Chapter 14

Chapter 14

ACCURACY AND PRECISION OF INTENSITY DETERMINATIONS IN QUANTITATIVE NMR Jean-Philippe Grivet Centre de Biophysique Moléculaire, C.N.R.S., et Université d'Orléans, Orléans, FRANCE 1 INTRODUCTION NMR, just as other spectroscopies, was at first concerned with interpreting spectra in terms of structure, then with species identification and later with dynamics. The determination of concentrations or more generally magnetisations associated with ensembles of isochronous nuclei, which is the purpose of quantitative NMR, is a more recent concern. Indeed, classic textbooks (Pople et al., 1959, Becker 1969) devote but a few pages to the subject, except when it bears on structure elucidation. More recently, quantitative determinations have become quite frequent. The following is a random (albeit biased by the author's involvement in biological NMR) sample of recent publications describing applications of quantitative NMR: amide proton exchange with solvent in a protein (Skelton et al.,1992), determination of cell volume (Hockings and Rogers 1994), measurement of metabolic fluxes (Pasternack et al., 1994), examination of isotopic fractionation in biosynthetic processes at natural abundance (Martin and Martin, 1990), determination of the solution structure of an inhibitor-protein complex (Spitzfaden et al., 15 1994), study of protein backbone dynamics by N relaxation (Akke et al., 1993), measurement of hydrogen diffusion in aqueous protein solutions by pulsed field-gradient methods (Kuchel et al., 1993). Many of these measurements now involve multidimensional NMR techniques. We will be mainly concerned with NMR spectroscopy in fluid solutions and its extensions to in vivo spectroscopy. Similar problems arise in solid-state NMR, in ESR and in magnetic resonance imaging, but these techniques fall outside of the scope of this review. Also, many questions concerning precision and accuracy also arise in other fields, such as infra-red, Raman, visible, and ultra-violet spectroscopies, chromatography, and

1

similar theories have been independently developed in these areas. Fourier transform NMR (and also Fourier IR) is characterised by its operating in two domains, connected by a Fourier transform: the measurement (or time) domain and the spectral (or frequency) domain. This circumstance leads to a great flexibility for signal processing. For other types of spectroscopies, the Fourier conjugate space is only a mathematical construct. In this introduction, it is perhaps useful to recall the classical definitions of the terms precision and accuracy (Bevington, 1969). Any experimental result, such as the frequency of an NMR signal is subject to an error. If the experiment is repeated n times, we get n different results f1, f2, ..., fn. If the errors were truly random, the average would * provide an unbiased estimator of the true result f . The sample variance can be taken as a measure of the precision of our measurement. Some errors are not random and averaging * several independent experiments will not yield a better answer: ≠ f . We say that non-random errors (bias or systematic errors) affect the accuracy of the result. In contrast to precision, accuracy can be exactly evaluated only if we can determine the true result by an independent method. These definitions are somewhat trivial for primary parameters (frequency, intensity), but their actual application can be difficult for derived quantities, such as interatomic distances in a macromolecule. Borderline cases are also known. When an animal moves inside an in vivo probe, the frequency of some metabolite changes. Whether this perturbation should be considered as a random or systematic error is a moot question. The answer depends in part on whether the correlation time of the animal's motion is shorter or longer than the time resolution of the experiment. The oldest and most frequently used method of quantitation in NMR is the computation of the integral of the signal of interest. It is interesting to speculate as to why NMR, in contrast to optical spectroscopy, uses line integrals rather than amplitude at any convenient frequency to determine concentrations. The same theoretical result applies in both cases: the integral of the absorption band shape is proportional to the number of absorbing centres. The current NMR practice is probably linked to the facts that the useful spectral width of the typical spectrum extends over several hundreds or thousands of linewidths and that natural widths of NMR lines vary widely from one signal to another. Another difference with optical spectroscopy is that absolute intensity measurements are almost never carried out; an internal (or external) standard is present or one is interested in comparing the intensities of the same signal at various times within the same experiment. This review is organised according to the following plan. We briefly examine requirements on the repetition times imposed by quantitative NMR. Assuming next that integrals will be used, we survey the causes of common sources of errors, baseline and phase anomalies, and the methods to remove them. After a brief description of spectrometer noise, we look at some peak-picking procedures, and then survey the accuracy and precision of integrals. The accuracy and precision of other methods of quantitation (least squares, linear prediction, maximum likelihood, maximum entropy) are then examined. We conclude with some indications on the precision of derived parameters (rate constants, interatomic distances).

