Efficient Nearest Neighbor Searching for Motion Planning

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Anna Yershova Department of Computer Science Duke
University February 5, 2010
Feb 5 2010, NC State University
Automated Protein Structure Determination using
RDCs
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Introduction
Motivation
Protein Structure Determination is Important
Amino acid sequences
Structures
Functions
Protein redesign

• High-resolution structures are needed for
• Determining protein functions
• Protein redesign

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Introduction
Motivation
What is Protein Structure Primary Structure
The sequence of amino acids forms the
backbone.Residues are sidechains attached to the
backbone.
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Dihedral angle
Side chain
Amino acid
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Introduction
Motivation
What is Protein Structure Secondary Structure
Elements
Local folding is maintained by short distance
interactions.
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Introduction
Motivation
What is Protein Structure 3D Fold
Global 3D folding is maintained by more distant
interactions.
Alpha-helix
Side chain
Loop
Beta-strands
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Introduction
Motivation
High-Throughput Structure Determination Is
Important
The gap between sequences and structures
http//www.metabolomics.ca/News/lectures/CPI2008-s
hort.pdf
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Introduction
Motivation
Current Approaches for Structure Determination

• X-ray crystallography
• Difficulty growing good quality crystals
• Nuclear Magnetic Resonance (NMR) spectroscopy
• Difficulty lengthy (expensive) time in
  processing and analyzing experimental data

Both require expressing and purifying proteins.
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Introduction
Motivation
Bruce Donalds Lab
Michael Zeng Chittu Tripathy
Lincong Wang
Pei Zhou
Bruce Donald
Cheng-Yu Chen John MacMaster
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Introduction
Motivation
Types of NMR Spectroscopy Data
4.2
R
Ha
NOE
133.1
172.1
B0
8.9

• Chemical shift (CS)
• Unique resonance frequency, serves as an ID
• Nuclear Overhauser effect (NOE)
• Local distance restraint between two protons
• Residual dipolar coupling (RDC)
• Global orientational restraint for bond vectors

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Introduction
Motivation
Resonance Assignment Problem
Assigning chemical shifts to each atom
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Bailey-Kellogg et al., 2000, 2004
http//www.pnas.org/content/102/52/18890/suppl/DC1
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Introduction
Motivation
NOE Assignment Problem
Obtain local distance restraints between protons
A famous bottleneck
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Bailey-Kellogg et al., 2000, 2004
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Introduction
Motivation
Structure Determination from NOEs
NOESY spectrum
Resonance assignments
NOE assignment
Assignment
Ambiguity
Distance Geometry
NP-Hard
Saxe 79 Hendrickson 92, 95
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Protein Structure Determination is Hard
Introduction
Motivation
Traditional Structure Determination Protocol
A famous bottleneck
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Protein Structure Determination is Hard
Introduction
Motivation
Traditional Structure Determination Protocol
error propagation
local minima
manual intervention for initial fold and for
evaluation of NOE assignments
A famous bottleneck
Can we have a poly-time algorithm using
orientational restraints?
Yes Wang and Donald, 2004 Wang et al, 2006
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Introduction
Motivation
Types of NMR Spectroscopy Data
4.2
R
Ha
NOE
133.1
172.1
B0
8.9

• Chemical shift (CS)
• Unique resonance frequency, serves as an ID
• Nuclear Overhauser effect (NOE)
• Local distance restraint between two protons
• Residual dipolar coupling (RDC)
• Global orientational restraint for bond vectors

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Background
RDCs
RDC Equation for a Single Bond
Alignment medium
?
b
B0
v
a
S Saupe Matrix S is traceless and symmetric S
contains 5 dofs
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Protein Structure Determination is Hard
Introduction
Motivation
Traditional Structure Determination VS RDC-Panda
RDC-PANDA Protocol
Constaint number of NOEs
RDCs
error propagation
RDC-ANALYTIC PACKER
local minima
Global Fold
manual intervention for initial fold and for
evaluation of NOE assignments
Sidechain Placement
NOE Assignments
XPLOR-NIH
NOE Assignments 3D Structures
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Zeng et al. (Jour. Biomolecular NMR,2009)
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Introduction
Motivation
Importance of Backbone Structure Determination
Global orientational restraints from RDCs
Sparce data (high-throughput, large proteins,
membraine proteins)
Compute initial fold using exact solutions to
RDC equations
Avoid the NP-Hard problem of structure
determination from NOEs
Resolve NOE assignment ambiguity
Automated side-chain resonance assignment
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Introduction
Motivation
Current Limitations of RDC-Panda

