URGENT HELP: Maxing Out on DB Connections

We are in the middle of a media campaign and maxed out on DB Connections with only 300+ users. Doesn't make sense. Our business partner configured an Amazon AWS infrastructure (4 large EC2 instances with auto-scaling) and a large database server instance (RDS). W3TotalCache is up and running.

What will cause this? Please help.

Next media spike will occur in a couple of hours.

Thanks Everyone