A one-dimensional reversed-phase gradient for peptides saturates, and it saturates at a level you can measure. Gilar and co-workers evaluated peak capacity experimentally for reversed-phase and cation-exchange columns under gradient conditions and compared it against reversed-phase gradient theory; both theory and experiment indicated that the number of peaks separated reaches an upper limit despite highly efficient columns or very shallow gradients, with the practical peak capacity of state-of-the-art reversed-phase columns placed at about several hundred. Doubling column length improved peak capacity by only about 40% while proportionally increasing both separation time and backpressure, and extremely shallow gradients helped at the cost of an unacceptably long run [1]. That arithmetic is what sends you to a second dimension. The same model predicted a two-dimensional peak capacity of 15,000 in 8 h, using multiple fraction collection in the first dimension followed by fast reversed-phase gradients on short but efficient second-dimension columns [1].
A prediction is a ceiling, not a promise. Two things stand between the product of the two dimensional peak capacities and what the instrument delivers: the dimensions are never fully orthogonal, and the first dimension is almost always undersampled. The traditional product rule ignores the second of these, and Li, Stoll and Carr describe it as usually overestimating true resolving power and biasing comparisons against optimised one-dimensional gradient work [6].
Orthogonality is the figure of merit, and it is estimated from retention data
Peak capacity in each dimension is necessary but not sufficient. Gilar and co-workers framed the problem structurally: 2D-LC performance depends on peak capacity in both chromatographic dimensions and on separation orthogonality, and at the time the selectivity of many LC modes for peptides was not well known and the mathematical description of orthogonality underdeveloped, which made the performance of a proposed 2D-LC system hard to estimate in advance [2]. They developed a geometric approach to describing orthogonality and applied it to estimate practical 2D-LC peak capacity across combinations of reversed-phase, strong cation exchange, size exclusion and HILIC. SCX-RP, HILIC-RP and RP-RP were all found to provide suitable orthogonality, and the RP-RP system employing significantly different pH in the two dimensions had the highest practical peak capacity of the systems investigated [2].
Which metric you compute matters, and the choice is not settled by consensus. Gilar, Fridrich, Schure and Jaworski took six peptide elution data sets acquired on different stationary phases, plotted each against reversed-phase retention data as a two-dimensional pair, added six data pairs prepared in silico, and used the set to challenge correlation coefficients, mutual information, box-counting dimensionality and surface fractional coverage computed with different hulls [3]. Correlation coefficients were found less suited to the intended purpose; box-counting dimensionality and fractional coverage turned out to be mathematically related [3]. Schure and Davis went wider, computing twenty orthogonality metrics over 47 experimental two-dimensional chromatograms drawn from two pooled sources, one a collection from Carr's laboratory and the other built from pairs of one-dimensional chromatograms previously published by Gilar and co-workers, then grading the same chromatograms subjectively with nine experienced practitioners [4]. Principal component analysis of the metrics showed none of them explaining the data variance clearly better than any other, and analysis of individual chromatograms showed different metrics favouring different chromatograms [4]. Products of two metrics correlated better with the expert percentages than single metrics did, and the combinations that performed well paired a global measure of how much of the separation space is occupied, the convex hull relative area, with a local measure of peak spacing such as the box-counting fractal dimension [4].
Read that as a constraint on what an orthogonality figure licenses you to say. It describes how the retention points of the sample you injected fill the separation plane. It is not a property of the column pair in the abstract, it does not transfer to a different peptide population, and it says nothing about whether your one critical pair lands on opposite sides of that plane.
Comprehensive and heart-cutting answer different questions
In comprehensive 2D-LC (LC×LC) the whole first-dimension effluent is cut into fractions and every fraction is separated in the second dimension. That is the configuration the product rule was written for, and the one undersampling penalises [6].
Heart-cutting (LC-LC) transfers only chosen first-dimension windows, and in both applications cited here it solves a detection problem rather than a resolving-power problem. Petersson, Haselmann and Buckenmaier used multiple heart-cutting 2D-LC-MS to obtain near-real-time mass spectra for bovine insulin related impurities present at low level, reported as much less than 0.03%, from a first dimension running a mobile phase containing high concentrations of sodium, sulphate and phosphate — conditions they note often give superior chromatographic performance but are not compatible with MS detection [18]. Against the usual alternative of offline fraction collection with solvent reduction or exchange, they report the workflow shortening from days to hours with reduced risk of poor recovery or degradation of the collected material [18]. Luo and co-workers built the same logic into an explicit desalting tool for peptides: fractions containing the compounds of interest are separated in a first dimension using an MS-incompatible mobile phase, then passed to a second-dimension method that performs fast desalting in an MS-compatible eluent before the peak reaches the Q-TOF, which removes the need to translate an established chromatographic method into MS-compatible conditions [19].
