Three-Way Decomposition and Nuclear Magnetic Resonance

Transcription

1 Three-Wy Decomposition nd Nucler Mgnetic Resonnce Mrtin Billeter 1 nd Vldislv Orekhov 2 1 Biochemistry nd Biophysics, Göteorg University, Box 462, Göteorg, Sweden mrtin.illeter@cp.gu.se 2 Swedish NMR Centre t Göteorg University, Box 462, Göteorg, Sweden orov@nmr.se Astrct. Nucler Mgnetic Resonnce (NMR) is widely used tool in functionl nd structurl genomics for the study of three-dimensionl structures of proteins. The experimentl dt otined y this method re multidimensionl spectr consisting of out 10 7 dt points. We demonstrte tht Three Wy Decomposition (TWD) provides n inherently suitle tool for the nlysis of these spectr. We pply here TWD for the first time to NOESY-NOESY spectrum, which in terms of numer of signls is mong the most complex spectr. The ppliction shows tht the three-dimensionl NMR spectr cn e fithfully descried y the components resulting from TWD, yielding mong other dvntges dt compression fctor of over 100. The inherent reltion etween NMR nd TWD is demonstrted on the NOESY-NOESY spectrum y deducing the TWD model from the mthemticl description of the NMR experiment. Applicility of TWD to vrious types of NMR spectr, the use of sprse experimentl dt sets in order to reduce instrument time nd other spects of the nlysis re discussed. 1 Introduction The current contriution concerns the processing nd nlysis of lrge dt sets (~10 7 individul mesurements) tht rise from nucler mgnetic resonnce (NMR [1]) experiments when pplied to iologicl mcromolecules, in prticulr proteins. The importnce of the prolem cn e illustrted on the one hnd y the interest in deciphering the humn genome, nd on the other hnd y the lrge NMR investments of phrmceuticl compnies for the development of new medicines. The lgorithm proposed here is three-wy decomposition (TWD), which hs n inherent reltion to multidimensionl NMR dt sets s shown elow. The TWD lgorithm hs een presented erlier s such [2] nd more recently s tool for the nlysis of NMR spectr [3 5]. Here, we complement erlier, in prt more technicl pulictions [6] y demonstrting the ppliction of TWD to the most complex NMR spectrum so fr (in terms of numer of signls). In the rest of this introduction, some generl spects of proteins nd of NMR re mentioned. Proteins nd DNA re very lrge molecules (mcromolecules) tht form the sis of life. DNA molecules serve mostly s medium for informtion storge, nd worldwide efforts to red the entire genomic informtion re currently eing completed. P.M.A. Sloot et l. (Eds.): ICCS 2003, LNCS 2657, pp , Springer-Verlg Berlin Heidelerg 2003

2 16 M. Billeter nd V. Orekhov Genes from the most importnt prt of this informtion: they encode the uilding plns for proteins. Proteins ply crucil role in lmost ny metolic process, nd the study of the function of these gene products is referred to s functionl genomics nd enjoys t present enormous ttention. Typiclly, proteins exert their function y intermoleculr interctions, nd thus n understnding of the function of protein requires knowledge of its three-dimensionl (3D) structure. In ddition, the description of the internl dynmics of these molecules my often provide significnt clues when explining function t moleculr level. Besides fundmentl questions out life, numerous prcticl pplictions rely on knowledge of structure nd dynmics of proteins, n exmple eing the design of cliniclly ctive regents ( drug discovery ). Proteins re molecules tht consist of thousnds of toms. Experimentl methods for 3D structure determintions tht yield coordintes for ech tom must therefore provide lrge mount of dt. NMR is one of few methods tht presently cn provide complete structures of mcromolecules t tomic resolution, nd it is the prime experimentl method for the chrcteriztion of internl moleculr dynmics. NMR provides