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Each group is then represented by a 'centroid' (the mean expression pattern of genes in the group), and the genes are repartitioned to the cluster whose centroid is most similar to their expression pattern.
Genes whose expression patterns are very similar to a given centroid will be assigned a high membership in that cluster, whereas genes that bear little similarity to the centroid will have a low membership.
The prototype centroids are refined in the subsequent steps: each gene is assigned a membership score to each of the prototype centroids, based on the Pearson correlation between the gene's expression pattern and the centroid in question.
Each of the centroid patterns is then recalculated as the weighted mean of all of the gene-expression patterns in the dataset, where each gene's weight is proportionate to its membership in the corresponding cluster (see Materials and methods for details regarding the calculations).
Genes that have a large membership to a given centroid will contribute more to the mean, and the new centroid will migrate in the direction of those genes.
The process of calculating gene-centroid memberships and updating the centroids is iterated until the centroid patterns become fixed (or until the termination criterion is met, as described in Materials and methods).
In the final step of the program, the membership of each gene to each centroid is calculated.
An example of this is a centroid that represents genes that were strongly induced by amino-acid starvation (cluster 2 in Figure 5b).
The fuzzy k-means algorithm [ 12] is based on the minimization of the objective function shown below, for a given fuzzy partition of the data, F, and a set of K cluster centroids, V where X i is the expression pattern of the i th gene in the dataset, V j is the centroid of the j th cluster, d is the Pearson distance between X i and V j , m XiVj is the membership of X i in cluster V j , N is the number of genes in the dataset, and K is the total number of clusters.
During the centroid refinement in each clustering cycle, new centroids were calculated on the basis of the weighted mean of all the gene-expression patterns in the dataset according to where each gene's membership m (a continuous variable from 0 to 1) was defined as and w was the gene weight: in the first clustering cycle, the gene weights used were those defined by the program Cluster, using a Pearson correlation cutoff of 0.
After each clustering cycle, the centroids were combined with those identified in previous cycles, and replicate centroids were averaged: each centroid was compared to all other centroids in the set, and centroid pairs correlated >0.
Genes that were correlated to any existing centroid with a Pearson correlation >0.
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