1. Face-centered cubic (FCC) lattice models for protein folding: energy function inference and biplane packing
Allan Stewart

2. Proteins carry out the work of the cell
Reményi A et al. Genes Dev. 2003;17:

3. Dogma of computational protein structure prediction (PSP)
The biological “native” has the minimum energy conformation over the entire fold landscape.
Controversial whether native is unique or there if there may generally be an ensemble

4. Protein folding is NP-hard in most formulations
Reduction to partition problem (Ngo & Marks '92)

5. The problem is hard under any reasonable model
The protein energy is minimized iff
there is an assignment of move vectors into two subsets st. the sum of the subsets are equal
We find NP-hardness results for other formulations: Ising model, bin packing, Hamiltonian path. (See Istrail & Lam, Combinatorial Algorithms for Protein Folding in Lattice Models: A Survey of Mathematical Results, 2009)
This sort of worst-case intractability analysis will not improve in even the simplest of models.

6. Biological basis of folding principally involves hydrophobic collapse
“Truth is much too complicated to allow anything but approximations.”
~ John von Neumann
Protein primary structure: N - AGECH... - C
The tertiary (3D) structure is dependent on primary structure alone (experimental evidence)

7. Biological basis of folding principally involves hydrophobic collapse
Suppose the protein sequence to be a string over the 20-letter amino acid alphabet
Avoiding the complexity (charge, size) of amino acids, we classify the residues as
'H': hydrophobic residues
'P': hydrophilic (polar) residues
The HP model (Ken Dill, 1985) is a simplest framework for folding, and prioritizes H-H interactions.

8. Biological basis of folding principally involves hydrophobic collapse
Right: red are hydrophobic, green hydrophilic
Reds form a core in the center → Long-range interactions

9. There are two camps in protein folding
Off-lattice (continuous mathematics)
More flexibility
Heuristic methods perform fairly well
Optimality of simulation is uncertain
On-lattice (discrete mathematics)
Exhaustive enumeration of space
Provably timely and near-optimal results
But a lot is yet unknown...
We don't know if the lattice gives good prediction

10. My Thesis discrete discrete PDB Repo Predict Native continuous LatFit
Conjecture/ theorems
LatFit
FCC SC
Energy
Structures
Model
Fit an
Energy
function
Statistical
Evaluation
of existing
methods
My Thesis

11. Face-centered cubic lattice
Lattices are discrete subgroups of R
distinguished by their basis vectors (connectivity) and coordination number 3

12. Folding is the minimization of the energy potential function
Protein conformations are Boltzmann distributed
Typical energy functions sum values for each of the pairs of amino acids in the protein sequence.
Caveat foldtor
Your fold is only as good as your potential function, and how hard you work is dependent on the function.
(Some don't)

13. Prior work shows that we can do with just a few parameters
HP model typically only scores H-H contacts
This corresponds to a symmetric interaction matrix

14. We look for empirical parameters which improve over the 'HP' matrix
Extract 1198 PDB structures
Generate decoys
decoys are natives which have been perturbed by roughly 16%
Count all types of contacts
Use gradient ascent to optimize choice of parameters
max

15. We found an optimum energy function, but not a universal one.H P S
-.13
-.05
-.04
-.06 X H P S
-.15
-.04 X
13997
13365
13997 → 72% successful prediction.
A large fraction of the decoys are very deceptive

16. Pairwise Function Impossibility Conjecture
In collaboration with Warren Schudy and Sorin Istrail, we conjecture that no linear function f which sums pairwise potential satisfies axioms (1) and (2)
We formulate it as an LP with the above as linear constraints.
(1)
(2)

17. Towards Realistic Models of Folding
“For me, the first challenge for computing science is to discover how to maintain order in a finite, but very large, discrete universe that is intricately intertwined.”
~ Edsgar Dijkstra

18. What do lattice algorithms look like?
We chop the protein into blocks and align blocks with high hydrophobicity.
(Hart and Istrail 1995)
Use inequalities to bound numbers of contacts
Approximation algorithms

19. What do lattice algorithms look like?
Hart and Istrail (1997) show an 86% approximation ratio for a 4x2 biplane on FCC sidechain model

20. The biplane is near-optimal, but is it realistic???
We found an optimal center cutting plane through each protein and annotated the hydrophobics lying within distance k.
There is high variance in biplane hydrophobicity.
Roughly 50:50 biplanar to non-biplanar

21. The biplane is near-optimal, but is it realistic???
set
alpha
beta
unstructured
solvent
biplanar
30%
19%
49%
46%
non-biplanar
37%
20%
44%
47%
Biplane corresponds best to a globular fold.
The alpha helix is a problem!

22. Rescuing the alpha-helix with the FCC
The alpha-helix is a right-handed helix, ~4 residues per turn.
The FCC places spheres at angles : the
dihedral angles of the helix.

23. Idea #1: Find a 4-tuple of alpha vectors in FCC
Alpha bundles from
Pokarowski et al. 2003

24. Idea #2: Assemble octahedrons in FCC lattice
Goal 73 69
Build an octahedron-like conformation with hydrophobics towards center.
Exploits angles of FCC: face angles = 120deg.

25. Conclusion: new frontiers for FCC sidechain folding
Implications for algorithm design
Block partitioning
Fold block into biplane or octahedron
Can we prove bounds for increasingly complex methods?
If we prove pairwise impossibility, how do we construct our energy function?

26. Questions?
Fire away.
“If you don't work on important problems, it's not likely that you'll do important work.”
~ Richard Hamming
Thank you for doing important work.

28. If I reach this slide, something went wrong
“There's no sense in being precise when you don't even know what you're talking about.”
– John von Neumann
“It is better to do the right problem the wrong way than the wrong problem the right way.
— Richard Hamming