BACKGROUND
gene expression profiling has gained a tremendous importance in functional genomic research.
the presently often used approach is to compare gene expression in discrete time points resulting for example of different genotypes or cell lines, morbid and healthy  objects or under different physiological conditions.
this type of static gene expression profiling can already give useful information on the patterns of significantly differentiated expression of genes.
however, in order to achieve a more complete picture of significantly differentiated gene expression, especially in order to capture and understand the dynamics of the altered gene expression, it is desirable to measure time-series gene expression  <cit> .
large efforts have been made in recent years to develop bioinformatics methods to study gene expression patterns  <cit>  and/or to infer functional linkages and even regulatory network from microarray data  <cit> .
for inference of functional associations among genes two major classes of methods are presently in use.
one class of the methods is based on graphical modeling, which include bayesian network  <cit>  and gaussian graphical model  <cit> .
recently, dynamic bayesian networks have been proposed to model temporal gene expression and represent a promising direction.
however, most of the current work in this area is limited to the analysis of a relatively small set of genes due to computational complexity  <cit> .
another class of methods infers functional association from large-scale gene expression data by defining a statistic threshold for the association.
the main measure used for defining association is the pearson correlation coefficient   <cit> .
pcc is widely used for detecting co-expressed genes from both static and time-series expression data.
however, several important issues are not specifically addressed in pcc based methods when applied to time-series expression data.
a major issue is that the pcc clustering method treats its input as a vector of independent samples and hence doesn't take into account the temporal relationship between consecutive time points.
in addition, time-shifted and/or inverted expression of certain gene pairs are not considered.
time-shifted and inverted relationships are important features of gene expression regulation  <cit> .
for example, a gene may activate or inhibit another gene or even several related genes downstream in a regulatory pathway, resulting in time-delayed positive or negative response in the transcription of the downstream gene.
to consider these phenomena, qian et al.
<cit>  proposed a local clustering  method.
as demonstrated with the expression profiling data of yeast cell cycle this method can identify new, biologically relevant interactions that could not be found by the conventional pcc clustering method.
however, the method of qian et al.
<cit>  is principally still based on a point-to-point comparison or local clustering of expression levels of genes although it explicitly considers time-shifted and inverted gene expression profile.
kwon et al.
<cit>  proposed an 'event-based' edge detection method to consider the change trend of gene expression between consecutive time points.
by simplifying a profile of time series into a sequence of decrease or increase events this method is more robust to noises.
however, it does not fully make use of the information contained in the gene expression levels in the original data.
filkov et al.
<cit>  proposed a similar method called 'edge detection'.
recently balasubramaniyan et al.
<cit>  proposed a method to use spearman rank correlation based on the rank of expressional values.
the rank of expressional values is more insensitive to noses or outliers but this method is still based on the point-to-point comparison per se.
to more comprehensively consider the temporal relationships of gene expression in time-series microarray data we propose here a new method that is based on extracting the main features of the change trend and the change level of gene expression between consecutive time points.
we not only consider the qualitative information  but also the quantitative information  in the original data.
we seek to make the methods more noise-tolerant and at the same time to keep more useful information in the expression values.
this new method, termed here as trend correlation , is demonstrated with the microarray data from cell cycle of yeast  <cit> .
the biological significance of functional associations of inferred gene pairs is examined with several large-scale yeast databases.
we also extensively compare our method with the lc method and the pcc based clustering method of eisen et al.
<cit> .
it is shown that a significant number of functionally associated gene pairs, which have well-documented biological interactions and relationships but cannot be significantly detected by the lc and pcc methods, can be inferred by the new method with high statistic significance.
the biological significance of the functional association pairs inferred by our method is generally higher than that of the lc method and similar to that of the pcc clustering method which detects however only simultaneous co-expression gene pairs.
furthermore, it is shown that the overlapping of gene pairs detected by the three methods is normally not very high, indicating a necessity of combining the different methods in search of functional association of genes from time-series microarray data.
principle and scheme of the proposed method
the principle of our method is to use information in the change trend and the change level of gene expression between consecutive time points for the inference of functional linkages among genes.
to consider not only the positive correlation but also time-shifted and/or inverted expression of certain gene pairs we employ a similar algorithm as the local sequence alignment  <cit>  to calculate a maximal local alignment of change trend  between each gene pair .
we noticed that a significant number of gene pairs that have a same sc value may in fact have a large difference in the change levels of their expression.
the distinction of their change levels may lead to the different degrees of similarity of functional association.
to solve this problem we make use of the quantitative information in the change levels between consecutive time points and calculate a correlation coefficient  between the maximal alignment.
