Fig 1.
Saturation curves for the three characteristic types of interactions between sites.
The fractional saturation of a macromolecule by a ligand was simulated using Eq 1 for identical and independent sites (green line), positive cooperativity (orange line) and negative cooperativity (blue line). Panel A shows a detail of the curves at low ligand concentrations, whereas Panel B includes saturation values for a wide range of ligand concentration in a logarithmic scale
Fig 2.
Microscopic reaction scheme for ligand binding to a macromolecule containing two binding sites.
Fig 3.
Site occupation for a macromolecule showing positive cooperativity.
Eqs 2 to 6 were numerically solved using the kinetic coefficients indicated in the main text for cooperative binding with ω = 10. (A) The average number of occupied sites (〈n〉t) was calculated using Eq 7 and represented as a function of time (in seconds). Arrow indicates increasing total ligand concentration. (B) The equilibrium values of 〈n〉 and [L] (μM) were obtained from each time course and plotted as a binding isotherm. (C) Wyman´s Hill plot of data from panel B. At both edges, the calculated data approaches to linear asymptotes with slope unity (dotted lines). The continuous red line represents a third order polynomial fitted to the data in the transition region with the best fitting parameters a3 = -0.016; a2 = 0.0807 a1 = 1.5085 and a0 = 2.718. From these values a Hill coefficient of 1.54 was obtained using Eq. B5 (S2 Appendix).
Fig 4.
Dependence of the Hill coefficient on the interaction and association free energies.
The Hill coefficient was calculated as described in Fig 3 from the binding isotherms obtained for identical sites with Ko = 1 (μM-1) and different values of the cooperativity factor ω (main plot) and for identical sites with Ko varying from 0.01 to 100 (μM-1) and a fixed value of the cooperativity factor ω = 8 (inset). ΔGoint (main plot) and ΔGoassoc (inset), were calculated using Eqs 10 and 9 respectively. Continuous lines are the graphical representation of polynomial functions fitted to the simulated data.
Fig 5.
Comparison between negative cooperativity and different sites at equilibrium conditions.
Eqs 2–6 were numerically solved using the following kinetic coefficients k1 = k2 = 0.55 (μM-1s-1) and k-1 = k-2 = 1 (s-1) and ω = 0.33058 for the case of two identical sites for a ligand with negative cooperativity and k1 = k1(2) = 1 (μM-1s-1); k2 = k2(1) = 0.1 (μM-1s-1) and k-1 = k-1(2) = k-2 = k-2(1) = 1 (s-1) for two classes of binding sites without interactions. Total concentration of macromolecule was in all cases 1 (μM) and the concentrations of ligand were varied from 0.01 to 300 (μM). The average number of occupied sites (〈n〉) was calculated using Eq 7 and equilibrium values of 〈n〉 and [L] were obtained from each curved and plotted as a binding isotherm. The binding isotherms of two identical sites for a ligand with negative cooperativity (black triangles) and two classes of binding sites without interactions (white circles) are shown. Inset shows a zoom of the main plot at low ligand concentrations.
Fig 6.
Comparison between negative cooperativity and different sites in pre-equilibrium conditions.
(A) Time course of site occupation for a macromolecule with two identical sites for negative cooperativity (continuous lines) and two classes of binding sites without interactions (dash-dotted lines) for the following initial ligand concentrations (μM): 300 (red), 100 (orange), 50 (green), 25 (turquoise), 10 (blue) and 5 (violet). Time courses simulations were obtained under the same conditions as described in Fig 5. (B) The difference Δ〈n〉 = 〈n〉coop—〈n〉diff sites was calculated for each ligand concentration and represented as a function of time. For clarity reasons only 6 representative time courses traces that gave rise to the equilibrium data points of Fig 5 are shown.
Fig 7.
Systems with strong negative cooperativity can mask binding sites.
Biding isotherms obtained from kinetic simulations of a macromolecule with two binding sites and high negative cooperativity (ω = 0.02 and Ko = 1 (μM-1)) for ligand concentrations up to 6 μM (A) or 300 mM (B). Insets show the simulated time courses used to generate the binding isotherms shown in both panels. Arrows indicate increasing ligand concentration. Notice the change in the kinetic behavior for high ligand concentrations in panel B.