Spectrum Method and Resonant Recognition Model for Cross-spectral Analysis of DNARNA Sequen

Some Problems in Application of Information Spectrum Method and Resonant Recognition Model for Cross-spectral Analysis of DNA/RNA SequencesVladimir B.Baji´c1and Ivan V.Baji´c21Centre for Engineering Research,Technikon Natal,P.O.Box953,Durban4000,South Africa E-mail:bajic.v@umfolozi.ntech.ac.za 2University of Natal,Department of Electronic Engineering,King George V Avenue,Durban4001,South AfricaAbstract-The Information Spectrum Method(ISM)and the Resonant Recogni-tion Model(RRM)represent a methodology based on signal processing that analyses char-acteristic spectral motifs in DNA/RNA and protein sequences.Some inconsistencies have been reported recently with regard to the spectral characterization of speciÞc biological sequences utilizing the ISM(RRM)and this motivated examination of the basics of the ISM(RRM).This note provides some discus-sions on the methodology on which the ISM and RRM are based and shows a potential reason why the ISM and the RRM produce inaccurate and unreliable results.Two ex-amples are given to backup the analysis and discussion.I.IntroductionIn the last13years a conceptually interesting methodology for the analysis of DNA/RNA and protein sequences,named initially the Information Spectrum Method(ISM)[1],[2],[3],has been devel-oped.In order to better correspond to the hypothet-ical selective resonant interaction of biomolecules, and after some further development,it was later re-named the Resonant Recognition Model(RRM)[4], [5].This theory treats biological sequences asÞnite length digital signals and attempts to apply meth-ods of signal processing to extract relevant spectral information from these signals.Utilizing the methodology proposed for the ISM (RRM),some inconsistencies have been reported in [6]with regard to the spectral characterization of speciÞc biological sequences.This gave rise to inter-est in examining the foundation of the ISM(RRM) which relates to signal processing and toÞnd pos-sible reasons for the discrepancy in results.In this note discussion of the background of the ISM(RRM) related to the utilization of spectral analysis is given.It is shown why,in the opinion of the authors,the ISM(RRM)will virtually always produce mislead-ing results.The discussion is supported by two ex-amples.II.ISM and RRMThe ISM(RRM)considers biological sequences (DNA/RNA and proteins)asÞnite length signals.DNA/RNA are sequences composed of only4dif-ferent types of nucleotides,while proteins are se-quences composed of20different types of amino-acids.Hence,in their conversion to the form suit-able for the application of signal processing,speciÞc numbers are allocated to each of the nucleotides and each of the amino-acids.This is normally done by means of the so-called Electron-Ion Interaction Po-tential(EIIP)which correlates some physical char-acteristics of nucleotides and amino-acids and some biological properties of organic molecules[7].These numerical values are precalculated and can be found in[4].After assigning the relevant EIIP values to the elements of biological sequences,they become simply numerical sequences ofÞnite length.From this point on,the extraction of the spectral infor-mational content of a selected group of biologically related sequences may be considered to be an engi-neering problem.The ISM(RRM)theory is aimed atÞnding the spectral characterization,the so-called consensus spectrum,of a group of sequences that have the same or similar type of biological functions.It is not known yet what are all the factors that deter-mine the biological function of a particular sequence.However,it is generally considered that the struc-ture of the sequence is one of the most important factors.Our interest here will be focused only on the structure of the sequence.The basis of the ISM(RRM)is the classical cross-spectral analysis.Because the sequences from one functional group,say promoters,normally have dif-1ferent lengths,the shorter sequences are extended to the length of the longest one in the examined group in order to enable the calculation of cross-spectrum.For this we cite [1],p.338,:”Comparison of sequences of di fferent length is done by padding out the shorter sequences with zeros