82 IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE JANUARY/FEBRUARY 2006

Finding Large Domains ofSimilarly Expressed Genes

A Novel Method Using the MDL Principle andthe Recursive Segmentation Procedure

BY DANIEL NICORICI,OLLI YLI-HARJA, ANDJAAKKO ASTOLA

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The advent of microarray technology enables us to mea-sure simultaneously levels of thousands of genes forentire genomes in a single experiment—producingdaunting amounts of data and genetic information.

After the Human Genome Project ended in 2003 with the suc-cessful completion of the human genetic sequence [1], atten-tion is turning to functional genomics. With gene expressiondata available for different organisms and their genomesalready sequenced, a major goal is to understand the regula-tion of genes at the gene level and at the chromosomal level.Thus, an essential task is to define the role of the regulationmechanism and to understand how the regulation of a set ofadjacent genes functions at the chromosomal level. In order tomake biological sense of the sequenced genomes and the geneexpression data that are available, all of the data must be orga-nized in a manner that allows 1) the discovery of patterns thatmay arise, and 2) the establishment of relations between thegene expressions and the regulation mechanisms, e.g., thetranscriptional regulation.

Recent studies [2]–[5] confirm that the genomes aredivided into large domains that are composed of adjacentgenes on chromosomes with similar expression profiles.There is evidence from budding yeast that some similarlyexpressed genes are found in adjacent pairs or triplets onchromosomes [3]. Larger domains are found to exist in thehuman genome [2], [4] due to the strong clustering of high-ly expressed genes in nearly all tissues. Also in their study,Spellman et al. [14] have found by analyzing theDrosophila genome and high-density oligonucleotidemicroarrays that its genes are clustered into coregulatedgroups of adjacent genes on chromosomes. The mechanismunderlying the large domains is not yet known, but theobserved similarities in the expression of adjacent genesare consistent with regulation at the level of chromatinstructure [5]. The method used by Spellman et al. [5] doesnot provide a very good criterion for evaluating the qualityof segmentation into large domains.

We describe a new method for finding large domains ofsimilarly expressed genes using the minimum descriptionlength (MDL) principle and a recursive segmentation pro-cedure. For the recursive segmentation, we introduce anew stopping criterion based also on the MDL principle.

Based on the MDL principle, we give a rigorous definitionof the quality of the segmentation of genomic profiles intolarge domains.

Intuitively, a large domain can be considered a group ofadjacent genes on a chromosome, where the expression profilesof the genes are similar. This can be described in a succinctway by using the MDL principle, which has been introducedby Rissanen [6], [7]. The MDL principle has been used in sta-tistics, machine learning, data mining [8], and genomic signalprocessing [9]–[11]. According to the MDL principle [6], themodel is selected based on its fitting performance, but it alsopenalizes a very high complexity of the model.

Genomes can be divided into large domains that are impor-tant in controlling the expression of groups of adjacent genes[5]. The recursive segmentation can be used for finding theirborders. The recursive segmentation methods have beenapplied to DNA segmentation into homogeneous domains; forfinding the borders between coding and noncoding regions inDNA; for detecting the existence of the isochores, CpG islands,and replication origin and terminus; for detecting complex pat-terns such as telomers; and for evaluating the genomic com-plexity [12]–[14]. The criterion for continuing the recursivesegmentation process can be based on 1) statistical significance[12], 2) the Bayesian information criterion (BIC) [12]–[14], or3) the MDL principle [11], [15]. Our approach uses only thegeneral properties of the large domains and, in this way, priortraining on data sets is not necessary. The training data sets thatcontain the positions of large domains are not available. Alsothroughout this study, we define the genome data as the datacontaining measurements of gene levels versus experimentalconditions, and the gene profiles are ordered according to theirposition on the chromosomes.