multidimensionl spectr with typicl sizes of severl million dt points. Their nlysis offers significnt computtionl chllenge. This is further ccentuted y vriety of experimentl rtifcts, y limited ccess to the expensive NMR equipment nd y prolems relted to the vilility of sufficient nd stle protein smples. The purpose of this contriution is twofold. TWD, n lgorithm for the nlysis of three- or higher dimensionl mtrices is pplied to NMR spectrum of type NOESY- NOESY [7]. This ppliction llows reltively simple illustrtion of the intimte reltion etween 3D NMR spectroscopy nd TWD y deducing the model ssumption of TWD directly from the description of the NMR experiment. The discussion includes nlyses of other types of NMR dt sets, including spectr with informtion on 3D structure [3], on moleculr dynmics [4] nd on intermoleculr inding s used in "drug discovery" [5]. It thus shows the wide pplicility of TWD in the field of high-resolution NMR. Other issues ddressed include spectrum reconstruction, dt compression, nd the hndling of sprse dt llowing significnt svings of instrument time, or lterntively optimiztion of spectrl resolution nd sensitivity. 2 Methods 2.1 Three-Wy Decomposition Three-wy decomposition (TWD) is mthemticl concept for the pproximtion of three- or higher dimensionl mtrix y lower-dimensionl mtrices (often onedimensionl) [2]. TWD hs een introduced s tool for dt nlysis in the erly seventies under vrious nmes such s prllel fctor nlysis or cnonicl decomposition. Theoreticl considertions concerned notly questions of uniqueness of optiml pproximtions nd convergence ehvior. Applictions include dt compression, chemometrics nd more recently the processing of multidimensionl NMR (nucler mgnetic resonnce) spectr. While the use in chemometrics, e.g. the nlysis of fluorescence dt, concerns decomposition of mtrices of out 10 3 elements into less thn ten components, dt compression nd NMR pplictions involves dt

3 Three-Wy Decomposition nd Nucler Mgnetic Resonnce 17 mtrices exceeding 10 7 elements nd requiring hundreds of components. TWD cn e formulted s follows. Given mtrix S with elements s ijk (i=1..i, j=1..j, k=1..k), find numers nd vectors F1, F2 nd F3 with elements f1 i, f2 j nd f3 k, respectively, such tht the norm S S ( F1 F2 F3 ) 2 (1) ecomes miniml. The sum in this expression represents the fundmentl model ssumption of TWD: Direct products of one-dimensionl vectors re sufficient to descrie ll fetures of high-dimensionl mtrix. In the following we refer to S s the (input) spectrum nd to the entities in the sum over s mplitudes nd shpes F1, F2 nd F3, while the summtion terms re clled (output) components. The mplitudes result from the use of normlized shpes F1, F2 nd F3. The summtion index runs over the numer of components used for the decomposition. The rnge for this index depends on the type of ppliction. For typicl 3D NMR spectr, which consist of severl millions of dt points, it is sufficient to use few hundred components. Consequently, description of the spectrum y components my yield significnt compression of the dt. The redundncy present in mny types of NMR spectr my thus e used to sve experiment time when solving modified prolem [6]: Minimize G œ [S S ( F1 F2 F3 ) ] 2 (2) where the mtrix G contins elements g ijk ³ {0,1} tht indicte the sence or presence of dt point in S, nd œ descries element-wise multipliction of mtrices. The product G œ S is used to denote sprse mtrix S s input for the decomposition. Note tht while the input sprse dt mtrix S lcks mny entries, the shpes F1, F2 nd F3 representing the output of TWD re complete, llowing to reconstruct full mtrix description. As shown lter, omission of elements of S results in svings of NMR experiment time, which is relevnt issue when considering the price of instruments (severl million dollrs) nd the durtion of experiments (for prcticl resons usully limited to out one week). An lterntive formultion of the gins chieved with TWD is tht with given totl experiment time one my improve spectrl resolution. Furthermore, compred to conventionl methods sed on Fourier trnsform, which requires uniform dt smpling, the use of sprse dt mtrices llows for optimized smpling nd provides improved spectrl sensitivity. 