1. Dead-End Elimination for Protein Design with Flexible Rotamers
Ivelin Georgiev
Donald Lab
02/19/2008

3. Computational Design energy function rotamer library protein design
wildtype
energy
function
input
structure
rotamer
library
protein design
algorithm
The input to a protein design algorithm typically includes the following: A wildtype fixed-backbone structure which is to be redesigned; A rotamer library of low-energy side-chain conformations that discretizes the continuous side-chain conformation space and makes the computational search feasible; and An energy function for scoring and ranking the candidate structures. The energy function typically consists of some of the standard molecular mechanics energy terms (such as vdW, electrostatics, and solvation energies), but may also include some statistical and other terms. Based on the input model, the protein design algorithm makes predictions about mutations to the wildtype sequence, in order to achieve a desired novel functionality, such as improving the thermostability of the protein, switching the enzyme specificity towards a novel substrate, or redesigning the protein to perform a completely novel function.
stability
specificity
novel function
drug design
mutant …

4. provable energy minimization ε-approximation algorithmC
MinDEE
provable energy minimization
ensembles
partition function
ε-approximation algorithm q q*
Contributions
redesign for Leu

5. Traditional-DEE it ir rotamer pruning O(q2n2) Enumerate E lower
Desmet et al., 1992 it ir
E lower
bound
E upper
bound
fixed backbone/side-chains ir
rotamer pruning
O(q2n2) E it
Dead-End Elimination (DEE) is a provably-accurate deterministic algorithm that reduces the search space for protein design problems by pruning rotamers that are provably not part of the GMEC. Specifically, for a given residue position i, DEE compares the pruning candidate rotamer i_r (here, the notation i_r means rotamer identity r at residue position i) against a competitor rotamer i_t. If a lower bound on the energy of any conformation with i_r is still greater than an upper bound on the energy of any conformation with i_t, then we can always obtain a lower-energy conformation by switching from rotamer i_r to rotamer i_t. Thus, rotamer i_r can be provably pruned from further consideration, since it cannot belong to the GMEC. This pruning step is repeated for all rotamers as pruning candidates, for all competitors, and for all residue positions, until no more rotamers can be provably pruned. This typically results in a significantly reduced set of conformations that have to be subsequently enumerated, making the mutation search computationally feasible. Unfortunately, DEE is provably-accurate only for a fixed backbone. The question then arises, can there be an algorithm for backbone flexibility that incorporates the same provable guarantees as DEE for a fixed backbone.
conformations
Enumerate
GMEC

6. Traditional-DEE with Rigid Rotamers/Backbone
Conformations
Energy it

7. Traditional-DEE with Side-chain Dihedral Flexibility
Conformations
min
Energy
max it

8. √ X √ C C C provably-correct not provably-correct MinDEE
Traditional-DEE C
rigid energies √
provably-correct
Traditional-DEE C
E minimization X
not provably-correct C √
MinDEE
E minimization
provably-correct

9. C √
MinDEE
E minimization
provably-correct

10. continuous side-chain dihedral space MinDEE voxels bound rotamer
movement
Instead of sampling the backbone dihedrals, we define restraining boxes around each residue that bound the displacement of this residue from its original pose. In effect, through kinematic constraints, these boxes bound the backbone movement. Instead of a discrete set of backbones, we thus have a continuous space of possible backbone conformations.
The restraining boxes can be obtained in many ways, but we use two bounding conditions. First, we bound the displacement of Ca atoms from their original position, and second, we bound the change in the phi and psi angles for each flexible residue.

11. E(ir , js) χir χjs lower / upper energy bounds ir MinDEE it js ir
We can then compute lower and upper energy bounds on the rotamer-to-backbone and pairwise rotamer-to-rotamer interaction energies within the restraining boxes for each rotamer. For example, we compute a lower energy bound for a rotamer pair i_r – j_s by allowing the backbone to flex in order to minimize the interaction energy between these two rotamers, within the rotamer restraining boxes. The minimization process in (phi, psi) space is shown by the red path in the figure on the right.
χir ir js
χjs

12. MinDEE: - - > 0 lower / upper energy bounds ir it js lower bound
pruning
candidate
lower / upper
energy bounds ir it
competitor js
witness
MinDEE: - -
> 0
The E(minus) terms represent a lower bound on the rotamer-to-template and rotamer-to-rotamer interaction energies that involve the pruning candidate rotamer i_r, and the E(plus) terms are the respective upper energy bounds that involve the competitor rotamer i_t. The main difference from the traditional-DEE criterion is the inclusion of the E(delta) term, which takes into account possible energy changes due to backbone movement.
lower bound
on ir conformation
energies
upper bound
on it conformation
energies
possible energy
changes due to
rotamer movement
not in trad-DEE

13. MinDEE: Side-chain Dihedral Flexibility
traditional-DEE
MinDEE
Also add some of Bruce Tidor’s summary of the MinDEE algorithm (appropriate manipulation of 1- and 2-body energy bounds) from the K* grant.

