Computing Protein Structures from Electron Density Maps: The Missing Loop Problem

1
Computing Protein Structures from Electron
Density Maps The Missing Loop Problem

• I. Lotan, H. van den Bedem, A. Beacon and J.C.
  Latombe

2
Protein Structure Experimental Techniques

• Nuclear Magnetic Resonance (NMR) spectroscopy
  limited to short sequences.
• X-ray crystallography

3
X-ray Crystallography
Crystallizing protein samples Collect X-ray
diffraction images
Calculate electronic charge a 3-D Electron
Density Map (EDM)
4
Electron Density Map

• 3-D image of atomic structure
• High value (electron density) at atom centers
• Density falls off exponentially away from center
• Limited resolution, sampled on 3D grid

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The End Goal Build Protein Model from EDM

• Completeness of automatically generated models
  varies with experimental data quality
• High Resolution ? 90 completeness.
• Low Resolution ? 2/3 completeness.
• Completing the missing fragments manually is time
  consuming.

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Experimental Data Quality Varies

• Recovering the phase of diffracted beam is
  associated with error.
• Resolution at which data were collected (High
  resolution images cannot be obtained for all
  proteins)
• Not all replicas of protein in the protein
  crystal are identical
• Mobility of molecule fragments
• Temperature dependent atomic vibration

7
Existing Techniques

• Existing software rely on
• Pattern recognition techniques
• Unambiguous density
• Elementary stereochemical constraints.

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Model Refinement

• Standard Maximum Likelihood (ML) algorithms
  exploit experimental and model phase information
  to build new refined models.
• Iterating model building and refinement steps
  improves completeness and quality of models.
• The problem missing fragments (Usually loops).
• The solution filling the gaps at early stage.

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Goal Propose Candidates to Missing Fragments

• Input
• EDM
• Known structure
• Anchor residues
• The amino acid sequence
• Output propose a structure that fall within the
  radius of convergence of existing refinement
  tools (1-1.5Å)

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Model

• Standard Phi-Psi model.
• Compute backbone, ignore side chains except Cß
  and O atoms.
• Loop closure
• Mobile anchor vs. stationary anchor.
• Closure is measured as the RMSD distance of the
  Mobile anchor atoms from stationary anchor atoms.

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IK EDM ? Loop Structure

• Two stages algorithm
• Guided by the EDM, sample closing conformation.
• Refine top-ranking conformation, using local
  optimization, while maintaining loop closure.
• Conformations Ranking density fit and
  conformational likelihood.

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Stage 1 Generating Loop Candidates

• Employ cyclic coordinate descent (CCD) method to
  obtain closing conformations, up to a tolerance
  distance dclose.
• Starting conformations are obtained by a random
  procedure, biased by PDB-derived distributions.
• Best scoring (95 percentile) conformations are
  submitted to stage 2.

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Cyclic Coordinate Descent (CCD)
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Adding the Electron Density Constraints

• We would like to guide the loop closing to fit
  the EDM.
• For residue i the CDD proposes a distance
  minimizing dihedral angles (F,?)ip.
• Find a pair (F,?)i in a square neighborhood of
  (F,?)ip that maximizes the local fit to the EDM.
  The neighborhoods size is reduced linearly with
  CCD iterations to allow closure.

Atoms that are changed by angle pair i and not
i1
Center of atom Aj
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Stage 2 Refining Loop Candidates

• Improve models fit to experimental data (This
  time the model as a whole, as opposed to local
  fit in stage 1).
• Maintain loop-closure constraint during
  optimization process.

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Target Function

• For conformation q, the target function T(q) is
  the sum of the squared differences between the
  observed density and the calculated density at
  each grid point in some volume V around the loop.

Scaling Factors
Grid Points in Volume
Calculated Density (sum of contributions of atoms
within a cutoff distance from gi)
Observed Density
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Optimization with Closure Constraints
Generic Approach Objective function optimization
(T(q)) while performing given task (loop-closure)
by taking advantage of manipulator redundancy
(DoFs).
f(q) forward kinematics equation. J(q) 6-by-n
Jacobian the change to the end of the
chain J(q) an approximation of J-1(q) N(q)
Orthonormal basis for the Null-Space (n-6
dimensions) y ?T(q)/?q gradient vector of
objective function T(q)
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Minimization Procedure Monte Carlo and Simulated
Annealing

• Choose a random sub-chain with at least 8 DoFs.
• Propose random move with magnitude proportional
  to current temperature
• High temperature use exact IK solver (Dill)
• Low temperature pick random direction in
  null-space
• Minimize resulting conformation (gradient decent)
• Accept using Metropolis criterion
• P(accept qnew) e( T(qprev) - T(qnew) ) /
  temp
• Use simulated annealing at each step decrease
  pseudo-temperature
• At each step verify closure constrained is
  satisfied within tolerance.

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Results High Resolution Data

• Applying RESOLVE to the data (high resolution)
  yielded 88 completed initial model .
• Applying the alg to a gap of 12 residues.
• Magenta the structure from the PDB
• Cyan Best scoring structure, RMSD 0.25Å.
• The lowest RMSD for 7 residues gap at the end of
  stage 1 is 0.35Å.

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Results Low Resolution Data

• Applying RESOLVE yielded a model with 61
  completeness.
• Applying the alg to a gap of 12 residues.
• Magenta the highest scoring, RMSD 0.6Å.
• Yellow starting conformation (end of stage 1),
  RMSD 2.1Å (the lowest)