to the length of the longest sequence before compar-ing the DFT.The padding operation may cause error in de Þning peak frequencies.It is interesting that in almost any group of biologically related sequences,the length of sequences do not vary by much.In the case of the great varying length of signals,component spectra are of a shape such that there is only a minor error resulting by padding operation.”The authors of this note are of the opinion that this is not correct in the context of the cross-spectral analysis of biological sequences.In what follows we will restrict our consideration only to the DNA/RNA sequences.For this reason we will refer to the biological sequences as genetic sequences.We point out that the analogous con-sideration holds for protein sequences.For some more details on the ISM and the RRM we will ex-plain the process of signal processing used in these methods.Let us consider a group Γof s numeri-cal sequences,Γ={y i },i =1,2,...,s,which relate to one speci Þc biological function.These sequences are Þrstly detrended,so that a new set ΓD of de-trended sequences {x i }is obtained.Let the longest sequence in ΓD be x L with the length N .Due to di fferent lengths of x i -s all those sequences x i that are shorter than x L are padded by zeros up to the length of x L in order to make a set of sequences of the same length so as to provide the possibility of calculating their cross-spectrum.Then the discrete N points Fourier transform (DFT)is computed for each of the sequences.If X i (n ),n =0,1,...,N −1,is the DFT of x i ,then the consensus magnitude spectrum of all sequences from ΓD is obtained asS (n )=s Yi =1|X i (n )|.The consensus spectrum is nor-malized with regards to frequency by assuming that the distance between the nucleotides in DNA/RNA is constant with the distance of d =1in suitably de-Þned units,so that the range of the normalized fre-quencies of non-redundant components in the spec-trum is [0,0.5].The spectrum is also normalized with regards to the spectral components magnitude,so that the maximal magnitude is of size 1.We will discuss only the magnitude consensus spectrum since it is considered relevant for the ISM (RRM)cross-spectrum calculation.After analyzing a number of functional groups of biological sequences by the above mentioned proce-dures,it is hypothesized in [1],[4],[5],for a num-ber of such groups,that each group in its consen-sus spectrum contains only one su fficiently sig-ni Þcant characteristic frequency component,which is unique for the group,but di ffers from group to group.In the ISM and the RRM a frequency component with the magnitude M is con-sidered signi Þcant if M ≥m,where m is the mean value of components in the consensus spectrum [4].The ISM (RRM)theory proposed is attractive and,if proved,will lead to the interesting implication that many of the biological functions are character-ized by one speci Þc frequency each in the normalized spectrum.However,applying this methodology in [6]to a group of promoter sequences,it was found that di fferent subsets of the same promoter region set have mutually di fferent characteristic frequen-cies,which were also di fferent from the characteris-tic frequency of the whole group.This necessitated the examination of the basics of the ISM and the RRM,the results of which are shown in the next sections.III.Problem backgroundThe signals (numerical sequences)obtained from the genetic sequences are discrete,have Þnite length,and the magnitudes of signal components take val-ues from two well de Þned sets S n and S a with Þnite number of elements each (the set S n has 4elements representing the values of EIIP for nucleotides,while the set S a has 20elements representing the EIIP val-ues for amino acids).The domain D of de Þnition of a genetic sequence s is Þnite and strictly discrete and is de Þned by D ={kd },where k =1,2,...,n,and d is the distance between the nucleotides in DNA/RNA (or the distance of neighboring amino acids in proteins).In the cross-spectral analysis of DNA/RNA se-quences,one of the main problems is that of com-paring the spectra of two Þnite-length sequences of di fferent lengths.Consider the sequences x 1(n ),n =0,1,...,N 1−1,of length N 1and x 2(n ),n =0,1,...,N 2−1,of length N 2,such that N 1<N 