Calculation of the Three Partition Coefficients logPow, logKoa and logKaw of Organic Molecules at Standard Conditions at Once by Means of a Generally Applicable Group-Additivity Method

1. Introduction

Environmental considerations of organic molecules as potential contaminants have become an important subject in recent years. Several descriptors have been applied to quantify their impact on the natural environment, among them the octanol/water partition coefficient P_ow, (more recently named K_ow) a standard model for the description of the lipophilicity of drugs in the medicinal and agricultural chemistry, whereby octanol is the substitute for the natural organic matter, and the octanol/air partition coefficient K_oa and the air/water partition coefficient K_aw which both indicate the role of the chemicals for air-breathing organisms [1,2,3]. In view of the time consumption and costs of their experimental determination, fast mathematical methods for the prediction of their value attributed to a molecule have been developed, all founded on the results of previous examinations on a relatively limited number of compounds. For the former coefficient P_ow a number of authors [4,5,6,7,8,9,10,11] have successfully carried out calculations for a large variability of compounds based on various group-additivity methods. On the other hand, many publications [12,13,14,15,16,17,18,19,20,21,22,23] dealing with the prediction of the coefficient K_oa, based on various QSPR methods, are limited to specific chemical families, thus lacking a general applicability. Li et al. [24] used a group-additivity method based on five fragment constants and one structural correction factor for the evaluation of K_oa, limited to halogenated aromatic pollutants. Recently, Ebert et al. [25] suggested a general-purpose fragment model for the calculation of the air/water partition coefficient K_aw resembling the atom-group additivity method presented in one of our earlier papers [11] for the calculation of – among several further descriptors – the octanol/water partition coefficient P_ow.

The goal of the present paper was to suggest the extension of a simple tool, which has already served well for the prediction of the octanol/water coefficient P_ow (in its logarithmic value logP_ow) described in [11], to enable it to calculate all three mentioned partition coefficients at once by means of a uniform computer algorithm based on the atom-group additivity method detailed in [11]. Since under common standard conditions any third partition coefficient can be directly calculated from the other two if we neglect the effect of the contamination of water in octanol (and vice versa) influencing the determination of the Pow values which will be addressed later on, it made sense to select those two coefficients for which any group parameters could be founded on the most reliable as well as the largest number of experimental data. It turned out that the experimental data for the partition coefficients Pow and Koa provided excellent basis sets for the evaluation of their respective tables of atom- and special group parameters. Accordingly, from the subsequently calculated values of a molecule´s P_ow and K_oa its air/water partition coefficient K_aw should easily be evaluable following the equation K_aw ≈ P_ow / K_oa.

2. Method

The calculation method is based on a regularly updated object-oriented database of more than 37100 compounds stored in their geometry-optimized 3D structure encompassing pharmaceuticals, herbicides, pesticides, fungicides, textile and other dyes, ionic liquids, liquid crystals, metal-organics, lab intermediates and many more, collecting – among further molecular experimental and calculated descriptors – a large set of experimental P_ow, K_oa and K_aw data in their logarithmic form, outlined in the respective sections below. Accordingly, their terms will henceforth be named logP_ow, logK_oa and logK_aw. It should be stressed that for the calculation of the partition coefficients the 3D geometry-optimized form of the compounds is not required - except for the algorithm-based determination of intramolecular hydrogen bridges, the impact of which will be discussed further down. In order to avoid structural ambiguities in the presentation of the chemical structures to the computer algorithm defining the molecules´ atom groups, a special algorithm ensured at the time of the input of a new compound that any six-membered aromatic ring system is defined by six aromatic bonds instead of alternating single-double-bonds.

