Longitudinal On-Column Thermal Modulation for Comprehensive Two-Dimensional Liquid Chromatography.

Longitudinal on-column thermal modulation for comprehensive two-dimensional liquid chromatography is introduced. Modulation optimization involved a systematic investigation of heat transfer, analyte retention, and migration velocity at a range of temperatures. Longitudinal on-column thermal modulation was realized using a set of alkylphenones and compared to a conventional valve-modulator employing sample loops. The thermal modulator showed a reduced modulation-induced pressure impact than valve modulation, resulting in reduced baseline perturbation by a factor of 6; yielding a 6-14-fold improvement in signal-to-noise. A red wine sample was analyzed to demonstrate the potential of the longitudinal on-column thermal modulator for separation of a complex sample. Discrete peaks in the second dimension using the thermal modulator were 30-55% narrower than with the valve modulator. The results shown herein demonstrate the benefits of an active focusing modulator, such as reduced detection limits and increased total peak capacity.

In Silico Screening of Two-Dimensional Separation Selectivity for Ion Chromatography × Capillary Electrophoresis Separation of Low-Molecular-Mass Organic Acids.

A prerequisite for ordered two-dimensional (2D) separations and full utilization of the enhanced 2D peak capacity is selective exploitation of the sample attributes, described as sample dimensionality. In order to take sample dimensionality into account prior to optimization of a 2D separation, a new concept based on construction of 2D separation selectivity maps is proposed and demonstrated for ion chromatography × capillary electrophoresis (IC×CE) separation of low-molecular-mass organic acids as test analytes. For this purpose, 1D separation selectivity maps were constructed based on calculation of pairwise separation factors and identification of critical pairs for four IC stationary phases and 28 levels of background electrolyte pH in CE. The derived IC and CE maps were then superimposed and the effectiveness of the respective 2D separations assessed using an in silico approach, followed by testing examples of one successful and one unsuccessful 2D combination experimentally. The results confirmed the efficacy of the predictions, which require a minimal number of experiments compared to the traditional one-at-a-time approach. Following the same principles, the proposed framework can also be adapted for optimization of separation selectivity in various 2D combinations and for other applications.

Dual-opposite injection capillary electrophoresis: Principles and misconceptions.

Dual-opposite injection capillary electrophoresis (DOI-CE) is a separation technique that utilizes both ends of the capillary for sample introduction. The electroosmotic flow (EOF) is suppressed to allow all ions to reach the detector quickly. Depending on the individual electrophoretic mobilities of the analytes of interest and the effective length that each analyte travels to the detection window, the elution order of analytes in a DOI-CE separation can vary widely. This review discusses the principles, applications, and limitations of dual-opposite injection capillary electrophoresis. Common misconceptions regarding DOI-CE are clarified.

Separation of weak acids, neutral compounds, and permanent anions using sequential elution liquid chromatography with tandem columns.

This paper discusses the development of an analytical method by an alternative separation approach, sequential elution liquid chromatography (SE-LC), to separate permanently charged ions (anions), weak acids, and neutral compounds using anion exchange and reversed-phase columns in tandem. SE-LC separates classes of compounds by group by employing two or more elution modes. Advantages to using SE-LC over conventional HPLC are a greater peak capacity and a reduced separation disorder. Importantly, the same HPLC as used for a conventional HPLC separation may be used to afford a successful SE-LC separation. Mobile phase selection and gradient optimization are integral for a successful SE-LC class separation of permanent anions, weak acids, and neutral compounds and will be discussed in detail in this paper. The most successful (best resolution and repeatability) SE-LC separation was achieved by applying isocratic elution at low pH to elute the weak acids, followed by an acetonitrile gradient to elute the neutral compounds, and last a sodium methanesulfonate gradient to elute the anionic compounds using a superficially porous C18 column coupled with a strong anion exchange (SAX) column. Repeatability (RSD) in the retention times and peak areas of the analytes was less than 0.25 % and 1.5 %, respectively.

A simulation-guided approach to the selection of RPLC columns for platform small molecule separations.