2.If X 1(k )and X 2(k )are the DFTs of x 1and x 2,respec-tively,then their respective domains of de Þnition in the normalized spatial frequency domain,are given byS 1={k/N 1|k =0,1,...,N 1−1}S 2={k/N 2|k =0,1,...,N 2−1}To make a meaningful comparison of the two spectra possible,as well as to compute the cross-spectrum of these two sequences,the sets S 1and S 2have to be the same.One possible way to achieve this is by extending the shorter sequence x 1by adding N 2−N 1components to its end,i.e.to create a sequencex (e )1(n )=½x 1(n ),n =0,1,...,N 1−1e (n ),n =N 1,...,N 2−12where e(n)denotes the extension.Suitable choice of e(n)will ensure that x(e)1retains all the signiÞcant characteristics of x1while at the same time mini-mizes the errors introduced by extension.Severalforms of extension,such as periodic,zero-padding,symmetric,etc.,are widely used in variousÞelds ofsignal processing[11].However,with any such ex-tension we also introduce(implicitly)the assump-tion that the new sequence will retain the samebiological function.Otherwise,Þnding the cross-spectrum to characterize a group of functionally re-lated sequences would lose meaning.Although ourproblem is analysis in the spectral domain,the al-terations that may be introduced to the form ofthe original sequence,to make the spectral analysismore convenient,are not allowed if the interpreta-tion of the biological functionality of the sequenceis to be retained.Any assumption about the na-ture(behavior)of the biological sequence s out ofthe original domain D of its deÞnition,which in-cludes also the points in between the known signalcomponents,is not correct.If we assume that thegenetic sequence s is altered in any way by adding to it new components for whatever reason,this im-plies the change of the domain D of deÞnition of thesequence.However,as soon as the original domainD of the sequence is changed,its original biologicalfunction is not guaranteed,i.e.it is most probablethat the new sequence will not have the same or even similar biological function,or it may not have anyparticular biological function at all.ISM(RRM)methodology proposes the zero-padding extension of the shorter sequence,i.e.e(n)=0,n=N1,...,N2−1,using the argument that the value of the EIIP is indeed zero outside ofthe original biological sequence.While this argu-ment might seem to be plausible for isolated biolog-ical sequences such as proteins(although it may bequestioned in view of the protein’s changing envi-ronment),it certainly cannot hold for segments of the DNA/RNA sequence,as they are not isolated from the rest of the DNA/RNA.We will show that there would be no use in making zero-padding for shorter sequences,since errors introduced by this operation might signiÞcantly modify theÞnal result of the cross-spectral analysis.IV.Effects of zero-paddingZero-padding operation is usually used to improvethe visual(or computational)resolution of the spec-trum.It enables the evaluation of the DFT at agreater number of points in the frequency domain than is allowed by the original length of the sig-nal.In view of the discussion in the preceding sec-tion,it would allow the calculation of the spectrum of the signal x1with resolution of1/N2instead of 1/N1,which is the physical resolution dictated bythe length of x1.However,due to the fundamental limitations summarized in the uncertainty principle of Fourier analysis[12],zero-padding by itself can-not improve the quality of the spectral estimate.It merely provides a way of interpolation in the fre-quency domain and,in the absence of any knowledge about the behavior of the signal outside its original domain of deÞnition(as in our case),we cannot de-termine whether or not the interpolation is done on the correct curve.Examples are given in[13]and[14]to show that zero-padding cannot help resolvetwo frequency components whose separation in the frequency domain is less than the limit set by the physical resolution,implying that in such cases it is of little use.The problem is further complicated by the fact that spectra obtained by zero-padding the shorter sequences in ISM(RRM)analysis are being multi-plied in the calculation of the cross-spectrum.Since zero-padding enforces the frequency domain convo-lution of the original frequency components with