High frequency difraction by a soft circular disc. I the plane wave at normal incidence

The far scattered field off the axis of symmetry of the disc is found for a high frequency, harmonic, normally incident, plane wave. The method used is due to Jones and involves the solution of a singular integral equation of the first kind for the field on the disc. This integral equation can be converted into an integral equation of the second kind which is of particular value at high frequencies. In the present work the known function in the equation is written in the form of a contour integral. A suitable change of unknown function then produces extensive cancellation and yields a single function fundamental to the problem. The detailed calculations of the far field give terms which are believed to be new. In executing these calculations some interesting relationships between the terms involved are demonstrated

Random intermittent search and the tug-of-war model of motor-driven transport

We formulate the tug-of-war model of microtubule cargo transport by multiple molecular motors as an intermittent random search for a hidden target. A motor-complex consisting of multiple molecular motors with opposing directional preference is modeled using a discrete Markov process. The motors randomly pull each other off of the microtubule so that the state of the motor-complex is determined by the number of bound motors. The tug-of-war model prescribes the state transition rates and corresponding cargo velocities in terms of experimentally measured physical parameters. We add space to the resulting Chapman-Kolmogorov (CK) equation so that we can consider delivery of the cargo to a hidden target somewhere on the microtubule track. Using a quasi-steady state (QSS) reduction technique we calculate analytical approximations of the mean first passage time (MFPT) to find the target. We show that there exists an optimal adenosine triphosphate (ATP)concentration that minimizes the MFPT for two different cases: (i) the motor complex is composed of equal numbers of kinesin motors bound to two different microtubules (symmetric tug-of-war model), and (ii) the motor complex is composed of different numbers of kinesin and dynein motors bound to a single microtubule(asymmetric tug-of-war model)

Filling of a Poisson trap by a population of random intermittent searchers

We extend the continuum theory of random intermittent search processes to the case of $N$ independent searchers looking to deliver cargo to a single hidden target located somewhere on a semi--infinite track. Each searcher randomly switches between a stationary state and either a leftward or rightward constant velocity state. We assume that all of the particles start at one end of the track and realize sample trajectories independently generated from the same underlying stochastic process. The hidden target is treated as a partially absorbing trap in which a particle can only detect the target and deliver its cargo if it is stationary and within range of the target; the particle is removed from the system after delivering its cargo. As a further generalization of previous models, we assume that up to $n$ successive particles can find the target and deliver its cargo. Assuming that the rate of target detection scales as $1/N$, we show that there exists a well--defined mean field limit $N\rightarrow \infty$, in which the stochastic model reduces to a deterministic system of linear reaction--hyperbolic equations for the concentrations of particles in each of the internal states. These equations decouple from the stochastic process associated with filling the target with cargo. The latter can be modeled as a Poisson process in which the time--dependent rate of filling $\lambda(t)$ depends on the concentration of stationary particles within the target domain. Hence, we refer to the target as a Poisson trap. We analyze the efficiency of filling the Poisson trap with $n$ particles in terms of the waiting time density $f_n(t)$. The latter is determined by the integrated Poisson rate $\mu(t)=\int_0^t\lambda(s)ds$, which in turn depends on the solution to the reaction-hyperbolic equations. We obtain an approximate solution for the particle concentrations by reducing the system of reaction-hyperbolic equations to a scalar advection--diffusion equation using a quasi-steady-state analysis. We compare our analytical results for the mean--field model with Monte-Carlo simulations for finite $N$. We thus determine how the mean first passage time (MFPT) for filling the target depends on $N$ and $n$

Local synaptic signaling enhances the stochastic transport of\ud motor-driven cargo in neurons