• Because it requires only 2 RDCs per residue
• Only SSE elements can be reliably determined,
  NOEs are needed to determine structure of loops
• Difficulty in handling missing data

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Introduction
Motivation
My Current Project

• Improve current protein structure determination
  techniques from our lab
• Design new algorithms for protein backbone
  structure determination using orientational
  restraints from RDCs

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Introduction
Motivation
Literature Overview

• Distance geometry based structure determination
• Braun, 1987
• Crippen and Havel, 1988
• More and Wu, 1999
• Heuristic based structure determination
• Brünger, 1992
• Nilges et al., 1997
• Güntert, 2003
• Rieping et al., 2005
• RDC-based structure determination
• Tolman et al., 1995
• Tjandra and Bax, 1997
• Hus et al., 2001
• Tian et al., 2001
• Prestegard et al., 2004
• Wang and Donald (CSB 2004)
• Wang and Donald (Jour. Biomolecular NMR, 2004)
• Wang, Mettu and Donald (JCB 2005)
• Donald and Martin (Progress in NMR Spectroscopy,
  2009 )

• Heuristic based automated NOE assignment
• Mumenthaler et al., 1997
• Nilges et al., 1997, 2003
• Herrmann et al., 2002
• Schwieters et al., 2003
• Kuszewski et al., 2004
• Huang et al., 2006
• Automated NOE assignment starting with initial
  fold computed from RDCs
• Wang and Donald (CSB 2005)
• Zeng et al. (CSB 2008)
• Zeng et al. (Jour. Biomolecular NMR,2009)
• Automated side-chain resonance assignment
• Li and Sanctuary, 1996, 1997
• Marin et al., 2004
• Masse et al., 2006
• Zeng et al. (In submission, 2009)

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Background
RDCs
RDC Equation for a Single Bond
Linear in S, A fixed v defines a hyperplane
Quadratic in v, A fixed S defines a hyperboloid
S
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Background
RDCs
RDC Equation for a Single Bond
1 RDC equation defines a collection of
hyperplanes, 7 variables
Linear in S, A fixed v defines a hyperplane
Quadratic in v, A fixed S defines a hyperboloid
S
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Background
RDCs
RDC Equations for a Protein Portion
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Background
RDCs
RDC Equations for a Protein Portion
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u1
v1
v2
1 L. Wang and B. R. Donald. J. Biomol. NMR,
29(3)223242, 2004. 2 J. Zeng, J. Boyles, C.
Tripathy, L. Wang, A. Yan, P. Zhou, and B. R.
Donald J. Biomol. NMR, Epub ahead of print
PMID19711185, 2009.
Too few equations, too many variables!
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Background
RDCs
Forward Kinematics Reduces the Number of Variables
v1
Fix coordinate system.
v2
u1
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Background
RDCs
RDC Equations for a Protein Portion
v1
v2
u1
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Background
RDCs
RDC Equations for a Protein Portion
Recursive representation is possible!
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Background
RDCs
One Equation Per Dihedral Angle is Not Enough!

• Each equation is linear in S, and quartic in
  either tan(?) or tan(?)
• To be able to solve this system there must be
  additional information
• Possible scenarios
• Additional RDC measurement(s) for each dihedral
  angle.
• Additional alignment media.
• Additional NOE data.
• Modeling (Ramachandran regions, steric clashes,
  energy function)
• Sampling (for alignment tensors)

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Background
RDC-Panda
The RDC-PANDA Structure Determination Package

• Current requirements
• 2 RDCs per residue to obtain SSE structures
• Sparse NOEs to pack the SSEs
• Current bottlenecks
• Missing data (even in long SSEs)
• Long loops
• Sampling for computing alignment tensor(s)
• Sampling for the orientation of the first pp

