Statistical Overlap Theory Research Articles

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.

Validation of Extended Statistical Overlap Theory for Application to Small Second Dimension Peak Capacity

Approximate equations for the numbers of singlet peaks and total peaks in two-dimensional (2D) separations of randomly distributed component zones predict values of zero as the second-dimension peak capacity 2nc goes to zero. This is the result of deriving the equations for an unbounded 2D space. More rigorous equations are needed to assess how large 2nc must be before the error in the approximate equations is acceptably small. An exact equation for the number of singlet peaks in a 2D separation with a bounded second dimension is derived in Online Resource 1. The equation accounts for the reduced likelihood of overlap of zones represented as circles or ellipses near the second-dimension boundaries and is the result of the lengthy derivation of different expressions of probability, with the appropriate expression depending on the value of 2nc. In this paper, the equation is used to modify the prediction of the Roach equation for the number of total peaks in the 2D separation. Computer simulations of the clustering of circles confirm the equation for singlet peaks over a wide range of first-dimension saturations α1D (0.05 ≤ α1D ≤ 5) and 2nc values (0.1 ≤ 2nc ≤ 15). The saturation α1D, equal to the ratio of the number of components requiring separation to the first-dimension peak capacity available for separation, is a metric of chromatographic crowdedness. Over the same α1D range, the equation for total peaks slightly underestimates the simulations when 2nc < 2.5 but agrees with them at larger 2nc. It is shown that the extended theory differs most from the approximate one when α1D is large and 2nc is small.

A systematic investigation of comprehensive two-dimensional gas chromatography time-of-flight mass spectrometry with dynamic pressure gradient modulation for high peak capacity separations

Dynamic pressure gradient modulation (DPGM) in full modulation mode is optimized for comprehensive two-dimensional (2D) gas chromatography (GC × GC) with time-of-fight mass spectrometry (TOFMS) detection to obtain high peak capacity separations and demonstrate broad applicability for complex samples. A pulse valve introduces an auxiliary carrier gas flow at a T-union connecting the first dimension (1D) column to the second dimension (2D) column. At a sufficiently high auxiliary pressure (Paux) the 1D flow is temporarily stopped. Then, during each modulation period (PM) the valve is turned off briefly, a period termed the pulse width (pw), allowing the 1D effluent to essentially be reinjected onto the 2D column for the modulated separations. Modifications to the modulator assembly are provided to improve performance. Method optimization is demonstrated for a 116-component test mixture by tuning the Paux and the pw. For a PM = 2 s and 1F of 0.10 ml/min, the optimal pw and initial Paux selected were 200 ms and 330.9 kPa (33 psig), respectively. The 30 min separation of the test mixture provided a 1D peak capacity of 1nc = 330 and a 2D peak capacity of 2nc = 15, hence an ideal 2D peak capacity nc,2D = 1nc × 2nc = 4950. Likewise, the 2D peak capacity corrected for undersampling of the 1D separation was 4500 and corrected for both undersampling and sampling variation via statistical overlap theory was 4090. These results provide a 2-fold improvement in peak capacity relative to the previous DPGM study in full modulation mode for GC × GC-TOFMS. The optimized conditions were applied for a variety of applications: diesel fuel, derivatized cow serum, solid phase microextraction (SPME) of coffee headspace, and SPME of river water headspace. Additionally, the fraction of 2D separation space utilized (fcoverage), as defined by the minimum convex hull method, ranged from 0.60 to 0.85. We observed that any fcoverage correction to 2D peak capacity is highly sample dependent, since all samples, except for the diesel sample, were run with the same separation conditions, and yet the fcoverage ranged from 0.60 to 0.80.

Development of an Enhanced Total Ion Current Chromatogram Algorithm to Improve Untargeted Peak Detection.

