Joint mapping of genes and conditions via multidimensional unfolding analysis

BioMed CentralBMC Bioinformatics

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Address: 1SymBioSys, Catholic University of Leuven, 3000 Leuven, Belgium, 2Department of Microbial and Molecular Systems, Catholic University of Leuven, 3000 Leuven, Belgium and 3Department of Psychology, Leiden University, 2300 RB Leiden, The Netherlands

Email: Katrijn Van Deun* - [email protected]; Kathleen Marchal - [email protected]; Willem J Heiser - [email protected]; Kristof Engelen - [email protected]; Iven Van Mechelen - [email protected]

* Corresponding author

AbstractBackground: Microarray compendia profile the expression of genes in a number of experimentalconditions. Such data compendia are useful not only to group genes and conditions based on theirsimilarity in overall expression over profiles but also to gain information on more subtle relationsbetween genes and conditions. Getting a clear visual overview of all these patterns in a single easy-to-grasp representation is a useful preliminary analysis step: We propose to use for this purposean advanced exploratory method, called multidimensional unfolding.

Results: We present a novel algorithm for multidimensional unfolding that overcomes bothgeneral problems and problems that are specific for the analysis of gene expression data sets.Applying the algorithm to two publicly available microarray compendia illustrates its power as atool for exploratory data analysis: The unfolding analysis of a first data set resulted in a two-dimensional representation which clearly reveals temporal regulation patterns for the genes and ameaningful structure for the time points, while the analysis of a second data set showed thealgorithm's ability to go beyond a mere identification of those genes that discriminate betweendifferent patient or tissue types.

Conclusion: Multidimensional unfolding offers a useful tool for preliminary explorations ofmicroarray data: By relying on an easy-to-grasp low-dimensional geometric framework, relationsamong genes, among conditions and between genes and conditions are simultaneously representedin an accessible way which may reveal interesting patterns in the data. An additional advantage ofthe method is that it can be applied to the raw data without necessitating the choice of suitablegenewise transformations of the data.

BackgroundComplex microarray experiments profile the expression ofa large number of genes under different conditions (envi-ronmental conditions, knockout experiments, patients),

and/or over time. Depending on the biological questionat hand, one may be interested in finding subsets of genesthat can be clustered together based on similarities in theiroverall expression profile, or in finding subsets of condi-

Published: 5 June 2007

BMC Bioinformatics 2007, 8:181 doi:10.1186/1471-2105-8-181

Received: 3 January 2007Accepted: 5 June 2007

This article is available from: http://www.biomedcentral.com/1471-2105/8/181

© 2007 Van Deun et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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tions (tissues, patients) that can be grouped togetherbased on similarities in their overall gene profile. Alsomore subtle relations between genes and conditions canbe envisaged, such as biclusters of genes being co-expressed over a subset of conditions only (modules) orgroups of genes being discriminative for subsets of condi-tions. However, the massive amount of information andrelations present in the data, pose a challenge for the dataanalyst: It is not trivial to know where to start looking forstructure and a priori choices can have the consequencethat something is missed. For instance, many cluster algo-rithms require defining in advance the number of clustersto be searched for, a parameter which is difficult to guessin advance. Therefore, having a rough idea on the mostprominent patterns present in the data and the (unex-pected) particularities, prior to performing a more pro-found analysis may be most useful. Exploratory methodsoffer the possibility to reduce the data to a manageableamount of information, for example by means of a clus-tering of the individual elements to a small number ofgroups or by means of reducing them to a small numberof dimensions (e.g., PCA/SVD). Often, such methodsyield insightful graphical representations. Ideally, suchrepresentations should display genes and conditionsjointly in a way that associations amongst genes, amongstconditions and between genes and conditions are all threeeasy to grasp.

From this perspective, multidimensional unfolding(MDU) seems a promising data exploration technique(for an introduction to MDU see [1] and [2]): Thismethod maps both genes and conditions into the samelow-dimensional space such that, 1) genes are locatedclosest to the conditions for which they exhibit the highestexpression levels, 2) genes (respectively conditions) witha more similar expression profile are located closer to eachother in the space. The resulting MDU configurations arevery easy to interpret and give a quick first insight into theoverall structure of the data and its particularities. Anadditional asset of the method, is that it can be applied toraw gene expression data: In contrast to results obtainedfrom other exploratory methods, results of MDU are inde-pendent of gene-specific transformations applied to theinput data.