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakqaabeqaaiabdifayjabd2gatjabdogajnaadaaaleaacqaixaqmaeaacqaiyagmaagccqgh9aqpcqaixaqmcqggsaalcqwgybawcqwgtbqbcqwgjbwydaqhaawcbagaegomaidabagaeg4mamdaaogaeyypa0jaegymaejaeiilawiaemiwaglaemyba0maem4yam2aa0baasqaaiabiodazaqaaiabisda0aaakiabg2da9iabgkhitiabigdaxiabcycasiabdifayjabd2gatjabdogajnaadaaaleaacqai0aanaeaacqai1aqnaagccqgh9aqpcqaiyagmcqggsaalcqwgybawcqwgtbqbcqwgjbwydaqhaawcbagaegynaudabagaegonaydaaogaeyypa0jaeginaqjaeiilawiaemiwaglaemyba0maem4yam2aa0baasqaaiabiada2aqaaiabieda3aaakiabg2da9iabgkhitiabikdayiabcycasiabdifayjabd2gatjabdogajnaadaaaleaacqai3awnaeaacqai4aaoaagccqgh9aqpcqaizawmaeaacqwgzbqwcqwgtbqbcqwgjbwydaqhaawcbagaegymaedabagaegomaidaaogaeyypa0jaegymaejaeiilawiaemywaklaemyba0maem4yam2aa0baasqaaiabikdayaqaaiabiodazaaakiabg2da9iabigdaxiabcycasiabdmfazjabd2gatjabdogajnaadaaaleaacqaizawmaeaacqai0aanaagccqgh9aqpcqghsislcqaixaqmcqggsaalcqwgzbqwcqwgtbqbcqwgjbwydaqhaawcbagaeginaqdabagaegynaudaaogaeyypa0jaegomaijaeiilawiaemywaklaemyba0maem4yam2aa0baasqaaiabiwda1aqaaiabiada2aaakiabg2da9iabiodaziabcycasiabdmfazjabd2gatjabdogajnaadaaaleaacqai2agnaeaacqai3awnaagccqgh9aqpcqghsislcqaizawmcqggsaalcqwgzbqwcqwgtbqbcqwgjbwydaqhaawcbagaeg4nacdabagaegioagdaaogaeyypa0jaeginaqdaaaa@ab67@ the change level between time points  <dig> and  <dig> is excluded because the change  trend is different from the main  change trend.
using the above matched change levels we can obtain a correlation coefficient cc =  <dig>  according to eq.
<dig>
the method consists of following major steps:
<dig>  generating a random dataset by shuffling the normalized expression levels at different time points among each gene expression profile in the original dataset;
<dig>  calculating a maximal local alignment of change trend  between each gene pair in the random dataset as illustrated in fig.
<dig> for a simple case;
<dig>  calculating a correlation coefficient  between the maximal alignment for each gene pair in the random dataset ;
<dig>  tabulating the frequency of sc ) as function of sc as shown in fig.
s1a ; followed by tabulating the distribution of cc for gene pairs which have the same sc;
<dig>  calculating the conventional p-values for the two scores sc and cc , pcc) through integration of the frequency distributions ;
<dig>  calculating sc and cc between each gene pair in the original dataset;
<dig>  extraction of functional linkages using procedure i proposed: extract gene pairs with significantly high sc values with a certain preset p-value.
the correlation coefficient cc is regarded as a second index when the gene pairs have the same score sc;
<dig>  extraction of functional linkages using procedure ii: extract gene pairs with statistically significantly high value of combined scores of sc and cc.
the key steps mentioned above are described in the section of methods in more detail.
the main programs tc_linkage_infer created in this work can be freely downloaded from our website   http://www.gbf.de/systemsbiology.
RESULTS
dataset of yeast cell-cycle
we tested our algorithm with the time series microarray data  generated by cho et al.
