Looking desperate doesn't really matter. Looking controlling does. I would say that you can write as often as you want, but when you start saying things like "Why don't you write?", or "Are you seeing somebody else?", you are now trying to control his life, and that is the problem.
Two days isn't so much time, so there really is no cause to get concerned yet. Perhaps he is on vacation with limited email access, or something similar.
Just continue writing him as often as before but don't expect any responses. Don't get accusatory - like "why don't you write me?" Instead, just talk about whatever it was you talked about before. You might talk about your life - "my dog went to the vet today and got a rabies shot". That's not controlling and is really a fun thing to read about. Also it gives him an opportunity to answer your letter without apologizing for his absences.
As for when you should get worried - unless he has a history to change this, I'd say this is a good rule of thumb: after seven days things are bleak, after two weeks its very unlikely he is interested in you anymore, and after one month you can forget about it. In that case it doesn't matter how often or when you write - you'll need to find another.
Just make sure you don't accuse him of anything or try to change him. Desperate girls are hot, controlling girls are like the Plague.