Stochastic modelling of reaction-diffusion processes: 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).Comment: 33 pages, submitted to Physical Biolog

Solving the chemical master equation using sliding windows

<p>Abstract</p> <p>Background</p> <p>The chemical master equation (CME) is a system of ordinary differential equations that describes the evolution of a network of chemical reactions as a stochastic process. Its solution yields the probability density vector of the system at each point in time. Solving the CME numerically is in many cases computationally expensive or even infeasible as the number of reachable states can be very large or infinite. We introduce the sliding window method, which computes an approximate solution of the CME by performing a sequence of local analysis steps. In each step, only a manageable subset of states is considered, representing a "window" into the state space. In subsequent steps, the window follows the direction in which the probability mass moves, until the time period of interest has elapsed. We construct the window based on a deterministic approximation of the future behavior of the system by estimating upper and lower bounds on the populations of the chemical species.</p> <p>Results</p> <p>In order to show the effectiveness of our approach, we apply it to several examples previously described in the literature. The experimental results show that the proposed method speeds up the analysis considerably, compared to a global analysis, while still providing high accuracy.</p> <p>Conclusions</p> <p>The sliding window method is a novel approach to address the performance problems of numerical algorithms for the solution of the chemical master equation. The method efficiently approximates the probability distributions at the time points of interest for a variety of chemically reacting systems, including systems for which no upper bound on the population sizes of the chemical species is known a priori.</p

PROTON RADIOGRAPHY EXAMINATION OF UNBURNED REGIONS IN PBX 9502 CORNER TURNING EXPERIMENTS

PBX 9502 Corner Turning Experiments have been used with various diagnostics techniques to study detonation wave propagation and the boosting of the insensitive explosive. In this work, the uninitiated region of the corner turning experiment is examined using Proton Radiography. Seven transmission radiographs obtained on the same experiment are used to map out the undetonated regions on each of three different experiments. The results show regions of high-density material, a few percent larger than initial explosive density. These regions persist at nearly this density while surrounding material, which has reacted, is released as expected. Calculations using Detonation Shock Dynamics are used to examine the situations that lead to the undetonated regions

Jet initiation thresholds of nitromethane

The initiation criterion for nitromethane and diethylenetriamine- sensitized solutions has been established over a confined range of jet diameters, velocities, and failure diameters. The data were normalized with the failure diameter that was chemically modified, and they support the hypothesis that the failure diameter should be made part of the critical initiation function. The difference between physically- and chemically sensitized NM in promptness of initiation, as measured by corner turning distance, was not statistically significant. The diameter of the Viper jet has been characterized over a wide range of velocities

Proton radiographic and numerical of colliding, diverging PBX-9502 detonations.

The Proton radiographic shot PRAD0077 was designed to study the interaction of colliding, diverging PBX-9502 detonations. The shot consisted of a 50 mm by 50 mm cylinder of PBX-9502 initiated on the top and bottom at the axis by a SE-1 detonator and a 12 mm by 12 mm cylinder of 9407. Seven radiographs were taken at times before and after the detonation collision. The system was modeled using the one-dimensional SIN code with C-J Burn in plane and spherically diverging geometry and using the two-dimensional TDL code with C-J Burn and Forest Fire. The system was also modeled with the recently developed AMR Eulerian reactive hydrodynamic code called NOBEL using Forest Fire. The system results in a large dead or nonreactive zone as the detonation attempts to turn the corner which is described by the model using Forest Fire. The peak detonation pressure achieved by the colliding diverging detonation is 50 gpa and density of 3.125 mg/ml which is about the same as that achieved by one-dimensional spherically diverging 9502 detonations but less than the one-dimensional plane 9502 peak colliding detonation pressure of 65 gpa and density of 3.4 mg/ml. The detonation travels for over 10 mm before it starts to expand and turn the corner leaving more than half of the explosive unreacted. The resulting diverging detonation is more curved than a one-dimensional spherical diverging detonation and has a steeper slope behind the detonation front. This results in the colliding pressure decaying faster than one-dimensional colliding spherical diverging pressures decay. The calculations using Forest Fire reproduce the major features of the radiograph and can be used to infer the colliding detonation characteristics