MDL Principle and Coding of Genome DataLet X be a genome data represented as a n × m matrix wherethe row i represents the activity of the gene i (gene profile i)over different experimental conditions, and the column j rep-resents the set of measurements for the experimental condi-tion j. The genes in the genome data X are ordered accordingto their positions along the chromosomes and its entries xi, j

take value in the set {0, 1, . . . , q − 1} due to quantization ofthe genome data to q levels.

IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE JANUARY/FEBRUARY 2006 83

MDL PrincipleThe MDL principle by Rissanen [6], [16] considers thedescription length of the data and the model as follows

L(M, X) = L(M) + L(X|M), (1)

where L(M) is the length of the description of the modeland L(X|M) is the length of the description of the data,where the data X is described using the model M. Accordingto the MDL principle, the best model that fits the data is themodel with the shortest length of the total descriptionL(M, X). Such a reduction indicates that the model M isable to capture the patterns and the dependencies within thedata X. The goal is not to write down the encoded data but tocompare the code length of the encoded data for a class ofmodels. Thus, the model with the best fitting performance,which gives the shortest overall code length, is selected butin a balanced way; the models with a very high complexityare penalized. The MDL principle has been used in variousapplications [6], [9], [10].

Coding of Genome DataThe compression of a given genome data Y is done using themodel M1 , which takes into consideration the similaritybetween all the gene profiles from Y. The genome data Y is an∗ × m∗ matrix that contains n∗ genes across m∗ differentexperimental conditions, where the entries yi, j take value in theset {0, 1, . . . , q − 1} due to quantization of Y to q levels. Thegenes are ordered within Y according to their position on thechromosomes. Further, Y is considered to be a submatrix ofthe matrix X. We apply several transformations to the matrix Ysuch that (matrix Y) → (matrix Z) → (string wn∗∗

). The proba-bility of the observed genome data Y is computed using thestring wn∗∗

, which takes into consideration the similaritybetween all gene profiles within the genome data Y.

We construct a q × m∗ matrix M such that entries mi, j arethe counts of the symbol (i − 1) within the column j of thematrix Y . Thus, one has

∑qi=1 mi, j = n∗ , where

j = 1, . . . , m∗ . The symbols observed at each column j ofthe matrix Y are reordered by their counts from the matrix Musing a permutation vj(·), where j = 1, . . . , m∗ . The permu-tations are used because their coding requires a relativelyshort code length. A permutation aligns the histograms ofsymbols of each Y’s column, such that all histograms aremonotonically decreasing, and collapses all histograms intoa single one [9]. A permutation vj(·) maps k → vj(k), wherek = 1, . . . , q, as follows

(0 1 . . . q − 1vj(0) vj(1) . . . vj(q − 1)

). (2)

The transformed matrix Z is obtained from the matrix Y byusing a set of permutations v = (v1(·), . . . , vm∗(·)) , wherezi, j = v(yi, j). Such a transformation is reversible due to the useof permutation [9], i.e., one can recover Y from Z knowing v.The matrix Z is transformed further into the string wn∗∗

of lengthn∗∗ , where n∗∗ = n∗ × m∗ , by concatenating its rows. Theentries W of the string wn∗∗

take value in the set {1, . . . , q − 1}.The entries of the matrices X, Y, and Z take value also in thesame set {1, . . . , q − 1}. The transformed string wn∗∗

is mod-eled further as a multinomial trial process with parametersP(W = 0) = θ0, . . . , P(W = q − 1) = θq−1 . The symbol l isobserved

∑m∗j=1 mvj(l), j times in the matrix Z and in the string

wn∗∗. Also, one can recover Y from wn∗∗

knowing v.For instance, one has for q = 2, n∗ = 3, m∗ = 2, and

Y =( 0 0

1 01 0

)that M =

(1 32 0

), v1 = (1 0), v2 = (0 1),

Z =( 1 0

0 00 0

), wn∗∗ = (1 0 0 0 0 0), and n∗∗ = 6.

