Simplified and Efficient Method of Computing Generalized Two-Dimensional Correlation Spectra
JEAN-RENI~ B U R I E Section de Biodnergdtique, CEA Saclay, 91191 GO" sur Yvette Cedex, France
A very powerful generalized two-dimensional correlation method applicable to various types of spectroscopy involving signals depen- dent on various physical var iables (e.g., t ime) was recent ly devel- oped and reported. In order to obtain an easy computational pro- cedure , the effect of the practical limitation of the data along the time axis to N is considered. This in fo rmat ion gives rise to a new fo rmula for the disrelation spec t rum. This modified disrelation spec t r um is shown to still be useful in the differentiation of over- lapping peaks, even when N is small.
Index Headings: Correlation spectra; Two-dimensional spectrosco- py; Spectroscopy technique.
I N T R O D U C T I O N
The two dimensional (2D) correlation method is a powerful tool used in the interpretation of t ime-dependent IR spectra. Since the first description of the concept in 1986] it has been the subject of increasing interest. 2 It was first applied to study a system subjected to external perturbations with sinusoidal waveforms. 3,4 Recently, a generalized 2D correlation scheme, applicable to various types of spectroscopy dependent on any physical vari- able, has been developed? (In the following discussion, the term " t ime" axis will be used to denote the depen- dency on any physical variable, since the conclusions are not dependent on the physical nature of this variable.) We will present below the development of a new formula leading to a much simplified computational procedure. Furthermore, in many experiments, only a limited num- ber of data are available along the " t i m e" axis. We will also study the effects of using only a very small number, N, of data (N < 20) along the " t i m e" axis on the results which can be expected from this method.
D I S C U S S I O N
Two-Dimensional Correlation Spectra. The 2D cor- relation analysis was first developed in the particular case of a study of the effect of a sinusoidal waveform pertur- bation on a system. 4 In this case, if the external pertur- bation applied to the studied system (e.g., oscillatory ten- sile strain on a polymers mixture film 3) is
e(t) = ~sin(oJt), (1)
y(v,t), the variation of the intensity (e.g., IR intensity), takes the form
y(v,t) = ¢9(v)sin[0~t + [3(v)]. (2)
It is important to note that in Eq. 2 the phase angle [3 is dependent on the wavenumber v.
The dynamic IR cross-relation function between two
Received 14 July 1995; accepted 10 February 1996.
wavenumber values, v~ and v2 (where T is the correlation period, generally assumed to be very long, i.e., T - ~ ~), defined as
1 fTya X~,2(r) = l i m - y(ul,t), y0,>t + r)dt (3) T--~ T d T/2
then reduces to
X i , i ( ' T ) = (I)(u1,I)2)COS(.0T + xtr(vl,v2)sinoo'r (4)
where the synchronous ~(vl,v2) and asynchronous (or quadrature) qt(vl,v2) correlation intensities are given by
(I)('1)1,'1)2) = 1 /~y( l J l ) " y( ' l )2) COS[[3(1)1) - - [3(-I)2) ] ( 5 )
and
qt(v~,v2) = akY(vl)..Y(v2)sin[[3(vO - [3(v0]. (6)
The 2D correlation analysis consists of the interpreta- tion of these intensities. 4 The synchronous 2D correlation spectrum characterizes the coherence between two sig- nals. It is constituted by autopeaks located on the diag- onal and cross peaks located at off-diagonal positions. Cross peaks present coupled changes (simultaneous in- crease or decrease) in spectral intensities. The asynchro- nous 2D correlation spectrum consists of only cross peaks. They arise f rom signals (at different wavenumber values) having different rates of change.
When the " t ime" dependency of y(v,t) is not sinusoi- dal, it has been shown 5 that a 2D correlation analysis is still possible by first applying a Fourier transform to y(v, t ) . The s y n c h r o n o u s ~(v~,v2) and a s y n c h r o n o u s • (v~,v2) 2D correlation spectra are then obtained as fol- lows:
1 Yl(O)) - Y~(o))dw ~(ut,v2) + iV(u1, v 2) - r r .T
(7)
where Y,(00) is the t ime-domain Fourier transform of y(vl,t), and Y~(t0) is the conjugate of the Fourier trans- form of y(va , t ) :
Yi(w) = y(vl, t ) . e -i~t dt
£7 Y](w) = Y(/"2, t)" e +i°°t dt. (8)
The 2D correlation spectra can thus be directly ob- tained f rom t ime-dependent spectra having an arbitrary waveform, as long as the Fourier transform of the time dependence can be calculated for the intensity changes at each wavenumber. 5 The synchronous and asynchronous 2D correlation spectra still have the same characteristics
as in the case of a sinusoidal external perturbation. This 2D analysis can also be performed for signals dependent on physical variables other than timeP
The use of Eq. 7 requires the t ime-domain Fourier transform of y(v,t). However, it has also been shown 5 that the 2D synchronous correlation spectrum can be calcu- lated as
• (v~, v2) = y(v~, t). y(v2, t)dt (9) T/2
and a special 2D asynchronous spectrum called the two- dimensional disrelation spectrum, A(vl,v2), considered as an approximation of the asynchronous 2D correlation spectrum, can then be calculated f rom a set of synchro- nous correlation intensities as
]A(vl,v2) ] = ~ / ( I ) ( l ) l , V l ) ' ( I )(P2,1)2) - - (I)2('1)1,'1)2). (10)
The sign of the disrelation spectrum is chosen in such a way that
The disrelation spectrum is very efficient as a means to differentiate overlapping peaks. In this case, only the ab- solute value is necessaryP With Eqs. l0 and l l, a 2D correlation analysis can thus be performed without Fou- rier transformation. However, the acquisition of the syn- chronous 2D correlation spectrum ~(v~,v2) still requires integration over the " t i m e" domain, T. We will examine below the possibility of obtaining formulas more easily converted in a computat ional procedure.