<cit>  for yeast cell cycle with a whole genome yeast oligonucleotide chip which included over  <dig> orfs.
after removing all the negative expression levels in the scaled measurements and all the dubious and genes now deleted in the sgd database  <cit> ,  <dig> genes were included in our calculation.
we examined all the possible pairs among them.
the values of the two scores sc and cc and the type of possible relationship  were calculated and assigned for each gene pair.
functional associations inferred by different methods
figs.
2a and 2b show the number of gene pairs with possible functional association inferred from the yeast time-series data by the tc method , the pcc clustering method and the lc method at two different cut-off p-values.
the number of gene pairs extracted by procedure ii of our method is generally higher than that from procedure i, especially at the lower p-value.
note that there is some overlapping between these two extraction procedures so that the total number of the gene pairs inferred is lower than the sum of the two procedures.
for comparing the tc method with the other two methods the combined results of the two extraction procedures are considered in the following if not otherwise mentioned.
p-values of  <dig> e- <dig> and 1e- <dig> in the tc method are equivalent to pearson correlated coefficients of  <dig>  and  <dig>  in the pcc clustering method and scores of  <dig>  <cit>  and  <dig>  in the local clustering method respectively.
mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqaiyagmcqgguaglcqai3awncqqglbqzdaahaawcbeqaaiabgkhitiabiodazaaakiabcqda6maalaaabagaegonayjaegimaajaegimaajaeginaqjaeginaqjaey4kasiaegynaujaegomaijaegynaujaegymaedabagaeginaqjaegyoakjaegynaujaegynaujaey4kasiaegymaejaegomaijaegonayjaegonayjaegymaejaegyoakdaaiabg2da9iabisda0iabimda5iabc6cauiabiada2iabcwcalaaa@4da3@.
c and d, number of process-identity pairs among the detected pairs from the three methods; e and f, number of known protein interactions  in collection dataset of protein interactions  <cit>  among the inferred gene pairs from the three methods.
as can be clearly seen in figs.
2a and 2b, the number of inferred gene pairs depends much on the p-value in all the three methods.
in general, the tc method infers a significantly lower number of gene pairs compared to the other two methods.
<dig> % and  <dig> % of the pairs inferred by the tc method are also found by the pcc and lc methods respectively.
the common part of the three methods is somewhat lower and accounts only  <dig> % for the lc method,  <dig> % for the pcc method and  <dig> % for the tc method.
the number of gene pairs merely inferred by the tc method is considerable compared with those merely from the lc method  and the pcc clustering method .
by decreasing the p-value threshold to 1e- <dig>  the number of inferred functional pairs decreases remarkably, especially for the tc and lc methods .
however, the shared part of gene pairs predicted by the tc method increases up to  <dig> % and  <dig> % compared to the lc and pcc methods respectively, indicating an increased reliability of the prediction .
the number of additional gene pairs merely inferred by the tc method amounts  <dig> and is still significant .
biological significance of the gene pairs inferred
to assess the biological significance of the inferred functional associations, the gene pairs are compared to known biological processes and protein functions, known protein interactions and regulatory interactions in yeast respectively.
the results from the three methods are also compared to each other.
the results show that the tc method can significantly enhance the lc and pcc methods to infer functional linkages and biological network, and is well suited to explore temporal relationships of gene expression in time-series data as detailed below.
biological process and protein function-similarity gene pairs
in order to generally assess the biological significance of the gene pairs inferred, we first use two databases of biological processes and protein functions classification  .
the s. cerevisiae genome database  mainly utilizes the gene ontology  annotations  <cit> .
we use  <dig> main biological processes .
in this work, if two genes in the pair inferred are involved in the same biological process, we consider the gene pair as a process-identity one.
