Stochastic modelling of reaction-diffusion processes:\ud algorithms for bimolecular reactions

Several stochastic simulation algorithms (SSAs) have been recently proposed for modelling reaction-diffusion processes in cellular and molecular biology. In this paper, two commonly used SSAs are studied. The first SSA is an on-lattice model described by the reaction-diffusion master equation. The second SSA is an off-lattice model based on the simulation of Brownian motion of individual molecules and their reactive collisions. In both cases, it is shown that the commonly used implementation of bimolecular reactions (i.e. the reactions of the form A+B → C, or A+A → C) might lead to incorrect results. Improvements of both SSAs are suggested which overcome the difficulties highlighted. In particular, a formula is presented for the smallest possible compartment size (lattice spacing) which can be correctly implemented in the first model. This implementation uses a new formula for the rate of bimolecular reactions per compartment (lattice site)

Exponentially slow transitions on a Markov chain: the frequency of Calcium Sparks

Calcium sparks in cardiac muscle cells occur when a cluster of Ca2+ channels open and release Ca2+ from an internal store. A simplified model of Ca2+ sparks has been developed to describe the dynamics of a cluster of channels, which is of the form of a continuous time Markov chain with nearest neighbour transitions and slowly varying jump functions. The chain displays metastability, whereby the probability distribution of the state of the system evolves exponentially slowly, with one of the metastable states occurring at the boundary. An asymptotic technique for analysing the Master equation (a differential-difference equation) associated with these Markov chains is developed using the WKB and projection methods. The method is used to re-derive a known result for a standard class of Markov chains displaying metastability, before being applied to the new class of Markov chains associated with the spark model. The mean first passage time between metastable states is calculated and an expression for the frequency of calcium sparks is derived. All asymptotic results are compared with Monte Carlo simulations

Time scale of random sequential adsorption

A simple multiscale approach to the diffusion-driven adsorption from a solution to a solid surface is presented. The model combines two important features of the adsorption process: (i) the kinetics of the chemical reaction between adsorbing molecules and the surface; and (ii) geometrical constraints on the surface made by molecules which are already adsorbed. The process (i) is modelled in a diffusion-driven context, i.e. the conditional probability of adsorbing a molecule provided that the molecule hits the surface is related to the macroscopic surface reaction rate. The geometrical constraint (ii) is modelled using random sequential adsorption (RSA), which is the sequential addition of molecules at random positions on a surface; one attempt to attach a molecule is made per one RSA simulation time step. By coupling RSA with the diffusion of molecules in the solution above the surface the RSA simulation time step is related to the real physical time. The method is illustrated on a model of chemisorption of reactive polymers to a virus surface

Arts curriculum implementation: Adopt and adapt as policy translation

This paper examines macro, meso and micro understandings of policy enactment within Western Australian primary school arts education where a new national arts curriculum is being revised and implemented through a process colloquially known as ‘adopt and adapt’. This paper focuses on how a government led implementation policy has influenced arts teaching and learning in unintended ways. It Includes a theoretical reflection and a consideration of the effects of such policies. Using policy enactment theory as the enquiry lens, four contextual variables are highlighted for their impact on teachers and schools. The variables include situated contexts, material contexts, professional cultures and external factors. Effects are discussed through the perspectives of eleven arts curriculum leaders drawn from in-depth semi-structured interviews. Marginalisation of the arts, the disconnection of schools and teachers to the arts and professional learning impacts are discussed as results of this policy translation

A hierarchy of models for superconducting thin films

A hierarchy of models for type-II superconducting thin films is presented. Through appropriate asymptotic limits this hierarchy passes from the mesoscopic Ginzburg--Landau model to the London model with isolated vortices as $\delta$-function singularities to vortex-density models and finally to macroscopic critical-state models. At each stage it is found that a key nondimensional parameter is $\Lambda = \lambda^2/d L$, where $\lambda$ is the penetration depth of the magnetic field, a material parameter, and d and L are a typical thickness and lateral dimension of the film,respectively. The models simplify greatly if this parameter is large or small

Glucocorticoid therapy for adrenal insufficiency: nonadherence, concerns and dissatisfaction with information

Objective: Appropriate self‐management of glucocorticoid therapy (GC) is crucial for patients with adrenal insufficiency (AI). We aimed to describe patients’ self‐reported nonadherence to GC, evaluate perceived doubts about need for GC, concerns about adverse effects, and dissatisfaction with information received about GC. Design: Cross‐sectional survey. Patients: Patients prescribed GC for AI (n = 81) from five European countries. Measurements: Online survey including the Medication Adherence Report Scale (MARS), Beliefs about Medicines Questionnaire© (BMQ Specific, adapted for AI) and Satisfaction with Information about Medicines Scale© (Prof Rob Horne; SIMS). Results: Most patients (85·2%) reported a degree of nonadherence to GC. The most frequent types of nonadherence concerned changing the timing of GC doses, for example taking a dose later in the day than advised (37·0%). Few patients doubted their personal need for daily GC, but most reported high concerns about GC including potential weight gain (50·6%), osteoporosis (53·6%) and the continuing risk of adrenal crisis (50·6%). Dissatisfaction with information about GC was frequent, with participants particularly dissatisfied with the amount of information they had received about potential problems with GC. People who expressed dissatisfaction with information about GC, and concerns about its adverse effects were also more likely to report nonadherence (P < 0·05). Conclusions: Nonadherence to treatment, concerns about potential adverse effects and dissatisfaction with the information provided about treatment were frequently reported by this European sample of AI patients. Many AI patients may need additional information about their GC and support to address concerns about GC and facilitate adherence

Modelling the measured local time evolution of strongly nonlinear heat pulses in the Large Helical Device

In some magnetically confined plasmas, an applied pulse of rapid edge cooling can trigger either a positive or negative excursion in the core electron temperature from its steady state value. We present a new model which captures the time evolution of the transient, non-diffusive local dynamics in the core plasma. We show quantitative agreement between this model and recent spatially localized measurements (Inagaki et al 2010 Plasma Phys. Control. Fusion 52 075002) of the local time-evolving temperature pulse in cold pulse propagation experiments in the Large Helical Device

The Two Regime method for optimizing stochastic reaction-diffusion simulations

The computer simulation of stochastic reaction-diffusion processes in biology is often done using either compartment-based (spatially discretized) simulations or molecular-based (Brownian dynamics) approaches. Compartment-based approaches can yield quick and accurate mesoscopic results but lack the level of detail that is characteristic of the more computationally intensive molecular-based models. Often microscopic detail is only required in a small region but currently the best way to achieve this detail is to use a resource intensive model over the whole domain. We introduce the Two Regime Method (TRM) in which a molecular-based algorithm is used in part of the computational domain and a compartment-based approach is used elsewhere in the computational domain. We apply the TRM to two test problems including a model from developmental biology. We thereby show that the TRM is accurate and subsequently may be used to inspect both mesoscopic and microscopic detail of reaction diffusion simulations according to the demands of the modeller

Continuum and discrete models of dislocation pile-ups. I Pile-up at a lock

A mathematical methodology for analysing pile-ups of large numbers of dislocations is described. As an example, the pile-up of n identical screw or edge dislocations in a single slip pane under the action of an external force in the direction of a locked dislocation in that plane is considered. As $n \rightarrow \infty$ there is a well-known formula for the number density of the dislocations, but this density is singular at the lock and it cannot predict the stress field there or the force on the lock. This poses the interesting analytical and numerical problem of matching a local discrete model near the lock to the continuum model further away