesian) inference studies of phylogenies.

Phylogenetic effective sample size

In this paper I address the question-how large is a phylogenetic sample? I propose a definition of a phylogenetic effective sample size for Brownian motion and Ornstein-Uhlenbeck processes-the regression effective sample size. I discuss how mutual information can be used to define an effective sample size in the non-normal process case and compare these two definitions to an already present concept of effective sample size (the mean effective sample size). Through a simulation study I find that the AICc is robust if one corrects for the number of species or effective number of species. Lastly I discuss how the concept of the phylogenetic effective sample size can be useful for biodiversity quantification, identification of interesting clades and deciding on the importance of phylogenetic correlations.

Phylogenetic confidence intervals for the optimal trait value

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

Phylogenetic confidence intervals for the optimal trait value

We consider a stochastic evolutionary model for a phenotype developing amongst n related species with unknown phylogeny. The unknown tree is modelled by a Yule process conditioned on n contemporary nodes. The trait value is assumed to evolve along lineages as an Ornstein-Uhlenbeck process. As a result, the trait values of the n species form a sample with dependent observations. We establish three limit theorems for the sample mean corresponding to three domains for the adaptation rate. In the case of fast adaptation, we show that for large n the normalized sample mean is approximately normally distributed. Using these limit theorems, we develop novel confidence interval formulae for the optimal trait value.

Inferring Sequential Order of Somatic Mutations during Tumorgenesis based on Markov Chain Model

Tumors are developed and worsen with the accumulated mutations on DNA sequences during tumorigenesis. Identifying the temporal order of gene mutations in cancer initiation and development is a challenging topic. It not only provides a new insight into the study of tumorigenesis at the level of genome sequences but also is an effective tool for early diagnosis of tumors and preventive medicine. In this paper, we develop a novel method to accurately estimate the sequential order of gene mutations during tumorigenesis from genome sequencing data based on Markov chain model as TOMC (Temporal Order based on Markov Chain), and also provide a new criterion to further infer the order of samples or patients, which can characterize the severity or stage of the disease. We applied our method to the analysis of tumors based on several high-throughput datasets. Specifically, first, we revealed that tumor suppressor genes (TSG) tend to be mutated ahead of oncogenes, which are considered as important events for key functional loss and gain during tumorigenesis. Second, the comparisons of various methods demonstrated that our approach has clear advantages over the existing methods due to the consideration on the effect of mutation dependence among genes, such as co-mutation. Third and most important, our method is able to deduce the ordinal sequence of patients or samples to quantitatively characterize their severity of tumors. Therefore, our work provides a new way to quantitatively understand the development and progression of tumorigenesis based on high throughput sequencing data.