Statistical Overlap Theory Research Articles

Statistical‐overlap theory for elliptical zones of high aspect ratio in comprehensive two‐dimensional separations

The low prediction by statistical-overlap theory of the numbers of singlets and peaks in two-dimensional separations containing zones represented by either circles of small number or eccentric ellipses of any number is shown to result from use of probability expressions for unbound spaces of infinite extent. An exact theory is derived for the probability of singlet formation in a reduced two-dimensional space of unit length, width, and area. The probability is a weighted sum of the probabilities of singlet formation in the interior, edge, and corner regions of the space, which depend only on saturation. The weighting factors are the fractions of area associated with each region and depend on the number of zones, the aspect ratio, the saturation, and the ellipse's spatial orientation. The average numbers of doublets, triplets, and peaks in the space are approximated by combining these results with Roach's equations describing the clustering of circles in an unbound two-dimensional space. Simulations show that theory predicts the number of singlets, doublets, triplets, and peaks, when the number of zones is 25 or more, the aspect ratio is 100 or less, and the saturation is 2 or less. The relationship is derived between the aspect ratios of ellipses in the reduced space and actual separation space. Calculations are presented for comprehensive two-dimensional gas chromatography.

Verification of statistical-overlap theory in micellar electrokinetic chromatography

The limited peak capacity of neutral compounds in micellar electrokinetic chromatography (MEKC) causes peak overlap in a simple 38-compound sample that is predicted by statistical-overlap theory (SOT). The low-concentration sample was prepared in-house from several compound classes to span the entire migration-time range and was resolved partially in a pH=7 phosphate buffer containing 50 mM sodium dodecyl sulfate. Peaks, singlets, doublets, and other multiplets were identified on the basis of known migration times and were counted at 13 voltages spanning 4 - 26 kV. These numbers agreed well with predictions of a simple SOT based on the assumption of an inhomogeneous Poisson distribution of migration times. Because the dispersion theory of MEKC is simple, the standard deviations of single-component peaks were modeled theoretically. As part of a new way to implement SOT, probability distributions of the numbers of peaks, singlets, and so on, were computed by Monte Carlo simulation. These distributions contain all theoretical information on peak multiplicity predictable by SOT and were used to evaluate the agreement between experiment and theory. The peak capacity of MEKC was calculated numerically and substituted into the simplest equations in SOT, affirming that peak overlap arises from limited peak capacity.

Automated analysis of individual particles using a commercial capillary electrophoresis system

Capillary electrophoretic analysis of individual submicrometer size particles has been previously done using custom-built instruments. Despite that these instruments provide an excellent signal-to-noise ratio for individual particle detection, they are not capable of performing automated analyses of particles. Here we report the use of a commercial Beckman P/ACE MDQ capillary electrophoresis (CE) instrument with on-column laser-induced fluorescence (LIF) detection for the automated analysis of individual particles. The CE instrument was modified with an external I/O board that allowed for faster data acquisition rates (e.g. 100 Hz) than those available with the standard instrument settings (e.g. 4 Hz). A series of eight hydrodynamic injections expected to contain 32 +/- 6 particles, each followed by an electrophoretic separation at -300 V cm(-1) with data acquired at 100 Hz, showed 28 +/- 5 peaks corresponding to 31.9 particles as predicted by the statistical overlap theory. In contrast, a similar series of hydrodynamic injections followed by data acquisition at 4 Hz revealed only 8 +/- 3 peaks suggesting that the modified system is needed for individual particle analysis. Comparison of electropherograms obtained at both data acquisition rates also indicate: (i) similar migration time ranges; (ii) lower variation in the fluorescence intensity of individual peaks for 100 Hz; and (iii) a better signal-to-noise ratio for 4 Hz raw data. S/N improved for 100 Hz when data were smoothed with a binomial filter but did not reach the S/N values previously reported for post-column LIF detection. The proof-of-principle of automated analysis of individual particles using a commercially available CE system described here opens exciting possibilities for those interested in the study and analyses of organelles, liposomes, and nanoparticles.

Assessment by monte carlo simulation of thermodynamic correlation of retention times in dual-column temperature programmed comprehensive two-dimensional gas chromatography.