2

2 RELAXATION TIMES AND RELAXATION DELAYS In order to obtain reliable quantitative results, we require that the signal be proportional to the solute concentration. Now, the sample magnetization evolves in time with a characteristic time constant, which is, for most cases of interest, the spin-lattice relaxation time T1. Further, since the sensitivity of the experiment is low, we need to average many scans. Lastly, in order to conserve spectrometer time, or to reach a sufficient time resolution, we wish to apply pulses as frequently as possible. The conditions for optimum sensitivity (in a 1D experiment, the repetition time Tr should be ≅ 1.3 T1) and quantitative results (Tr ≥ 3 T1) are contradictory. This problem has been extensively studied in the case 13 of 1D C NMR, where relaxation times of 30 s are common (Sotak et al., 1983, Daubenfeld et al., 1985). Solutions include: the use of relaxation reagents (Wenzel et al., 1982), the computation or the experimental determination of correction factors (Gard et al., 1985, van Dobbenburgh et al., 1994), the use of polarisation transfer sequences (DEPT, INEPT) where the intrinsic sensitivity is higher and the characteristic time is the much shorter proton T1 (Bardet et al., 1985, Netzel 1987, Lambert et al., 1992). Similar problems arise in saturation transfer and in NOESY-type experiments, although no systematic study of the relaxation constraints has been reported. 3 BASELINE AND PHASE ANOMALIES When quantitative data must be extracted by integration of a spectral region, it is found that even slight departures from a perfect baseline (or base surface) have dramatic effects on the quality of the results. The causes of distorted baselines, and the methods proposed to correct them, are so numerous that it would be impossible to present a complete review (see, however, chapters by L. Eveleigh, K. Cross, and É. Kupče). We shall only give a few leading references. 3.1 Pulse feedthrough and acoustic ringing Instrumental baseline distortions can be due, amongst other causes, to excitation pulse feed-through or acoustic ringing of the probe. Introducing a preacquisition delay produces unmanageable phase shifts. Elegant methods introduce a carefully designed spin-echo refocusing period in the measurement sequence (Lippens and Hellenga, 1990, Froystein 1993), effectively removing the signal from the dead time, at the cost of some reduction in sensitivity. Post-acquisition data manipulations can also correct these defects. Heuer and Haeberlen (1989) have shown that the first data points of the FID may be zeroed and then reconstructed from the experimental base-line, using the knowledge that the base-line perturbation is approximately a phase shifted 'sinc' function. The "FLATT" procedure of Güntert and Wüthrich (1992) is somewhat similar. The missing signal, which should have been recorded during the spectrometer dead time, can also be reconstructed using linear prediction algorithms (Marion and Bax, 1989). Other instrumental imperfections give rise to so-called t1 ridges or t1 noise, which can be effectively suppressed by reference

3

deconvolution (chapter by G. Morris). Manolera and Norton (1992) have proposed an adaptative smoothing method. An average value of the t1 noise is computed for each column of a 2D map (each value of f2); smoothing coefficients are then derived in such a way that points along the f2 axis subject to the highest noise receive the strongest smoothing. 3.2 Sequential sampling and the Redfield trick Sequential signal sampling is another instrumental cause of artifacts. Since the introduction of quadrature detection (Ellet et al., 1971, Redfield and Gupta, 1971), users are accustomed to think that their spectrometer operates according to theory, sampling the complex FID at regular intervals and storing its in phase (x k) and out of phase (yk) components at each time point tk. However, such is not the case for some instruments (notably Bruker machines) where a different scheme, sequential acquisition along with the so-called 'Redfield trick' (Redfield and Kunz, 1975) is used. This method allows for the use of a single digitizer, the transmitter frequency being set in the middle of the spectral region of interest. It relies on the following theorem: if the Fourier transform of f(t) is F(ν), then the transform of exp(2iπβt)f(t) is F(ν-β), i.e. the same spectrum shifted by β Hz towards positive frequencies. Using a digitization rate equal to twice the spectral width, the computer records the signal values x0,y 1,-x2, -y3, x4....This phase rotation is equivalent to a multiplication by exp(iπk/2), k = 0,1,...,N- 1, and is exactly what is required to shift the resulting spectrum by one half the spectral width towards positive frequencies, thus allowing the use of a single digitizer, while preventing confusion between positive and negative frequencies. A consequence of this procedure is that the data can be interpreted by the computer as real, and can be Fourier transformed by a real discrete Fourier transform (DFT) algorithm. A similar scheme is used in many multidimensional NMR experiments, under the name TPPI (Bodenhausen et al., 1980). Anomalies due to sequential sampling have been described (Marion and Bax, 1988). 3.3 First point problem The DFT algorithm itself is at the origin of lineshape and baseline distortions, some of which can be conveniently grouped under the term "first point problem" and which have been lucidly analyzed recently (Otting et al., 1986, Abilgaard et al., 1988, Gesmar et al., 1990, Zhu et al. 1993, Starčuk et al. 1994, Tang 1994). Part of the problem is algebraic in nature. The function f(t) = 0, t0 has the Fourier transform F(ν) 2 = (1 - 2iπν)/[1 + (2iπν) ]; f(t) is undefined at t=0, but it is convenient to set f(0)=1/2 (Bracewell 1986): the theorem that the integral over the spectrum is equal to the value of the time-dependant function at the origin is then valid. Thus the first point of the FID should be divided by 2. A similar conclusion is reached when one considers sequences of discrete values and their discrete transform. In that case, we in effect compute a finite Fourier series. According to Fejer's theorem, the Fourier series of a discontinuous + function converges to the average of function values before (t = 0 ) and after(t = 0 ) the

