Let me first say that i KNOW that i didn't play this right. This is more of a general question about a situation i see quite often. I have a hand that i know is beat but i keep being strung by small bets because the math of the situation has me by the balls.

villain is a very standard 22/ten micro player.

i guess my question is FIRST: concretely, when do i fold here? i'm assuming preflop once he four-bets, but it's a min bet and if we say that AK, AA and KK is his range than AK is more than half of his range and i have position on him. by the time he bet 2 dollars i knew what he was up to but didn't know what to do about it because a missed AK is still in his range.

SECOND: more abstractly, how do i handle similar situations (e.g. i have a missed AQ and villain bets .40 at a $2 pot on every street)

Full Tilt Pot-Limit Hold'em, $0.10 BB (6 handed) - Full-Tilt Converter Tool from FlopTurnRiver.com

UTG ($6.76)
MP ($2.16)
CO ($4)
Button ($9.85)
SB ($13.14)
Hero (BB) ($10.51)

Preflop: Hero is BB with 10, 10
1 fold, MP checks, CO checks, Button calls $0.10, SB bets $0.65, Hero raises to $2.30, 3 folds, SB raises to $4.77, Hero calls $2.47

Flop: ($9.84) 2, 5, 5 (2 players)
SB bets $2, Hero calls $2

Turn: ($13.84) 6 (2 players)
SB bets $3.60

Total pot: $13.84