Limit Theorems for Hybridization Reactions on Oligonucleotide Microarrays

We derive herein the limiting laws for certain stationary distributions of birth-and-death processes related to the classical model of chemical adsorption-desorption reactions due to Langmuir. The model has been recently considered in the context of a hybridization reaction on an oligonucleotide DNA microarray. Our results imply that the truncated gamma- and beta- type distributions can be used as approximations to the observed distributions of the fluorescence readings of the oligo-probes on a microarray. These findings might be useful in developing new model-based, probe-specific methods of extracting target concentrations from array fluorescence readings

Multicultural Literature: An Overview of Best Practices

The value of using multicultural literature in the educational setting has gained much support in the last few decades. At the same time, the exact meaning of multicultural literature, while having been debated and discussed, has not reached "consensus." These two facts create an interesting dilemma: while many educators want to incorporate literature from diverse cultures into their curricula, they are unsure of how best to accomplish this integration. Perhaps, initially reluctant because of their unfamiliarity with the representative cultures, teachers' hesitations are further fueled by the dynamic nature of the genre. This is all very understandable. And while I will not cover this continuing debate over definition, for the purposes of this article, my definition of multicultural literature will be borrowed from Glazier and Seo -- that is, those writings "that represent voices typically omitted from the traditional canon." The terms "multicultural literature" and "culturally diverse literature" will also be used interchangeably

Asymptotic analysis of multiscale approximations to reaction networks

A reaction network is a chemical system involving multiple reactions and chemical species. Stochastic models of such networks treat the system as a continuous time Markov chain on the number of molecules of each species with reactions as possible transitions of the chain. In many cases of biological interest some of the chemical species in the network are present in much greater abundance than others and reaction rate constants can vary over several orders of magnitude. We consider approaches to approximation of such models that take the multiscale nature of the system into account. Our primary example is a model of a cell's viral infection for which we apply a combination of averaging and law of large number arguments to show that the ``slow'' component of the model can be approximated by a deterministic equation and to characterize the asymptotic distribution of the ``fast'' components. The main goal is to illustrate techniques that can be used to reduce the dimensionality of much more complex models.Comment: Published at http://dx.doi.org/10.1214/105051606000000420 in the Annals of Applied Probability (http://www.imstat.org/aap/) by the Institute of Mathematical Statistics (http://www.imstat.org

Equivalence of Mass Action and Poisson Network SIR Epidemic Models

This brief note highlights a largely overlooked similarity between the SIR ordinary differential equations used for epidemics on the configuration model of a Poisson network and the classical mass-action SIR equations introduced nearly a century ago by Kermack and McKendrick. We demonstrate that the decline pattern in susceptibles is identical for both models. This equivalence carries practical implications: the susceptibles decay curve, often referred to as the epidemic or incidence curve, is frequently used in empirical studies to forecast epidemic dynamics. Although the curves for susceptibles align perfectly, those for infections do differ. Yet, the infection curves tend to converge and become almost indistinguishable in high-degree networks. In summary, our analysis suggests that under many practical scenarios, it is acceptable to use the classical SIR model as a close approximation to the Poisson SIR network model.Comment: 2 figure