Abraham Descriptors by Andrew Lang

Abraham et al., have developed equations, the Abraham solvation equations, that can be used to predict the solubility of organic compounds in 84 different organic solvents [1]. The Abraham method assumes that the partition coefficient between water and a solvent, P_s, is given by the ratio of solubilities of a solute in the solvent, S_s, and in water, S_w, P_s = S_s / S_w. (1)

If this assumption is reasonable for the solute and solvent in question, then the solubility of the solute in the solvent can be calculated from the predicted partition coefficient,

log P_s = c + e E + s S + a A + b B + v V, (2)

where the continually refined (as more experimental data becomes available) coefficients c, e, s, b, and v vary by solvent and E, S, A, B, and V are solute descriptors, described as follows [11]:
E is the solute excess molar refractivity in units of (cubic cm per mol)/10. It represents the solute's polarizability and gives a measure of the ability of a solute to interact with a solvent through n- and π- electron pairs.
S is the solute dipolarity/polarizability. It gives a measure of the solute's ability to stabilize a charge or dipole
A is the overall (summation) hydrogen bond acidity. The hydrogen bond acidity descriptor measures the extent of hydrogen bonding by the solute in a basic solvent.
B is the overall (summation) hydrogen bond basicity. The hydrogen bond basicity descriptor measures of the extent of hydrogen bonding by the solute in an acidic solvent.
V is the McGowan characteristic volume in units of (cubic cm per mol)/100.

Descriptor Calculation Methods

Various methods are used to determine the solute descriptors either from first principles or by working backwards from known logP and solubility values. Abraham et al. [2] have published a review of methods, though some other more recent methods are discussed below.

V - The McGowan characteristic volume

V is the McGowan approximation for the molecular volume. It is calculated from fragment contributions. It can be calculated from the solute's molecular formula and the number of bonds. Remember to divide by 100.

E - The excess molar refractivity

E is the molar refraction, MR, of the compound calculated using McGowan’s volume less the molar refraction of an alkane with the same McGowan volume,

E = (MR) − 2.83195 V + 0.52553, (3)

where MR = 10 (η^2 −1)/(η^2 + 2) V where η is the refractive index. (4)

Note: Predicted values for η, MR, and molar volume can be found on ChemSpider but since Abraham descriptors are calculated using the McGowan volume, they must be converted before use using V (intrinsic) = 0.597 + 0.6823 V. (5) [3]

A few examples from of using equation (5) with the ChemSpider molar volume value compared to the published McGowan volume [6].
solute
Calculated V
Published V
4-methoxybenzoic acid
0.9702
1.1313
4-nitrobenzoic acid
0.7929
1.1059
2-methylbenzoic acid
0.8662
1.0726
4-chloro-3-nitrobenzoic acid
0.9673
1.2283
ibuprofen
2.0607
1.7771

S, A, and B

S, A, and B can be calculated from GLC [2], RP-HPLC [4], or simple linear regression from known solubility data or known logP values which have been determined and recorded in the Med-Chem data base for a large number of systems [5]. There is interestingly a method for calculating A using NMR [6],

"A nuclear magnetic resonance (NMR) method based on the difference in the 1H NMR
chemical shift of a protic hydrogen in dimethylsulfoxide and CDCl3 can be used to
determine the overall, or summation, hydrogen bond acidity descriptor of the solute.
The NMR method can measure the overall hydrogen bond acidity of individual
functional groups. Which are then summed to give the value for the entire molecule.
The estimated error in the solute descriptor A determined by the NMR method is around
± 0.05 units. Solutes that dimerize in CDCl3 cannot be studied with the NMR method." p. 31.

Absolv [8], part of the ACD/ADME Suite, can calculate S, A, and B from SMILES by presumably summing fragment contributions, see [9], though the results are apparently disappointing [10].

Clustering

By giving each solvent descriptors (c, e, s, a, b, v) it becomes possible to determine which solvents are similar by comparing their descriptors - much like as is done with Hansen solubility parameters . Using recently updated values [1], we used HappieClust to perform a cluster analysis (agglomerative hierarchical clustering) to see which solvents are considered similar in the Abraham general solvation method. The results are depicted in the figure below. abraham.png


Example - Benzoic acid

The Abraham solute descriptors for benzoic acid have been calculated [4] as E = 0.73, S = 0.9, A = 0.59, B = 0.4, V = 0.9317. We can compare the predicted solubility values with the measured values for the solubility of benzoic acid in various organic solvents . Taking the solubility of bezoic acid in water to be 0.03 M [7] and using the the solubility coefficients for known solvent, as published in [1], we arrive at the following.

Solvent
Predicted (M)
Measured (M)
THF
4.04
3.37
acetonitrile
0.46
0.76
ethanol
2.54
2.65
methanol
2.28
2.84
benzene
0.09
0.48
toluene
0.07
0.65

The discrepancy for benzene and toluene is likely due to the fact that carboxylic acids are known to dimerize in saturated hydrocarbon and aromatic hydrocarbon solvents. When this happens, the predicted value will be lower than the measured value [6].

References

[1] Abraham MH, et al. 2009. Prediction of Solubility of Drugs and Other Compounds in Organic Solvents. Journal of Pharmaceutical Sciences. DOI: 10.1002/jps.21922
[2] Abraham MH, Ibrahim A, Zissimos AM. 2004. The determination of sets of solute descriptors from chromatographic measurements. J Chromatogr A 1037:29–47.
[3] Abraham MH and McGowan JC. 1987. The use of characteristic volumes to measure cavity terms in reversed phase liquid chromatograph. Chromatographia, Volume 23, Number 4. pp. 243-246. DOI: 10.1007/BF02311772
[4] Andreas MZ, Abraham MH, et al. 2002. Calculation of Abraham descriptors from experimental data from seven HPLC systems; evaluation of five different methods of calculation. J. Chem. Soc. Perkin Transactions 2. pp. 2001-2010. ISSN 1472-779X
[5] The MedChem Database 2002, BioByte Corp. and Pomona Col- lege, Daylight Chemical Information Systems, 27401 Los Altos, CA. [http://www.daylight.com/products/medchem.html]
[6] Stovall DM, 2006. Thermodynamics of the Abraham genrel solvation model: Solubility and partition aspects. Masters Thesis. University of Texas [http://digital.library.unt.edu/permalink/meta-dc-5342:1]
[7] Wikipedia article on benzoic acid [http://en.wikipedia.org/wiki/Benzoic_acid]
[8] Absolv [http://www.acdlabs.com/products/admet/adme/capabilities.html]
[9] Platts, J. 1999. Rapid, Automated Prediction of Abraham LSER Descriptors [http://www.documentarea.com/qsar/jamie_p.pdf ]
[10] Drug Bioavailability. Estimation of Solubility, Permeability, Absorption and Bioavailability [http://books.google.com/books?id=lFYgJbYHZi0C&lpg=PA236&ots=I33NfHutbl&dq=absolv%20abraham&pg=PA236#v=onepage&q=&f=false ]
[11] Christina Mintz. 2009. Predicting Chemical and Biochemical Properties Using the Abraham General Solvation Model [http://proquest.umi.com/pqdlink?did=2126292181&Fmt=14&VType=PQD&VInst=PROD&RQT=309&VName=PQD&TS=1295326606&clientId=79356]