DISTANCE MATRIX-BASED APPROACH TO PROTEIN STRUCTURE PREDICTION

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DISTANCE MATRIX-BASED APPROACH TO PROTEIN
STRUCTURE PREDICTION

• Andrzej Kloczkowski, Robert L. Jernigan, Zhijun
  Wu, Guang Song, Lei Yang - Iowa State University,
  USA
• Andrzej Kolinski, Piotr Pokarowski - Warsaw
  University, Poland

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Matrices containing structural information

• Distance matrix (dij)
• Matrix of square distances D (dij2)
• Contact matrix C (cij)
• cij 1 if dij gt dcutoff
• otherwise cij 0
• Laplacian of C (Kirchhoff matrix)
• Lc diag(Scij) - C

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• Lc-1 generalized inverse of Lc in elastic network
  models defines covariance between fluctuations
• Similarly we can define Laplacian of D LD and
  generalized inverse LD-1

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Spectral decomposition of structural matrices

• A S lk vk vkT
• is expressed by eigenvalues and corresponding
  eigenvectors of A

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Spectral decomposition of a square distance matrix

• Spectral decomposition of a square distance
  matrix is a complete and simple description of a
  system of points. It has at most 5 nonzero,
  interpretable terms
• A dominant eigenvector is proportional to r2 -
  the square distance of points to the center of
  the mass, and the next three are principal
  components of the system of points.

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• CN contact number
• PECM principal eigenvector of the contact
  matrix
• GNM fluctuations of residues computed from the
  Gaussian Network Model (Bahar et al. 1997)
• SVR Support Vector Regression variant of SVM
  for continuous variables
• B-factor temperature factor from X-ray
  crystallography

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• B-factor correlates with the distance from the
  center of mass r2 Petsko 1980
• Correlation between fluctuations of residues and
  the inverse of their contact number Halle 2002

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Approximation of distance matrices

• A S lk vk vkT
• We used a nonredundnt database of 680 structures
  from the ASTRAL database
• r2 itself approximates structures with DRMS 7.3Å
• r2 combined with first principal component
  approximates structures with DRMS 4.0Å

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• Current work
• Prediction of r2 from the sequence with SVR
• Prediction of the first structural component
  from the sequence

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Principal Component Analysis of Multiple HIV-1
Proteases Structures

• 164 X-ray PDB structures and 28 NMR PDB
  structures and 10,000 structures (snapshots) from
  the Molecular Dynamics simulations were analysed.
• The Principal Component Analysis of these three
  different datasets were performed.
• The results were compared with normal modes
  computed from the Anisotropic Network Model an
  Elastic Network Model that considers anisotropy
  of fluctuations of residues in protein.

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The a-carbon trace of the HIV-1 structure
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Elastic network models

• Rubber elasticity
• (polymers - Flory)
• Intrinsic motions of structures
• (Tirion 1996)
• Simple elastic networks of uniform material
• Appropriate for largest, most important domain
  motions of proteins - independent of many
  structure details
• High resolution structures not needed to learn
  about important motions

Rubbery Bodies with Well Defined, Highly
Controlled Motions
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Elastic Network Models Calculating Protein
Position Fluctuations

• Vtot(t) (g/2) tr DR(t)T G DR(t)
• ltDRi . DRjgt (1/ZN) ? (DRi . DRj) exp
  -Vtot/kT dDR
• (3kT/g) G-1ij
  
• G Kirchhoff matrix of contacts

G
Compute Normal Modes for Fluctuations and
Correlations
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HIV Reverse Transcriptase Slowest Motion
Push-pull Hinge
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Modes of Motion HIV Protease
Mode 1 Mode 2 Mode 3
Three Ways to Open the Flaps
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NMR Structures Fit Elastic Networks Better than
X-Ray Structures
HIV Protease Overlaps between directions of
motions (dot products of vectors)
Includes Many Drug Bound Structures Distortions
for Drug Binding Are Intrinsic to Protein
Structure
Results for 164 X-ray and 28 NMR HIV Protease
Structures
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Cumulative Overlaps with NMR Motions
NMR Agreement Better than X-ray
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Structural Refinement Using Distribution of
Distances

• We have developed a method of refining NMR
  structures using derived distance constraints and
  mean-force potentials.
• The original NMR experimental constraints for the
  structures were downloaded from BioMagResBank.
• The structures were refined using the default
  dynamic simulated annealing protocol implemented
  in CNS software (Brunger et al. Yale Univ).
• We used also mean-force potentials E kT ln P(r)
  by adding them into the energy function of the
  NMR modeling software CNS. The structures have
  been improved significantly (in terms of RMSD,
  their energy, NOEs, etc.) after refinement with
  the database-derived mean-force potentials.

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CASPR 2006

• We have successfully used this method in CASPR
  2006 structure refinement experiment.
• Figure below shows application of our method for
  a model of 1WHZ (70 residues) a refinement from
  2.19 Å to 1.80 Å has been obtained.

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Distance Intervals
The distances are given with their possible
ranges.
i
j
NP-hard!
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A Generalized Distance Geometry Problem
i
Dri
Root mean square fluctuations B-factors
di,j
j
rj
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Protein 1AX8
Data generation fi the rms fluctuation of
atom i. S (i,j) di,j yi yj lt
5Å li,j di,j fi fj ui,j di,j fi fj

Original
Problem solved ri the fluctuation radius of
atom i. maxx, r ?D ri3 di,j xi
xj li,j di,j Dri Drj ui,j di,j Dri
D rj, (i,j) in S
Computed
RMSD (x, y) 3.6 e -07
1017 atoms
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Atomic Fluctuations
Original
fi
Dri
Computed
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Acknowledgments

• NIH support
• 1R01GM081680-01 (AKlo)
• 1R01GM073095-01A2 (RLJ) 1R01GM072014-01 (RLJ)