Two-Dimensional Correlation Spectra with N Data along the " T i m e " Axis. From a practical point of view, only N data are experimentally available along the " t ime" axis. ~(v~,v:), given in Eq. 9, is then replaced by an approximation ~(v~,v:) defined by
- - - y(u I, O.y(v2 , ti). (13) - ~ - ( V l ' P2) N 1 i=1
When N --~ ~, Eq. 13 reduces to Eq. 9, and ~(v~, v2) = ~(Vl, v2). With Eq. 13, it is possible to obtain A(v~, v2), an approximation of IA(v~, v2)[, by using ~(u~, v2) instead of ~(Vl, v2) in Eq. 10. It can then be shown (see Appen- dix) that A(u~, v2) can be obtained as follows:
A(v,, v9
1 , ; 2 ~ {y(v,, t3 ' y(v2, t 2) - y(v,, tj). y(v2, t,)} 2 ) i=l j>i
(14)
which is easily used for a computational procedure. When only discrimination of overlapping peaks is re-
quired in a 2D correlation analysis, Eq. 14 provides an easy computational way of obtaining such information.
Case of Smal l N. Because of experimental conditions, it might happen that only a very small number of data along the physical variable axis are available. 6-8 The question then arises whether it is possible to use 2D cor- relation spectra with only very few data (e.g., N < 20)
along this axis since the Fourier transform normally re- quires a sum from 0 to +~ . Similarly, the question whether _~(vl,v2) and A(vj,v2) still bear information when N is small (--<20) arises. We will examine below whether it is possible to obtain useful information from these 2D correlation spectra.
An important characteristic of the synchronous 2D cor- relation spectrum is that a large number of data along the " t ime" axis are normally needed to obtain a clear dis- tinction between peaks that are correlated together and peaks that are not correlated. This characteristic is easily understood if one considers the case of two peaks (cen- tered at VA and vB) of low intensities that are correlated together and a peak (centered at Vc) that is not correlated with them but has a high intensity. In the case of a limited number of data, the integrals are replaced by the sums
and as Vti and Vtj, y(vc,ti) >> y(VA,tj) and y(voti) >> y(vB,(j), then ~(VA,VC) >> ~(VA,VB) and ~(VB,VC) >> .~_("DA, ])B) •
The conclusion is, then, that peak C seems correlated with peaks A and B, while these latter peaks do not seem to be correlated, which is not the case by definition.
In contrast, the 2D disrelation spectrum's characteristic feature [i.e., when two peaks (centered at VA and %) are correlated, the value of the 2D disrelation spectrum A(VA,VB) is zero, while when two peaks (centered at v A and VB) are not correlated, the value of the 2D disrelation spectrum A(VA,VB) is not zerol is conserved in A when only a few data along the " t ime" axis are used.
Indeed, if two peaks centered at VA and Vg are corre- lated, then Vti, y(vB,ti)=a • y(vA,t i) and
_~_(VA,VB) = (Y.' ( I ) (VA,VA) ( 1 6 )
~(VB,VB) = Od'~(VA,VA). (17)
Thus, even with a limited number of data,
A(VA,VB) = ~V/_~_(VA,I)A) " a 2" ~ ( I ) A , P A ) - - [0£" _~_('I)A,PA)I 2 = 0 (18)
which is also easily found with Eq. 14. If the peaks centered at u A and vB are not correlated, it
appears in Eq. 14 that A(v A, VB) is obtained by summing only the square of the differences between the product of the intensities at different " t ime" . As no correlation ex- ists between the peaks at VA and vB, the intensities be- tween time i and j will have evolved differently and the two products will not cancel respectively in the differ- ence. [Example: the peak centered at VA decreases rapidly and the peak centered at VB decreases slowly
y(VB, t) > y(v A, t) f o r j > i,
y(VB, ti) y(va, ti)
and y(VA, t~). y(VB, t) > y(VB, tl)" y(VA, t), which makes
862 Volume 50, Number 7, 1996
C //•• .\
/ ' \ D /
A B / /
//' ' \ / '
Spectral Variable
C
A D
Spectral variable
FIG. 1. Theoretical spectrum obtained as the sum of the four peaks A, B, C, and D of Gaussian shape. The choice was made so that strong peak overlapping exists.
