1. In silico Protein Design: Implementing Dead-End Elimination algorithm
CS273
Tyrone Anderson, Yu Bai &
Caroline E. Moore-Kochlacs
May 31st 2005

2. Computational protein design
Backbone scaffold
New sequence
Iterative refinement
Native structure
Given backbone
coordinates, find the best sequence(s) with which the protein is stable.

3. Components of the problem
The protein design problem can be roughly divided into searching procedure and scoring function.
The searching procedure samples the sequence space AND side-chain conformational space to create conformations.
The scoring function evaluates each conformation created by the searching procedure. The evaluation scores are used to rank the conformations (and therefore the sequences) and pick the best one to be the final model.

4. Why is searching procedure difficult ?
Consider a short protein with 20 amino acids. Possible sequence:
S = 2020 ~1026
Each side chain has on average 2 dihedral angles (χ angles). Assuming that we will sample every 40º in the dihedral angle space,
N = (360/40)(202) ~1038
This number S*N is too large to be naively sampled
Algorithms that find good solutions by screening only parts of the search space are needed

5. Rotamer libraries
Already in the 70s, Janin et al. showed that different side chain conformations are not found in equal distribution over the dihedral angle space but tend to cluster at specific regions of the space, much as in the Ramachandran plot.
In the 80’s, this observation was used to improve modeling of side chain conformations.
MAY BE SKIP
Today, essentially all programs that model side chain conformations use rotamer libraries.

6. What do rotamer libraries provide?
Rotamer libraries reduce significantly the number of conformations that need to be evaluated during the search.
This is done with almost no risk of missing the real conformations.
Even small libraries of about rotamers cover about 96-97% of the conformations actually found in protein structures.
The probabilities of each rotamer in the library can be applied to estimate the potential energy due to interactions within the side chain and with the local backbone atoms, using the Boltzmann distribution. (Not applied in this project)
E  ln(P)

7. Rotamer Library Creation
Source:
Parsing:
Select all Nitrogens (N), Oxygens (O), Alpha Carbons (CA) , & all other Carbons i.e CD, CZ, etc.
Exclude all other elements and the end of file
Store in a 3D array: Residues (1D)  Rotamers (2D)

8. Rotamer Library Creation
Original Rotamer: ('R', 1)
Atom
Residue X Y Z N R
4.986
-6.494
10.983 CA
3.99
-5.511
11.369 C
2.774
-6.288
11.848 O
2.209
-7.039
11.068 CB
3.528
-4.689
10.203 CG
4.563
-3.671
9.815 CD
4.228
-2.905
8.546 NE
5.119
-1.768
8.418 CZ
5.354
-1.136
7.268
NH1
4.736
-1.573
6.181
NH2
6.164
-0.097
7.202 HN
4.75
-7.184
10.299 HA
4.344
-4.953
12.119
1HB
3.357
-5.292
9.424
2HB
2.682
-4.216
10.45
1HG
4.656
-3.012
10.561
2HG
5.434
-4.141
9.675
1HD
4.338
-3.509
7.756
2HD
3.281
-2.586
8.592 HE
5.618
-1.384
9.195
1HH1
4.882
-1.115
5.304
2HH1
4.123
-2.361
6.237
1HH2
6.314
0.365
6.328
2HH2
6.628
0.228
8.026
HEAD 1
7.156
-7.614
11.118 2
6.202
-6.499
11.516 3
6.564
-5.625
12.293
Rotamer Library Creation
Example:
Black: Include in array
Red: Exclude from array
Blue: *Not part of the array

9. Aligning with the Backbone
Translate backbone and rotamer to origin
CA atom of ‘R’, 1 and backbone = (0, 0, 0)
Rotate rotamer around X-axis
Rotate rotamer around Z-axis
Translate rotamer back to original position based on original position of CA atom
i.e. CA atom of ‘R’, 1 = (3.99, , )
       
                               
       
                                

