## Zr lrlifi

(r) from k to k L - r and define the It can be shown, that for a long enough sequence L of uncorrected values s k) (i.e., C(r) 0 for r > 1) with finite mean and variance C(0), we must have ifl(r) -> (r+ 3)C(0) 15. Thus the graph of Fo r) for such a sequence on a log-log plot is a straight line with slope the a 1 2 if plotted versus r + 3. Any deviation from the straight line behavior indicates the presence of correlations or anti-correlations. It can be also shown that for a sequence with...

## Chapter

Power Law Correlations in DNA Sequences A wide variety of natural phenomena is characterized by power law behavior of their parameters. This type of behavior is also called scaling. The first observation of scaling probably goes back to Kepler1 who empirically discovered that squares of the periods of planet revolution around the Sun scale as cubes of their orbits radii. This empirical law allowed Newton to discover his famous inverse-square law of gravity. In the nineteenth century, it was...

## Linear Stochastic BDIM and Its Applications

The probability of formation of a family of size n starting from a family of size i before getting to extinction can be computed with the help of known formulas for the birth-and-death process to reach state n before reaching state 0. For the linear 2nd order balanced BDIM, the probability that a singleton expands to a family of size n before dying, Pl ,ri), is where y 1 + h - a, with the same power J as the equilibrium frequencies of the families.37 The values of probabilities Pl ,n) for...

## Metabolic Networks and Planetary Atmospheres

While the above speculation makes a weak case for a selectionist explanation of broad-tailed degree distributions in metabolic networks, another line of evidence makes a more solid case against it. One can ask whether power-law degree distributions might not be features of many or all large chemical reaction networks, whether or not part of an organism, whether or not they have a biological function which benefits from a robust network diameter. If so, then metabolic network degree...

## Conclusion and Discussion

We have studied the structural properties of the yeast protein interaction networks and the transport phenomena along the shortest pathways on biocomplex networks from the graph theoretic viewpoint. Thanks to recent development of data collection and graph analysis methods, the structural properties of the yeast protein interaction networks have been unveiled rapidly. Here we analyzed the degree distribution, the degree-degree correlation, and the clustering coefficient of the yeast interaction...

## Ntl

Vertex degree distributions and fits. counts used for the fit. Thus, the network obtained from highly-normalized preys is described well by power-law decay. The interactions that contribute to the prey degree distribution are not all biologically relevant. Some may reflect assay-specific artifacts having to do with the two-hybrid reporter system other false-positive interactions may arise from weak or nonspecific interactions other interactions may be highly reproducible in vitro but...

## The Importance of Independent Functional Hierarchical Description

The simplistic divergent evolution model8 that explains the nonrandom behavior of the PDUG is based solely on the premise that a protein has an ancestor that is its closest structural homologue. This model fits the data observed on the PDUG. The model characterizes the oldest proteins as those having the largest number of descendants and consequently the number of descendants for each protein depends on the protein's evolutionary age. We can therefore argue from our divergent evolution model...

## Srp1

Network of physical interactions between nuclear proteins in yeast. Here we show the subset of protein-protein physical interactions reported in the full set of reference 8 consisting of 318 interactions between proteins that are known to be localized in the yeast nucleus.5 The resulting network involves 329 proteins. Note that most neighbors of highly connected proteins have rather low connectivity. This feature will be later quantified in the correlation profile of this network...

## Properties of the Protein Domain Universe Graphs

We computed the size of the largest cluster in PDUG and random control graph as a function of Zm n.8 We found a pronounced transition of the size of the largest cluster in PDUG at Zm a Zc 9. The random graphs feature a similar transition, but at a higher value of Zmm ZC 11. The distribution of cluster sizes depends significantly on whether Zmm > Zc or Zmm < Zc for both the PDUG and random graphs. We also found that the probability density P(M) of cluster sizes M for both the PDUG and random...

## Introduction

The tremendous progress in the natural sciences we witnessed in the last century was based on the reductionist approach, allowing us to predict the behavior of a system from the understanding of its (often identical) elementary constituents and their individual interactions. However, our ability to understand simple fundamental laws governing individual building blocks is a far cry from being able to predict the overall behavior of a complex system.5 Additionally, the building blocks of most...