To conclude, the probability of the genome data Y is given by

P(Y; θ̂θθ, v̂ ) = P(wn∗∗( v̂ ); θ̂ (wn∗∗

), v̂ (Y))

= θ̂

∑m∗j=1

mvj(0), j

0 · . . . · θ̂∑m∗

j=1mvj (q−1), j

q−1 , (3)

where the set of permutations ̂v (Y) = {̂vi(·) : i = 1, . . . , m∗}determines the string wn∗∗

, and the multinomial parameters ofthe string wn∗∗

( v̂ ) are θ̂ (wn∗∗) = (θ̂0(wn∗∗

), . . . , θ̂q−1(wn∗∗)) .

The string wn∗∗contains m∗

i values of i, whereθ̂i(wn∗∗

) = m∗i

/n∗∗ and i = 0, . . . , q − 1. Clearly, one has

m∗i = ∑m∗

j=1 mvj(i), j.The overall code length of the encoded n∗ × m∗ matrix Y

using the model M1 based on the MDL principle is as follows

L(M1, Y) = − log2 P(Y; θ̂θθ, v̂) + m∗ log2(q!)

+ log2 Nθ (n∗∗, q), (4)

where the first part encodes the data Y given the model M1

with the parameters ̂θθθ and ̂v, the second part encodes the opti-mal permutations ̂v, and the third part encodes the maximumlikelihood (ML) estimates of ̂θθθ and ̂v. In order to encode effi-ciently, the set of probabilities, Nθ (n∗∗, q) is used because theset of pairs (θ̂θθ , v̂) is redundant, and one can restrictθ̂θθ = (θ̂0, . . . , θ̂q−1) such that θ̂0 � θ̂0 � . . . � θ̂q−1 [9]. Thelength of the list containing all possible q-tuples

Our new segmentation method allows us

to find large domains of similarly expressed

genes without any a priori data for training.

84 IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE JANUARY/FEBRUARY 2006

(n∗∗0 , . . . , n∗∗

q−1) is Nθ (n∗∗, q) such that n∗∗0 + · · ·+

n∗∗q−1 = n∗∗ and n∗∗

0 � n∗∗1 � . . . � n∗∗

q−1 [9]. The model M1

considers that the entire given genome data Y is encoded as asingle part, and no large domains of similarly expressed genesexist within Y. Thus encoding the genome data Y with fewsimilarities between genes profiles will be penalized with alarger code length than in the case when there are more simi-larities between the gene profiles. We note that our approachof computing the code length of encoded data Y, especiallyof (3) and (4), is a modification of the approach used byTabus et al. [9] for computing the code length of encoded

class labels as a two-part code. The code length computed byTabus et al. [9] is based on gene expressions of each patient,and it is used in the problem of class discrimination. Themajor difference between our approach and the approach ofTabus et al. [9] is that we compute the code length of theencoded genome data Y in such way to take into considerationthe similarity between the gene profiles (the matrices M and Zare constructed in this way). Furthermore, the computed codelength is used in the problem of segmentation into largedomains of similarly expressed genes.

Coding of Genome Data with LargeDomain of Similarly Expressed GenesThe encoding of a given genome data X is done also using themodel M3(i, j), which considers the existence of a largedomain (i, j) containing similarly expressed genes. A largedomain (i, j) starts with the ith gene profile and ends with thejth gene profile within matrix X, and it splits the matrix X intothree submatrices X(a), X(b), and X(c), which contain the X’sgene profiles from 1 to i − 1, i to j, and j+ 1 to n, respectively.The submatrix X(b) is considered to be the only one that repre-sents the large domain (i, j), and it cannot contain more thanthe a priori established maximum number of gene profiles.The number of genes contained in a large domain has beendetermined previously using biological experiments or data[2]–[5]. A large domain with similarly adjacent gene profilesgives a submatrix X(b) that is encoded very effectively using(4) based on the MDL principle.