Simulations were conducted to evaluate the potential of several hundred reversed-phase columns to separate small molecules. By calculating the retention factor of compounds in randomly generated virtual mixtures via the HSM (hydrophobic subtraction model) and applying basic chromatography theory, the simulation can estimate the retention time and peak width of every virtual compound and calculate the resolution between every adjacent pair of compounds. A preferred column set based on the number of successful separations of randomly generated virtual mixtures was developed. The tandem-column liquid chromatography (TC-LC) approach can separate 53.2 % of the 16-compound samples using 20 tandem-column pairs, while a single-column approach can only separate 42.6 % of the 16-compound samples with 20 single columns. The preferred set of columns obtained from the simulation was almost the same as the empirical set of columns previously obtained. In screening applications, TC-LC can achieve a comparably successful separation factor (selectivity) with a smaller column inventory (nine 50-mm columns) compared to the larger inventory needed by single-column LC (twenty-one 100-mm columns).

Development of tandem-column liquid chromatographic methods for pharmaceutical compounds using simulations based on hydrophobic subtraction model parameters.

The liquid chromatography (LC) analysis of small molecule pharmaceutical compounds and related impurities is crucial in the development of new drug substances, but developing these separations is usually challenging due to analyte structural similarities. Tandem-column LC (TC-LC) has emerged as a powerful approach to achieve alternative separation selectivity compared to conventional single column separations. However, one of the bottlenecks associated with use of tandem column approaches is time-consuming column pair screening and selection. Herein, we compared critical resolution (Rc) in single column vs. TC-LC separations for a given set of small molecule pharmaceutical compounds and developed a column selection workflow that uses separation simulations based on parameters from the hydrophobic subtraction model (HSM) of reversed-phase selectivity. In this study, HSM solute parameters were experimentally determined for a small molecule pharmaceutical (Linrodostat) and ten of its related impurities using multiple linear regression of their retentions on 16 selected RPLC columns against in-house determined HSM column parameters. Rc values were calculated based on HSM database column parameters for a pool of about 200 available stationary phases in both single-phase column (2.1 mm i.d. × 100 mm) or tandem column paired (two 2.1 mm i.d. × 50 mm) formats. Four column configurations (two single and two tandem) were predicted to achieve successful separations under isocratic HSM separation conditions, with a fifth tandem pair predicted to have a single co-elution. Of these five potential candidates, one tandem pair yielded compete baseline resolution of the 11-component mixture in an experimental separation. In this specific case, the tandem column pairs outperformed single-phase columns, with better predicted and experimental Rc values for the Linrodostat mixture under the HSM separation conditions. The results reported in this study demonstrated the enormous selectivity potential of TC-LC in pharmaceutical compound separations and are consistent with our previous study that examined the potential of tandem column approaches using purely computational means, though there is room for substantial improvement in the prediction accuracy. The proposed workflow can be used to prioritize a small number of column combinations by computational means before any experiments are conducted. This is highly attractive from the point of view of time and resource savings considering over 200,000 different tandem column pairings are possible using columns for which there are data in the HSM database.

Are two liquid chromatography columns in tandem better than one?: Answers from the hydrophobic subtraction model.

An approach is described for determining if there is an intrinsic advantage, from a selectivity and resolution perspective, of using two different UHPLC/HPLC reversed-phase columns in tandem for a separation of a given sample compared to a single U/HPLC reversed-phase column that provides the same plate number. Retention data for 16 compounds extracted directly from the hydrophobic subtraction model (HSM) database at HPLCColumns.org are used to simulate and then compare the critical resolution of those compounds obtained using HSM conditions (isocratic elution at 35°C using 50% acetonitrile, 50% aqueous phosphate buffer at pH 2.8 or 7) for each of 662 U/HPLC single columns or 218,791 combinations of tandem columns and assuming a modest plate number of 8000. The critical resolution obtained for 16 additional "n-1" samples created by the systematic removal of one of the original 16 compounds was also compared using single- and tandem-column LC, as was the critical resolution obtained for thousands of synthetic samples generated by randomly varying HSM solute descriptors for each synthetic compound. When all possible single-column or tandem-column results were compared, a significant advantage was observed with tandem-column liquid chromatography (TC-LC), with an average increase in critical resolution of 0.63 (pH 2.8) or 0.75 (pH 7) units observed for the synthetic samples with the smallest number of components (m = 5). As the number of components in a sample increased, the average improvement in critical resolution (∆Rs,crit) using TC-LC gradually decreased from about 0.70 for m = 5 to 0.18 for m = 32 components. The average improvement in critical resolution achieved by switching from SC-LC to TC-LC was also lower when a smaller number of columns and column combinations were available to explore, as would be the case for a finite column inventory in a real laboratory. Nevertheless, on average there does appear to be an intrinsic advantage of tandem-column liquid chromatography, however small, which can be amplified by using high efficiency columns.