Although theoretically suitable as a data exploration tech-nique, current MDU algorithms cannot readily be applieddue to problems of a general kind and of problems thatare specific for the case of microarray gene expressiondata. As regards problems of a general kind: first, somealgorithms do not converge to a local minimum and yieldunstable results; second, in many cases MDU representa-tions are not well interpretable due to a sticking togetherof a majority of gene and condition points implying thatthey cannot be discriminated from one another. As

regards problems that are specific for the case of geneexpression data, first, existing MDU algorithms have notbeen designed for the analysis of data sets of the typicalsizes of microarray data as they require a large amount ofmemory; second, existing MDU algorithms also are com-putationally very intensive (e.g., because they rely onmatrix inversions). To deal with these problems, in thepresent paper we propose a novel MDU algorithm. A sub-sequent application of it to two publicly available micro-array datasets, each of which serving a different biologicalpurpose, will demonstrate its exploratory power.

ResultsMethodThe purpose of a multidimensional unfolding of geneexpression data is to find coordinates in a low-dimen-sional space, both for the genes and the conditions, in away that the (Euclidean) distance of a gene point to a con-dition point is shorter the higher the gene is expressed inthat condition. Note that MDU can be considered as anextension of multidimensional scaling (MDS) to the rec-tangular case (see chapter 14 of [1]). To formalize MDUwe will use the following notation: Genes are indexed byi = 1 ... n, the conditions by j = 1 ... m, and the dimensionsof the low-dimensional space by r = 1 ... p. Also, let E bethe n × m expression matrix for the n genes measured in mconditions with eij representing the expression of gene i incondition j and ei the m-sized vector representing theexpression profile of gene i.

To map E to a p-dimensional space, a n × p matrix of genecoordinates X and a m × p matrix of condition coordinatesY are sought such that the Euclidean distances for gene i,contained in the m-sized vector di and with

, reflect the expression profile ei

of gene i.

To find X and Y such that n vectors of distances di reflectthe expression profiles ei, we will maximize

the average squared correlation between the expressionprofiles and the distances in the low-dimensional repre-sentation. Because higher expression levels are to corre-spond to shorter distances, the summation runs only overthose genes for which there is a negative correlationbetween the expression levels and the Euclidean distances(denoted by r(ei, di) < 0). In order to maximize (1), thecoordinate matrices X and Y have to be such that the dis-tance vectors correlate as negatively as possible with the

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profiles while positive correlations are to be avoided. Animportant aspect of the optimization criterion, is that anyset of positive genewise linear transformations of the rawexpression data E, will yield the same optimal X, Y,because the correlation is insensitive to linear transforma-tions; as such, tough questions about preprocessing, inso-far they pertain to gene-specific linear transformations arebypassed.

AlgorithmTo find X and Y that maximize (1), we reformulate thisoptimization problem to an equivalent one, namely min-imizing

(with var and std denoting variance and standard devia-tion respectively) with respect to ai, X, and Y under theconstraint that ai ≥ 0 for all i; see the appendix for a proofof the equivalence. For reasons given below, two moreconstraints are added to the optimization problem: First,the ai weights are bounded by an upper bound u (suchthat 0 ≤ ai ≤ u); and second, ||xi|| ≤ 1 for all i in a space cen-tered at the point of gravity for the condition coordinatesyj. Note that centering can be done without loss of gener-ality.

For the minimization of (2) with respect to the ai's, X, andY under the constraints 0 ≤ ai ≤ u and ||xi|| ≤ 1, we proposethe algorithm GENEFOLD (which may be considered amajor upgrade of the algorithm proposed in [3]). Adetailed description of it along with a MATLAB imple-mentation can be found at [4]. GENEFOLD is of an alter-nating least squares type. In each iteration the ai's, X, andY are updated each in turn while the remaining parame-ters are kept fixed. The (constrained) update of the aiweights can be done on the basis of a closed form expres-sion (see appendix). The update of the gene coordinates Xunder the constraint ||xi|| ≤ 1 for all i, as well as the updateof the condition coordinates Y, are based on a numericaltechnique called iterative majorization (see [5-7] for theuse of iterative majorization in the context of multidimen-sional data analysis). Briefly said, iterative majorizationrelies on surrogate objective functions with the followingproperties: The surrogate function is easier to minimizethan the original, it lies above the original function, andthe surrogate function touches the original function in theso-called supporting point. By choosing the supportingpoint equal to the minimum of the surrogate function inthe previous iteration, the sequence of loss-values will benon-increasing.

