# Differential co-expression framework to quantify goodness of biclusters and compare biclustering algorithms

- Burton Kuan Hui Chia
^{1, 3}and - R Krishna Murthy Karuturi
^{2}Email author

**5**:23

https://doi.org/10.1186/1748-7188-5-23

© Hui and Karuturi; licensee BioMed Central Ltd. 2010

**Received: **21 January 2010

**Accepted: **28 May 2010

**Published: **28 May 2010

## Abstract

### Background

Biclustering is an important analysis procedure to understand the biological mechanisms from microarray gene expression data. Several algorithms have been proposed to identify biclusters, but very little effort was made to compare the performance of different algorithms on real datasets and combine the resultant biclusters into one unified ranking.

### Results

In this paper we propose differential co-expression framework and a differential co-expression scoring function to objectively quantify quality or goodness of a bicluster of genes based on the observation that genes in a bicluster are co-expressed in the conditions belonged to the bicluster and not co-expressed in the other conditions. Furthermore, we propose a scoring function to stratify biclusters into three types of co-expression. We used the proposed scoring functions to understand the performance and behavior of the four well established biclustering algorithms on six real datasets from different domains by combining their output into one unified ranking.

### Conclusions

Differential co-expression framework is useful to provide quantitative and objective assessment of the goodness of biclusters of co-expressed genes and performance of biclustering algorithms in identifying co-expression biclusters. It also helps to combine the biclusters output by different algorithms into one unified ranking i.e. meta-biclustering.

## Background

The inception of microarrays has facilitated quantification of expression of genes at genomic scale in large sets of conditions in time and cost effective manner resulting in a wealth of massive gene expression datasets. Appropriate analysis of these datasets lead to the understanding of the roles of various genes and pathways at genomic-scale.

Biclustering plays an important role in microarray gene expression analysis. Expression of a cluster of genes may be modulated only in a small subset of conditions demonstrating interesting biology of the condition dependent transcriptional co-regulation and potentially leading to understanding of the underlying mechanisms. For example, in knock out studies, certain groups of genes are activated or suppressed only in a small subset of knock-out conditions. Similarly, in cancer studies, due to heterogeneity of the tumors, certain groups of genes involving in a certain pathway may be co-expressed only in a subset of tumors. In the traditional clustering, the genes co-expressed over all conditions dominate the clustering analysis and the genes co-expressed only in a small subset of conditions may not be elicited.

*Cheng and Church's algorithm*(

*CC*) [5] minimizes mean squared error in linear model fit.

*Iterative Signature Algorithm (ISA)*[7] finds biclusters by maximizing z-scores of expression.

*Order Preserving Sub Matrix (OPSM)*[8] elicits biclusters by finding order preserving co-expression submatrices with highest statistical significance support.

*Statistical Algorithmic Method for Bicluster Analysis (SAMBA)*[12] is based on finding heavy subgraphs in the gene-condition bipartite graph. The algorithms are summarized in Table 1 for a quick reference.

*X*

_{ ij }=

*N*(0, 1) with co-expression for

*s*∈

*S*(all conditions) for

*|s|*«

*|S|*. The application of anyone of CC, ISA, OPSM and SAMBA algorithms can find this bicluster partly or fully as its genes are not co-expressed in the non bicluster conditions |

*S-s*| » |

*s*|. Whereas, application of same algorithms on lung [14], liver [15] and breast cancer [16] datasets resulted in biclusters (belonged to the top 10 biclusters output by each algorithm) with genes showing co-expression in non bicluster groups of conditions, see the Figure 2. This problem is not unique to any one algorithm but holds true for all biclustering algorithms as their scoring functions mainly depend on the bicluster conditions only. The presence of co-expression at comparable or better levels in the non-bicluster conditions show that the co-expression and biology of the bicluster genes is not limited to the conditions in the bicluster but it is a global effect. Therefore, evaluation on idealized simulated bicluster data may not be sufficient to reveal true effectiveness of a biclustering algorithm.

On real data, Prelic et al's [11] evaluation was based on the number of gene ontology (GO) terms enriched for the biclusters. It may not be a good measure for four reasons: (1) it solely depends on the genes in the biclusters and does not account for the conditions involved; (2) GO terms may be highly enriched even for normal clusters of genes which may not lack co-expression in any subset of the conditions; (3) it does not distinguish between good biclusters from traditional clusters; and, (4) it may be subjective owing to the hierarchical structure of the GO.