2.1. Definition of the Atom and Special Groups

The details of the atom-group additivity model applied in the present study have been outlined in [11]. Accordingly, the definition of the atom types and their immediate atomic neighbourhood and meaning are retained as described in Table 1 of [11] and are also valid for both the logP_ow and logK_oa descriptors. However, since these atom groups are not able to cover certain additional structural effects such as intramolecular hydrogen-bridge bonds and the influence of saturated cyclic compared to saturated noncyclic systems, a number of additional special groups had to be introduced. In a paper applying a different group-additivity method for the calculation of logP_ow, Klopman et al. [6] discovered that the inclusion of a correction value per carbon atom in pure saturated and unsaturated hydrocarbons improved the compliance with experiment. This has indeed been confirmed in the present study.

In order to take account of these and further potential structure-related peculiarities, the list of atom groups has been extended by “special groups” for which the column-title terms “atom type” and “neighbours” in the subsequent tables should not be taken literally, but which the computer algorithm treats in the same way as ordinary atom groups. In Table 1, the respective special groups, their nomenclature and meaning are detailed. In order to enable a future comparison of the contributions of the special-group parameter sets within this study, the same special groups have been applied for the calculation of both descriptors logP_ow and logK_oa.

At present, the list of elements is limited to H, B, C, N, O, P, S, Si, and halogen, but an extension is always possible, provided that corresponding molecules with experimental descriptor data are available.

2.2. Calculation of the Atom- and Special-Group Contributions

Since the algorithm for the evaluation of the parameter values of the atom groups has been outlined in detail in [11], its four steps may just be summarized as follows: the first step encompasses the selection from the database of at present more than 37100 compounds of all the compounds for which the experimental descriptor data in question are known and their storage in a temporary compounds list. In the second step, the molecules in the temporary list are broken down into their constituting atom groups, whereby their central atoms, called “backbone atoms”, are characterized in that they are bound to at least two covalently bound neighbour atoms. The atom groups´ atom types and neighbour terms are generated according to the rules described in [11] and their occurrence is registered. Any molecule carrying an atom group that is not found in the pre-defined group-parameters list is discarded from the temporary compounds list. The third step generates an M x (N + 1) matrix, wherein M is the number of molecules, N + 1 is the number of pre-defined atom groups plus the container for the molecule´s descriptor value, and wherein the matrix element (i,j) contains the the number of registered occurrences of the jth atom group in the ith molecule. Atom groups and their related jth column, which are not present in any molecule of the temporary molecules list, are removed from the M x (N + 1) matrix. In the final step, this adjusted matrix is normalized into an Ax = B matrix, followed by its balancing by means of a fast Gauss–Seidel calculus [26] to receive the atom- and special-group parameters x. These parameters are then added to their related atom and special group in the corresponding parameters table assigned to the specific descriptor.

The group-parameters calculation is then immediately followed by the computation of each molecule’s descriptor value in question on the basis of these group parameters according to equation (1) outlined in the next section and compared with its experimental value to receive the related statistics data which are finally added at the bottom of the parameters table. Following the above-mentioned procedure resulted in the two parameters sets in Table 2 and Table 3, designed for the calculation of the molecules’ logP_ow and logK_oa values, respectively.

2.3. Calculation of Descriptors logP_ow and logK_oa

Based on the aforementioned respective atom-group parameters tables, the descriptors logP_ow and logK_oa of a molecule can now easily be calculated by simply summing up the contribution of each atom and special group occurring in the molecule, following the equation (1), wherein i and j is the number of atom groups A_i and special groups B_j respectively, a_i is the contribution of atom group A_i, b_j the contribution of special group B_j, and c the constant listed at the top of the respective parameters table.

Descriptor calc = ∑a_iA_i + ∑b_jB_j + c

(1)

In Table 4, a typical example is presented with endosulfan sulfate, demonstrating the ease of the calculation of logK_oa for which the experimental value was 9.68 [25]. Note that the term "endocyclic bonds" only concerns C-C single bonds.

Figure 1. Endosulfan Sulfate (graphics by ChemBrain IXL).

Figure 1. Endosulfan Sulfate (graphics by ChemBrain IXL).

