Base pairing probabilities
McCaskill's algorithm [31] computes the base pairing probabilities from the partition functions of subsequences. Again, it seems easier to first perform the backtracking recursions on the "raw" partition functions that do not take into account the initiation contribution. This yields pairing probabilities Pkl for an ensemble of structures that does not distinguish between true dimers and isolated structures for A and B and ignores the initiation energy. McCaskill's backwards recursions are formally almost identical to the case of folding a single linear sequence. We only have to exclude multiloop contributions in which the cut-point u between components coincides with the cut point c. All other cases are already taken care of in the forward recursion.
Thus:
The "raw" values of Pij, which are computed without the initiation term, can now be corrected for this effect. To this end, we separately run the backward recursion starting from Z1,n and from  to obtain the base pairing probability matrices  and  for the isolated molecules. Note that equivalently we could compute  and  directly using the partition function version of RNAfold.
In solution, the probability of an intermolecular base pair is proportional to the (concentration dependent) probability that a dimer is formed at all. Thus, it makes sense to consider the conditional pair probabilities given that a dimer is formed, or not. The fraction of structures without intermolecular pairs in our partition function Z (i.e. in the cofold model without initiation contributions) is ZAZB/Z, and hence the fraction of true dimers is
Now consider a base pair (i, j). If i ∈ A and j ∈ B, it must arise from the dimeric state. If i, j ∈ A or i, j ∈ B, however, it arises from the dimeric state with probability p* and from the monomeric state with probability 1 - p*. Thus the conditional pairing probabilities in the dimeric complexes can be computed as
The fraction of monomeric and dimeric structures, however, cannot be directly computed from the above model. As we shall see below, the solution of this problem requires that we explicitly take the concentrations of RNAs into account.