the rectangular window function W R,it creates signiÞ-cant’noise’sidelobes in regions between the original spectral components(the term’noise’is used in the context of the previous paragraph,to indicate that we cannot determine the exact shape of the spec-trum in these regions).In the example of the pre-vious section,each component of X1(k),when con-volved with W R,would add some’noise’to each of the intervals(k/N1,(k+1)/N1),k=0,1,...,N1−2.’Noise’contribution from each speciÞc component of X1(k)may be relatively insigniÞcant,but the com-bined effect of’noise’produced by all original spec-tral components may easily result in the creation of’noise’components that are of the same order of magnitude or higher than the average of X1(k),as will become obvious from the example in the fol-lowing section.All of these newly created spectral ’noise’components,for whose existence there is no support in the original spectrum of the signal x1, now enter the process of cross-spectral multiplica-tion,and become candidates to be recognized as fre-quency components common to signals x1and x2.Eventually this may lead to obtaining a characteris-tic frequency of a genetic sequence that is a product only of the spectral’noise’components.V.ExamplesA.Example1Let us assume that we want toÞnd the frequency that characterizes the common biological function of a group of sequences that include only sequences whose length is at most32.Let us also assume that two hypothetical biological sequences,x1and x2, each of length32,whose respective spectra are given in Fig.1,belong to the group.Each spectrum in3Figure1—Magnitude spectra of sequence x1and x2 Figure2—Magnitude spectra of zero-padded se-quences x1p and x2p,and their cross-spetrum ob-tained with512points DFThas only3non-zero components.It is easy tothat the cross spectrum of x1and x2is identi-zero,and thus the cross-spectrum of the wholewill yield only components of zero magni-Consequently,the frequency to characterize common biological function of these sequences be found(or,in the worst case,one of the x1or x2does not possess the expectedfunction of the group).Let us now as-that a new sequence x3of the length512isthat has the same biological function as the in the examined group.If it was known that such a sequence exist it would be in-into the cross-spectrum calculation.So,wethe process of cross-spectrum determination consider only the cross-spectrum of x1,x2and .According to the ISM(RRM),zero-padding ofand x2has to be done up to the length of512the new sequences obtained in this way are de-as x1p and x2p.The envelope of the respec-spectra for x1p and x2p,as well as the envelopetheir cross-spectrum are given in Fig.2.As canexpected,the cross-spectrum of zero-padded se-produced a lot of spectral components that not exist in the original cross-spectrum of x1 and x2.These new components can inßuence sig-niÞcantly theÞnal cross-spectrum.Moreover,the strongest spectral components in the cross-spectrum Fig.2are not on the frequency positions of the components of the spectra for x1and x2 in Fig.1.Thus the inconsistency.Example2set of40human promoter sequences was extracted GenBank and EMBL databases.Promoter re-lengths varied from87to606.The sequencesthen divided into several groups according Ta-1.Groups A and B are distinct subsets of the originalof sequences,sorted according to length.Group was formed by combining some of the sequences group A with some of the sequences from groupwhile group D was formed by combining all of the from groups A and B.In groups B,C and the longest sequence is of length606.We applied the ISM(RRM)procedure to groupsC andD from Table1.The resulting consensusare shown in Figs.3-5.Results are summa-in Table2.The results obtained show that the calculatedf RRM varies signiÞcantly depending on the selectionof sequences included in the analysis.Characteris-tic frequencies f RRM obtained for different groups of sequences differ by an amount much larger than the computational resolution of the spectrum in each 4Group No.of seq.Length Rounded av.length A 16≤200141B 24200−650348C 10<650300D40<650265Table 1.Four groups of sequences from theset of 40sequencesGroup f RRM SNR Comp.res.B 0.0604141.241.65×10−3C 0.085885.961.65×10−3D 0.0099151.681.65×10−3Table 2.Results of