What they wanted to determine was the effect of point mutations in the peptide on the inhibition. They performed saturation mutagenesis between positions 2 and 9 of the peptide and generated 152 mutants whose activities they tested in a minimum inhibitory concentration (MIC) assay. They found that 8 point mutants especially resulted in more potent analogs.
Now there can be several reasons why the potency went up, but one potential reason is entropy. Macrocycles, while often more rigid than their corresponding linear analogs, are still quite flexible. In fact, my own work with the macrocycle dictyostatin in graduate school showed how flexible even a supposedly constrained molecule can be. What this paper finds out is that in cases where the mutant lost activity, there was a corresponding increase in flexibility and entropy as measured by the number and distinctive nature of conformations from a conformational search technique which they have developed. Particularly striking changes in potency occurred when a single residue was modified from having a planar sp2 carbon to a non-planar sp3 carbon: in that case the saturated analog had many more conformations than the unsaturated one.
As someone who has always been partial to the impact of entropy and conformational flexibility on molecular activity, I like this kind of work. But I am not quite convinced yet that it is decreased flexibility that leads to more potent inhibition. For one thing, inhibition is not direct binding, and there are a variety of factors including changes in cell permeability and off target effects that could lead to the observed changes in inhibitory - not binding - affinity. Secondly, there were 152 mutants, and it's not clear to me how many were tested for flexibility: in other words, I am not sure there were enough controls to determine whether the flexibility-inhibition correlation really holds up. For instance, many of the mutants were inactive: were there instances in which some of these were actually less flexible and challenged the hypothesis? Another way to put it is to ask what the right null model for this dataset is.
Thirdly, decreased or increased inhibition can be a result of both more conformations as well as conformational selection. For instance, two macrocycles can have similar conformations, but in one case a particular conformation more suitable for binding could be more stable (perhaps because of an intramolecular hydrogen bond) and represented to a higher degree in solution, making it easier for a protein target to pick it out. Lastly, it is not clear whether the improved affinity could simply have been a result of better interactions: although that seems unlikely for the sp2 vs sp3 pair above, it is nonetheless a factor that could be operating in other cases.
Entropy is an important consideration in drug design, but it's also trickier than it sounds to both understand its effects and implement its benefits. To their credit the authors acknowledge that rigidity is a necessary but not sufficient condition for increased affinity, and other studies seem to bear it out. Macrocyclization can also be counterintuitive: for instance in my own studies I found out that dictyostatin which is a macrocycle seems more flexible than its corresponding acyclic counterpart discodermolide. In that case it was fairly straightforward syn-pentane interactions which made the acyclic molecule rigid. In other cases it could be the opposite. In any case, this study serves as an interesting starting point for exploring the impact of flexibility on drug affinity, but it also serves to illustrate how thick the jungle of SAR really is.