0.8 Motion planning for proteins: biophysics and applications  (Page 7/10)

The transition probabilities are defined as they are to be consistent with a Boltzmann-like distribution of energies, and therefore with standard Monte Carlo simulation probabilities. The authors demonstrated that each continuous path in the roadmap may be interpreted as a Monte Carlo simulation, and that, if a very large number of samples and edges are made, the aggregate behavior of these Monte Carlo simulations can be analyzed to estimate properties of the protein such as folding rates and transition states . Essentially, SRS is a way to generate large amounts of Monte Carlo simulation data in a short time. The developers or this method have provided a proof that, for a sufficiently large SRS and a sufficiently long Monte Carlo simulation, the distribution of conformations is expected to be equal.

To study protein folding using SRS, we observe that some set of nodes in the roadmap represent conformations in or very close to the folded state (native structure). We will refer to this set of nodes as F. For every node in the roadmap, we can compute an expected number of state transitions (or Monte Carlo steps) to go from that state to a node in F, with the base case that any node in F is defined to be at distance 0 from F. Given a precomputed SRS, we can compute this statistic for each node as follows:

We can also define a set of nodes representing conformations close to the stable denatured state of the protein as the unfolded state, U. Given both of the sets U and F, we can define a quantity called the transmission coefficient, τ, for each node. The transmission coefficient expresses the probability that a structure at a particular node will proceed to a state in F before it reaches a state in U--in other words, it is the probability that a given structure will fold before it unfolds. This is often called the folding probability, or Pfold, in more recent research. The quantity, τ, is calculated for each node using the following relation: