Each requisition can get several hundred responses. Recruiters usually do a quick scan of incoming resumes and either disqualify them or put them in a holding pattern. Once they have enough prescreened resumes (which can take a few weeks in an urgent situation, longer for non-urgent) they'll usually phone screen or pass to the hiring manager to review.
Phone call follow ups, unless you were referred in and given a specific contact, is more likely to make you look like you don't understand corporate well. If they are ready to talk to you, they'll call. If you call, they'll just say they're reviewing resumes and will call you if you're a fit. You're taking up their time and if you do that more than once or twice they'll remember you... as being kind of a pest.
Apply for whatever interests you and you're qualified for, but don't spam the application process. Every recruiting system out there allows the recruiter to view your profile along with a list of what you've applied to, and applying to 200 jobs makes you look desperate and non-discriminating.
A week is really short, and follow up doesn't really help much in a large corporate environment, though it can be worthwhile in a smaller company.
<----- worked in corporate recruiting, now implements applicant tracking systems