First, ask the question, "Is it possible that this type of player would take these actions with AK?" We know that from his vantage point, AK is losing to AA, KK, QQ, TT, KQ...but does he think about it this way or does he simply say, "**** it, I have TPTK in a 4bet pot, I'm pushing the action and I'm not folding."
If the answer to that question is, "Yes," then our next step should be to break down the problem combinatorically.
With you holding two aces, AK combos decrease from 16 to 8.
With a K out there, there are 3 combos of KK available for villain.
With a Q out there, there are 3 combos of QQ available for villain.
So with an assumption that V will 4b QQ+, AK, we are beating 8 combos and losing to 6 combos. We have a 5% chance to suck out against KK/QQ, and AK has a 7% chance to suck out against us. We are never ever folding (regardless of bet size) when we beat more than half of his plausible range.
If we modify the assumption and suppose that he 4b KQ suited, this actually only changes our result to 8 combos we beat, 8 combos we are behind. We would also still be calling it off here.
If we modify the assumption to suppose that he 4b KQs and TT+, then we are beating 8 combos and are behind 11 combos. Getting something like 2.7 to 1 on our money on the turn, we will also be calling it off in this instance.
So yeah, I see the turn as a mathematically justified call.
As for the flop...I think I like a shove better than a call. I don't think this player will b/f his AK at this point given the description you offer. Possible he even shrug calls it off with AQ, deciding that this is the one time we've squeezed with J
T
and have now decided to semi-bluff. I'm value raising to prevent him from x/f his AK on future streets when something like the 9
falls on the turn and he decides that we now must have him beat.
If I think he might b/f AK on the flop, I like a call down better, but against him I think shoving the flop is optimal. Again, if you run into KKK, well, you just hit the top of his range and need to river AAA. Make the best plays against villain's range.