The request was for links to your purported motherboards. The only to 785 boards I'm finding on Newegg are the E45 and E65 and only the E51 and E65 on MSI's web site.
I don't see why it would not support the ATI HD5770. It's a dual width card, so you'll lose one PCI slot. It card may sit atop the south bridge heatsink.
I would assume (since I can't find the E53) that the main differences between the two boards is the number of memory slots/SATA connectors (since that seems to be the difference between the two boards I could compare on Newegg).
-Wolf sends