The tug-of-war model of motor-driven cargo transport is formulated as an intermittent trapping process. An immobile trap, representing the cellular machinery that sequesters a motor-driven cargo for eventual use, is located somewhere within a microtubule track. A particle representing a motor-driven cargo that moves randomly with a forward bias is introduced at the beginning of the track. The particle switches randomly between a fast moving phase and a slow moving phase. When in the slow moving phase, the particle can be captured by the trap. To account for the possibility the particle avoids the trap, an absorbing boundary is placed at the end of the track. Two local signaling mechanisms—intended to improve the chances of capturing the target—are considered by allowing the trap to affect the tug-of-war parameters within a small region around itself. The first is based on a localized adenosine triphosphate (ATP) concentration gradient surrounding a synapse, and the second is based on a concentration of tau—a microtubule-associated protein involved in Alzheimer’s disease—coating the microtubule near the synapse. It is shown that both mechanisms can lead to dramatic improvements in the capture probability, with a minimal increase in the mean capture time. The analysis also shows that tau can cause a cargo to undergo random oscillations, which could explain some experimental observations

Stochastic models of intracellular transport

The interior of a living cell is a crowded, heterogenuous, fluctuating environment. Hence, a major challenge in modeling intracellular transport is to analyze stochastic processes within complex environments. Broadly speaking, there are two basic mechanisms for intracellular transport: passive diffusion and motor-driven active transport. Diffusive transport can be formulated in terms of the motion of an over-damped Brownian particle. On the other hand, active transport requires chemical energy, usually in the form of ATP hydrolysis, and can be direction specific, allowing biomolecules to be transported long distances; this is particularly important in neurons due to their complex geometry. In this review we present a wide range of analytical methods and models of intracellular transport. In the case of diffusive transport, we consider narrow escape problems, diffusion to a small target, confined and single-file diffusion, homogenization theory, and fractional diffusion. In the case of active transport, we consider Brownian ratchets, random walk models, exclusion processes, random intermittent search processes, quasi-steady-state reduction methods, and mean field approximations. Applications include receptor trafficking, axonal transport, membrane diffusion, nuclear transport, protein-DNA interactions, virus trafficking, and the self–organization of subcellular structures

Living on the margin: Assessing the economic impacts of Landcare in the Philippine uplands

In the Philippines, about 38 per cent of the population resides in rural areas where poverty remains a significant problem. In 2006, 47 per cent of all households in Bohol Province fell below the national poverty line, with the percentage even higher in upland communities. These households often exist in marginal landscapes that are under significant pressure from ongoing resource degradation and rising input costs. This paper first explores whether the adoption of Landcare practices in a highly degraded landscape has resulted in improved livelihood outcomes for upland farming families in Bohol. Second, it analyses the potential for the piecemeal adoption of these measures to deliver tangible benefits at the watershed scale. Finally, using a BCA approach, these outcomes are compared to the costs of the research and extension projects that have helped achieve them.Landcare, Philippines, livelihoods, poverty, watershed, ACIAR,

Isolating intrinsic noise sources in a stochastic genetic switch

The stochastic mutual repressor model is analysed using perturbation methods. This simple model of a gene circuit consists of two genes and three promotor states. Either of the two protein products can dimerize, forming a repressor molecule that binds to the promotor of the other gene. When the repressor is bound to a promotor, the corresponding gene is not transcribed and no protein is produced. Either one of the promotors can be repressed at any given time or both can be unrepressed, leaving three possible promotor states. This model is analysed in its bistable regime in which the deterministic limit exhibits two stable fixed points and an unstable saddle, and the case of small noise is considered. On small time scales, the stochastic process fluctuates near one of the stable fixed points, and on large time scales, a metastable transition can occur, where fluctuations drive the system past the unstable saddle to the other stable fixed point. To explore how different intrinsic noise sources affect these transitions, fluctuations in protein production and degradation are eliminated, leaving fluctuations in the promotor state as the only source of noise in the system. Perturbation methods are then used to compute the stability landscape and the distribution of transition times, or first exit time density. To understand how protein noise affects the system, small magnitude fluctuations are added back into the process, and the stability landscape is compared to that of the process without protein noise. It is found that significant differences in the random process emerge in the presence of protein noise