2.1. Construction of conformation-dependent chiral descriptors

1. a. Triple products of vectors denoting distributions of atomic properties

As briefly mentioned in the introduction, chiral descriptors in this class are based on scalar triple products of three different first-order moment vectors. A well-known example of such a vector is the molecular dipole moment (Eq 2):

(2)

With the atomic partial charge and the position vector of atom i. Similar vectors were constructed by replacing the partial charge by another atomic attribute, , which may, for example, be a measure for the atom’s polarizability, its ability to form hydrogen bonds,… (Eq 3).

(3)

It is important to note that, when , the resulting moment becomes translationally variant; this is a well-known problem when attempting to calculate the dipole moment of a molecule with a nonzero net charge [22]. In this work, we mitigate this issue by translating the center of mass of any given molecular geometry to the origin prior to calculating its moment vectors. The center of mass is one of the more physically meaningful options in this respect, because it is the only choice that yields a dipole moment that is proportional to the torque an unrestrained molecule will undergo in an electric field [23].

An overview of the proposed vectors is given in Table 1. However, vector is not calculated according to Eq 3. Indeed, , as obtained by setting , is simply the position vector of the center of mass, which in our case will always be zero due to the method chosen to mitigate the translational variance of all other moments. Instead, we use a measure for the asymmetry of the mass distribution as defined in Eq 4:

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Table 1. Overview of atomic attribute-based vectors used in computing the scalar triple product to obtain the chiral descriptor.

https://doi.org/10.1371/journal.pone.0333635.t001

(4)

This is inspired by our pilot study [18], where Eq 4 was used more systematically but the robustness of the resulting descriptors with respect to small conformational changes was observed to be poor. Therefore, in the present study, we limit ourselves to the first-order moments defined in Eq 3, with the exception for .

As mentioned above, when partial charges are used for , the molecular dipole moment is obtained. By taking the absolute values of these charges instead, a vector corresponding to the distribution of polar groups is obtained. Since atomic partial charges are not rigidly defined, we chose to compute these vectors using either the partial charges from version 1.1.0 of the Charmm General Force Field (CGenFF) program [24] or the Gasteiger-Marsili charges obtained from OpenBabel version 3.1.1 [25]. This resulted in two different values for µ, the molecular dipole moment ( and ) and for the vector for polarity ( and ). To include a proxy for the strength of the London dispersion interactions, the fitted atomic polarizabilities from Wang et al. (AA-models) [26] were used for calculating . Since π interactions between the analyte and the CSP are important, a vector was introduced. The attribute x_i,pi used for this purpose was set to 1 for atoms with an aromatic CGenFF 1.1.0 atom type [27] and 0 for all other atoms. Similarly, four vectors were defined based on the hydrogen bond donating and accepting character of atoms as determined by their CGenFF atom type. In the case of , hydrogen atoms that are capable of forming hydrogen bonds (HB) get a value for x_i,hd equal to 1. Similarly, the x_i,ha of HB-accepting atoms was set to 1 for the purpose of constructing . The attribute x_i,hb was set to 1 for both HB-donating and -accepting atoms . Finally, was constructed the same way but with a value of −1 for HB-accepting atoms instead .

Based on the 11 vectors defined in Table 1, 11!/ (8! 3!) = 165 unique triple products could be constructed, each of which would constitute a chiral descriptor. To keep track of the physical meaning of all these descriptors, the 2-letter abbreviations of Table 1 were applied to indicate which vectors they consist of. For instance, descriptor msagpi is obtained by taking the scalar triple product of , , and . More generally, an arbitrary descriptor named abcdef is defined as

(5)

where the denominator is inspired by Dervarics et al. [19] and serves to reduce the dimension of the final descriptor value to a length, with which we hope to further mitigate the lack of robustness that was encountered with the more naive triple product descriptors in our pilot study [18]. Finally, it should be noted that some combinations of vectors are not expected to yield sensible results, for example those including two copies of the same vector based on different charge models (CGenFF vs. Gasteiger). Excluding such combinations, a total of 143 pure triple products of distributions of atomic properties were available for model building in the present paper. To this, 24 descriptors were added that contain shape-based properties, as outlined in the next subsection.

1. b. Shape-based descriptors

A second class of descriptors, introduced in the present study, is loosely based upon a combination of (1) the chiral recognition model proposed by Booth et al. in 1997 [20], and (2) the notion that polysaccharide-based chiral selectors present the analytes with a helical binding groove, as first proposed by Yamamoto et al. in 2002 [21]. The picture emerging from these references is that the chiral recognition process is not necessarily driven by the analyte, exhibiting different non-bonded properties when viewed from different angles, but may (also) be the result of general shape complementarity of the analyte with the aforementioned helical binding groove (or “ravine” in Booth’s terminology). Accordingly, we sought to develop descriptors that capture the degree to which the general shape of an arbitrary organic molecule exhibits a “helix-like” twist. As measures of helicity are intrinsically linked to dihedral angles, the most straightforward measure of twist would be a dihedral angle of the whole shape of the molecule. However, mathematically spoken, the calculation of a dihedral angle in a molecular context requires four points in space, necessitating a representation of the analyte as a chain of four “blobs” (which conceptually bear some degree of analogy to particles in Coarse Grained force fields [28]). This was accomplished by clustering the atomic coordinates of an arbitrary molecular conformation into four clusters, upon which our shape-based descriptors are built.