According to the MDL principle, the overall code length ofthe encoded genome data X using model M3(i, j) that consid-ers the existence of a large domain (i, j) is

L(M3(i, j), X) =L(M1, X(a)

) + L(M1, X(b)

)+ L

(M1, X(c)

) + 2 · log2 n, (5)

Fig. 2. A 3-D representation of code length L(M1, T) − L(M3(i, j), T)based on the MDL principle, computed for all possible can-didate large domains (maximum length of 30) for synthetic genome data of 100 genes from Figure 1. The maximum value forthe computed code length is circled on the graph and it corresponds to the large domain (51, 70).

Fig. 1. Synthetic genome data containing 100 gene profilesacross 100 experimental conditions. The first 50 gene profilesare randomly generated, the next 20 gene profiles are iden-tical, and the last 30 gene profiles are randomly generated.The expression of genes have binary values.

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where the first three terms L(M1, X(a)), L(M1, X(b)), andL(M1, X(c)) are computed using (4) based also on the MDLprinciple, and they give the cost in bits of encoding the sub-matrices X(a) , X(b) , and X(c) , respectively. The last term2 · log2 n is needed to encode in bits the two positions wherethe large domain starts and ends within the matrix X. When itis assumed that no large domain exists within the genome thedata X, the code length of the encoded matrix X is L(M1, X),and it is computed using the model M1 and relation (4).

Recursive Segmentation into Large DomainsIn this study we use the recursive segmentation method pro-posed by Bernaola-Galvan et al. [12] and Li [13] for findinglarge domains of similarly expressed genes in the given genomedata. The recursive segmentation of agiven genome data X proceeds as fol-lows. We sweep through the gene pro-files of X and compute at every position iand j, where i < j, i = 1, . . . , n − h andj = i + 1, . . . , i + h , that divide thematrix X into the upper submatrix X(a),the middle submatrix X(b), which repre-sents the large domain (i, j), and thelower submatrix X(c), the code lengthsof the whole matrix, the upper, the mid-dle, and the lower submatrices.According to Spellman et al. [5], wechoose the maximum length of a largedomain to be h = 30 genes. The posi-tions i and j are accepted as cuttingpoints, representing the large domain(i, j) , when the code length,L(M1, X) − L(M3(i, j), X), computedusing (4) and (5), reaches its maximum.Further, we recursively apply thesegmentation to the upper submatrixX(a) and to the lower submatrix X(b)

unti l maximized code lengthL(M1, X) − L(M3, X) is above acertain threshold. In this approach,the threshold is based on the MDLprinciple, where

L(M1, X) − L(M3, X)

= max(i, j)

(L(M1, X) − L(M3(i, j), X))

= L(M1, X) − min(i, j)

L(M3(i, j), X). (6)

Clearly, one has L(M3, X) = min(i, j) L(M3(i, j), X) . If themaximized code length L(M1, X) − L(M3, X) is above thethreshold, the genome data is segmented, and if not, the seg-mentation is stopped for the respective data. We note the

Fig. 3. Expression profiles of 200 adjacent genes on the rightarm of the Drosophila chromosome 3 (3R).

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Fig. 4. Expression profiles of 200 adjacent genes (from Figure3) on the right arm of the Drosophila chromosome 3 (3R)quantized to binary values. For each square, a black colordenotes a lower relative expression than a white color for agene in an experiment.

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Fig. 5. 3-D representation of code length L(M1, X̃) − L(M3(i, j), X̃)based on the MDLprinciple, computed for all possible candidate large domains (maximum length of30) for data of 200 adjacent genes, from the right arm of Drosophila chromosome 3(3R), shown in Figure 4. The maximum value for the computed code length is circledon the graph, and it corresponds to the large domain (83, 104).

(Start Position of CandidateLarge Domain)[Gene Index]

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86 IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE JANUARY/FEBRUARY 2006

similarity of this method with the approach when Jensen–Shannon divergence is used [13], [14].