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-

discontinuity. Because of the implicit periodicity of the signal, the value at 0 is the same as that at the end of the FID, often zero. In the case of NMR, this problem must be compounded with the facts that the first data point is not recorded at t = 0 (Otting et al., 1986) and that the signal is corrupted by the analog filters (Hoult et al., 1983). The spectrum will seem decent enough in the case of well separated lines of similar amplitudes, but will show baseline offsets, a rolling baseline and unphasable peaks in the case of a very intense (solvent) line and/or severely overlapping lines. The quoted references show how these defects may be mitigated by choosing the time of the first sample and scaling its value. A further problem is that the end of the FID may be truncated (Abilgaard et al., 1988). The result is often that some lines look poorly phased, with subsequent errors in the integrals. It becomes apparent as soon as the acquisition time * is less than 3 T2 , the decay time of the FID. 3.4 Baseline flattening algorithms Despite all precautions that can be taken during acquisition, the baseline can still be distorted, and the signal phase can be quite different from pure absorption. For many applications, such as NOESY spectroscopy or in vivo kinetics, where a large number of spectra must be recorded and integrated, there is a need for fast, automatic and reliable phase and baseline corrections. New algorithms for these purposes appear at a high rate: their very number is an indication that none completely solves these problems. Baseline (or base surface) fitting by various polynomials have been described: Lagrange (Barsukov and Arseniev, 1987), Golay-type least squares (Dietrich et al., 1991), spline (Zolnai et al., 1989). It is well known that high degree polynomial fits of a noisy curve pose an ill conditioned problem (Hamming 1962). Thus, a piece-wise linear fit (Saffrich et al., 1993) is probably the most robust and effective method. Algorithms for automatic phase correction of in vivo spectra have been proposed (van Waals and van Gerven, 1990). They are claimed to be fast and rather immune to noise and overlap. Balacco (1994) offers a dissenting view and another algorithm, dedicated to high resolution spectra of liquids. 3.5 Solvent suppression The presence of a large solvent signal (water in most biomedical applications) is the cause of many difficulties. For quantitative NMR, the main problems are rolling or sloping baselines due to the dispersive tails of the solvent line, ghost lines due to harmonic mixing, poor digitization of low intensity signals. Numerous counter-measures have been proposed, both of the pre-acquisition (such as low power water saturation) and postacquisition varieties (such as subtracting a 1/(ν - ν0) background, similar to the water dispersion signal). Readers should consult the many reviews available (Guéron et al., 1991, chapter by É. Kupče). 3.6 Oversampling Recent developments in the applications of oversampling and digital filtering (Nuzillard

5

and Freeman, 1994, chapter by K. Cross) to NMR will contribute to render many of the above considerations obsolete. For effective oversampling, the incoming signal is sampled 10 or 20 times faster than provided for by the Nyquist criterion, with two beneficial consequences. The dynamic range is increased, digital noise is decreased (Delsuc and Lallemand, 1986) and baseline distortions are minimized (Wider, 1990). Digital filtering with downsampling is used to reduce the amount of data to be stored after oversampling (Rosen, 1994, Redfield and Kunz, 1994). However, it may be used also to implement band rejection filters that can efficiently remove intense solvent lines, improving the baseline (Cross, 1993). Because of the much sharper cut-off of digital filters as compared to analog filters, less noise is folded into the pass-band, and the sensitivity is improved. 4. INTEGRATION ALGORITHMS AND INTEGRATION RANGE Assuming that we have at hand a spectrum free of systematic errors unfortunately does not imply that we will certainly compute accurate integrals. Two possible sources of error remain in our way: the integration algorithm and the integration interval. 4.1 Algorithmic errors Since an algorithm uses a finite number of arithmetic operations, its result is always subject to a truncation error (Hamming 1962). In the case of numerical integration of magnetic resonance spectra, truncation errors have been investigated (Herring and Phillips, 1985) and found to be less than 1% of the total area as soon as the density of data points exceeded 3 or 4 points per full width at half height. These authors, however, only considered digitized lorentzian lines, and not spectra obtained by DFT of time domain signals. The line-shape is then as given by Abilgaard et al. (1988). The use of windows will lead to other band-shapes of unknown analytical form. The effects of 'algorithmic truncation' are difficult to disentangle from those of 'FID truncation', as the following qualitative argument shows. The narrowest line that can ever be observed occupies a single data point in the frequency domain, that is a full width of (W s)/N Hz, N being the number of spectral points and Ws the spectral width. This corresponds to an effective * relaxation time of T2 = N/π(Ws) = Ta/π s (Ta is the acquisition time) and is precisely the limit for which truncation effects of the FID become important. The author has performed some computer simulations using Abilgaard's lineshape and either the trapezoidal rule or Simpon's rule for integration for even numbered points and Simpson's half-interval rule for odd numbered points as advocated by Hamming (1962). This formalism gives only slightly improved results, indicating that algorithmic truncation is of minor importance. Similar problems must arise in multidimensional NMR, since digital resolution is severely limited, but they have not been investigated. 4.2 Error due to a limited integration range The integration range for NMR lines is obviously finite, specially for crowded spectra. It has been pointed out long ago (Ernst 1966) that restricting the integration interval has