GENEFOLD solves both the general MDU problems andthe problems that are specific for gene expression data.

With respect to the general problems, first convergence isguaranteed because the proposed algorithm yields a non-increasing sequence of loss values for a function which isbounded below (by zero). Second, the problem of a lackof discrimination of the coordinates such that a majorityof points stick together (which is known as the degeneracyproblem in MDU), is solved by the constraints ai ≤ u and||xi|| ≤ 1. 1) Due to the restriction ||xi|| ≤ 1 in a space cen-tered at the point of gravity of the condition coordinates,the gene points lie on the unit sphere, and 2) limiting ai tovalues smaller than or equal to u with the value of u wellchosen (see the appendix), pulls the variance of the dis-tances di to the variance of the distances on the unit spherewith uniformly distributed points. With regard to prob-lems that are specific for the analysis of (large) geneexpression data sets; first, GENEFOLD works on consider-ably smaller matrices than the (n + m) × (n + m) used inclassical procedures for MDU (in which MDU is treated asa special case of MDS); second, GENEFOLD does not relyon computationally intensive methods like matrixinverses [1,8]. As an illustration of the computationalspeed of GENEFOLD: with 100 iterations, the analysis ofa 517 × 12 matrix takes about a second and of a 6075 ×173 matrix about 6 minutes (on a desktop, Pentium 2.80GHz4 CPU with 0.99 GB RAM).

ApplicationsWe applied multidimensional unfolding to two publiclyavailable data sets, one situated in an experimental con-text [9] where the study aimed at characterizing the tem-poral program of gene expression in human fibroblastsand one situated in a clinical setting [10] where the aimwas to classify two tissue types on the basis of the geneexpression levels.

Time-course gene expression dataThe data discussed in [9] pertain to the temporal changeof genes in human fibroblasts that had been deprivedfrom serum for 48 hours which causes them to enter anondividing state. The deprivation was ended by additionof a medium containing fetal bovine serum (FBS) andmicro-array hybridization was performed at severalmoments during the 24 hours following serum stimula-tion. We will analyze the 517 genes that were also retainedby [9] and that can be obtained at [11].

Because our unfolding algorithm relies on an iterativeprocedure with a non-convex solution space and a pre-specified dimensionality, some consideration has to begiven to the choice of a stopping rule, to the problem oflocal minima, and to the dimensionality of the configura-tion. With respect to the stopping rule, we chose to termi-nate the iterative procedure when the difference in lossbetween the current and previous solution was less than10-5; our experience with this value is that it yields stable

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solutions (in the sense that more iterations result inalmost the same configuration and loss) in a reasonableamount of time. The problem of local minima wasaccounted for by restarting the algorithm 101 times, using100 semi-rational starts and a rational start for the initialcoordinates, the solution with the lowest loss beingretained. The dimensionality of the configuration is deter-mined by a comparison of loss values: For the one up tofive-dimensional solution, loss was respectively 0.55,0.25, 0.19, 0.15, 0.11, which suggests a two-dimensionalconfiguration (one dimension less results in a considera-ble increase in loss while more dimensions barely reducethe loss). For the two-dimensional configuration, a verygood fit was obtained: The average genewise correlationbetween the distances and the raw data is -0.86. A visualrepresentation of the solution is depicted in Figure 1where the genes are denoted by dots and the time pointsby self-explanatory labels.

