This is on-line version of our paper:

Nature Methods,2006,3:605-607

Removal of a time barrier for high-resolution multi-dimensional NMR spectroscopy Victor Jaravine1, Ilgis Ibraghimov2, Vladislav Yu. Orekhov1* 1

The Swedish NMR Centre at Gothenburg University, Box 465, 40530 Gothenburg, Sweden 2 Saarbrücken University, Mathematical Department, Saarbrücken, D-66041, Germany

Keywords: MDD, fast methods, high-resolution, missing data, sparse, non-uniform sampling, PARAFAC Abbreviations: ppm, parts per million; FID, free induction decay; DFT, discrete Fourier transform; nD, ndimensional; MDD, multidimensional decomposition. *correspondent author, [email protected]

We

introduce

decomposition

a

recursive

(R-MDD)

multi-dimensional

method

to

by uniform sampling of the data points needed to fill

speed-up

the huge space of multidimensional spectra. The total

recording of high-resolution NMR spectra. This

time of the traditional M-dimensional experiment with

method has a logarithmic dependence of measurement

N points in all indirect dimensions is estimated as: TDFT(M, N) = NM-1 2M-1 TFID

time on the size of indirect spectral dimensions, enjoys sensitivity and resolution advantages of optimized non-uniform acquisition schemes, and is applicable to all types of spectra of biomolecules. We demonstrate performance for several triple resonance experiments recorded on three globular proteins with molecular weights 8-22 kDa.

where TFID is the time for a single data point. It is clear (Fig. 1) that for dimensionality higher than two it is practically impossible to sample more than 100 points using conventional approach. This situation is known as the “sampling barrier”, which limits typical dimension sizes for 3D and 4D spectra to within 20-50

Since the definitive introduction by Richard Ernst of

points

Fourier Transformation and pulsed NMR spectroscopy

dimensional spectroscopy. In most cases, information

1

, molecular biologists had benefited from a dramatic

content of spectra can be substantially higher would

increase in sensitivity and resolution. Currently NMR

the larger sizes be feasible. For example, for a

spectroscopy plays an essential role in structural

deuterated 30 kDa protein measured at spectrometer

genomics as a tool for determining the spatial structure

magnetic field of 21.1 Tesla, the optimal sizes for

of proteins with atomic resolution

2,3

. However,

15

and

prohibits

Nx-TROSY and

13

practical

use

of

higher

Cx2Hz, coherences were estimated

throughput was limited by long measurement times4 as

to be 554 and 1395, respectively5. For 10 kDa and 100

conventional multidimensional NMR can take two to

kDa proteins the numbers are approximately twice as

six weeks of measurements per structure. With modern

large and small, respectively.

sensitive instrumentation, this time is mostly defined

A general solution for overcoming the sample-limiting

superposition of sine waves decaying with time. This

barrier is seen in unconventional sampling and signal

behavior of the NMR signal is exploited, for example,

processing schemes, several of which were recently

in the linear prediction 9, the method routinely applied

demonstrated6-8. In an approach referred as non-

to rescue digital resolution from short time series.

uniform or sparse sampling data points are measured here

and

there

with

non-equal

pseudo-random

intervals. By employing ideas of matched acquisition

9

this sparse sampling schedule can be optimized (Supplementary Figure 1 online) for providing resolution and sensitivity that are superior to uniformsampling Fourier transform based methods. Until now, however, Maximum Entropy (ME)10 and Multidimensional decomposition (MDD)

11,12

, which are the

two methods capable of dealing with the sparse sampling, both required at least 20% of the full data set for successful spectra reconstruction. The main result of this work is a substantial improvement in the MDD analysis, which allows spectra reconstruction with a drastically smaller proportion of the data.

