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

Prediction by statistical overlap theory of fraction of baseline occupied by chromatographic peaks

The average minimum resolution required for separating adjacent single-component peaks (SCPs) in one-dimensional chromatograms is an important metric in statistical overlap theory (SOT). However, its value changes with changing chromatographic conditions in non-intuitive ways, when SOT predicts the average number of peaks (maxima). A more stable and easily understood value of resolution is obtained on making a different prediction. A general equation is derived for the sum of all separated and superposed widths of SCPs in a chromatogram. The equation is a function of the saturation α, a metric of chromatographic crowdedness, and is expressed in dimensionless form by dividing by the duration of the chromatogram. This dimensionless function, f(α), is also the cumulative distribution function of the probability of separating adjacent SCPs. Simulations based on the clustering of line segments representing SCPs verify expressions for f(α) calculated from five functions for the distribution of intervals between adjacent SCPs. Synthetic chromatograms are computed with different saturations, distributions of intervals, and distribution of SCP amplitudes. The chromatograms are analyzed by calculating the sum of the widths of peaks at different relative responses, dividing the sum by the duration of the chromatograms, and graphing the reduced sum against relative response. For small values of relative response, the reduced sum approaches the fraction of baseline that is occupied by chromatographic peaks. This fraction can be identified with f(α), if the saturation α is defined with the average minimum resolution equaling 1.5. The identification is general and is independent of the saturation, the interval distribution, or the amplitude distribution. This constant value of resolution corresponds to baseline resolution, which simplifies the interpretation of SOT.

Theory of the probability of total resolution in chromatograms with systematic variation of average peak spacing and peak width

An equation is proposed for the probability that all mixture constituents are separated, when the density (i.e., average number of eluting constituents per time) and width of single-component peaks (SCPs) vary systematically. The probability Pr that m SCPs are separated is modeled as the product of the m - 1 probabilities that adjacent pairs of SCPs are separated. Pr is then expressed as the geometric mean of the probability product raised to the power of m - 1. This geometric mean is approximated by an arithmetic mean equaling the probability that adjacent SCPs are separated, as calculated from previously developed statistical overlap theory (SOT) for variable SCP density and width. The theory is tested using previously reported and current in-house simulations of isocratic chromatograms of SCPs with random differences in standard chemical potential. In such chromatograms, more SCPs elute at short times than long times, and their widths are less at short times than long times. The average difference between 179 previously reported and currently predicted values of 100 x Pr is about 0.6, when 100 x Pr > 5. The theory requires numerical computation of one integral but can be approximated by an analytic equation for SOT probabilities close to one. For SCPs having retention times exceeding twice the void time, this equation simplifies to a previous SOT expression, with the gradient peak capacity replaced by the isocratic peak capacity. The versatility of the Pr theory is tested using three other models of chromatograms, in which the density and width of SCPs vary. The Pr predictions agree with simulation for all three models.

Estimation of migration-time and mobility distributions in organelle capillary electrophoresis with statistical-overlap theory

The separation of organelles by capillary electrophoresis (CE) produces large numbers of narrow peaks, which commonly are assumed to originate from single particles. In this paper, we show this is not always true. Here, we use established methods to partition simulated and real organelle CEs into regions of constant peak density and then use statistical-overlap theory to calculate the number of peaks (single particles) in each region. The only required measurements are the number of observed peaks (maxima) and peak standard deviation in the regions and the durations of the regions. Theory is developed for the precision of the estimated peak number and the threshold saturation above which the calculation is not advisable due to fluctuation of peak numbers. Theory shows that the relative precision is good when the saturation lies between 0.2 and 1.0 and is optimal when the saturation is slightly greater than 0.5. It also shows the threshold saturation depends on the peak standard deviation divided by the region's duration. The accuracy and precision of peak numbers estimated in different regions of organelle CEs are verified by computer simulations having both constant and nonuniform peak densities. The estimates are accurate to 6%. The estimated peak numbers in different regions are used to calculate migration-time and electrophoretic-mobility distributions. These distributions are less biased by peak overlap than ones determined by counting maxima and provide more correct measures of the organelle properties. The procedure is applied to a mitochondrial CE, in which over 20% of peaks are hidden by peak overlap.

Evaluation of peak overlap in migration-time distributions determined by organelle capillary electrophoresis: Type-II error analogy based on statistical-overlap theory

Organelles commonly are separated by capillary electrophoresis (CE) with laser-induced-fluorescence detection. Usually, it is assumed that peaks observed in the CE originate from single organelles, with negligible occurrence of peak overlap. Under this assumption, migration-time and mobility distributions are obtained by partitioning the CE into different regions and counting the number of observed peaks in each region. In this paper, criteria based on statistical-overlap theory (SOT) are developed to test the assumption of negligible peak overlap and to predict conditions for its validity. For regions of the CE having constant peak density, the numbers of peaks (i.e., intensity profiles of single organelles) and observed peaks (i.e., maxima) are modeled by probability distributions. For minor peak overlap, the distributions partially merge, and their mergence is described by an analogy to the Type-II error of hypothesis testing. Criteria are developed for the amount of peak overlap, at which the number of observed peaks has an 85% or 90% probability of lying within the 95% confidence interval of the number of peaks of single organelles. For this or smaller amounts of peak overlap, the number of observed peaks is a good approximation to the number of peaks. A simple procedure is developed for evaluating peak overlap, requiring determination of only the peak standard deviation, the duration of the region occupied by peaks, and the number of observed peaks in the region. The procedure can be applied independently to each region of the partitioned CE. The procedure is applied to a mitochondrial CE.

Validation of theory ofn-column separations with gas chromatograms predicted by commercial software

A probability theory for the average number of compounds resolved by the partial separation of complex mixtures on n columns was tested using commercial-software predictions of gas chromatograms. Such n-column separations are traditional means for addressing peak overlap, in which one chooses additional columns of different selectivity to separate compounds that cannot be separated by a single column. Gas chromatograms of five types of complex mixtures containing from 99 to 283 compounds were predicted for eight stationary phases using both optimized and other temperature programs. The number n of columns for different mixtures varied from 2 to 5. The numbers of compounds separated as singlet peaks at different resolution thresholds were compared to predictions, as evaluated with point-process statistical-overlap theory based on a Poisson distribution. A good agreement between theory and results was found in all cases corresponding to low saturation. Both good and poor agreements were found for cases corresponding to high saturation. A good agreement also was found for results based on resolving complex mixtures by a single column subject to two temperature programs. The moments and distribution of the number of resolved compounds were computed by Monte Carlo simulation, thus gauging the significance of departures between results and theory. The potential of such simulations to explore the limitations of theory was briefly investigated.

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.