Coarsening of Sand Ripples in Mass Transfer Models with Extinction

Coarsening of sand ripples is studied in a one-dimensional stochastic model, where neighboring ripples exchange mass with algebraic rates, $\Gamma(m) \sim m^\gamma$, and ripples of zero mass are removed from the system. For $\gamma < 0$ ripples vanish through rare fluctuations and the average ripples mass grows as \avem(t) \sim -\gamma^{-1} \ln (t). Temporal correlations decay as $t^{-1/2}$ or $t^{-2/3}$ depending on the symmetry of the mass transfer, and asymptotically the system is characterized by a product measure. The stationary ripple mass distribution is obtained exactly. For $\gamma > 0$ ripple evolution is linearly unstable, and the noise in the dynamics is irrelevant. For $\gamma = 1$ the problem is solved on the mean field level, but the mean-field theory does not adequately describe the full behavior of the coarsening. In particular, it fails to account for the numerically observed universality with respect to the initial ripple size distribution. The results are not restricted to sand ripple evolution since the model can be mapped to zero range processes, urn models, exclusion processes, and cluster-cluster aggregation.Comment: 10 pages, 8 figures, RevTeX4, submitted to Phys. Rev.

Self-similarity and power-like tails in nonconservative kinetic models

In this paper, we discuss the large--time behavior of solution of a simple kinetic model of Boltzmann--Maxwell type, such that the temperature is time decreasing and/or time increasing. We show that, under the combined effects of the nonlinearity and of the time--monotonicity of the temperature, the kinetic model has non trivial quasi-stationary states with power law tails. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution. The same idea is applied to investigate the large-time behavior of an elementary kinetic model of economy involving both exchanges between agents and increasing and/or decreasing of the mean wealth. In this last case, the large-time behavior of the solution shows a Pareto power law tail. Numerical results confirm the previous analysis

On a kinetic model for a simple market economy

In this paper, we consider a simple kinetic model of economy involving both exchanges between agents and speculative trading. We show that the kinetic model admits non trivial quasi-stationary states with power law tails of Pareto type. In order to do this we consider a suitable asymptotic limit of the model yielding a Fokker-Planck equation for the distribution of wealth among individuals. For this equation the stationary state can be easily derived and shows a Pareto power law tail. Numerical results confirm the previous analysis

Mesoscopic modelling of financial markets

We derive a mesoscopic description of the behavior of a simple financial market where the agents can create their own portfolio between two investment alternatives: a stock and a bond. The model is derived starting from the Levy-Levy-Solomon microscopic model (Econ. Lett., 45, (1994), 103--111) using the methods of kinetic theory and consists of a linear Boltzmann equation for the wealth distribution of the agents coupled with an equation for the price of the stock. From this model, under a suitable scaling, we derive a Fokker-Planck equation and show that the equation admits a self-similar lognormal behavior. Several numerical examples are also reported to validate our analysis

Inelastically scattering particles and wealth distribution in an open economy

Using the analogy with inelastic granular gasses we introduce a model for wealth exchange in society. The dynamics is governed by a kinetic equation, which allows for self-similar solutions. The scaling function has a power-law tail, the exponent being given by a transcendental equation. In the limit of continuous trading, closed form of the wealth distribution is calculated analytically.Comment: 8 pages 5 figure

Spatial organization in cyclic Lotka-Volterra systems

We study the evolution of a system of $N$ interacting species which mimics the dynamics of a cyclic food chain. On a one-dimensional lattice with N<5 species, spatial inhomogeneities develop spontaneously in initially homogeneous systems. The arising spatial patterns form a mosaic of single-species domains with algebraically growing size, $\ell(t)\sim t^\alpha$, where $\alpha=3/4$ (1/2) and 1/3 for N=3 with sequential (parallel) dynamics and N=4, respectively. The domain distribution also exhibits a self-similar spatial structure which is characterized by an additional length scale, ${\cal L}(t)\sim t^\beta$, with $\beta=1$ and 2/3 for N=3 and 4, respectively. For $N\geq 5$, the system quickly reaches a frozen state with non interacting neighboring species. We investigate the time distribution of the number of mutations of a site using scaling arguments as well as an exact solution for N=3. Some possible extensions of the system are analyzed.Comment: 18 pages, 10 figures, revtex, also available from http://arnold.uchicago.edu/~ebn