2.2 Nucler Mgnetic Resonnce NMR is sed on the interction of spins with mgnetic fields 1. Avoiding physicl explntions s much s possile, the following description concentrtes on spects required elow when showing the intimte reltion etween 3D NMR nd TWD using NOESY-NOESY spectr [7]. We need only consider the spins of the nuclei of hydrogen toms (i.e. simple protons). A typicl protein contins few hundred hydrogen toms nd thus lrge numer of proes for NMR mesurements. In strong, sttic mgnetic field, spins ssume preferred sttes (orienttions), corresponding to different mgnetiztions. An nlogy of this effect is the lignment of compss needle in the mgnetic field of the erth. In NMR experiment, short pulses of electromgnetic ir-

4 18 M. Billeter nd V. Orekhov rdition cn mnipulte the spin mgnetiztion. For exmple, pulses cn e designed to flip the mgnetiztion from n initil stte prllel to the strong sttic mgnetic field to perpendiculr orienttion. Mgnetiztion perpendiculr to the mgnetic field will precess round the field direction. A third effect tht we need to mention is the exchnge of mgnetiztion etween nery spins. This trnsfer occurs only cross short distnces (<0.5 nm). It is populr to view NMR experiments s series of steps clled "pulse sequence", nd the individul steps cn e chrcterized y set of "recipes" [1]. The following is simplified nd incomplete list of recipes, reduced to meet our purposes. Spin mgnetiztion is descried y vector I. () The initil (equilirium) orienttion of ll spin mgnetiztion is long the sttic mgnetic field, which defines the direction of the z-xis. Thus only the components Iz re different from zero. () A 90x-pulse rottes the mgnetiztion y 90 round the x-xis: Iz Iy. (c) Precession with time t of mgnetiztion perpendiculr to the z-xis occurs with frequency W I chrcteristic for ech spin: Iy Iy cos(w I t) + Ix sin(w I t). (d) Different spins (leled nd ) oriented long the z-xis nd seprted y short distnce interct with ech other with mixing efficiency m : Iz m Iz. (e) Oservtion long the y-xis yields sclr (mplitude),, s the sum of contriutions from ll spins: Sg I g, where g enumertes ll spins. In 3D NMR experiment, successive mgnetiztion trnsfers etween three different spins I, J nd K provide signl tht crries informtion on ll three chrcteristic frequencies W I, W J nd W K. This signl cn e reported in 3D spectrum t the position given y the coordinte triple (W I, W J, W K ). The complete spectrum will include signls for ll possile mgnetiztion pthwys, i.e. for ll spin triples for which the trnsfer defined y the experiment tkes plce. 2.3 Experiments A three-dimensionl NOESY-NOESY 7 experiment ws collected t 30 o C on smple with the protein uiquitin (1 mm in 50 mm potssium phosphte uffer ph 5.8, H2O/D2O 9:1). The experiment required 144 hours on 600 MHz Vrin Unity Inov spectrometer (mixing times: 100 ms for oth NOESY steps). The experimentl dt formed mtrix of 120*180*608 complex dt points (sizes in the dimensions t1, t2 nd t3). The spectrum ws Fourier trnsformed in ll dimensions yielding dt mtrix with 240*810*901 entries. For n illustrtion, section of this spectrum with mtrix size 240*28*901 ws selected (rnge ppm in the second dimension), nd this su-spectrum ws sujected to TWD using 27 components. 3 Results 3.1 Appliction to NOESY-NOESY Spectr The NOESY-NOESY su-spectrum descried in Methods ws chosen s input to TWD; plne perpendiculr to the second dimension is shown in Fig. 1 (left side).