Hence, it is important to develop an objective scoring function that works well on real data to assess the quality or goodness of biclusters and hence the reliability of the biclustering algorithms. It will also be helpful in combining the results of applying different biclustering algorithms on a data into a single unified ranking, i.e. a meta-biclustering, which has not been addressed so far. It would be of great help as it facilitates best utilization of all biclustering algorithms as different algorithms may behave differently on different datasets.

In this paper we propose to develop such a scoring function based on differential co-expression framework similar to that proposed by Kostka and Spang [17]. In this framework, for a given bicluster, we fit two linear models for the expression of genes in the bicluster for the conditions in the bicluster and for the remaining (the non-bicluster) conditions separately. The resultant models are used together to assess goodness of the bicluster using our differential co-expression scoring function. Note that the aim of this paper is not to assess the efficiency of the biclustering algorithms in retrieving underlying biclusters in the data, but to assess how good the identified biclusters are and how to provide a good unified ranking of the biclusters (meta-biclustering algorithm) output by them. Using our scoring function we compare the performance of different biclustering algorithms on six real datasets.

## Results

### Differential co-expression framework for biclustering

_{1}conditions: one is obtained from J

_{1}bicluster conditions (aka group G

_{1}) and the other is obtained from J

_{2}non-bicluster conditions (aka group G

_{2}); J

_{1}+J

_{2}= M, the total number of conditions in the study. Each row corresponds to a gene and each column corresponds to a condition. Note that I is used to indicate both gene set and its cardinality, similar interpretation holds for the other sets of genes and conditions. The task is to find how well I genes form a bicluster on J

_{1}conditions compared to the J

_{2}conditions. If is a good bicluster then there should be a co-expression of I in J

_{1}and a clear differential co-expression of I between J

_{1}and J

_{2}conditions. To find it, we employ the framework developed for differential co-expression by Kostka and Spang [17], based on the linear modeling used by Cheng and Church [5], for both groups of conditions G

_{1}and G

_{2}. Specifically, the linear model for the expression of I genes in the condition group G

_{k}is as follows:

1 ≤ i ≤ I; 1 ≤ j ≤ J_{k}; 1 ≤ k ≤ 2

*X*

_{ ijk }is the log-expression of gene g

_{i}in condition p

_{jk}belonged to group G

_{k}. It is modeled as a summation of four factors:

*μ*

_{k}, effect of group (overall effect) G

_{k};

*τ*

_{ik}, effect of gene

*g*

_{ i }in

*G*

_{ k };

*β*

_{ jk }, effect of condition p

_{jk}in G

_{k}; and,

*ε*

_{ijk}, an iid random error or residual of

*g*

_{ i }in p

_{jk}. Based on this model, Kostka and Spang's procedure obtains the mean of the squared residuals (

*E*

_{ k }) to score a set of genes I on J

_{k}conditions as follows:

,
,
are the estimates of *τ*_{ik}, *β*_{jk}, and -*μ*_{k} respectively.

*τ*

_{ik},

*β*

_{ jk }and

*μ*

_{ k }) shown by four heatmaps in the Figure 3: (1)

**T**-type co-expression; (2)

**B**-type co-expression; and (3)

**μ**-type co-expression. T-type co-expression is depicted by strong gene only effects resulting in strong

*τ*

_{ik}s only as the effect of any condition over I is weak leading to weak or near-zero

*β*

_{jk}s and

*μ*

_{k}. B-type co-expression results from strong condition only effects leading to strong

*β*

_{jk}s only as the overall expression of a gene across the bicluster conditions is weak leading to weak or near zero

*τ*

_{ik}s and

*μ*

_{k}. But,

*μ*-type co-expression results due to the presence of strong gene as well as strong condition effects (strong

*τ*

_{ik}s and

*β*

_{jk}s) leading to strong

*μ*

_{k}. We use the coefficients

*τ*

_{ik}s,

*β*

_{jk}s to quantify different types of co-expression, which is the first step to quantifying differential co-expression, of I genes in J

_{1}and J

_{2}conditions. T

_{k}(b) and B

_{k}(b) quantify the T-type and B-type co-expression of genes in a bicluster

*b*:

I(b) is the number of genes in b and J_{k}(b) is number of conditions in G_{k} for b. Similar interpretation holds for the other variables also.

