Modularity Enhances the Rate of Evolution in a Rugged Fitness Landscape

Biological systems are modular, and this modularity affects the evolution of biological systems over time and in different environments. We here develop a theory for the dynamics of evolution in a rugged, modular fitness landscape. We show analytically how horizontal gene transfer couples to the modularity in the system and leads to more rapid rates of evolution at short times. The model, in general, analytically demonstrates a selective pressure for the prevalence of modularity in biology. We use this model to show how the evolution of the influenza virus is affected by the modularity of the proteins that are recognized by the human immune system. Approximately 25\% of the observed rate of fitness increase of the virus could be ascribed to a modular viral landscape.Comment: 45 pages; 7 figure

Self-normalized Cram\'{e}r type moderate deviations for the maximum of sums

Let $X_1,X_2,...$ be independent random variables with zero means and finite variances, and let $S_n=\sum_{i=1}^nX_i$ and $V^2_n=\sum_{i=1}^nX^2_i$. A Cram\'{e}r type moderate deviation for the maximum of the self-normalized sums $\max_{1\leq k\leq n}S_k/V_n$ is obtained. In particular, for identically distributed $X_1,X_2,...,$ it is proved that $P(\max_{1\leq k\leq n}S_k\geq xV_n)/(1-\Phi (x))\rightarrow2$ uniformly for $0<x\leq\mathrm{o}(n^{1/6})$ under the optimal finite third moment of $X_1$.Comment: Published in at http://dx.doi.org/10.3150/12-BEJ415 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm