Without loss of generality, we use the term "tag" to refer to the transcripts' representation, as usual in the SAGE field (this is equivalent to the term "signature" in MPSS analysis or "contigs" in EST analysis). The theoretical model used here to describe the transcript enumeration process is the usual uniform sampling of interchangeable colored balls from an infinite urn model. Given the total number n of counted tags and the abundance vector π of all transcripts, this model leads to a probabilistic description of the observed result: x|π, n ~ Multi(π, n), i.e., the counts x follow a Multinomial distribution 16. It is also possible to model x as Poisson distributed 17 since it is an approximation for the Multinomial. Regardless of the specificities of the theoretical probabilistic model, it is well known that, as with other counting or voting processes, the natural space for dealing with this kind of data is the simplex space. The unitary simplex space, having d dimensions, is defined as 1819:

S d − 1 = { π | π ∈ ℝ + d , π 1 ′ = 1 } MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaacqWGtbWudaWgaaWcbaGaemizaqMaeyOeI0IaeGymaedabeaakiabg2da9iabcUha7HGadiab=b8aWjabcYha8jab=b8aWjabgIGioprr1ngBPrwtHrhAYaqeguuDJXwAKbstHrhAGq1DVbaceaGae4xhHi1aa0baaSqaaiabgUcaRaqaaiabdsgaKbaakiabcYcaSiab=b8aWHqabiqb9fdaXyaafaGaeyypa0JaeGymaeJaeiyFa0haaa@4E6F@

where 1 is a vector of ones. In the gene expression context, d is the number of unique tags observed. An example of a simplex vector is p = [π|x] and applying a standard Bayesian approach, one obtains from x|π, n, using a Dirichlet prior density π ~ Dir(α), the posterior density: π|x ~ Dir(x + α).

It is known that clustering analysis is inherently dependent on the choice of a distance measure between the considered objects. This, in turn, is connected to the structure of the underlying space. A metric Δ, measuring the distance between two objects a and b, must respect the properties:

(i) Δ(a, b) = Δ(b, a);

(ii) Δ(a, b) = 0 ⇔ a = b;

(iii) Δ(a, c) ≤ Δ(a, b) + Δ(b, c).

One may also consider additional reasonable properties such as:

(iv) scale invariance Δ(xa, yb) = Δ(a, b), x, y ∈ ℝ₊; and

(v) translational invariance Δ(a + t, b + t) = Δ(a, b).

These commonly required additional properties guarantee that distance measurements are not affected by the definition of arbitrary scale or measurement units and that more importance is given to the actual difference between the objects being measured rather than commonalities (more details can be found in the appendix Additional File 1).

Translations on the simplex space are defined by 19:

p ⊕ t = ( p ⋅ t ) ( p ⋅ t ) 1 ′ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaaieWacqWFWbaCcqGHvksXcqWF0baDcqGH9aqpdaWcaaqaaiabcIcaOiab=bhaWjabgwSixlab=rha0jabcMcaPaqaaiabcIcaOiab=bhaWjabgwSixlab=rha0jabcMcaPGqabiqb+fdaXyaafaGae4hiaacaaaaa@4204@

where · is the usual Hadamard product and the division is vector-evaluated.

Well known distances, such as Euclidean, Manhattan, and correlation-based distances, do not exhibit the properties (i)-(v) if the measured objects belong to the simplex space, as is the case of transcript enumeration data. A possible metric that obeys (i)-(v) on the simplex space is the Aitchisonean distance 19:

Δ ( p , q ) = l n ( p − d / p d q − d / q d ) ( I + 1 ′ × 1 ) − 1 l n ( p − d / p d q − d / q d ) ′ MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbwvMCKfMBHbqedmvETj2BSbqee0evGueE0jxyaibaieIgFLIOYR2NHOxjYhrPYhrPYpI8F4rqqrFfpeea0xe9Lq=Jc9vqaqpepm0xbbG8FasPYRqj0=yi0lXdbba9pGe9qqFf0dXdHuk9fr=xfr=xfrpiWZqaaeaabiGaaiaacaqabeaabeqacmaaaOqaaiabfs5aejaacIcatCvAUfKttLearyWrP9MDH5MBPbIqV92AaGabdiab=bhaWjaacYcacqWFXbqCcaGGPaGaeyypa0ZaaOaaaeaacaWGSbGaamOBamaabmaabaWaaSaaaeaacqWFWbaCdaWgaaWcbaGaeyOeI0IaamizaaqabaGccaGGVaGaamiCamaaBaaaleaacaWGKbaabeaaaOqaaiab=fhaXnaaBaaaleaacqGHsislcaWGKbaabeaakiaac+cacaWGXbWaaSbaaSqaaiaadsgaaeqaaaaaaOGaayjkaiaawMcaaiaacIcacqWFjbqscqGHRaWkiqqacuGFXaqmgaqbaiabgEna0kab+fdaXiaacMcadaahaaWcbeqaaiabgkHiTiaaigdaaaGccaWGSbGaamOBamaabmaabaWaaSaaaeaacqWFWbaCdaWgaaWcbaGaeyOeI0IaamizaaqabaGccaGGVaGaamiCamaaBaaaleaacaWGKbaabeaaaOqaaiab=fhaXnaaBaaaleaacqGHsislcaWGKbaabeaakiaac+cacaWGXbWaaSbaaSqaaiaadsgaaeqaaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGccWaGqBOmGikaaaWcbeaaaaa@706D@

where I is the identity matrix, × is the Kronecker product, -d subscript is a notation for "excluding the d^th element", and elementary operations are vector-evaluated.

Clustering procedures coherent with this theoretical background are suitable for transcript enumeration data.

In short, Simcluster's method can be described as the use of a Bayesian inference step (currently with a uniform prior) to obtain the expected abundance simplex vectors given the observed counts E MathType@MTEF@5@5@+=feaafiart1ev1aaatCvAUfKttLearuWrP9MDH5MBPbIqV92AaeXatLxBI9gBaebbnrfifHhDYfgasaacH8akY=wiFfYdH8Gipec8Eeeu0xXdbba9frFj0=OqFfea0dXdd9vqai=hGuQ8kuc9pgc9s8qqaq=dirpe0xb9q8qiLsFr0=vr0=vr0dc8meaabaqaciaacaGaaeqabaqabeGadaaakeaatuuDJXwAK1uy0HMmaeHbfv3ySLgzG0uy0HgiuD3BaGabaiab=ri8fbaa@388C@[π|x], and the use of the Aitchisonean distance in the following algorithms: k-means, k-medoids and self-organizing maps (SOM) for partition clustering, PCA for inferring the number of variability sources present, and common variants of agglomerative hierarchical clustering.

Currently, the Simcluster package is comprised of: Simtree, for hierarchical clustering; Simpart, for partition clustering; Simpca for Principal Component Analysis (PCA); and several utilities such as TreeDraw, a program to draw hierarchical clustering dendrograms with user-defined colored leaves. Simcluster's modularity allows relatively simple extension and addition of new modules or algorithms. Increasing the coverage of algorithms and validity assessment methods 20 are envisioned in future updates. Simcluster can be used, modified and distributed under the terms of the GPL license 21. The software was implemented in C for improved performance and memory usage, assuring that even large datasets can be processed on a regular desktop PC (Additional File 2).

To increase source code reuse, established libraries were used: Cluster 3 22 for clustering, GNU Scientific library 23 for PCA, Cairo 24 and a modification of TreeDraw X 25 for colored dendrogram drawing. The input data set can be a matrix of transcript counts or general simplex vectors. Some auxiliary shell and Perl scripts are available to: automatically download data from the GEO database 26, convert GEO files to Simcluster input format, and filter out low-count tags.

The Linux-based installation and compilation is facilitated by a configuration script that detects all the prerequisites for Simcluster compilation. Missing libraries are automatically downloaded from the Simcluster website and compiled by the Simcluster compilation process.

To broaden usability, a user-friendly web interface was developed and is made available at http://xerad.systemsbiology.net/simcluster_web/. Figure 1 shows a screenshot of an analysis session using Simcluster's web-based interface.