<dig> % of the  <dig> gene pairs inferred with the tc method at a p-value threshold of  <dig> e- <dig>  are found to be process-identity pairs .
the detailed distribution of the process-identity pairs in each biological process is listed in table s <dig>  a similar ratio  of process homology gene pairs is found for the  <dig> gene pairs inferred by the pcc method.
only  <dig> % of the  <dig> gene pairs inferred by the lc method with the same p-value cutoff are process-identity pairs.
if the results inferred by procedure i and procedure ii of the tc method are separately considered,  <dig> % of the  <dig> gene pairs detected by procedure i are process-identity pairs.
the percentage  of process-identity pairs among the  <dig> gene pairs detected by procedure ii is slightly lower than that by procedure i but still somewhat higher than that  of the lc method.
the additional number  of process-identity pairs inferred by the tc method is considerable compared to those resulted only from the lc method  and the pcc method  respectively.
among the  <dig> gene pairs  predicted by the tc method with a p-value threshold of 1e- <dig> there are  <dig>  process-identity pairs .
separately, only  <dig> % of the gene pairs  detected by procedure i are process-identity pairs, in contrast to as high as  <dig> % for the gene pairs by procedure ii.
considering the results from p-value ≤  <dig> e- <dig> mentioned above no general conclusion can be drawn with regard to the question which extraction procedure is more relevant.
the percentage of the process-identity pairs among the genes pairs resulted from the lc method and the pcc method is 49% and 42% respectively.
thus, the lower p-value threshold can significantly increase the portion of gene pairs involved in the same biological processes in all the three methods.
at this low p-value the additional number  of process-identity pairs merely inferred by the tc method amounts to  <dig> % of those only resulted from the lc method.
compared with those resulted by the pcc method this number declines to  <dig> , suggesting that the gene pairs with a higher ranked functional association inferred by the tc method is more similar to those resulted by the pcc method.
with the percentage of process-identity pairs in the range of  <dig> – <dig>  at p-value ≤  <dig> e- <dig> the gene pairs inferred by the three methods seem to have a relatively low biological significance.
if the common part of the gene pairs inferred by all the three methods is considered , the percentage of process-identity pairs  increases to  <dig> %, resulting in fairly good biological significance for an in silico method of biological function inference.
the biological significance can be significantly increased by lowering the p-value.
at p-value ≤ 1e- <dig> the percentage of process-identity pairs ranges from  <dig> to 49% for the three methods.
if the common part of the gene pairs inferred by the three methods at this p-value is considered , the percentage of process-identity pairs  increases to  <dig> %, resulting in a satisfactorily high biological significance.
the second source used in this work for assessing biological relevance of the gene pairs is the munich information center for protein sequences  functional catalogue database.
for protein functional classification, the mips database contains up to  <dig> different levels within the hierarchy .
we use here the second level of mips  as qian et al.
<cit>  did.
altogether  <dig> function classes are used.
it should be mentioned that some functional classes merely belong to protein cellular functions of plants and animals.
here if two genes in the pair inferred have the same protein cellular function, we term the pair as a function-similarity pair.
the results of comparison among the three methods  are similar to those of process-identity pairs, and detailed distribution of the function-similarity pairs in each protein cellular function by the trend correlation method is provided in table s <dig> .
as in the case of function-similarity pairs, if the common part of the inferred gene pairs  from the three methods is considered the percentage of function-similarity pairs  increases to  <dig> % and  <dig> % at the two p-values respectively, resulting in a good basis for further experimental and functional studies of the gene pairs inferred.
comparison of inferred gene pairs with known protein interactions
to further assess the biological significance of the gene pairs inferred and especially for comparing the three methods we examine here the known protein-protein interactions  in current databases of yeast among the gene pairs inferred from the yeast cell cycle data by the different methods.
four databases and published high quality datasets  are chosen for this purpose.
the protein-protein interactions collection of yu et al  <cit>  integrates datasets from the databases of mips  <cit> , the database of interacting proteins , the biomolecular interaction network database  and the experimental datasets of yeast two-hybrid  <cit>  and high-throughput mass spectrometry measurements  <cit> .
many of the interactions are manually curated beyond the experimentally derived protein-protein interactions in the three databases mentioned above  <cit> .
with a p-value threshold of  <dig> e- <dig> the tc method detects  <dig> gene pairs among the  <dig> known protein interactions.
separately, extraction procedure i detects  <dig> protein interaction pairs, compared to  <dig> pairs by extraction procedure ii.
with the same p-value threshold  <dig> and  <dig> such gene pairs can be detected by the lc method and the pcc method respectively.
the relatively low number of protein interactions among the gene pairs inferred by the tc method is a consequence of the overall lower number of gene pairs inferred by the tc method .
in fact, the percentage of protein interaction pairs  in the overall gene pairs  inferred by the tc method  is comparable to that of the pcc method  but higher than that of lc method .
especially the extraction procedure ii of the tc method achieves a relatively higher percentage .
it should be mentioned that the low percentage in all these cases is not surprising since only data from cell cycle is used in the inference and the cell cycle represents only a part of cellular functions involving protein-protein interactions in yeast.
the results presented in this section are thus more suitable for comparing the different methods rather than for quantitatively assessing the biological significance of the gene pairs inferred.
fig.