The correlation of retention times in comprehensive two-dimensional gas chromatography caused by correlation of enthalpy and entropy changes between two stationary phases, methylsilicone and poly(ethylene glycol), was examined using commercial GC software and in-house Monte Carlo simulation. The enthalpy change, deltaH0, and entropy change, deltaS0, of 93 compounds were predicted from isothermal one-dimensional gas chromatograms predicted by the software. These values then were mimicked by Monte Carlo simulation, which removed the strong correlation of deltaH0 and modest correlation of deltaS0 between the two phases. Retention times in a comprehensive two-dimensional gas chromatogram (GC x GC) and in simulations of it were predicted for typical dual-capillary temperature-programmed conditions using the actual, correlated values of deltaH0 and deltaS0 and their uncorrelated Monte Carlo counterparts, respectively. The uncorrelated deltaH0 and deltaS0 values caused the retention-time range of the simulations' second dimension to expand substantially beyond that in the GC x GC. Other simulations were developed using a restricted range of uncorrelated deltaH0 and deltaS0 values to mimic more closely the retention-time range of the GC x GC's second dimension. The intervals between nearest neighbor retention-time coordinates were calculated in both the latter simulations and the GC x GC. The intervals were larger in the simulations than in the GC x GC because the former contained uncorrelated coordinates and the latter contained correlated ones, which clustered along or near the diagonal. The retention times in the first dimension of the GC x GC were Poisson distributed, as assessed by statistical-overlap theory. In contrast, the two-dimensional reduced retention-time coordinates in the GC x GC were not Poisson distributed, because retention times were correlated. However, the reduced retention-time coordinates in the simulations were Poisson distributed.

Statistical-overlap theory of column switching in gas chromatography: applications to flavor and fragrance compounds.

First-column gas chromatograms (GCs) of hundreds of flavor and fragrance compounds, and second-column GCs of specific regions of these GCs, are predicted using thermodynamic databases in commercial software. A statistical-overlap theory of column switching with cryogenic focusing then is developed by mimicking the predicted GCs by two kinds of Monte Carlo simulations. In the first kind, a probability distribution is calculated for the number of compounds in a region of the first-column GC, based on the number of observed peaks in the region, the number of observed peaks in the second-column GC, and the retention-time distributions and breadths of single-component peaks in both GCs. In the second kind, criteria are established for the theory's application. The theory is applied to 12 regions of first-column GCs. The theory predicts the number of compounds in all of them and shows that separation rarely is complete in second-column GCs, when 10 or more compounds are transferred between columns. The theory also rationalizes the tedious search required to find good separation conditions by showing that column-switching gas chromatography with cryogenic focusing is inherently statistical. The number of peaks in the second-column GC can be greater than, less than, or equal to the number of peaks in the relevant region of the first-column GC, and the good conditions sought by researchers to substantially improve separation correspond to favorable "rolls of the dice" found only by trial and error.

Justification of statistical overlap theory in programmed temperature gas chromatography: thermodynamic origin of random distribution of retention times.

A specific distribution of compounds' standard-state changes of enthalpy and entropy between mobile and stationary phases in programmed temperature gas chromatography (PTGC) is shown to produce the Poisson distribution of retention times often postulated in statistical-overlap theory (SOT). A three-part model is proposed, in which the enthalpy change is Poisson distributed, the average entropy change depends on the enthalpy change, and the actual entropy change varies in a uniformly random manner about the average entropy change. To test the model, the entropy and enthalpy changes of 350 aliphatic and aromatic hydrocarbons in petroleum were calculated with commercial GC software. These changes are shown to follow the three-part model. The model then was used with Monte Carlo methods to mimic the enthalpy and entropy changes. The substitution of the mimicked enthalpy and entropy changes into an equation for the retention temperature in PTGC is shown to produce a Poisson distribution of retention times that is statistically significant. This finding establishes a scientific link between the thermodynamics governing retention in PTGC and the superficially ad hoc assumption of the Poisson retention time distribution in SOT. Similar thermodynamic distributions are found for flavors and fragrances and for tetrachlorodibenzo-p-dioxins and furans, which follow SOT based on the Poisson distribution, but not for polychloronaphthalenes, which do not follow that SOT.

The Need for Two-Dimensional Gas Chromatography: Extent of Overlap in One-Dimensional Gas Chromatograms

The need for two-dimensional gas chromatography is justified by the extent of peak overlap in one-dimensional gas chromatograms (GCs) of complex mixtures. Such overlap was predicted long ago by statistical-overlap theory (SOT). In this paper, SOT is conceptually reviewed and its predictions are shown to be quantitatively accurate. GCs of complex mixtures of polychlorinated biphenyls, pyridine- and nitrogen-containing polynuclear aromatic hydrocarbons, tetrachlorodibenzo-p-dioxins and dibenzofurans, fatty acid methyl esters, flavors and fragrances, and naphtha were simulated by commercial GC software on DB-1, DB-5, and Stabilwax stationary phases. The numbers of peak maxima in the GCs agreed with predictions of SOT, when the interval of time between successive peaks of pure compounds was described by Poisson statistics. This agreement was realized even though the time intervals actually are deterministic, not statistical. In addition, the numbers of mixture components were predicted with accuracy by regression of peak numbers against SOT. Similar regressions have been reported before, but the theory used here is more sophisticated and its predictions consequently are more accurate. Future directions for finalizing SOT are suggested.