Figures 1–2 shown synthetic data T of 100 gene profilesand the three-dimensional (3-D) representation of the codelength L(M1, T) − L(M3(i, j), T). The synthetic data T con-sist of 100 gene profiles that take binary values, where thefirst 50 gene profiles of data T are randomly generated, thenext 20 gene profiles are identical, and the last 30 gene pro-files are randomly generated. The maximum value for thecomputed code length is circled on Figure 2, and it corre-sponds to the large domain (51, 70). Figure 2 shows that the

maximum value of the computed code length,L(M1, T) − L(M3, T), finds exactly the start position andthe length of the large domain with 20 identical gene profilesin the synthetic data T.

Stopping Criterion for Recursive SegmentationThe stopping criterion, in the case when relation (6) isused, can be considered from the point of view of hypothe-sis testing and the model selection framework. For thehypothesis testing framework, the probability that the valueof L(M1, X) − L(M3, X) can be obtained by chance is

Fig. 6. Large domains of similarly expressed genes—gray or black rectangles in (b)—found using recursive segmentationapplied to data of 200 adjacent genes from the right arm of Drosophila chromosome 3 (3R), shown in Figure 4.

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The model with the best fitting performance,

which gives the shortest overall code length,

is selected but in a balanced way.

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computed by the null hypothesis that the genome data ishomogeneous. The exact form of the null distribution isdifficult to find [13].

The stopping criterion, based on Bayesian information crite-rion (BIC), for segmentation using Jensen–Shannon diver-gence has been introduced by Li in his study [13]. One can seethe subtle relations between (6) and the Jensen–Shannondivergence or the Kullback-Leibler divergence [6]–[8], andalso between the BIC and the MDL principle, which give anidentical formula for certain cases [8].

In this study, we introduce a new stopping criterion for therecursive segmentation, when using the code lengthL(M1, X) − L(M3, X), based on model selection using theMDL principle. Thus, the stopping criterion tests if a three-random–submatrices model M3 gives a shorter code lengththan the one-random–matrix model M1 . If the three-ran-dom–submatrix model has a shorter code length (it better fitsthe data), then the cuts are accepted (the large domain isaccepted); otherwise, it is not. The MDL principle assures usof balancing the goodness-of-fit of the model with the com-plexity of the model in relations (4) and (5), and a very highcomplexity of the model is penalized. In order to continuethe recursive segmentation procedure and to decide if thecuts i and j are significant or not (if the large domain (i, j) issignificant), the three-random-submatrices model must fit thedata better than the one-random model. This leads to a stop-ping criterion that is as follows

L(M1, X) − L(M3, X) > 0, (7)

where L(M1, X), L(M3, X) are computed using (4) and (6),respectively. Thus, the recursive segmentation continues, orthe cuts i and j, which represent the large domain (i, j), areaccepted as significant as long as criterion (7) is fulfilled.

Experimental ResultsWe illustrate the finding of largedomains of similarly expressed genesbased on the MDL principle and recur-sive segmentation using theDrosophila genome data of Spellmanet al. [5], publicly available [17], andhuman genome data.

The microarray genome data ofDrosophila contains 13,165 geneexpression profiles, covering 89 dis-tinct experimental conditions from 267Affymetrix GeneChip DrosophilaGenome Arrays [5]. The experimentalconditions consist of adults andembryos which are visible in Figures 3and 4 as a vertical line. The data are inlog2 ratio format, all replicates areaveraged, and the values are time zerocorrected [5]. Data preprocessing andexperimental conditions are describedin detail in [5]. The genes in thisdataset are organized according to theirpositions along the chromosome.Visual inspection of the data, as shownin Figures 3–4, reveals that groups ofadjacent genes with similar expression

patterns, which are not otherwise functionally related in anyobvious way, appear frequently [5].

The starting point is the gene expression data X, also calledgenome data, where each entry xi, j indicates the expressionlevel of gene i for experimental condition j. We make theassumption that the transcription machinery of a gene usesthe expressed/not expressed or upregulated/downregulated states [18]. More precisely, we quantize eachgene profile independently to binary states [18] by applyingthe Lloyd algorithm [9]. The quantization to discrete valuesof genome data can be viewed also as removing the noisefrom data [9]. In this study, the entries in X are quantized toq = 2 levels, but the newly introduced method for findinglarge domains can use more than two levels of quantization.For the remainder of the article, we assume that the genomedata X is quantized to binary values (how many quantizationlevels are chosen is outside the scope of this article). TheMDL principle can also be used to select an optimum q valueas suggested in [9].