14. MinDEE Applications

15. ∫ MinDEE Applications GMEC-based Ensemble-based min JCB’05 1 Z single
lowest-energy
conformation
Ensemble-based
JCB’05 ∫ 1 Z
weighted
average
sequence K*
TIAAIC
7.3
GIRMQM
3.1
TGIAIV
2.9
LMLAIS
1.7
TWAIGY
0.3 a
K*: provably-accurate approximation to the binding constant via conformational ensembles

16. MinDEE/A*: GMEC-based Method

17. MinDEE/A*: GMEC-based Method
pruning
O(n2r2) C’
A* search
(E lower bounds)
full E
minimization

18. MinDEE/A*: GMEC-based Method
pruning
O(n2r2) C’
A* search
(E lower bounds)
full E
minimization

19. MinDEE/A*: GMEC-based Method
pruning
O(n2r2) C’
A* search
(E lower bounds) …
full E
minimization …

20. MinDEE/A*: GMEC-based Method
pruning
O(n2r2) C’
A* search
(E lower bounds) … …
full E
minimization …
minGMEC
B(c) > E(best)

21. Hybrid-K*: Ensembles Method

22. Hybrid-K*: Ensembles Method
Volume
filter
seq1 C
DEE
pruning C’
A* search
(E lower bounds) … …
full E
minimization p’ q*

23. Hybrid-K*: Ensembles Method
Volume
filter
seq1 C
DEE
pruning C’
A* search
(E lower bounds) … …
full E
minimization p’ q*

24. Hybrid-K*: Ensembles MethodC
DEE
pruning C’
A* search
(E lower bounds) …
full E
minimization p’ q*
q* < (1-ε)q

25. Hybrid-K*: Ensembles Method
repeat search C C
DEE
pruning
DEE
pruning C’ C’
A* search
(E lower bounds)
A* search
(E lower bounds) … … …
full E
minimization
full E
minimization p’ p’ q* q*
q* < (1-ε)q
q* ≥ (1-ε)q

26. Hybrid-K*: Ensembles Method
Volume
filter
seq1
seqn C … C
DEE
pruning
DEE
pruning C’ C’
A* search
(E lower bounds)
A* search
(E lower bounds) … … …
full E
minimization p’
full E
minimization q* q*
q* ≥ (1-ε)q
q* ≥ (1-ε)q

27. Hybrid-K*: Inter-mutation Pruning
Volume
filter
seq1
seqn C C
DEE
pruning C’
A* search
(E lower bounds) …
full E
minimization p’ q*
q* ≥ (1-ε)q

28. Hybrid-K*: Inter-mutation Pruning
Volume
filter
seq1
seqn C … C
DEE
pruning
DEE
pruning C’ C’
A* search
(E lower bounds)
A* search
(E lower bounds) … …
full E
minimization p’
full E
minimization p’ q* q*
Ķ*n
K*i
<

29. Hybrid-K*: Inter-mutation Pruning
Volume
filter
seq1
seqn C … C
DEE
pruning
DEE
pruning C’ C’
A* search
(E lower bounds)
A* search
(E lower bounds) … … …
full E
minimization p’ p’
full E
minimization q* q*
q* < (1-ε)q
K*i
>>>
Ķ*n

30. Hybrid-K*: Intra-mutation Pruning
Volume
filter
seq1
seqn C … C
DEE
pruning
DEE
pruning C’ C’
A* search
(E lower bounds)
A* search
(E lower bounds) … … …
full E
minimization p’
full E
minimization q* q*
q* ≥ (1-ε)q
q* ≥ (1-ε)q

31. Results MinDEE: GMEC-based MinDEE: Ensembles Traditional-DEE
this work
previous
Results
Traditional-DEE
single structure
rigid energies
MinDEE: GMEC-based
K* (RECOMB’04)
E minimization
ensembles
MinDEE: Ensembles

32. Structural Model 1AMU (Conti et al., 1997) Residues: 39 flexible: 9
steric shell: 30
Flexible ligand
AMP
Richardsons’ rotamer library
AMBER (vdW,elect,dihed) + EEF1
2-point mutation search for Leu
GAVLIFYWM allowed
235
236
239
278
299
301
322
330
331 D A W T I C
GrsA-PheA Active Site

33. Comparison to Traditional-DEE
minGMEC
trad-GMEC
minGMEC:
235
236
239
278
299
301
322
330
331 D M W T I A* C 2 5 3 6 - 9
* minGMEC rotamer pruned by traditional-DEE
trad-GMEC ranked 397th
E(minGMEC) < E(rigid-GMEC) by ≈ 6 kcal/mol

34. Hybrid-K* Computational 9 hrs. on 24 processors
Conf. Remaining
Pruning Factor (%)
Initial
6.8 x 108 -
Volume Filter
2.04 x 108
3.33 (70.0)
MinDEE Filter
4.13 x 106
49.43 (98.0)
Steric Filter
3.86 x 106
1.07 (6.5)
A* Filter
7.82 x 104
49.41 (98.0)
Computational
9 hrs. on 24 processors
Original K* fully-evaluated 30% more conformations
K* w/o filters: ≈ 3,263 days
Top 40 Mutations – Hybrid-K*
Predictions
T278M/A301G (Stachelhaus et al., 1999) ranked 3rd
G301 in all known natural Leu adenylation domains
Experimental verification

35. MinDEE/A*  Ew = 12.5 kcal/mol 4 days on a single processor
Top 40 Mutations – MinDEE/A*
Ew = 12.5 kcal/mol
4 days on a single processor
206 of 421 rotamers pruned
over 60,000 extracted conformations
7,261 conformations (221 unique sequences) within Ew
minGMEC: A236M/A322M
Rotamer Diversity for A236M/A322M
Conf Energies vs. RMSD for A236M/A322M 

36. Conclusions and Future Work
Top 40 Mutations – Hybrid-K*
Traditional-DEE not correct with energy minimization
MinDEE provably-correct and efficient
MinDEE capable of returning lower-energy conformations
Ensemble-based and GMEC-based redesign predictions are substantially different
MinDEE: Ensembles Method successfully predicts both known and novel redesigns
Improve MinDEE pruning efficiency
Improve model accuracy
Marriage of MinDEE and BD

37. Acknowledgments Funding: Bruce Donald Ryan Lilien Amy Anderson
Serkan Apaydin
John MacMaster
Tony Yan
All members of Donald Lab
Funding:
NIH
NSF