we will soon be starting a new round of runs in which each trajectory is directed towards low energy regions of the landscape identified in the runs you are doing now

This would be a great approach to solving the problem, as it would reduce the combinatorial complexity enormously. It turns out, however, that even though proteins are synthesized one amino acid at a time, folding probably does not occur until most of the protein is made because special helper proteins, called "chaperones" prevent folding from occuring prematurely. The problem is that most pieces of proteins do not fold into stable structures (this is part of why the prediction problem is difficult!), and so the growing peptide chain would likely make incorrect and possibly dangerous interactions with other molecules unless kept in check by the chaperones. So unfortunately, a chain growing algorithm would probably not work well here--the global minimum is only clearly defined in the context of the whole protein chain.

Is it true that proteins are built in the same order every time?