5 Three-Wy Decomposition nd Nucler Mgnetic Resonnce 19 Crowded groups of signls long the horizontl nd verticl xes correspond to digonl plnes in the 3D mtrix. Genuine 3D peks connecting three different hydrogen toms, nd crrying structurl informtion, re locted outside ny digonl plnes. The verticl stripe of peks visile in the middle of the spectrum is n rtifct cused y the strong signl of wter. The min result of the decomposition with 27 components is shown in Fig. 1 (right side), where reconstruction of the plne from the left side is presented. The reconstruction corresponds to the sum of ll 27 components ccording to expression (1). All importnt fetures of the originl spectrum, including strong nd wek signls, re preserved in the reconstruction, thus illustrting the good fit of the model given y expression (1) to the experiment. Note tht the numer of used components is severl times smller thn the numer of peks in the su-spectrum, which totlly contins 28 plnes like the one shown in Fig. 1. This justifies the vlidity of the model nd proves the redundncy of the experimentl dt. A mesure of the ltter is given y the compression fctor. The originl dt consisting of 240*28*901 dt points is pproximted y the ( )*27 prmeters of the model. The compression fctor, given y the rtio of the two vlues, is s high s 185. This compression cn e used directly to simplify hndling of dt set of hundreds of megytes. Alterntively, redundncy of the dt cn e used to sve experiment time y pplying sprse detection. Fig. 1. Selected plne long the F1 nd F3 dimensions from 3D NOESY-NOESY spectrum (left side). The strong verticl line of signls in the middle is n rtifct cused y strong signls from wter. For the decomposition, stripe covering this rtifct ws leched out. Other strong signls long lines descrie digonl plnes of the 3D spectrum. Genuine cross peks defined y frequencies from three hydrogen nuclei lie off these digonls, together with weker noise peks. A su-spectrum with 28 plnes (including the one shown here) ws decomposed using 27 components. Reconstruction of this su-spectrum y summing the resulting components ccording to expression (1) yields for the selected plne the result shown on the right side. In ddition to significnt dt compression, noise is strongly suppressed. 3.2 Reltion etween 3D NMR nd TWD TWD nd 3D NMR spectroscopy re intimtely relted, s cn e shown y deducing the model ssumption for TWD (expression (1)) from the description of 3D NMR experiments. This connection cn e shown on generl level; however, in the following we use for this purpose the experiment presented ove, NOESY-NOESY. Fig.

6 20 M. Billeter nd V. Orekhov 2A schemticlly summrizes this experiment. Initilly, ll hydrogen nuclei re irrdited; these nuclei re referred to s spins I nd enumerted with the index. Mgnetiztion is then trnsferred to neighoring nuclei, clled J with the index, provided tht the distnce I to J is short. A second trnsfer step rings the mgnetiztion to neighors of J, referred to s nuclei K with the index g. The individul steps of the experiment together with description of the resulting spin sttes nd the "recipes" used for the steps (see Methods) re listed in Tle 1. The initil sttes with mgnetiztion vectors prllel to the mgnetic field, Iz, re rotted to the y-xis y the first pulse. During time t1 the spins process round the z-xis, so tht their components long the y-xis re given y cos(w I t1). After further pulse, the mgnetiztion cn e trnsferred to neighoring spins J in step clled "mixing". Since this trnsfer occurs only over short distnces, the sum over is reduced to smll numer of terms where the trnsfer efficiency m differs significntly from zero. We cn revite the sum over s function F1 ; note tht this function is unique for ech spin J, nd it depends on the precession time t1. Repeting the sequence pulseprecession-pulse-mixing we hve leled the mgnetiztion in ddition with the frequency of the nucleus J, nd trnsferred the mgnetiztion to nuclei referred to s K g. The short nottion F2 for cos(w J t2) cnnot include the sum over since F1 depends s well on this summtion index. F2 is function of the precession time t2. A finl pulse-precession comintion lels the mgnetiztion with the frequencies of the nuclei K g. The oservtion, which yields sclr given y n mplitude, is the sum of ll existing mgnetiztion. Thus, we cn define third function F3 tht collects the terms in the summtion over g, nd tht depends on t3. Fig. 2. (A) Schemtic summry of the NOESY-NOESY experiment. Spheres denote individul spins with different shding for the spins I, J nd K (see text) nd Greek lettering for their enumertion. Arrows indicte mgnetiztion trnsfers. (B) Fourier trnsformed shpes corresponding to one component, J 1, nd I nd K spins tht interct with it. (C) Schemtic result of the direct product of the Fourier trnsformed shpes from (B) ccording to expression (1).

7 Three-Wy Decomposition nd Nucler Mgnetic Resonnce 21 Tle 1. NOESY-NOESY experiment nd connection to TWD Step Spin stte description 1 Recipe 2 Comments initil stte S Iz () enumertes hydrogens 90x-pulse S Iy () 90 rottion round x-xis t1-evolution S Iy cos(w I t1) (c) precession during t1 3 90x-pulse - S Iz cos(w I t1) () 90 rottion round x-xis mixing - S Jz S m cos(w I t1) (d) short distnce interction 90x-pulse - S Jy F1 () 90 rottion round x-xis define F1 =S m cos(w I t1) t2-evolution - S Jy cos(w J t2) F1 (c) precession during t2 3 90x-pulse S Jz F2 F1 () 90 rottion round x-xis define F2 =cos(w J t2) mixing S S g Kz g m g F2 F1 (d) short distnce interction 90x-pulse S S g Ky g m g F2 F1 () 90 rottion round x-xis t3-evolution S S g Ky g g m g cos(w K t3) F2 F1 (c) precession during t3 3 oservtion S F3 F2 F1 (e) define F3 =S g m g cos(w K gt3) In the present simplified explntion of n experiment, terms re dropped fter some of the steps; these terms re not essentil for the outcome of the resulting spectrum. See Methods for list of the recipes used. Terms in sin(wt) re not ffected y the following step nd therefore not further considered. The finl dt recorded forms mtrix with the three xes given y the precession times t1, t2 nd t3. To this end, discrete vlues for these three precession times re selected. The pulse sequence of Tle 1 is repeted for ll comintions of selected t1 nd t2 vlues. At the end of ech pulse sequence, oservtion occurs for ll selectedvlues of t3. With the resulting 3D grid of oservtions, the functions F1, F2 nd F3 ecome vectors tht re equivlent to the shpes used in expression (1). Components re enumerted y the index, nd ech component corresponds to one spin J. The shpes consist of (discrete) cosine function with the chrcteristic frequency W J in F2, nd sum of cosines for the chrcteristic frequencies of the spins I in F1 nd K g in F3 g tht interct with the spin J s follows: F1 (t1) = S m cos(w I t1) F2 (t2) = cos(w J t2) F3 (t3) = S g m g cos(w K gt3) (3)

8 22 M. Billeter nd V. Orekhov Fourier trnsform of these three discrete functions yields the 1D-spectr of Fig. 2B (for one of the -nuclei of Fig. 2A). The direct product of these three 1D-spectr provides the 3D spectrum of Fig. 2C, nd summtion over the 3D spectr of ll nuclei J corresponds to the experimentl dt fter Fourier trnsform in ll three dimensions ccording to expression (1). Anlysis of 3D NMR dt set y TWD thus provides () wy of dt compression nd () seprtion of the suset of signls tht elong to individul nuclei J. 4 Discussion 4.1 Applicility of TWD to NMR Dt of Proteins The ove derivtion of the model expression (1) for TWD for the NOESY-NOESY illustrtes for prticulr 3D NMR experiment the close reltion etween NMR dt sets nd TWD. Other exmples re heteronucler NMR experiments tht rely on dditionl nuclei with spins ½, nmely the isotopes cron-13 ( 13 C) nd/or nitrogen-15 ( 15 N). Thus, the ppliction of TWD to 3D NMR spectr ws first demonstrted for nother experiment, 15 N-NOESY-HSQC 3. The connection etween NMR nd TWD is, however, much more generl; in fct it holds lso for dt sets tht re not strictly the result of 3D NMR experiments. Since no requirement is mde on the form of the shpes F1, F2 nd F3 of expression (1), one my consider ny type of modultion. Two such exmples hve een presented erlier [4,5]; oth rely on the sme simple two-dimensionl NMR experiment yielding 15 N-HSQC spectrum. In this spectrum, every hydrogen-nitrogen pir connected y single chemicl ond gives rise to signl defined y the chrcteristic frequencies of the two nuclei, nd these frequencies define the shpes F1 nd F2. At first glnce this my look little informtive; however, the usefulness of these spectr ecomes pprent when considering series of such spectr with prticulr modultion. Thus, one my record 15 N-HSQC spectr where the signl intensity is modulted y n dditionl prmeter tht descries relxtion. The third function, F3, follows n exponentil expression, nd curve fitting yields decy time for ech hydrogen-nitrogen pir, i.e. dense chrcteriztion of the different