Fluorescence Assay for Polymerase Arrival Rates

To engineer complex synthetic biological systems will require modular design, assembly, and characterization strategies. The RNA polymerase arrival rate (PAR) is defined to be the rate that RNA polymerases arrive at a specified location on the DNA. Designing and characterizing biological modules in terms of RNA polymerase arrival rates provides for many advantages in the construction and modeling of biological systems. PARMESAN is an in vitro method for measuring polymerase arrival rates using pyrrolo-dC, a fluorescent DNA base that can substitute for cytosine. Pyrrolo-dC shows a detectable fluorescence difference when in single-stranded versus double-stranded DNA. During transcription, RNA polymerase separates the two strands of DNA, leading to a change in the fluorescence of pyrrolo-dC. By incorporating pyrrolo-dC at specific locations in the DNA, fluorescence changes can be taken as a direct measurement of the polymerase arrival rate.

Convergence Rates of Approximation by Translates

In this paper we consider the problem of approximating a function belonging to some funtion space Φ by a linear comination of n translates of a given function G. Ussing a lemma by Jones (1990) and Barron (1991) we show that it is possible to define function spaces and functions G for which the rate of convergence to zero of the erro is 0(1/n) in any number of dimensions. The apparent avoidance of the "curse of dimensionality" is due to the fact that these function spaces are more and more constrained as the dimension increases. Examples include spaces of the Sobolev tpe, in which the number of weak derivatives is required to be larger than the number of dimensions. We give results both for approximation in the L2 norm and in the Lc norm. The interesting feature of these results is that, thanks to the constructive nature of Jones" and Barron"s lemma, an iterative procedure is defined that can achieve this rate.