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

Matrices containing structural information

• Distance matrix (dij)

• Matrix of square distances D = (dij2)

• Contact matrix C = (cij)

cij = 1 if dij > dcutoff

otherwise cij = 0

• Laplacian of C (Kirchhoff matrix)

Lc = diag(cij) - C

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

Spectral decomposition of structural matrices

A = k vk vkT

is expressed by eigenvalues and corresponding eigenvectors of A

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.

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

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

Approximation of distance matrices

• A = k 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

Current work:

Prediction of r2 from the sequence with SVR

Prediction of the first structural component from the sequence

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.

The -carbon trace of the HIV-1 structure

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

Elastic Network Models

Calculating Protein Position Fluctuations

Vtot(t) = (/2) tr [R(t)T R(t)]

<Ri . Rj> = (1/ZN) ∫ (Ri . Rj) exp {-Vtot/kT} d{R}

= (3kT/) [-1]ij

= Kirchhoff matrix of contacts

=

Compute Normal Modes for Fluctuations and Correlations

HIV Reverse Transcriptase – Slowest Motion

Push-pull Hinge

Modes of Motion – HIV Protease

Mode 1 Mode 2 Mode 3

Three Ways to Open the Flaps

NMR Structures Fit Elastic Networks Better than X-Ray Structures

Results for 164 X-ray and 28 NMR HIV Protease Structures

HIV ProteaseOverlaps between directions of motions

(dot products of vectors)

Includes Many Drug Bound Structures

Distortions for Drug Binding Are Intrinsic to Protein Structure

Cumulative Overlaps with NMR Motions

NMR Agreement Better than X-ray

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.

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.

Distance Intervals

i

j

The distances are given with their possible ranges.

Sj)i,(,u||xx||l

such that xall find

ji,jiji,

j

NP-hard!

A Generalized Distance Geometry Problem

j

i

n3j

x,rj 1

i j i j i,j

i j i j i,j

max r

subject to

||x x || r r u

||x x || r r l , (i,j) S

ri

rj

di,j

Root mean square fluctuationsB-factors

Data generation:

fi : the rms fluctuation of atom i.

S = {(i,j) : di,j = ||yi – yj|| < 5Å}

li,j = di,j – fi – fj

ui,j = di,j + fi + fj

Problem solved:

ri : the fluctuation radius of atom i.

maxx, r ∑ ri3

di,j = ||xi – xj||

li,j ≤ di,j – ri – rj

ui,j ≥ di,j + ri + rj, (i,j) in S

Original:

Computed:

RMSD (x, y) = 3.6 e -07

Protein 1AX8

1017 atoms

0 200 400 600 800 1000 12000

0.05

0.1

0.15

0.2

0.25

Atomic Fluctuations

Original

Computed

fi

ri

Acknowledgments:

• NIH support:

• 1R01GM081680-01 (AKlo)

• 1R01GM073095-01A2 (RLJ) 1R01GM072014-01 (RLJ)