1 L. Wang and B. R. Donald. J. Biomol. NMR,
29(3)223242, 2004. 2 J. Zeng, J. Boyles, C.
Tripathy, L. Wang, A. Yan, P. Zhou, and B. R.
Donald J. Biomol. NMR, Epub ahead of print
PMID19711185, 2009.
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Background
RDC-Panda
When Saupe Matrix is Known Solution Can Be Found
Exactly!
Ellipse equations for CH bond vector
Wang Donald, 2004
Donald Martin, 2009.
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Solution Structure of FF Domain 2 of human
transcription elongation factor CA150 (FF2) using
RDC-PANDA
Background
RDC-Panda
Solution Structure Deposited Using RDC-Panda
PDB ID 2KIQ
In collaboration with Dr. Zhous Lab
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Current Project
Problem Formulation NH, CH RDCs in 2 Media
We require measurements for at least 9
consecutive bond vectors (4.5 residues) in 2
media. The goal is to handle more equations and
errors.
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Current Project
Relationship to Minimization
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Current Project
Relationship to Minimization and SVD
Solving an over constrained system of linear
equations is equivalent to finding a projection
of the b vector on the A hyperplane. This is also
equivalent to minimizing the least square
function of the terms.
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Current Project
Relationship to Minimization
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Current Project
Relationship to Minimization and SVD
b
A(?i ?i)
s
Solving such a system of non-linear equations is
not trivial! There are multiple local minima in
the corresponding minimization problem.
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Current Project
Advantages

• If the minimization problem is solved then
• Computation of packed SSEs and loops is possible
  without additional NOE data.
• Saupe matrices for each of the alignment medium
  can be computed without sampling.
• Robust handling of missing values

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Current Project
The Algorithm Initialization Using Helix
Initialize (?i,?i) for a helix
Compute initial approximation for Si using SVD
Compute (?i,?i) using tree search and
minimization
Update Si using SVD
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Current Project
The Algorithm Protein Portion
Initialize Si to computed approximations
Compute (?i,?i) using tree search and
minimization
Update Si using SVD
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Current Project
The Algorithm Computing Dihedrals
?1
Minimize each of the RMSD terms as a univariate
function.
?1
x
x
?n
x
?n
Iteratively minimize the RMSD function
x
Compute the list of best solutions.
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Current Project
Advantages

• The algorithm is converging, since every step
  minimizes RMSD function
• If the data was perfect then the solution to
  the minimization problem would be the roots of
  the polynomials in the RMSD terms, and the
  algorithm would find ALL of them.
• The minima of the RMSD terms give a good
  collection of initial structures for finding
  local and global minima
• Robust handling of missing values

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Preliminary Results
Preliminary Results Ubiquitin Helix
Conformation of the portion 25-31 of the helix
for human ubiquitin computed using NH and CH RDCs
in two media (red) has been superimposed on the
same portion from high-resolution X-ray structure
(PDB Id 1UBQ) (green). The backbone RMSD is 0.58
Å.
Protein RMSD (Hz) Alignment Tensor (Syy, Szz)
Ubq ?25-31 C?H? 0.32 NH 0.24 (23.66, 16.48) (53.25, 7.65)
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Preliminary Results
Preliminary Results Ubiquitin Strand
Conformation of the portion 2-7 of the
beta-strand for human ubiquitin computed using NH
and CH RDCs in two media has been superimposed on
the same portion from high-resolution X-ray
structure (PDB Id 1UBQ). The backbone RMSD is
1.151 Å.
Protein RMSD (Hz) Alignment Tensor (Syy, Szz)
Ubq beta 2-7 C?H? NH (53.32, 4.83) (48.03, 14.32)
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Conclusions

• Complete and exhaustive search over the space of
  all structures minimizing the RDC fit function
  seems feasible due to understanding the structure
  of the solution.
• Possible and exiting extensions to more/different
  data

Funding NIH
Thank you!
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Comparison
Sparse
Accuracy
Data requirements vs. Accuracy (Ubiquitin)
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