In this work, we show that the autoregressive assumption can be incorporated into the MDD processing. In particular, each of the time domain shapes of length N in the original MDD model is recursively

subjected

to

additional

MDD

decomposition into a product of K vectors of length d so that N = dK. This converts the original Mdimensional decomposition to (M-1+K)-dimensional one, which we entitle R-MDD. The aim of the second MDD decomposition is to reduce the number of unknowns in the least-square minimization in the MDD model. For small d values (we use d = 2) the original vectors with N unknown elements are defined by much smaller number of parameters (d-1)Logd(2N). This provides the basis for further timesaving,

Reconstruction of a spectrum from a non-uniformly

considering that number of measurements should at

sampled time signal is the equivalent to prediction of

least be equal to number of adjustable parameters.

data lacking actual measurements. The only way to

Assuming that not more than half of signals (total Nc)

solve this problem is to introduce assumptions or, in

have the same position in the directly detected

other words, employ prior knowledge about the

dimension, we make an estimate of the measurement

spectrum. The mathematical model of the MDD,

time needed for a R-MDD reconstruction:

which was introduced three decades ago, assumes that

TR-MDD(M, N) = Nc (M-1)(d-1) logd(2N) TFID

all essential features of a M-dimensional matrix can be described as the sum of a small number of tensor products of one-dimensional vectors 13. When MDD is applied to NMR, the matrix corresponds to an input spectrum, whereas the vectors represent output lineshapes. The original MDD approach does not apply any constraints on the appearance of the line-shapes. However, using a model with a smaller number of adjustable parameters has an advantage that a bigger fraction of missing data can be predicted. In most practical cases, the NMR line-shapes are known. Theory predicts the autoregressive property of the NMR

time

domain

signal,

Removal of the time barrier in NMR

exemplified

2

by

a

While equation 1 gives a clear illustration of the sampling barrier in conventional multidimensional NMR, equation 2 gives solution to this problem. Logarithmic dependence on the dimension sizes and linear

dependence

on

spectrum

dimensionality

essentially eliminates any practical restrictions on these parameters. For example, a 5D experiment with all indirect dimensions sizes of 100 points would take 101 years of measurement time using conventional uniform sampling scheme. R-MDD using matched non-uniform sampling obtains such spectrum in 2.4 hours, which is comparable with time offered by the

methods based on 2D projections7,8 and 13 times faster

both for weak and strong peaks (Supplementary

than the original MDD (Supplementary Methods

Figure 3a online) with the correlation factor R of

online).

0.9984. The accuracy of the peak positions in the

Precise spectral R-MDD reconstructions with 91-94% missing data for the systems with molecular weights ranging from 8 to 22 kDa (Table 1) demonstrate general applicability of the method. We verify accuracy of the reconstruction by comparing it with the conventional “full” spectrum referred hereafter as

reconstructed spectrum with respect to the reference is 0.0056 ppm. The value should be compared with five times poorer value (0.0270 ppm) obtained for the conventional (“truncated”) spectrum (Fig. 2a) that was recorded during the same measurement time. The accuracy in

13

Cα frequencies can be important, for

“reference”. The time estimates from Eq. 2 for each

example, during sequential assignment, as it defines

protein compare well with the actual minimal

how many sequential connectivity links need to be

measurement

all

considered for a particular resonance. For the

backbone resonances, and are related to the number of

“truncated” experiment 43% of all resonances have

peaks in the spectra. Below we focus on describing the

ambiguous connectivities within the range defined by

most demanding case: 3D HNCA spectrum of a 14

the accuracy, while for the R-MDD reconstruction

kDa protein azurin recorded with the large number of

only 9% resonances have non-unique connectivity. In

times,

required

for

obtaining

increments to obtain high resolution in the

13

Cα

dimension. The reconstructions (Fig. 2c) exhibit higher noise compared to the reference (Fig. 2b), since sensitivity intrinsically decreases with reduction of measurement time. Analysis of this spectrum has additional complexity because of resolved J(Cα-Cβ)coupling, which almost doubles the number of peaks. A total of the 246 signals (singlet or duplet) corresponding to the published backbone assignment of 123 residues of azurin