Evidently, the group-additivity method is limited to the calculation of a molecule‘s logP_ow or logK_oa for which a parameter value in the Tables 2 or 3 respectively is known for each atom group that is found in the molecule. Beyond this, since the reliability of these parameter values increases with the number of independent molecules upon which their calculation is based, the lowest reliability limit for these parameters was set to three molecules which as a consequence excluded any atom group being based on less than three molecules from further calculations. Accordingly, only atom groups for which the number of molecules is three or more, shown in the rightmost column of Table 2 and Table 3, have been accepted as “valid” for descriptor calculations. This explains the lower number of molecules for which the logP_ow and logK_oa have been calculated (lines B, C and D in Table 2 and Table 3) than the number upon which the evaluation of the complete set of parameters is based (line A).

2.4. Cross-Validation Calculations

Plausibility of the group-parameters calculations has immediately been checked applying a 10-fold cross-validation algorithm, which comprises 10 recalculations of the complete set of group parameters, whereby before each recalculation every other 10th compound of the total compounds‘ list has temporarily been removed from calculation and separated into a test list, thus ensuring that each molecule has played once the role of a test sample. The combined test data have then been statistically worked up and their results added to the Table 2 and Table 3 at the bottom in lines E, F, G and H. It may be noticed that the total number of test compounds shown in the right-most column of the statistics lines is lower than that in the training set in lines B, C and D; this is a consequence of the requirement that only “valid” atom groups are to be used for descriptor calculations, and due to the 10% lower number of training samples in each recalculation, the number of “valid” atom groups (as defined in the prior section) tends to decrease to an unpredictable degree. Atom groups, which are represented by less than three molecules as shown in the rightmost column and are thus not “valid” for descriptor calculations, are therefore remnants of the parameters calculation based on the complete compounds set (line A in the Table 2 and Table 3). Nevertheless, they have deliberately been left in the Table 2 and Table 3 for use in future calculations with additional molecules potentially carrying under-represented atom groups in this ongoing project.

3. Sources

3.1. Sources of logP_ow Values

The majority of the experimental logP_ow data originates from the comprehensive collection of Klopman et al. [6], supplemented by works of Sangster [27] and Lipinski et al. [28], already cited in [11]. Additional data have been provided for unsubstituted and substituted, saturated and unsaturated hydrocarbons, alcohols and esters in the works of Tewari et al. [29], for heterocycles, hetarenes and carboxylic acids by Ghose et al. [4,5], complemented for amines, amides and nitro derivatives by Leo [8]. Further data for the aforementioned compound classes have been found in papers of Abraham et al. [30], for certain drugs by Hou and Xu [10] and Wang et al. [9], for organophosphorus derivatives by Czerwinski et al. [31], for the specific energetic compound 2,4-dinitroanisole by Boddu et al. [32], for a number of fluorobenzenes, -anilines and -phenols by Li et al. [33] and finally for a number of pesticides and oil constituents in a paper of Saranjampour et al. [34].

3.2. Sources of logK_oa Values

Recently, Ebert et al. [35] published a comprehensive collection of more than 2000 experimental logK_oa values upon which the present study is essentially based. This set of data has been complemented with data for 75 chloronaphthalene derivatives by Puzyn et al. [36], for 14 PAHs by Odabasi et al. [37], for some methylsiloxanes and dimethylsilanol by Xu and Kropscott [38] and for ethyl nitrate by Easterbrook et al. [39].

3.3. Sources of logK_aw Values

Ebert’s paper [25], cited in the introductory section, presented in their supplementary information a large collection of experimental logK_aw data, which served as reference values for the calculated data. Sander [40] provided an extensive library of Henry’s law constants for more 2600 compounds which, after translation into logK_aw values at 298.15K, complemented Ebert’s data set.

