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Crystallography -- lecture 21 Sidechain chi angles Rotamers Dead End Elimination Theorem Sidechain chi angles Rotamers Dead End Elimination Theorem

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Sidechain space is discrete, almost A random sampling of Phenylalanine sidechains, when superimposed, fall into three classes: rotamers. This simplifies the problem of sidechain modeling. All we have to do is select the right rotamers and we're close to the right answer.

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Rotamer

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What determines rotamers CG H H H O=CO=C N CA CB CG H H H O=CO=C N CA CB CG H H H O=CO=C N CA CB "m" "p" "t" -60° gauche 180° anti/trans+60° gauche 3-bond or 1-4 interactions define the preferred angles, but these may differ greatly in energy depending on the atom groups involved.

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Measure a rotamer

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Rotamer Libraries Rotamer libraries have been compiled by clustering the sidechains of each amino acid over the whole database. Each cluster is a representative conformation (or rotamer), and is represented in the library by the best sidechain angles (chi angles), the "centroid" angles, for that cluster. Two commonly used rotamer libraries: *Jane & David Richardson: Roland Dunbrack:

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Protein cores are tightly packed The environment of a buried leucine in 1A07. The interior of a protein is tightly packed. Bad packing produces voids or collisions.

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Sidechain modeling Given a backbone conformation and the sequence, can we predict the sidechain conformations? Energy calculations are sensitive to small changes. So the wrong sidechain conformation will give the wrong energy. ≠

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Goal of sidechain modeling Desmet et al, Nature v.356, pp (1992) Given the sequence and only the backbone atom coordinates, accurately model the positions of the sidechains. fine lines = true structure think lines = sidechain predictions using the method of Desmet et al.

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Dead end elimination theorem There is a global minimum energy conformation (GMEC), where each residue has a unique rotamer. In other words: GMEC is the set of rotamers that has the lowest energy. Energy is a pairwise thing. Total energy can be broken down into pairwise interactions. Each atom is either fixed (backbone) or movable (sidechain). fixed-movablemovable-movablefixed-fixed E is a constant, =E template E depends on rotamer, but independent of other rotamers E depends on rotamer, and depends on surrounding rotamers

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Theoretical complexity of sidechain modeling The Global Minimum Energy Configuration (GMEC) is one, unique set of rotamers. How many possible sets of rotamers are there? n 1 n 2 n 3 n 4 n 5 … n L where n 1 is the number of rotamers for residue 1, and so on. Estimated complexity for a protein of 100 residue, with an average of 5 rotamers per position: = 8*10 69 DEE reduces the complexity of the problem from 5 L to approximately (5L) 2

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Dead end elimination theorem Each residue is numbered (i or j) and each residue has a set of rotamers (r, s or t). So, the notation i r means "choose rotamer r for position i". The total energy is the sum of the three components: NOTE: E global ≥ E GMEC for any choice of rotamers. E global = E template + i E(i r ) + i j E(i r,j s ) where r and s are any choice of rotamers. fixed-fixed fixed-movable movable-movable

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Dead end elimination theorem If i g is in the GMEC and i t is not, then we can separate the terms that contain i g or i t and re-write the inequality. E(i r ) + j min s E(i r j s ) > E(i t ) + j max s E(i t,j s ) E GMEC = E template + E(i g ) + j E(i g,j g ) + j E(j g ) + j k E(j g,k g ) E notGMEC = E template + E(i t ) + j E(i t,j g ) + j E(j g ) + j k E(j g,k g )...is less than... E(i r ) + j E(i r j s ) > E(i g ) + j E(i g,j s ) Canceling all terms in black, we get: So, if we find two rotamers i r and i t, and: Then i r cannot possibly be in the GMEC.

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Dead end elimination theorem E(i r ) + j min s E(i r j s ) > E(i t ) + j max s E(i t,j s ) If the "worst case scenario" for rotamer t is better than the "best case scenario" for rotamer r, then you can eliminate r. This can be translated into plain English as follows:

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Exercise: Dead End Elimination Using the DEE worksheet: (1) Find a rotamer that satisfies the DEE theorem. (2) Eliminate it. (3) Repeat until each residue has only one rotamer. What is the final GMEC energy?

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DEE exercise abcabc Three sidechains. Each with three rotamers. Therefore, there are 3x3x3=27 ways to arrange the sidechains. Each rotamer has an energy E(r), which is the non-bonded energy between sidechain and template. Each pair of rotamers has an interaction energy E(r 1, r 2 ), which is the non-bonded energy between sidechains.

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DEE exercise r1r1 r2r2 E(r 1,r 2 ) abcabc abcabc abcabc abcabcabcabcabcabc E(r 2 ) E(r 1 )

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DEE exercise: instructions (1) The best (worst) energies are found using the worksheet: Add E(r 1 ) to the sum of the lowest (highest) E(r 1,r 2 ) that have not been previously eliminated. (2) There are 9 possible DEE comparisons to make: 1a versus 1b, 1a versus 1c, 1b versus 1c, 2a versus 2b, etc. etc. For each comparison, find the minimum and maximum energy choices of the other rotamers. If the maximum energy of r 1 is less than the minimum energy of r 2, eliminate r 2. (3) Scratch out the eliminated rotamer and repeat until one rotamer per position remains. If the “best case scenario” for r 1 is worse than the “worst case scenario” for r 2 you can eliminate r 1.

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