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

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Face-centered cubic (FCC) lattice models for
protein folding energy function inference and
biplane packing

• Allan Stewart

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Proteins carry out the work of the cell
Reményi A et al. Genes Dev. 2003172048-2059
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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

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Protein folding is NP-hard in most formulations

• Reduction to partition problem (Ngo Marks '92)

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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.

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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)

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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.

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Biological basis of folding principally involves
hydrophobic collapse

• Right red are hydrophobic, green hydrophilic
• Reds form a core in the center ? Long-range
  interactions

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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

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My Thesis
PDB Repo
Predict Native
continuous
discrete
discrete
Conjecture/ theorems
LatFit
FCC SC
Model
Energy
Structures
Statistical Evaluation of existing methods
Fit an Energy function
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Face-centered cubic lattice

• Lattices are discrete subgroups of R
• distinguished by their basis vectors
  (connectivity) and coordination number

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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)
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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

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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
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We found an optimum energy function, but not a
universal one.
H P S
H -.13 0 -.05
P -.04 -.06
S X
H P S
H -.15 0 0
P -.04 0
S X
13997
13365

• 13997 ? 72 successful prediction.
• A large fraction of the decoys are very deceptive

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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)
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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

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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

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What do lattice algorithms look like?

• Hart and Istrail (1997) show an 86 approximation
  ratio for a 4x2 biplane on FCC sidechain model

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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 5050 biplanar to non-biplanar

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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!

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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.
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Idea 1 Find a 4-tuple of alpha vectors in FCC

• Alpha bundles from
• Pokarowski et al. 2003

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

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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?

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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.

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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