A bacterial chemoreceptor cluster is modelled as a protein with identical…
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 bacterial chemoreceptor cluster is modelled as a protein with n=3 identical ligand-binding sites that never interact with one another directly. The protein exists in two conformations, active (A) and inactive (I). A bound ligand contributes a statistical weight x=c/KA per site when the protein is in state A, and ϵx per site (with ϵ=KA/KI<1) when it is in state I; the unliganded I state carries a weight L relative to the unliganded A state, which is taken as the reference state of weight 1.
((a)) Build the states-and-weights table over the number of ligands bound (0,1,2,3) in each conformation, sum each row with the binomial theorem, and hence show that the partition function is Z(x)=(1+x)n+L(1+ϵx)n with n=3. Differentiate lnZ to obtain the fractional occupancy θ(x), and show that it collapses to the simple hyperbola θ=x/(1+x) if either L=0 or ϵ=1. In one or two sentences explain why the three sites can still show a cooperative, S-shaped binding curve for general L and ϵ=1 even though no site ever senses the occupancy of another.
((b)) A separate synthetic polypeptide of 400 residues has Zimm-Bragg nucleation parameter σ=1.0×10−3. Starting from the cooperative length ℓ=σ−1/2 and the midpoint slope dlnsdθs=1=4σ1, derive a single relation between the slope and ℓ, then evaluate ℓ and the slope for this peptide. State whether the whole chain melts as one cooperative unit.