Physics 213 Problem 2 Week 5
Proteins
We depend on protein-folding for life processes. Some proteins are long chain molecules. A
simple model consists of N molecular segments, like the three illustrated schematically below.
Each triangular segment can be oriented in one of three possible orientations with respect to its
neighbor. Weaker bonds allow parts of the chain to stick together and fold up. Each segment can
either be i) open in 3 possible orientations or ii) folded in just 1 orientation. (Note that this is
rather the opposite behavior than in polymers, where there are more ways to be folded than to be
stretched out.)
a) How many configurations are there for the protein to be completely folded (Ωfolded) and to be
completely unfolded (Ωopen)?
Ωfolded = 1 ; Ωopen = 3N
b) In the open state the molecule has an energy Δ higher than in the typical folded state. Recall
that the probability of a particular configuration ‘c’ (open or closed) is given by pc ∝ Ωc Pc,
where Ωc is the number of states in that configuration and Pc is the probability that each state is
occupied. Complete the following equation in terms of N, Δ and kT:
c) Consider a molecule with N = 10 segments. If Δ = 5 x 10-20 J and the molecule is located in
your body (T = 310 K), what is the ratio of the time it spends open to the time it spends folded?
t(open)/t(folded) = Prob(open)/Prob(folded) = 0.5
molecular segments
(very schematic)
Folded protein Unfolded (open) protein
Solution