14

were observed in the

region from 5.75 to 10.75 ppm in the amide proton dimension of the reference 3D HNCA spectrum. All of these

signals

(their

line-shapes, intensities

and

positions) are correctly reproduced in the reconstructed azurin spectrum (Supplementary Figure 2a online), with the exception of the three weakest inter-residue sequential correlations (Gly45, Gly88, and Phe114). It should be emphasized that the distribution of the peak intensities spans two orders of magnitude including a large number of weak signals that define sensitivity and final time allocation. The signal intensities in the reconstruction and the reference are highly correlated Removal of the time barrier in NMR

3

general, precise data allows for high degree of automation and is vital for obtaining accurate and unambiguous spectral NMR information

15

. To

summarize, we present a new method that solves the sampling limit problem. This opens an avenue for routine usage of higher spectroscopic dimensions in biomolecular NMR spectroscopy. Thus, we expect it to become a “single-stop solution” for highly automated resonance assignment and structure determination. We implemented non-uniform sampling for R-MDD processing

on

commercially

available

NMR

spectrometers (Varian). The processing package will be available from the authors. ACKNOWLEDGEMENTS This work was supported by grants from the Swedish Foundation for Strategic Research (A3 04:160d), the Swedish National Allocation Committee (SNIC 3/0444), Swedish Research Council (621-2005-2951), Wenner-Gren Foundation. The authors are grateful to G. Karlsson and S. Grzesiek for the azurin and ubiquitin samples.

REFERENCES 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15.

Ernst, R.R. Angew. Chem. Int. Edit. 31, 805-823 (1992). Montelione, G.T., Zheng, D.Y., Huang, Y.P.J., Gunsalus, K.C. & Szyperski, T. Nat. Struct. Biol. 7, 982-985 (2000). Yee, A.A. et al. J. Am. Chem. Soc. 127, 1651216517 (2005). Liu, G. et al. Proc. Natl. Acad. Sci. USA 102, 10487-10492 (2005). Rovnyak, D., Hoch, J.C., Stern, A.S. & Wagner, G. J. Biomol. NMR 30, 1-10 (2004). Mandelshtam, V.A., Taylor, H.S. & Shaka, A.J. J. Magn. Reson. 133, 304-312 (1998). Atreya, H.S. & Szyperski, T. Proc. Natl. Acad. Sci. USA 101, 9642-9647 (2004). Freeman, R. & Kupce, E. 23A, 63-75 (2004). Hoch, J.C. & Stern, A.S. NMR data processing, xi, p.196 (Wiley-Liss, New York, 1996). Stern, A.S. & Hoch, A.S. Maximum Entropy Reconstruction in NMR. in Encyclopedia of NMR, Vol. 8 (ed. Harris, D. & Grant, R.) (John Wiley & Sons, Chichester, 1996). Orekhov, V.Y., Ibraghimov, I. & Billeter, M. J. Biomol. NMR 27, 165-173 (2003). Tugarinov, V., Kay, L.E., Ibraghimov, I. & Orekhov, V.Y. J. Am. Chem. Soc. 127, 2767-2775 (2005). Kruskal, J.B. Linear Algebra Appl. 18, 95–138 (1977). van de Kamp, M. et al. Biochemistry 31, 1019410207 (1992). Altieri, A.S. & Byrd, R.A. Curr. Opin. Struct. Biol. 14, 547-553 (2004).

Removal of the time barrier in NMR

4

Tables & Figures

Table 1. Results of R-MDD analysis of the protein spectra. Ubiquitin Spectrum type Sample concentration, mM time per one real data point, TFID, sec Mol. Weight, kDa Dimensions sizes, 13C,15N, complex points Number of spectral signals All peak 13C position r.m.s.d., R-MDD vs. reference, ppm Time estimates from eq. 2, hours Actual time needed to acquire all backbone correlations, hours, (% sparse)

azurin

HNCO 1.7 2.4 8 32 32

HNcoCA 1.7 4 8 64 64

HNCA 1.0 4 14 400 62

barstar-barnase complex HNCO 1.2 2.8 22 64 64

70 0.0015

70 0.010

481 0.006

101 0.005

0.5 0.2 (7%)

