BACKGROUND
genes interact with each other as basic building blocks of life, forming a complicated network.
the relationship between groups of genes with different functions can be represented as gene networks.
recent developments in microarray technology revolutionized research in the life sciences, allowing researchers to measure tens of thousands of genes simultaneously  <cit> .
with the deposition of huge microarray data sets in public domains, study on gene networking is now possible.
reconstructing gene networks from the microarray data will facilitate cellular function dissection at the molecular level.
hence the study will have a profound impact on biomedical research, ranging from cancer research to disease prevention  <cit> .
there has been an increasing interest in the reconstruction of gene networks from gene expression data.
recent works include linear models  <cit> , boolean network models  <cit> , and bayesian networks  <cit> .
bayesian networks seem to be very effective in the construction of gene networks.
they can incorporate prior knowledge from biology into their models and handle missing data effectively.
in particular, dynamic bayesian networks can learn a gene network from time-course gene expressions.
as noted in  <cit> , a major problem with bayesian networks is the computation problem.
our motivation is to overcome this limitation of bayesian networks in gene interactions.
for this purpose, we introduce a simple method for constructing gene networks based on copulas.
note that copulas can model a variety of interactions.
in statistical literature, the general way to describe dependence between correlated random variables is to use copulas  <cit> .
copulas are multivariate distribution functions whose one-dimensional margins are uniform on the  <cit>  interval  <cit> .
copulas are useful for constructing joint distributions, especially with nonnormal random variables.
the design, features, and some implementation details of the r package copula can be easily extended in multivariate modeling in many fields  <cit> .
in finance, copula functions are adopted to handle the interaction between the markets and risk factors in a flexible way  <cit> .
in biology, a gaussian copula has been applied in quantitative trait linkage.
copulas play an important role in developing a unified likelihood framework to analyze discrete, continuous, and censored traits  <cit> .
in principle, copulas can be used to model the joint distributions of any discrete or continuous gene and even mixed continuous and discrete genes.
in  <cit> , several measures of directional dependence in regression based on copula functions were proposed.
recently, a sieve maximum likelihood estimation procedure for semiparametric multivariate copula models has been proposed in  <cit> .
the proposed estimation achieved efficiency gains in finite samples, especially when prior information of the marginal distribution is incorporated.
in this paper, we adopt a measure of directional dependence to investigate the gene interactions for yeast cell cycle data.
one advantage of our approach is that dependence between genes is not assumed to be linear.
moreover, our approach can detect directional dependence.
hence our approach can provide valuable biological information on the presence of directional dependence between genes.
RESULTS
in this section, we analyze yeast cell cycle regulation  <cit> .
the data set is composed of measurements on  <dig> genes observed at  <dig> time points.
<dig> genes regulated by cell cycle were identified.
to compare our results with other results in the literature, we selected two groups of genes with known interaction patterns.
note that known interactions are still incomplete at present.
the first group includes eight histone genes-hht <dig>  hht <dig>  hhf <dig>  hhf <dig>  hta <dig>  hta <dig>  htb <dig> and htb <dig>  these eight genes encode for the four histones .
the histones are used to form the fundamental packaging unit of chromatin, called the code of nucleosome.
chromosomes, consisting of dna and histones, need to be replicated before cell division.
expression of the histone genes should be regulated tightly for the proper functioning of the replication process.
figure  <dig> shows the time-series plot of genes in the histone group.
it can be easily seen that the eight genes in the histone group are highly correlated with each other.
looking at table  <dig> and figure  <dig> for group i dataset, we can find that those aic values have pretty low values.
it means that our copula method for group i dataset is appropriate.
¿from figure  <dig> and  for group ii dataset, we also find that those aic values have relatively inconsistent low values compared to group i dataset.
it still means that our copula method for group ii dataset is also appropriate.
because of the small number of gene data sets, the estimates of fgm parameters and proportions for directional dependence in table  <dig> do not strongly support our claim that each pair of these  <dig> histone genes are dependent on each other in both directions.
figure  <dig> shows 3-dimensional and contour plots for hta <dig> vs htb <dig>  hta <dig> vs htb <dig>  hta <dig> vs htb <dig>  and htb <dig> vs htb <dig>  irregularly shaped contours indicate the existence of directional dependence, i.e., the asymmetry of dependence.
from the plots, we see that the asymmetry of dependence is not clear for each pair of genes.
contour plots for other pairs of histone genes show similar patterns.
figure  <dig> together with table  <dig> tells us that the 3d and contour plots are relatively symmetric which means a weak directional dependence in this gene data set.
to further evaluate the performance of the fgm copula model, we selected another group  which is comparatively larger than the first group.
this group consisted of  <dig> genes which include dna polymerases , dna helicase , type b cyclin genes , dna primases , radiation sensitive genes , repaire related genes , replication protein a encoding gene , dna replication initiation factor , securin gene , nucleosome assembly factor , and a subunit of the cohesin complex .
