Modern Spectrum Analysis in NMR

Only method capable of determining the structure
of biological macromolecules in solution at atomic
resolution
Uniquely able to probe dynamics
Useful for investigating biomolecular interactions
(affinities and strucctures)
Useful for investigating biomolecular rate processes
(e.g. protein folding, enzyme kinetics)
Important applications in drug discovery,
metabolomics

Very few mathematical tools have had the
impact of the Fourier Transform on science
and engineering
Jean Baptiste Joseph Fourier (1768 – 1830)

Fourier Transform NMR
Impulse
FT
Ernst & Anderson, 1963
SW = 1/ Δt
Δt

• Exactly invertible: iDFT
• Linear: DFT(A+B) = DFT(A) + DFT(B)
implies S/N ∝
N
• Assumes signal is periodic (period = NΔt)
• Efficient algorithms: FFT (~NlogN operations)
(Cooley & Tukey, 1965)

Jean Jeener, 1971
Parametric sampling
of indirect time dimensions
Δt 1
Δt 2

• High resolution requires sampling at long
times
• Short times for high sensitivity
• DFT requires uniformly spaced samples

Super-resolution
Higher sensitivity
Shorter experiments

• “Matched filter” for sensitivity
enhancement (van Vleck et al; Ernst
introduced to NMR)
• Emphasis on long times for resolution
enhancement (sine-bell, Lorentzian-to-
Gaussian, etc.)

LP extrapolation reduces truncation
artifacts, but introduces errors
(false peaks, frequency shifts)

Modern spectrum analysis in NMR,
an incomplete time line
1983
LP extrapolation Ni & Scheraga (1983)
MaxEnt reconstruction Sibisi et al. (1983,1984)
Burg MEM Martin; Hoch (1985)
LPSVD van Ormondt et al. (1985)
Exponential sampling Laue, Skilling et al. (1987)
Iterative thresholding (Jansson-Van Cittert) Scheraga et al. (1989)
Complex entropy Hore et al., Hoch et al. (1990)
Bayesian, MLM Bretthorst; Chylla & Markley (1990; 1993)
GFT Szyperski et al. (1993)
Wavelet transform, iterative soft thresh Hoch & Stern (1996)
Filter diagonalization Mandelshtam, Shaka et al. (2000)
Multiway decomposition Billeter, Orekhov et al. (2001)
Back projection Kupče & Freeman 2002
Covariance NMR Brüschweiler et al. (2004)
“Multidimensional FT” Kozminski et al., 2006
Now Compressed sensing Donoho et al. (“2004”)

Classifying methods
periodicity
random noise
Robustness
symmetry
exponential
decay
model
Assumptions

LP, LPSVD, HSVD, Burg maximum entropy
d j
=
K
∑
i =1
a i
d j −i
Bayesian, MLM, FDM
€ d k
=
L
∑
j =1
a j
e iφ j
e −kΔt / τ j
e 2πikΔtω j
+ ε k
Being parametric, they yield frequency lists directly
Prone to bias, false positives when S/N is low or signals non-ideal

S i
(f 1
,f 2
,f 3
) = s i
(f 1
) ⊗ s i
(f 2
) ⊗ s i
(f 3
)
Also known as PARAFAC; PCA is a 2D equivalent
€
Billeter, Orekhov et al.
(Denk & Wagner;
Rinaldi et al.)
Less restrictive than the exponential decay assumption
Random noise does not have a decomposition
Unique decomposition restricted to 3+ dimensions

Brüschweiler et al. (2004)
Frequency correlations in nD via covariance rather than spectrum
analysis
C is the correlation matrix, F is the 2DFT or the f 2 -t 1 interferogram
Affords the resolution of the direct (f 2 ) dimension in the indirect
dimension
€
C = (FF T )
Scaling the intensities appropriately requires matrix square root; can
use SVD
Extensions to indirect correlations enable application to
unsymmetrical spectra

Back projection, by analogy to CT
Kupče & Freeman 2002
t 2
f 2
t 1
f 1
Radial sampling in the time domain yields tilted projections in frequency

Data d Trial spectrum f Mock data ˆ (= iDFT(f))
Maximize
N
S(f) = ! " f k logf k
k = 1
S=(entropy, l 1 -norm, l 2 -norm, etc.)
Subject to
C(f,d) =
M
" ( d j ! ˆ ) 2 #≈ noise
j=1
d j
Fixed-point, convex optimization, interior point

Entropy functionals for NMR
Shannon entropy inappropriate for spectra with +/- components,
or complex (phase-sensitive)
Entropy of the power spectrum has multiple critical points, thus no
unique solution; other problems
Skilling et al.: 2-channel entropy (f = f + - f - ; S = S + + S - )
4-channel for complex spectra; computational cost
due to additional unknowns; entropy depends on phase
Entropy for an ensemble of spin-1/2 particles (Hore et al.; Hoch,
Donoho et al.) is insensitive to phase, convex

