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

Dependence of Effective Peak Capacity in Comprehensive Two-Dimensional Separations on the Distribution of Peak Capacity between the Two Dimensions

“…Many of the reported trends can be explained by the systematic decrease in the number of observed peaks with increasing saturation factor ( α ) [7,8,15] defined below: $$\alpha =4\pi \stackrel{false‒}{m}<\beta >{}^1\sigma {}^2\sigma {\left(R_s^{\ast }\right)}^2∕\left({}^{1}D{}^{2}D\right)=\frac{\pi }{4}\frac{{\left(R_s^{\ast }\right)}^2}{{}^{1}R_s{}^{2}R_s}\frac{\stackrel{‒}{m}}{n_{c,2D}^{\prime }}$$ where m̄ is a statistical approximation to the number of constituents and $R_s^{\ast }$ is the average minimum resolution required for two observed peaks to be formed [8,24]. $R_s^{\ast }$ itself depends on the saturation factor [8]. The final expression in Eq.…”

“…The basis of 2D SOT is discussed in detail elsewhere [5-8]. Geometrically, peak overlap is modeled in two dimensions by the overlap of ellipses representing the peak contours.…”

“…1. Subsequent theoretical work by Davis and coworkers showed that 1) the rectangular grid model underestimates the true total peak capacity, because peaks can fall anywhere in the bed and not just at ‘bin’ centers [5,6], and 2) the edge effect, which is influenced by the ratio of the first and second dimension peak capacities, significantly increases the probability of observing peaks near the edges and corners of the bed [7,8]. The reason why edges and corners show more observed peaks and less peak overlap is that unlike the central regions of the grid there is no possibility of overlap from peaks whose retentions lie outside the periphery as such peaks simply don't exist.…”

“…It is not clear that the product rule, which does not distinguish between the first and second dimensions, connects the asymmetric data matrix to usual metrics of interest. Thus far only for the case of completely randomly distributed peaks has the very highly developed statistical overlap theory (SOT) [5-8] allowed the derivation of an exact mathematical relationship between the average number of observed peaks and the total peak capacity. We propose here that for each type of distribution of peak coordinates (strong correlation, weak correlation, or no correlation; see Figure 1), a relationship exists between the average number of observed peaks and the total peak capacity.…”

Relationship between selectivity and average resolution in comprehensive two-dimensional separations with spectroscopic detection

“…In the absence of additional spectral data, these attributesthe average minimum resolution and the peak-broadening factor -have been characterized in multi-component separations [16,20,21] by the use of statistical-overlap theory (SOT), which describes the expected amount of peak overlap in a large ensemble of separations. In this work we will show the important unifying result that the SEL can be interpreted as the fractional singlet character of a peak.…”