Determination of residual cell culture media components by MEKC.

Folic acid, hypoxanthine, mycophenolic acid, nicotinic acid, riboflavin, and xanthine are widely used as cell culture media components in monoclonal antibody manufacturing. These components are subsequently removed during the downstream purification processes. This article describes a single MEKC method that can simultaneously determine all the listed compounds with acceptable LOD and LOQ. All the analytes were successfully separated by MEKC using running buffer containing 40 mM SDS, 20 mM sodium phosphate, and 20 mM sodium borate at pH 9.0. The MEKC method was compared to the corresponding CZE method using the same running buffer containing no SDS. The effect of SDS concentration on separation, the pH of the running buffer, and the detection wavelength were studied and optimal MEKC conditions were established. Good linearity was obtained with correlation coefficients of more than 0.99 for all analytes. Specificity, accuracy, and precision were also evaluated. The recovery was in the range of 89-112%. The precision results were in the range of 1.7-4.8%. The experimentally determined data demonstrated that the MEKC method is applicable to the determination of the six analytes in in-process samples from monoclonal antibody manufacturing processes.

The effect of co-surfactant-modified micelles on chiral separations in EKC.

The use of chiral pseudostationary phases in EKC provides high efficiencies and excellent resolution for enantiomeric separations. The chiral pseudostationary phases of interest in this study are alcohol-modified ("swollen") micelles, in which a co-surfactant (medium chain-length alcohol) is added with the surfactant. In this study, the chiral surfactant, dodecoxycarbonylvaline (DDCV), along with the co-surfactant, 2-hexanol, has been prepared as swollen micelle in order to investigate the chiral separation of enantiomeric pairs. Three sets of experiments were investigated in which swollen micelle systems contained: chiral surfactant and racemic co-surfactant; chiral surfactant and chiral co-surfactant; and phase ratio increases, in which both chiral surfactant and chiral co-surfactant were employed. In the first two sets of experiments, co-surfactant concentration was held constant and the surfactant concentration was increased. In the third set of experiments, both surfactant and chiral surfactant concentrations were increased proportionally. The chromatographic figures of merit for each enantiomeric pair were investigated and compared with various chiral aggregate systems. In swollen micelle compositions using constant racemic 2-hexanol concentration, when DDCV concentration increased, enantioselectivity and resolution increased; whereas, efficiency remained constant for most of the test compounds. Compositions using constant S-2-hexanol concentration reached a maximum in all chromatographic figures of merit when DDCV concentration was increased from 2 to 3%. An increase in both surfactant and co-surfactant concentrations led to noisy baselines and chiral aggregates that were generally unstable in solution.

Simultaneous separation of small interfering RNA and lipids using ion-pair reversed-phase liquid chromatography.

RNA interference offers a novel approach for the development of new therapeutics for targets that are otherwise "undruggable" using traditional modalities. The safety and efficacy of siRNA-based therapy mainly rely on lipid or polymer-based nanocarriers to overcome inherent barriers to a systemic delivery of siRNA. A multicomponent lipid nanoparticle (LNP) system is a promising delivery platform, typically consisting of a cationic lipid, phospholipid, PEG-containing short-chain lipid, and cholesterol. Characterization and chemical analysis of the LNP formulation is important to assure drug product stability, a key consideration for chemistry, manufacturing and control strategy. Here we report an ion-pair reversed phase UHPLC method capable of simultaneously separating both siRNA and functional lipids in LNPs with a minimal retention gap for two classes of biologically essential yet chemically distinct molecules. Key chromatographic parameters critical to the separation are discussed, including the structure of the ion-pair agent, stationary phase chemistry, column temperature and an organic additive. The results showed that the retention time of siRNA is tunable by using various ion-pair reagents. The retention factor of the siRNA exhibited a first order relationship with the number of carbons in the alkyl chain of the ion-pair reagents. In contrast, the type of ion-pair reagent has no significant impact on the separation of phospholipids. Separations using a BEH phenyl column and dibutylammonium acetate as the ion-pair reagent showed satisfactory selectivity for a range of double-stranded siRNAs and phospholipids, key components for lipid nanoparticle formulations. Furthermore, the method was applied to the separation of an experimental LNP formulation, demonstrating good selectivity for siRNA, functional lipids and their potential degradation products.