4. Results

4.1. Partition Coefficient logP_ow

As shown at the bottom of Table 2, the number of molecules upon which the present group-parameters set is based is with 3332 substantially larger than the 2780 samples in our earlier paper [11]. Beyond this, the significantly better statistical results in Table 2 (lines B to H) with e.g. a cross-validated standard deviation S of 0.42 (line H) vs. the earlier value of 0.51 is the result of the removal of molecules from the parameters computation for which the experimental value deviates by more that three times the value of S. The thus 122 removed molecules (3.5% of the total set) have been collected in an outliers list, available in the Supplementary Materials. The larger number of compounds for the group-parameters computation not only significantly improved the statistical results but also enlarged the list of "valid" atom groups from 195 to 214, enabling the calculation of the logP_ow value of at present 29462 molecules (79.4% of the total dataset). The correlation coefficients R² of 0.9648 and (cross-validated) Q² of 0.9599, based on 3246 and 3164 molecules respectively, are significantly better than in our earlier paper [11] and clearly outperform Klopman’s [6] results, which are based on less than half the number of molecules used in the present case and reveal an R² and Q² value of 0.93 and 0.926 respectively, applying a group-additivity method which is comparable to ours. As shown in the correlation diagram of Figure 2 and the histogram for Figure 3, the experimental logP_ow values range from -4.6 to +12.53 with a cross-validated standard error S of 0.42 with a fairly even Gaussian error distribution.

It is worth mentioning that the observation discussed in our earlier paper (see Table 9 in [11]) concerning the two forms of amino acids (nonionic or zwitterionic) is not only confirmed by the new and extended group-parameters set of Table 2, but that the logP_ow differences in nearly all cases even more clearly distinguish the two forms. On the other hand, the ambiguous results concerning the keto/enol forms of the compounds listed in Table 10 in [11] could not be lifted by the new parameters set, which is not surprising in view of the sometimes strong solvent dependence of the equilibrium, as exemplified with acetylacetone [41]. In view of the discussion of certain particularities concerning the subsequent calculation of the third partition coefficient logK_aw in section 4.3., it should be stressed at this point that the calculated logP_ow values for the hydrocarbons do not show any abnormal or systematic deviations from experimental values.

4.2. Partition Coefficient logK_oa

The calculation of the group-parameters set of Table 3 used for the prediction of the logK_oa values essentially based on the curated data set provided in Ebert’s paper [35], whereby compounds with just one "backbone atom" such as the halomethanes or hydrocyanide had to be omitted as they are obviously not calculable by the present method. After the removal of another 129 compounds as outliers (6.36% of the total), following the same exclusion criterion as in the previous section, 1900 samples with their experimental data (line A in Table 3) remained for the computation of the group-parameter values. Again, the outliers have been collected in a separate list available in the Supplementary Materials for readers who might want to re-evaluate their logK_oa values.

The subsequent calculation of the logK_oa values of 1829 training and 1765 test molecules based on 167 "valid" atom and special groups (line A) revealed excellent statistical results with a correlation coefficient R² of 0.9765, a standard deviation s of 0.44 (lines B and D), and a cross-validated Q² of 0.9717 with a corresponding S of 0.48 (lines F and H), visualized in the correlation diagram on Figure 4 and the histogram on Figure 5. These statistics data even outperform those given in Ebert’s paper and thus also their competing methods mentioned therein such as COSMOtherm [42] and EPI-Suite KOAWIN [43], not only confirming the versatility but also the reliability of the present group-additivity approach, which allowed the calculation of the logK_oa value for 27044 molecules (72.9% of the entire database). Again, it should be kept in mind that just like in section 4.1. there could not be observed any particularly large or systematic deviations between the experimental and calculated logK_oa values for the hydrocarbons.

