WordNet (r) 3.0 (2006):

bear upon
    v 1: have an effect upon; "Will the new rules affect me?" [syn:
         affect, impact, bear upon, bear on, touch on,
         touch]

Moby Thesaurus II by Grady Ward, 1.0:

102 Moby Thesaurus words for "bear upon":
   abut on, act on, affect, answer to, appertain to, apply to,
   approach, assault, be based on, bear, bear on, belong to,
   bestraddle, bestride, boost, buck, bull, bulldoze, bump,
   bump against, bunt, butt, butt against, concern, connect,
   correspond to, cram, crowd, deal with, dig, draw, draw on, drive,
   elbow, exert influence, force, get cozy with, goad,
   have connection with, hurtle, hustle, interest, involve, jab, jam,
   jog, joggle, jolt, jostle, lead on, lean on, liaise with, lie on,
   link with, lobby, lobby through, magnetize, make advances,
   make overtures, make up to, nudge, perch, pertain to, pile drive,
   poke, press, press down, prod, pull strings, punch, push, ram,
   ram down, rattle, refer to, regard, relate to, rely on, repose on,
   respect, rest on, ride, run, run against, shake, shoulder, shove,
   sit on, stand on, straddle, stress, stride, tamp, thrust,
   tie in with, touch, touch upon, treat of, weigh on, weigh upon,
   wire-pull, work on