It is shown that the density of modes of the vibrational spectrum of globular proteins is universal, i.e., regardless of the protein in question it closely follows one universal curve. The present study, including 135 proteins analyzed with a full at
omic empirical potential (CHARMM22) and using the full complement of all atoms Cartesian degrees of freedom, goes far beyond previous claims of universality, confirming that universality holds even in the high-frequency range (300- 4000 1/cm), where peaks and turns in the density of states are faithfully reproduced from one protein to the next. We also characterize fluctuations of the spectral density from the average, paving the way to a meaningful discussion of rare, unusual spectra and the structural reasons for the deviations in such outlier proteins. Since the method used for the derivation of the vibrational modes (potential energy formulation, set of degrees of freedom employed, etc.) has a dramatic effect on the spectral density, another significant implication of our findings is that the universality can provide an exquisite tool for assessing and improving the quality of various models used for NMA computations. Finally, we show that the input configuration too affects the density of modes, thus emphasizing the importance of simplified potential energy formulations that are minimized at the outset.
We introduce a formulation for normal mode analyses of globular proteins that significantly improves on an earlier, 1-parameter formulation (M. Tirion, PRL 77, 1905 (1996)) that characterized the slow modes associated with protein data bank structure
s. Here we develop that empirical potential function which is minimized at the outset to include two features essential to reproduce the eigenspectra and associated density of states over all frequencies, not merely the slow ones. First, introduction of preferred dihedral-angle configurations via use of torsional stiffness constants eliminates anomalous dispersion characteristics due to insufficiently bound surface sidechains. Second, we take into account the atomic identities and the distance of separation of all pairwise interactions. With these modifications we obtain stable, reliable eigenmodes over a wide range of frequencies.
We study the Krapivsky-Redner (KR) network growth model but where new nodes can connect to any number of existing nodes, $m$, picked from a power-law distribution $p(m)sim m^{-alpha}$. Each of the $m$ new connections is still carried out as in the KR
model with probability redirection $r$ (corresponding to degree exponent $gamma_{rm KR}=1+1/r$, in the original KR model). The possibility to connect to any number of nodes resembles a more realistic type of growth in several settings, such as social networks, routers networks, and networks of citations. Here we focus on the in-, out-, and total-degree distributions and on the potential tension between the degree exponent $alpha$, characterizing new connections (outgoing links), and the degree exponent $gamma_{rm KR}(r)$ dictated by the redirection mechanism.
We study a one-dimensional particles system, in the overdamped limit, where nearest particles attract with a force inversely proportional to a power of their distance and coalesce upon encounter. The detailed shape of the distribution function for th
e gap between neighbouring particles serves to discriminate between different laws of attraction. We develop an exact Fokker-Planck approach for the infinite hierarchy of distribution functions for multiple adjacent gaps and solve it exactly, at the mean-field level, where correlations are ignored. The crucial role of correlations and their effect on the gap distribution function is explored both numerically and analytically. Finally, we analyse a random input of particles, which results in a stationary state where the effect of correlations is largely diminished.
We study Kleinberg navigation (the search of a target in a d-dimensional lattice, where each site is connected to one other random site at distance r, with probability proportional to r^{-a}) by means of an exact master equation for the process. We s
how that the asymptotic scaling behavior for the delivery time T to a target at distance L scales as (ln L)^2 when a=d, and otherwise as L^x, with x=(d-a)/(d+1-a) for a<d, x=a-d for d<a<d+1, and x=1 for a>d+1. These values of x exceed the rigorous lower-bounds established by Kleinberg. We also address the situation where there is a finite probability for the message to get lost along its way and find short delivery times (conditioned upon arrival) for a wide range of as.
We study a new class of networks, generated by sequences of letters taken from a finite alphabet consisting of $m$ letters (corresponding to $m$ types of nodes) and a fixed set of connectivity rules. Recently, it was shown how a binary alphabet might
generate threshold nets in a similar fashion [Hagberg et al., Phys. Rev. E 74, 056116 (2006)]. Just like threshold nets, sequence nets in general possess a modular structure reminiscent of everyday life nets, and are easy to handle analytically (i.e., calculate degree distribution, shortest paths, betweenness centrality, etc.). Exploiting symmetry, we make a full classification of two- and three-letter sequence nets, discovering two new classes of two-letter sequence nets. The new sequence nets retain many of the desirable analytical properties of threshold nets while yielding richer possibilities for the modeling of everyday life complex networks more faithfully.