relxtion ehvior over the protein [4]. Relxtion dt in turn provide informtion on the internl moleculr dynmics nd thus llow etter description of the iologicl function. A second ppliction of 15 N-HSQC spectr concerns lignd inding to protein. In drug discovery, the gol is to detect lignds with good inding properties to given trget protein. These lignds (or leds ) my fter further improvement ecome drug with selective clinicl ctivity tht trgets the given protein. For such serch, lirries with sometimes smll molecules re screened. Although severl lignds my e tested simultneously, the lrge numer of spectr recorded requires some utomted nd relile high-throughput screening method. The spectrl feture to recognize inding in given mixture is chnge of position of the signls from those hydrogen-nitrogen pirs tht re prt of the inding site. With TWD pplied to set of 15 N-HSQC spectr recorded for different mixture of the trget protein nd potentil lignds, the third function, F3, simply indictes the presence or sence of signl for the hydrogen-nitrogen pir 5. For given signl, the function dopts non-zero vlue for ll spectr where no inding oc-

9 Three-Wy Decomposition nd Nucler Mgnetic Resonnce 23 curs ner the corresponding tom pir, nd drops to zero when the signl is shifted wy due to lignd inding. (Other components my descrie the complementry ehvior for signls t the new position.) 4.2 Sprse Dt Sets So fr the discussion ws limited to the description of complete, lrge 3D mtrix with experimentl NMR dt y sum of direct products of 1D vectors. This resulted in severl dvntges. First, the use of lower rnk mtrices yields significnt compression of the dt, which due to the tight reltion etween the NMR dt nd the TWD model cn e chieved without relevnt loss of ccurcy. The second dvntge is consequence of the NMR-TWD reltion. The decomposition produces nturl seprtion of the NMR dt, i.e. the individul terms of the sum in expression (1) typiclly correspond to individul nuclei nd the shpes F1, F2 nd F3 descrie interctions of these nuclei. A further dvntge ws very recently suggested 8 ; it is sed on expression (2) nd concerns incomplete experimentl dt. Significnt svings of exmple 75% in NMR instrument time cn e chieved. The processing of the 25% experimentl dt y the lgorithm corresponding to expression (2) provides complete shpes F1, F2 nd F3 for ll components. Multipliction of these functions nd ddition s in expressions (1) nd (2) susequently yields the reconstruction of complete spectrum. Tests show tht the reconstructed spectrum closely corresponds to reference spectrum tht would result from the recording of full NMR dt set. 4.3 Mixing nd Regulriztion The decomposition ccording to expression (1) is for dt sets with dimensionlity exceeding three in generl unique s long s the shpes differ in ll three dimensions. If signls overlp in one dimension, phenomenon referred to s mixing occurs (this mixing hs nothing to do with the mixing mentioned in Tle 1). Both the ppernces of mixing nd utomted procedures for removing it hve een descried in detil [3,9]. In short, mixed component contins signl intensity from two or more components. With dditionl informtion, e.g. tht signls must e strictly nonnegtive, one cn determine trnsformtion tht "demixes" such components. Consider for exmple two mixed components with ner-identicl shpes F3. The following trnsformtions yield non-mixed components F1 1 cos(f) + F1 2 sin(y) nd F2 1 cos(f) + F2 2 sin(y). Signls tht overlp in two dimensions cn e comined into single component. Sometimes, due to non-perfect ppernce of the experimentl dt with respect to the model of expression (1), two components result from TWD tht descrie essentilly only one component. Often, these two components hve very lrge mplitudes with different signs such tht they mostly cncel ech other. To void this phenomenon, Tikhonov regulriztion term is introduced tht penlizes mplitudes tht re significntly lrger thn the others [10]. The term, which is dded to expression (1), hs the following form R S ( ) 2, where R is the regulriztion fctor.