Theorem: T_{k} and B_{k} are the unbiased estimators of
and
respectively under the assumption that the noise in *X*_{
ijk
}follows *N*(0,
)

As *E*_{
k
}is an unbiased estimator of
,
is an unbiased estimator of *β*_{
k
}. Similarly
is an unbiased estimator of *Γ*_{
k
}. ■

In the above proof,
is a non-central Chi-square distribution with 'n' degrees of freedom and 'c' being the non-centrality parameter; ⟨Z⟩ is the expectation of the random variable *Z*.

#### Scoring goodness of biclusters

where 0<a<<1, it is a small fudge factor to offset large ratios based on very small co-expression in both groups of a bicluster. Strong positive SB(b) indicates strong co-expression in G_{1} and weaker or no co-expression in G_{2} vice versa.

Though we score a bicluster based on its differential co-expression, our quantification of differential co-expression by SB(b) is different from that used by Kostka and Spang, the S(b) = LOG(E_{1}(b)/E_{2}(b)), and their variance standardization approach for two reasons: (1) S(b) accounts mainly for B-type co-expression; and, (2) variance standardization does not account for different signal variances in the two groups.

#### Stratifying biclusters

_{k}(b) on the k

^{th}group which is declared to be co-expressed by SB(b):

where k = 1 if SB(b) > 0

= 2 if SB(b) < 0

_{b}(I) means the bicluster is of T-type (strong gene effects only), large negative score means the bicluster is of B-type (strong condition effects only) and small score close to 0 means they are of

*μ*-type (strong gene as well as condition effects). Therefore, user can define a parameter

*φ*> 0 to identify these three groups as follows:

### Evaluating Biclustering Algorithms and Combining Bicluster Lists

*BiCAT*toolbox [18] and SAMBA was applied using

*EXPANDER*package [19]. The ranking of the biclusters by each algorithm is the ranking generated by the respective package. The biclusters with fewer than 5 conditions were filtered out from the evaluation as they appear to be strong because of the small number of conditions and may not be significant.

Datasets used in the analysis

S. No | Dataset | Experiment | References | No. of Genes | No. of Samples |
---|---|---|---|---|---|

1 | Breast | Breast Cancer | Wang et al. [16] | 22283 | 286 |

2 | Liver | Liver Cancer | Chen et al. [15] | 10200 | 203 |

3 | Yeast | Knock Out in Yeast | Gasch, et al. [20] | 2993 | 173 |

4 | Lymphoma | Lymphoma and Normal | Alizadeh, et al. [21] | 4026 | 96 |

5 | Lung | Lung Cancer | Broët et al. [14] | 54837 | 79 |

6 | Path_Metabolic | Plant | Wille et al. [22] | 734 | 69 |

We have evaluated the biclustering algorithms based on four criteria: (1) number of biclusters found; (2) median number of conditions in the biclusters; (3) ranking of the biclusters generated by an algorithm in the combined ranking of all biclusters generated by all algorithms; and, (4) types of biclusters generated.

*Path_Metabolic*. Median number of conditions used by CC strongly depends on the number of conditions in the dataset as seen by the trends; whereas ISA and SAMBA show a weak dependency on the same. Interestingly, OPSM does not show any dependency on the number of conditions in the dataset. Notably SAMBA, OPSM and ISA output biclusters of similar size.

_{1}(b). We plot the cumulative distribution of the TS

_{1}(b) score of the biclusters output by each algorithm for each dataset as shown in the Figures 8 and 9, we set

*φ*= 1. The behaviour of the algorithms does appear to be dependent on the dataset. ISA output ~60% of the biclusters of B-type for Breast, only 15%-20% for the other datasets. Apart from B-type, it output only

*μ*-type biclusters and no T-type biclusters can be seen from ISA on any dataset. SAMBA output ~90% B-type in Breast and Lung, 40-50% in the remaining datasets. Strikingly, OPSM output only one type of biclusters for any dataset: only B-type biclusters were output on Breast, Liver and Lung datasets; only

*μ*-type biclusters for Yeast, Lymphoma and Path_Metabolic datasets. This could be because OPSM identifies order preserving biclusters of B-type. Like ISA and SAMBA, OPSM also have not output any T-type biclusters on any dataset. Interestingly, only CC output biclusters of T-type and it output more of

*μ*-type and T-type biclusters compared to B-type biclusters except on Breast data. On the whole it appears that all algorithms favoured B-type biclusters on Breast and Lung datasets and

*μ*-type biclusters on Liver, Yeast and Lymphoma datasets.

