Random model for RNA interference yields scale free network

We introduce a random bit-string model of post-transcriptional genetic regulation based on sequence matching. The model spontaneously yields a scale free network with power law scaling with $\gamma=-1$ and also exhibits log-periodic behaviour. The in-degree distribution is much narrower, and exhibits a pronounced peak followed by a Gaussian distribution. The network is of the smallest world type, with the average minimum path length independent of the size of the network, as long as the network consists of one giant cluster. The percolation threshold depends on the system size.Comment: 9 pages, 13 figures, submitted to Midterm Conference COSIN on ``Growing Networks and Graphs in Statistical Physics, Finance, Biology and Social Systems'', Rome, 1-5 September 200

A Publish-Subscribe Model of Genetic Networks

We present a simple model of genetic regulatory networks in which regulatory connections among genes are mediated by a limited number of signaling molecules. Each gene in our model produces (publishes) a single gene product, which regulates the expression of other genes by binding to regulatory regions that correspond (subscribe) to that product. We explore the consequences of this publish-subscribe model of regulation for the properties of single networks and for the evolution of populations of networks. Degree distributions of randomly constructed networks, particularly multimodal in-degree distributions, which depend on the length of the regulatory sequences and the number of possible gene products, differed from simpler Boolean NK models. In simulated evolution of populations of networks, single mutations in regulatory or coding regions resulted in multiple changes in regulatory connections among genes, or alternatively in neutral change that had no effect on phenotype. This resulted in remarkable evolvability in both number and length of attractors, leading to evolved networks far beyond the expectation of these measures based on random distributions. Surprisingly, this rapid evolution was not accompanied by changes in degree distribution; degree distribution in the evolved networks was not substantially different from that of randomly generated networks. The publish-subscribe model also allows exogenous gene products to create an environment, which may be noisy or stable, in which dynamic behavior occurs. In simulations, networks were able to evolve moderate levels of both mutational and environmental robustness

İçerik-temelli ağlar üzerinde analitik hesaplar

Content-based networks have been proposed (Balcan and Erzan, 2004; Mungan et al., 2005) to model the topological properties of complex networks built on the principle of information sharing, where the interactions between system components assume the simultaneous fulfillment of a series of constraints (Mezard et al., 2002). In content-based networks, the constraint-satisfaction problem is realized by means of a sequence-matching rule between sequences associated with the nodes of a network. In the case of transcriptional gene regulation, the transcription factors recognize special subsequences of DNA and bind them. This is one instance of constraint-satisfaction, which can be realized with a sequence-matching rule between two different classes of sequences (Balcan et al., 2006). Another example is the so called the RNA interference (Balcan and Erzan, 2004), where sequence-specific gene silencing occurs at the level of post-transcriptional gene regulation. In our content-based networks, n linear codes are associated with each node of the network. For n=2, one of the sequences associated with the node represents the key-sequence through which the node recognizes other nodes, whereas the second sequence represents the lock-sequence through which the same node is recognized. An interaction between a pair of nodes is established if the key-sequence associated with the first node is repeated as an uninterrupted subsequence in the lock-sequence associated with the second node. Thus, the length distributions of these sequences are the most important parameters determining the topological properties of the content-based networks. In this article we will discuss the validity of analytical calculations performed on the topological properties of content-based networks in the mean-field approximation (Balcan and Erzan, 2007), by means of two examples. In this mean field approach (Mungan et al., 2005) the pair-wise connectivity probabilities are only functions of the respective lengths of the sequences which must satisfy an inclusion requirement, and of the size r of the alphabet from which the symbols are drawn. This approximation ignores the correlations between the overlapping subsequences within a sequence. Moreover the fluctuations in the information content of finite sequences are neglected. In Balcan and Erzan (2007), the correlations between the edges co-incident on the same node were also ignored. In the first example, the key- sequences of unit length (thus, they consist of single letters) are searched in lock-sequences of an arbitrary fixed length. Via this simple example it is possible to show that the probability that lock-sequences will be recognized by a key-sequence depends not only on the length of the lock-sequence but also on the number of distinct subsequences embedded in it. At this point the coarse grained approximation neglecting the fluctuations in the information content of the finite lock sequences about their mean information content, misses the behavior of the in-degree distribution. This error is in fact identical to neglecting the correlations between edges incident upon a given node. In the second example, the lengths of the key sequences are fixed at an arbitrary value l, and the lock-sequences are chosen to be of length k=l+1, one character longer than the key-sequences. In this example, it is clear that the correlations between the two subsequences of length l cannot be neglected. It has already been shown (Guibas and Odlyzko, 1981; Mungan et al., 2005; Mungan, 2007; Bilge et al., 2004) that the connection probability of a key-sequence depends on the ?shift-match number? which measures the auto-correlations within a sequence, in other words, the degree to which successive subsequences are correlated with each other. We show here by an explicit and rather transparent calculation that, neglecting this correlation yields out- and in-degree distributions that are totally in error. The mean-field approximations used in the calculation of the topological properties of the double-string model (Balcan and Erzan, 2007) yield results that are in good agreement with the simulations, since i) the lengths k of the lock sequences far exceed r, ii) the number of distinct substrings contained in any given lock string is large ( k-l >> rl ) and iii) the fine structure of the topological properties are determined by the fact that there is a disribution of lock- and key-string lengths. Keywords: Complex networks, content-based networks, mean-field approach.Bu makalede içerik-temelli ağlar üzerinde, ağın topolojik özelliklerini belirlemek için, ortalama-alan yaklaşımlarıyla yapılan analitik hesapların güvenilirliği tartışılacaktır. İçerik-temelli ağları, “tanıma ve bağlanma” mekanizmalarının belirleyici olduğu kontrol çizgelerinin topolojik özelliklerini tasvir etmek için önermiştik. Birçok karmaşık ağ yapısının bu tür enformasyon paylaşımına dayalı bir prensibe göre inşa edildiğini söyleyebiliriz. Örneğin gen ifadesinin düzenlemesinde, anahtar/kilit olarak niteleyebileceğimiz elemanların özelleşmiş etkileşimleri söz konusudur. Bu sebeple modelimizin biyolojik çizgeler de dahil olmak üzere, birçok gerçek ağ yapısının tasviri için uygun olduğunu düşünüyoruz. İçerik-temelli ağımızda, ağın düğümlerini bir ya da birden fazla rastgele dizi ile eşleştirip, düğümler arasındaki etkileşimleri onlara atanan dizilerin birbirleri içinde tekrarlanma koşulu altında inşa ediyoruz. Böylece, bu dizilerin uzunlukları ve içerikleri, ortaya çıkacak olan çizgenin tüm topolojik özelliklerini belirlemektedir. Düğüm çiftleri arasındaki bağlanma olasılıklarının hesabında yapılan ortalama-alan yaklaşımlarının ise, dizilerin uzunluk dağılımlarına bağlı olarak, varılan sonuçlarda ağın gerçek özelliklerinden önemli farklılaşmalara yol açabileceği görülüyor. Bu yaklaşımlarda, dizilerin farklı enformasyon içerikleri ihmal edilmekte ve olasılıklar sadece dizilerin uzunlukları cinsinden elde edilmektedir. Halbuki her sonlu dizi için, dizinin içerdiği farklı sembol sayısı ek bir enformasyon içermektedir. Burada sergilemeye çalışacağımız, kabalaştırılmış ortalama-alan türünden yaklaşımların, belli ekstrem durumlarda, tasvir etmeyi amaçladıkları ağın özelliklerinden uzak sonuçlar verebileceğidir. Ancak gerçek biyolojik ağ yapılarının modellenmesinde karşımıza çıkan uzunluk dağılımlarında ortaya çıkan hatalar hiçbir zaman burada sergileyeceğimiz örneklerde olduğu kadar büyük olmamış, bilakis ortalama-alan  yaklaşımı simülasyon sonuçlarına oldukça yakın sonuçlar vermiştir. Anahtar Kelimeler: Karmaşık ağ yapıları, içerik-temelli ağlar, ortalama-alan yaklaşımı

