Standard state free energies, not pKas, are ideal for describing small molecule protonation and tautomeric states

Abstract

The pK_a is the standard measure used to describe the aqueous proton affinity of a compound, indicating the proton concentration (pH) at which two protonation states (e.g. A^- and AH) have equal free energy. However, compounds can have additional protonation states (e.g. AH₂⁺), and may assume multiple tautomeric forms, with the protons in different positions (microstates). Macroscopic pK_as give the pH where the molecule changes its total number of protons, while microscopic pK_as identify the tautomeric states involved. As tautomers have the same number of protons, the free energy difference between them and their relative probability is pH independent so there is no pK_a connecting them. The question arises: What is the best way to describe protonation equilibria of a complex molecule in any pH range? Knowing the number of protons and the relative free energy of all microstates at a single pH, ∆G°, provides all the information needed to determine the free energy, and thus the probability of each microstate at each pH. Microstate probabilities as a function of pH generate titration curves that highlight the low energy, observable microstates, which can then be compared with experiment. A network description connecting microstates as nodes makes it straightforward to test thermodynamic consistency of microstate free energies. The utility of this analysis is illustrated by a description of one molecule from the SAMPL6 Blind pK_a Prediction Challenge. Analysis of microstate ∆G°s also makes a more compact way to archive and compare the pH dependent behavior of compounds with multiple protonatable sites.

Keywords: Multiprotic; Protonation state; SAMPL6; Tautomer; pH titration; pKa.

Fig 1.. Network of protonation and tautomeric microstates for SAMPL6 pK_a challenge target SM07

Only the subset of possible microstates identified as being of interest in the SAMPL6 challenge are shown. The protons of interest are denoted by blue balls on top of the nitrogen which is the proton acceptor. Each column has a different number of dissociable protons, indicated at the top. All microstates in a column are tautomers with the same total charge; their vertical order is arbitrary. All the tautomers in a column contribute to the macrostates states with 4 (4H) to zero (0H) dissociable protons. Black double-headed arrows indicate pK_as that were reported in the SAMPL6 Challenge; red arrows are the transitions between tautomers. Twenty four pK_as can be defined between all microstates in one column and all in the neighboring columns as they differ by one proton. Some transitions, such as between 13 and 7 required tautomer changes. Likewise, transitions between tautomers in the top and bottom rows (e.g. between microstates 2 and 3) are also well-defined, but are not shown here for clarity. The numbers associated with each microstate simplify the microstate IDs assigned by the SAMPL6 pKa Challenge, which have the form: SM07_microXXX, where XXX are three digits. For example, microstate 4 in this figure corresponds to SM07_micro004.

Figure 2.. Relative standard state free energies (ΔG°) and relative number of protons (Δm) is all that is needed to completely describe the pH dependence of a system with multiple protonation states.

2A: The relative free energies of the states A,B,C, and D as a function of pH given the pK_as in Table 1 transformed to standard state free energies, ΔG° (Table 2). Relative free energies of microstates change linearly with respect to pH (eqn. 2b). Squares are input pK_as which can be experimentally observable. Circles mark ΔG°_jB, the free energies at pH 0 of the other three states relative to B. Horizontal arrows at the bottom show the state at lowest free energy (dominant population) in each pH range. 2B): states A, B, C and D with state C as the reference; 2C): Titration showing relative state populations vs pH predicted using ΔG°s in Table 2 and equations 3 and 4. This plot is the same independent of which state is used as the reference. ΔG is given in unitless free energies where a unit change in ΔG yields a 10-fold population change.

Figure 3.. Graphical depiction of the microstate energy as a function of pH and resultant titration curve for the eight microstates of SM07 described in Table 2

a) Graphical representation of the 8 microstate energies as a function of pH using ΔG°s and Δms from Table 3b. The squares show pK_as that would be seen experimentally as they connect the states that are at low energy at that pH and the triangles show pK_as that were the input to the calculation. There is an inconsistency in the relative energy of states 2 and 3 calculated when state 12 or state 6 or 7 are used to obtain the free energy difference from reference state 4 (Table 3b). b); The microstate network of 8 microstates of SM07 connected by pK_as calculated with Epik [24]. Microstates predicted by Epik are a subset of those shown in Fig 1. Dark blue arrows are the ΔG° between the two microstates; Red arrows are ΔG° between tautomers. The standard deviations for ΔG°_2,4 and ΔG°_3,4 represent the standard error for the free energy calculated around the two nearest closed triangular loops. Green numbers under microstate identifiers are ΔG°, the free energy relative to state 4 at pH=0._. c) The probability of each state as a function of pH. Note that while state 6 is the predominant microstate between pH −5 and 5, a small amount of tautomer microstate 7 is seen. ΔG represents unitless free energies where a unit change in ΔG yields a 10-fold population change. Python scripts and interactive Jupyter notebooks to generate networks and graphs of the relative free energy as a function of pH from a list of microscopic pKas can be found at http://github.com/choderalab/titrato

Fig 4:. Graphical analysis of a network of 11 microstates SM07.

(a) Network of 11 SM07 microstates considered by calculation ECRISM-13 (ID 0xi4b) [23], their connecting pK_a (black 2-headed arrows); the resultant pairwise ΔG°_j4 (blue arrows) and microstate ΔG°_j4 relative to microstate 4 (green numbers). (b) ΔG_j4 as a function of pH for all microstates. ΔG°_j4 is ΔG_j4 at pH 0. ΔG represents unitless free energies where a unit change in ΔG yields a 10-fold population change. (c) Predicted titration curves given the microstate ΔGs as a function of pH. It should be noted that the pK_as shown in Fig 4c match the crossing points in Fig 4b that occur as one lowest energy state is replaced by another as the pH changes.

Figure 5.. Overview of relative free energies for individual microstates for individual submissions to the SAMPL6 blind pK_a challenge.

All submissions that provided information about all 11 microstates are included, ordered by the free energy difference between the 4H microstate (16) and the 0H microstate (12). Data from Table 4; definition of microstates from Figure 1. a) Blue: microstate 16, Red: 12; b) Green: microstate 2; Black: 3; Red: 4 (reference state); c) Green: microstate 11; Black: 7; Red: 6; d) Green: microstate 15; Black: 14; Red: 13.