Bounding the convergence time of local probabilistic evolution

International audienceIsoperimetric inequalities form a very intuitive yet powerful characterization of the connectedness of a state space, that has proven successful in obtaining convergence bounds. Since the seventies they form an essential tool in differential geometry [1, 2], graph theory [4, 3] and Markov chain analysis [8, 5, 6]. In this paper we use isoperimetric inequalities to construct a bound on the convergence time of any local probabilistic evolution that leaves its limit distribution invariant. We illustrate how this general result leads to new bounds on convergence times beyond the explicit Markovian setting, among others on quantum dynamics. This paper is concerned with the discrete-time spreading of a distribution along the edges of a graph. In essence we establish that even by exploiting global information about the graph and allowing a very general use of this information, this spreading can still not be accelerated beyond the conductance bound. Before providing more ample context, we start with a motivating example ascribed to Eugenio Calabi, but which came to our attention through the seminal 1969 paper by Jeff Cheeger [1]. Whereas the original example concerns differential geometry, we will apply it to a graph setting. W V\W Fig. 1. (in black) Dumbbell graph K n-K n for n = 6. (in blue) superimposed cycle of length 4n in a construction towards faster mixing. Consider a locality structure (discrete geometry) prescribed by the " dumbbell " graph family K n − K n shown in Figure 1, consisting of two complete graphs over n nodes, connected by a single edge. The diameter of this graph is three, but a random walk over this graph converging to the uniform distribution has an expected convergence time in O(n 2). This convergence time can be improved with a " global design " but without violating locality of the evolution, by adding some memory to the walker. In Figure 1 (in blue), the system designer has superimposed a cycle over the dumbbell graph. By adding subnodes that allow to conditionally select different subflows through the graph (formally we " lift " the walk [9]), the walker can be restricted to walk along this cycle. Using this cycle, we can impose a strategy by Diaconis, Holmes and Neal [18, 9] to efficiently speed-up mixing over this cycle: let the walker cycle in the same direction with a probability 1 − 1/n, and switch direction with probability 1/n. This way the walk will mix over the graph in O(n), i.e. quadratically faster than the original random walk. But this is still order n times slower than the diameter. Nevertheless, we show in our paper that this improvement is the best possible for any local probabilistic process that leaves the target distribution invariant. So mixing in diameter time may be possible, but not without loosening any of these constraints. 1. Problem description and main result: Consider a graph G with nodes V and edges E ⊆ V × V. We use the convention that (i, i) ∈ E ∀i ∈ V. We define " states " X as probability distributions over V. Given an initial state X 0 , some system " → " propagates it over t time steps as X

RNA-Seq-based analysis of the physiologic cold shock-induced changes in Moraxella catarrhalis gene expression

BACKGROUND: Moraxella catarrhalis, a major nasopharyngeal pathogen of the human respiratory tract, is exposed to rapid downshifts of environmental temperature when humans breathe cold air. The prevalence of pharyngeal colonization and respiratory tract infections caused by M. catarrhalis is greatest in winter. We investigated how M. catarrhalis uses the physiologic exposure to cold air to regulate pivotal survival systems that may contribute to M. catarrhalis virulence. RESULTS: In this study we used the RNA-seq techniques to quantitatively catalogue the transcriptome of M. catarrhalis exposed to a 26 °C cold shock or to continuous growth at 37 °C. Validation of RNA-seq data using quantitative RT-PCR analysis demonstrated the RNA-seq results to be highly reliable. We observed that a 26 °C cold shock induces the expression of genes that in other bacteria have been related to virulence a strong induction was observed for genes involved in high affinity phosphate transport and iron acquisition, indicating that M. catarrhalis makes a better use of both phosphate and iron resources after exposure to cold shock. We detected the induction of genes involved in nitrogen metabolism, as well as several outer membrane proteins, including ompA, m35-like porin and multidrug efflux pump (acrAB) indicating that M. catarrhalis remodels its membrane components in response to downshift of temperature. Furthermore, we demonstrate that a 26 °C cold shock enhances the induction of genes encoding the type IV pili that are essential for natural transformation, and increases the genetic competence of M. catarrhalis, which may facilitate the rapid spread and acquisition of novel virulence-associated genes. CONCLUSION: Cold shock at a physiologically relevant temperature of 26 °C induces in M. catarrhalis a complex of adaptive mechanisms that could convey novel pathogenic functions and may contribute to enhanced colonization and virulence