Anna Yershova Department of Computer Science Duke University February 5, 2010

Anna YershovaDepartment of Computer Science

Duke University

February 5, 2010

Automated High-Resolution Protein Structure Determination using

Residual Dipolar Couplings

Feb 5 2010, NC State UniversityFeb 5 2010, NC State University Automated Protein Structure Determination using Automated Protein Structure Determination using RDCsRDCs

1

High-resolution structures are needed for:

Determining protein functions Protein redesign

IntroductionIntroduction Motivation

Protein Structure Determination is Protein Structure Determination is ImportantImportant

Protein Structure Determination is Protein Structure Determination is ImportantImportant

2

Amino acid sequences

Structures

Functions Protein redesign


What is Protein Structure: Primary What is Protein Structure: Primary StructureStructure

What is Protein Structure: Primary What is Protein Structure: Primary StructureStructure

3

1 2 3 4

The sequence of amino acids forms the backbone.Residues are sidechains attached to the backbone.

Amino acidSide chain Dihedral angle


What is Protein Structure: Secondary What is Protein Structure: Secondary Structure ElementsStructure Elements

What is Protein Structure: Secondary What is Protein Structure: Secondary Structure ElementsStructure Elements

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Local folding is maintained by short distance interactions.


What is Protein Structure: 3D FoldWhat is Protein Structure: 3D FoldWhat is Protein Structure: 3D FoldWhat is Protein Structure: 3D Fold

5

Global 3D folding is maintained by more distant interactions.

Side chain

Beta-strands

Alpha-helix

Loop


High-Throughput Structure High-Throughput Structure DeterminationDeterminationIs ImportantIs Important

High-Throughput Structure High-Throughput Structure DeterminationDeterminationIs ImportantIs Important

6 http://www.metabolomics.ca/News/lectures/CPI2008-short.pdf

The gap between sequences and structures


Current Approaches for Structure Current Approaches for Structure DeterminationDetermination

Current Approaches for Structure Current Approaches for Structure DeterminationDetermination

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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.


Bruce Donald’s LabBruce Donald’s LabBruce Donald’s LabBruce Donald’s Lab

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Bruce Donald

Cheng-Yu Chen

John MacMaster

Michael Zeng Chittu Tripathy Lincong Wang

Pei Zhou


Types of NMRTypes of NMR Spectroscopy DataSpectroscopy DataTypes of NMRTypes of NMR Spectroscopy DataSpectroscopy Data

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

R

133.1

8.9

Ha

4.2

B0

172.1

NOE

Bailey-Kellogg et al., 2000, 2004http://www.pnas.org/content/102/52/18890/suppl/DC1

Assigning chemical shifts to each atom


Resonance Assignment ProblemResonance Assignment ProblemResonance Assignment ProblemResonance Assignment Problem

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Obtain local distance restraints between protons

Bailey-Kellogg et al., 2000, 2004


NOE Assignment ProblemNOE Assignment ProblemNOE Assignment ProblemNOE Assignment Problem

11

A famous bottleneck

a1 a2 a3

a1

a2

a3

4

?

3

?

3

4

an

an

. . .

.

.

.

. . .

. . .

. . .

.

.

....

.

.

..

..

Distance Geometry

Assignment Ambiguity

NP-Hard

NOESY spectrum

NOE assignment

Resonance assignments


Structure Determination from NOEsStructure Determination from NOEsStructure Determination from NOEsStructure Determination from NOEs

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[Saxe ’79; Hendrickson ’92, ’95]

Protein Structure Determination is Hard

A famous bottleneck


Traditional Structure Determination Traditional Structure Determination ProtocolProtocol


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NOE Assignments

NOE Assignments 3D Structures

Resonance assignments NOESY spectra

RDCs

SA/MD

Initial fold

XPLOR-NIH

Structure Refinement


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




NOE Assignments



RDCs

SA/MD

Initial fold

XPLOR-NIH


14


Types of NMRTypes of NMR Spectroscopy DataSpectroscopy DataTypes of NMRTypes of NMR Spectroscopy DataSpectroscopy Data