1.1 0.6 (6%)

9.2 8.7 (6%)

1.1 1.5 (9%)

Figure 1. Theoretically estimated times Texp required for experimental measurement of 2D, 3D, 4D and 5D spectra using RMDD reconstruction (solid lines, Eq.2 for d = 2, Nc = 70, TFID = 4 sec) and conventional method (dash lines, Eq.1) as a function of indirect dimensions sizes N.

Figure 2. High resolution in the R-MDD reconstructions.

13

Cα -1HN regions (top) and

13

Cα 1D cross-sections (bottom) are

compared for the signals of Gly123, Val31 in (a) “truncated”, i.e. conventional 3D HNCA with 40(13Cα)x40(15N) points, experimental time 8.9 hours; (b) “reference”, i.e. conventional spectrum with 400(13Cα)x64(15N) points, 138 hours ; (c) RMDD reconstruction from 6% of data, 8.9 hours.

Removal of the time barrier in NMR

6

Suppl. Tables & Figures

Supplementary Methods – R-MDD theory and experimental details find scalar numbers βa and normalized vectors Calculation of measurement time for the case of uniformly sampled experiment. The total duration

β

Fm, with elements βFm(nm) (nm=1..Nm), such that

the following norm becomes minimal.

of an M-dimensional NMR experiment is defined

| G • [S – Σβ (βa β F1 ⊗ β F2 …⊗ β FM)]|2 +

by three factors: (i) TFID - time required per single

λΣβ (βa)2

transient of a one-dimensional spectrum (or FID), times the number of transients in a phase cycle used for eliminating spurious signals; (ii) Nm number of sample points in each of M indirect dimensions; (iii) Np=2M-1 - the number of FID’s required for phase discrimination in the indirect dimensions. Without loosing generality of this presentation we assume equal number of points Nm=N in all indirect dimension. Thus, the time in seconds of a conventional experiment with uniformly sampled points can be expressed as following: TDFT(M, Nm) = NM-1 2M-1 TFID

Here, the symbol ⊗ denotes tensor product operation;

the

matrix

S

corresponds

to

experimental M-dimensional NMR spectrum in time and/or frequency domain representation. In the case of sparse sampling only a fraction of elements in S is measured and the matrix G, which

contains

elements

gn1,n2,…,nM ∈ {0,1},

indicates the absence or presence of a particular data point. Accordingly, the symbol • describes element-wise multiplication of matrices. The last term represents a Tikhonov regularization3, which is parameterized with the factor λ and may be (S1) used for improving the convergence of the MDD algorithm 4. The summation index β runs over the

The general MDD model. Multidimensional

number

of

components

decomposition (MDD) model assumes that all

decomposition. The range for this index depends

essential features of M-dimensional matrix can be

on the type of spectrum. For example, for a 3D

described as a sum of small number of tensor

HNCO spectrum it is roughly equal to the number

products of one-dimensional vectors 1,2. MDD has

of protein amide groups.

been used in different fields as a tool for data

Each vector

analysis and signal processing since the early

elements. We make an estimate of measurement

seventies under various names such as the parallel

time needed for MDD reconstruction from the

factor analysis, canonical decomposition, and

notion that number of measurements should

three-way decomposition. When applied to NMR

exceed number of adjustable parameters with a

spectra the MDD can be formulated as follows.

certain redundancy. We also need to make an

Given a matrix S with sizes Nm of its M

assumption about maximal number of peaks that

dimensions (m=1…M) and elements Sn1,n2,…,nM ,

have the same position in the directly detected

β

used

for

the

Fm is defined by Nm unknown

dimension. Altogether, the redundancy and signal

Note that Eq. S4 is a generalization of Eq. S5,

overlap give a factor, which is approximately

when d is any integer number and vector elements

equal to half of the number of peaks in the

are not restricted to integer powers of exp(-Δt).