Firstly, a complete linkage spatial clustering [29] of all atomic coordinates is performed, setting the clustering cut-off such that exactly four clusters are obtained. Next, the geometric means of all atomic coordinates in each cluster are computed. As an example, the cluster centers for (an arbitrary conformation of) ibuprofen are displayed in Fig 2. Subsequently, the centers which are the furthest away from one another are labelled A and D. Finally, the two remaining centers are labelled B and C such that . This ordering of the cluster centers is generally rigorous, except that the choice between ABCD and the inverse (DCBA) is arbitrary; how this is handled is discussed below.

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Fig 2. A conformation of ibuprofen obtained from the MD simulation with its four cluster centers A-D (blue).

https://doi.org/10.1371/journal.pone.0333635.g002

A descriptor was defined that aims at representing the three-dimensional twist of the compound more explicitly. As hinted above, this descriptor is based on the dihedral angle of cluster centers A, B, C and D (Fig 2). For two mirror-image geometries, this dihedral angle will have the same absolute value but opposite sign. In addition, the closer it approaches 90°, the greater the out-of-plane angle. Combining these two properties, the sine of this dihedral angle seems a natural factor to be incorporated into a chiral descriptor, representing the molecular twist. Note that inverting the sequence (DCBA) will not affect this sign. In order for our final descriptor stwist to have the dimension of a length (which should be advantageous for robustness as argued above), we multiplied the sine of the dihedral angle ABCD, θ_ABCD, with (both) the distances of A and D to the BC axis (denoted as v and w respectively) and then divided by the magnitude of the vector q from point B to point C (q = ). This mathematical process is described by Equation 6, with the relevant quantities illustrated in Fig 2 displayed on an arbitrary conformation of ibuprofen.

(6)

Finally, Booth et al. [20] specifically assume a combination of a hydrophilic anchor point with hydrophobic interactions between the analyte and the binding ravine/groove. This assumption is confirmed by the work of Yamamoto et al. [21] and Wang et al. [30]. Accordingly, a variant of the “twist” descriptor was calculated based on a clustering of only the hydrophobic atomic coordinates, rather than all atoms (as inferred from their CGenFF atom types). This resulted in the descriptor ftwist.

1. c. Triple product-based descriptors combining molecular shape and atomic properties

A last manner we propose to develop chiral descriptors is by combing the strategy from the triple product with the shape based one. By exploiting the clustering of the atomic coordinates, we proposed an additional pair of vectors which describes the molecule’s shape. Based on the ordering described in the previous paragraph, an outer and inner shape vector could be defined as and , respectively. These two shape vectors can then be combined with a third vector from Table 1, giving rise to 11 additional descriptors of the form sosixx (i.e., scalar triple products of , and a vector from Table 1). Note how inverting the sequence ABCD inverts the signs of both and , so that the final value of the triple product is unchanged. An additional pair of vectors was defined but based on the hydrophobic shape of the molecule using the same methodology as in which only the hydrophobic atoms were clustered. More specifically, 11 descriptors of the form xxfifo were constructed using vectors and , but using the cluster centers of only the hydrophobic atoms.

2.2. Development of averaged and windowed conformation-independent chiral descriptors

As mentioned in the introduction, two variants of conformation-independent descriptors are proposed. The “averaged” descriptors are simple ensemble averages of the corresponding conformation-dependent descriptors. Conversely, to calculate a “windowed” version of an arbitrary descriptor abcdef, the descriptor values abcdef(t) of the individual snapshots t are first scaled by a factor s such that max|s.abcdef(t)| = 1. Consequently, the (normalized) histogram H(x) of the scaled descriptor x = s.abcdef(t)) is horizontally bound between −1 and 1, as can be seen on the example in Fig 3.

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Fig 3. Horizontally scaled histogram of the descriptor stwist of (R)-ibuprofen (obtained from the explicit solvent simulation. See §2.1b).

The averaged descriptor value, i.e., the average value, is situated in the shaded box. The (+) and (-) windowed descriptor values are obtained when the product between H(x) and the positive (red) or negative (blue) windowed window functions, f+(x) or f−(x), respectively, is integrated.

https://doi.org/10.1371/journal.pone.0333635.g003

To this histogram, either the positive window function or the negative window function may be applied via Eq 7, giving rise to the final positive- and negative-windowed descriptors abcdef+ and abcdef − , respectively.

