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We determine distributions and correlation properties of offshore wind speeds and wind speed increments by analyzing wind data sampled with a resolution of one second for 20 months at different heights over the sea level in the North Sea. Distributions of horizontal wind speeds can be fitted to Weibull distributions with shape and scale parameters varying weakly with the vertical height separation. Kullback-Leibler divergences between distributions at different heights change with the squared logarithm of the height ratio. Cross-correlations between time derivates of wind speeds are long-term anticorrelated and their correlations functions satisfy sum rules. Distributions of horizontal wind speed increments change from a tent-like shape to a Gaussian with rising increment lag. A surprising peak occurs in the left tail of the increment distributions for lags in a range $10-200\,{\rm km}$ after applying the Taylor's hypothesis locally to transform time lags into distances. The peak is decisive in order to obtain an expected and observed linear scaling of third-order structure functions with distance. This suggests that it is an intrinsic feature of atmospheric turbulence., Comment: 20 pages, 14 figures

We introduce an epidemic spreading model on a network using concepts from percolation theory. The model is motivated by discussing the standard SIR model, with extensions to describe effects of lockdowns within a population. The underlying ideas and behavior of the lattice model, implemented using the same lockdown scheme as for the SIR scheme, are discussed in detail and illustrated with extensive simulations. A comparison between both models is presented for the case of COVID-19 data from the USA. Both fits to the empirical data are very good, but some differences emerge between the two approaches which indicate the usefulness of having an alternative approach to the widespread SIR model.

We review statistical properties of models generated by the application of a (positive and negative order) fractional derivative operator to a standard random walk and show that the resulting stochastic walks display slowly-decaying autocorrelation functions. The relation between these correlated walks and the well-known fractionally integrated autoregressive (FIGARCH) models, commonly used in econometric studies, is discussed. The application of correlated random walks to simulate empirical financial times series is considered and compared with the predictions from FIGARCH and the simpler FIARCH processes. A comparison with empirical data is performed., Comment: 10 pages, 14 figures

We review and further develop an analytical model that describes how thermodynamic constraints on the stability of the native state influence protein evolution in a site-specific manner. To this end, we represent both protein sequences and protein structures as vectors: Structures are represented by the principal eigenvector (PE) of the protein contact matrix, a quantity that resembles closely the effective connectivity of each site; Sequences are represented through the ``interactivity'' of each amino acid type, using novel parameters that are correlated with hydropathy scales. These interactivity parameters are more strongly correlated than the other hydropathy scales that we examine with: (1) The change upon mutations of the unfolding free energy of proteins with two-states thermodynamics; (2) Genomic properties as the genome-size and the genome-wide GC content; (3) The main eigenvectors of the substitution matrices. The evolutionary average of the interactivity vector correlates very strongly with the PE of a protein structure. Using this result, we derive an analytic expression for site-specific distributions of amino acids across protein families in the form of Boltzmann distributions whose ``inverse temperature'' is a function of the PE component. We show that our predictions are in agreement with site-specific amino acid distributions obtained from the Protein Data Bank, and we determine the mutational model that best fits the observed site-specific amino acid distributions. Interestingly, the optimal model almost minimizes the rate at which deleterious mutations are eliminated by natural selection., Comment: 11 pages, 5 figures

We derive an analytic expression for site-specific stationary distributions of amino acids from the Structurally Constrained Neutral (SCN) model of protein evolution with conservation of folding stability. The stationary distributions that we obtain have a Boltzmann-like shape, and their effective temperature parameter, measuring the limit of divergent evolutionary changes at a given site, can be predicted from a site-specific topological property, the principal eigenvector of the contact matrix of the native conformation of the protein. These analytic results, obtained without free parameters, are compared with simulations of the SCN model and with the site-specific amino acid distributions obtained from the Protein Data Bank. These results also provide new insights into how the topology of a protein fold influences its designability, i.e. the number of sequences compatible with that fold. The dependence of the effective temperature on the principal eigenvector decreases for longer proteins, a possible consequence of the fact that selection for thermodynamic stability becomes weaker in this case., Comment: 9 pages, 4 figures, references added

With the aim to study the relationship between protein sequences and their native structures, we adopt vectorial representations for both sequence and structure. The structural representation is based on the Principal Eigenvector of the fold's contact matrix (PE). As recently shown, the latter encodes sufficient information for reconstructing the whole contact matrix. The sequence is represented through a Hydrophobicity Profile (HP), using a generalized hydrophobicity scale that we obtain from the principal eigenvector of a residue-residue interaction matrix and denote it as interactivity scale. Using this novel scale, we define the optimal HP of a protein fold, and predict, by means of stability arguments, that it is strongly correlated with the PE of the fold's contact matrix. This prediction is confirmed through an evolutionary analysis, which shows that the PE correlates with the HP of each individual sequence adopting the same fold and, even more strongly, with the average HP of this set of sequences. Thus, protein sequences evolve in such a way that their average HP is close to the optimal one, implying that neutral evolution can be viewed as a kind of motion in sequence space around the optimal HP. Our results indicate that the correlation coefficient between N-dimensional vectors constitutes a natural metric in the vectorial space in which we represent both protein sequences and protein structures, which we call Vectorial Protein Space. In this way, we define a unified framework for sequence to sequence, sequence to structure, and structure to structure alignments. We show that the interactivity scale is nearly optimal both for the comparison of sequences with sequences and sequences with structures., Comment: 13 pages, 3 figures, 2 tables