2e presents a comparison of the numbers of unique and common pairs of protein-protein interactions inferred by the three methods.
the additional pairs  merely  detected by the tc method are significant compared with those resulted merely from the lc method  and the pcc method .
remarkably,  <dig> protein-protein interactions among  <dig>  protein components of the large ribosomal subunit of yeast are only significantly detected by the tc method.
the fact that the  <dig> proteins connect with each other by the detected interactions is consistent with the assembling phenomena of components of the large ribosomal subunit.
similarly,  <dig> interactions among  <dig> protein components of the small ribosomal subunit of yeast are also only significantly detected by the tc method.
the  <dig> proteins also connect with each other by the  <dig> interactions .
therefore, to have a more complete coverage of the protein interactions it is obvious that the tc method should be combined with the lc and pcc methods.
by loosing the p-value threshold from  <dig> e- <dig> to  <dig> e- <dig> the tc method can detect significantly more pairs  of known protein-protein interactions .
a p-value of  <dig> e- <dig> in the tc method is equivalent to a pearson correlated coefficient of  <dig>  in the pcc method and a score of  <dig> in the lc method  <cit> .
with this p-value threshold,  <dig> and  <dig> pairs can be detected by the lc method and the pcc respectively.
the number of protein-protein interactions additionally inferred by the tc method is also considerable .
when the mips, dip and bind updated databases are used individually for comparison similar results are obtained and summarized in table  <dig>  though the majority of the three databases is consistent some differences exist among them.
we chose them to cover more known protein interactions and could therefore obtain more reliable statistics results.
comparison with known regulatory interactions
to examine the biological significance of functional associations inferred by the different methods it is also interesting to know if the gene pairs inferred cover some of the known regulatory interactions, especially those involved in the regulation of cell cycle.
for this purpose, we use two regulatory datasets  as a comparison basis.
we first use the dataset including regulatory interactions which are confirmed with a p-value threshold of 1e- <dig> by genome wide location analysis   <cit> .
with a p-value threshold of  <dig> e- <dig>  only a relative low number of the regulatory interactions is detected by the lc method , the pcc method  and the tc  .
regulatory interactions detected by the tc method are significantly less than those by the other two methods.
nevertheless, the additional number of regulatory interactions detected by the tc method is considerable compared with those only resulted from the lc method  and the pcc method .
similar results are obtained when the regulatory interaction collection dataset of luscombe et al.
<cit>  is used .
the additional number of interactions predicted by the tc method is  <dig>  and  <dig>  compared with those resulted only from the lc method and the pcc with a p-value threshold of  <dig> e- <dig>  with a p-value threshold of  <dig> e- <dig>  the results of comparison are similar to those at a p-value cutoff of  <dig> e- <dig>  some of the typical interactions between transcriptional regulators and target genes which are detected by the tc method but cannot be significantly detected by the lc and/or the pcc method are summarized in table s <dig>
a detailed investigation of the results from a p-value threshold of  <dig> e- <dig> shows that the transcriptional factors and/or the targeted genes of  <dig>  among  <dig> known regulatory interactions inferred by the tc method are involved in cell cycle regulation according to luscombe et al.
<cit> .
among the  <dig> interactions only six interactions have both the activated regulator and the differentially expressed targeted gene according to the gene set of cell cycle regulation given in luscombe et al.
<cit> .
the fact that  <dig> of the known interactions have either an inactivated regulator or a non-differentially expressed targeted gene in the cell cycle condition as detected by the tc method with a high p-value cutoff seems to indicate controversies in the designation of activated regulators and differentially expressed genes in the cell cycle condition  <cit> .
for example, the gene fkh <dig> is annotated in hollenhorst et al.
