Figure 2.

Example of public goods with differently shaped benefit functions. Proteases which works extracellularly by degrading polymers that are too large to be transported over the cell membrane can either break the peptide bonds of the polymer starting from the end of the polymer (exoprotease) or target a range of specific types of peptide bond within the chain, effectively breaking bonds at random (endoprotease). In this scenario benefit is proportional to the probability of yielding “edible” pieces of polymer. (A1,A2) Benefit increases approximately linearly with exoprotease concentration resulting in an initially concave benefit function. (B1,B2) The probability of breaking the polymer at a site producing an piece small enough to transport over the cell membrane is low when enzyme concentration is low, but accelerates as endoprotease concentration increases, resulting in a initially convex benefit function.