An allosteric enzyme is a hexamer with identical catalytic sites that never interact…
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
An allosteric enzyme is a hexamer with n=6 identical catalytic sites that never interact with one another directly. The enzyme exists in two conformations, relaxed (R, high substrate affinity) and tense (T, low affinity). A bound substrate contributes weight x=c/KR per site in R, and ϵx per site in T (with ϵ=KR/KT<1); the unliganded T state has weight L relative to the unliganded R state (weight 1).
((a)) Build the states-and-weights table over the number of substrates bound (0 to 6) 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 saturation 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 enzyme can still show a cooperative substrate-binding curve without any direct coupling between its six sites.
((b)) A designed 250-residue helical peptide has Zimm-Bragg nucleation parameter σ=6.25×10−5. 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. State roughly how many independent cooperative domains the chain contains at its midpoint.