<cit>  as a transcriptional factor of the forkhead family that regulates the cell cycle according to sgd.
furthermore, many interactions known to involve fkh <dig> are detected by at least one of the three methods.
however, fkh <dig> is not regarded as a regulator in the cell cycle condition by luscombe et al.
<cit> .
we further present two examples of predicted functional linkages which have well-documented biological relationships but cannot be significantly detected by the pcc method and the lc method.
the regulatory relationship between rcs <dig> and gcn <dig>  was confirmed by gwla with a p-value of  <dig> e- <dig>  the pearson correlation coefficient for this gene pair is as low as  <dig>  and the score of the lc is only  <dig>  .
we detect this gene pair with a maximal matched inverted change trend sc of  <dig> after rcs <dig> is shifted forward by one time point and obtain a cc value of  <dig>  between the matched change levels by the tc method.
another example  is the proteins of the genes sac <dig> and hrp <dig> that form a protein complex  <cit> .
though the score of the lc method is only  <dig>  and the pearson correlation coefficient is  <dig> , we could detect this gene pair with a maximal matched similar change trend sc of  <dig> and obtained a cc value of  <dig>  by the tc method.
the inference of regulatory circuit and network
as shown above, functional linkages inferred by the individual method do not give a complete picture.
it is conceivable that this would be especially the case for inferring regulatory circuits and further for reconstructing regulatory network.
fig.
<dig> shows comparisons of inferred functional relationships by the individual methods and their combination which are mapped into regulatory circuits  and networks based on the datasets  <cit> .
in consistence with the results of detected regulatory interactions  the lc method is superior to the other two methods but still gives a very much incomplete picture.
the tc and pcc methods can augment the lc method significantly.
to infer more complete regulatory circuits and network the three methods should be therefore combined.
this is exemplified with two relatively simple regulatory circuits.
for the regulatory chain  from the regulator tos <dig> to the target gene cct <dig> the negative relationship between tos <dig> and swi <dig> can only be detected by the lc method with a shift of two-time-points.
the interaction between swi <dig> and abf <dig> can be detected by all the three methods.
however, the interaction between abf <dig> and cct <dig> can only be detected by the tc method.
the three interactions can form one regulatory chain among the complex combination of several motifs .
another example about a single input motif is explained in fig.
s <dig> .
discussion
from a practical point of view there are two key general issues in the analysis of gene expression data.
first, we ought to infer functional relationships with high statistic significance and as completely as possible.
second, the functional relationships inferred should have a high biological significance.
we proposed a new method in this work and evaluated it mainly regarding these two aspects with the microarray data of yeast cell cycle  <cit> .
the new method is also compared with the local clustering method and the pearson correlation coefficient based clustering method.
the number of functional gene pairs inferred depends very much on the p-value threshold in all the three methods .
with the two p-value cutoffs  applied the tc method detects general significantly lower number of gene pairs.
nevertheless, a considerable number of gene pairs is only detected by the tc method, ranging from  <dig> % of the number merely detected by the lc method to as high as  <dig> % of that merely detected by the pcc method at p-value ≤  <dig> e- <dig>  the ratio of additional gene pairs inferred by our method remains similar  at p-value ≤ 1e- <dig> compared with the lc method but decreases to a relatively lower value  compared with the pcc method.
this can be explained by the fact that the statistically higher ranked correlations have mostly simultaneous relationships that can be well detected by all the three methods.
since p-value ≤  <dig> e- <dig> represents a relatively high statistic significance it can be concluded that the new method detects a significantly high portion of additional functional relationships compared to the other two methods.
since the shared part of gene pairs of the three methods at p-value ≤  <dig> e- <dig> is less than 50%  it is also obvious that these methods should be combined to have a more complete exploitation of functional associations of genes buried in time-series expression data.
concerning the biological significance  <dig> % and  <dig> % of the gene pairs  inferred by our method  are process-identity ones  by comparing with the known biological processes in the s. cerevisiae genome database .
this is compared with a process-identity ratio in the range of  <dig> –49% for the lc method and  <dig> – <dig> % for the pcc method.
similar results are obtained when the mips protein functional catalogue database is used to assess the biological significance .
these results suggest that the gene pairs detected by the three methods achieve fairly good and comparable biological significance.
as in the case of the total gene pairs the additional gene pairs with biological significance inferred merely by the trend correlation method is considerable compared to the other two methods .