15

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

R

133.1

8.9

Ha

4.2

B0

172.1

NOE

BackgroundBackground RDCs

RDC Equation for a Single BondRDC Equation for a Single BondRDC Equation for a Single BondRDC Equation for a Single Bond

16

2

1cos3

4

2

3,

20

ba

ba

rD

B0 v

a

b

S – Saupe MatrixS is traceless and symmetricS contains 5 dofs

Alignment medium

Sxx Syy

Szz

vD


NOE Assignments



RDCs

SA/MD

Initial fold

XPLOR-NIH


RDCsConstaint number of NOEs

Global Fold

RDC-ANALYTIC PACKER

Sidechain Placement

NOE Assignments

XPLOR-NIH


RDC-PANDA Protocol


Traditional Structure Determination VS RDC-Traditional Structure Determination VS RDC-PandaPanda

Traditional Structure Determination VS RDC-Traditional Structure Determination VS RDC-PandaPanda

error propagation

local minima

manual intervention for initial fold and for evaluation of NOE assignments

17Zeng et al. (Jour. Biomolecular

NMR,2009)

Global orientational restraints from RDCs

Compute initial fold using exact solutions to RDC equations

Resolve NOE assignment ambiguity

Sparce data (high-

throughput, large proteins,

membraine proteins)

Automated side-chain resonance assignment

Avoid the NP-Hard problem of structure determination from

NOEs


Importance of Backbone Structure Importance of Backbone Structure DeterminationDetermination

Importance of Backbone Structure Importance of Backbone Structure DeterminationDetermination

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Current Limitations of RDC-PandaCurrent Limitations of RDC-PandaCurrent Limitations of RDC-PandaCurrent 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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My Current ProjectMy Current ProjectMy Current ProjectMy 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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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 ) Ruan et al., 2008 Zeng et al. (Jour. Biomolecular

NMR,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)


Literature OverviewLiterature OverviewLiterature OverviewLiterature Overview

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22

Linear in S,

A fixed v defines a hyperplane

Quadratic in v,

A fixed S defines a hyperboloid

S

Sxx Syy

Szz

vD



23

1 RDC equation defines a collection of hyperplanes, 7 variables

S

Linear in S,

A fixed v defines a hyperplane

Quadratic in v,

A fixed S defines a hyperboloid


RDC Equations for a Protein PortionRDC Equations for a Protein PortionRDC Equations for a Protein PortionRDC Equations for a Protein Portion

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1 2 3 4



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Too few equations, too many variables!

[1] L. Wang and B. R. Donald. J. Biomol. NMR, 29(3):223–242, 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] PMID:19711185, 2009.

1 2 3 4

v1

u1

v2


Forward Kinematics Reduces the Number of Forward Kinematics Reduces the Number of VariablesVariables

Forward Kinematics Reduces the Number of Forward Kinematics Reduces the Number of VariablesVariables

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v1

u1

v2

Fix coordinate system.



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v1

u1

v2



28

Recursive representation is possible!


One Equation Per Dihedral Angle is One Equation Per Dihedral Angle is Not Enough!Not Enough!

One Equation Per Dihedral Angle is One Equation Per Dihedral Angle is Not Enough!Not Enough!

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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:1. Additional RDC measurement(s) for each dihedral angle.2. Additional alignment media.3. Additional NOE data.4. Modeling (Ramachandran regions, steric clashes, energy function)5. Sampling (for alignment tensors)

BackgroundBackground RDC-Panda

The RDC-PANDA Structure Determination The RDC-PANDA Structure Determination PackagePackage

The RDC-PANDA Structure Determination The RDC-PANDA Structure Determination PackagePackage

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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):223–242, 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] PMID:19711185, 2009.

Ellipse equations for CH bond vectorEllipse equations for CH bond vector

Wang & Donald, 2004; Donald & Martin, 2009.


When Saupe Matrix is Known Solution When Saupe Matrix is Known Solution Can Be Found Exactly!Can Be Found Exactly!