(7)

As for the mathematical description of the window function and its derivation, we refer to the Supporting Information (S1 File). Note that for a converged simulation of an achiral molecule, abcdef(t) is distributed symmetrically around zero, giving rise to averaged descriptors of 0 and windowed descriptors for which the positive and negative versions have the same value. Conversely, for converged simulations of two enantiomers, the abcdef(t) distributions are each other’s opposite so that their averaged descriptors have the same values but opposite signs, and the positive and negative versions of the windowed descriptors are swapped.

3.1. Conformational sampling

3.2. Vector and conformation-independent chiral descriptor calculation

First, the attributes in Table 1 were assigned to each atom based on its CGenFF atom type. This enabled the calculation of the 11 vectors listed in Table 1 and the 4 shape-based vectors described in §2.1b for each MD trajectory snapshot. Next, the triple product and shape-based conformation-dependent descriptors were calculated (S1 Table). For the “averaged” descriptors, the arithmetic mean of the per-frame conformation-dependent descriptor values was used for the explicit solvent simulations and the “unweighted” implicit solvent-based conformation-independent descriptors. Conversely, the “weighted” variant of the descriptors included the per-conformation SGLD weight factors [36] as calculated using the awk script from the SGLD chapter of the CHARMM documentation (discarding the first 200 samples, i.e., the first 100 ps, when calculating the descriptor’s weighted averages). The “windowed” versions of the descriptors were calculated according to Eq 7 and the SI, again applying the SGLD weight factors for the “weighted” variant. All steps in the present paragraph were performed in an automated fashion using in-house scripts.

The convergence of the descriptors was monitored; their values stabilized within 100 ns of simulation time for both implicit (SGLD) and explicit solvent simulations. As an illustration, the evolution of the averaged descriptor values of ftwist, msagda and pihahb is plotted in S2 Fig for the rigid molecule ibuprofen and the flexible molecule verapamil. Note that the descriptor data were standardized using Z-scores to facilitate comparison across different descriptors or systems.

3.3. Construction of QSER models

Retention factors k were calculated as k = (t_R – t₀)/t₀, with t_R the retention time of the corresponding compound and t₀ the dead time.

Eight series of chiral descriptors (see Table 2) were considered for modelling. They included 5 series of descriptors for the molecules at their pH 9.0 protonation state: 1 set of chiral descriptors from the explicit solvent (water/ACN) simulations and 4 sets comprising both weighted and unweighted descriptors calculated from implicit solvent simulations with both solvent models (water and water/ACN). Conversely, the implicit solvent simulations of the molecules in their uncharged state were only performed using the water/ACN model, yielding a weighted and an unweighted set in addition to a 3^rd descriptor set based on the explicit solvent simulations (also in water/ACN).

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Table 2. Overview of the 8 descriptor sets used for modelling.

https://doi.org/10.1371/journal.pone.0333635.t002

QSER models were built with only chiral descriptors as explanatory variables and an alternative selectivity factor, α_RS, as response. The latter was obtained by dividing k of the R enantiomer by that of the S enantiomer, regardless of their actual elution order. Consequently, an α_RS above 1 means that the R enantiomer elutes last whereas a value below 1 implies that it elutes first. Since the same descriptor value with different signs is obtained for the two enantiomers when applying the averaged descriptors and the values for the negative and positive version are swapped for the two enantiomers when applying the windowed descriptors, the chiral descriptor values calculated on a single enantiomer can be used for modelling. Specifically, α_RS and log α_RS were linked to the chiral-descriptor values calculated on the R enantiomer. A detailed explanation of the modelling approach is provided below and illustrated in Fig 4.

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Fig 4. Overview of the different steps in QSER model building.

Prior to model building, the descriptor values were autoscaled. As a result, the contributions of the descriptors to the model can directly be assessed by comparing their respective coefficients.

https://doi.org/10.1371/journal.pone.0333635.g004

MATLAB® 2021b (The Mathworks, Natick, MA, USA) was used to build stepwise multiple linear regression (sMLR) and partial least squares (PLS) regression models [44,45]. Both sMLR and PLS models with different complexities were built and the best model was determined from leave-one-out cross validation (LOOCV) results. LOOCV was chosen because the dataset was relatively small, and this approach ensured maximal use of the available data for model development.

In the sMLR algorithm, α was set by default to 5%. Where necessary, it was subsequently increased to be able to build more than 15 models, which was required to select the final model.

The complexity of selected models was determined as follow: initially, a graph was constructed of the root mean squared error of calibration (RMSEC) and root mean squared error of cross validation (RMSECV) as a function of the model complexity. Optimal complexity is found where RMSECV goes through a minimum (RMSECV_min). However, often this is not the case and a behavior as for RMSEC (continuously decreasing tendency) is seen. Therefore, an alternative for the complexity at RMSECV_min has to be determined.