spectrum. Finally, the measurement time is

The aim of the second MDD decomposition is to

estimated as:

reduce number of unknowns in the least-square

TMDD(M, Nm) = Nc (M-1) Nm TFID where Nc is number of spectral signals. For example, for HNCO spectrum of ubiquitin Nc is ca 70.

minimization of Eq S2. This is (S3)achieved by replacing vectors β Fm for all β in Eq. S2 by the right side of Eq. S4. This merely converts Mdimensional

decomposition

to

(M-1+Km)-

dimensional one. In case of phase sensitive detection, the complex vector

The Recursive MDD model. Each of the time m

β

Fm with 2Nm

unknown elements is defined by much smaller

of length Nm in Eq. S2 is

number of real parameters 2(d-1)*Logd(2Nm).

represented as a product of Km vectors of length

Finally, using the same arguments as for Eq. S3,

dm so that Nm = (dm)Km:

total measurement time needed for robust R-MDD

domain shapes β F

β

Fm= β V1 ⊗ β V2 …⊗ β VK

reconstruction of the spectrum can (S4)be estimated

This can be interpreted as applying second

as, TRMDD(M, Nm) = Nc (M-1) (d-1) Logd(2Nm)

decomposition to the shapes obtained in the first decomposition, thus the name “recursive”. To

TFID

rationalize the additional decomposition of each

This equation with Nm=N is plotted for M=2,3,4,5

k

(S6)

vector F into a set of subvectors V , (we can omit

as solid curves in Figure 1 of the paper using d=2,

indices m, for simplicity), we shall illustrate the

TFID =4 sec, and Nc=70 (number of backbone

recursive decomposition for d=2. The points of

amides in ubiquitin).

exponential decay function E(t)=exp(-t) sampled

logarithmic dependences on number of increments

at

t=nΔt,

in the indirect dimensions, Nm. The exponential

(n=0,…,N), constitute vector E={exp(-Δt) , exp(-

dependence (Nm)M-1 on the number of dimensions

Δt)1,… exp(-Δt)N-1} of length N. The vector can be

M in Eq. S1 is reduced to linear term (M-1) in Eq.

“folded” into a hypercube with N elements, where

S6. From the plot it is clear that all of the

each side has length 2. The hypercube can be

experimental times needed for the MDD 2D-5D

uniformly

spaced

time

points 0

k

decomposed into K=log2N vectors V = {1, exp(k

Δt) } of length two

The curves exhibit

spectral processing are below 5 hours, which is generally the 2D range of standard Fourier

E= {1, exp(-Δt)1} ⊗ {1, exp(-Δt)2} ⊗ {1, exp(-Δt)4… }

spectroscopy. Eq. S6 provides the following enhancement in (S5) time saving relative to eq. S3:

Removal of the time barrier in NMR

8

TMDD(Nm) / TRMDD(Nm) = Nm / {(d-1) Logd(2Nm) }

shapes as well as ability of the method (S7) to deal

The ratio increases with larger Nm (e.g. equals 13

with signals in wide range of intensities.

for Nm = 100). It does not depend on spectra dimensionality and number of components. Equations S4 – S7 were derived in the assumption

Materials and Methods

that the recursion, and correspondingly the

NMR spectroscopy. All spectra were recorded

autoregressive constraint on the time domain line

using room temperature triple resonance probes

shapes, is applied in all indirect dimensions.

with pulse field gradients. The proteins were

However, the constraint can be used in a subset of

13

C,15N labeled (except for barstar): ubiquitin14;

dimensions. Example is the 3D 15N NOESY

barstar-barnase complex12,13; reduced azurin11.

experiment, where for reducing number of

The key experimental parameters are listed

components and preserving sensitivity, all peaks

(Supplementary Table 1).

