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Gottingen, Sep 2010 Vladislav Orekhov

High resolution and sensitivity in nD-spectra by non-uniform sampling �

  Multi-Dimensional Decomposition (MDD) �

  Incremental NUS as an optimization method �

  Process or co-process? �

  Non-Uniform Sampling (NUS) as an inverse problem�

"Outline�

"High resolution with NUS �

Regular uniform� Non-Uniform (NUS)�

1-5% of 3D data for backbone assignment !�10-30% for 3D,4D NOESY !�

t2

t1

t3

"FT is a linear inverse problem�

Spectrum s is a solution of the system of linear equations

Φ s – f = 0

s = Φ-1 f where •  Φ, Φ-1 – iFT and FT, resp. •  solution is unique

Spectrum FID

|| Φ’ s – f’ ||2 = 0

The linear system is underdetermined (matrix Φ’ is fat) Solution (spectrum) is NOT unique

Additional assumptions about the signal (spectrum) are needed

"FT & NUS�

•  min || s ||2

•  min Σi |si| log|si|

•  sj = 0, j ∈ Ω

•  min || s ||0 ��� min || s ||1

→  Mimimal Power Non-Unifom Fourier transform, K. Kazimierczuk & W. Kozminski, P. Zhou, et al

→  Maximum Entropy E. Laue, A. Stern, J. Hoch., G. Wagner, S., Hyberts, et al

→  SIFT Judith Herzfeld

→  Compressed Sensing (max sparseness) David Donoho, (2004) A. Stern, J. Hoch., S., Hyberts, et al

|| Φ’ s – f’ ||2 = 0 "Assumptions�

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•  Solution (spectrum) is unique •  Calculations are very fast (use FFT) •  S/N is often poor

|| Φ’ s – f’ ||2 = 0, min|| s ||2

Non-Unifom Fourier transform: Kazimierczuk, Kozminski, et al, I. J. Biomol. NMR 2006, 36, 157-168. �

"Minimal power�

s’ = Φ”*f’, where Φ”* - pseudo-inverse of Φ”

•  Reduced number of basis functions •  Solution (spectrum) is unique •  Calculations are fast, O(K’3) •  Result is good if K’ ≤ K. If K’ K, the minimal norm solution

SIFT: Matsuki, Eddy, and Herzfeld, J. Am. Chem. Soc., 2009, 131, 4648-56. �

Known zeros in the spectrum�

P

ν (Hz)

Nyquist/Shannon theorem NNS ~ 1 / SW SW

P

SW ν (Hz)

Candes, Romberg & Tao (2006)

NCS ≥ 4α NNS log(NNS)

100 %�6 %�

Spectrum is α % sparse

Spectrum is SW – band limited

"Compressed Sensing �

Φ is known – Linear Least Squares Wagner, G. et al., 1986; Moonen, et al., 1987

"Linear least squares & MDD �Reduced set of basis functions. Matrix Φ is skinny, i.e. the linear system is overdetermined

Φ is NOT known ���solution is unique for ≥ 3D

Kruskal, J. B. 1977 ���– Alternating Least Squares ���

PARAFAC, MDD

K<<N

K<<N NxK

NxK

F ≈ Σβ aβ ⊗ bβ ⊗ cβ �MDD �

F ≈ Σβ aβ ⊗ bβ�

(SVD) �

"Multi-Dimensional Decomposition (MDD)also PARAFAC, Canonical Decomposition, etc�

Solution is unique for ≥ 3D, i.e. modes a, b and c are defined without additional assumptions (Kruskal, J. B. 1977).

F ≈ Σβ aβ�

Solution �Unique

Rotational ambiguity

not unique

F = ABT �

F = Σβ aβ ⊗ bβ ⊗ cβ�

MDD algorithm: Alternating Least Squares (ALS)�

ALS iterations converge monotonously

F1 = AΦA, solve for A, with fixed C, B; ΦA = (CB)T�

F2 = BΦB, solve for B, with fixed C, A; ΦB = (CA)T �

F3 = CΦC, solve for C, with fixed B, A; ΦC = (BA)T�

⊗ - outer product! - Khatri-Rao product

F �

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Assumption: �NMR signal is completely defined by its line shapes in all spectral dimensions

S ≈ Σβ F1β ⊗ F2β ⊗ F3β �MDD �

"Multi-Dimensional Decomposition (MDD) �

Orekhov, Ibraghimov, & Billeter (2003) J. Biomol NMR 27, 165-173

1D shapes are complete and describe signals in 3D spectrum

"Non-uniform sampling with MDD �

NUS 4D 13C 3D

1H 3D

Methyls of Val, Ile, Leu are 13C

labeled and back protonanted

4D FID, sparse 30.8%. Acquisition times & SW: 1H 27 ms, 960 Hz 13C 22 ms, 2211 Hz 13C 16 ms, 2211 Hz

Out of 359424 FIDs 110592 were randomly selected for detection.