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striking consequences for lorentzian lines. Starting from the expression for a normalized 2 2 lorentzian: L(ν) = (λ/π)/(ν + λ ), with full width 2λ, we find that the integral from -2kλ to 2kλ is Ik = (2/π)arctan(2k).This is quite different from unity for any reasonable value of k. For instance, I3 = 0.895. This may be of consequence when comparing the intensities of peaks having widely different widths. The case of two-dimensional spectra was examined by Weiss et al. (1992). These authors showed that errors on peak-volume estimates could easily exceed 100%. This disturbing result was obtained, however, with a simple lorentzian line-shape and direct summation. Ernst (1966) also showed how resolution could be improved by line deconvolution; the most common of these transformations involves a change of line shape from lorentzian to 2 gaussian: the FID is multiplied by a window function of the form exp(αt - βt ), with α,β + > 0. As long as the value of the FID at t = 0 is not modified by the window, integrals are not changed, but the very wide "wings" of the Lorentz shape are removed. For a given integration interval, the bias is much diminished. Thus, I1 = 0.98 for a pure gaussian shape. When the parameter α is large, the line acquires a characteristic shape, with negative "feet": even in that case, the integral value is not seriously altered. 4.3 The full width-height estimator When base-line flattening or deconvolution are not applicable, a simple and accurate estimation of the area of a lorentzian line can still be obtained by computing the full widthamplitude product. For the lorentzian of full width 2λ, the amplitude is L(0) = A/πλ, and the area is A = (π/2)(2λ)(A/πλ). This amplitude-width estimator has been shown to be 0 rather immune to phase errors (Weiss et al., 1983), up to an error of 60 , but its behaviour in the presence of noise was not investigated. At this point, one should take up the problem of overlapping lines. In the author's opinion, it has no convincing solution when straightforward integration is to be used. An expert system has been described (Chow et al., 1993). It is claimed that this program can 31 automatically quantify in vivo P NMR spectra as reliably as a human spectroscopist. The program fits each line to an asymetrical triangle shape. Holak et al.(1987) showed that NOESY cross-peak volumes could be well approximated from the scaled product of two integrals, one along f1, the other along f2; by a suitable choice of row and column, the interference from a second peak may be minimized. 5. NOISE IN NMR SPECTROMETERS Noise in an NMR spectrometer is almost always taken as additive, stationary, gaussian and of zero mean. These hypotheses are very convenient for theoretical derivations, but experimental support is scant. In fact, the first demonstration of the gaussian property is quite recent (Rouh et al., 1994). There are some indications that spectrometer noise is not stationary: the uncertainty on peak heights was found to depend on the recovery delay used in a T1 measurement (Skelton et al., 1993). Except in Ernst's classic paper (Ernst, 1966), the effects of non-white noise have never been investigated for any data processing scheme

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in NMR even though 1/f noise, for instance, occurs often and can be easily simulated (Laeven et al., 1984). In fact, the noise is both band-limited (by an input filter of pass-band approximately equal to the spectral width Ws ) and time limited (by the acquisition time Ta). Taking these features rigorously into account leads to computational difficulties. Authors have therefore assumed that the noise is unbounded in time or frequency or both.

5.1 Peak-picking routines A knowledge of the statistical properties of the noise is necessary in order to design automatic peak-picking routines. Commercial software packages usually assume that no signals are present in the first and last 5 or 10 percent of the spectral width, compute the standard deviation of noise in these regions and label as "peak" any feature that rises more than three standard deviations above the baseline. A parabola may then be fit by leastsquares to the central part of the line, to obtain an accurate value of the resonance frequency. Such a procedure will not perform satisfactorily without a clean base-line and can hardly be generalized to multidimensional spectra, because spikes, ghosts, and t1 ridges will give rise to a large number of false resonances. The first defect can be addressed by first removing base-line artifacts as described above or by using algorithms that correct the base-line simultaneously with peak recognition (Chylla and Markley, 1993). The second problem is partially solved using additional information to reject spurious peaks. Statistical criteria may be used (Mitschang et al., 1991, Rouh et al., 1993, Rouh et al., 1994). Information about the width or the symmetry of the cross-peaks can be used (Kleywegt et al., 1990, Rouh et al., 1994). 6. PRECISION OF INTEGRALS In this paragraph, we consider random errors in the integral caused by noise superimposed on the spectrum. This topic was first treated by Ernst (1966) in the case of continuous wave NMR, then independently by Smit and coworkers for chromatography signals (Smit and Walg, 1975, Laeven and Smit, 1985, Smit 1990), by Weiss and collaborators (Weiss and Feretti, 1983, Weiss et al., 1987, 1988, Ferretti and Weiss, 1991) and by Nadjari and Grivet (1991, 1992). 6.1 Frequency domain approach Either of two approaches can be taken to compute the standard deviation of the integral of noise, in the frequency or in the time domain. Let us begin with the more direct method, using the frequency domain. We wish to integrate some line between the limits νinf = ν0 - ∆/2 and νsup = ν0 + ∆/2.The integral of the region of interest is then I(∆) = J(∆) + K(∆), the sum of signal and noise contributions. The expectation value of I is J, because we assume that the noise has zero mean. The standard deviation of I is entirely 2 2 due to the fluctuating quantity K, that is σI = σK . It can be shown (Papoulis 1966, Nadjari 1992) that σK reduces to:

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∆

σ

2 K

=

∫Γ

R

( µ )( ∆ − µ dµ

(1)

−∆

where ΓR is the real part of the complex noise correlation function; it may be obtained from the real part of the Fourier transform of the noise intensity q(t) in the time domain. The effect of windows appears through the function q(t). If we use no filter, q(t) = q 0 times the unit rectangle from 0 to T a. Then ΓR(µ) = q0Tasinc(2πµTa) for white noise and 2 σK = (q0∆/π)Si(∆Ta), where Si stands for the sine integral. The quantity ∆Ta is often 2 much larger than unity, and σK reduces to q0∆/2. 6.2 Time domain approach Another relation may be obtained (Ernst, 1966) by working in the time domain. The integral of the region of interest can be considered as the integral of x(ν) rec(ν - ν0; ∆), where the second factor is a unit rectangle extending from νinf to νsup. The integral of any function in the frequency domain is the value, at t = 0, of its inverse Fourier transform. In the present case: FT {x(ν) rec(ν - ν0; ∆)} = s(t)⊗[∆ sinc(∆t) exp(-2iπν0t)] -1