belonging to one amide group are collected in one

Sampling schedules. Optimal non-linear sampling

MDD component5. In this case, the recursion can

schedules are devised to match the envelopes of

be applied only to the

15

N dimension. Thus,

signal coherences in all indirect dimensions for a

although NOESY type spectra will also profit

particular NMR experiment 9. Since 15N chemical

from R-MDD, the expected timesaving relative to

shift evolves in all our experiments during a

original MDD in this case is moderate.

constant delay, the envelope in this dimension is a

MDD is a true multidimensional signal processing

constant function. The envelope of the signals in

method, as it analyses entire multidimensional

13

C dimension is the exponential decay with

data array simultaneously. An additional potential

transverse relaxation time T2 (Supplementary

offered by the method is in its ability to process

Table 1). The convolution of the two envelopes

non-uniform data, when valuable spectrometer

produces a two-dimensional probability density

time

function in two acquisition dimensions t1 and t2.

is

preferentially

spent

on

obtaining

information on signals of interest rather than

The sampling

noise. This insures highest sensitivity for given

accordance with the density function. For

digital resolution and total measurement time.

example, in the sampling schedule optimized for

High sensitivity delivered by the MDD processing

the 3D HNCA of azurin (Supplementary Figure

have

been

addressed

7

generated

in

and

1) in addition to the T2-decay, the sampling

. A related

density is modulated by cosine function due to

specifically

demonstrated in practical cases

was randomly

6

technique 8, which is designed for dealing with

J(13Cα-13Cβ)-coupling of 35 Hz.

GFT projections, was also proved successful in identifying weak signals. Likewise, sensitivity of the R-MDD is reflected in demonstrated high accuracy of signal amplitudes, positions, and line Removal of the time barrier in NMR

9

Data processing and reconstruction for all spectra were done using the same procedure. It is

exemplified below for the azurin case. Initially

were merged to obtain the whole spectrum

time domain data along the directly detected

reconstruction.

dimension, t3 of the reference spectrum were

components for each of the 48 segments was

multiplied with a squared cosine-bell window

estimated as following: the number of 1HN-15N

function, zero-filled to 2K points, and Fourier

correlations in the corresponding 1H strip of the

transformed with nmrPipe package

10

. The

15

The

number

of

the

MDD

N HSQC spectrum was multiplied by 4 to

“truncated” data set was derived from the

account for the intra- and inter-residue duplets in

“reference” by truncation after initial 40 (t1), 40

the HNCA spectrum; the number was further

(t2) points. The “sparse” data set was extracted

increased by 30% to account for presence of

from the “reference” according to the sampling

possible minor peaks and some of the large noise

schedule (Supplementary Figure 1) containing

features

1600 points (t1,t2). A complete time domain

reconstruction, which produced a regular 3D data

signal containing 29=512 (t1), 26=64 (t2) complex

set in the format of nmrPipe, the time domain

points was reconstructed from the sparse data set

signal was multiplied with a squared cosine-bell

using the R-MDD procedure (Eqs. S2, S4)

window function, zero-filled to double size and

implemented in a homebuilt software mddNMR.

Fourier transformed. This gave digital resolution

Note that the final size in

13

a

C dimension (t1) in

or

spectral

artefacts.

of 0.029 ppm (5.17 Hz) for

13

After

the

Ca dimension. The

the reconstruction is 25% larger than that in the

peaks positions in the reference spectrum are

reference spectrum. This shows that R-MDD does

determined using ‘pkFindROI’ subroutine of the

not only fill gaps between the measured points,

nmrPipe package

but also can effectively extrapolate time domain

all peak lists were refined by three-point

data. Application of the recursion (Eq. S4) to

interpolation. In the analysis the parameters of

dimensions t1 and t2 but not to t3 resulted finally

duplets were determined as average of two

in 9+6+1 = 16 – dimensional decomposition in

singlets. To check stability of the algorithm

Eq.

and

convergence we repeated the R-MDD calculation

reconstruction were performed individually for 48

with the same experimental input but different

overlapped 3D segments covering the range from

initial approximation for the shapes β VK (Eq. S4).