Improved efficiency in micellar liquid chromatography using triethylamine and 1-butanol as mobile phase additives to reduce surfactant adsorption.

The effect of triethylamine as a mobile phase modifier on chromatographic efficiency in micellar liquid chromatography (MLC) is reported for nine different columns with various bonded stationary phases and silica pore sizes, including large-pore short alkyl chain, non-porous, and perfluorinated. Reduced plate height (h) versus reduced velocity (nu) plots were constructed for each column and the A' and C' terms calculated using a simplified Van Deemter equation introduced in our previous work. To further explore the practicality of using triethylamine in the micellar mobile phase, the efficiency of nine polar and non-polar substituted benzenes was studied on seven columns. Surfactant adsorption isotherms were measured for five columns with three micellar mobile phases to understand the relationship between adsorbed surfactant, mobile phase additive, and column efficiency. Clear improvements in efficiency were observed with the addition of 2% (v/v) triethylamine to a 1-butanol modified aqueous micellar mobile phase. This finding is supported by the lower amount of surfactant adsorbed onto the stationary phase when TEA is present in the mobile phase compared to an SDS only or a 1-butanol modified SDS mobile phase.

Aqueous size-exclusion chromatography of polyelectrolytes on reversed-phase and hydrophilic interaction chromatography columns.

The size-exclusion separation of a water-soluble polyelectrolyte polymer, sodium polystyrene sulfonate (NaPSS), was demonstrated on common reversed-phase (C18, C4, phenyl, and cyano) and hydrophilic interaction chromatography (HILIC) columns. The effect of common solvents - acetonitrile (ACN), tetrahydrofuran (THF), and methanol (MeOH), used as mobile phase modifiers - on the elution of NaPSS and the effect of column temperature (within a relatively narrow range corresponding to typical chromatographic conditions, i.e., 10 °C-60 °C) on the partition coefficient, KSEC, were also investigated. Non-size-exclusion chromatography (non-SEC) effects can be minimized by the addition of an electrolyte and an organic modifier to the mobile phase, and by increasing the column temperature (e.g., to 50 °C or 60 °C). Strong solvents such as THF and ACN are more successful in the reduction of such effects than is the weaker solvent MeOH. The best performance is seen on medium polarity and polar stationary phases, such as cyanopropyl- and diol-modified silica (HILIC), where the elution of the NaPSS polyelectrolyte is by a near-ideal SEC mechanism. Hydrophobic stationary phases, such as C18, C4, and phenyl, require a higher concentration of a strong solvent modifier (THF) in the mobile phase to reduce non-SEC interactions of the solute with the stationary phase.

Organic solvent modifier and temperature effects in non-aqueous size-exclusion chromatography on reversed-phase columns.

Common reversed-phase columns (C18, C4, phenyl, and cyano) offer inert surfaces suitable for the analysis of polymers by size-exclusion chromatography (SEC). The effect of tetrahydrofuran (THF) solvent and the mixtures of THF with a variety of common solvents used in high performance liquid chromatography (acetonitrile, methanol, dimethylformamide, 2-propanol, ethanol, acetone and chloroform) on reversed-phase stationary phase characteristics relevant to size exclusion were studied. The effect of solvent on the elution of polystyrene (PS) and poly(methyl methacrylate) (PMMA) and the effect of column temperature (within a relatively narrow range corresponding to typical chromatographic conditions, i.e., 10°C-60°C) on the SEC partition coefficients KSEC of PS and PMMA polymers, were also investigated. The bonded phases show remarkable differences in size separations when binary mixtures of THF with other solvents are used as the mobile phase. The solvent impact can be two-fold: (i) change of the polymeric coil size, and possible shape, and (ii) change of the stationary phase pore volume. If the effect of this impact is properly moderated, then the greatest benefit of optimized solute resolution can be achieved. Additionally, this work provides an insight on solvent-stationary phase interactions and their effects on column pore volume. The only effect of temperature observed in our studies was a decreased elution volume of the polymers with increasing temperature. SEC partition coefficients were temperature-independent in the range of 10°C-60°C and therefore, over this temperature range elution of PS and PMMA polymers is by near-ideal SEC on reversed-phase columns. Non-ideal SEC appears to occur for high molar mass PMMA polymers on a cyano column when alcohols are used as mobile phase modifiers.

Efficiency enhancements in micellar liquid chromatography through selection of stationary phase and alcohol modifier.