Accurate analyte peak detection from the background noise is a fundamental step in data analysis. Often, this is initially performed on the total ion current chromatogram (TIC), which is the summed signal from all mass spectral channels. Despite the detection of many of the most abundant peaks within a chromatogram, a large fraction of peaks remains undetected in the standard TIC due to their signal being below the limit of detection. To find peaks obscured by background noise, an untargeted peak detection method termed the "enhanced TIC algorithm" was developed for comprehensive two-dimensional gas chromatography coupled with time-of-flight mass spectrometry (GC × GC-TOFMS). The reported algorithm utilizes the entire mass spectral dimension to find regions of analytical signal above a threshold while zeroing the background noise. The resulting chromatographic data is summed together to create the enhanced TIC. The utility of the enhanced TIC algorithm is demonstrated using serial dilutions from a 10 parts-per-thousand (ppth) test mixture. For the chromatograms collected at 1 and 10 parts-per-million (ppm), the enhanced TIC algorithm recovered 62% and 93%, respectively, of the original peaks observed in the 10 ppth mixture, while the standard TIC recovered only 0% and 45%, respectively. The improvement in signal enhancement was also shown on a separation of a yeast cell metabolite extract, where the enhanced TIC found 33-64% more peaks than the standard TIC. Chromatographic simulations with increasing levels of background noise were also conducted to compare the enhanced and standard TICs in the context of statistical overlap theory (SOT). Simulated chromatograms with lower signal-to-noise were more accurately modeled by the SOT after enhanced TIC processing compared to those processed by the standard TIC. The enhanced TIC method demonstrates an immense benefit in peak discovery to improve data analysis efforts.

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.

Probabilistic Model for Untargeted Peak Detection in LC-MS Using Bayesian Statistics.

We introduce a novel Bayesian probabilistic peak detection algorithm for liquid chromatography-mass spectroscopy (LC-MS). The final probabilistic result allows the user to make a final decision about which points in a chromatogram are affected by a chromatographic peak and which ones are only affected by noise. The use of probabilities contrasts with the traditional method in which a binary answer is given, relying on a threshold. By contrast, with the Bayesian peak detection presented here, the values of probability can be further propagated into other preprocessing steps, which will increase (or decrease) the importance of chromatographic regions into the final results. The present work is based on the use of the statistical overlap theory of component overlap from Davis and Giddings (Davis, J. M.; Giddings, J. Anal. Chem. 1983, 55, 418-424) as prior probability in the Bayesian formulation. The algorithm was tested on LC-MS Orbitrap data and was able to successfully distinguish chemical noise from actual peaks without any data preprocessing.

Probabilistic peak detection for first-order chromatographic data

We present a novel algorithm for probabilistic peak detection in first-order chromatographic data. Unlike conventional methods that deliver a binary answer pertaining to the expected presence or absence of a chromatographic peak, our method calculates the probability of a point being affected by such a peak. The algorithm makes use of chromatographic information (i.e. the expected width of a single peak and the standard deviation of baseline noise). As prior information of the existence of a peak in a chromatographic run, we make use of the statistical overlap theory. We formulate an exhaustive set of mutually exclusive hypotheses concerning presence or absence of different peak configurations. These models are evaluated by fitting a segment of chromatographic data by least-squares. The evaluation of these competing hypotheses can be performed as a Bayesian inferential task. We outline the potential advantages of adopting this approach for peak detection and provide several examples of both improved performance and increased flexibility afforded by our approach.

The statistical overlap theory of chromatography using power law (fractal) statistics

The chromatographic dimensionality was recently proposed as a measure of retention time spacing based on a power law (fractal) distribution. Using this model, a statistical overlap theory (SOT) for chromatographic peaks is developed that estimates the number of peak maxima as a function of the chromatographic dimension, saturation and scale. Power law models exhibit a threshold region whereby below a critical saturation value no loss of peak maxima due to peak fusion occurs as saturation increases. At moderate saturation, behavior is similar to the random (Poisson) peak model. At still higher saturation, the power law model shows loss of peaks nearly independent of the scale and dimension of the model. The physicochemical meaning of the power law scale parameter is discussed and shown to be equal to the Boltzmann-weighted free energy of transfer over the scale limits. The scale is discussed. Small scale range (small β) is shown to generate more uniform chromatograms. Large scale range chromatograms (large β) are shown to give occasional large excursions of retention times; this is a property of power laws where "wild" behavior is noted to occasionally occur. Both cases are shown to be useful depending on the chromatographic saturation. A scale-invariant model of the SOT shows very simple relationships between the fraction of peak maxima and the saturation, peak width and number of theoretical plates. These equations provide much insight into separations which follow power law statistics.

