NL25; facing turn check/min-raise

[Octopus]

I know the answer: it depends on your opponent. Never-the-less:

I open for $1 in middle position with AA. There is a late position poster in the hand. He and everyone else folds to the BB who calls. I have only 20 hands on him, but he has been reasonably tight over that time, seeing only 4 flops. The only hand of his I have seen was AKo which he bet OOP on the flop and turn on a QT58 board; each time betting only 1/4 the pot. He started this hand with $23.55. I have him covered.

Pot is $2.35. Flop comes QJ4; Q4 suited; I do not have the suited ace. He checks and I bet $2. He calls.

Pot is $6.35. Turn is 9, no flush. He checks and I bet $4. He min-raises to $8. I wonder if I am drawing dead. I look at all the possible draws. I think about how much I suck at this game. I call.

The river is another 4, no flush, and he bets $9.75 (into a $22.35 pot with leaving only $2.80 behind). I add all those two pair hands to the list of those I beat and I call.