Non-reversible Metastable Diffusions with Gibbs Invariant Measure II: Markov Chain Convergence

This article considers a class of metastable non-reversible diffusion processes whose invariant measure is a Gibbs measure associated with a Morse potential. In a companion paper [25], we proved the Eyring-Kramers formula for the corresponding class of metastable diffusion processes. In this article, we further develop this result by proving that a suitably time-rescaled metastable diffusion process converges to a Markov chain on the deepest metastable valleys. This article is also an extension of [32], which considered the same problem for metastable reversible diffusion processes. Our proof is based on the recently developed partial differential equation (PDE) approach to metastability. To the best of our knowledge, this study is the first to propose a robust methodology for applying the PDE approach to non-reversible models.Comment: 43 pages, 4 figure

Metastability and time scales for parabolic equations with drift 1: the first time scale

Consider the elliptic operator given by $$\mathscr{L}_{\epsilon}f= {b} \cdot \nabla f + \epsilon \Delta f$$ for some smooth vector field $b\colon \mathbb R^d \to\mathbb R^d$ and a small parameter $\epsilon>0$. Consider the initial-valued problem $$\left\{ \begin{aligned} &\partial_ t u_\epsilon = \mathscr L_\epsilon u_\epsilon,\\ &u_\epsilon (0, \cdot) = u_0(\cdot), \end{aligned} \right.$$ for some bounded continuous function $u_0$. Denote by $\mathcal M_0$ the set of critical points of $b$, $\mathcal M_0 =\{x\in \mathbb R^d : b(x)=0\}$, assumed to be finite. Under the hypothesis that $b = -(\nabla U + \ell)$, where $\ell$ is a divergence-free field orthogonal to $\nabla U$, the main result of this article states that there exist a time-scale $\theta^{(1)}_\epsilon$, $\theta^{(1)}_\epsilon \to \infty$ as $\epsilon \rightarrow 0$, and a Markov semigroup $\{p_t : t\ge 0\}$ defined on $\mathcal M_0$ such that $$\lim_{\epsilon\to 0} u_\epsilon (t\theta^{(1)}_\epsilon, x) =\sum_{m'\in \mathcal M_0} p_t(m, m')\, u_0( m'),$$ for all $t>0$ and $x$ in the domain of attraction of $m$ for the ODE $\dot{x}(t)= b( x(t))$. The time scale $\theta^{(1)}$ is critical in the sense that, for all time scale $\varrho_\epsilon$ such that $\varrho_\epsilon \to \infty$, $\varrho_\epsilon/\theta^{(1)}_\epsilon \to 0$, $$\lim_{\epsilon\to 0} u_\epsilon (\varrho_\epsilon, x)=u_0(m)$$ for all $x \in \mathcal D(m)$. Namely, $\theta_\epsilon^{(1)}$ is the first scale at which the solution to the initial-valued problem starts to change. In a companion paper [Landim, Lee, Seo, forthcoming] we extend this result finding all critical time-scales at which the solution $u_\epsilon$ evolves smoothly in time and we show that the solution $u_\epsilon$ is expressed in terms of the semigroup of some Markov chain taking values in sets formed by unions of critical points of $b$.Comment: 60 pages, 3 figure

Asymptotic stability and cut-off phenomenon for the underdamped Langevin dynamics

In this article, we provide detailed analysis of the long-time behavior of the underdamped Langevin dynamics. We first provide a necessary condition guaranteeing that the zero-noise dynamical system converges to its unique attractor. We also observed that this condition is sharp for a large class of linear models. We then prove the so-called cut-off phenomenon in the small-noise regime under this condition. This result provides the precise asymptotics of the mixing time of the process and of the distance between the distribution of the process and its stationary measure. The main difficulty of this work relies on the degeneracy of its infinitesimal generator which is not elliptic, thus requiring a new set of methods

The Impact of Job Retention on Continuous Growth of Engineering and Informational Technology SMEs in South Korea