For this purpose, the relative change in RMSECV (ΔRMSECV) between consecutive models differing by one term, i.e., (RMSECV_A-1 – RMSECV_A)/RMSECV_A-1, in which A represents the model complexity, was calculated. The minimal complexity for which ΔRMSECV becomes ≤ 0.02 or that of the bending point in the curve, is determined [46]. Then the average variance (RMSECV²_average) of the subsequent 10 model complexities is calculated, as well as the variance RMSECV²_Crit, which is not significantly larger than RMSECV²_average, as determined by the following equation [47]:

(8)

with F(α, N, N) determined at α = 0.05 and N, N degrees of freedom (with N the number of compounds in the training set). The final model was then selected as the simplest model with an RMSECV² not exceeding RMSECV²_Crit.

For all selected models, RMSECV and RMSEC were calculated to validate the models. The RMSEC and RMSECV were normalized (represented as RMSEC_N and RMSECV_N), by dividing by the experimental range of the response considered, to become comparable for models on different responses. In addition, the determination coefficient r² between the experimental and predicted responses, and q², the determination coefficient of the cross validation results, which can be related to the predictive abilities of the models, were determined. Subsequently, the average relative prediction error between the experimental and predicted responses was calculated to evaluate the model performance. For log α_RS, the percentage average prediction error was backcalculated to α_RS, i.e., calculating the absolute difference of the residues between the experimental and predicted α_RS, divided by the experimental α_RS and multiplied by 100.

For log α_RS and α_RS, the predicted versus experimental responses were plotted (log α_RS was first back calculated to α_RS). Three types of prediction were evaluated. In a first instance, the prediction of the enantioselectivity is evaluated. The α_RS of a molecule was arbitrarily considered as accurately predicted when the experimental α_RS (further called x) and the predicted α_RS (further called y) do not differ by more than 0.05, thus y Є [x ± 0.05].

Secondly, it is evaluated whether the model correctly predicts that a molecule is separated or not. In this approach, a molecule was considered unseparated when x Є [0.95, 1.05], otherwise it is considered separated. When for a molecule x and y are both Є [0.95, 1.05], it is correctly predicted as unseparated. Otherwise, when x and y are both outside [0.95, 1.05], it is correctly predicted as separated.

Thirdly, the prediction of the elution sequence is evaluated, which was performed by dividing the above plot into four quadrants (See §4.1). The elution sequence was considered correctly predicted when x and y are both higher or lower than 1.0. Consequently, molecules that remained experimentally unresolved were excluded.

GraphPad Prism (GraphPad Software, San Diego, CA, USA, Version 10.0.0) was used to draw the plots. Dendrograms were created using the Scikit-learn Python package (version 1.6.0) [48].

3.4. k-Nearest Neighbours based applicability domain

An applicability domain was initially defined by applying the k-Nearest Neighbours (kNN) algorithm to the descriptor matrix. Specifically, the average distance between test set molecules and their k-nearest neighbours was calculated, and an overall average and standard deviation were determined to set a threshold. This threshold, calculated as the average plus twice the standard deviation, indicates whether a new molecule falls within the applicability domain of the model (with about 95% confidence) [49]. As distance measure, the Euclidian distance was used, and the value of k was set to 4 based on the empirical formula k = n^1/3, where n represents the number of test set compounds [49]. All analyses were conducted in Matlab® 2021b.

3.5. Reagents

Methanol (MeOH) and ACN (VWR Chemicals, Leuven, Belgium) were HPLC grade. A 0.02 M borate buffer was prepared with boric acid (Merck, Darmstadt, Germany), of which the pH was adjusted to pH 9.0 with 1 M sodium hydroxide (Fisher Scientific, Pittsburgh, PA, USA). Ultrapure water was provided by an Arium Pro UV system (Sartorius Stedim Biotech, Göttingen, Germany). The buffer was vacuum-filtered through a 0.20 µm membrane (Sartorius Stedim Biotech), mixed with the organic modifier, and the mobile phase was degassed for 15 min in an ultrasonic bath (Branson, Brookfield, CT, USA) before use.

The test set consists of 40 racemates: acenocoumarol from Novartis (Basel, Switzerland), aminogluthetimide, baclofen, cetirizine, clopidogrel, equol, medetomidine and tipifarnib from Cayman Chemical (Ann Arbor, MI, USA), atenolol, etiracetam, ibuprofen, ketoprofen, mandelic acid, metalaxyl, ofloxacin, razoxane, sulpiride, tetramisole, tolterodine, verapamil and warfarin from Sigma Aldrich (St. Louis, MO, USA), blebbistatin and bupivacaine from Enzo (Farmingdale, NY, USA), fluoxetine from USP (Rockville, MD, USA), isoxanthohumol from HWI Group (Rülzheim, Germany), laudanosine, modafinil and tamsulosine (European Pharmacopoeia Reference Standards), lisofylline from Hoechst (Frankfurt am Main, Germany), omeprazole from RTC (Steinheim, Germany), propranolol from Certa (Braine l’Alleud, Belgium), carbinoxamine, indapamide, piperitone and praziquantel from TCI (Tokyo, Japan), ondansetron from Thermo Fisher Scientific, rolipram from Apollo Scientific (Stockport, UK), salbutamol from Glaxo Wellcome (London, United Kingdom), thalidomide from MP Biomedicals (Irvine, CA, USA), lansoprazole (gift from unknown origin). The minimum purity of these compounds was 94%. For pramipexole, carvone and rivaroxaban, no racemic mixture was available. Therefore, their R and S enantiomers were purchased.