we also examined the gene pairs inferred by the three methods with regard to known protein interactions  and known regulatory interactions  in yeast.
in general, the percentage of protein interaction pairs in the overall gene pairs is low and comparable for all the methods ,  <dig> % for the lc method and  <dig> % for the pcc method).
this can be understood in view of the fact that only cell cycle data under very specific conditions are used here and the cell cycle represents only a small portion of the cellular activities.
furthermore, only protein interaction pairs with significantly changed expression levels of all the involved partner proteins can be theoretically detected.
this applies also to the known regulatory interactions.
it is shown that most of the detected regulatory interactions are indeed involved in the cell cycle regulation.
nevertheless, the number of protein and regulatory interactions additionally inferred merely by the trend correlation method is significant in all the cases.
it is obvious that more molecular interactions can be obtained if more microarray datasets under different conditions are considered  <cit> .
given the large number of functionally associated gene pairs inferred by the different methods an important question arises as to how we can find gene pairs which have really a high biological significance and thus would be best candidates for further experimental and functional studies.
to this end, the shared part of gene pairs inferred by all the three methods is of particular interest, especially at low p-values.
it is found for example that the percentage of process and function-similarity gene pairs of the shared pairs at p-value ≤ 1e- <dig> can be as high as  <dig> % and  <dig> %, building a very good basis for experimental study.
the common part of the tc and lc methods would be also of particular interest for finding time-delayed and/or inverted functional relationships which have received so far less attention.
it should be mentioned that all the functional associated pairs predicted have a high probability to be true and can thus serve as hypotheses for further study in view of the high statistic significance criteria applied.
we would also like to emphasize that the combined use of the different methods is not only useful for finding more and highly possible potential candidates of functional association but also very important to infer more complete regulatory circuits and network as exemplified in this study .
CONCLUSIONS
the major difference between our method and the other current methods is that the trend correlation method is based on the main change trend and comprehensively considers correlation coefficient between the main change trend of two genes, whereas the other methods are mainly based on the correlation of point-to-point expression levels of two genes.
hence the trend correlation method can reveal additional gene pairs with same function or in the same biological process but yet not significantly co-expressed.
it therefore also can infer additional protein-protein and regulatory network as demonstrated.
as mentioned above, a combined use of different methods is presently necessary for the analysis of time-series microarray data.
as clearly demonstrated in this work the proposed new method can significantly augment the currently major methods and is well suitable for exploring temporal relationships of gene expression in time-series data.

 gene expression profiling has gained a tremendous importance in functional genomic research. the presently often used approach is to compare gene expression in discrete time points resulting for example of different genotypes or cell lines, morbid and healthy objects or under different physiological conditions. this type of static gene expression profiling can already give useful information on the patterns of significantly differentiated expression of genes. however, in order to achieve a more complete picture of significantly differentiated gene expression, especially in order to capture and understand the dynamics of the altered gene expression, it is desirable to measure time-series gene expression. large efforts have been made in recent years to develop bioinformatics methods to study gene expression patterns and/or to infer functional linkages and even regulatory network from microarray data. for inference of functional associations among genes two major classes of methods are presently in use. one class of the methods is based on graphical modeling, which include bayesian network and gaussian graphical model. recently, dynamic bayesian networks have been proposed to model temporal gene expression and represent a promising direction. however, most of the current work in this area is limited to the analysis of a relatively small set of genes due to computational complexity. another class of methods infers functional association from large-scale gene expression data by defining a statistic threshold for the association. the main measure used for defining association is the pearson correlation coefficient. pcc is widely used for detecting co-expressed genes from both static and time-series expression data. however, several important issues are not specifically addressed in pcc based methods when applied to time-series expression data. a major issue is that the pcc clustering method treats its input as a vector of independent samples and hence doesn't take into account the temporal relationship between consecutive time points. in addition, time-shifted and/or inverted expression of certain gene pairs are not considered. time-shifted and inverted relationships are important features of gene expression regulation. for example, a gene may activate or inhibit another gene or even several related genes downstream in a regulatory pathway, resulting in time-delayed positive or negative response in the transcription of the downstream gene. to consider these phenomena, qian et al. proposed a local clustering method. as demonstrated with the expression profiling data of yeast cell cycle this method can identify new, biologically relevant