pKa calculations for tautomerizable and conformationally flexible molecules: partition function vs. state transition approach

  • Nicolas Tielker
  • Lukas Eberlein
  • Christian Chodun
  • Stefan Güssregen
  • Stefan M. KastEmail author
Original Paper
Part of the following topical collections:
  1. Tim Clark 70th Birthday Festschrift


Calculations of acidities of molecules with multiple tautomeric and/or conformational states require adequate treatment of the relative energetics of accessible states accompanied by a statistical-mechanical formulation of their contribution to the macroscopic pKa value. Here, we demonstrate rigorously the formal equivalence of two such approaches: a partition function treatment and statistics over transitions between molecular tautomeric and conformational states in the limit of a theory that does not require adjustment by empirical parameters correcting energetic values. However, for a frequently employed correction scheme, linear scaling of (free) energies and regression with respect to reference data taking an additive constant into account, this equivalence breaks down if more than one acid or base state is involved. The consequences of the resulting inconsistency are discussed on our datasets developed for aqueous pKa predictions during the recent SAMPL6 challenge, where molecular state energetics were computed based on the “embedded cluster reference interaction site model” (EC-RISM). This method couples integral equation theory as a solvation model to quantum-chemical calculations and yielded a test set root mean square error of 1.1 pK units from a partition function ansatz. For all practical purposes, the present results indicate that a state transition approach yields comparable accuracy despite the formal theoretical inconsistency, and that an additive regression intercept, which is strictly constant in the limit of large compound mass only, is a valid approximation.

Graphical abstract

Embedded cluster reference interaction site model-derived vs. experimental pKa for the test set calculated with either the partition function (blue) or the state transition approach (red), using m as a free parameter


pKa prediction Solvation model Quantum chemistry Integral equation theory EC-RISM Tautomers/conformers 



This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Germany’s Excellence Strategy – EXC-2033 – Projektnummer 390677874, and under the Research Unit FOR 1979. We also thank the IT and Media Center (ITMC) of the TU Dortmund for computational support and, of course, Tim Clark for the continuous fruitful collaborations and discussions over the years.

Supplementary material

894_2019_4033_MOESM1_ESM.txt (0 kb)
ESM 1 (TXT 31 bytes)
894_2019_4033_MOESM2_ESM.txt (0 kb)
ESM 2 (TXT 511 bytes)
894_2019_4033_MOESM3_ESM.csv (5 kb)
ESM 3 (CSV 4 kb)
894_2019_4033_MOESM4_ESM.csv (1 kb)
ESM 4 (CSV 1 kb)


  1. 1.
    Alongi KS, Shields GC (2010) Theoretical calculations of acid dissociation constants: a review article. Ann Rep Comput Chem 6:113–138CrossRefGoogle Scholar
  2. 2.
    Kloss T, Heil J, Kast SM (2008) Quantum chemistry in solution by combining 3D integral equation theory with a cluster embedding approach. J Phys Chem B 112:4337–4343CrossRefGoogle Scholar
  3. 3.
    Tielker N, Tomazic D, Heil J, Kloss T, Ehrhart S, Güssregen S, Schmidt KF, Kast SM (2016) The SAMPL5 challenge for embedded-cluster integral equation theory: solvation free energies, aqueous pK a, and cyclohexane–water log D. J Comput Aided Mol Des 30:1035–1044CrossRefGoogle Scholar
  4. 4.
    Tielker N, Eberlein L, Güssregen S, Kast SM (2018) The SAMPL6 challenge on predicting aqueous pK a values from EC-RISM theory. J Comput Aided Mol Des 32:1151–1163CrossRefGoogle Scholar
  5. 5.
    Pracht P, Wilcken R, Udvarhelyi A, Rodde S, Grimme S (2018) High accuracy quantum-chemistry-based calculation and blind prediction of macroscopic pK a values in the context of the SAMPL6 challenge. J Comput Aided Mol Des 32:1139–1149CrossRefGoogle Scholar
  6. 6.
    Tissandier MD, Cowen KA, Feng AY, Gundlach E, Cohen MH, Earhart AD, Coe JV (1998) The Proton’s absolute aqueous enthalpy and Gibbs free energy of solvation from cluster-ion solvation data. J Phys Chem A 102:7787–7794CrossRefGoogle Scholar
  7. 7.
    Zhang H, Jiang Y, Yan H, Cui Z, Chunhua Y (2017) Comparative assessment of computational methods for free energy calculations of ionic hydration. J Chem Inf Model 57:2763–2775CrossRefGoogle Scholar
  8. 8.
    Heil J, Tomazic D, Egbers S, Kast SM (2014) Acidity in DMSO from the embedded cluster integral equation quantum solvation model. J Mol Model 20:2161CrossRefGoogle Scholar
  9. 9.
    Klamt A, Eckert F, Diedenhofen M, Beck ME (2003) First principles calculations of aqueous pK a values for organic and inorganic acids using COSMO-RS reveal an inconsistency in the slope of the pK a scale. J Phys Chem A 107:9380–9386CrossRefGoogle Scholar
  10. 10.
    Beck ME, Bürger T (2003) Predicting acidity for agrochemicals. In: Ford M, Livingstone D, Dearden J, Van deWaterbeemd H (eds) Euro-QSAR 2002: designing drugs and crop protectants. Blackwell, Oxford, pp 446–450Google Scholar
  11. 11.
    Bochevarov AD, Watson MA, Greenwood JR (2016) Multiconformation, density functional theory-based pK a prediction in application to large, flexible organic molecules with diverse functional groups. J Chem Theory Comput 12:6001–6019CrossRefGoogle Scholar
  12. 12. Accessed 13 February 2019; see also special issue of J Comput Aided Mol Design (2018) 32(10)
  13. 13.
    Rebollar-Zepeda A, Galano A (2016) Quantum mechanical based approaches for predicting pK a values of carboxylic acids: evaluating the performance of different strategies. RSC Adv 6:112057CrossRefGoogle Scholar
  14. 14.
    Gilson MK, Given JA, Bush BL, McCammon JA (1997) The statistical-thermodynamic basis for computation of binding affinities: a critical review. Biophys J 72:1047–1069CrossRefGoogle Scholar
  15. 15.
    Klicić JJ, Friesner RA, Liu SY, Guida WC (2002) Accurate prediction of acidity constants in aqueous solution via density functional theory and self-consistent reaction field methods. J Phys Chem A 106:1327–1335CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Physikalische Chemie IIITechnische Universität DortmundDortmundGermany
  2. 2.R&D Integrated Drug DiscoverySanofi-Aventis Deutschland GmbHFrankfurt am MainGermany

Personalised recommendations