FIG. 2. Representation of the exponential decay of the four peaks of Fig. 1 as a function of time. Peaks A and C, and peaks B and D, have the same rate of decay, respectively.
the difference nonzero.] The sum, as it is a sum of squares, will then increase rapidly as N increases, which leads t o A( / /A, /}B) ~ 0.
When only a few data are available along the " t ime" axis, the synchronous 2D correlation spectrum ~(Vl, vz) is difficult to use, while A(vl, v2) is still useful in the differentiation of overlapping peaks.
Exponential Decay. In order to confirm the conclusion drawn above, we will consider the particular case of four peaks (A, B, C, and D), which follow an exponential decay:
y(v,t) = A(v). e -k~)t. (19)
We will consider A and C correlated together [i.e., k(VA) = k(vc)], B and D correlated together [k(vB) = k(vo)], and k(vA) ~ k(vB). Figure 1 represents the four peaks at t = 0. These peaks have been chosen in such a way that some overlap exists, as the main use of the simplified disrelation spectrum is in discriminating overlapping peaks. Figure 2 represents the time dependency of these peaks.
C
f
FIG. 3. Synchronous CI~('I)i,V2) 2D correlation spectrum obtained with the use of Eq. 13 withThe data of Fig. 2 (N = 20 time data points).
Figure 3 represents ~(vl,v2) obtained with 20 data points along the time axis. In this figure, it appears that no clear conclusion can be drawn concerning the corre- lation between the different peaks. The autocorrelation peak corresponding to peak A is hardly observable be- cause of its low amplitude and its fast decay.
Figure 4A represents A(vl,v2) obtained with 20 data along the t ime axis. In the case of exponential decay, it is possible to obtain mathematical formulae for dP(vl,v2) with T ---) ~.5 Figure 4B represents A(v~,v2) obtained with these formulae. It is almost impossible to detect any dif- ferences between A and A. Even peak A, which has a low intensity and a fast decay, gives rise to very clear cross peaks. The conclusion based on these spectra is that peaks A and C are correlated together and are anticor- related to peaks B and D, which are correlated t oge the r - - which is the case by definition. It is thus possible to use A(vl,v2) obtained with Eq. 14 instead of A(v~, v2) in this case to differentiate overlapping peaks.
C O N C L U S I O N
The 2D correlation method can easily (from a com- putational point of view) be used in order to resolve over- lapping peaks using the disrelation spectrum A(v~,v2) de- fined by Eq. 14. In the particular case where a few data are available along the " t ime" axis, it appears that the disrelation spectrum A(Vl,V2) obtained with Eq. 14 can be used, while the 2D synchronous correlation spectrum might require a larger number of data to give clear con- clusions. The reliability of this method with the use of A(vl,v2) on a complicated experimental case (differenti- ation of two almost identical ytterbium binding sites in a protein) with a very small number of data points (N = 4) has been demonst ra ted)
APPLIED SPECTROSCOPY 863
S
D
A
Fro. 4. (A) A(v,v2) 2D disrelation spectrum obtained using Eq. 14 with the data of Fig. 2 (N = 20 time data points). (B) A(Vl,V2) 2D disrelation spectrum rigorously calculated for the peaks of Fig. 1 when the time of integration T---~.
ACKNOWLEDGMENTS
The author thanks C6cile Roselli, Tony Mattioli, and Alain Boussac for their help and for useful and stimulating discussions.
1. I. Noda, Bull. Am. Phys. Soc. 31, 520 (1986). 2. I. Noda, A. E. Dowrey, and C. Marcott, Appl. Spectrosc. 47, 1317
(1993). 3. I. Noda, J. Am. Chem. Soc. 111, 8116 (1989). 4. I. Noda, Appl. Spectrosc. 44, 550 (1990). 5. I. Noda, Appl. Spectrosc. 47, 1329 (1993). 6. T. Nakano, S. Shimida, R. Saitoh, and I. Noda, Appl. Spectrosc. 47,
1337 (1993). 7. K. Ebihara, H. Takahashi, and I. Noda, Appl. Spectrosc. 47, 1343
(1993). 8. C. Roselli, J.-R. Burie, T. Mattioli, and A. Boussac, Biospectroscopy
1, 329 (1995).
APPENDIX
D e t e r m i n a t i o n o f the d i s re la t ion spec t rum A(vl,v2) in the case o f a l imi ted n u m b e r o f po in t s a long the " t i m e " axis p rocedes as fo l lows :
y is def ined as a func t ion o f v and t. The s u m m a t i o n ind ices are n a m e d i, j , k, and l. A c c o r d i n g to Noda , 5 the synch ronous 2D cor re la t ion spec t rum is g iven by
l ( "2 = Y(/11, t) 'y(/12, t) dt O(/11, /12) T .,-TJ2 (A1)
and the 2D d i s re la t ion spec t rum by
1A(/11, z'2)l = X/qb(/11, v , ) ' ~ ( u 2 , /12) - IfflD2(/11, //2)" (A2)
In the case o f a l imi ted n u m b e r o f poin ts a long the " t i m e " axis , the synch ronous co r re la t ion spec t rum can be a p p r o x i m a t e d by ~(v~ , /12), def ined as