A striking feature of Figure 1 is the clocklike organizationof the time points, which is characterized by the followingfeatures: First, the points lie approximately on an elon-gated circle; second, they are ordered according to time;and third, the last time points fold somewhat back to theearliest. Note that no information on the order of the timepoints is included in the unfolding analysis; the orderedoutcome is therefore not a trivial finding. The unfoldinganalysis also reveals that there is little differentiationbetween some time points; for example, the time points 0hr and 15 min are clustered together, which means thatexpression 15 minutes after stimulation is barely differentfrom expression during the nondividing state (0 hr); thesame holds for the time points 30 min, 1 hr, 2 hr and 16hr, 20 hr, 24 hr, while the time points 4 hr, 6 hr, 8 hr, 12hr are more spread out. The large gaps between 2 hr and 4hr and between 12 hr and 16 hr suggest a biological eventoccurring within these time intervals.

Taking a look at the genes, we see that these, too, areorganized in a circular way with a blank spot in the mid-dle. Another feature to look at, is the location of themajority of the genes: Most are located close to the earliestand latest time points, whereas only a few genes arelocated at intermediate time points. The expression of agene at the different time points is reflected by the dis-tances from this gene to the time points: The closer a timepoint is located to a gene, the higher the expression levelor, conversely, the more distant a time point is from agene, the lower the expression level. Note that for thesedata, we know that the expression at 0 hr corresponds toa neutral state, such that higher expression levels indicateinduction and lower expression levels repression. Thismeans that induction occurs at time points close by whilerepression occurs at distant time points. For ease of inter-pretation, we used distinctive labels for genes with an

induction peak and with a repression peak: Genes with aninduction peak being those for which the difference indistance between the reference time point and the timepoint closest to the gene point is larger then the differencein distance between the reference time point and the timepoint furthest from the gene point (i.e., genes for whichthe largest difference in expression level from the expres-sion level at 0 hr is positive, respectively negative).Remember further that the distances between gene andcondition points inversely reflect the expression level.Consider, for example, the gene represented by the squarenumbered one in Figure 1 (this is close to 0 hr): From theunfolding configuration, it can be derived that this genewill have its highest expression at 0 hr, the time point thatis closest to it; continuing in a time-wise direction, expres-sion decreases up to 12 hr as is reflected by the increasingdistance; from that point on, the distances become pro-gressively shorter, which suggests that the expression lev-els steadily increase. The resulting expression profile isplotted in the upper part of the right panel of Figure 1 (theconnected dots represent the profile as modeled by thedistances; the non-connected dots the original data pro-file); the horizontal axis represents time, the vertical axisthe expression levels as they are modeled by the unfoldingrepresentation. The time axis is proportional to real time(e.g., the tick mark for 12 hr is placed 48 times furtherthan the tick mark for 15 min); the modeled expressionlevels are obtained (per gene) by subtracting the distancefrom the distance at time 0 hr and multiplying these dis-tances by minus one. In the expression profiles in Figure1, the neutral state is indicated by the horizontal (refer-ence) line. Thus the configuration suggests that gene one,and all those that are close to it, will be repressed soonafter stimulation with the highest repression occurringfrom 8 to 12 hours. Therefore, for genes close to gene one,there appears to be little or no induction. Otherwise, thegene labeled one (gene number 21 in the original dataset), belongs to the first cluster of genes found by [9] andis an inhibitor of the progression of the cell-cycle division(p57 Kip2). Profiles for other genes can be derived in asimilar way: In Figure 1, four additional profiles are givenfor the genes numbered two to five in the left panel. Notethat the gene numbered five, which is located in the centerof the clock, badly fits the original data: The correspond-ing derived profile is irregular and unstable in that smallchanges in the location of this gene would result inanother profile, given the fact that all time points arealmost equally distant.

Taguchi and Oono [12] analyzed the same data, leavingout the preset expression level at time 0 hr. They appliednonmetric multidimensional scaling on the matrix of dis-similarities between genes where dissimilarities weremeasured by the Pearson correlation coefficient with thesign flipped. As a result they obtained a two-dimensional