Computation of distribution of minimum resolution for log-normal distribution of chromatographic peak heights

General equations are derived for the distribution of minimum resolution between two chromatographic peaks, when peak heights in a multi-component chromatogram follow a continuous statistical distribution. The derivation draws on published theory by relating the area under the distribution of minimum resolution to the area under the distribution of the ratio of peak heights, which in turn is derived from the peak-height distribution. Two procedures are proposed for the equations’ numerical solution. The procedures are applied to the log-normal distribution, which recently was reported to describe the distribution of component concentrations in three complex natural mixtures. For published statistical parameters of these mixtures, the distribution of minimum resolution is similar to that for the commonly assumed exponential distribution of peak heights used in statistical-overlap theory. However, these two distributions of minimum resolution can differ markedly, depending on the scale parameter of the log-normal distribution. Theory for the computation of the distribution of minimum resolution is extended to other cases of interest. With the log-normal distribution of peak heights as an example, the distribution of minimum resolution is computed when small peaks are lost due to noise or detection limits, and when the height of at least one peak is less than an upper limit. The distribution of minimum resolution shifts slightly to lower resolution values in the first case and to markedly larger resolution values in the second one. The theory and numerical procedure are confirmed by Monte Carlo simulation.

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

The average value of the multivariate selectivity ( SEL) of randomly positioned peaks in a multi-component separation is shown to equal the average fraction of peaks that are singlets, as predicted by statistical-overlap theory (SOT). This equality is the basis for proposing a useful metric, specifically the average minimum resolution of nearest-neighbor peaks, for the performance of comprehensive two-dimensional (2D) separations. Furthermore this metric was computed both without ancillary spectroscopic information and with the assistance of such help, specifically multi-wavelength UV–vis spectra, acquired during the separation. Separations are simulated with randomly positioned peaks over wide ranges of total number of peaks, first- and second-dimension peak capacity, dimensionless first-dimension sampling time, and spectral diversity. The specific version of the general multivariate selectivity concept that is used here – identified as SEL – gives the relative precision of quantification when using the PARAFAC (parallel factor analysis) method, a popular curve resolution algorithm. The SEL values of all peaks were calculated, averaged, and compared to the predictions of SOT. In the absence of auxiliary spectral data, the SEL-based average minimum resolution required to separate two peaks in a 2D separation is 0.256, compared to resolution of 0.5 if no chemometric assistance is available. This was found to be valid over a wide range of conditions and is essentially independent of peak crowding. With the assistance of the spectral data, the requisite minimum resolution substantially improves, that is, it decreases, especially when peak crowding is severe. The requisite minimum resolution decreases even further, up to a limit, as the spectral diversity is increased. In contrast, the SEL-based average under-sampling correction factor is virtually independent of the presence of the additional spectral data, and additionally is about the same as calculated with SOT from the average number of maxima in closely analogous simulations. The use of selectivity greatly increases the fraction of peaks that are singlets, relative to the number of singlet maxima, especially when spectral assistance is added. The insensitivity of the under-sampling correction factor to either the use of selectivity or added spectral data simplifies optimization of the corrected peak capacity in on-line comprehensive 2D separations.

