Limit profiles for reversible Markov chains

In a recent breakthrough, Teyssier (Ann Probab 48(5):2323–2343, 2020) introduced a new method for approximating the distance from equilibrium of a random walk on a group. He used it to study the limit profile for the random transpositions card shuffle. His techniques were restricted to conjugacy-invariant random walks on groups; we derive similar approximation lemmas for random walks on homogeneous spaces and for general reversible Markov chains. We illustrate applications of these lemmas to some famous problems: the k-cycle shuffle, sharpening results of Hough (Probab Theory Relat Fields 165(1–2):447–482, 2016) and Berestycki, Schramm and Zeitouni (Ann Probab 39(5):1815–1843, 2011), the Ehrenfest urn diffusion with many urns, sharpening results of Ceccherini-Silberstein, Scarabotti and Tolli (J Math Sci 141(2):1182–1229, 2007), a Gibbs sampler, which is a fundamental tool in statistical physics, with Binomial prior and hypergeometric posterior, sharpening results of Diaconis, Khare and Saloff-Coste (Stat Sci 23(2):151–178, 2008).

mariner-Based Transposon Mutagenesis of Rickettsia prowazekii

ABSTRACT Rickettsia prowazekii, the causative agent of epidemic typhus, is an obligate intracellular bacterium that grows directly within the cytoplasm of its host cell, unbounded by a vacuolar membrane. The obligate intracytoplasmic nature of rickettsial growth places severe restrictions on the genetic analysis of this distinctive human pathogen. In order to expand the repertoire of genetic tools available for the study of this pathogen, we have employed the versatile mariner-based, Himar1 transposon system to generate insertional mutants of R. prowazekii. A transposon containing the R. prowazekii arr-2 rifampin resistance gene and a gene coding for a green fluorescent protein (GFPUV) was constructed and placed on a plasmid expressing the Himar1 transposase. Electroporation of this plasmid into R. prowazekii resulted in numerous transpositions into the rickettsial genome. Transposon insertion sites were identified by rescue cloning, followed by DNA sequencing. Random transpositions integrating at TA sites in both gene coding and intergenic regions were identified. Individual rickettsial clones were isolated by the limiting-dilution technique. Using both fixed and live-cell techniques, R. prowazekii transformants expressing GFPUV were easily visible by fluorescence microscopy. Thus, a mariner-based system provides an additional mechanism for generating rickettsial mutants that can be screened using GFPUV fluorescence.