Micellar liquid chromatography (MLC) remains hindered by reduced chromatographic efficiency compared to reversed phase liquid chromatography (RPLC) using hydro-organic mobile phases. The reduced efficiency has been partially explained by the adsorption of surfactant monomers onto the stationary phase, resulting in a slow mass transfer of the analyte within the interfacial region of the mobile phase and stationary phase. Using an array of 12 columns, the effects of various bonded stationary phases and silica pore sizes, including large-pore short alkyl chain, non-porous, superficially porous and perfluorinated, were evaluated to determine their impact on efficiency in MLC. Additionally, each stationary phase was evaluated using 1-propanol and 1-butanol as separate micellar mobile phase alcohol additives, with several columns also evaluated using 1-pentanol. A simplified equation for calculation of A' and C' terms from reduced plate height (h) versus reduced velocity (nu) plots was used to compare the efficiency data obtained with the different columns and mobile phases. Analyte diffusion coefficients needed for the h versus nu plots were determined by the Taylor-Aris dispersion technique. The use of a short alkyl chain, wide-pore silica column, specifically, Nucleosil C4, 1000A, was shown to have the most improved efficiency when using a micellar mobile phase compared to a hydro-organic mobile phase for all columns evaluated. The use of 1-propanol was also shown to provide improved efficiency over 1-butanol or 1-pentanol in most cases. In a second series of experiments, column temperatures were varied from 40 to 70 degrees C to determine the effect of temperature on efficiency for a subset of the stationary phases. Efficiency improvements ranging from 9% for a Chromegabond C8 column to 58% for a Zorbax ODS column were observed over the temperature range. Based on these observed improvements, higher column temperatures may often yield significant gains in column efficiency, assuming the column is thermally stable.

Multidimensional liquid-phase separations combining both chromatography and electrophoresis - A review.

Described as intrinsically powerful building blocks for two-dimensional separations by Giddings [Anal. Chem., 56 (1984) 1258A-1270A], the coupling of chromatography and electrophoresis has been proven to enhance the resolution of a wide array of molecules in complex biological, environmental and food samples. This review provides a comprehensive overview of multidimensional chromato-electrophoretic (LC - E) and electrophero-chromatographic (E - LC) separation systems from inception to the most recent published examples. LC separation modes include reversed phase, ion exchange, and size exclusion. Electromigration separation modes include capillary, microchip or free flow electrophoresis; micellar electrokinetic chromatography; electrochromatography; and isoelectric focusing. The advantages and disadvantages of various non-gel based off-line and on-line hyphenation technologies of LC - E and E - LC are discussed, with conditions and system characteristics also provided.

Peak capacity and peak capacity per unit time in capillary and microchip zone electrophoresis.

The origins of the peak capacity concept are described and the important contributions to the development of that concept in chromatography and electrophoresis are reviewed. Whereas numerous quantitative expressions have been reported for one- and two-dimensional separations, most are focused on chromatographic separations and few, if any, quantitative unbiased expressions have been developed for capillary or microchip zone electrophoresis. Making the common assumption that longitudinal diffusion is the predominant source of zone broadening in capillary electrophoresis, analytical expressions for the peak capacity are derived, first in terms of migration time, diffusion coefficient, migration distance, and desired resolution, and then in terms of the remaining underlying fundamental parameters (electric field, electroosmotic and electrophoretic mobilities) that determine the migration time. The latter expressions clearly illustrate the direct square root dependence of peak capacity on electric field and migration distance and the inverse square root dependence on solute diffusion coefficient. Conditions that result in a high peak capacity will result in a low peak capacity per unit time and vice-versa. For a given symmetrical range of relative electrophoretic mobilities for co- and counter-electroosmotic species (cations and anions), the peak capacity increases with the square root of the electric field even as the temporal window narrows considerably, resulting in a significant reduction in analysis time. Over a broad relative electrophoretic mobility interval [-0.9, 0.9], an approximately two-fold greater amount of peak capacity can be generated for counter-electroosmotic species although it takes about five-fold longer to do so, consistent with the well-known bias in migration time and resolving power for co- and counter-electroosmotic species. The optimum lower bound of the relative electrophoretic mobility interval [µr,Z, µr,A] that provides the maximum peak capacity per unit time is a simple function of the upper bound, but its direct application is limited to samples with analytes whose electrophoretic mobilities can be varied independently of electroosmotic flow. For samples containing both co- and counter-electroosmotic ions whose electrophoretic mobilities cannot be easily manipulated, comparable levels of peak capacity and peak capacity per unit time for all ions can be obtained by adjusting the EOF to devote the same amount of time to the separation of each class of ions; this corresponds to µr,Z=-0.5.