(2)

where FT {...} stands for the inverse Fourier transform and ⊗ is the symbol for a convolution product; s(t) is the FID (signal plus noise, possibly multiplied by a window function) and sinc(x) = sin(x)/x. The value of the convolution product at t=0 is the integral that we seek. Here again, we are interested in the standard deviation of I or of K, the integral of the noise, which can be obtained by retaining only the noise component b(t) of s(t). In the case of white noise, multiplied by the window function f(t), the variance of K is (Nadjari 1992): -1

σ

2 K

=

∫q

0

f 2 (t ) ∆ 2 sinc 2 ( ∆ t ) dt

(3)

If we do not use a filter, f(t) = 1, the integral of sinc is 1/2, and σK = q0∆/2, identical to the previous result. The standard deviation of the integral is proportional to the square root of the integration interval. It is useful to define the signal to noise ratio for the integral, r I = I(∆)/σK. On the other hand, the (usual) signal to noise ratio is r S = A/σν, if A is the amplitude of the peak and σν the standard deviation of the noise in the frequency domain. For an unfiltered lorentzian line, r S = (A/πλ)/√(q0Ta). The ratio r I/r s is then: 2

rI = rs

π Ta / T2*

arctan( ∆ / 2λ ) ∆ / 2λ 9

2

(4)

The function arctan(x)/√x has a maximum value of 0.8 for x = 1.4. Several points of interest emerge from a consideration of r I or of the ratio r I/r s. (i) Since Ta is often several * times T2 , r I/r s is often slightly larger than unity, showing that peak integration performs somewhat better than measuring the peak height. (ii) There is an optimum integration range, ∆ = 2.8λ. However, this value will entail a large systematic error. (iii) The above formula can be used to determine the signal to noise ratio (i.e. the number of scans) required in order to achieve a given precision on the integral. As mentioned previously, the area of a constant-width signal can often be replaced with advantage by the signal amplitude. It has been reported that peak amplitudes, rather than integrals, lead to better precision in T 1 determinations (Akke et al., 1993), although no explanation was offered. It can be surmised that systematic errors due to a faulty baseline or to very low frequency ("flicker" or 1/f) noise will be less important for amplitude than for area measurements. 6.3 The effects of windows The previous formalism is easily extended to any type of windows, exponential, gaussian or sinusoidal for instance. Theoretical results have often been supplemented by computer simulations. Some authors have chosen to incorporate uncertainty due to noise and systematic error due to a finite integration interval into a single "error term" obtained either by summing the two contributions (Smit and Walg, 1975) or by a mean square combination (Ferretti and Weiss, 1991). The main conclusion to be drawn is that the use of filters is ineffective in increasing the precision of an integral. The main reason is that computing an integral is of itself equivalent to filtering the signal. The transfer function is H(ω) = 1/iω and the high frequency components of noise are effectively strongly attenuated. Another equivalent explanation is that the integral of the whole spectrum is equal to the value of the FID at + time 0 , which is kept constant for all filters. Various windows differ in the way they modify linewidths. Best results will be obtained with procedures that diminish linewidths and prevent peak overlap: the Lorentz-Gauss deconvolution is thus recommended. 6.4 Two-dimensional spectra Some two-dimensional experiments are designed to provide quantitative information as a function of two frequencies. The volume of a NOESY cross-peak depends on the distance between the two relevant nuclei. In the case of the EXSY experiment, the peak volume is a function of the rate constants of reactions carrying the nucleus between the corresponding sites. Relative concentrations can also be obtained from COSY cross-peaks. The two-dimensional signal is made up of a series of FID's, recorded as functions of t 2, for various values of the parameter t1. This array is then interpreted as a sampled version of a function of two variables, s(t1,t 2). Each FID is corrupted by the noise b(t2). This series of noise traces is then interpreted as a sampled realisation of a two-dimensional random

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process b(t1,t 2). Just as in the one-dimensional case, we assume that the noise is stationary, white and gaussian. In fact, in the first dimension (t 1), successive noise samples are uncorrelated, since they are recorded far apart in real time. Further, there is no correlation between the two dimensions. These hypotheses allow the separation of the two dimensions, t1,ν1 on one hand and t2,ν2 on the other. The precision of volume estimation for two-dimensional spectra can then be computed (Nadjari and Grivet, 1992) as a product of two factors, one for each dimension. These factors are identical to the results of the previous paragraph. Thus, when no filters are used: σK = q1δ1q2δ2 / 4 = (q/4W1)∆1∆2 2

(5)

The ∆i are the integration ranges in each direction, q is the noise intensity (in the physical t2 direction), W1 is the spectral width in the first direction, and the qi are the mathematical noise intensities for the two-dimensional random process. Under the hypotheses that pure absorption lineshapes are obtained and that an exponential filter is used in both dimensions, the signal to noise ratio for the peak volume can be written as:

rv =

2A π2

∏

i=1,2

1 ∆i arctan( ) 2λ i qi ∆ i

(6)

A is the peak amplitude, the λi are the half-widths of the peak in each direction, after broadening by the exponential filter. The ratio of r v to r s, the usual signal to noise ratio, is:

4 rv = π rs

∏

i=1,2

∆i λi arctan( ) 2λ i ∆ iδ f i

(7)

where the filter broadening is called δfi. This ratio is not very different from unity, for any reasonable choice of parameters: we verify again that filtering is ineffective in improving the precision of volume integrals. Just as in 1D NMR, windowing can be helpful in reducing overlap and truncation artefacts. 7 LEAST SQUARES METHODS The previous sections have shown that intensity determinations are fraught with many difficulties: the first points of the FID can be corrupted, the last points may be missing, lines may overlap. Further, spectral processing must often rely on subjective judgment by the spectroscopist, in order to choose the baseline or the integration limits. Therefore, there is strong interest in methods that can use incomplete data, provide an objective estimate of the intensity and its probable error, and possibly can be used in an automatic

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mode, with minimal operator intervention. The least squares (LS) fit of either free precession signals or spectra is one such method. The maximum likelihood formalism is also a powerful data processing method (Gelfgat et al.,1993); it has been applied to solid state NMR spectra (Kosarev, 1990). In most implementations, the noise is assumed to be white and gaussian. In that case, the maximum likelihood approach is completely equivalent to least squares. The two formalisms will be considered together.The precision of LS-derived parameters was first investigated by Posener (1974) and Chen et al. (1986), for continuous wave NMR. In the case of FT NMR, one can work either in the time domain ("measurement domain" ) or in the frequency domain. The two approaches are theoretically equivalent. The quality of the fit is more easily assessed in the frequency domain (Martin, 1994) but the model function in the time domain (sum of damped sinusoids) is simpler than in the frequency domain (sum of Abilgaard functions). Further the properties of exponential functions make some algebraic shortcuts possible, thus saving computer time. It is also worth noticing that frequencies, phases and relaxation times enter the model in a non-linear manner, while amplitudes are linear parameters. It is possible to make use of this difference in the algorithm (Golub and Pereyra, 1973), albeit with a much more complex formalism. The minimisation of the sum of squared residuals is usually accomplished with the Levenberg-Marquardt algorithm, although simulated annealing has been used (Sekihara and Ohyama, 1990); this method is attractive in principle, since it seeks the global minimum, however it is computer intensive. Since each line is described by four parameters, an n-line spectrum results in a problem with 4n unknowns. The EM algorithm (Miller and Greene, 1989, Miller et al., 1993, Chen et al., 1993) attempts to fit each line sequentially. On the other hand, it was observed (Montigny et al., 1990) that small artefacts (or lines) distant from the region of interest could be omitted from the fit, without hindering the convergence. This favourable property has been used to perform a frequency selective fit in the time domain, allowing for a large reduction in the problem size (Knijn et al., 1992). In an original approach, Webb et al. 5 (1992) proposed to represent the spectrum as a large (several 10 ) number of lorentzian "spectral elements" of constant amplitude but varying frequency and width. The probable errors on line intensities were very close to the Cramer-Rao lower bounds (see below). The large number of recent reports on the use of least-squares makes an exhaustive survey difficult, but recent reviews are available (de Beer and van Ormondt, 1992, Gesmar et al., 1990). The performance of a least squares method can be assessed by computing the Cramer-Rao lower bounds on the statistical errors of the parameter estimates. In the case of uncorrelated gaussian noise and well-resolved signals, the estimated variance on the signal amplitude is nearly equal to the variance of noise in the time domain (Macovski and Spielman, 1986). Not surprisingly, the method performs better the higher the signal to noise ratio. Thus, it proved possible to retrieve an accurate 13 13 C abundance from a fit to C satellites of a high-sensitivity proton spectrum (Montigny et al., 1990). When the signal to noise ratio is too small, the least-squares method breaks down, the uncertainties rising much above the Cramer-Rao limits (de Beer and von

12

Ormondt 1992). 7.1 The importance of previous knowledge Another approach that can be taken to improve the robustness and precision of an LS fit is the incorporation of previous knowledge. Obviously, the use of an accurate model (exact lineshape, true number of lines) amounts to incorporation of a vast amount of such knowledge, but information on phases, relative intensities, frequency and width of some lines will also lead to a better result (Decanniere et al., 1994). Incorporation of the number of resonances among the unknown parameters has been attempted (Sekihara et al., 1990). The shear size of two-dimensional data sets and parameter spaces have probably prevented the appearance of complete least-squares treatments, except for some exploratory or model problems (Jeong et al., 1993, Miller et al., 1993, Chen et al., 1993). However, the LS method can be put to good use for a less ambitious goal, the determination of volumes of cross-peaks. The method of Denk et al.(1986), incorporated in the program EASY (Eccles et al., 1991), implements an LS fit of a cross-section to a sum of reference line-shapes; these are derived empirically from isolated peaks. This procedure is in effect an LS version of the reference line deconvolution method. Another approach (Gesmar et al., 1994) combines the volume decomposition of Holak et al.(1987) (see §4) with an LS fit to exact line-shapes in the f 1 and f2 dimensions. A very efficient deconvolution of overlapping peaks results, as well as very accurate values of linewidths (Led and Gesmar, 1994). 8 LINEAR PREDICTION Recent years have seen a flurry of papers describing the applications of various linear prediction (LP) algorithms to the processing of magnetic resonance signals. Among the reasons that justify this intense interest are the wide applicability of such methods and their ability to handle truncated signals. It is also very important that LP algorithms do not require any starting values and thus no operator intervention, hence the qualifiers "automatic" or "blackbox" often applied to such methods. Readers can find an introduction to LP in the chapter by Lupu and Todor or in the review by Led and Gesmar (1991); a somewhat older but more general survey (Lin and Wong, 1990) lists more than four hundred references. On the negative side, one must mention the complex theory, the intricate algebra and the heavy computer load. It would take us too far afield to describe the algorithms. Suffice it to say that the LP method models the FID as a sum of damped sinusoids, retrieves first the frequencies and damping factors by a non linear procedure, then determines the amplitudes and phases of each sinusoid by a linear least-squares fit. Since the number of spectral lines is unknown at the outset, the model must incorporate more components than can be possibly present in the FID. This number, often called the order of the model, also influences the resolution (Gesmar and Led, 1988). Further, using more sinusoids than strictly necessary will allow the algorithm to accommodate lines of non lorentzian shape (Pijnappel et al., 1992, de Beer and van Ormondt,1994). The