The Need for Two-Dimensional Gas Chromatography: Extent of Overlap in One-Dimensional Gas Chromatograms

The need for two-dimensional gas chromatography is justified by the extent of peak overlap in one-dimensional gas chromatograms (GCs) of complex mixtures. Such overlap was predicted long ago by statistical-overlap theory (SOT). In this paper, SOT is conceptually reviewed and its predictions are shown to be quantitatively accurate. GCs of complex mixtures of polychlorinated biphenyls, pyridine- and nitrogen-containing polynuclear aromatic hydrocarbons, tetrachlorodibenzo-p-dioxins and dibenzofurans, fatty acid methyl esters, flavors and fragrances, and naphtha were simulated by commercial GC software on DB-1, DB-5, and Stabilwax stationary phases. The numbers of peak maxima in the GCs agreed with predictions of SOT, when the interval of time between successive peaks of pure compounds was described by Poisson statistics. This agreement was realized even though the time intervals actually are deterministic, not statistical. In addition, the numbers of mixture components were predicted with accuracy by regression of peak numbers against SOT. Similar regressions have been reported before, but the theory used here is more sophisticated and its predictions consequently are more accurate. Future directions for finalizing SOT are suggested.

Application of statistical-overlap theory to gas chromatograms simulated on intermediate-polarity and polar stationary phases

Application of statistical-overlap theory to gas chromatograms simulated on intermediate-polarity and polar stationary phases

Application of statistical-overlap theory to gas chromatograms simulated on nonpolar stationary phases with commercial software

Scores of gas chromatograms of complex nonpolar mixtures were simulated on nonpolar stationary phases with commercial software. The mixtures included petroleum, alkanes, pesticides, flavors and fragrances, drugs, volatiles, industrial solvents, phenoxy acids, semivolatiles, C-, N-, and S-containing polynuclear aromatic hydrocarbons, Arochlors, and polychlorinated biphenyls, naphthalenes, dibenzo- p-dioxins, and dibenzofurans. With commercial software, the retention times of mixture components were calculated from Kováts retention indices, and the standard deviations of single-component peaks were calculated from the modified Golay–Giddings equation. These peak parameters then were used to predict the numbers of peaks expected in the chromatograms, using laboratory software based on the point-process statistical theory of overlap and the Poisson distribution. In most cases, the predictions agreed very well with the numbers of peaks found in chromatograms simulated with the peak parameters. This agreement verified that the interval between successive single-component peaks in most gas chromatograms of complex nonpolar mixtures developed on nonpolar stationary phases can be represented by a random variable. Statistical-overlap theory can be applied to such chromatograms.

Prediction of numbers of singlets, doublets, and triplets in poorly resolved separations by statistical-overlap theory

A recently improved theory for the distribution of resolution in complex separations described by statistical-overlap theory is used to predict the average numbers of singlet, doublet, and triplet peaks in such separations, even when the peak capacity is much less than the number of components requiring separation. The theory is fundamental and modifies previously published equations for these numbers by causing the average minimum resolution that defines saturation to depend on saturation itself. Interestingly, theory shows that the resolutions describing the separation and overlap of single-component peaks in singlets, doublets, and triplets actually differ. The theory predicts correctly the numbers of singlets, doublets, and triplets in computer simulations of separations, even at high saturation. Example calculations are provided to show the ease with which the theory can be used.

Extension of Statistical Overlap Theory to Poorly Resolved Separations

A general theory of statistical overlap is developed that enables prediction of the number of observable peaks in separations, even when the peak capacity is much less than the number of single-component peaks requiring separation. The theory is applicable to any separation in which the distribution of intervals between adjacent single-component peaks is governed by homogeneous statistics. The theory shows that the average resolution required to separate single-component peaks varies with saturation. This variation occurs because multicomponent peaks, and not single-component peaks, overlap as separation efficiency decreases. Previously reported equations relating the average number of observable peaks to the average number of single-component peaks are shown to be good even when the former is only 10−15% or so of the latter, as long as this variation is addressed. The theory is verified further by its application to experimental gas chromatograms. This paper consequently represents a significant advance ...

Effects of different organic modifiers in optimization of reversed-phase high-performance liquid chromatographic gradient elution of a mixture of natural secoiridoid compounds

Optimization of the linear gradient elution of a crude extract of Swertia herb (Gentianaceae) was performed by means of sequential methods (Fibonacci and simplex) using a reversed-phase partition system and three different organic modifiers. The optimum gradient elution separations obtained with methanol, acetonitrile and tetrahydrofuran as different organic modifiers are discussed and compared by using the linear solvent strength theory. The Poisson character of the optimized separations was checked and the Davis-Giddings statistical overlap theory was applied in order to calculate the number of components. The value obtained is independent of both the organic modifier and column type. The quality of the best separation obtained was close to the absolute optimum expected from the Davis-Giddings theory. The usefulness of dealing with two or more optimized gradient separations is discussed. Solvent strengths are reported for selected compounds (amarogentin, amaroswerin and gentiopicroside).