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configuration in which the genes were arranged on theedge of a circular structure. To detect the temporal regula-tion, the authors subsequently drew the configuration ateach time point, plotting only those genes that exceeded apreset expression peak: As shown in [12], the expressionpeaks move gradually around the circle in a timely fash-ion. These authors also take up the discussion on the peri-odicity of genes in relation to the cell-cycle. They arguethat the ring-like structure is in favor of periodicity in thedata. Yet, undoubtedly, the unfolding approach is a muchbetter technique to tackle this substantive issue: In case ofperiodicity, the time points will fall approximately on acircle with time points that are separated k periods fallingtogether. As illustrated by Figure 1, the clockwise organi-zation of the time points suggests some periodicity but thefact that the last time points do not coincide with the ear-liest ones, does not fit within a periodic frame. Given theexperimental difficulties encountered in studies thatinvolve temporal regulation of genes, we do not wish todraw any conclusions concerning the presence or absenceof periodicity in this particular data set. It should be clear,however, that multidimensional unfolding is a particu-larly suitable analysis technique to deal with such anissue.

Colon cancer dataMany applications of gene expression profiling can befound in clinical settings where genome-wide expressionis measured for different patient or tissue groups. A chal-lenge for the exploratory MDU tool within this settingmay be to retrieve useful information beyond a mere iden-tification of genes that optimally discriminate betweenthe different groups.

We consider gene expression data for 62 colon tissues, 40of which are tumorous (colon adenocarcinoma) and 22normal [10] (see [13] for the data in MATLAB format). Toobtain those genes that discriminate optimally betweenthe groups, we selected of the 2000 available genes those400 that have the highest correlation with the binary clas-sifier (normal versus tumor). The 400 × 62 expressionmatrix was subjected to a MDU analysis yielding the fol-lowing loss values for the one up to five dimensional solu-tions: 0.80, 0.38, 0.31, 0.27, and 0.24. The two-dimensional solution offered the best trade off between fitand sparseness with an average correlation between dis-tances and expression profiles amounting to -0.78.

Unfolding configuration for time experiment dataFigure 1Unfolding configuration for time experiment data. Unfolding configuration (left panel) and derived expression profiles (right panel) for five selected genes. Genes labeled by a dot show a repression peak, while genes labeled by a cross show an induction peak compared to the initial time point 0 hr. Some genes are labeled by a black square and also have a numbered label: their derived expression profile is given in the right panel (the connected black dots), together with the original profile (the unconnected blue dots).

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TissuesAs to be expected, the normal and colon cancer tissues areseparated in the MDU configuration, see the left panel ofFigure 2: At the left, we find the normal tissues (labeled byN) and at the right the tumorous ones (labeled by Tu);two normal tissues are erroneously grouped with thetumors and, conversely, five tumor samples are placedwith the normal samples (these results are comparablewith the results in [10]). However, the right panel of Fig-ure 2 in which the tissues are labeled according to thepatient number (such that the same number correspondsto the same patient) includes a clear indication that incase of patient number 36 the tissues have been misla-beled rather than misclassified. A further aspect of theMDU configuration that jumps to the eye, is the separa-tion of the cancer tissues in two groups, one located in theupper right of Figure 2 and one in the lower right. We can-not be certain of the cause of this separation, but based onthe available patient descriptions, it might be related tothe contamination of tumor tissue with normal tissue, thecancer stage, or both. In Figure 3 some of the tissues arelabeled in function of the percentage of contaminationwith normal tissue, and the stage of the cancer (rangingfrom A, early stage, to D, metastasis). It can be seen thatthe top group contains tissues of a more developed cancerstage and are less contaminated than the bottom group(this information was found in [14]). The fact that morecontaminated tissues do not shift towards the group ofnormal tissues, might suggest that the degree of contami-nation is not the prime cause for the observed subdivi-sion. Related to this, all misclassifications are situated inthe region containing the more contaminated tissue sam-ples.

GenesTaking a look at the genes in Figure 2 (left panel), there aretwo clearly separated groups, one associated to the normaltissues and one associated to the tumor tissues. To retrievethe functional annotation of the genes in these groups, weused DAVID [15] on a selection of genes that are fittedwell by the MDU representation (r(ei,di) ≤ -0.80). For thegenes closer to the normal tissues, consistent with theresults in [10], we found a cluster of genes annotated withmuscle contraction. As to the genes associated to thetumor tissues, several functional groups were discernedwhich are further labeled using different symbols andcolors in the right panel of Figure 3. These functionalgroupings seem to hint at an elevated cellular metabolismreflecting the higher metabolic activity and division ratetypical for cancer cells: a group of ribonucleoprotein genes(red dots), a group of ribosomal protein genes (greensquares), a group of proteasome genes (blue triangles), agroup of protein folding genes (yellow stars), and a groupof kinases related to cell cycle regulation.

