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Saturday, April 16, 2016

Computing pKa values for molecules with several ionizable groups

We're working on pKa prediction using semiempirical methods and need to compute pKa values  for molecules with several ionizable groups. Here are my current thoughts so far.

Background: one ionizable group
If there is only one tritrateable site
\mathrm{BH \rightleftharpoons B + H^+} \ \ \  K=\mathrm{\frac{[B][H^+]}{[BH]}}

then the fraction of \mathrm{BH} molecules f_{\mathrm{BH}} is \begin{split}  f_{\mathrm{BH}} & =\mathrm{\frac{[BH]}{[B]+[BH]} } \\ & = \mathrm{\frac{[B]}{[B]}\frac{[BH]/[B]}{1+[BH]/[B]} } \\ & = \mathrm{\frac{[H^+]/K}{1+[H^+]/K} } \\ & = \mathrm{\frac{10^{p\textit{K}-pH}}{1+10^{p\textit{K}-pH}} } \end{split}
where \mathrm{pH = -log[H^+] \implies [H^+] = 10^{-pH}}

and similarly for K.

From this we can see that the pK value is the pH value for which f_{\mathrm{BH}} = 1/2. So you compute the pK value from the standard free energy difference
\text{p}K =\left( \Delta G^\circ(\mathrm{B})+\Delta G^\circ(\mathrm{H^+})- \Delta G^\circ(\mathrm{BH})\right)/RT\ln(10)

and you're done.

Two ionizable groups
For a molecule with two titrateable groups ( \mathrm{HB_\alpha B_\beta H}) and the following equilibrium constants \mathrm{HBBH \rightleftharpoons BBH + H^+} \ \ K_{\alpha1}
\mathrm{HBBH \rightleftharpoons HBB + H^+} \ \ K_{\beta1}
\mathrm{HBB \rightleftharpoons BB + H^+} \ \ K_{\alpha0}
\mathrm{BBH \rightleftharpoons BB + H^+} \ \ K_{\beta0}
The probability of, for example, \mathrm{BBH} is f_{\mathrm{BBH}} =\mathrm{\frac{[BBH]}{[BB]+[BBH]+[HBB]+[HBBH]}= \frac{[BBH]}{\textit{P}}}
f_{\mathrm{BBH}} can be rewritten in terms of pK values f_{\mathrm{BBH}} = \mathrm{\frac{[BBH]/[BB]}{\textit{P}/[BB]} = \frac{10^{p\textit{K}_{\beta0}-pH}}{\textit{P}/[BB]}}
where \mathrm{ \textit{P}/[BB] = 1+10^{p\textit{K}_{\alpha0}-pH}+10^{p\textit{K}_{\beta0}-pH}+ 10^{p\textit{K}_{\alpha0}+p\textit{K}_{\beta1}-2pH}}  
Similarly, f_{\mathrm{HBB}} = \mathrm{\frac{10^{p\textit{K}_{\alpha0}-pH}}{\textit{P}/[BB]}}
and f_{\mathrm{HBBH}} = \mathrm{\frac{10^{p\textit{K}_{\alpha0}+p\textit{K}_{\beta1}-2pH}}{\textit{P}/[BB]}}
 The apparent pK value of the \alpha group (\mathrm{p}K_{\alpha}) is the pH at which its protonation probability f_\alpha =f_{\mathrm{HBB}} + f_{\mathrm{HBBH}}
is 1/2 and similarly for the \beta group.  So compute the microscopic pK values (Eq 4-7), then f_\alpha and f_\beta as a function of pH, and then \mathrm{p}K_{\alpha} and \mathrm{p}K_{\beta}

If one of the groups (say \alpha) titrates at a significantly lower pH than the other (\mathrm{p}K_{\alpha1} << \mathrm{p}K_{\beta1}) then \mathrm{p}K_{\alpha}=\mathrm{p}K_{\alpha1} and \mathrm{p}K_{\beta}=\mathrm{p}K_{\beta0} and it is not necessary to compute the free energy of \mathrm{HBB}, but it can be hard to determine this in advance.  Similarly, if there is no significant interaction between the sites then \mathrm{p}K_{\alpha}=\mathrm{p}K_{\alpha1}=\mathrm{p}K_{\alpha0} and \mathrm{p}K_{\beta}=\mathrm{p}K_{\beta1}=\mathrm{p}K_{\beta0} and one can skip one of the protonation states.

For N ionizable groups one has to determine 2^N microscopic pKa values, which quickly gets out of hand if one has to do a conformational search for each protonation state and the molecule is large.

Related post
Generating protonation states and conformations


This work is licensed under a Creative Commons Attribution 4.0

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