5.75

S2.

to

The

10.75

R-MDD

ppm

calculation

(w3-direct

1

H),

each

corresponding to 32 frequency-domain points. The segmentation doesn’t affect the algorithm convergence and quality of reconstruction. It was done only to save computational time by parallel processing, since segments can be processed independently. To account for possible incomplete reconstruction of peak tails on the segments borders (in w3) only the central 16 point parts Removal of the time barrier in NMR 10

10

. The exact peak positions in

The result was essentially the same as for the first run described in the main text: the all-signal r.m.s.d. to reference was 0.0062 ppm in the CA dimension and the correlation of intensities was 0.998. These values for the difference between the two solutions were: 0.0036 ppm and 0.9985, respectively.

8. 9.

References 1. 2. 3. 4. 5. 6. 7.

Malmodin, D. & Billeter, M. J. Am. Chem. Soc. 127, 13486-7 (2005). Hoch, J.C. & Stern, A.S. NMR data processing, xi, 196 p. (Wiley-Liss, New York, 1996). Delaglio, F., Grzesiek, S., Vuister, G.W., Zhu, G., Pfeifer, J. & Bax, A. J. Biomol. NMR 6, 277-93 (1995). Leckner, J. Folding and structure of azurin the influence of a metal PhD thesis, Chalmers Technical University (2001). Korzhnev, D.M. et al. Appl. Magn. Reson. 21, 195-201 (2001). Zhuravleva, A.V., Korzhnev, D.M., Nolde, S.B., Kay, L.E., Arseniev, A.S., Billeter, M. & Orekhov, V.Y. manuscript in preparation (2006). Wang, A.C., Grzesiek, S., Tschudin, R., Lodi, P.J. & Bax, A. J. Biomol. NMR 5, 376-382 (1995).

Kruskal, J.B. Linear Algebra Appl. 18, 95–138 (1977). Beylkin, G. & Mohlenkamp, M.J. Proc. Natl. Acad. Sci. USA 99, 10246-10251 (2002). Tikhonov, A.N.a.S., A.A. Equations of mathematical physics, (Dover Publ., New York, 1990). Ibraghimov, I. Num. Linear Algebra Appl. 9, 551-565 (2002). Orekhov, V.Y., Ibraghimov, I. & Billeter, M. J. Biomol. NMR 27, 165-173 (2003). Luan, T., Jaravine, V., Yee, A., Arrowsmith, C.H. & Orekhov, V.Y. J. Biomol. NMR 33, 114 (2005). Tugarinov, V., Kay, L.E., Ibraghimov, I. & Orekhov, V.Y. J. Am. Chem. Soc. 127, 27672775 (2005).

10. 11. 12. 13.

14.

Supplementary Table 1. The samples and spectral parameters of the 3D experiments.

ubiquitin

Barstar-barnase

azurin

complex Experiment type

HNCO

HNcoCA

HNCO

HNCA

1

800

600

800

900

Number of transients,

2

4

4

4

Interscan delay, sec

1.2

1

0.75

1.25

Sample concentration, mM

1.7

1.4

1.2

1.0

Mol. Weight, kDa

8

8

22

14

dimensions, complex points 13C,15N

32 32

64 64

64 64

400 64

Spectral width, Hz, 13C,15N

2000 2000

4500 1500

2350 2800

6800 2500

Limits 1H, ppm

6.047 9.702

6.742 10.397

5.75 10.75

Bias in sparse sampling, 13C

T2=50 ms

T2=50 ms

T2=30ms

T2=50ms, 1J=35 Hz

Number of points in the sparse

72

246

369

1600

3.0

10

16.8

138

0.16

0.23

0.092

0.029

H spectrometer frequency, MHz

sampling schedule Full experiment measurement time, hours Digital resolution 13C, ppm/pnt

Removal of the time barrier in NMR

11

Supplementary Figure 1 Optimized sampling schedule for acquisition of high-resolution 3D HNCA spectrum of azurin. 1600 points are distributed non-linearly to match optimal sensitivity. Uniform random distribution of sampled points is modulated by homo-nuclear 13Cα-13Cβ coupling of 35 Hz and exponential decay of 50 ms in 13Cα dimension; constant amplitude signal is assumed in 15N dimension.