In order to illustrate the segmentation procedure, we applythe new segmentation method on a group, chosen arbitrarilyfrom the chromosome 3R of Drosophila, of 200 adjacentgenes, noted as X̃. Figure 3 shows the original expression pro-files of the group of 200 gene profiles from X̃ that are orderedaccordingly to their position along the chromosome. Also,Figure 4 shows the same 200 gene profiles from X̃ after quan-tization to binary states using Lloyd algorithm, where a whitecolor indicates a higher relative expression of a gene in anexperiment than a black color. In Figure 3, and especially inFigure 4, are visible groups of adjacent genes that have similarexpression profiles and the vertical separation betweenembryos and adults of Drosophila.

Figure 5 illustrates the 3-D representation of the code lengthL(M1, X̃) − L(M3(i, j), X̃) , where the other two axes

Fig. 7. The results of a recursive segmentation of Drosophila genome into largedomains of similarly expressed genes.

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represent i (position where the candidate large domainstar ts) and j+ i − 1 (length of the candidate largedomain given as number of genes). The maximum valueof L(M1, X̃) − L(M3(i, j), X̃) is circled on Figure 5 andit corresponds to the large domain (83, 104), which ismarked in Figure 4 on the right-hand side as a gray-filled rectangle. This large domain is accepted as signifi-cant because L(M1, X̃) − L(M3(83, 104), X̃) = 987.14 band L(M1, X̃) − L(M3(83, 104), X̃) > 0, where criterion(7) is fulfilled.

Figure 6 shows the results of the recursive segmenta-tion applied to the same 200 adjacent gene profiles X̃from the Drosophila chromosome 3R. The new methodintroduced in this study, based on the MDL principleand recursive segmentation, is able to find successfullythe large domains of similarly expressed genes in X̃ asshown in Figure 6.

The new recursive method based on the MDL principle isapplied to the genome of Drosophila [5] containing 13,615genes, and it finds 750 large domains of similarly expressedgenes together with their exact positions on the chromosomes.From these 750 large domains, 223 large domains are thedomains that contain between 10–30 similar gene profiles.Figure 7 shows a histogram of the sizes of similar gene-profile

segments that result when the new recursive segmentationmethod, based on the MDL principle, is applied to theDrosophila genome data.

The human genome data contain 21,810 genes orderedaccording to their position on the chromosomes versus 50patients with colorectal tumors. The Affymetrix HG-133Achips have been used for gene measurements. The gene profiles are quantized to binary using the Lloyd algorithm asdone for the Drosophila genome data. The new recursivemethod finds 160 large domains of similarly expressed genes inhuman genome data. From these 160 large domains, 40 largedomains are the large domains that contain between 10–30 simi-lar gene profiles. Figure 8 shows a histogram of the sizes ofsimilar gene-profile segments that result when the new recursivesegmentation method is applied to the Human genome data.

Even though the biological significance of the detailedresults for the Drosophila genome data and the humangenome data remains to be later investigated and the underly-ing mechanism of the large domains is unknown [5], our newsegmentation method permits us to find successfully largedomains of similarly expressed genes without any use of a pri-ori data for training.

The novel method introduced in this study, based on theMDL principle, for finding large domains of similarly

expressed genes is different in sever-al aspects from the method intro-duced in [11] for finding the largedomains based on the MDL principleand normalized maximum likelihood(NML) model. The major differencesare that in the current method, thequantization is done using the Lloydalgorithm and the similarity betweenall genes from a large domain aretaken into consideration, which iscloser to the biological knowledgeavailable. In our previous study [11],only the similarities between the firstgene profile and the rest of the geneprofiles from a given large domainwere taken into consideration.

