(A) The allostery graph, , which implements the choices of effective higher-order cooperativities (HOCs) in Figure 7, shown as the product of the vertical subgraph of binding patterns at conformation , , and the horizontal subgraph of empty conformations, . As required in the proof of the flexibility theorem, both conformations and binding subsets are indexed by subsets of , where is the number of binding sites. Since for the effective HOCs in Figure 7, there are 16 binding subsets and 16 conformations, . (B) Intrinsic bare association constants, , in each conformation, in arbitrary units of (concentration)−1, and the probability distribution on the subgraph of empty conformations, , for the allostery graph in (A), giving the three choices of effective parameters in Figure 7A to an accuracy of 0.01 (Materials and methods), colour coded on a log scale as shown in the respective legends below. (C) Overall binding functions for the three parameterised ensembles in (B) (black curves), overlaid on the overall binding functions from Figure 7B (red curves), which were calculated from the effective parameters. The match is too close for the red curves to be visible. Numerical values are given in the Materials and methods. Calculations were undertaken in a Mathematica notebook, available on request.