these genes play important role in the process of cell cycle which conducts dna replication initiation, dna damage-induced checkpoint arrest, dna damage repair, formation of mitotic spindle, and so on.
however, similar to the histone genes, their expression is also strictly regulated for the normal cellular process  <cit> .
the estimates of fgm parameters and proportions for directional dependence  clearly support our claim that each pair of  <dig> genes are dependent on each other in both directions, which is consistent with the observation from figure  <dig> and figure  <dig>
note that the measures of dependence ρc <dig>  ρu→v, and ρv→u have different scales from usual correlation coefficient.
since pearson's correlation coefficient is based on the assumption of normality and linearity of random variables x and y, the range of pearson's correlation is usually wider than that of our measures of directional dependence.
furthermore, pearson's correlation coefficient depends on random variables x and y, while the measures of directional dependence depend on the joint function of their cumulative distribution functions.
therefore, depending on the copula function adopted, the scales of the measures can be different.
also, when we use the uniform distribution or exponential distribution for the transformation of the marginal cumulative distribution functions of x and y, the measure of dependence can be smaller than pearson's correlation coefficient.
for a comparison of the measure of dependence of our fgm copula model, we used the normal copula model which is one of the representative copula models.
if we look at the fgm type and normal type in table  <dig> and , we find that depending on the gene data pair, the measures of dependence using the normal copula has more variation then the measures of dependence using our proposed fgm copula.
in light of these facts, our results are valid and consistent.
to support our results, we also provided the matematical derivations of our proposed fgm copula model in the method section.
the results from our method have been compared with those from other methods such as pathwayassist and chen's method  <cit> .
pathwayassist  is based on a comprehensive gene  interaction database compiled by a text mining tool from the entire pubmed  <cit> .
our method found  <dig> edges among these  <dig> genes.
from table  <dig>  we find that a pathwayassist search identified  <dig> edges and chen's method identified  <dig> edges.
however, because two copies of each core histone i.e., h2a, h2b, h <dig> and h <dig>  are assembled into an octamer, all  <dig> core histones can interact with each other.
the  <dig> edges we found indicate that each histone gene is connected with the remaining  <dig> histone genes.
all possible pairs of interaction genes from the group ii .
the reason is that by using the fgm copula model, we are better able to investigate the better directional interaction dependence compared to pathwayassist and chen's method  <cit> .
although microarray results show a transcriptional co-regulation pattern and do not imply that the gene products are physically interactive, this tight genetic connection may suggest that each gene product has either direct or indirect connections between the other gene products.
indeed, recent comprehensive analysis of a protein interaction map revealed that those histone genes are physically connected with each other  <cit> , supporting the results obtained by our method.
the findings of this study may help to design artificial drug candidates, which can block or activate biologically meaningful pathways.
furthermore, our copula approach can be extended to investigate the effects of local environments on protein-protein interactions.
the copula mutual information approach will help to propose a new variant of aracne: an algorithm for the reconstruction of gene regulatory networks.
CONCLUSIONS
in this paper, we presented a new methodology for analyzing gene interactions based on copula functions.
our method is shown to be useful in the construction of gene networks through the analysis of yeast cell cycle data.
our method may be able to overcome the limitation of bayesian network method for gene-gene interaction, i.e.
information loss due to binary transformation.
since a copula represents a way of extracting the dependence structure of the random variables from the joint distribution function, it is a useful approach to understanding and modeling dependent structure for random variables.
in our future works on gene directional dependence, we will develop hypothesis testing for directional dependence and formulate a network construction process using false discovery rate.