10 24 M. Billeter nd V. Orekhov 5 Conclusions TWD is very verstile tool for the processing nd nlysis of NMR experiments. It hs n intimte reltion to 3D NMR spectr, nd it is pplicle to mny different types of NMR dt. Besides spects tht re specific to individul NMR dt sets, e.g. grouping of signls tht involve one prticulr nucleus s in the exmple of Tle 1, TWD offers two mjor dvntges. First, the lrge NMR spectr with typiclly severl million dt points cn e efficiently compressed while mintining high similrity to the originl dt. Second, the originl dt, ut lso selected susets thereof, cn e reconstructed y multiplying shpes nd summing components ccording to expression (1). These reconstructions llow visuliztion nd nlysis with the norml tools used for NMR spectr. A specil cse relying on reconstruction is the decomposition of sprsely recorded dt sets, sving costly instrument time, which yields complete shpes nd thus llows reconstruction of full dt mtrix. Acknowledgements. This work ws supported y grnts nd from the Swedish Reserch Council. The uthors would like to thnk Ilghiz Irgimov for fruitful discussions. The NMR experiment ws performed t the Swedish NMR Centre. References 1. Levitt, M.H.: Spin Dynmics. Wiley, New York (2001) 2. Kruskl, J. B.: Three-wy rrys: rnk nd uniqueness of triliner decomposition, with ppliction to rithmetic complexity nd sttistics. Liner Alger Appl. 18 (1977) Orekhov, V., Irghimov, I.V., Billeter, M.: MUNIN: new pproch to multidimensionl NMR spectr interprettion. J. Biomol. NMR 20 (2001) Korzhnev, D. M., Irghimov, I. V., Billeter, M., Orekhov, V.: MUNIN: ppliction of three-wy decomposition to the nlysis of heteronucler NMR relxtion dt. J. Biomol. NMR 21 (2001) Dmerg, C. S., Orekhov, V., Billeter, M.: Automted nlysis of lrge sets of heteonucler correltion spectr in NMR-sed drug design. J. Med. Chem. (2002) in press 6. Irghimov, I.: Appliction of the three-wy decomposition for mtrix compression. Numer. Liner Alger Appl. 9 (2002) Boelens, R., Vuister, G.W., Koning, T.M.G., Kptein R. Oservtion of spin duffusion in iomolecules y three.dimensionl NOE-NOE Spectroscopy. J. Am. Chem. Soc. 111 (1989) Orekhov, V., Irghimov, I, Billeter, M.: Optimizing resolution in multidimensionl NMR y three-wy decomposition. Sumitted 9. Gutmns, A., Jrvoll, P., Orekhov, V., Billeter, M.:Three-wy decomposition of complete 3D 15N-NOESY-HSQC. J. Biomol. NMR. 24 (2002) Tikhonov, A.N., Smrskij, A.A.: Equtions of mthemticl physics. Dover, New York (1990)