We show that the contact map of the native structure of globular proteins can be reconstructed starting from the sole knowledge of the contact map's principal eigenvector, and present an exact algorithm for this purpose. Our algorithm yields a unique contact map for all 221 globular structures of PDBselect25 of length $N \le 120$. We also show that the reconstructed contact maps allow in turn for the accurate reconstruction of the three-dimensional structure. These results indicate that the reduced vectorial representation provided by the principal eigenvector of the contact map is equivalent to the protein structure itself. This representation is expected to provide a useful tool in bioinformatics algorithms for protein structure comparison and alignment, as well as a promising intermediate step towards protein structure prediction., Comment: 4 pages, 1 figure

We simulate neutral evolution of proteins imposing conservation of the thermodynamic stability of the native state in the framework of an effective model of folding thermodynamics. This procedure generates evolutionary trajectories in sequence space which share two universal features for all of the examined proteins. First, the number of neutral mutations fluctuates broadly from one sequence to another, leading to a non-Poissonian substitution process. Second, the number of neutral mutations displays strong correlations along the trajectory, thus causing the breakdown of self-averaging of the resulting evolutionary substitution process., Comment: 4 pages, 2 figures

Neutral evolution is the simplest model of molecular evolution and thus it is most amenable to a comprehensive theoretical investigation. In this paper, we characterize the statistical properties of neutral evolution of proteins under the requirement that the native state remains thermodynamically stable, and compare them to the ones of Kimura's model of neutral evolution. Our study is based on the Structurally Constrained Neutral (SCN) model which we recently proposed. We show that, in the SCN model, the substitution rate decreases as longer time intervals are considered, and fluctuates strongly from one branch of the evolutionary tree to another, leading to a non-Poissonian statistics for the substitution process. Such strong fluctuations are also due to the fact that neutral substitution rates for individual residues are strongly correlated for most residue pairs. Interestingly, structurally conserved residues, characterized by a much below average substitution rate, are also much less correlated to other residues and evolve in a much more regular way. Our results could improve methods aimed at distinguishing between neutral and adaptive substitutions as well as methods for computing the expected number of substitutions occurred since the divergence of two protein sequences., Comment: 17 pages, 11 figures

Protein structures are much more conserved than sequences during evolution. Based on this observation, we investigate the consequences of structural conservation on protein evolution. We study seven of the most studied protein folds, finding out that an extended neutral network in sequence space is associated to each of them. Within our model, neutral evolution leads to a non-Poissonian substitution process, due to the broad distribution of connectivities in neutral networks. The observation that the substitution process has non-Poissonian statistics has been used against the original Kimura's neutral theory, while our model shows that this is a generic property of neutral evolution with structural conservation. Our model also predicts that the substitution rate can strongly fluctuate from one branch to another of the evolutionary tree. The average sequence similarity within a neutral network is close to the threshold of randomness, as observed for families of sequences sharing the same fold. Nevertheless, some positions are more difficult to mutate than others. We compare such structurally conserved positions to positions conserved in protein evolution, suggesting that our model can be a valuable tool to distinguish structural from functional conservation in databases of protein families. These results indicate that a synergy between database analysis and structurally-based computational studies can increase our understanding of protein evolution., Comment: 15 pages, 8 figures, revised

We study the asymptotic shape of self-avoiding random walks (SAW) on the backbone of the incipient percolation cluster in d-dimensional lattices analytically. It is generally accepted that the configurational averaged probability distribution function for the end-to-end distance r of an N step SAW behaves as a power law for r -> 0. In this work, we determine the corresponding exponent using scaling arguments, and show that our suggested `generalized des Cloizeaux' expression for the exponent is in excellent agreement with exact enumeration results in two and three dimensions., Comment: 7 pages, 1 postscript figure

We simulate the evolution of a protein-like sequence subject to point mutations, imposing conservation of the ground state, thermodynamic stability and fast folding. Our model is aimed at describing neutral evolution of natural proteins. We use a cubic lattice model of the protein structure and test the neutrality conditions by extensive Monte Carlo simulations. We observe that sequence space is traversed by neutral networks, i.e. sets of sequences with the same fold connected by point mutations. Typical pairs of sequences on a neutral network are nearly as different as randomly chosen sequences. The fraction of neutral neighbors has strong sequence to sequence variations, which influence the rate of neutral evolution. In this paper we study the thermodynamic stability of different protein sequences. We relate the high variability of the fraction of neutral mutations to the complex energy landscape within a neutral network, arguing that valleys in this landscape are associated to high values of the neutral mutation rate. We find that when a point mutation produces a sequence with a new ground state, this is likely to have a low stability. Thus we tentatively conjecture that neutral networks of different structures are typically well separated in sequence space. This results indicates that changing significantly a protein structure through a biologically acceptable chain of point mutations is a rare, although possible, event., Comment: added reference, to appear on European Physical Journal B

