Cooperativity in ligand binding: For a protein with n binding sites that exhibits “infinite” cooperativity, which outcome best describes its binding behavior and the Hill coefficient?

Introduction / Context:
Cooperativity describes how ligand binding at one site on a multimeric protein influences ligand binding at other sites. The Hill model is a classical way to summarize this behavior. Understanding the extreme case of “infinite” cooperativity clarifies the conceptual link between all-or-none binding and the Hill coefficient used in biochemistry and biophysics.

Given Data / Assumptions:

• The protein has n identical binding sites for the same ligand.
• “Infinite cooperativity” means binding is concerted and perfectly coupled across sites.
• Hill analysis is used to summarize overall cooperativity with nH.

Concept / Approach:
In positive cooperativity, initial ligand binding increases the affinity of remaining sites. The Hill coefficient nH reflects the steepness of the binding curve in a log form. In the limiting case of infinite cooperativity, intermediate partially liganded states are effectively unpopulated under equilibrium conditions, yielding an all-or-none transition between empty and fully saturated states.

Step-by-Step Solution:
Define infinite cooperativity → only two populated macrostates: all sites empty or all sites filled.Relate to Hill slope → the steepness approaches the number of sites; in the ideal limit, nH = n.Interpret binding behavior → system appears in either unliganded or fully liganded forms, reflecting a digital switch-like response.Therefore, both statements “only unliganded or fully liganded” and “nH = n” are true.

Verification / Alternative check:
In the Monod–Wyman–Changeux framework, the limit of maximal coupling collapses intermediate states. The Hill plot slope approaches n at half-saturation, matching the theoretical upper bound for nH.

Why Other Options Are Wrong:

• nH = 0.0 indicates no binding response and is incompatible with the scenario.
• Picking only one of the two correct properties is incomplete; both hold in the infinite limit.

Common Pitfalls:
Confusing nH with the number of sites in real systems (usually nH < n); assuming infinite cooperativity exists in practice rather than as an instructive limit.

Final Answer:
both (b) and (c)