Dependence on effective saturation of numbers of singlet peaks in one- and two-dimensional separations

The average numbers of singlet peaks in one-dimensional (1D) and two-dimensional (2D) separations of randomly distributed peaks are predicted by statistical-overlap theory and compared against the effective saturation. The effective saturation is a recently introduced metric of peak crowding that is more practitioner-friendly than the usual metric, the saturation. The effective saturation absorbs the average minimum resolution of statistical-overlap theory, facilitating the comparison of 1D and 2D separations by traditional metrics of resolution and peak capacity. In this paper, singlet peaks are identified with maxima produced by a single mixture constituent. Their effective saturations are calculated from published equations for the average minimum resolution of 1D singlet peaks, and from equations derived here for the average minimum resolution of 2D singlet peaks. The fractions of peaks that are singlets in 1D and 2D separations are predicted by statistical-overlap theory as functions of saturation but are compared as functions of effective saturation. The two fractions differ by no more than 0.033 at any effective saturation between 0 and 6, when the distribution of peak heights is exponential and the edge effect is neglected. This result shows that 1D and 2D separations of randomly distributed peaks are about the same in their ability to separate singlet peaks as maxima, when assessed relative to effective saturation. Empirical equations in effective saturation are reported for the fractions of peaks that are singlets. It is argued that the effective saturation is a good metric for comparing separations having different average minimum resolutions.

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.

Gas chromatography of 209 polychlorinated biphenyl congeners on an extremely efficient nonselective capillary column

The gas chromatographic–mass spectrometric (GC–MS) separation of all 209 polychlorinated biphenyl (PCB) congeners was studied on an extremely efficient 80 m × 0.1 mm i.d. capillary column coated with a 0.1 μm film of poly(5%-phenyl methyl)siloxane stationary phase. The quality of the separation and the number of resolved and coeluting peaks were compared to predictions according to the statistical overlap theory (SOT) and to literature data on PCB separations obtained by one-dimensional and comprehensive two-dimensional GC (GC × GC) and GC–MS. Mass spectral and chemometric deconvolution procedures were used to resolve overlapping peaks. On the highly efficient column, 195 PCB congeners were resolved in 96 min separation time using spectral and chemometric deconvolution. This number is comparable to the best separations described in GC × GC–MS mode. The novel method was developed for spectral deconvolution of overlapped PCB congeners which was verified determining the most toxic, dioxin-like PCBs both in the model mixture of 209 PCBs as well as in the Aroclor 1242 and Aroclor 1254 formulations.

Effective Saturation: A More Informative Metric for Comparing Peak Separation in One- and Two-Dimensional Separations

A theoretical comparison is made of the numbers of observed peaks in one-dimensional (1D) and two-dimensional (2D) separations having the same peak capacity, as calculated from the traditional metric of resolution. The shortcoming of the average minimum resolution of statistical overlap theory (SOT) for this comparison is described. A new metric called the "effective saturation" is introduced to ameliorate the shortcoming. Unlike the "saturation", which is the usual metric of peak crowding in SOT, the effective saturation is independent of the average minimum resolution and can be determined using traditional values of resolution and peak capacity. Our most important finding is that, under a wide range of practical conditions, 1D and 2D separations of the same mixture produce almost equal numbers of observed peaks when the traditional peak capacities of the separations are the same, provided that the effective saturation and not the usual saturation is used as the measure of crowding. This is the case when peak distributions are random and when edge effects are minor. The numerical results supporting this finding can be described by empirical functions of the effective saturation, including one for the traditional peak capacity needed to separate a given fraction of mixture constituents as observed peaks. The near equality of the number of observed peaks in 1D and 2D separations based on the effective saturation is confirmed by simulations. However, this equality is compromised in 2D separations when edge effects are large. The new finding does not contradict previous predictions by SOT of differences between 1D and 2D separations at equal saturation. Indeed, the simulations reaffirm their validity. Rather, the usual metric, i.e., the saturation, is just not as simple a metric for comparing 1D and 2D separations as is the new metric, i.e., the effective saturation. We strongly recommend use of the new metric for its great simplifying effect.