Logan’s Theorem
perfect reconstruction is possible from incomplete and noisy
data by maximizing the l 1 -norm under certain conditions
Noisy debates still erupt
see Times of London, J. Roy. Stat. Soc.
John Skilling:
“If you just want a nice picture for the brochure, or to keep your
sponsor happy, maximise a nice functional like entropy (or 1-norm).
If you want to be properly professional and take uncertainty and
error bars seriously, then you need an id prior, and those posteriors
are a lot harder to compute.”

When N = M, Parseval’s Theorem
MaxEnt: analytic solution
Donoho, Johnstone, Stern, and Hoch PNAS 87, 5066-5068 (1990)
in essence allowing us to compute C(d) in the frequency domain, and
finding the critical point of the objective function Q = S - λC is easy.
The MaxEnt reconstruction ( ˆ f ) then reduces to applying the
transformation
that is the solution to
to the f (the DFT of d)
! " z
0 = ! log" +1 ! 2# " ! Re z
N
2
2
" d ˆ i ! d i = " f ˆ j ! f j
i
M
j

Consequences of nonlinearity
• Obvious implications for difference spectroscopy
• Nonlinearities depend on both the method and the
data; accurate quantification may require empirical
calibration
• S/N no longer reflective of sensitivity
• Order of processing becomes important
• Rules of thumb (that are based on linear response) for
apportioning acquisition time in nD experiments may
not be optimal

• DFT can be applied to NUS data - obvious from definition of DFT
(Kozminski et al.)
• Formal equivalence of “radial DFT” and BPR (sum projection) for
radial sampling (Coggins & Zhou)
• For NUS samples restricted to Nyquist grid ( a good approx.),
nuDFT is equivalent to FFT with non-sampled time samples
set to zero
• Weighting (e.g. numerical quadrature on an irregular mesh)
improves nuDFT for a small subset of NUS schemes

DFT ⇒ radial DFT
Weighted DFT
Fourier transformation is an integration.
By analogy with Simpsons rule, etc., the values
of the function (at discrete intervals) are weighted
by the area/volume of the interval.
Transformation of units under integration(e.g. xy -> rθ)
involves the Jacobian,
dxdy=
"
$
#
! !r x %"
' $
!
&#
!( y % "
' ) $
!
& #!( x %
&
' "
$
! # !r y %
' drd(= rdrd(
&
Jacobian
weights for polar FT

Weighted DFT: General case
Radial FT:
analytic weights
Radial FT:
Voronoi weights
Voronoi weights
for irregular sampling
30
20
t 1
10
0 10 20 30
0 10 20 30 0 10 20 30
t 2
t 2
t 2
Voronoi polyhedra:
See qhull.org

No standard data sets
Lack of consensus metrics
LP extrapolation vs. MaxEnt reconstruction
Stern, Li, Hoch JACS 2002
Back projection vs. MaxEnt reconstruction
Mobli, Stern, Hoch JMR 2006

Different methods frequently give comparable results when applied
to the same data
The biggest differences (for a number of metrics) between methods
result when data samples are distributed differently in time
Those methods compatible with nonuniform sampling (NUS)
consistently yield better resolution than methods requiring uniform
sampling

NUS in NMR, an approximate time line
1981
Bodenhausen & Ernst accordion spectroscopy 4
Barna, J.C.J., Laue, E.D., Mayger, M.R., Skilling, J. and Worrall, S.J.P. (1987)
Exponential sampling: an alternative method for sampling in two dimensional
NMR experiments. J. Magn. Reson., 73, 69.
1
Schmieder, Hoch, Wagner et al.
Delsuc et al
Szyperski et al. , Ding & Gronenborn
random, cosine/exponential random
triangular
coupled evolution (RD/GFT)
2002
Kupce & Freeman
back projection
2
Zhou et al.
elliptical
Now
Marion et al., Kozminski et al. random
G. Larry Bretthorst (2001)
Nonuniform Sampling: Bandwidth and Aliasing
in Maximum Entropy and Bayesian Methods in Science and Engineering
Joshua Rychert, Gary Erickson and C. Ray Smith (eds.)
pp. 1-28. (AIP conference proceedings)
3

BPR and MaxEnt give similar results
-800
A
B
-400
0
400
800
F 1 [Hz]
-800
C
800
400
0
-400
-800
-800
D
A) BPR and B) MaxEnt
reconstruction of ongrid
radially sampled
datA
C) BPR off-grid
D) MaxEnt on-grid
approximation
-400
-400
0
0
400
400
800
800
F 2 [Hz]
800
400
0
-400
-800

t 1
t
t 1
t 2
t 2
t 3
1D
1 FID
2 sec.
2D
t 1 : 128 FIDs
8 min.
(hypercomplex)
3D
t 1 x t 2 : 128x128=16,384 FIDs
36 hrs.
4D
t 1 x t 2 x t 3 : 32x32x32=32,768 FIDs
6 days
High resolution: a year!