10. Rotamer Library Manipulation
Retrieve a specific rotamer:
Provide the residue and the rotamer number
i.e. ‘R’, 1  Gives you the 1st rotamer related to the Arginine residue
Rotamer is already aligned with the backbone
Only the coordinates of the atoms are returned in a 2D array
Aligned Rotamer: ('R', 1)
3.99
-5.511
11.369

11. Now,
Consider again our protein of 20 amino acids. Each side chain has on average 9 rotamers. Assuming that we search now in the space of rotamers:
N = 920 ≈ 1019
The searching space is restricted and oriented but the number of conformations is still too large for a naive search

12. Algorithms in searching (side-chain) conformational space
Greed search (systematically scans the search space)
DEE (Algorithmic approaches to reduce the search space)
Self consistent algorithms (iterative sequential procedure)
Monte Carlo algorithms (random search)

13. DEE (Dead-End Elimination)
Aims to safely eliminates (clusters of) rotamers without loosing the GMEC (Global Minimum Energy Conformation).
rotamer ir
in force field
of backbone only
rotamer ir with
rotamer(s) of other
residues
Given residue i, eliminate a rotamer ir if the minimum energy it can obtain by interaction with conformational background (js) is higher (worse) than the maximum possible energy that another rotamer it (of the same residue) can have

14. E(i,j) is it js
rotamer background
Desmet et al., 1992

15. The Goldstein improvement
Rotamer ir can be safely eliminated when some other rotamer it exists with lower (better) energy for a certain environment that mostly favors ir.
This criteria is much less restrictive and therefore more powerful. It requires though more computational time.

16. The Goldstein improvementis
E(i,j) it js
rotamer background

17. Scoring function: Energy function
Terms:
Van der Waals
represents packing specificity
Hydrogen bonding
typically represented by an angle dependent, hydrogen bond potential
Electrostatics
Guard against destabilizing interactions between like charged residues
Internal coordinate terms
‘bonded’ energies
Solvation energy
Protein-solvent interactions
Entropy
Assumes conformational space is completely restricted in the folded state
Gordon et al, 1999

18. Scoring function: Energy function
Terms:
Van der Waals
represents packing specificity
Hydrogen bonding
typically represented by an angle dependent, hydrogen bond potential
Electrostatics
Guard against destabilizing interactions between like charged residues
Internal coordinate terms
‘bonded’ energies
Solvation energy
Protein-solvent interactions
Entropy
Assumes conformational space is completely restricted in the folded state
Gordon et al, 1999

19. Van der Waals Interaction between two uncharged atoms
Mildly attractive as two atoms approach from a distance
Repulsive as they approach too close
Represents packing specificity
Prefers native-like folded states with well-organized cores over disordered or molten-globule states
Gordon et al, 1999

20. Van der Waals 12-6 Lennard-Jones potential Standard approximation
12-6 Lennard-Jones potential
Standard approximation
R = distance between atoms
R0 = van der Waals radii
Dij = well depth
Variation from Kuhlman and Baker, 2004
Erep is dampened to account for the fixed backbone and rotamer set being used.

21. Electrostatics Stability Specificity
Moderate temperatures: favorable electrostatic interactions not thought to be strong enough to compensate for the energy of desolvation
Extreme conditions: salt bridges may stabilize
Specificity
folding and functional interactions
maybe the more significant role of electrostatics
Currently, term guards against destabilizing interactions between like-charged residues
Gordon et al, 1999

22. Electrostatics Approximations: Coulomb’s Law (Gordon et al, 1999)
Qi,Qj = charge on amino acid
R = distance
ε= dielectric constant = 40
Bayesian version (Kuhlman & Baker, 2004)
Probability of two amino acids close together given environment and distance (from PDB)
aa=amino acid, d = distance, env =environment

23. Solvation
Hydrophobic effects drive folding, modeling solvation effects is critical to a protein design force field
Computationally expensive
Solvent model from Lazaridis and Karplus, 1999
dij = distance between atoms, rij = van der Waals radii, Vi = atomic volume
ΔGref = reference solvation free energy, ΔGfree = solvation free energy of free (isolated) group
λ = correlation length