Concluding RemarksIn this study, we have introduced anew method for finding and defininglarge domains of adjacent genes on achromosome with similar expressionprofiles based on the use of the MDLprinciple and the recursive segmenta-tion procedure. For the recursive seg-mentation, we used a newly introduced

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Fig. 8. The results of a recursive segmentation of the human genome into largedomains of similarly expressed genes.

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IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE JANUARY/FEBRUARY 2006 89

stopping criterion using the MDL principle. Together theyoffer a novel method to view the large domains of similarlyexpressed genes in genome data. The description of thegenome data and of the large domain is done according tothe MDL principle, which selects the model based on its fit-ting performance and also penalizes a very high complexityof the model. The success of segmentation comes from theobservation that the more similar the gene-expression pro-files are in a large domain, the shorter the description of thedata that represents the large domain. We have applied thenew recursive segmentation method to the microarraymeasurements of the Drosophila genome and humangenome in order to demonstrate the ability of the newmethod to find large domains successfully.

AcknowledgmentsThe authors thank Prof. Lauri A. Altonen and Diego Arangofrom the Department of Medical Genetics, BiomedicumHelsinki, Finland, for their valuable support and for providingthe human genome data. Special thanks are also addressed toChristophe Roos from MediCel Ltd., Finland, for insightfuldiscussions and references on large domains of similarlyexpressed genes.

Daniel Nicorici received his B.S. and M.S.degrees in electrical engineering from theTechnical University of Cluj-Napoca,Romania, in 1999 and 2000, respectively.He received the Ph.D. degree in signal pro-cessing from Tampere University ofTechnology, Finland, in 2005. Since 2001,he has been with the Institute of Signal

Processing, Tampere University of Technology, as aresearcher. He is currently pursuing his postdoctoral studies atTampere University of Technology. His research interestsinclude genomic signal processing, bioinformatics, and com-putational systems biology.

Olli Yli-Harja received an M.Sc. in energytechnology (1985) and a Ph.D. in computerscience and applied mathematics (1989)from Lappeenranta University ofTechnology, Finland. His professionalexperience involves research and teachingin signal and image processing, computerscience, and computational systems biolo-

gy. Currently, he is a professor at the Institute of SignalProcessing, Tampere University of Technology, Finland, lead-ing a research group in computational systems biology. Hisresearch interests involve signal processing methods for sys-tems biology, nonlinear signal processing, computational sys-tems biology, discrete dynamic networks, image analysis, andcomputational analysis of music.

Jaakko Astola received his B.Sc., M.Sc.,Licentiate, and Ph.D. degrees in mathemat-ics (specializing in error-correcting codes)from Turku University, Finland, in 1972,1973, 1975, and 1978, respectively. From1976–1977 he was with the ResearchInstitute for Mathematical Sciences ofKyoto University, Kyoto, Japan. Between

1979 and 1987, he was with the Department of InformationTechnology, Lappeenranta University of Technology, Finland,holding various teaching positions in mathematics, appliedmathematics, and computer science. In 1984, he worked as avisiting scientist in Eindhoven University of Technology, TheNetherlands. From 1987–1992 he was an associate professorin applied mathematics at Tampere University, Tampere,Finland. Since 1993, he has been professor of signal process-ing and director of Tampere International Center for SignalProcessing, leading a group of about 60 scientists. He wasnominated as an academy professor by the Academy ofFinland (2001–2006). His research interests include signalprocessing, coding theory, spectral techniques, and statistics.He is a Fellow of the IEEE.

Address for Correspondence: Daniel Nicorici, Institute ofSignal Processing, Tampere University of Technology, P.O.Box 553, FIN-33101 Tampere, Finland. E-mail:Daniel.Nicorici@tut.fi.

