Dependence on saturation of average minimum resolution in two-dimensional statistical-overlap theory: Peak overlap in saturated two-dimensional separations

A theory is proposed for the dependence on saturation of the average minimum resolution R* in point-process statistical-overlap theory for two-dimensional separations. Peak maxima are modelled by clusters of overlapping circles in hexagonal arrangements similar to close-packed layers. Such clusters exist only for specific circle numbers, but equations are derived that facilitate prediction of equivalent cluster properties for any number of circles. A metric is proposed for the average minimum resolution that separates two such clusters into two maxima. From this metric, the average minimum resolution of the two nearest-neighbor single-component peaks (SCPs) - one in each cluster - is calculated. Its value varies with the number of SCPs in both clusters. These resolutions are weighted by the probability that the two clusters contain the postulated numbers of SCPs and summed to give R*, which decreases with increasing saturation. The dependence of R* on saturation is combined with a theory correcting the probability of overlap in a reduced square for boundary effects. The numbers of maxima in simulations of 75, 150, and 300 randomly distributed bi-Gaussians having exponential heights and aspect ratios of 1, 30, and 60 are compared to predictions. Excellent agreement between maxima numbers and theory is found at low and high saturation. Good estimates of the numbers of bi-Gaussians in simulations are calculated by fitting theory to numbers of maxima using least-squares regression. The theory is applied to mimicked GC x GCs of 93 compounds having many correlated retention times, with predictions that agree fairly well with maxima numbers. (c) 2006 Elsevier B.V. All rights reserved.