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We study the capture of a diffusing "lamb" by diffusing "lions" in one dimension. The capture dynamics is exactly soluble by probabilistic techniques when the number of lions is very small, and is tractable by extreme statistics considerations when the number of lions is very large. However, the exact solution for the general case of three or more lions is still not known.
Molecular networks guide the biochemistry of a living cell on multiple levels: its metabolic and signalling pathways are shaped by the network of interacting proteins, whose production, in turn, is controlled by the genetic regulatory network. To address topological properties of these two networks we quantify correlations between connectivities of interacting nodes and compare them to a null model of a network, in which al links were randomly rewired. We find that for both interaction and regulatory networks, links between highly connected proteins are systematically suppressed, while those between a highly-connected and low-connected pairs of proteins are favored. This effect decreases the likelihood of cross talk between different functional modules of the cell, and increases the overall robustness of a network by localizing effects of deleterious perturbations.
Spatially or chemically isolated functional modules composed of several cellular components and carrying discrete functions are considered fundamental building blocks of cellular organization, but their presence in highly integrated biochemical networks lacks quantitative support. Here we show that the metabolic networks of 43 distinct organisms are organized into many small, highly connected topologic modules that combine in a hierarchical manner into larger, less cohesive units, their number and degree of clustering following a power law. Within Escherichia coli the uncovered hierarchical modularity closely overlaps with known metabolic functions. The identified network architecture may be generic to system-level cellular organization.
We study the dynamical behaviour of the proton transfer in the hydrogen bonds in the base-pairs of the double helices of the DNA type. Under the assumption that the elastic and the tunnelling degrees of freedom may be coupled, we derive a non-linear and non-local Schrodinger equation (SNLNL) that describes the concerted motion of the proton tunnelling. Rough estimates of the solutions to the SNLNL show an intimate interplay between the concerted tunnelling of protons and the symmetry of double helix.
An 1d model with time-dependent random hopping is proposed to describe charge transport in DNA. It admits to investigate both diffusion of electrons and their tunneling between different sites in DNA. The tunneling appears to be strongly temperature-dependent. Observations of a strong (exponential) as well as a weak distance dependence of the charge transfer in DNA can be explained in the framework of our model.
The physical idea of the natural origin of diseases and deaths has been presented. The fundamental microscopical reason is the destruction of any metastable state by thermal activation of a nucleus of a nonreversable change. On the basis of this idea the quantitative theory of age dependence of death probability has been constructed. The obtained simple Death Laws are very accurately fulfilled almost for all known diseases.
The linking number (topological entanglement) and the writhe (geometrical entanglement) of a model of circular double stranded DNA undergoing a thermal denaturation transition are investigated by Monte Carlo simulations. By allowing the linking number to fluctuate freely in equilibrium we see that the linking probability undergoes an abrupt variation (first-order) at the denaturation transition, and stays close to 1 in the whole native phase. The average linking number is almost zero in the denatured phase and grows as the square root of the chain length, N, in the native phase. The writhe of the two strands grows as the square root of N in both phases.
Recently Garel, Monthus and Orland (Europhys. Lett. v 55, 132 (2001)) considered a model of DNA denaturation in which excluded volume effects within each strand are neglected, while mutual avoidance is included. Using an approximate scheme they found a first order denaturation. We show that a first order transition for this model follows from exact results for the statistics of two mutually avoiding random walks, whose reunion exponent is c > 2, both in two and three dimensions. Analytical estimates of c due to the interactions with other denaturated loops, as well as numerical calculations, indicate that the transition is even sharper than in models where excluded volume effects are fully incorporated. The probability distribution of distances between homologous base pairs decays as a power law at the transition.
A feedback mechanism that involves the proteins p53 and mdm2, induces cell death as a controled response to severe DNA damage. A minimal model for this mechanism demonstrates that the respone may be dynamic and connected with the time needed to translate the mdm2 protein. The response takes place if the dissociation constant k between p53 and mdm2 varies from its normal value. Although it is widely believed that it is an increase in k that triggers the response, we show that the experimental behaviour is better described by a decrease in the dissociation constant. The response is quite robust upon changes in the parameters of the system, as required by any control mechanism, except for few weak points, which could be connected with the onset of cancer.
We present a detailed analytical and numerical study for the spreading of infections in complex population networks with acquired immunity. We show that the large connectivity fluctuations usually found in these networks strengthen considerably the incidence of epidemic outbreaks. Scale-free networks, which are characterized by diverging connectivity fluctuations, exhibit the lack of an epidemic threshold and always show a finite fraction of infected individuals. This particular weakness, observed also in models without immunity, defines a new epidemiological framework characterized by a highly heterogeneous response of the system to the introduction of infected individuals with different connectivity. The understanding of epidemics in complex networks might deliver new insights in the spread of information and diseases in biological and technological networks that often appear to be characterized by complex heterogeneous architectures.
