A ligand-gated ion channel gate is a tetramer with identical regulatory sites that…
PHYSICS · Multi-part · Fermi University Physics · Biophysics & Medical Physics · Macromolecules, membranes and cell architecture · Macromolecules, membranes and cell architecture
Forces, Machines and Signals, Ch Cooperativity: zippers, melting and allosteric binding
Problem
A ligand-gated ion channel gate is a tetramer with n=4 identical regulatory sites that never interact with one another directly. The gate exists in two conformations, closed (C) and open (O). A bound regulatory ligand contributes weight x=c/KO per site when the gate is in O, and ϵx per site when it is in C (with ϵ=KO/KC<1); the unliganded C state has weight L relative to the unliganded O state (weight 1).
((a)) Build the states-and-weights table over the number of ligands bound (0 to 4) in each conformation, sum each row with the binomial theorem to obtain Z(x)=(1+x)n+L(1+ϵx)n, and use the logarithmic derivative of Z to derive the fractional occupancy θ(x). Show that θ reduces to the hyperbola x/(1+x) when L=0 or when ϵ=1, and explain in words how the gate can still open cooperatively with rising ligand concentration without any direct coupling between its four sites.
((b)) A short 15-residue α-helical peptide has Zimm-Bragg nucleation parameter σ=2.0×10−2. Combine the built relations for the cooperative length ℓ=σ−1/2 and the midpoint slope 1/(4σ) into a single relation between the two, then evaluate ℓ and the slope for this peptide, and compare the sharpness of its transition with that of the residues taken independently (slope 41).