Forty-six enantiopure compounds were analyzed besides the racemates to determine the elution sequence of the enantiomers: S(-)-acenocoumarol, R(+)-baclofen, R(+)-blebbistatin, S(-)-bupivacain, S(-)-isoxanthohumol, R(+)-lansoprazole, S(+)-laudanosine, R(-)-lisofylline, R(-)-modafinil, R(-)-salbutamol, R(+)-ofloxacine, S(-)-omeprazole, R(+)-pramipexole, R(-)-rolipram, S(-)-warfarine (Cayman Chemical), S(-)-aminogluthetimide, R(+)-atenolol, S(-)-equol, S(-)-etiracetam, R(-)-fluoxetine, S(+)-medetomidine, S(+)-metalaxyl, S(-)-pramipexole, S(-)-propranolol, S(+)-razoxane, S-rivaroxaban, S(-)-sulpiride, S(-)-tetramisole, R(+)-thalidomide, R(+)-tipifarnib, R(+)-tolterodine and S(-)-verapamil from Sigma Aldrich, S(+)-carvone and R(-)-carvone from Alfa Aesar, R(-)-cetirizine, R(-)-piperitone and R(-)-tamsulosine from TCI (Tokyo, Japan), S(+)-clopidogrel from Thermo Fisher Scientific, S(+)-ibuprofen and R(-)-mandelic acid from Acros Organics (Geel, Belgium), S(+)-ketoprofen from Enzo (Antwerp, Belgium) and R-rivaroxaban from USP. The minimum purity of these compounds was 94%. For carbinoxamine, indapamide, ondansetron and praziquantel, no enantiopure compound was available. These racemic mixtures were separated into their enantiomers by means of preparative supercritical fluid chromatography, in collaboration with Prof. E. Lipka, Department of Analytical Chemistry, University of Lille, France.

Racemates and enantiopure compounds of the test set were dissolved in MeOH at a concentration of 0.5 mg/mL. The samples were protected from light and kept in the fridge until analysis.

3.6. Chromatographic conditions

The analyses were performed on a LaChrom Elite HPLC system from VWR-Hitachi (Radnor, PA, USA), composed of an L-2200 autosampler, L-2350 column oven, L-2130 pump and L-2455 DAD detector. The separations were carried out in isocratic elution mode at a flow rate of 0.5 mL/min. The injection volume was 5 µL and the detection wavelength 220 nm. The column temperature was kept at 20 °C. The system was operated by EZChrom Elite software (version 3.3.2.SP2, VWR, 2017). The first disturbance of the baseline signal was used as the dead time.

The mobile phase consisted of 0.02 M borate buffer pH 9.0 and ACN in a 60/40 (V/V) ratio. A Lux amylose-2 CSP (250 x 4.6 mm i.d., 5 µm particle size) from Phenomenex (Torrance, CA, USA) was used. Under these conditions, mandelic acid was not retained and is therefore excluded from the model building and discussion.

4.1. QSER models of α_RS and log α_RS

The experimentally determined t_R, k, α_RS and log α_RS values from the analysis of the test set compounds on Lux amylose-2 with the basic mobile phase are given in S2 Table. For each racemic mixture, the enantiomeric elution sequence was determined with the corresponding enantiopure compound. The aforementioned eight sets of descriptors (sets I – VIII in Table 2) were applied individually and combined as explanatory variables to build sMLR and PLS models for log α_RS and α_RS. These models can be found in S3–S4 Tables. As the PLS models do not perform better than the sMLR models, we will limit ourselves to discussing the latter.

In a first instance, descriptor sets I – V (Table 2) were applied individually and together to build models (Eqs S3 – S12), with the averaged and windowed descriptors in sets III, IV and V also being evaluated separately, allowing us to investigate the influence of the implicit and explicit solvent descriptors on the modelled response. No significant difference was observed between the models based on implicit water and implicit water/ACN descriptors. However, the model with explicit solvent descriptors (Eq S7) show better performance parameters and predictions than those with implicit solvent descriptors (Eqs S3-S4). The observation that a more realistic solvent model appears to yield better separation models might be an indication that the conformation-dependent descriptors as defined in this paper exhibit a physically meaningful correlation with the separation. Additionally, heat maps were made to compare the correlations between sets of descriptors (see S5 Table for which descriptor corresponds to which number). From the heat map in S3 Fig, it can be inferred that the descriptors from the implicit water simulations are strongly correlated with their implicit water/ACN counterparts (S3A Fig), while the correlation was somewhat lower between implicit and explicit solvent descriptors (S3B Fig). Considering the solvent system used in the chromatographic experiments, the explicit and implicit water/ACN descriptors were retained in our QSE(R)R modelling. Furthermore, the model’s performance was examined using either averaged or windowed descriptors, specifically focusing on their effects when applying unweighted or weighted implicit solvent descriptors. The models built with the weighted descriptors (Eqs S9-S10) showed slightly better performance compared to those using the unweighted descriptors (Eqs S11-S12).