 genes interact with each other as basic building blocks of life, forming a complicated network. the relationship between groups of genes with different functions can be represented as gene networks. recent developments in microarray technology revolutionized research in the life sciences, allowing researchers to measure tens of thousands of genes simultaneously. with the deposition of huge microarray data sets in public domains, study on gene networking is now possible. reconstructing gene networks from the microarray data will facilitate cellular function dissection at the molecular level. hence the study will have a profound impact on biomedical research, ranging from cancer research to disease prevention. there has been an increasing interest in the reconstruction of gene networks from gene expression data. recent works include linear models, boolean network models, and bayesian networks. bayesian networks seem to be very effective in the construction of gene networks. they can incorporate prior knowledge from biology into their models and handle missing data effectively. in particular, dynamic bayesian networks can learn a gene network from time-course gene expressions. as noted in, a major problem with bayesian networks is the computation problem. our motivation is to overcome this limitation of bayesian networks in gene interactions. for this purpose, we introduce a simple method for constructing gene networks based on copulas. note that copulas can model a variety of interactions. in statistical literature, the general way to describe dependence between correlated random variables is to use copulas. copulas are multivariate distribution functions whose one-dimensional margins are uniform on the interval. copulas are useful for constructing joint distributions, especially with nonnormal random variables. the design, features, and some implementation details of the r package copula can be easily extended in multivariate modeling in many fields. in finance, copula functions are adopted to handle the interaction between the markets and risk factors in a flexible way. in biology, a gaussian copula has been applied in quantitative trait linkage. copulas play an important role in developing a unified likelihood framework to analyze discrete, continuous, and censored traits. in principle, copulas can be used to model the joint distributions of any discrete or continuous gene and even mixed continuous and discrete genes. in, several measures of directional dependence in regression based on copula functions were proposed. recently, a sieve maximum likelihood estimation procedure for semiparametric multivariate copula models has been proposed in. the proposed estimation achieved efficiency gains in finite samples, especially when prior information of the marginal distribution is incorporated. in this paper, we adopt a measure of directional dependence to investigate the gene interactions for yeast cell cycle data. one advantage of our approach is that dependence between genes is not assumed to be linear. moreover, our approach can detect directional dependence. hence our approach can provide valuable biological information on the presence of directional dependence between genes.  in this section, we analyze yeast cell cycle regulation. the data set is composed of measurements on genes observed at time points. genes regulated by cell cycle were identified. to compare our results with other results in the literature, we selected two groups of genes with known interaction patterns. note that known interactions are still incomplete at present. the first group includes eight histone genes-hht hht hhf hhf hta hta htb and htb these eight genes encode for the four histones. the histones are used to form the fundamental packaging unit of chromatin, called the code of nucleosome. chromosomes, consisting of dna and histones, need to be replicated before cell division. expression of the histone genes should be regulated tightly for the proper functioning of the replication process. figure shows the time-series plot of genes in the histone group. it can be easily seen that the eight genes in the histone group are highly correlated with each other. looking at table and figure for group i dataset, we can find that those aic values have pretty low values. it means that our copula method for group i dataset is appropriate. ¿from figure and for group ii dataset, we also find that those aic values have relatively inconsistent low values compared to group i dataset. it still means that our copula method for group ii dataset is also appropriate. because of the small number of gene data sets, the estimates of fgm parameters and proportions for directional dependence in table do not strongly support our claim that each pair of these histone genes are dependent on each other in both directions. figure shows 3-dimensional and contour plots for hta vs htb hta vs htb hta vs htb and htb vs htb irregularly shaped contours indicate the existence of directional dependence, i.e., the asymmetry of dependence. from the plots, we see that the asymmetry of dependence is not clear for each pair of genes. contour plots for other pairs of histone genes show similar patterns. figure together with table tells us that the 3d and contour plots are relatively symmetric which means a weak directional dependence in this gene data set. to further evaluate the performance of the fgm copula model, we selected another group which is comparatively larger than the first group. this group consisted of genes which include dna polymerases, dna helicase, type b cyclin genes, dna primases, radiation sensitive genes, repaire related genes, replication protein a encoding gene, dna replication initiation factor, securin gene, nucleosome assembly factor, and a subunit of the cohesin complex. these genes play important role in the process of cell cycle which conducts dna replication initiation, dna damage-induced checkpoint arrest, dna damage repair, formation of mitotic spindle, and so on. however, similar to the histone genes, their expression is also strictly regulated for the normal cellular process. the estimates of fgm parameters and proportions for directional dependence clearly support our claim that each pair of genes are dependent on each other in both directions, which is consistent with the observation from figure and figure note that the measures of dependence ρc ρu→v, and ρv→u have different scales from usual correlation coefficient. since pearson's correlation coefficient is based on the assumption of normality and linearity of random variables x and y, the range of pearson's correlation is usually wider than that of our measures of directional dependence. furthermore, pearson's correlation coefficient depends on random variables x and y, while the measures of directional dependence depend on the joint function of their cumulative distribution functions. therefore, depending on the copula function adopted, the scales of the measures can be different. also, when we use the uniform distribution or exponential distribution for the transformation of the marginal cumulative distribution functions of x and y, the measure of dependence can be smaller than pearson's correlation coefficient. for a comparison of the measure of dependence of our fgm copula model, we used the normal copula model which is one of the representative copula models. if we look at the fgm type and normal type in table and, we find that depending on the gene data pair, the measures of dependence using the normal copula has more variation then the measures of dependence using our proposed fgm copula. in light of these facts, our results are valid and consistent. to support our results, we also provided the matematical derivations of our proposed fgm copula model in the method section. the results from our method have been compared with those from other methods such as pathwayassist and chen's method. pathwayassist is based on a comprehensive gene interaction database compiled by a text mining tool from the entire pubmed. our method found edges among these genes. from table we find that a pathwayassist search identified edges and chen's method identified edges. however, because two copies of each core histone i.e., h2a, h2b, h and h are assembled into an octamer, all core histones can interact with each other. the edges we found indicate that each histone gene is connected with the remaining histone genes. all possible pairs of interaction genes from the group ii. the reason is that by using the fgm copula model, we are better able to investigate the better directional interaction dependence compared to pathwayassist and chen's method. although microarray results show a transcriptional co-regulation pattern and do not imply that the gene products are physically interactive, this tight genetic connection may suggest that each gene product has either direct or indirect connections between the other gene products. indeed, recent comprehensive analysis of a protein interaction map revealed that those histone genes are physically connected with each other, supporting the results obtained by our method. the findings of this study may help to design artificial drug candidates, which can block or activate biologically meaningful pathways. furthermore, our copula approach can be extended to investigate the effects of local environments on protein-protein interactions. the copula mutual information approach will help to propose a new variant of aracne: an algorithm for the reconstruction of gene regulatory networks.  in this paper, we presented a new methodology for analyzing gene interactions based on copula functions. our method is shown to be useful in the construction of gene networks through the analysis of yeast cell cycle data. our method may be able to overcome the limitation of bayesian network method for gene-gene interaction, i.e. information loss due to binary transformation. since a copula represents a way of extracting the dependence structure of the random variables from the joint distribution function, it is a useful approach to understanding and modeling dependent structure for random variables. in our future works on gene directional dependence, we will develop hypothesis testing for directional dependence and formulate a network construction process using false discovery rate. 
