A Comparison of Bimolecular Reaction Models for Stochastic Reaction Diffusion Systems

Stochastic reaction-diffusion models have become an important tool in studying how both noise in the chemical reaction process and the spatial movement of molecules influences the behavior of biological systems. There are two primary spatially-continuous models that have been used in recent studies: the diffusion limited reaction model of Smoluchowski, and a second approach popularized by Doi. Both models treat molecules as points undergoing Brownian motion. The former represents chemical reactions between two reactants through the use of reactive boundary conditions, with two molecules reacting instantly upon reaching a fixed separation (called the reaction-radius). The Doi model uses reaction potentials, whereby two molecules react with a fixed probability per unit time, $\lambda$, when separated by less than the reaction radius. In this work we study the rigorous relationship between the two models. For the special case of a protein diffusing to a fixed DNA binding site, we prove that the solution to the Doi model converges to the solution of the Smoluchowski model as $\lambda \to \infty$, with a rigorous $O(\lambda^{-1/2 + \epsilon})$ error bound (for any fixed $\epsilon > 0$). We investigate by numerical simulation, for biologically relevant parameter values, the difference between the solutions and associated reaction time statistics of the two models. As the reaction-radius is decreased, for sufficiently large but fixed values of $\lambda$, these differences are found to increase like the inverse of the binding radius.Comment: 21 pages, 3 Figures, Fixed typo in titl

Walkable Gettysburg— How Pedestrian Friendly is the Borough of Gettysburg, Pennsylvania?

Walkability is a measure of how easily pedestrians can reach a variety of destinations via walking. Greater walkability has been linked to several benefits, including improvements in human health, economic stimulus, and improved air quality. We surveyed 37 blocks in Gettysburg, Pennsylvania to record the presence of 13 design factors such as street trees and pedestrian oriented amenities that have been shown to encourage walking. These results were then compared with the Walk Score from walkscore.com, a common measurement tool of walkability. Based on the surveys, we calculated a design quality score (DQI) for each block. There was no correlation between DQI and Walk Score. The highest scores for aesthetics were recorded near Gettysburg College, the highest scores for ease of use were recorded around the traffic circle at the center of town, and the highest scores for safety were recorded near the traffic circle and the College. We believe that this discrepancy can be attributed to the focus of walkscore.com on the proximity of a location to various destination while our DQI score considered aesthetics, ease of use, and safety. Based on the results of this study, we recommend that the borough of Gettysburg invest in alternatives to automobile transport such as bicycle oriented amenities in order to increase walkability

Hiring Costs of Skilled Workers and the Supply of Firm-Provided Training

This paper analyzes how the costs of hiring skilled workers from the external labor market affect a firm's supply of training. Using administrative survey data with detailed information on hiring and training costs for Swiss firms, we find evidence for substantial and increasing marginal hiring costs. However, firms can invest in internal training of unskilled workers and thereby avoid costs for external hiring. Controlling for a firm's training investment, we find that a one standard deviation increase in average external hiring costs increases the number of internal training positions by 0.7 standard deviations.hiring costs, apprenticeship training, firm-sponsored training