DiscussionMultidimensional unfolding can be a useful tool whendealing with the challenging task of extracting usefulinformation from microarray gene expression data: Asshown in this paper, MDU yields easy-to-grasp represen-tations and appears to be a versatile tool for data explora-tion that may reveal many kinds of interesting patternspresent in the data. For example, in the first application,an intriguing clock-like structure for the time points wasrevealed, a pattern that has not been uncovered in a directway up to present for these well-studied data; in the sec-ond application, the unfolding analysis revealed anintriguing subdivision of the cancer tissue groups, beyonda mere discrimination of normal and tumor tissues. A pos-sible limitation of the unfolding approach as presentedhere, is that in case of a large number of heterogeneousconditions, low-dimensional configurations can beobtained that are mainly blurred due to the actual high-dimensional structure of the data. Also, a huge number ofgenes can result in a configuration that provides littleinsight into the data. A possible way to overcome this lim-itation could be the use of a hybrid approach that resultsin low-dimensional distance-based representations ofclustered data. Such an approach, has already been pro-posed for multidimensional scaling [16] and for the clus-tering of row elements in metric multidimensionalunfolding [17].

AppendixEquivalence of loss functionsWe show the equivalence of minimizing loss function (2)and maximizing (1). Consider the loss function for onegene,

Equation (3) is a plain quadratic form in ai which, underthe constraint ai ≥ 0, reaches its minimum at

if r(ei, di) < 0, and at 0 if r(ei, di) ≥ 0. Substituting the opti-mal ai's in loss function (2) yields

Obviously minimizing (5) is equivalent to maximizing(1).

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Unfolding configuration for colon cancer data: Normal versus tumorFigure 2Unfolding configuration for colon cancer data: Normal versus tumor. Unfolding configuration for the colon cancer data: The left and right panels only differ in the labels used for the tissues. Genes are labeled by blue dots in both panels; tissues in the left panel are labeled with 'N' for the normal tissues and with 'Tu' for the tumor tissues while in the right panel they are labeled in function of the patient number with positive numbers indicating a normal tissue for that patient and negative num-bers a tumor tissue.

Unfolding configuration for colon cancer data: Two tumor typesFigure 3Unfolding configuration for colon cancer data: Two tumor types. Unfolding configuration for the colon cancer data: In the left panel, tissues are labeled for the normal tissues with 'N' and for the tumor tissues either with 'Tu' or a label indicating both the Duke stage and the percentage of contamination with normal tissue. In the right panel, a detail of the unfolding config-uration is given that zooms in on the region containing genes that are more highly expressed in the tumor tissues. Different colors and symbols are used to discern the different functional gene groups: Red dots for the ribonucleoprotein genes, green squares for the ribosomal protein genes, blue triangles for the proteasome genes, orange asterisks for genes involved in pro-tein folding, and black asterisks for protein kinase genes.

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Choice upper bound on ai's and constrained updateFrom Equation (4), it can be seen that subjecting the ais toan upper bound u attracts the solution space to configura-tions with a positive lower bound on std(di), the spread ofthe distances, and this will be the more so the stronger thedistances correlate with the expression profiles. A suitablevalue for u depends on the range of the coordinates: Wepropose to work in a reference space centered at the pointof gravity of the condition coordinates and with ||xi|| ≤ 1(such that the gene coordinates lie within the unitsphere).

For this reference space, the upper bound for ai is set equalto u = (mv)-0.5 with v the variance of the Euclidean distancefrom a point i to points sampled uniformly in the unitsphere of dimensionality p with v calculated using MonteCarlo simulation. Using this upper bound for ai will, for amaximal (absolute) correlation r(ei, di) = -1, pull the var-iance of the distances towards v or larger values. The con-strained update for ai becomes then

Authors' contributionsKVD developed the algorithm and applied it to the twogene expression data sets discussed. IVM, KM, and KVDdrafted the manuscript. KM and KE made substantial con-tributions to the biological background. WJH and IVMscrutinized the unfolding method and its application togene expression data. All authors read and approved thefinal manuscript.