4.3. Partition Coefficient logK_aw

Once the partition coefficients logP_ow and logK_oa have been calculated by means of the group-additivity method based on Table 2 and Table 3 respectively, it was easy to determine the logK_aw values, applying equation (2) on each molecule in the database for which both descriptors have been calculated, adding up to 26220 molecules. In order to assess the quality of the logK_aw values, it is important to recognize the flaws of this approach: while the logP_ow values have been experimentally measured in a mixture of water-saturated octanol and octanol-saturated water, the logK_oa measurements occurred in dry octanol, an aspect that has been discussed in detail by Ebert et al. [35]. Hence, equation (2) serves only as an aproximation. In addition, since both descriptors on the right side of the equation appear with their own standard error, the error-propagation rule stipulates a standard error of logK_aw that is clearly larger than any of the two constituting descriptors. Entering the standard errors S for the test molecules of 0.42 (for logP_ow) and 0.48 (for logK_oa) into an error-propagation calculation, the expected standard error S for logK_aw is 0.638.

logK_aw (calc) = logP_ow (calc) - logK_oa (calc)

(2)

In order to test the reliability of the thus calculated logK_aw values, a representative number of experimentally determined logK_aw data, extracted from the comprehensive databases of Ebert et al. [25] and Sander [40], have been added to the database. In the latter case, the Henry’s law solubility constants H_s^cp have been translated into the corresponding logK_aw values at 298.15K. The comparison of the calculated with the experimental logK_aw values is visualized in the correlation diagram of Figure 6 and the histogram in Figure 7.

The complete set of experimental data has been separated from the outliers, applying the same exclusion conditions as for the logP_ow and logK_oa values, and the outliers have been collected in a corresponding list, available in the Supplementary Material. Comparison of the remaining dataset with the calculated values yielded a standard error of 0.67, slightly higher than predicted by the error-propagation calculation. A detailed analysis of the experimental data revealed two potential explanations for the inordinate scatter: 1) Within a series of substitution isomers, e. g. the tetra- or hexachlorobiphenyls, the tri- or pentachlorodiphenyl ethers or the dichloroanisoles, the experimental logK_aw values varied in a range of up to and over 1 unit, which is hard to assign to the specific positioning of the substituents. At any rate, the group-additivity-based calculation of the logP_ow and logK_oa values is not able to distinguish between these substitution isomers. 2) Sander’s comprehensive database of Henry’s law constants [40], listing the experimental H_s^cp values for a compound originating from various authors, showed for many compounds large differences of their H_s^cp values, in some cases exceeding one unit after translation into logK_aw, e. g. for undecane, acetylacetone or anthraquinone.

A thourough analysis of the correlation diagram in Figure 6 and the histogram in Figure 7 revealed an interesting peculiarity, visible as an indentation at the upper end of the correlation diagram and as a weak hump on the right side of the histogram: except for some siloxanes with experimental logK_aw values above 1.6 and normal scatter about calculated values, the predicted logK_aw for the remaining compounds having experimental logK_aw values above -1.0 are nearly systematically too low by ca. 0.5 - 1 units. It turned out that they are all pure hydrocarbons, in particular alkanes, alkenes and alkynes. The correlation diagram of the logK_aw data in Figure 8, focussing on these hydrocarbons, confirms this observation.

Since, as was mentioned in section 4.1. and 4.2., no particularly large or systematic deviations between the experimental and calculated logP_ow and logK_oa data for the hydrocarbons could be detected, a potential explanation for this peculiarity might be based on the experimental conditions for the determination of the logP_ow values as mentioned by Ebert et al. [35]: since water-saturated octanol is a more polar solvent than pure octanol, whereas octanol-saturated water is less polar than pure water, the experimental logP_ow values, measured in an octanol/water mixture, tend to be shifted to smaller absolute values than theory would predict. While this is true for all measured solutes, it is possibly most effective for the least polar solutes such as the mentioned hydrocarbons, thus leading to experimental logP_ow values that are particularly low for the hydrocarbons. As a consequence, their calculation based on the group-additivity method predicts equally low logP_ow values, which again leads to low logK_aw data when equation (2) is applied and then compared with experimental logK_aw values that are determined under pure air/water conditions.