13

computation load increases roughly as the third power of the order. 8.1 Tests of accuracy and precision of LP Some authors have examined the accuracy and precision of frequency determinations for the simple cases of one or two sinusoids at high signal to noise ratios (Okhovat and Cruz, 1989, Rao and Hari, 1989, Li et al., 1990), but intensities were not considered. LP methods do not perform well for noisy signals (Joliot et al., 1991, Mazzeo and Levy 1991). This means that the algorithm will sometime yield meaningless values of frequencies, and consequently of intensities, the proportion of failures increasing with the noise level (Zaim-Wadghiri et al., 1992). Further, and apparently in contradiction to theoretical results (Li et al., 1990), amplitude values are biased for spectra of low signal to noise ratio (Diop et al., 1992, Koehl et al., 1994a). Simultaneously, the probable errors in the parameters deviate more and more from the optimal Cramer-Rao bounds (Diop et al., 1994). A particular variant of LP, called total least squares (Tirendi and Martin, 1989, van Huffel et al., 1994) recovers a higher proportion of the spectral components. LP is specially useful for truncated signals, such as those recorded in multinuclear experiments; an interesting application to the quantitation of a two-dimensional heteronuclear NOESY map was presented recently (Mutzenhardt et al., 1993).

8.2 Improving the reliability of LP Several approaches may be taken to improve the performance and reliability of LP data processing. It is known that true quadrature detection yields better results than sequential sampling and Redfield pseudo-quadrature (Zaim-Wadghiri et al.,1992). Cadzow (1988) presented a signal conditioning algorithm that has proved very useful for noisy NMR signals (Diop et al., 1992, Lin and Hwang, 1993, Diop et al., 1994a). A computationally more efficient variant has been proposed that is claimed to have superior robustness in the presence of noise (Chen et al.,1994). Regularization techniques may also be successful (Kölbel and Schäfer, 1992, Diop et al.,1994b). Oversampling was shown to improve the robustness of LP, probably because artifacts are thereby reduced (Koehl et al., 1994b). In stark contrast to signal preconditioning techniques, Barone et al. (1994) proposed that LP processing be applied with many different model orders. The resulting frequencies and amplitudes are considered as random variables whose mean values are taken to be the best estimates of the actual spectral parameters. It is claimed that intensities of closely spaced lines are reliably estimated. 9 MAXIMUM ENTROPY METHODS In the previous sections, we examined data processing methods that made full use of the information available about the NMR signal. It may therefore seem odd that we now turn to a data processing algorithm which is, in principle, indifferent to any previous knowledge. However, the maximum entropy method (MEM) has proved successful for

14

NMR data processing (See chapter by K.M. Wright). We refer here to the Jaynes-Skilling MEM (Sibisi et al., 1984) and not to the Burg algorithm, which belongs to the class of linear prediction methods. In mere outline, the MEM method is a constrained optimization algorithm. One starts with a trial spectrum, s(ν), obtains, by inverse Fourier transform, a trial FID f(t), which is compared to the actual data F(t). Many different s(ν) functions will give an f(t) "close" to F(t) in a mean-square sense. One retains that s(ν) which maximises an entropy function, for a given maximum mean square distance between f and F. Lineshape information can be incorporated by multiplying f(t) by some weighting function. The algorithm easily accommodates truncated FIDs and other forms of signal degradation (Laue et al., 1987). It is again a computer intensive method. There is but a single systematic study on the performance of MEM for quantitation purposes (Jones and Hore, 1991). This Monte-Carlo investigation concluded that the probable error on line integrals was comparable to what could be obtained using a conventional least-squares fit. The advantages of MEM therefore do not lie in the precision or accuracy of the resulting intensities, but rather in its automatic or "blackbox" operation and its robustness (Hodgkinson et al., 1993). 10 THE USE OF MODULUS OR POWER SPECTRA Motion of the "sample" can occur during some in vivo NMR spectroscopy experiments, with specially deleterious effects for those comprising echo sequences. Motion causes an attenuation of the FID and random phase shifts, which in turn produce further signal losses through averaging. Therefore, a gain in sensitivity is expected if one averages spectra rather than FIDs. This is born out by experiment (Ziegler and Decorps, 1993): the coaddition of phased spectra (block averaging) improves T 2 measurements. In a similar vein, modulus or squared modulus spectra can be averaged, also with beneficial results. 11 PRECISION OF DERIVED PARAMETERS Some intensity measurements are used directly to determine concentrations, and the uncertainty in peak area translates directly into probable error on the concentration. In contrast, reaction rates or internuclear distances determinations incorporate intensities (or amplitudes) in a complicated and non-linear manner. The question then arises as to what are the probable errors (or confidence intervals) for such derived parameters. The propagation of errors for parameters which are known analytic function of the intensities is given by a well-known if computationally tedious formula (Bevington, 1969). This formalism has been applied to reaction rate constants, as derived from two-dimensional EXSY experiments (Kuchel et al., 1988, Perrin and Dwyer, 1990). In small, rigid molecules, the dipolar relaxation rates are known functions of a few geometric parameters. Thus Trudeau et al. (1993) used the results of an extensive series of Monte Carlo simulations to determine the uncertainties of relaxation parameters in the case -of the phosphite anion (PHO3 ). The relative probable error on the internuclear distance was found to be a linear function of the noise standard deviation. Macura (1994,1995) has

