This is an awkward question, but can be answered using free energies. First note that a positive change in pKa means a decrease in Ka (since pKa = -logKa). Using the relation between the standard free energy and the equilibrium constant,
We see a decrease in Ka results in a more positive free energy. Therefore, the protein is destabilized energetically speaking. Reversely, a negative change in pKa will result in more negative free energy, and thus stabilization.
Now, realistically this is a stretch because this is only representative of the free energy of the acid dissociation of an amino acid. Many more factors are contributory. But since free energies are additive, and simply making the assumption that all other interactions are far less significant, then this tends to be an okay approximation.
You're getting your equilibria mixed up. Ka is an acid dissociation equilibrium, which is HA ---> H+ + A-. Once you realize this, then all you're saying is that a negative change in pKa (more acidic) makes the HA ---> H+ + A- equilibrium favor the right-hand side, or acid dissociation. In other words, it's a tautology: making something more acidic makes dissociation more likely.
But conceptually, you're on the right path. The delta G and K that you should be interested in here is the one related to protein folding: unfolded ---> folded. Thus, K = [F]/ at equilibrium. So how do these relate? Well, you're missing a key element that's in the passage:
the change in pKa mimics the change in free energy between the folded and unfolded states. This is not a general phenomenon but rather specific to specific classes of proteins.
For instance, say you're measuring the change in pKa of a lysine residue between the folded and unfolded states. In the folded state, say this lysine residue lies right next to a glutamate residue. You would expect then, that in the folded state, the lysine residue will want to be protonated, or be more basic (higher pKa), because it will then be able to participate in ion pairing. So its pKa would be higher in the folded than in the unfolded states, or a positive delta pKa. This would stabilize the protein, or result in a
negative free energy change for the U ---> F equilibrium, because of the additional stabilization imparted by the ion pairing. Therefore, a
positive change in pKa here results in a
negative free energy change in the protein.
The opposite effect is also possible. That is, say you're measuring the pKa of a glutamate buried inside the protein instead of the lysine. Which direction would the pKa be shifted in the folded state? Well, it would want to remain deprotonated because that would allow ion pairing with lysine. So in other words, it would want to be more acidic, or have a lower pKa than in the unfolded state. Therefore, the delta pKa here would be
negative, and that would be correlated with a
negative free energy change in the protein.