Modeling protein network evolution under genome duplication and domain shuffling

<p>Abstract</p> <p>Background</p> <p>Successive whole genome duplications have recently been firmly established in all major eukaryote kingdoms. Such <it>exponential </it>evolutionary processes must have largely contributed to shape the topology of protein-protein interaction (PPI) networks by outweighing, in particular, all <it>time-linear </it>network growths modeled so far.</p> <p>Results</p> <p>We propose and solve a mathematical model of PPI network evolution under successive genome duplications. This demonstrates, from first principles, that evolutionary conservation and scale-free topology are intrinsically linked properties of PPI networks and emerge from <it>i) </it>prevailing <it>exponential </it>network dynamics under duplication and <it>ii) asymmetric divergence </it>of gene duplicates. While required, we argue that this asymmetric divergence arises, in fact, spontaneously at the level of protein-binding sites. This supports a refined model of PPI network evolution in terms of protein domains under exponential and asymmetric duplication/divergence dynamics, with multidomain proteins underlying the combinatorial formation of protein complexes. Genome duplication then provides a powerful source of PPI network innovation by promoting local rearrangements of multidomain proteins on a genome wide scale. Yet, we show that the overall conservation and topology of PPI networks are robust to extensive domain shuffling of multidomain proteins as well as to finer details of protein interaction and evolution. Finally, large scale features of <it>direct </it>and <it>indirect </it>PPI networks of <it>S. cerevisiae </it>are well reproduced numerically with only two adjusted parameters of clear biological significance (<it>i.e</it>. network effective growth rate and average number of protein-binding domains per protein).</p> <p>Conclusion</p> <p>This study demonstrates the statistical consequences of genome duplication and domain shuffling on the conservation and topology of PPI networks over a broad evolutionary scale across eukaryote kingdoms. In particular, scale-free topologies of PPI networks, which are found to be robust to extensive shuffling of protein domains, appear to be a simple consequence of the conservation of protein-binding domains under asymmetric duplication/divergence dynamics in the course of evolution.</p

Bayesian statistical modelling of human protein interaction network incorporating protein disorder information

<p>Abstract</p> <p>Background</p> <p>We present a statistical method of analysis of biological networks based on the exponential random graph model, namely p2-model, as opposed to previous descriptive approaches. The model is capable to capture generic and structural properties of a network as emergent from local interdependencies and uses a limited number of parameters. Here, we consider one global parameter capturing the density of edges in the network, and local parameters representing each node's contribution to the formation of edges in the network. The modelling suggests a novel definition of important nodes in the network, namely <it>social</it>, as revealed based on the local <it>sociality </it>parameters of the model. Moreover, the sociality parameters help to reveal organizational principles of the network. An inherent advantage of our approach is the possibility of hypotheses testing: <it>a priori </it>knowledge about biological properties of the nodes can be incorporated into the statistical model to investigate its influence on the structure of the network.</p> <p>Results</p> <p>We applied the statistical modelling to the human protein interaction network obtained with Y2H experiments. Bayesian approach for the estimation of the parameters was employed. We deduced <it>social </it>proteins, essential for the formation of the network, while incorporating into the model information on protein disorder. <it>Intrinsically disordered </it>are proteins which lack a well-defined three-dimensional structure under physiological conditions. We predicted the fold group (ordered or disordered) of proteins in the network from their primary sequences. The network analysis indicated that protein disorder has a positive effect on the connectivity of proteins in the network, but do not fully explains the interactivity.</p> <p>Conclusions</p> <p>The approach opens a perspective to study effects of biological properties of individual entities on the structure of biological networks.</p

Simulated Evolution of Protein-Protein Interaction Networks with Realistic Topology

We model the evolution of eukaryotic protein-protein interaction (PPI) networks. In our model, PPI networks evolve by two known biological mechanisms: (1) Gene duplication, which is followed by rapid diversification of duplicate interactions. (2) Neofunctionalization, in which a mutation leads to a new interaction with some other protein. Since many interactions are due to simple surface compatibility, we hypothesize there is an increased likelihood of interacting with other proteins in the target protein’s neighborhood. We find good agreement of the model on 10 different network properties compared to high-confidence experimental PPI networks in yeast, fruit flies, and humans. Key findings are: (1) PPI networks evolve modular structures, with no need to invoke particular selection pressures. (2) Proteins in cells have on average about 6 degrees of separation, similar to some social networks, such as human-communication and actor networks. (3) Unlike social networks, which have a shrinking diameter (degree of maximum separation) over time, PPI networks are predicted to grow in diameter. (4) The model indicates that evolutionarily old proteins should have higher connectivities and be more centrally embedded in their networks. This suggests a way in which present-day proteomics data could provide insights into biological evolution

Predicted binding site information improves model ranking in protein docking using experimental and computer-generated target structures

BACKGROUND: Protein-protein interactions (PPIs) mediate the vast majority of biological processes, therefore, significant efforts have been directed to investigate PPIs to fully comprehend cellular functions. Predicting complex structures is critical to reveal molecular mechanisms by which proteins operate. Despite recent advances in the development of new methods to model macromolecular assemblies, most current methodologies are designed to work with experimentally determined protein structures. However, because only computer-generated models are available for a large number of proteins in a given genome, computational tools should tolerate structural inaccuracies in order to perform the genome-wide modeling of PPIs. RESULTS: To address this problem, we developed eRank(PPI), an algorithm for the identification of near-native conformations generated by protein docking using experimental structures as well as protein models. The scoring function implemented in eRank(PPI) employs multiple features including interface probability estimates calculated by eFindSite(PPI) and a novel contact-based symmetry score. In comparative benchmarks using representative datasets of homo- and hetero-complexes, we show that eRank(PPI) consistently outperforms state-of-the-art algorithms improving the success rate by ~10 %. CONCLUSIONS: eRank(PPI) was designed to bridge the gap between the volume of sequence data, the evidence of binary interactions, and the atomic details of pharmacologically relevant protein complexes. Tolerating structure imperfections in computer-generated models opens up a possibility to conduct the exhaustive structure-based reconstruction of PPI networks across proteomes. The methods and datasets used in this study are available at www.brylinski.org/erankppi