Supplementary Figure 2 Comparison with the reference of individual resonance reconstructions for 3D HNCA of azurin. All peaks with reported 15N -1HN backbone assignment are shown: 1st column of each cell is the reference spectrum; 2rd column is the R-MDD reconstruction from the sparse data.

13

C -1HN

regions for the amides of the backbone resonances are shown above the 13C 1D cross-sections, taken at the peak centers. The 15N, 1H, and 13C peak positions are indicated in ppm ; the cell sizes in ppm are indicated of the 1st page. The plots have the same intensity threshold and contour multiplication factor of 1.3. For better visibility, the 1D cross-sections of weak signals were scaled up. The corresponding scaling factors (1x or 3x) are indicated for all peaks.

Removal of the time barrier in NMR

12

Removal of the time barrier in NMR

13

Removal of the time barrier in NMR

14

Removal of the time barrier in NMR

15

Removal of the time barrier in NMR

16

Removal of the time barrier in NMR

17

Removal of the time barrier in NMR

18

Removal of the time barrier in NMR

19

Supplementary Figure 3 Comparison with the reference of individual resonance reconstructions for 3D HNCO of barnase in complex with barstar. All peaks with reported

15

N -1HN backbone assignment are

shown: 1st column of each cell is the reference spectrum; 2rd column is the R-MDD reconstruction from the sparse data. 13C -1HN regions for the amides of the backbone resonances are shown above the 13C 1D cross-sections, taken at the peak centers. The 15N, 1H, and 13C peak positions are indicated in ppm ; the cell sizes in ppm are indicated of the 1st page. The plots have the same intensity threshold and contour multiplication factor of 1.3. For better visibility, the 1D cross-sections of weak signals were scaled up. The corresponding scaling factors (1x or 3x) are indicated for all peaks.

Removal of the time barrier in NMR

20

Removal of the time barrier in NMR

21

Removal of the time barrier in NMR

22

Removal of the time barrier in NMR

23

Supplementary Figure 4 Comparison with the reference of individual resonance reconstructions for (a) 3D HNCO and (b) 3D HNcoCA of ubiquitin. All peaks with reported 15N -1HN backbone assignment are shown: 1st column of each cell is the reference spectrum; 2rd column is the R-MDD reconstruction from the sparse data. 13C -1HN regions for the amides of the backbone resonances are shown above the 13C 1D cross-sections, taken at the peak centers. The 15N, 1H, and 13C peak positions are indicated in ppm ; the cell sizes in ppm are indicated of the 1st page. The plots have the same intensity threshold and contour multiplication factor of 1.3. For better visibility, the 1D cross-sections of weak signals were scaled up. The corresponding scaling factors (1x or 3x) are indicated for all peaks.

Removal of the time barrier in NMR

24

a

Removal of the time barrier in NMR

25

Removal of the time barrier in NMR

26

b

Removal of the time barrier in NMR

27

Removal of the time barrier in NMR

28

Supplementary Figure 5 Accuracy of intensities and peak positions (a-b) between 6% data reconstruction and reference spectra for all backbone 3D HNCA correlations of protein azurin. (a) High level of “reconstruction-reference” correlation of intensities (normalized to the max intensity). (b) All-peak histograms of differences of bars, “6% sparse”) for the “reconstruction-reference” and

13

Cα chemical shifts is plotted (filled

(“40x40 truncated”, gray bars) for “truncated-reference”.

Reconstruction and truncated spectra had total of 1600 pts in two indirect dimensions, with equal experimental times of ca 8.9 hours each. Reference spectrum was recorded with 400x64 points with experimental time of 138 h.

Removal of the time barrier in NMR

29