References[1] F.C. Collins, M. Morgan, and A. Patrinos, “The human genome project:Lessons from large-scale biology,’’ Sci., vol. 300, no. 5617, pp. 286–290, 2003.[2] H. Caron et al., “The human transcriptome map: Clustering of highly expressedgenes in chromosomal domains,’’ Sci., vol. 291, no. 5507, pp. 1289–1292, 2001.[3] B.A. Cohen, R.D. Mitra, J.D. Hughes, and G.M. Church, “A computationalanalysis of whole genome expression data reveals chromosomal domains of geneexpression,’’ Nature Genetics, vol. 26, no. 2, pp. 183–186, 2000.[4] M.J. Lercher, A.O. Urrutia, and L.D. Hurst, “Clustering of housekeeping genesprovides a unified model of gene order in the human genome,” Nature Genetics,vol. 31, no. 2, pp. 180–183, 2002.[5] P.T. Spellman and G.M. Rubin, “Evidence for large domains of similarlyexpressed genes in Drosophila genome,’’ J. Biol., vol. 1, no. 5, pp. 1–5, 2002.[6] J. Rissanen, “Modeling by shortest data description,’’ Automatica, vol. 14, pp. 465–471, 1978.[7] J. Rissanen, “Strong optimality of the normalized ML codes as universal codesand information in data,’’ IEEE Trans. Information Theory, vol. IT-47, no. 5, pp.1712–1717, 2001.[8] M.H. Hansen and B. Yu, “Model selection and the principle of minimumdescription length,’’ J. Amer. Statistical Assoc. , vol. 96, no. 454, pp. 746–774, 2001.[9] I. Tabus, J. Rissanen, and J. Astola, “Classification and feature gene selectionusing the normalized maximum likelihood model for discrete regression,’’ SignalProcessing, vol. 83, no. 4, pp. 713–727, 2003.[10] I. Tabus, G. Korodi, and J. Rissanen, “DNA sequence compression using thenormalized likelihood model for discrete regression,’’ in Proc. Data CompressionConf., Snowbird, UT, 2003, pp. 253–262.[11] D. Nicorici, O. Yli-Harja, and J. Astola, “An MDL method for finding largedomains of similarly expressed genes,” in Proc. Workshop Genomic SignalProcessing and Statistics (GENSIPS), Baltimore, Maryland, 2004.[12] P. Bernaola-Galvan, I. Grosse, P. Carpena, J.L. Oliver, R. Roman-Roldan, andH.E. Stanley, “Finding borders between coding and noncoding DNA regions by anentropic segmentation method,’’ Physical Rev. E , vol. 85, no. 6, pp. 1342–1345, 2000.[13] W. Li, P. Bernaola-Galvan, F. Haghighi, and I. Grosse, “Applications ofrecursive segmentation to the analysis of DNA sequences,’’ Computers andChemistry, vol. 26, pp. 491–510, 2002.[14] D. Nicorici and J. Astola, “Segmentation of DNA into coding and noncodingregions based on recursive entropic segmentation and stop-codon statistics,’’ J.Applied Signal Processing, vol. 1, no. 1, pp. 81–91, 2004.[15] M. Koivisto, M. Perola, T. Varilo, W. Hennah, J. Ekelund, M. Lukk, L.Peltonen, E. Ukkonen, and H. Mannila, “An MDL method for finding haplotypeblocks and for estimating the strength of haplotype block boundaries,” in Proc.Pacific Symposium Biocomputing 2003 (PSB’03), Hawaii, 2003, pp. 502–513.[16] A. Barron, J. Rissanen, and B. Yu, “The minimum description length princi-ple in coding and modeling,’’ IEEE Trans. Inform. Theory, vol. 44, no. 6, pp. 2743–2760, 1998.[17] P.T. Spellman, and G.M. Rubin, Web supplement to: “Evidence for largedomains of similarly expressed genes in Drosophila genome,” J. Biol. [Online].Available: http://www.fruitfly.org/expression/dse/[18] L.A. Soinov, M.A. Krestyaninova, and A. Brazma, Web supplement for“Towards reconstruction of gene networks from expression data by supervisedlearning,’’ Genome Biol., vol. 4, no. R4 [Online]. Available: http://genomebiolo-gy.com/2003/4/1/R6