4.4. Interpretation of the Special-Groups Contributions for logP_ow and logK_oa, and ultimately for logK_aw

While the atom-group parameters are descriptor-specific and their comparison between descriptors does not make sense, special groups serve as differentiators of molecules that carry these groups from those that do not. Therefore, their meaning is descriptors-overlapping, their values however must be viewed in the context of the value range of the descriptors. In the present case, the value ranges of logP_ow and logK_oa are similar (ca. 17 log units) and in the same area, a direct comparison of the special-group contributions in Table 2 and Table 3 is permissible and led to a few interesting observations: while the groups "(COH)n", "Alkane", "Unsaturated HC" and "Endocyclic bonds" in both tables only contributed to a minor degree (but nevertheless improved the statistical results) and consequently showed only minor differences between the two tables, a significant differentiation was found for the groups "H / H Acceptor" and "(COOH)n". The former special group, taking account of intramolecular hydrogen bridges, indicates a small but clear higher tendency of a compound carrying an intramolecular H-bridge towards the octanol side in an octanol/water mixture than without, thus raising the logP_ow value. In contrast, the same H-bridge-carrying molecule has its inclination significantly shifted more to the air side in an octanol/air environment than without H-bridge, expressed in a lower logK_oa value. The reason may be found in the lower solvent-solute interaction caused by the H-bridge being bound intramolecularly, leading in both cases to a preference of the less polar of the respective two media. A typical example is the compounds couple 2- and 3-nitroaniline, sampled in Table 5, where the former molecule carries a H-bridge between an amino-H and an oxygen of the nitro group.

An inverse effect can be found with molecules carrying two or more carboxylic acid functions: while the additional contribution of a second or third COOH function shows little effect in an octanol/water environment with a slightly increased shift towards water, leading to a lower logP_ow value, in an octanol/air environment each additional COOH group drastically tilts the equlibrium towards the octanol side, thus strongly raising the logK_oa value. This may be demonstrated by the couple of hexanoic/1,6-hexanedioic acid, where both have the same carbon-chain length but where the second molecule carries two carboxylic acid functions, which tilts the octanol/air equilibrium by a factor of more than 10000 towards the octanol side as shown in Table 6. Now, it is well known that monocarboxylic acids usually exist as dimeric associates in all three aggregate states. This association effect on the solubility is inherently taken account of in the atom-group parameters evaluation for the COOH function. On the other hand, dicarboxylic acids do not only form dimers but also cyclical and linear oligomeric associates, with drastic consequences on their solubility in the various solvents. It is these additional associations that are considered by the special group "(COOH)n".

As a consequence, solutes with a low tendency to interact with solvents, either inherent or induced by intramolecular hydrogen bridges, show a trend to higher logK_aw values; the additional intermolecular association of di- and tricarboxylic acids, on the other hand, results in a significantly lower logK_aw value, as is exemplified in Table 7, where the respective calculated data of the Table 5 and Table 6 have been applied in equation (2). The experimental logK_aw values have been extracted from Ebert et al. [25].

5. Conclusions

The present study, which is part of an ongoing project, put to use a tool for the simple and reliable calculation of the two partition coefficients logP_ow and logK_oa, that has proven its unmatched versatility in the equally reliable prediction of up to now 19 physical, thermodynamic, solubility-, optics-, charge-, and environment-related molecular descriptors [11,44,45,46,47,48,49,50], based on a common group-additivity method. The large database of more than 3300 and 1900 experimental data, respectively, upon which the group parameters for the logP_ow and logK_oa calculations are founded enabled their prediction for nearly 29500 and more than 27000 molecules, respectively, of the presently more than 37100 compounds in ChemBrain’s database. In addition, these results also allowed the trustworthy calculation of the third partition coefficient logK_aw for more than 26000 compounds. The big advantage of the present approach is its ease of use by simply adding by means of paper and pencil the parameters of the atoms and groups found in a particular molecule that are listed in the respective Table 2 and Table 3.

The mentioned project’s software is called ChemBrain IXL, available from Neuronix Software (www.neuronix.ch, Rudolf Naef, Lupsingen, Switzerland).