This study aims to explore what factors are critically associated with job retention in Engineering and Information Technology small- and medium-sized enterprises (SMEs) in South Korea, and how employers think about sta retention policy in relation to business growth. This contrasts with previous studies that mainly focus on employee motivation, job retention, and turnover. Qualitative semi-structured interviews were conducted face-to-face with founder Chief Executive O cers (CEOs). The results suggest that an important factor influencing job retention policies of these SMEs was to motivate employees to make greater voluntary e ort, including through developing a collaborative organizational culture, rather than solely o ering them additional financial rewards or using other Human Resource Management (HRM) practices to improve individual performances. Interviewees believed that job retention and business growth were closely related, and they discussed various ways of eliciting emotional commitment from employees. Unlike research on larger firms, these suggestions did not involve immediate financial rewards. How employers thought that the roles played by employees strongly influenced their firm’s productivity and competitiveness. This study suggests SME employers adjust their retention policy specifically to improve their firm’s survival and long-term growth

A high-accuracy consensus map of yeast protein complexes reveals modular nature of gene essentiality

<p>Abstract</p> <p>Background</p> <p>Identifying all protein complexes in an organism is a major goal of systems biology. In the past 18 months, the results of two genome-scale tandem affinity purification-mass spectrometry (TAP-MS) assays in yeast have been published, along with corresponding complex maps. For most complexes, the published data sets were surprisingly uncorrelated. It is therefore useful to consider the raw data from each study and generate an accurate complex map from a high-confidence data set that integrates the results of these and earlier assays.</p> <p>Results</p> <p>Using an unsupervised probabilistic scoring scheme, we assigned a confidence score to each interaction in the matrix-model interpretation of the large-scale yeast mass-spectrometry data sets. The scoring metric proved more accurate than the filtering schemes used in the original data sets. We then took a high-confidence subset of these interactions and derived a set of complexes using MCL. The complexes show high correlation with existing annotations. Hierarchical organization of some protein complexes is evident from inter-complex interactions.</p> <p>Conclusion</p> <p>We demonstrate that our scoring method can generate an integrated high-confidence subset of observed matrix-model interactions, which we subsequently used to derive an accurate map of yeast complexes. Our results indicate that essentiality is a product of the protein complex rather than the individual protein, and that we have achieved near saturation of the yeast high-abundance, rich-media-expressed "complex-ome."</p

Broad network-based predictability of Saccharomyces cerevisiae gene loss-of-function phenotypes

Loss-of-function phenotypes of yeast genes can be predicted from the loss-of-function phenotypes of their neighbours in functional gene networks. This could potentially be applied to the prediction of human disease genes

Circular Kinks on the Surface of Granular Material Rotated in a Tilted Spinning Bucket

We find that circular kinks form on the surface of granular material when the axis of rotation is tilted more than the angle of internal friction of the material. Radius of the kinks is measured as a function of the spinning speed and the tilting angle. Stability consideration of the surface results in an explanation that the kink is a boundary between the inner unstable and outer stable regions. A simple cellular automata model also displays kinks at the stability boundary

A single gene of a commensal microbe affects host susceptibility to enteric infection

Indigenous microbes inside the host intestine maintain a complex self-regulating community. The mechanisms by which gut microbes interact with intestinal pathogens remain largely unknown. Here we identify a commensal Escherichia coli strain whose expansion predisposes mice to infection by Vibrio cholerae, a human pathogen. We refer to this strain as 'atypical' E. coli (atEc) because of its inability to ferment lactose. The atEc strain is resistant to reactive oxygen species (ROS) and proliferates extensively in antibiotic-treated adult mice. V. cholerae infection is more severe in neonatal mice transplanted with atEc compared with those transplanted with a typical E. coli strain. Intestinal ROS levels are decreased in atEc-transplanted mice, favouring proliferation of ROS-sensitive V. cholerae. An atEc mutant defective in ROS degradation fails to facilitate V. cholerae infection when transplanted, suggesting that host infection susceptibility can be regulated by a single gene product of one particular commensal species.