Analytical Solution of a Stochastic Content Based Network Model

We define and completely solve a content-based directed network whose nodes consist of random words and an adjacency rule involving perfect or approximate matches, for an alphabet with an arbitrary number of letters. The analytic expression for the out-degree distribution shows a crossover from a leading power law behavior to a log-periodic regime bounded by a different power law decay. The leading exponents in the two regions have a weak dependence on the mean word length, and an even weaker dependence on the alphabet size. The in-degree distribution, on the other hand, is much narrower and does not show scaling behavior. The results might be of interest for understanding the emergence of genomic interaction networks, which rely, to a large extent, on mechanisms based on sequence matching, and exhibit similar global features to those found here.Comment: 13 pages, 5 figures. Rewrote conclusions regarding the relevance to gene regulation networks, fixed minor errors and replaced fig. 4. Main body of paper (model and calculations) remains unchanged. Submitted for publicatio

Information content based model for the topological properties of the gene regulatory network of Escherichia coli

Gene regulatory networks (GRN) are being studied with increasingly precise quantitative tools and can provide a testing ground for ideas regarding the emergence and evolution of complex biological networks. We analyze the global statistical properties of the transcriptional regulatory network of the prokaryote Escherichia coli, identifying each operon with a node of the network. We propose a null model for this network using the content-based approach applied earlier to the eukaryote Saccharomyces cerevisiae. (Balcan et al., 2007) Random sequences that represent promoter regions and binding sequences are associated with the nodes. The length distributions of these sequences are extracted from the relevant databases. The network is constructed by testing for the occurrence of binding sequences within the promoter regions. The ensemble of emergent networks yields an exponentially decaying in-degree distribution and a putative power law dependence for the out-degree distribution with a flat tail, in agreement with the data. The clustering coefficient, degree-degree correlation, rich club coefficient and k-core visualization all agree qualitatively with the empirical network to an extent not yet achieved by any other computational model, to our knowledge. The significant statistical differences can point the way to further research into non-adaptive and adaptive processes in the evolution of the E. coli GRN.Comment: 58 pages, 3 tables, 22 figures. In press, Journal of Theoretical Biology (2009)

