Analysis of intragenomic variation of 16S rRNA genes is a unique approach to examining the concept of ribosomal constraints on rRNA genes; the degree of variation is an important parameter to consider for estimation of the diversity of a complex microbiome in the recently initiated Human Microbiome Project (http://nihroadmap.nih.gov/hmp). The current GenBank database has a collection of 883 prokaryotic genomes representing 568 unique species, of which 425 species contained 2 to 15 copies of 16S rRNA genes per genome (2.22 ؎ 0.81). Sequence diversity among the 16S rRNA genes in a genome was found in 235 species (from 0.06% to 20.38%; 0.55% ؎ 1.46%). Compared with the 16S rRNA-based threshold for operational definition of species (1 to 1.3% diversity), the diversity was borderline (between 1% and 1.3%) in 10 species and >1.3% in 14 species. The diversified 16S rRNA genes in Haloarcula marismortui (diversity, 5.63%) and Thermoanaerobacter tengcongensis (6.70%) were highly conserved at the 2°structure level, while the diversified gene in B. afzelii (20.38%) appears to be a pseudogene. The diversified genes in the remaining 21 species were also conserved, except for a truncated 16S rRNA gene in "Candidatus Protochlamydia amoebophila." Thus, this survey of intragenomic diversity of 16S rRNA genes provides strong evidence supporting the theory of ribosomal constraint. Taxonomic classification using the 16S rRNA-based operational threshold could misclassify a number of species into more than one species, leading to an overestimation of the diversity of a complex microbiome. This phenomenon is especially seen in 7 bacterial species associated with the human microbiome or diseases.

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We study semi-linear elliptic PDEs with polynomial non-linearity and provide a probabilistic representation of their solution using branching diffusion processes. When the non-linearity involves the unknown function but not its derivatives, we extend previous results in the literature by showing that our probabilistic representation provides a solution to the PDE without assuming its existence. In the general case, we derive a new representation of the solution by using marked branching diffusion processes and automatic differentiation formulas to account for the non-linear gradient term. In both cases, we develop new theoretical tools to provide explicit sufficient conditions under which our probabilistic representations hold. As an application, we consider several examples including multi-dimensional semi-linear elliptic PDEs and estimate their solution by using the Monte Carlo method.

We develop a new unbiased estimation method for Lipschitz continuous functions of multi-dimensional stochastic differential equations with Lipschitz continuous coefficients. This method provides a finite variance estimator based on a probabilistic representation which is similar to the recent representations obtained through the parametrix method and recursive application of the automatic differentiation formula. Our approach relies on appropriate change of variables to carefully handle the singular integrands appearing in the iterated integrals of the probabilistic representation. It results in a scheme with randomized intermediate times where the number of intermediate times has a Pareto distribution.