ISM (RRM)analysis for three groups of sequencesin group BFigure 4—RRM consensus spectrum for sequences in group CFigure 5—RRM consensus spectrum for sequences in group Dcase.SNR in each group is found to be larger than 20,which is the minimal value con-to be signi Þcant.Moreover,in each of the examined shorter sequences are zero-padded length of 606what is the length of the longest in the group.Thus,any of the frequencies can,according to the ISM (RRM),be con-as a characteristic RRM frequency for the set of sequences.The problem,however,is all these f RRM are di fferent and hence unus-for the characterization of the selected group.obvious discrepancy appears and the authors of the opinion that it is due to the e ffects on the content of spectra of individual zero-padded C.Commentsexamples are made only for the illustration pur-to directly point out some problems in cross-calculation if its determination is based on of shorter genetic sequences.We also our opinion on the request that the original of genetic sequence de Þnition must not be due to the demands for the convenience of analysis.conclusion based on these examples is that other techniques have to be developed to han-the problem of cross-spectrum calculation for ge-sequences in a consistent and physically inter-manner.One such technique based on the of ’blurred’spectrum that is suitable for the of biological sequences has been proposed in [10],[8],and has been successfully applied to the problem of promoter region recognition in [8]and [9].5VI.ConclusionsThe examples presented show without doubt that the concept of padding by zeros of shorter sequences in the ISM(RRM)method for the convenience of calculation of the cross-spectrum of different length sequences produces results that may be misleading. At the same time,the biological interpretation of the zero-padded sequences is destroyed.For this reason, such technology of cross-spectrum determination for genetic sequences cannot be accepted.References[1]V.Veljkovi´c,I.´Cosi´c,B.Dimitrijevi´c and D.Lalovi´c,Is It Possible to Analyze DNA and Pro-tein Sequences by the Methods of Digital Signal Processing,IEEE Transactions on Biomedical Engineering,Vol.BME-32,No.5,pp.337-341, May1985.[2]I.´Cosi´c,D.Neši´c,M.Pavlovi´c and R.Williams,Enhancer binding proteins predicted by infor-mational spectrum method,Biophys.Biochem.m.,Vol.141,pp.831-839,1986. [3]I.´Cosi´c,M.Pavlovi´c and V.Vojisavljevi´c,Pre-diction of’hot spots’in interleukin-2based on informational spectrum characteristics of growth regulating factors,Biochimie,Vol.71, pp.333-342,1989.[4]I.´Cosi´c,Macromolecular Bioactivity:Is It Res-onant Interaction Between Macromolecules?-Theory and Applications,IEEE Transactions on Biomedical Engineering,Vol.41,No.12, pp.1101-1114,December1994.[5]I.´Cosi´c,The Resonant Recognition Model ofMacromolecular Bioactivity-Theory and Ap-plications,Birkhäuser,1997.[6]V.B.Baji´c,I.V.Baji´c and W.Hide,Appli-cation of Resonant Recognition Model to a set of human promoters:A word of warning,to appear in Proceedings of the International Sys-tems,Signals,Control,Computers(SSCC’98) Conference,Durban,South Africa,1998. [7]V.Veljkovi´c and I.Slavi´c,General models ofpseudo-potentials,Phys.Rev.Lett.,Vol.29, pp.105-108,1972.[8]V. B.Baji´c,I.V.Baji´c and W.Hide,Ad-vantages and drawbacks of spectral analysis of DNA sequences,Plenary Lecture,to appear in Proceedings of The First International Confer-ence on Bioinformatics of Genome Regulation and Structure,(BGRS’98)Novosibirsk,Russia, 1998[9]V.B.Baji´c,I.V.Baji´c and W.Hide,Spec-tral characterization of human promoters,toappear in Proceedings of the International Sys-tems,Signals,Control,Computers(SSCC’98)Conference,Durban,South Africa,1998.[10]V.B.Baji´c and I.B.Baji´c,Spectral meth-ods for analysis of DNA/RNA and protein se-quences,Technical Report,Technikon Natal,1998.[11]G.Strang and T.Nguyen,Wavelets andÞlterbanks,Wellesley-Cambridge Press,Wellesley,MA,USA,1996.[12]A.Papoulis,Signal Analysis,McGraw-HillBook Company,New York,1977.[13]J.G.Proakis and D.G.Manolakis,Digital Sig-nal Processing:Principles,Algorithms and Ap-plications,Second Ed.,Macmillan PublishingCompany,New York,1992[14]S.J.Orfanidis,Introduction to Signal Process-ing,Prentice Hall,Upper Saddle River,NewJersey,1996.6。