We consider the mean distribution functions Phi(r|l), Phi(B)(r|l), and Phi(S)(r|l), giving the probability that two sites on the incipient percolation cluster, on its backbone and on its skeleton, respectively, connected by a shortest path of length l are separated by an Euclidean distance r. Following a scaling argument due to de Gennes for self-avoiding walks, we derive analytical expressions for the exponents g1=df+dmin-d and g1B=g1S-3dmin-d, which determine the scaling behavior of the distribution functions in the limit x=r/l^(nu) much less than 1, i.e., Phi(r|l) proportional to l^(-(nu)d)x^(g1), Phi(B)(r|l) proportional to l^(-(nu)d)x^(g1B), and Phi(S)(r|l) proportional to l^(-(nu)d)x^(g1S), with nu=1/dmin, where df and dmin are the fractal dimensions of the percolation cluster and the shortest path, respectively. The theoretical predictions for g1, g1B, and g1S are in very good agreement with our numerical results., Comment: 10 pages, 3 figures

We simulate the evolution of model protein sequences subject to mutations. A mutation is considered neutral if it conserves 1) the structure of the ground state, 2) its thermodynamic stability and 3) its kinetic accessibility. All other mutations are considered lethal and are rejected. We adopt a lattice model, amenable to a reliable solution of the protein folding problem. We prove the existence of extended neutral networks in sequence space -- sequences can evolve until their similarity with the starting point is almost the same as for random sequences. Furthermore, we find that the rate of neutral mutations has a broad distribution in sequence space. Due to this fact, the substitution process is overdispersed (the ratio between variance and mean is larger than one). This result is in contrast with the simplest model of neutral evolution, which assumes a Poisson process for substitutions, and in qualitative agreement with biological data., Comment: 17 pages, 8 figures, submitted to Journal of Theoretical Biology

The probability distribution of random walks on linear structures generated by random walks in $d$-dimensional space, $P_d(r,t)$, is analytically studied for the case $\xi\equiv r/t^{1/4}\ll1$. It is shown to obey the scaling form $P_d(r,t)=\rho(r) t^{-1/2} \xi^{-2} f_d(\xi)$, where $\rho(r)\sim r^{2-d}$ is the density of the chain. Expanding $f_d(\xi)$ in powers of $\xi$, we find that there exists an infinite hierarchy of critical dimensions, $d_c=2,6,10,\ldots$, each one characterized by a logarithmic correction in $f_d(\xi)$. Namely, for $d=2$, $f_2(\xi)\simeq a_2\xi^2\ln\xi+b_2\xi^2$; for $3\le d\le 5$, $f_d(\xi)\simeq a_d\xi^2+b_d\xi^d$; for $d=6$, $f_6(\xi)\simeq a_6\xi^2+b_6\xi^6\ln\xi$; for $7\le d\le 9$, $f_d(\xi)\simeq a_d\xi^2+b_d\xi^6+c_d\xi^d$; for $d=10$, $f_{10}(\xi)\simeq a_{10}\xi^2+b_{10}\xi^6+c_{10}\xi^{10}\ln\xi$, {\it etc.\/} In particular, for $d=2$, this implies that the temporal dependence of the probability density of being close to the origin $Q_2(r,t)\equiv P_2(r,t)/\rho(r)\simeq t^{-1/2}\ln t$., Comment: LATeX, 10 pages, no figures submitted for publication in PRE

A simple lattice model describing the recombination dynamics in visible light emitting porous Silicon is presented. In the model, each occupied lattice site represents a Si crystal of nanometer size. The disordered structure of porous Silicon is modeled by modified random percolation networks in two and three dimensions. Both correlated (excitons) and uncorrelated electron-hole pairs have been studied. Radiative and non-radiative processes as well as hopping between nearest neighbor occupied sites are taken into account. By means of extensive Monte-Carlo simulations, we show that the recombination dynamics in porous Silicon is due to a dispersive diffusion of excitons in a disordered arrangement of interconnected Si quantum dots. The simulated luminescence decay for the excitons shows a stretched exponential lineshape while for uncorrelated electron-hole pairs a power law decay is suggested. Our results successfully account for the recombination dynamics recently observed in the experiments. The present model is a prototype for a larger class of models describing diffusion of particles in a complex disordered system., Comment: 33 pages, RevTeX, 19 figures available on request to pavesi@science.unitn.it