6.5 instead of 21 days of measurements

Tugarinov et al (2005) PNAS 102, 622-627

!4D 13C Methyl HMQC NOESY of malate synthase (81.4 kDa) �

Malate Synthase G 82 kDa �

Hiller, Wagner et al, 2008, Science,

Integral membrane Volt Depend Anion Channel, VDAC-1 �

Tugarinov, Kay, et al.,2005, PNAS

Hsp70 substrate-binding domain, DnaK�64 kDa �

Zhuravleva, Gierasch, et al., 2009, Keystone Conf.

"Applications of NUS-MDD 3/4D NOSEY’s �

A. Gutmanas, C. Arrowsmith, et al

•  Experiments are collected in sets�•  Signal frequencies and line-shapes are

redundant �•  Use of interdependencies between

signals in different experiments may significantly improve results of the spectra processing and analysis.�

"Process or co-process?�!Hyper Dimensional Spectroscopy�

"Co-processing: 4D NOESY’s�

"2D Met "HMQC Σβ βFC ⊗ βFH �"4D Met-Met "NOESY Σβ βFHC ⊗ βFC ⊗ βFC ⊗ βFH "4D HN -Met "NOESY Σβ βFHN ⊗ βFN ⊗ βFC ⊗ βFH

4D NOESY’s

Hiller, Wagner et al, 2008, Science, 321, 1206-10

Integral membrane Volt Depend Anion Channel, VDAC-1 �

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  13% NUS schedule �  6 days on 900 MHz Bruker�  mddNMR software �  Full experiment 46 days�

"VDAC - integral membrane ion channel � ! !4D NUS-MDD Methyl NOESY, Bruker 900 MHz�

Hiller, Wagner et al, 2008, Science, 321, 1206-10

co- MDD � MDD � NUS

DFT �False peaks� - � - � - �

Missing peaks� - � +- � - �

S/N � + � + � +- �

NUS DFT = regular DFT processing with missing data points set to zero

co-MDD vs MDD and NUS DFT processing �4D NUS 13C HMQC - NOESY - 13C HMQC, VDAC-1 �

Hiller, Wagner, Orekhov et al, 2008, JACS

VDAC-1 �

Contact statistics from the 4D NOESY spectra �

VDAC-1 �

"3D HNCO Σβ βFN ⊗ βFH ⊗ βFC’ "3D iHNCA "Σβ βFN ⊗ βFH ⊗ βFCA "3D iHNCB "Σβ βFN ⊗ βFH ⊗ βFCB "3D HN(CO)CA "Σβ βFN ⊗ βFH ⊗ βFCA-1 �"3D HN(COCA)CB Σβ βFN ⊗ βFH ⊗ βFCA-1CB-1 "3D HN(CA)CO "Σβ βFN ⊗ βFH ⊗ βFCOCO-1

"Co-processing: backbone assignment �

Triple resonance BB experiemnts

Igα

" Back-bone assignment in 1D shapes�All peaks related to an amide group are collected into one component

L24 � Y25� E26 �H

N

CO,CO-1

CO-1

CA

CA-1

CB

CB-1

Igα disordered, 63 aa, 50 µM

HNCACO

HNCO

iHNCA

HNcoCA

iHNCB

HNCOCACB

"Co-processing: less data is needed �

HN(CO)CA �

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•  Incremental non-uniform sampling (iNUS) scheme�•  MDD processing possible at each step �•  Automated analysis of spectra at each step �

Data are randomly sampled according to selected probability distribution

"Targeted Acquisition with iNUS�TA �

50 µM Iga�63 aa, disordered �

"TA: automated collection of peaks �

NUS @ 90% assignment

NUS @ 90% assignment

2 mM Azurin �128 aa, globular�

"TA: assignment validation, bootstrapping �

NUS @ 90% assignment

Assignment is calculated automatically many times using different experimental data �

•  Incremental non-uniform sampling (iNUS) scheme�•  Co-MDD processing at each step �•  Automated peak-picking in 1D shape�•  Automated assignment, AutoAssign (Moseley & Montelione)

"TA assignment with iNUS & co-MDD �

TA �

Acknowledgments

FUNDING: �""Swedish Foundation of Strategic Research �""Swedish Research Council�""NIH, FP6 (EU), Wenner-Gren Foundation (Stockholm) �

Swedish NMR Centre, GU "" " "Harvard Medical School�"Mayzel Maxim "Hiller Sebastian "�"Isaksson Linnea "Wagner Gerhard �"Kazimierczuk Krzysztof�"Jaravine Victor�"Zhuravleva Anastasia "�

University Health Network, Toronto "University of Toronto �"Gutmanas Aleksandras "" "Tugarinov Vitali�"Arrowsmith Cheryl "" ""Kay Lewis�

Elegant Mathematics Ltd, Germany "Ibraghimov Ilghiz ! "�Inst. Biotechnology, Helsinki !Permi Perttu "�University of Massachusetts "Sigalov Alexander "