Separation of small interfering RNA stereoisomers using reversed-phase ion-pairing chromatography.

Small interfering RNA (siRNA) therapeutics have received significant interest in recent years owing to their ability to elicit sequence-specific gene knockdown and subsequent suppression of protein expression. Chemical modifications can improve the hydrolytic stability and susceptibility of siRNAs to enzymatic degradation. One commonly used modification to improve hydrolytic stability, the replacement of the native phosphodiester linkage with a phosphorothioate moiety, introduces an additional phosphorous stereocenter into the molecule, resulting in the formation of diastereomers. The chromatographic separation of stereoisomeric siRNAs containing such phosphothioates is a challenging problem, especially when multiple phosphothioates are present within a modified siRNA duplex giving rise to multiple stereoisomers. In this study, we report an investigation into the use of an ion-pairing reversed phase UHPLC (or IP-RP UHPLC) method for the baseline separation of closely related diastereomers of an Apo-B gene targeting siRNA duplex under denaturing conditions. The related siRNA species consist of two diastereomers from the sense strand and the two pairs of diastereomers from the antisense strand. Key chromatographic parameters critical to stereoisomer separation are highlighted, including the structure of the ion-pairing agent, chemical composition of the stationary phase, and organic modifier. The method was applied to the separation of an siRNA stressed with iodine and demonstrated satisfactory selectivity for the parent siRNA and the desulfurization product.

Size-exclusion chromatography using deuterated mobile phases.

The effect of deuterated solvents in size-exclusion chromatography (SEC) was studied by comparing intrinsic viscosity measurements, SEC calibration curves, and column efficiency using water-soluble polymers. For aqueous SEC, the use of deuterium oxide slightly increases the SEC elution volume. To verify that adsorption onto the packing was absent, data from exclusion experiments were compared at 35 and 50 degrees C. Our results indicate that adsorption is not occurring for pullulan or polyethylene glycol (PEG)/poly(ethylene oxide) (PEO); for the latter, however, the elution volume increased using both D2O and H2O, indicative of slight hydrodynamic volume contraction of PEG/PEO at higher temperatures. A moderate increase in band broadening (moderate decrease in column efficiency) was observed using D2O. Finally, the effects of chloroform versus deuterated chloroform were evaluated, but no hydrodynamic volume changes were observed.

Direct determination of amino acids by hydrophilic interaction liquid chromatography with charged aerosol detection.

A chromatographic analytical method for the direct determination of amino acids by hydrophilic interaction liquid chromatography (HILIC) was developed. A dual gradient simultaneously varying the pH 3.2 ammonium formate buffer concentration and level of acetonitrile (ACN) in the mobile phase was employed. Using a charged aerosol detector (CAD) and a 2(nd) order regression analysis, the fit of the calibration curve showed R(2) values between 0.9997 and 0.9985 from 1.5mg/mL to 50µg/mL (600ng to 20ng on column). Analyte chromatographic parameters such as the sensitivity of retention to the water fraction in the mobile phase values (mHILIC) were determined as part of method development. A degradation product of glutamine (5-pyrrolidone-2-carboxylic acid; pGlu) was observed and resolved chromatographically with no method modifications. The separation was used to quantitate amino acid content in acid hydrolysates of various protein samples.

Stochastic approach for an unbiased estimation of the probability of a successful separation in conventional chromatography and sequential elution liquid chromatography.

A stochastic approach was utilized to estimate the probability of a successful isocratic or gradient separation in conventional chromatography for numbers of sample components, peak capacities, and saturation factors ranging from 2 to 30, 20-300, and 0.017-1, respectively. The stochastic probabilities were obtained under conditions of (i) constant peak width ("gradient" conditions) and (ii) peak width increasing linearly with time ("isocratic/constant N" conditions). The isocratic and gradient probabilities obtained stochastically were compared with the probabilities predicted by Martin et al. [Anal. Chem., 58 (1986) 2200-2207] and Davis and Stoll [J. Chromatogr. A, (2014) 128-142]; for a given number of components and peak capacity the same trend is always observed: probability obtained with the isocratic stochastic approach