What is the entropy difference between $n$ free-floating amino acids and $n$ amino acids that have been translated into a protein? Is there any known calculation for the (~ average) change in entropy between $n$ free-floating amino acids in a cell, at typical physiological concentrations, versus those same $n$ amino acids after they have been translated into a protein by the ribosome?
I imagine calculating the conformational entropy of a protein can be quite involved, so a simplified calculation for this is fine. In general, I would be satisfied with a rough order-of-magnitude answer, or even a bound.
There are several difficulties that I can see in performing this calculation.  First, the $n$ amino acids that go into a given protein might not represent $n$ independent samples of free-floating amino acids, but have correlations between their spatial locations (i.e., if the ribosome sucks up amino acids faster than amino acids can diffuse across the cell). Second, there is a change in the excluded volume of the $n$ free-floating amino acids vs. the $n$ polymerized amino acids, which affects translational entropy of surrounding water.
 A: There are several aspects in this problem, each of which can give a rise to an independent calculation.
Amino-acids are joined in a chain
In a protein the amino-acids are joined in a chain, unlike the free-floating amino-acids. From the estimation point of view, such a chain can be modeled as a random walk on a cubic lattice, whereas the unbound amino-acids can occupy any of the lattice sites.
Amino acids have identities
Another aspect of this calculation is that amino-acids have specific identity (usually one of the 20 canonical amino-acids), and, moreover, they are arranged in a certain way in a protein. One could thus consider a ordered protein with specific amino-acid sequence
$$
X_1X_2...X_N
$$ and count all the possible re-arrangements of the amino-acids having the same identities ($X_i=X_j$. This is to be compared with all the possible rearrangements of all the amino-acids (i.e., in an arbitrary order). This way we literally calculate the information encoded in the protein (although this might be a more logical interpretation when dealing with DNA or RNA).
Other factors
The correlations between the floating amino-acids, their propensity to bind to each other, the effects related to protein adopting a preferred spatial confformation are further factors complicating such a calculation.
References
A couple of interesting papers dealing with the the entropy in protein/RNA/DNA replication process:

*

*England, Statistical physics of self-replication

*Andrieux & Gaspard, Nonequilibrium generation of information in copolymerization processes