When Saupe Matrix is Known Solution When Saupe Matrix is Known Solution Can Be Found Exactly!Can Be Found Exactly!

Solution Structure of FF Domain 2 of human transcription elongation factor CA150 (FF2) using RDC-PANDA

PDB ID: 2KIQ

In collaboration with Dr. Zhou’s Lab


Solution Structure Deposited Using RDC-Solution Structure Deposited Using RDC-PandaPanda

Solution Structure Deposited Using RDC-Solution Structure Deposited Using RDC-PandaPanda

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Current ProjectCurrent Project

Problem Formulation: NH, CH RDCs in 2 Problem Formulation: NH, CH RDCs in 2 MediaMedia

Problem Formulation: NH, CH RDCs in 2 Problem Formulation: NH, CH RDCs in 2 MediaMedia

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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.


Relationship to MinimizationRelationship to MinimizationRelationship to MinimizationRelationship to Minimization

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Relationship to Minimization and SVDRelationship to Minimization and SVDRelationship to Minimization and SVDRelationship to Minimization and SVD

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b

sA

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.


Relationship to MinimizationRelationship to MinimizationRelationship to MinimizationRelationship to Minimization

36


Relationship to Minimization and SVDRelationship to Minimization and SVDRelationship to Minimization and SVDRelationship to Minimization and SVD

37

b

sA(i i)

Solving such a system of non-linear equations is not trivial!

There are multiple local minima in the corresponding minimization problem.

AdvantagesAdvantagesAdvantagesAdvantages

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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

The Algorithm: Initialization Using The Algorithm: Initialization Using HelixHelix

The Algorithm: Initialization Using The Algorithm: Initialization Using HelixHelix

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Compute initial approximation for Si using SVD

Initialize (i,i) for a helix

Compute (i,i) using tree search and minimization

Update Si using SVD

The Algorithm: Protein PortionThe Algorithm: Protein PortionThe Algorithm: Protein PortionThe Algorithm: Protein Portion

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Initialize Si to computed approximations

Compute (i,i) using tree search and minimization

Update Si using SVD

The Algorithm: Computing DihedralsThe Algorithm: Computing DihedralsThe Algorithm: Computing DihedralsThe Algorithm: Computing Dihedrals

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ψn

n

x

x

x

x

1

ψ1

Minimize each of the

RMSD terms as a

univariate function.

Compute the

list of best

solutions.

Iteratively

minimize the

RMSD function

AdvantagesAdvantagesAdvantagesAdvantages

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• 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

Preliminary Results: Ubiquitin HelixPreliminary Results: Ubiquitin HelixPreliminary Results: Ubiquitin HelixPreliminary Results: Ubiquitin Helix

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Preliminary ResultsPreliminary Results

Protein RMSD (Hz) Alignment Tensor (Syy, Szz)

Ubq :25-31

CH : 0.32

NH: 0.24

(23.66, 16.48)

(53.25, 7.65)

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.

-60

-40

-20

0

20

40

60

-60 -40 -20 0 20 40 60

back-computed RDCs

exp

erim

enta

l RD

Cs

NH RDCs CH RDCs

Preliminary Results: Ubiquitin StrandPreliminary Results: Ubiquitin StrandPreliminary Results: Ubiquitin StrandPreliminary Results: Ubiquitin Strand

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Preliminary ResultsPreliminary Results

Protein RMSD (Hz)

Alignment Tensor (Syy, Szz)

Ubq: beta 2-7 CH :

NH:

(53.32, 4.83)

(48.03, 14.32)

-60

-40

-20

0

20

40

-60 -40 -20 0 20 40

back-computed RDCsex

peri

men

tal R

DC

s

NH RDCs CH RDCs

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.

ConclusionsConclusionsConclusionsConclusions

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Thank you!

Thank you!

• 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

FundingFunding: : NIHNIH

Comparison

Data requirements vs. Accuracy (Ubiquitin):

Accuracy:Sparse

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Documents

Anna Yershova Department of Computer Science Duke University February 5, 2010