I'd certainly close the whole roof.I would be leery of leaving that eastern section of roof open, even if you do only get 5" of rain a year. When it does rain all the rain is going to sluice off the covered part and dump into the coop in that one area, leaving a much bigger mess in a small area than if the rain fell naturally into that space. If your area is like mine, with most of the rain concentrated in a few big storm events, you are going to want very good drainage out of that coop litter so the water doesn't wick into everything. Or gutters to redirect the runoff.
Also, if you leave food in the coop you have far fewer options for keeping the food dry if you have that big open area in the roof. I say this from experience. My coop/run is 12' deep. I left the south 2' of roof open (protected by hardware cloth and a shadecloth cover) for "ventilation." Never again. It is really hard to find a place to keep food dry in that coop. Most of our storms involve wind which can come from any direction, and the wind usually moves around a lot during the course of the storm. In a single storm it can blow in from the front (all wire) and also through the windows in one or more of the other sides. Even in the most protected part of the coop/run I have had to throw out a bunch of feed because rain got into the tray of the hanging feeder and wicked up into the rest of the feed. The birds have a lot of perching options and can always find a place to get out of the wind and rain, but the feeder can't figure out how to stay dry. Guess I need a smarter feeder.
Sarah
Someone else suggested leaving a small gap in the roof panels at the peak, and putting a ridge cap over it allowing for ventilation out the peak. I second that suggestion. It is an excellent idea.