Subsequently, the averaged and windowed descriptors from sets VII and VIII (Table 2) were applied both individually and together (Eqs S13-S15) to evaluate their performance in models developed for predicting log α_RS. Additionally, the averaged and windowed descriptors from sets VI and VIII (Table 2) were used together (Eq S16) to compare the model’s performance when employing either weighted or unweighted implicit solvent descriptors. This analysis also aimed to determine whether model improvement may be observed when applying the descriptors calculated from the uncharged molecules. The best model (Eq S14) was based on 9 descriptors (6 from set VIII and 3 from set VII) and predicts log αRS accurately for 22/42 molecules. However, it performs only slightly better than the model with both averaged and windowed descriptors (Eq S15). Notably, this model contains only one averaged descriptor and is less complex. Because this model performs best, it highlights the significance of chiral descriptors derived from uncharged molecules. This can be justified by the fact that the analyte exists in an equilibrium between its protonated and unprotonated forms when moving through the chromatographic system, where the latter form will preferably interact with the selector.

Finally, descriptor sets III – VIII were applied together, leading to models (Eqs 9–12 in Table 3) with better performance parameters than before. Here, the “averaged” and “windowed” schemes for calculating conformation-independent descriptors from a time series of conformation-dependent descriptor values (as explained in the introduction) are compared based on 4 models. In model A (Eq 9 in Table 3), log α_RS was modelled using four sets (IV, V, VII and VIII) of averaged descriptors as explanatory variables. Model B (Eq 10) used the same four sets of windowed descriptors. Models C and D (Eqs 11 and 12) used a combination of the four sets (IV, V, VII and VIII) averaged and windowed descriptors, modelled for log α_RS and α_RS, respectively. For models C and D, plots of the predicted α_RS as a function of the experimental are given in Fig 5.

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Table 3. Overview of models built from four descriptor sets (IV, V, VII and VIII given in Table 2), their performance parameters (see §3.3) and equations.

https://doi.org/10.1371/journal.pone.0333635.t003

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Fig 5. sMLR models: predicted α_RS as a function of the experimental one.

Used descriptors: sets IV, V, VII and VIII (Table 2). Modelled responses: (A) log α_RS, and (B) α_RS. The dashed lines divide the graph into four quadrants (indicated by numbers 1, 2, 3 and 4) and the black full line is the bisector. The red lines are the limits for what is considered an accurate α_RS prediction. Forty-two molecules were involved in the modelling and the red dots correspond to the experimentally unseparated molecules.

https://doi.org/10.1371/journal.pone.0333635.g005

A somewhat better model appears to be obtained when the sMLR algorithm is applied to a broader selection of descriptor sets. This is observed when comparing models A and B, because nearly all performance parameters improved. Conversely, when the sMLR algorithm was provided with all averaged and windowed descriptor sets to build model C, a significant improvement in RMSECV_N (but not in the other performance parameters) is seen in comparison with model B, thus indicating that model C would yield similar prediction errors but is more robust with regard to different selections of test set compounds. When we take a closer look at the selected descriptors in model C, it is observed that seven out of eight are from the windowed set. The dominance of the windowed descriptors in these models can be justified by the use of window functions. For the sake of argument, let us consider a single bar j in the histogram H(x) of the scaled conformation dependent descriptor x given Fig 3. The width w of this bar in x represents a (small) range in descriptor values, corresponding to the subset j of analyte conformations for which the descriptor value falls within this range. Let us now define a “constrained binding constant” with the chiral stationary phase K_i,j for only the conformational (sub-)ensemble j. Assuming that the histogram is normalized, we also know that sub-ensemble j represents a fraction of the total conformational ensemble in solution proportional with H_j: the height of bar j. This yields the following expression for the unconstrained macroscopic binding constant K_i for the full solution-phase conformational ensemble:

(13)

where the function K_i(x) is the continuous counterpart of the hitherto discrete variable K_i,j. Doing so reveals a strong analogy with the integral in Eq 7, wherein we try to obtain descriptors abcdef± that are linearly correlated with binding constants K_i^± (associated with the retention factors of the R-enantiomer (k_R) and the S-enantiomer (k_S)) and where f ^±(x) functions as a proxy for K_i(x) or K_i,j.