A
B
Barna, Laue, et al., 1987
C

1D NUS (exponential)
LP 512->2048
DFT
LP 64->2048
DFT
MaxEnt 64 out of 512
->2048
Factor of 8 time saved

Influence of sample distribution
radial
number of samples
fixed
exp. weighted
random

• “Coherences” in the sampling scheme
• Nyquist theorem does not apply: basis
functions are non-orthogonal
• Convolution of the sampling spectrum
(PSF) with the true spectrum

Problem is tatamount to deconvolution
of the sampling spectrum or point-spread function (PSF)
A
DFT of trial data
B
Sampling spectrum
(DFT of sampling schedule:
ones for sampled times,
zero for not sampled)
Dynamic range
C
DFT of NUS trial data
D
MaxEnt of NUS trial data

Point Spread Function (PSF)
Relative dynamic range
128
t 2
t 1 t 1
1
1 128
t 1
1 128 1 128
Contours:
3%
0.5%

Improving dynamic range via additional sampling
# of samples: 382 634 1132
128
t 2
t 1 t 1
1
1 128
t 1
1 128 1 128
HNCO: ubiquitin
Contours: 3%, 0.5% of maximum

On-grid vs. off-grid sampling
A) B) C)
t 1
t 2 t 2 t 2
A) On-grid radial sampling:
BPR and MaxEnt can be applied
to the same data
B) Off-grid radial sampling:
not amenable to existing MaxEnt
codes
C) “Anti-aliased” radial sampling:
on-grid approximation to arbitrary
radial sampling

Coupled evolution periods:
radial sampling
!
• GFT, RD, BPR
• Tilted planes

“Off-grid” sampling
LSD
0.013 sec
0.027
0.041
0.058
0.070
0.079
…
Smallest interval
Can always find an underlying grid that is bounded by
“1” in the LSD of the ;me(s) and by the smallest interval
between samples. Here the bounds are 0.001 and 0.009.
The grid is a mul;ple of the greatest common divisor (GCD).
Implies that the effec;ve bandwidth of a sampling scheme
can be maximized by ensuring that the sample ;mes
are rela;vely prime, i.e. have no common divisor.
G. L. BreNhorst

NUS artifacts can be viewed
as aliases (Nyquist no longer
applies)
Intra-band alias

RMS blurring 0 0.625 1.25
128
Every 4 th sample
from the Nyquist grid
t 2
t 1 t 1
1
1 128
t 1
1 128 1 128
5 real peaks
Contours: 3%, 0.5% of maximum

RMS blurring 0 0.625 1.25
128
t 2
t 1 t 1
1
1 128
t 1
1 128 1 128
HNCO: ubiquitin
Contours: 3%, 0.5% of maximum

Oversampling moves sampling artifacts to higher frequency

NUS on oversampled grid
15
N chemical shift (ppm)
Figure 6
1X 4X 8X

Metrics for spectral quality
Public test data repository
Tower of Babel
Optimal sampling theory
Data formats

Depends on the aim (metric)
Depends on the data (noise, dynamic
range, sparsity (“blackness”), etc.)
Depends on the analysis method
No formal theory

Many methods are nonlinear
How are the nonlinearities manifest?
Some assume prior knowledge, e.g. Lorentzian lineshapes
What happens if the assumptions are violated?
Many methods are numerically unstable
How does the method respond to noise, crowding?
Methods are frequently compared to a “straw man”, or “validated”
on synthetic data
How does the method compare to established practices for real data?
Many are computationally costly
How long does it take to compute?
Assessing novel methods:
A skeptics manifesto

Special Thanks
David Donoho
Berkeley, Stanford, hedge fund
Alan Stern
Rowland Institute

Acknowledgements
Mehdi Mobli
Mark Maciejewski
Blagoje Filipovic
Rowland
Gerhard Wagner
Peter Schmieder
David Rovnyak
Jim Sun
Sven Hyberts
Mallika Sastry
Dominique Frueh
Stanford
UConn Health Center
Alan Stern
Kuo-Bin Li
Peter Connolly
Andrew Chi
Harvard Medical School
David Donoho
Iain Johnstone
$$$ NIH, NSF, State of Connecticut