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

One of the basic tenets of comprehensive two-dimensional chromatography is that the total peak capacity is simply the product of the first- and second-dimension peak capacities. As formulated, the total peak capacity does not depend on the relative values of the individual dimensions but only on the product of the two. This concept is tested here for the experimentally realistic situation wherein the first-dimension separation is undersampled. We first propose that a relationship exists between the number of observed peaks in a two-dimensional separation and the effective peak capacity. We then show here for a range of reasonable total peak capacities (500-4000) and various contributions of peak capacity in each dimension (10-150) that the number of observed peaks is only slightly dependent on the relative contributions over a reasonable and realistic range in sampling times (equal to the first-dimension peak standard deviation, multiplied by 0.2-16). Most of this work was carried out under the assumption of totally uncorrelated retention times. For uncorrelated separations, the small deviations from the product rule are due to the "edge effect" of statistical overlap theory and a recently introduced factor that corrects for the broadening of first-dimension peaks by undersampling them. They predict that relatively more peaks will be observed when the ratio of the first- to the second-dimension peak capacity is much less than unity. Additional complications are observed when first- and second-dimension retention times show some correlation, but again the effects are small. In both cases, deviations from the product rule are measured by the relative standard deviations of the number of observed peaks, which are typically 10 or less. Thus, although the basic tenet of two-dimensional chromatography is not exact when the first dimension is undersampled, the deviations from the product rule are sufficiently small as to be unimportant in practical work. Our results show that practitioners have a high degree of flexibility in designing and optimizing experimental comprehensive two-dimensional separations.

Determining under‐ and oversampling of individual particle distributions in microfluidic electrophoresis with orthogonal laser‐induced fluorescence detection

This report investigates the effects of sample size on the separation and analysis of individual biological particles using microfluidic devices equipped with an orthogonal LIF detector. A detection limit of 17 +/- 1 molecules of fluorophore is obtained using this orthogonal LIF detector under a constant flow of fluorescein, which is a significant improvement over epifluorescence, the most common LIF detection scheme used with microfluidic devices. Mitochondria from rat liver tissue and cultured 143B osteosarcoma cells are used as model biological particles. Quantile-quantile (q-q) plots were used to investigate changes in the distributions. When the number of detected mitochondrial events became too large (>72 for rat liver and >98 for 143B mitochondria), oversampling occurs. Statistical overlap theory is used to suggest that the cause of oversampling is that separation power of the microfluidic device presented is not enough to adequately separate large numbers of individual mitochondrial events. Fortunately, q-q plots make it possible to identify and exclude these distributions from data analysis. Additionally, when the number of detected events became too small (<55 for rat liver and <81 for 143B mitochondria) there were not enough events to obtain a statistically relevant mobility distribution, but these distributions can be combined to obtain a statistically relevant electrophoretic mobility distribution.

Effect of First-Dimension Undersampling on Effective Peak Capacity in Comprehensive Two-Dimensional Separations

The objective of this work is to establish a means of correcting the theoretical maximum peak capacity of comprehensive two-dimensional (2D) separations to account for the deleterious effect of undersampling first-dimension peaks. Simulations of comprehensive 2D separations of hundreds of randomly distributed sample constituents were carried out, and 2D statistical overlap theory was used to calculate an effective first-dimension peak width based on the number of observed peaks in the simulated separations. The distinguishing feature of this work is the determination of the effective first-dimension peak width using the number of observed peaks in the entire 2D separation as the defining metric of performance. We find that the ratio of the average effective first-dimension peak width after sampling to its width prior to sampling (defined as <beta>) is a simple function of the ratio of the first-dimension sampling time (t(s)) to the first-dimension peak standard deviation prior to sampling (1sigma): <beta> = square root1+0.21(t /(s)(1) sigma(2) This is valid for 2D separations of constituents having either randomly distributed or weakly correlated retention times, over the range of 0.2 </= t(s)/1 sigma < or = 16. The dependence of <beta> on t(s)/1 sigma from this expression is in qualitative agreement with previous work based on the effect of undersampling on the effective width of a single first-dimension peak, but predicts up to 35% more broadening of first-dimension peaks than is predicted by previous models. This simple expression and accurate estimation of the effect of undersampling first-dimension peaks should be very useful in making realistic corrections to theoretical 2D peak capacities, and in guiding the optimization of 2D separations.

Validation of theory of n‐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.

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 × GCs of 93 compounds having many correlated retention times, with predictions that agree fairly well with maxima numbers.