Experimental studies of the thermal denaturation of DNA yield a strong indication that the transition is first order. This transition has been theoretically studied since the early sixties, mostly within an approach in which the microscopic configurations of a DNA molecule are given by an alternating sequence of non-interacting bound segments and denaturated loops. Studies of these models neglect the repulsive, self-avoiding, interaction between different loops and segments and have invariably yielded continuous denaturation transitions. In this study we exploit recent results on scaling properties of polymer networks of arbitrary topology in order to take into account the excluded-volume interaction between denaturated loops and the rest of the chain. We thus obtain a first-order phase transition in d=2 dimensions and above, in agreement with experiments. We also consider within our approach the unzipping transition, which takes place when the two DNA strands are pulled apart by an external force acting on one end. We find that the unzipping transition is also first order. Although the denaturation and unzipping transitions are thermodynamically first order, they do exhibit critical fluctuations in some of their properties. For instance, the loop size distribution decays algebraically at the transition and the length of the denaturated end segment diverges as the transition is approached. We evaluate these critical properties within our approach.
We examine a model of biological evolution of Eigen's quasispecies in a holey fitness landscape, where the fitness of a site is either 0 (lethal site) or a uniform positive constant (viable site). So, the evolution dynamics is determined by the topology of the genome space. It is modeled by the random Bethe lattice. We use the effective medium and single-defect approximations to find the criteria, under which the localized quasispecies cloud is created. We find that shorter genomes, which are more robust to random mutations than average, represent a selective advantage which we call ``topological''. A way of assessing empirically the relative importance of reproductive success and topological advantage is suggested.
This paper describes how realistic neuromorphic networks can have their connectivity properties fully characterized in analytical fashion. By assuming that all neurons have the same shape and are regularly distributed along the two-dimensional orthogonal lattice with parameter $Δ$, it is possible to obtain the accurate number of connections and cycles of any length from the autoconvolution function as well as from the respective spectral density derived from the adjacency matrix. It is shown that neuronal shape plays an important role in defining the spatial spread of network connections. In addition, most such networks are characterized by the interesting phenomenon where the connections are progressively shifted along the spatial domain where the network is embedded. It is also shown that the number of cycles follows a power law with their respective length. Morphological measurements for characterization of the spatial distribution of connections, including the adjacency matrix spectral density and the lacunarity of the connections, are suggested. The potential of the proposed approach is illustrated with respect to digital images of real neuronal cells.
Recent experiments on unzipping of RNA helix-loop structures by force have shown that about 40-base molecules can undergo kinetic transitions between two well-defined `open' and `closed' states, on a timescale = 1 sec [Liphardt et al., Science 297, 733-737 (2001)]. Using a simple dynamical model, we show that these phenomena result from the slow kinetics of crossing large free energy barriers which separate the open and closed conformations. The dependence of barriers on sequence along the helix, and on the size of the loop(s) is analyzed. Some DNAs and RNAs sequences that could show dynamics on different time scales, or three(or more)-state unzipping, are proposed.
We calculate the induced elastic-interaction between pointwise membrane inclusions that locally interact up to quadratic order with the membrane curvature tensor. For isotropic inclusions, we recover the usual interaction proportional to the inverse fourth power of the separation, however with a prefactor showing a non-trivial dependence on the rigidity $Γ$ of the quadratic potential. In the large $Γ$ limit, corresponding to ``hard'' inclusions, we recover the standard prefactor first obtained by Goulian et al. [Europhys. Lett. \textbf{22}, 145 (1993)]. In the small $Γ$ limit, corresponding to "soft" inclusions, we recover the recent result of Marchenko and Misbah [Eur. Phys. J. E \textbf{8}, 477 (2002)]. This shows that the latter result bears no fundamental discrepancy with previous works, but simply corresponds to the limit of soft inclusions. We discuss how the same inclusion can be depicted as hard or soft according to the degree of coarse-graining of the pointwise description. Finally, we argue that conical transmembrane proteins should be fundamentally considered as hard inclusions.
We analyze the nonlinear relaxation of a complex ecosystem composed of many interacting species. The ecological system is described by generalized Lotka-Volterra equations with a multiplicative noise. The transient dynamics is studied in the framework of the mean field theory and with random interaction between the species. We focus on the statistical properties of the asymptotic behaviour of the time integral of the i-th population and on the distribution of the population and of the local field.
We propose a new model of neural network. It consists of spin variables to describe the state of neurons as in the Hopfield model and new gauge variables to describe the state of synapses. The model possesses local gauge symmetry and resembles lattice gauge theory of high-energy physics. Time dependence of synapses describes the process of learning. The mean field theory predicts a new phase corresponding to confinement phase, in which brain loses ablility of learning and memory.
In this review some simple models of asexual populations evolving on smooth landscapes are studied. The basic model is based on a cellular automaton, which is analyzed here in the spatial mean-field limit. Firstly, the evolution on a fixed fitness landscape is considered. The correspondence between the time evolution of the population and equilibrium properties of a statistical mechanics system is investigated, finding the limits for which this mapping holds. The mutational meltdown, Eigen's error threshold and Muller's ratchet phenomena are studied in the framework of a simplified model. Finally, the shape of a quasi-species and the condition of coexistence of multiple species in a static fitness landscape are analyzed. In the second part, these results are applied to the study of the coexistence of quasi-species in the presence of competition, obtaining the conditions for a robust speciation effect in asexual populations.