24. Energy Function: Incomplete model
Current standard models include Bayesian terms based on PDB statistics
Several terms have not been thoroughly validated as useful for design (Gordon et al, 1999)
Hydrogen bonding
Electrostatics
Internal coordinates
Current standard models are ad hoc, physical quantities and variables are weighted based on “what works best”

25. Integrated algorithm schema
..N1 I2L3D2E1F2. ..
D1. . .
..N1 L2L3K2N1V1. ..
..W7L3D2K9K10G1. ..
Best seq
D2. . .
2nd
DEE
Exhaustive
search
1st order
DEE
..N D2...
Iterating algorithm till no more rotamers can be eliminated.
only one rotamer is left for each of several side chains (i.e., these are part of the GMEC). For several others, only a (hopefully) few are left.
DEE is extended to the Second order: cluster size =2, only core design finished
An additional exhaustive search is then applied to obtain the final model D … N
N1 N2 N3
. . . N

26. Design cold-shock protein (core) & Trp-Cage protein
Cold-shock protein (1MJC.pdb)
10 residues (core)
Trp-Cage(1L2Y.pdb)
20 residues

27. cold-shock protein (core)
After 1st-order DEE 2 3 7 8 9 5 4 1 6
Residue 0 A 1 F 2 3 I L V W 4 5 6 7 …
Residue 3
Residue 8
Residue 1 A 1 F 2 I 3 L V W 5 6 …
Residue 3 7
Residue 8
Hydrophobic
Amino acids:
A (1),
F (3),
I (3),
L (2),
V (2),
W(7)

28. Trp-Cage protein All 20 AA After 1st-order DEE . . . Residue 9
H: 1...8
I: 1,2,3 K:
L: 1,2 M:
N: 1...9
P: 1 Q: R:
S: 1,2
T: 1
V: 1,2
W:1...7
Y: 1,2,3
After 1st-order DEE
. . .
Residue 9
A: 1
C: 2
D: 6
E: 6,15
F: 1
G: 1
H: 7
I: 2
K: 18,22,59
L: 1
M: 1,12
N: 6
P: 1
Q: 4
R: 7,107
S: 2
T: 1
V: 1,2
W:6
Y: 1
All 20 AA
Both finished the first order DEE, then go to 2nd order DEE, Trp-Cage is still running. core design finished two rounds, and further Evaluation for the best sequences are applied.

29. Results for cold-shock protein (core)
Seq EScore
N: V F I V V I L V F V 1. 2. 3. 4. 5. 6. 7. 8. 9.
10.
. . .
I F I I I I L I F V
I F I I V I L I F V
V F I I I I L I F V
I F I I I I L V F V
V F I I V I L I F V
I F I I V I L V F V
I F V V I I L I F V
V F V I I I L I F V
I F V I I I L V F V
V F I I I I L V F V
Cold-shock protein (1MJC.pdb)
10 residues (core)
We are very happy to see the predicted sequences are at minimum 50% identical to the native one, and the rest residues are limited to 2 amino acids which include the native ones. Secondly, there is high similarity of the energies of the sequences which suggests the global E minima of conformational space is smooth and our solution is likely to be stable.

30. Summary & Future Speed Accuracy
Achievement: Naïve ~ 107 sequence X 104 rotamers
DEE ~ 3000 sequences X 200 rotamers
BioX-cluster(~ GHz Xeon CPUs) 26 hrs
Future: Rotamers ordering (by self-energies) (Gordon 1998)
Comparison cluster focusing (Looger 2001)
Stronger elimination criteria (Looger 2001)
Accuracy
Achievement: 50 % identical with native sequence
High similarity in total energy
Future: Additional energy terms (H-bond, solvation)
Incorporate rigorous force field calculators(Gromacs)
Structure relaxation
Have not had chance to compare with other algorithms( greedy, MC), but simplest demenstration is naïve search needs

31. Thanks !