# How to get a protein's dihedral angles ordered by variance?

The setting: I want to simulate protein docking and let some dihedral angles vary, but in order to keep it low-dimensional I have to select those which are most likely to change.

What I've thought of so far:

1. The easiest solution would be some tool/database that simply returns a list of dihedral angles ordered by their variance (something like http://www.ifm.liu.se/compchem/msi/doc/life/cerius46/qsar/theory_descriptors.html#515786, but that only returns the number of "meaningful" rotamers)
2. Take a set of different conformations and use principal component analysis on the phase space consisting of the dihedral angles.
3. Take Ramachandran plots and try to combine all those conditional variances (if φ is …, then ψ has some amount of freedom) into a more "holistic" statement.

The problems I encountered:

1. I couldn't find such a list.
2. Searching rcsb.org, I only found about ~100 conformations (for albumin), but if I'm not mistaken the format isn't the exactly the same, so it'd be kinda hard to compare. Besides, ~100 samples won't yield good results for >100 dimensions (=angles).
3. The only approach for which data seems to be available, but it just looks like a real pain.

If I understand you, I can't imagine how to solve the your problem with this approach. Also I don't get the physical meaning of the proposed solution.

If you have to deal with a molecule with more than 100 rotatable bonds, you are just in problems. You could find the rotatable bonds with some tools (see for example OpenBabel). But the problem will just start.

At first, you can not just try one by one the $N$ angles ($\{\theta _i \}$) (by changing the angle value $n$ times and trying each of them), because for exploring all the conformational space you will just have $n^N$ points. And you just can't know a priori which are the most important angles (in case (I think not) that effectively exist the most important angles).

Also note that not every point ($\theta _1, \theta _2, \cdots,\theta _N)$ is acceptable, just because parts of the molecule can superpose with other parts.

"Take a set of different conformations and use principal component analysis on the phase space consisting of the dihedral angles."

I think that this is not possible. The values of the rotatable angles will be just chosen by you. Even more, suppose that you have two rotatable bonds very close each other, in a linear part of the molecule. If not a large superposition ocurrs, you will have no diference in the efect of them. But the effect of them will be very correlated. So, you can get useful info. with PCA.

A possible approach

As I know very few of biology, I would make a research about if there are prefered configuration, that are much more common than others (because of biological facts).

If you are forced to work with the entire configurational space, the computational approach that comes to my mind is:

• Use very fast method for energy computations that allow you set the connectivity of the atoms. This will prevent changes of the covalent bonds due to superpositions.
• Use the Monte Carlo method, this should explore the configurational space giving you a reasonable idea of this space.

Final note

I find extrange doing docking with a large protein and ¿without a ligand? If there is a ligand for binding to the protein that we are talking about, I won't take too much care of the protein. The scoring functions or energies are few accurate, I find hard to believe that just one found configuration would represent accurately the problem (considering the large degeneracies involved)

• First of all, thank you very much for the extensive answer! —**3rd paragraph**: That's what I meant with "conditional probabilities" for the 3rd point, like "assuming all angles except $\Theta_i$ fixed, $\Theta_i$ can take $x$% of all possible angles without creating some overlapping" (this can be read off Ramachandran plots)—however, the task of combining all those conditional probabilities into a single one looks pretty tedious —**PCA**: I wanted to feed it with real world data, not randomly sampled conformations which may be invalid. I'm not sure why there's a problem with correlated atoms. – Peter Apr 3 '15 at 20:48
• —**possible approach**: Do you happen to know where I can find a list of preferred configurations or should I try to construct them myself by ranking Monte Carlo generated conformations? —**final note**: To be honest, I have no idea about the chemistry; I'm doing this for a mathematics seminar—I thought protein-protein docking would be the way to go, so that's why I want to let at least one protein stay somewhat flexible. – Peter Apr 3 '15 at 20:51
• a) "assuming all angles except..." . As the overlapping depends on many angles, you should do, for each $\theta _i$, $n^{N-1}$ configurations. b) I bet that the data is not available. About correlated bonds: Suppose that here is a contiguous rotatable bonds. A rotation about one of them will do an almost equal effect on the "surface" of the molecule (where the dispersion forces are important for binding). So, I will show a similar importance in PCA (so, even if it is would be possible, PCA can't give you useful info.). Now, imagine that there are 2 rotatable bonds not too close each other ... – user1420303 Apr 3 '15 at 22:02
• For each of them, you could get $n^{N-1}$ values. A distribution of values for each bond, that will then to be very close to the other one. Can PCA detect this? I bet no, just because both angles are almost equally important. c) "or should I try to construct them myself by ranking Monte Carlo generated conformations?" Yes! d) "I thought protein-protein docking would be the way to go" I would be difficult to predict protein-protein binding omiting solvent (water) effect, for the energy variations and specially for entropy changes. Ligand protein would be much more simple and physical meaning – user1420303 Apr 3 '15 at 22:09
• I would be interesting to hear a biochemistry (or similar) opinon – user1420303 Apr 3 '15 at 22:10