Cooperativity Henderson-Hasselbalch

Experience shows marked deviation from the Henderson-Hasselbalch equation. There are examples of acid-base titration curves with curves broader than predicted by the Henderson-Hasselbalch equation, an effect called negative cooperativity, and there are examples of acid-base titration curves with curves sharper than predicted by the Henderson-Hasselbalch equation, an effect called positive cooperativity. These deviations from Equation (5.16) can be demonstrated by cooperative interactions involving the charged functional group. 

The formalism for introducing negative and positive cooperativity into the Henderson-Hasselbalch equation. Equation (5.16), follows. 

Introduction of Cooperativity into the Henderson-Hasselbalch Equation for Polymers of Linear Repeats... 

Figure 5.34. Acid-base titration curves of the series of elastic Model Proteins I and i through v of Table 5.5 that exhibit systematic increases in hydrophobic-induced pK shifts and positive cooperativity resulting from competition for hydration between apolar and polar groups. (Inset) Slope of the Henderson-Hasselbalch equation with n = 1, and the slopes for...

Although the ApK and ApKap values are of the same sign and add, the partial derivatives are of the opposite sign and counter each other. In particular, [(3AG/3a)T]c is the result of a negative cooperativity with a curve that is broader than given by the Henderson-Hasselbalch equation, and [(3AG/3a)T]a-p is the result of a positive cooperativity with curve that is steeper than given by the Henderson-Hasselbalch equation. 

Figure 5.35. Visualization of the relative efficiencies of the electrostatic charge-charge repulsion and the apolar-polar repulsion mechanisms by comparison of acid-base titration curves for poly(methacrylic acid), which exhibits charge-charge repulsion (negative cooperativity), and for the Model Proteins i and V, which exhibit apolar-posar repulsion (positive cooperativity). The polymers are each compared with the Henderson-Hasselbalch curve as reference. Chemical energy is Ap>An, where Ap = 2.3RTApH for the change in pH to go from one state to the other, and An is the number of moles to go from a degree...

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