So the configurations are not competing implementations of one idea. Comprehensive mode buys separation space across the whole sample; heart-cutting buys identity for a handful of peaks you have already located, and buys it without rebuilding the method that located them.
Undersampling, and where the penalty actually falls
Cutting a first-dimension peak into fractions broadens it. Davis, Stoll and Carr simulated comprehensive 2D separations of hundreds of randomly distributed sample constituents and used 2D statistical overlap theory to back out an effective first-dimension peak width from the number of peaks observed across the entire two-dimensional separation [5]. The ratio of the average effective first-dimension peak width after sampling to its width before sampling, β, came out as β = √(1 + 0.21(ts/1σ)2), where ts is the first-dimension sampling time and 1σ the first-dimension peak standard deviation prior to sampling. The expression held for constituents with randomly distributed or weakly correlated retention times over 0.2 ≤ ts/1σ ≤ 16, and predicted up to 35% more first-dimension broadening than earlier models built on the effect of undersampling a single first-dimension peak [5].
Divide the first-dimension peak capacity by β and you have a corrected figure. Li, Stoll and Carr folded that correction into a closed-form equation for effective two-dimensional peak capacity, and the consequence runs against intuition: the speed of the second-dimension separation is not merely a way to shorten the run, it plays a vital role in setting overall peak capacity when the first dimension is undersampled, and for 2D separations much shorter than a couple of hours they found the first-dimension peak capacity far less important than is commonly believed, not needing long columns or very small particles to be worth running [6]. Potts and colleagues reached the same place from the sampling-time side, concluding that in 2D separations under an hour the second dimension matters more than the first in determining overall peak capacity and that extreme measures to enhance first-dimension peak capacity are usually unwarranted; they also examined how sample type, peptides against small molecules, influences second-dimension peak capacity and the optimum second-dimension gradient and re-equilibration times [7].
Vivó-Truyols, van der Wal and Schoenmakers quantified the total loss in a Pareto-optimality analysis that optimised analysis time, total peak capacity and total dilution together. Detection band broadening from undersampling reduced the original peak capacity by about half; the second effect, large injection volumes onto the second-dimension column, could remove about half of what remained, so the total loss in peak capacity may reach 75% of its theoretical value [8]. Their optimal modulation times corresponded to between two and three second-dimension runs per first-dimension peak [8].
That two-to-three window is not a constant. Sarrut, Rouvière and Heinisch optimised on-line RPLC×RPLC for complex peptide samples using a Pareto approach with maximum peak capacity and minimum dilution as competing objectives, and found the optimum sampling rate falling as low as one fraction per peak for very short first-dimension gradient times, below about 10 min; for first-dimension gradient times under an hour, sampling rate was a key parameter alongside column dimensions, particle size, flow rate and gradient conditions in both dimensions [10]. The sampling optimum moves with the analysis time you have chosen. Re-derive it rather than inherit it.
Solvent incompatibility, two remedies, and the salt that neither fixes
The first-dimension effluent arrives at the second column as an injection solvent you did not choose. Stoll and co-workers report that users most frequently cite this incompatibility between the solvent systems used in the two dimensions as a major obstacle to 2D-LC method development, and that solvent-strength-related incompatibility can lead to severe peak distortion and loss of resolution and sensitivity in the second dimension [11].
Dilution is the first remedy. Active solvent modulation is a valve-based approach that dilutes the first-dimension effluent with weak solvent before transfer to the second-dimension column without additional instrument hardware, and differs from fixed solvent modulation in that the diluent stream can be toggled during each second-dimension cycle [11]. Stoll and co-workers report that this eliminates the major drawbacks of the fixed arrangement, naming complex elution solvent profiles, baseline disturbances and slow second-dimension re-equilibration, and demonstrate the improvement in second-dimension separation quality using simple small-molecule probes and degradants of heat-treated bovine insulin as case studies [11].
Trapping is the second. Stationary-phase-assisted modulation replaces the storage loop with a small retentive bed that captures the fraction and releases it into the second-dimension gradient. Vonk and colleagues used it to break the usual geometric constraint in which a narrow, long first-dimension column running shallow gradients at low flow has to feed a wide, short second-dimension column running fast at high flow, a mismatch that limits first-dimension sample capacity, dilutes heavily and makes splitless coupling to MS difficult [12]. With active modulation they coupled a 0.30 mm internal diameter first-dimension column, chosen for its higher loadability, to a 0.075 mm internal diameter second-dimension column feeding a nanoelectrospray high-resolution mass spectrometer; applied to a yeast proteome digest the system generated about 20,000 MS/MS spectra in 24 h and identified 701 proteins [12].
