From Protein Structure to Function with Bioinformatics.pdf

2 Fold Recognition 33distribute themselves according to a Boltzmann distribution. Second, one can calculatethe potential of mean force responsible for the observed statistics via theBoltzmann equation. The ‘energy’ associated with a given property p is:nE(p) = -log ⎡ ⎢⎣⎢n(p) ⎤⎥(p) ⎦⎥where n obs(p) is the observed value of p and n exp(p) is the ‘expected’ value of p in areference state that assumes there are no specific interactions or preferences.Implementing this usually means discretizing distances and producing a look-uptable of force-field values rather than the continuous differentiable functions usedin molecular mechanics (with some recent exceptions). From a threading perspective,this look-up table permits one to assign an ‘energy’ to a given threaded structure.Each amino acid in the model will have some degree of exposure/burial.Depending on the amino acid type in question, one can reference the look-up tablefor a value for having, say, a valine that is 30% exposed. One can assign an energyto the entire model by simply summing these ‘energies’ across all residues in themodel. (Note that summing can be used due to the log term in the equation).A more interesting and powerful energy function can be derived if one considersinteracting pairs of amino acids. One may count the frequency with which oneamino acid type is found in close proximity to another, e.g. how often do weobserve a leucine residue 4 Å from a valine residue? As above, one gathers suchstatistics for every different pairing of the 20 amino acid types. One then calculatesthe expected frequency based on the number of observations of each of the aminoacid types at any distance separation.In fact the typical pair-potentials in widespread use are usually rather moreelaborate. For those more mathematically inclined readers I present below the fulltreatment of a widely used pair potential. Others may feel free to skip thissection.Contacts can be classified into distance bins up to some ceiling (say 30 Å).Distance bins can then be further subdivided into sequence separation bins for closerange (say three to nine residues apart) and long range (>9 residues apart). In addition,even with 50,000 protein structures in the database, data sparsity can be aproblem with this many subdivisions and so an observation weighting scheme isintroduced (the M ijkσ term) which essentially only counts an observation if it hasbeen seen 1/σ times. The residue energy E kijfor pair ij separated by k residues indistance bin l is then calculated as:obsexp⎡tjEk=RTln [1 +Mtjks] -RTln ⎢1+M⎣tjksijf (l) ⎤kxx ⎥fk(l) ⎦where M ijkis the number of occurrences for pair ij separated by k residues, σ is theijobservation weight (often set to 1/50), fk(l) is the relative frequency of pair ij separatedby k residues in distance class l: