Diffusive Boundary Layers in the Free-Surface Excitable Medium Spiral

Spiral waves are a ubiquitous feature of the nonequilibrium dynamics of a great variety of excitable systems. In the limit of a large separation in timescale between fast excitation and slow recovery, one can reduce the spiral problem to one involving the motion of a free surface separating the excited and quiescent phases. In this work, we study the free surface problem in the limit of small diffusivity for the slow field variable. Specifically, we show that a previously found spiral solution in the diffusionless limit can be extended to finite diffusivity, without significant alteration. This extension involves the creation of a variety of boundary layers which cure all the undesirable singularities of the aforementioned solution. The implications of our results for the study of spiral stability are briefly discussed.Comment: 6 pages, submitted to PRE Rapid Com

Discrete stochastic modeling of calcium channel dynamics

We propose a simple discrete stochastic model for calcium dynamics in living cells. Specifically, the calcium concentration distribution is assumed to give rise to a set of probabilities for the opening/closing of channels which release calcium thereby changing those probabilities. We study this model in one dimension, analytically in the mean-field limit of large number of channels per site N, and numerically for small N. As the number of channels per site is increased, the transition from a non-propagating region of activity to a propagating one changes in nature from one described by directed percolation to that of deterministic depinning in a spatially discrete system. Also, for a small number of channels a propagating calcium wave can leave behind a novel fluctuation-driven state, in a parameter range where the limiting deterministic model exhibits only single pulse propagation.Comment: 4 pages, 5 figures, submitted to PR

Mechanisms underlying sequence-independent beta-sheet formation

We investigate the formation of beta-sheet structures in proteins without taking into account specific sequence-dependent hydrophobic interactions. To accomplish this, we introduce a model which explicitly incorporates both solvation effects and the angular dependence (on the protein backbone) of hydrogen bond formation. The thermodynamics of this model is studied by comparing the restricted partition functions obtained by "unfreezing" successively larger segments of the native beta-sheet structure. Our results suggest that solvation dynamics together with the aforementioned angular dependence gives rise to a generic cooperativity in this class of systems; this result explains why pathological aggregates involving beta-sheet cores can form from many different proteins. Our work provides the foundation for the construction of phenomenological models to investigate the competition between native folding and non-specific aggregation.Comment: 20 pages, 5 figures, Revtex4, simulation mpeg movie available at http://www-physics.ucsd.edu/~guochin/Images/sheet1.mp

Analytic approach to the evolutionary effects of genetic exchange

We present an approximate analytic study of our previously introduced model of evolution including the effects of genetic exchange. This model is motivated by the process of bacterial transformation. We solve for the velocity, the rate of increase of fitness, as a function of the fixed population size, $N$. We find the velocity increases with $\ln N$, eventually saturated at an $N$ which depends on the strength of the recombination process. The analytical treatment is seen to agree well with direct numerical simulations of our model equations

Embryonic Pattern Scaling Achieved by Oppositely Directed Morphogen Gradients

Morphogens are proteins, often produced in a localised region, whose concentrations spatially demarcate regions of differing gene expression in developing embryos. The boundaries of expression must be set accurately and in proportion to the size of the one-dimensional developing field; this cannot be accomplished by a single gradient. Here, we show how a pair of morphogens produced at opposite ends of a developing field can solve the pattern-scaling problem. In the most promising scenario, the morphogens effectively interact according to the annihilation reaction $A+B\to\emptyset$ and the switch occurs according to the absolute concentration of $A$ or $B$. In this case embryonic markers across the entire developing field scale approximately with system size; this cannot be achieved with a pair of non-interacting gradients that combinatorially regulate downstream genes. This scaling occurs in a window of developing-field sizes centred at a few times the morphogen decay length.Comment: 24 pages; 11 figures; uses iopar

Two State Behavior in a Solvable Model of β\beta-hairpin folding

Understanding the mechanism of protein secondary structure formation is an essential part of protein-folding puzzle. Here we describe a simple model for the formation of the $\beta$-hairpin, motivated by the fact that folding of a $\beta$-hairpin captures much of the basic physics of protein folding. We argue that the coupling of ``primary'' backbone stiffness and ``secondary'' contact formation (similar to the coupling between the ``secondary'' and ``tertiary'' structure in globular proteins), caused for example by side-chain packing regularities, is responsible for producing an all-or-none 2-state $\beta$-hairpin formation. We also develop a recursive relation to compute the phase diagram and single exponential folding/unfolding rate arising via a dominant transition state.Comment: Revised versio

Evolution on a smooth landscape

We study in detail a recently proposed simple discrete model for evolution on smooth landscapes. An asymptotic solution of this model for long times is constructed. We find that the dynamics of the population are governed by correlation functions that although being formally down by powers of $N$ (the population size) nonetheless control the evolution process after a very short transient. The long-time behavior can be found analytically since only one of these higher-order correlators (the two-point function) is relevant. We compare and contrast the exact findings derived herein with a previously proposed phenomenological treatment employing mean field theory supplemented with a cutoff at small population density. Finally, we relate our results to the recently studied case of mutation on a totally flat landscape.Comment: Revtex, 15 pages, + 4 embedded PS figure