Roca, Gargano and Schoenmakers applied the same principle at low flow with HILIC in the first dimension and reversed phase in the second, using C18 traps to focus and automatically transfer every component and so handle solvent incompatibility and dilution together. On 1–5 µg of sample with micro-electrospray at 1 µL min-1, the two-dimensional setup gave a 60% increase in peak capacity over a one-dimensional RPLC separation and a 17–34% increase in the number of proteins identified, without increasing the 2 h analysis time [13]. Sommella and colleagues combined both remedies in an on-line HILIC×RP peptidomic method, adding a dilution flow to reduce first-dimension mobile phase strength alongside trapping columns, and compared in-house packed 1.9 µm C18 traps against commercial C18 guard cartridges: the custom traps gave a fourfold signal enhancement and 58 more peptides detected, with the gain falling especially on short sequences of six residues or fewer [14].
Salt is the third problem, and dilution does not touch it. Where the first dimension runs an ion-exchange or buffered method, the fraction carries non-volatile electrolyte that the second dimension has to strip before the peak reaches the source; that is precisely the job the second dimension performs in the heart-cutting desalting configurations above [18, 19].
The pairings, and what each one buys
Reversed phase at high pH into reversed phase at low pH is the pairing that came out ahead in the peptide comparisons cited here. Gilar and co-workers evaluated reversed-phase columns in both dimensions, varying stationary phase type, ion-pairing agent and mobile phase pH, and found pH the most significant influence on reversed-phase selectivity; the greatest orthogonality was achieved with C18 columns at pH 10 in the first dimension and pH 2.6 in the second, run off-line with partial evaporation of the collected fractions, and the achievable peak capacity and overall performance compared favourably with SCX-RP [15]. The lever there is the mobile phase, not the bonded phase, which is what allows two C18 columns to behave as two selectivities.
Wang and colleagues tested the offline version of that idea against the traditional first dimension on a trypsin-digested human MCF10A cell sample: concatenated high-pH (pH 10) reversed-phase fractionation ahead of low-pH (pH 3) reversed-phase separation gave 1.8-fold and 1.6-fold increases in the numbers of identified peptides and proteins, counting proteins supported by two or more unique peptides, relative to strong cation exchange, together with improved protein sequence coverage, simplified sample processing and reduced sample losses [16].
Strong cation exchange into reversed phase remains the reference pairing that newer combinations are measured against, and it is the one Wang and colleagues describe as the more traditional method for two-dimensional shotgun work [16]. HILIC into reversed phase is the third established option; Gilar and co-workers found it among the systems providing suitable orthogonality for peptides [2], and Sommella and colleagues reported a BEH amide first dimension into a C18 peptide column second dimension delivering a peak capacity of 957 in 60 min with an orthogonality of 0.70 on the A0 scale for microalgal digest peptides [14]. If you want the three run side by side on one sample rather than across three papers, Vanhoenacker and colleagues ran SCX×RP, RP×RP and HILIC×RP as comprehensive peptide maps of the same tryptic digest of a monoclonal antibody with both diode-array and MS detection, and reported the method precise in peak volume and second-dimension retention time [17].
What the recovered peak capacity is worth
Sarrut, D'Attoma and Heinisch optimised on-line RPLC×RPLC for peptides against both peak capacity and dilution, established the need for very short and efficient second-dimension columns, typically 30 mm in length packed with 1.3 µm particles, whatever the first-dimension geometry, and then ran the predicted conditions on a protein digest [9]. Compared with the top results reported in the literature for one-dimensional RPLC separations of peptides, they obtained more than a fourfold increase in peak capacity at 200 min, 5100 against 1150, and reached an effective peak capacity of 1800 in 1 h where the one-dimensional route needed 40 h [9].
The gain has a price, and the currency is dilution. In the sub-hour comparison, Sarrut, Rouvière and Heinisch found on-line RPLC×RPLC could outperform one-dimensional RPLC for gradient times longer than 5 min; at 60 min the same peak intensity, meaning the same dilution, was observed for both techniques, but with a threefold lower injected amount in RPLC×RPLC, and the signal-to-noise ratio increased significantly in RPLC×RPLC-MS mainly through noise reduction rather than signal gain [10].
Which sets the reporting standard for a two-dimensional peptide method. State the sampling time and the first-dimension peak standard deviation so β can be computed, state the orthogonality metric by name and the data it was computed on, and report effective peak capacity rather than the product of two catalogue numbers. A bare product is an upper bound that the undersampling correction, the second-dimension injection volume and the unoccupied corners of the separation plane have already spent.