we analyzed the gene interactions for two gene data sets by adopting a measure of directional dependence based on a copula function. we have compared our results with those from other methods in the literature. although microarray results show a transcriptional co-regulation pattern and do not imply that the gene products are physically interactive, this tight genetic connection may suggest that each gene product has either direct or indirect connections between the other gene products. indeed, recent comprehensive analysis of a protein interaction map revealed that those histone genes are physically connected with each other, supporting the results obtained by our method. the results illustrate that our method can be an alternative to bayesian networks in modeling gene interactions. the copula mutual information approach will help to propose the new variant of aracne: an algorithm for the reconstruction of gene regulatory networks. in recent years, there has been an increasing interest in the reconstruction of gene networks from gene expression data. genes interact with each other as basic building blocks of life, forming a complicated network. we expect that our study may help to design artificial drug candidates, which can block or activate biologically meaningful pathways. with the deposition of huge microarray data sets in public domains, study on gene networking is now possible. recent work includes linear models, boolean network models, and bayesian networks. among them, bayesian networks seem to be the most effective in constructing gene networks. a major problem with the bayesian network approach is the excessive computational time. another advantage is that our approach can detect directional dependence. moreover, our copula approach can be extended to investigate the effects of local environments on protein-protein interactions. discreteness of the characteristics is not required in the copula approach which involves use of uniform representation of the continuous random variables. the relationship between groups of genes with different functions can be represented as gene networks. information loss due to binary transformation. our method is able to overcome the limitation of bayesian network method for gene-gene interaction, i.e.
 genes interact with each other as basic building blocks of life, forming a complicated network. the relationship between groups of genes with different functions can be represented as gene networks. with the deposition of huge microarray data sets in public domains, study on gene networking is now possible. in recent years, there has been an increasing interest in the reconstruction of gene networks from gene expression data. recent work includes linear models, boolean network models, and bayesian networks. among them, bayesian networks seem to be the most effective in constructing gene networks. a major problem with the bayesian network approach is the excessive computational time. this problem is due to the interactive feature of the method that requires large search space. since fitting a model by using the copulas does not require iterations, elicitation of the priors, and complicated calculations of posterior distributions, the need for reference to extensive search spaces can be eliminated leading to manageable computational affords. bayesian network approach produces a discretely expression of conditional probabilities. discreteness of the characteristics is not required in the copula approach which involves use of uniform representation of the continuous random variables. our method is able to overcome the limitation of bayesian network method for gene-gene interaction, i.e. information loss due to binary transformation.  we analyzed the gene interactions for two gene data sets by adopting a measure of directional dependence based on a copula function. we have compared our results with those from other methods in the literature. although microarray results show a transcriptional co-regulation pattern and do not imply that the gene products are physically interactive, this tight genetic connection may suggest that each gene product has either direct or indirect connections between the other gene products. indeed, recent comprehensive analysis of a protein interaction map revealed that those histone genes are physically connected with each other, supporting the results obtained by our method.  the results illustrate that our method can be an alternative to bayesian networks in modeling gene interactions. one advantage of our approach is that dependence between genes is not assumed to be linear. another advantage is that our approach can detect directional dependence. we expect that our study may help to design artificial drug candidates, which can block or activate biologically meaningful pathways. moreover, our copula approach can be extended to investigate the effects of local environments on protein-protein interactions. the copula mutual information approach will help to propose the new variant of aracne: an algorithm for the reconstruction of gene regulatory networks. 