Solution pH and the map window

Every site at once (16.2)

A molecule with several ionisable groups does not titrate one group at a time. Once one group has lost its proton, its charge holds the next proton back (citric acid: 3.13, 4.76, 6.40), or it helps an opposite charge stay (the glycine zwitterion). Bjerrum explained this in 1923 for dicarboxylic acids.

The pH slider therefore titrates all sites of the molecule together (up to six, so at most 64 protonation microstates). Every microstate h, with hᵢ = 1 when site i holds its proton, has the weight

\log_{10} w(h) = \sum_i h_i\,(\mathrm{p}K_i - \mathrm{pH}) - \sum_{i<j} W_{ij}\, q_i q_j ,

qᵢ is the site's charge: −1 for a deprotonated acid, +1 for a protonated base, otherwise 0.
pKᵢ is the site's value with every other site neutral.
W_ij is the pair term: 2.9 · 0.78^(d−1) pKa units for groups d bonds apart.

Both come from /api/pi-model's speciation block, fitted on 2832 molecules of the IUPAC Digitized pKa Dataset. See pKa of every site, tested against IUPAC data.

What you see. For one site the model reduces to Henderson–Hasselbalch:

f_{A^-}=\frac{1}{1+10^{\mathrm{p}K_a-\mathrm{pH}}},\qquad f_{BH^+}=\frac{1}{1+10^{\mathrm{pH}-\mathrm{p}K_a}} .

For several sites the page shows:

the fraction of each net charge from pH 0 to 14;
every macroscopic step: the pH at which the molecule as a whole loses its next proton. This is what a titration measures. Each step lists the sites that take part, with their share in it;
each site's charged fraction at the chosen pH.

Click or drag on the chart to set the pH.

The maps. Density and potential are mixed over all sites. With F₀ the field of the molecule as entered, θᵢ the deprotonated fraction of site i at this pH, and θᵢ⁰ its value as entered:

F(\mathrm{pH}) = F_0 + \sum_i \bigl(\theta_i(\mathrm{pH}) - \theta_i^{0}\bigr)\,\bigl(F_{\mathrm{base},i} - F_{\mathrm{acid},i}\bigr).

This is the population average whenever the sites' effects on the field add. That holds exactly for the potential of point charges. It holds for the density of separate π systems (citric acid's three carboxyl groups). It holds approximately when two sites share one π system.

All forms of one molecule share one conformer, frame and grid, so their fields add point by point.

Wavefunctions are not averaged: distinct species in solution have no common phase. The orbital view shows the dominant form of the selected site.

Curated molecules keep their one measured pKa.

Adaptive map window

The curated validation set stays on the fixed 10 Å window, so its maps can be compared directly.

For an arbitrary molecule, the browser projects the heavy atoms into the map frame, measures their span, and chooses

W = \max(10\ \text{Å},\ \lceil 2(\mathrm{span}+4\ \text{Å})\rceil/2).

The grid stays 168 × 168. A larger molecule trades spatial resolution for a complete view instead of being clipped. The scale bar uses the actual window.

Solution pH and the map window

#Every site at once (16.2)

#Adaptive map window

Every site at once (16.2)

Adaptive map window