Monte Carlo Renormalization Group for Entanglement Percolation

We use a large cell Monte Carlo Renormalization procedure, to compute the critical exponents of a system of growing linear polymers. We simulate the growth of non-intersecting chains in large MC cells. Dense regions where chains get in each others' way, give rise to connected clusters under coarse graining. At each time step, the fraction of occupied bonds is determined in both the original and the coarse grained configurations, and averaged over many realizations. Our results for the fractal dimension on three dimensional lattices are consistent with the percolation value.Comment: 5 pages, 5 figure

Modeling vaccination campaigns and the Fall/Winter 2009 activity of the new A(H1N1) influenza in the Northern Hemisphere

The unfolding of pandemic influenza A(H1N1) for Fall 2009 in the Northern Hemisphere is still uncertain. Plans for vaccination campaigns and vaccine trials are underway, with the first batches expected to be available early October. Several studies point to the possibility of an anticipated pandemic peak that could undermine the effectiveness of vaccination strategies. Here we use a structured global epidemic and mobility metapopulation model to assess the effectiveness of massive vaccination campaigns for the Fall/Winter 2009. Mitigation effects are explored depending on the interplay between the predicted pandemic evolution and the expected delivery of vaccines. The model is calibrated using recent estimates on the transmissibility of the new A(H1N1) influenza. Results show that if additional intervention strategies were not used to delay the time of pandemic peak, vaccination may not be able to considerably reduce the cumulative number of cases, even when the mass vaccination campaign is started as early as mid-October. Prioritized vaccination would be crucial in slowing down the pandemic evolution and reducing its burden.Comment: Paper: 19 Pages, 3 Figures. Supplementary Information: 10 pages, 8 Table

Estimate of Novel Influenza A/H1N1 cases in Mexico at the early stage of the pandemic with a spatially structured epidemic model

Determining the number of cases in an epidemic is fundamental to properly evaluate several disease features of high relevance for public health policies such as mortality, morbidity or hospitalization rates. Surveillance efforts are however incomplete especially at the early stage of an outbreak due to the ongoing learning process about the disease characteristics. An example of this is represented by the number of H1N1 influenza cases in Mexico during the first months of the current pandemic. Several estimates using backtrack calculation based on imported cases from Mexico in other countries point out that the actual number of cases was likely orders of magnitude larger than the number of confirmed cases. Realistic computational models fed with the best available estimates of the basic disease parameters can provide an ab-initio calculation of the number of cases in Mexico as other countries. Here we use the Global Epidemic and Mobility (GLEaM) model to obtain estimates of the size of the epidemic in Mexico as well as of imported cases at the end of April and beginning of May. We find that the reference range for the number of cases in Mexico on April 30th is 121,000 to 1,394,000 in good agreement with the recent estimates by Lipsitch et al. [M. Lipsitch, PloS One 4:e6895 (2009)]. The number of imported cases from Mexico in several countries is found to be in good agreement with the surveillance data

Phase transitions in contagion processes mediated by recurrent mobility patterns

Human mobility and activity patterns mediate contagion on many levels, including the spatial spread of infectious diseases, diffusion of rumors, and emergence of consensus. These patterns however are often dominated by specific locations and recurrent flows and poorly modeled by the random diffusive dynamics generally used to study them. Here we develop a theoretical framework to analyze contagion within a network of locations where individuals recall their geographic origins. We find a phase transition between a regime in which the contagion affects a large fraction of the system and one in which only a small fraction is affected. This transition cannot be uncovered by continuous deterministic models due to the stochastic features of the contagion process and defines an invasion threshold that depends on mobility parameters, providing guidance for controlling contagion spread by constraining mobility processes. We recover the threshold behavior by analyzing diffusion processes mediated by real human commuting data.Comment: 20 pages of Main Text including 4 figures, 7 pages of Supplementary Information; Nature Physics (2011

Towards a characterization of behavior-disease models

The last decade saw the advent of increasingly realistic epidemic models that leverage on the availability of highly detailed census and human mobility data. Data-driven models aim at a granularity down to the level of households or single individuals. However, relatively little systematic work has been done to provide coupled behavior-disease models able to close the feedback loop between behavioral changes triggered in the population by an individual's perception of the disease spread and the actual disease spread itself. While models lacking this coupling can be extremely successful in mild epidemics, they obviously will be of limited use in situations where social disruption or behavioral alterations are induced in the population by knowledge of the disease. Here we propose a characterization of a set of prototypical mechanisms for self-initiated social distancing induced by local and non-local prevalence-based information available to individuals in the population. We characterize the effects of these mechanisms in the framework of a compartmental scheme that enlarges the basic SIR model by considering separate behavioral classes within the population. The transition of individuals in/out of behavioral classes is coupled with the spreading of the disease and provides a rich phase space with multiple epidemic peaks and tipping points. The class of models presented here can be used in the case of data-driven computational approaches to analyze scenarios of social adaptation and behavioral change.Comment: 24 pages, 15 figure