In pilot studies with smaller test sets and only averaged descriptors, modelling log α_RS resulted in better fitting models (results not shown). This was in line with the fact that log α_RS is directly linked with the difference in binding free energy between the enantiomers (Eq 1), which was assumed to correlate linearly with the averaged descriptor values (as also hinted by the fact that both quantities retain their values but switch signs when swapping enantiomers). However, when a model D was built for α_RS instead of log α_RS, the performance parameters further improved compared to models B and C. This can be explained by the dominance of the windowed descriptors in these models which as rationalized above would be expected to correlate with (ratios of) binding constants rather than (differences in) binding free energies, and thus with α_RS rather than log α_RS.

In the final model, 27 out of 42 α_RS (Fig 5B) values were predicted in the range of α_{RS,experimental} ± 0.05. Concerning the weighted and unweighted implicit solvent descriptors, heat maps were constructed (S4 Fig). Since these sets were strongly correlated, either the weighted or unweighted implicit solvent descriptors should be retained in our QSER models, but not both. S6 Table and S5 Fig present the performance parameters and plots, respectively, when modelling with the unweighted implicit solvent descriptors. When comparing S6 Table in the SI to the performance parameters using the weighted descriptors (see Table 3: models C and D), it is observed that the latter perform better. Specifically, the RMSEC_N and RMSECV_N obtained with the weighted implicit solvent descriptors are lower, and the r² and q² are closer to 1.0, indicating they show a better fit and predictive abilities. This is substantiated by the better prediction of the enantiomeric separation and the enantioselectivity.

Since the windowed descriptors from the explicit solvent simulations (sets V and VIII) are particularly prominent in our models, we tried using only these descriptors to build models for both log α_RS (Eq S19 and S6A Fig) and α_RS (Eq S20 and S6B Fig). The model for α_RS showed better performance parameters and predictive abilities compared to the log α_RS model (S7 Table), but overall, the models with wider descriptor sets discussed above performed even better.

Models C and D (Table 3) exhibited the highest performance of all models evaluated in this study. They show the best performance parameters and provided reliable predictions regarding elution sequence, separation and enantioselectivity.

An applicability domain for the test set was established based on the chiral descriptors (averaged and windowed descriptors from sets IV, V, VII and VIII in Table 2) retained in the final models (models C and D in Table 3). This was initially done using the kNN algorithm as detailed in §3.4; the results and threshold are provided in S2 File. Additionally, principal component analysis (PCA) and robust PCA, following Hubert et al. [51], were also used to define the applicability domain. According to the kNN-based threshold, two molecules (tipifarnib and cetirizine) fell outside the domain or were borderline. PCA identified three (tipifarnib, clopidogrel and cetirizine), and robust PCA four (tipifarnib, clopidogrel, cetirizine and blebbistatine). However, these molecules were not removed from the test set, because excluding them did not improve the model performance compared to model D (Table 3 versus SI) and neither could their outlying character be substantiated with experimental or structural arguments. However, the descriptor vectors of new molecules can be tested against the applicability domain to verify the validity of their predictions.

In conclusion, satisfactory models were obtained for the α_RS of a chemically diverse set of analytes by applying the averaged and windowed descriptors calculated in the native protonation state in the mobile phase as well as the uncharged state. Such models may potentially be useful for the prediction of the enantioselectivity, separation and elution sequence of chiral molecules.

4.2. Evaluation of the chiral descriptors

4.2.1. Distribution of chiral descriptor values.

The averaged chiral descriptors calculated from MD simulations in implicit water, implicit water/ACN and explicit water/ACN were compared for two molecules with a different conformational flexibility, i.e., ibuprofen and verapamil (Fig 6). As representative examples, the descriptors msagda and ftwist were selected because they are present in some models, indicating their importance for predicting the enantioselectivity, and because msagda is a triple product descriptor while ftwist is a twist descriptor. For all solvent models, the distribution of the descriptor values is wider for verapamil than for ibuprofen, in line with the former’s higher conformational flexibility.

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Fig 6. Chiral descriptor value box plots for msagda (see Table 1) and ftwist (see §2.1b).

The descriptors are obtained from MD simulations in explicit water/ACN (10,000 snapshots), for ibuprofen and verapamil.

https://doi.org/10.1371/journal.pone.0333635.g006

4.2.2. Correlation between chiral descriptor values.

In this section, the correlation between implicit and/or explicit (averaged and/or windowed) solvent descriptors was plotted to assess whether complementarity exists between the descriptors. Strong correlations between descriptors can negatively impact modelling. Therefore, it is important to identify which descriptor sets are highly correlated.

To visualize the correlation between the averaged chiral descriptors, 2D heat maps of their correlation coefficients r were constructed. In addition, a hierarchical (complete linkage) clustering was performed using (1 - |r|) as a distance function, giving rise to dendrograms. In S7 Fig, the heatmaps are shown from the averaged chiral descriptors calculated from the molecules at their pH 9.0 state simulated in implicit water/ACN (S7A Fig) and explicit water/ACN (S7B Fig). The lower amount of yellow in the former plot suggests that the implicit solvent descriptors are slightly more diverse, which is confirmed by the higher number of branches below 1 - |r| < 0.2 in dendrogram S9 Fig compared to S8 Fig, and is explained by the use of an enhanced sampling method (SGLD) in the implicit solvent simulations, which gave rise to higher conformational diversity. Upon closer examination of the dendrograms, it was observed that chiral descriptors containing at least one HB vector and at least one non-HB vector (e.g. ), often appear together at (1 - |r| < 0.2), thus showing a high correlation (see S5 Table for which descriptor corresponds to which number).