## Scale Free Evolution

Shakhnovich Introduction One of the most intriguing problems in molecular biology is the origin of the vast population diversity of protein families.14 Following the assumption that the protein families are populated at random, one would expect a multinomial distribution of the family populations.5 However, it has been discovered69 that distribution of the family populations is by far nonexponential, but has a long tail, which signifies that some specific...

## Info

The upper and lower parts of the table show the phylogenetic distribution of 15 arbitrarily chosen high and low degree proteins from publicly available yeast protein interaction data.38 Gapped BLAST was used tosearchforhomologs to these yeast proteins in the GenBankdatabase(www.ncbi.nlm.nih.gov). Columns in the table correspond to the following broad taxonomic groups. Metazoa (M), Protists (Pr), Plants (P), Fungi (F, exclusive of the genus Saccharomyces), Eubacteria (E) and Archaea (Ar). A '+'...

## Topological Properties of Protein Networks Single Node Topological Properties

An interesting property of many biological networks that was recendy brought to attention of the scientific community13 is an extremely broad distribution of nodes' degrees (often called connectivities in the network literature) defined as the number of immediate neighbors of a given node in the network. While the majority of nodes have just a few edges connecting them to other nodes in the network, there exist some nodes, that we will refer to as hubs, with an unusually large number of...

## J

So we can see that the diagonal elements of this matrix, p(r) 1 2 + (2p - 1)72 exponentially converge to 1 2 for r The speed of convergence determines the correlation length exp(2 el kBT) +1 exp(2e kBT) -1 . (15) For T 0 the correlation length diverges exp(2e 7) while for T the correlation length approaches zero 4 1 ln( 77e) 0. For finite temperature the correlation length is finite. Hence for one-dimensional Ising model, there is no critical point at positive temperatures, however the absolute...

## Preface

The Genomic Revolution, Systems Biology, Power Laws, and Scale-Free Networks The decade between 1995 and 2004 witnessed an ongoing revolution in biology. Certainly, sequencing of the human genome1 serves as a legitimate symbol of this new revolution whereas its beginning is marked by the appearance of the first complete genome sequence of a cellular life form, the bacterium Haemophilus influenzae Indeed, comparative genomics is the core and foundation of the new biology. It brought about the...

## X

i > efifnonl4t FkA of CLC Lipi flfJ Figure 5. Flux distribution for the metabolism of E. coli. A) Flux distribution for optimized biomass production on succinate (black) and glutamate (red) rich uptake substrates. The solid line corresponds to the power law fit P(v) - (v + Vo)a with Vo 0.00003 and a 1.5. B) The distribution of experimentally determined fluxes (seerefH) from the central metabolism of coli also displays power-law behavior with a best fit to P(y) - Vawitha 1. A color version of...

## The Drosophila Protein Interaction Network May Be neither Power Law nor Scale Free

Scale-free networks have become a topic of intense interest because of the potential to develop theories universally applicable to networks representing social interactions, internet connectivity, and biological processes. Scale-free topology is associated with power-law distributions of connectivity, in which most network components have only few connections while a very few components are extremely highly-connected. Here we investigate the power-law and scale-free properties of the network...

## Discussion What It May All Mean

The large-scale organization of molecular networks deduced from correlation profiles of protein interaction and transcription regulatory networks in yeast is consistent with compart-mentalization and modularity characteristic of many cellular processes.22 Indeed, the suppression of connections between highly-connected proteins (hubs) suggests the picture of semi-independent modules centered around or regulated by individual hubs. On the other hand, the very fact that these molecular networks do...

## Birth and Death Models of Genome Evolution

Wolf and Eugene V. Koonin* Abstract Gene duplication is the primary avenue of genome evolution. The gene repertoire of any species can be described as an ensemble of paralogous gene families, ranging in size from one to large numbers that amount to a substantial fraction of genes in the respective genome. Evolution of such an ensemble is naturally represented by a birth-and-death process, the birth of a gene being duplication, and death being gene inactivation and...

## Power Laws in Network Topology

The complex network representation of different systems as networks has revealed surprising similarities, many of which are intimately tied to power laws. The simplest network measure is the average number of nearest neighbors of a node, or the average degree. However, this is a rather crude property, and to gain further insight into the topological organization of real networks, we need to determine the variation in the nearest neighbors, given by the degree distribution. For a surprisingly...