AcknowledgementsThis research was supported by the Centre of Excellence SymBioSys (Research Council KU Leuven EF/05/007) and by the Research Council KU Leuven (grant GOA/2005/04).

References1. Borg I, Groenen PJF: Modern Multidimensional Scaling: Theory and Appli-

cations 2nd edition. Springer series in statistics, New York: Springer-Verlag; 2005.

2. Heiser WJ, Busing FMTA: Multidimensional Scaling and Unfold-ing of Symmetric and Asymmetric Proximity Relations. InThe SAGE Handbook of Quantitative Methodology for the Social SciencesEdited by: Kaplan D. Thousand Oaks, California: Sage; 2004:25-48.

3. Van Deun K, Groenen PJF, Heiser WJ, Busing FMTA, Delbeke L:Interpreting degenerate solutions in unfolding by use of thevector model and the compensatory distance model. Psy-chometrika 2005, 70:23-47.

4. GENEFOLD: A GENE expression data multidimensionalunFOLDing MATLAB toolbox [http://ppw.kuleuven.be/okp/genefold/]

5. De Leeuw J: Applications of convex analysis to multidimen-sional scaling. In Recent developments in statistics Edited by: Barra JR,

Romier G, van Cutsem B. Amsterdam, The Netherlands: North-Hol-land; 1977:133-145.

6. De Leeuw J, Heiser WJ: Multidimensional scaling with restric-tions on the configuration. In Multivariate Analysis Volume 5. Editedby: Krishnaiah PR. Amsterdam, The Netherlands: North-Holland Pub-lishing Company; 1980:501-522.

7. Heiser WJ: Convergent computation by iterative majoriza-tion: theory and applications in multidimensional data anal-ysis. In Recent advances in descriptive multivariate analysis Edited by:Krzanowski WJ. Oxford: Oxford University Press; 1995:157-189.

8. Takane Y, Young FW, De Leeuw J: Nonmetric individual differ-ences multidimensional scaling: an alternating least squaresmethod with optimal scaling features. Psychometrika 1977,42:7-67.

9. Iyer VR, Eisen MB, Ross DT, Schuler G, Moore T, Lee JC, Trent JM,Staudt LM, James Hudson J, Boguski MS, Lashkari D, Shalon D, Bot-stein D, Brown PO: The Transcriptional Program in theResponse of Human Fibroblasts to Serum. Science 1999,283(5398):83-87.

10. Alon U, Barkai N, Notterman DA, Gish K, Ybarra S, Mack D, LevineAJ: Broad patterns of gene expression revealed by clusteringanalysis of tumor and normal colon tissues probed by oligo-nucleotide arrays. Proc Natl Acad Sci USA 1999, 96:6745-6750.

11. Data underlying Figure 2 in Iyer et al., Science, 1999 [http://genome-www.stanford.edu/serum/fig2data.txt]

12. Taguchi Yh, Oono Y: Relational patterns of gene expression vianon-metric multidimensional scaling analysis. Bioinformatics2005, 21:730-740.

13. MACBETH: MicroArray Classification BEnchmarking Tool[http://tomcat.esat.kuleuven.be/MACBETH/]

14. Notterman DA, Alon U, Sierk AJ, Levine AJ: Transcriptional geneexpression profiles of colorectal adenoma, adenocarcinoma,and normal tissue examined by oligonucleotide arrays. Can-cer Research 2001, 61:3124-3130.

15. DAVID: Database for Annotation, Visualization, and Inte-grated Discovery [http://david.abcc.ncifcrf.gov/]

16. Heiser WJ, Groenen PJF: Cluster differences scaling with awithin-clusters loss component and a fuzzy succesiveapproximation strategy to avoid local minima. Psychometrika1997, 62:63-83.

17. De Soete G, Heiser WJ: A latent class unfolding model for ana-lyzing single stimulus preference ratings. Psychometrika 1993,58:545-565.

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Page 8 of 8(page number not for citation purposes)