## Discussion and Conclusions

Our study on real data has shown that evaluation of biclustering algorithms on idealized simulated data may not reflect the actual performance on real data owing to its complexity. So we proposed a conceptually and statistically sound framework based on the concept of differential co-expression to objectively compare the performance of the biclustering algorithms on real data and combine their output into a single unified ranking. This is based on the observation that a bicluster is revealed because the grouping of the bicluster genes could be strong only based on the bicluster conditions. As several biclustering algorithms do not consider the effect of non-bicluster conditions in the scoring and discovery of the biclusters, we found several biclusters with a strong grouping of genes based on the non-bicluster conditions also. This does not qualify them to be biclusters as the genes could be grouped nearly strongly even with all conditions together i.e. co-expression is more of a global effect. The strength of grouping can be represented by condition and gene effects and their differential between bicluster and non-bicluster conditions for the bicluster genes indicate true biclusters. We considered three types of co-expression unlike in a typical differential co-expression study and the ranking is based on the model coefficients rather than the model errors to reflect different types of co-expression. In this formulation, we explicitly estimate the effects of genes, conditions in bicluster conditions and non bicluster conditions. Strong effects of either genes or conditions would indicate co-expression of genes in the given group of conditions. Taking ratio of the co-expression scores between bicluster and non bicluster conditions gives us the measure of the goodness of the biclusters. Further we proposed a bicluster stratification score to classify the biclusters based on their co-expression patterns: high score means genes are co-expressed similarly across conditions in the bicluster, but the genes could be divided into two groups one with induction and the other with repression; low score means genes are co-expressed across conditions, conditions can be divided into two groups - one with induction of all genes and the other with repression; medium or near-zero score means all genes are either induced or repressed but not a combination in all conditions. The framework we used is analogous to ANOVA with T_{k}, B_{k} and *μ*_{k} being similar to the variance terms with null centrality parameter being '0'.

We have compared four well known biclustering algorithms: ISA, OPSM, CC and SAMBA. Their application on six different datasets revealed that ISA outputs the best biclusters but its performance is dependent on the number of conditions in the dataset; SAMBA performs well on all datasets of the varying number of conditions; though OPSM does not perform well on most datasets, it is still useful on certain datasets like Lung cancer data; whereas CC outputs least goodness biclusters with high stratification scores. Further, there is a data dependency on the types of co-expression present in the biclusters: all algorithms output predominantly B-type biclusters on Breast and Lung datasets and a mix of B-type and *μ*-type biclusters for Liver, Yeast and Lymphoma datasets, though *μ*-type biclusters are slightly more in number. Strikingly, OPSM output mostly B-type biclusters and CC is the only algorithm output T-type biclusters.

However, the evaluation presented in the paper may vary with a change in parameter settings of the individual algorithms. But it is helpful even to compare different parameter settings for a given algorithm and choose suitable parameter settings. Hence, the scoring function is useful, as shown here, to get unified ranking of the biclusters (i.e. meta-biclustering algorithm) produced by different algorithms for different parameter settings. However, we are working on devising an algorithm based on the differential co-expression framework as it may find novel biclusters with strong differential co-expression.

Moreover, though the proposed goodness scoring function is tailored to assess the goodness of the biclusters of co-expressed genes, the general framework of differential co-expression can be extended to evaluate the goodness of the other types of biclusters such as low error in the expression which requires a scoring function proposed by Kostka & Spang i.e. ratio of error variances = E_{2}(b)/E_{1}(b).

## Declarations

### Acknowledgements

We thank Ian, Huaien and Juntao for their valuable comments during the work. We also thank the anonymous reviewers for their valuable constructive comments which helped improve the manuscript. The research was funded by Genome Institute of Singapore, BMRC, Agency for Science, Technology and Research (A-STAR), Singapore.

## Authors’ Affiliations

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