Allan Stewart

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

2. Proteins carry out the work of the cell Reményi A et al. Genes Dev. 2003;17:2048-2059

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

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

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. • 13997 → 72% successful prediction. • A large fraction of the decoys are very deceptive 13997 13365

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??? • 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 • Build an octahedron-like conformation with hydrophobics towards center. • Exploits angles of FCC: face angles = 120deg. 69

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