Subsequently, the bisecting line on a correlation heat map for the chiral descriptors (implicit versus explicit) of uncharged chiral molecules (S10 Fig) shows that descriptors from implicit solvent simulations are highly correlated with their explicit-solvent counterparts. This aligns with the observation that Eq S9 (including both implicit and explicit solvent descriptors) has predictive abilities similar to Eq S7 (only explicit solvent descriptors).

Finally, heat maps for chiral descriptors from both the molecule’s native protonation state in the mobile phase and its uncharged state show modest correlation in implicit solvent (Fig 7A) and almost none in explicit solvent (Fig 7B). This demonstrates the strong impact of the protonation state on a molecule’s conformational preference and noncovalent interaction profile. In addition, it might suggest a shortcoming of the implicit solvent model to fully capture this impact, although the difference between the implicit and explicit solvent simulations might alternatively be due to the use of the SGLD enhanced sampling method in the former. Either way, it can be concluded that descriptors obtained from simulations on (de)protonated (charged) and uncharged molecules differ substantially.

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Fig 7. Heat maps from the correlation coefficients calculated between the averaged chiral descriptors obtained from different MD simulations from the test set molecules in their actual state at pH 9 and their uncharged state.

The correlation is calculated between descriptors from charged and uncharged molecules in (A) implicit water/ACN and (B) explicit water/ACN. Numbers 1–167: number of a chiral descriptor, given in S5 Table.

https://doi.org/10.1371/journal.pone.0333635.g007

As explained in the introduction, the windowed descriptors come in pairs consisting of a negatively and a positively windowed version, which are directly related to the amount of sampling of negative and positive values of the respective conformation-dependent descriptor during the simulation. S11 Fig indicates a high correlation when comparing both versions for the same MD simulation approach; it would appear that high values of the (+)-windowed descriptor correspond to low values of its (-)-windowed counterpart and vice versa. Consequently, for the purpose of assessing the correlations between the averaged and windowed descriptors, it suffices to only consider their (-)-windowed versions. Examining the correlation between these (-)-windowed descriptors resulting from simulations of (de)protonated analytes yields similar heat maps and the same conclusions as for the averaged descriptors: the protonation state makes a substantial difference, and this seems somewhat more pronounced in explicit solvent (though the latter difference is smaller than for the averaged descriptors). Finally, the correlation is compared between the windowed and averaged chiral descriptors (S12 Fig). When comparing these descriptors calculated from the same simulation, they show a high similarity (yellow-colored diagonal).

In summary, the descriptors calculated from implicit water and implicit water/ACN MD simulations show a high correlation, while they show a lower correlation with those calculated from explicit solvent simulations. Furthermore, it was observed that the windowed descriptors appear to perform better than the averaged ones if only one approach is included in the models. However, they show complementarity: the best models were obtained using both types of descriptors simultaneously. In addition, descriptors obtained from molecules in their uncharged state are important for the modelling and show complementarity with the descriptors derived from molecules in their pH 9 state. Finally, it was observed that the implicit solvent descriptors calculated using the SGLD weight factors yielded somewhat better models than their unweighted counterparts. Combining all these observations, we expect and see that the best performing models would be obtained by using the averaged and windowed descriptors from molecules in both their charged and uncharged state calculated from implicit and explicit MD simulations in a mixture of water/ACN.

It is apparent from Table 3 that the conformation-dependent descriptors agsiso, achdda, gsalhb and mschag occur with prominent coefficients in all three models B, C, and D, which supports the consistent nature of our modeling methodology. To further interpret the models from Table 3 in terms of the selected descriptors, it is helpful to divide the vectors that constitute the triple product descriptors into the following rough categories:

• ms, al: related to strength of London Dispersion force;
• ch, gs, da: related to (signed) charge;
• ac, ag, hb: related to hydrophilicity (regardless of positive or negative charge).

(Note that pi, hd, ha, siso and fifo are left uncategorized because they are not as closely related to other properties.) In the entirety of the final models, only 1 descriptor was selected that contained two vectors in the same category (acgshb-, containing both ac and hb, in Model C, albeit with a relatively low coefficient). This further supports the idea that the modeling methodology gravitates toward meaningful triple products (which combine significantly different vectors) and also opens the future perspective of more aggressively eliminating descriptors containing “similar” vectors (which we are currently doing only for [ch, gs], [ac, ag] and a few particularly redundant combinations of properties related to hydrogen bonds).