15

derived error propagation formulas for model 3- and 4-spin systems. He used expressions for the partial derivatives of cross-peak volumes presented by Yip and Case (1989). The future will show wether this work can be practically applied to a large biomolecule. Let us first remark that it is usually not possible to derive probable errors on atomic coordinates from crystallographic data, due to the extensive computations required. An average uncertainty for all coordinates can be estimated from a Luzzati plot (Luzzati, 1952). The situation is both less clear-cut and less favourable in the case of NMR. We refer the reader to reviews describing the derivation of structures from NMR data (Clore and Gronenborn, 1991, James and Basus, 1991, Sherman and Johnson, 1993). Using simulated data, several authors have addressed the accuracy and precision question (Liu et al.,1992, Nibedita et al., 1992, Clore et al.,1993, Brunger et al.,1993, Zhao and Jardetzky, 1994, McArthur et al., 1994). We will be content to quote a few salient remarks from these papers. It appears that results (atomic coordinates) are always somewhat biased, whatever the algorithm used. One reason is that the theory relating the most useful data (NOESY cross peak intensities) to interatomic distances is oversimplified: it generally assumes a rigid molecule, with a single correlation time. Another reason is that the information provided by NMR is only of short range, in the form of distances smaller than 6 Å and dihedral angles. Then, the NMR data are not really translated into coordinates by a mathematical transformation. Instead, the macromolecule conformation space is searched for those conformations compatible with the experimental constraints. This step is sensitive to the researcher's personal bias. For instance, was a large enough part of the conformational space sampled ? The end result is usually an ensemble of structures. The quality of a model can be estimated from the R factor for which a possible definition is (Gonzalez et al., 1991, McArthur et al., 1994):

∑ ( [V ] calc ij

Rnmr =

i, j

1/ 6

[ ]

- V ijobs

∑ [V ]

obs 1/ 6 ij

1/ 6

)

(8)

i, j

obs

calc

where Vij and Vij are the observed and calculated cross-peak volumes for the spin pair i,j. There is as yet not much experience in the use of these factors. Taking for granted that NMR will provide a family of structures, it appears that the accuracy depends on the quality and quantity of the data but can hardly be better than 1 Å, while the precision is almost insensitive to the quality of the data and can be of the order of 0.5 Å: it is limited 1/6 by the experimental errors on peak volumes, but only through a highly favourable V dependence. In fact, because of the flexible or disordered nature of a protein in solution, it has been advocated that a statistical description be used (McArthur et al., 1994).

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12 CONCLUSION At the present time, there is no universal data handling procedure that will lead to clean baselines and to accurate and precise integrals for every type of NMR experiment. It is in fact fortunate that the application which is the most "integral-intensive" (NOESY spectra) is also the most tolerant to errors. Semi-automatic (least-squares) or automatic (linear prediction or maximum entropy) methods should be used whenever possible, because they relieve the spectroscopist of much tedious work, can be reproducible and objective, and provide estimates of probable errors. The main argument raised against them is the long computation times involved. This time should be compared to the time spent in preparing the sample, recording the data and assigning the spectra. The ready availability of fast work stations will also make these methods accessible to a growing number of spectroscopists. 13 REFERENCES Abilgaard F., Gesmar H., and Led J.J., 1988. Quantitative analysis of complicated nonideal Fourier transform NMR spectra. J. Magn. Reson., 79, 78-89. Akke M., Skelton N. J., Kördel J., Palmer A.G. III, and Chazin W.J., 1993. Effects of 15 ion binding on the backbone dynamics of calbindin D 9k determined by N NMR relaxation. Biochemistry, 32, 9832-9844. Balacco G., 1994. New criterion for automatic phase correction of high resolution NMR spectra which does not require isolated or symmetrical lines. J. Magn. Reson., A110, 19-25. Bardet M., Foret M.-F., and Robert D., 1985. Use of the DEPT pulse sequence to 13 facilitate the C NMR structural analysis of lignins. Makromol. Chem., 186, 14951504. Barone P., Guidoni L., Ragona R., Viti V., Furman E., and Degani H., 1994. Modified 31 Prony method to resolve and quantify in vivo P NMR spectra of tumors. J. Magn. Reson., B105, 137-146. Barsukov I.L. and Arseniev A.S., 1987. Base-plane correction in 2D NMR. J. Magn.Reson., 73, 148-149. Becker E.D., 1969. High resolution nuclear magnetic resonance. Academic Press, NewYork. Beer R. de and Ormondt, van D., 1992. Analysis of NMR data using time domain fitting procedures. NMR basic principles and progress, 26, 201-248. Beer R. de and Ormondt, van D., 1994. Background features in magnetic resonance signals, addressed by SVD-based state space modelling. Appl. Magn. Reson., 6, 379390. Bevington P.R., 1969. Data reduction and error analysis for the physical sciences. McGrawHill, New York. Bodenhausen G., Vold R.L., and Vold R.R., 1980. Multiple quantum spin-echo

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