@article{848, abstract = {The nature of factors governing the tempo and mode of protein evolution is a fundamental issue in evolutionary biology. Specifically, whether or not interactions between different sites, or epistasis, are important in directing the course of evolution became one of the central questions. Several recent reports have scrutinized patterns of long-term protein evolution claiming them to be compatible only with an epistatic fitness landscape. However, these claims have not yet been substantiated with a formal model of protein evolution. Here, we formulate a simple covarion-like model of protein evolution focusing on the rate at which the fitness impact of amino acids at a site changes with time. We then apply the model to the data on convergent and divergent protein evolution to test whether or not the incorporation of epistatic interactions is necessary to explain the data. We find that convergent evolution cannot be explained without the incorporation of epistasis and the rate at which an amino acid state switches from being acceptable at a site to being deleterious is faster than the rate of amino acid substitution. Specifically, for proteins that have persisted in modern prokaryotic organisms since the last universal common ancestor for one amino acid substitution approximately ten amino acid states switch from being accessible to being deleterious, or vice versa. Thus, molecular evolution can only be perceived in the context of rapid turnover of which amino acids are available for evolution.}, author = {Usmanova, Dinara and Ferretti, Luca and Povolotskaya, Inna and Vlasov, Peter and Kondrashov, Fyodor}, journal = {Molecular Biology and Evolution}, number = {2}, pages = {542 -- 554}, publisher = {Oxford University Press}, title = {{A model of substitution trajectories in sequence space and long-term protein evolution}}, doi = {10.1093/molbev/msu318}, volume = {32}, year = {2015}, } @article{8495, abstract = {In this note, we consider the dynamics associated to a perturbation of an integrable Hamiltonian system in action-angle coordinates in any number of degrees of freedom and we prove the following result of ``micro-diffusion'': under generic assumptions on $ h$ and $ f$, there exists an orbit of the system for which the drift of its action variables is at least of order $ \sqrt {\varepsilon }$, after a time of order $ \sqrt {\varepsilon }^{-1}$. The assumptions, which are essentially minimal, are that there exists a resonant point for $ h$ and that the corresponding averaged perturbation is non-constant. The conclusions, although very weak when compared to usual instability phenomena, are also essentially optimal within this setting.}, author = {Bounemoura, Abed and Kaloshin, Vadim}, issn = {0002-9939}, journal = {Proceedings of the American Mathematical Society}, number = {4}, pages = {1553--1560}, publisher = {American Mathematical Society}, title = {{A note on micro-instability for Hamiltonian systems close to integrable}}, doi = {10.1090/proc/12796}, volume = {144}, year = {2015}, } @article{8498, abstract = {In the present note we announce a proof of a strong form of Arnold diffusion for smooth convex Hamiltonian systems. Let ${\mathbb T}^2$ be a 2-dimensional torus and B2 be the unit ball around the origin in ${\mathbb R}^2$ . Fix ρ > 0. Our main result says that for a 'generic' time-periodic perturbation of an integrable system of two degrees of freedom $H_0(p)+\varepsilon H_1(\theta,p,t),\quad \ \theta\in {\mathbb T}^2,\ p\in B^2,\ t\in {\mathbb T}={\mathbb R}/{\mathbb Z}$ , with a strictly convex H0, there exists a ρ-dense orbit (θε, pε, t)(t) in ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ , namely, a ρ-neighborhood of the orbit contains ${\mathbb T}^2 \times B^2 \times {\mathbb T}$ . Our proof is a combination of geometric and variational methods. The fundamental elements of the construction are the usage of crumpled normally hyperbolic invariant cylinders from [9], flower and simple normally hyperbolic invariant manifolds from [36] as well as their kissing property at a strong double resonance. This allows us to build a 'connected' net of three-dimensional normally hyperbolic invariant manifolds. To construct diffusing orbits along this net we employ a version of the Mather variational method [41] equipped with weak KAM theory [28], proposed by Bernard in [7].}, author = {Kaloshin, Vadim and Zhang, K}, issn = {0951-7715}, journal = {Nonlinearity}, keywords = {Mathematical Physics, General Physics and Astronomy, Applied Mathematics, Statistical and Nonlinear Physics}, number = {8}, pages = {2699--2720}, publisher = {IOP Publishing}, title = {{Arnold diffusion for smooth convex systems of two and a half degrees of freedom}}, doi = {10.1088/0951-7715/28/8/2699}, volume = {28}, year = {2015}, } @article{8499, abstract = {We consider the cubic defocusing nonlinear Schrödinger equation in the two dimensional torus. Fix s>1. Recently Colliander, Keel, Staffilani, Tao and Takaoka proved the existence of solutions with s-Sobolev norm growing in time. We establish the existence of solutions with polynomial time estimates. More exactly, there is c>0 such that for any K≫1 we find a solution u and a time T such that ∥u(T)∥Hs≥K∥u(0)∥Hs. Moreover, the time T satisfies the polynomial bound 0