interactions that could not be found by the conventional pcc clustering method. however, the method of qian et al. is principally still based on a point-to-point comparison or local clustering of expression levels of genes although it explicitly considers time-shifted and inverted gene expression profile. kwon et al. proposed an'event-based' edge detection method to consider the change trend of gene expression between consecutive time points. by simplifying a profile of time series into a sequence of decrease or increase events this method is more robust to noises. however, it does not fully make use of the information contained in the gene expression levels in the original data. filkov et al. proposed a similar method called'edge detection'. recently balasubramaniyan et al. proposed a method to use spearman rank correlation based on the rank of expressional values. the rank of expressional values is more insensitive to noses or outliers but this method is still based on the point-to-point comparison per se. to more comprehensively consider the temporal relationships of gene expression in time-series microarray data we propose here a new method that is based on extracting the main features of the change trend and the change level of gene expression between consecutive time points. we not only consider the qualitative information but also the quantitative information in the original data. we seek to make the methods more noise-tolerant and at the same time to keep more useful information in the expression values. this new method, termed here as trend correlation, is demonstrated with the microarray data from cell cycle of yeast. the biological significance of functional associations of inferred gene pairs is examined with several large-scale yeast databases. we also extensively compare our method with the lc method and the pcc based clustering method of eisen et al.. it is shown that a significant number of functionally associated gene pairs, which have well-documented biological interactions and relationships but cannot be significantly detected by the lc and pcc methods, can be inferred by the new method with high statistic significance. the biological significance of the functional association pairs inferred by our method is generally higher than that of the lc method and similar to that of the pcc clustering method which detects however only simultaneous co-expression gene pairs. furthermore, it is shown that the overlapping of gene pairs detected by the three methods is normally not very high, indicating a necessity of combining the different methods in search of functional association of genes from time-series microarray data. principle and scheme of the proposed method the principle of our method is to use information in the change trend and the change level of gene expression between consecutive time points for the inference of functional linkages among genes. to consider not only the positive correlation but also time-shifted and/or inverted expression of certain gene pairs we employ a similar algorithm as the local sequence alignment to calculate a maximal local alignment of change trend between each gene pair. we noticed that a significant number of gene pairs that have a same sc value may in fact have a large difference in the change levels of their expression. the distinction of their change levels may lead to the different degrees of similarity of functional association. to solve this problem we make use of the quantitative information in the change levels between consecutive time points and calculate a correlation coefficient between the maximal alignment. mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=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@ab67@ the change level between time points and is excluded because the change trend is different from the main change trend. using the above matched change levels we can obtain a correlation coefficient cc = according to eq. the method consists of following major steps: generating a random dataset by shuffling the normalized expression levels at different time points among each gene expression profile in the original dataset; calculating a maximal local alignment of change trend between each gene pair in the random dataset as illustrated in fig. for a simple case; calculating a correlation coefficient between the maximal alignment for each gene pair in the random dataset; tabulating the frequency of sc ) as function of sc as shown in fig. s1a; followed by tabulating the distribution of cc for gene pairs which have the same sc; calculating the conventional p-values for the two scores sc and cc, pcc) through integration of the frequency distributions; calculating sc and cc between each gene pair in the original dataset; extraction of functional linkages using procedure i proposed: extract gene pairs with significantly high sc values with a certain preset p-value. the correlation coefficient cc is regarded as a second index when the gene pairs have the same score sc; extraction of functional linkages using procedure ii: extract gene pairs with statistically significantly high value of combined scores of sc and cc. the key steps mentioned above are described in the section of methods in more detail. the main programs tc_linkage_infer created in this work can be freely downloaded from our website http://www.gbf.de/systemsbiology.  dataset of yeast cell-cycle we tested our algorithm with the time series microarray data generated by cho et al. for yeast cell cycle with a whole genome yeast oligonucleotide chip which included over orfs. after removing all the negative expression levels in the scaled measurements and all the dubious and genes now deleted in the sgd database, genes were included in our calculation. we examined all the possible pairs among them. the values of the two scores sc and cc and the type of possible relationship were calculated and assigned for each gene pair. functional associations inferred by different methods figs. 2a and 2b show the number of gene pairs with possible functional association inferred from the yeast time-series data by the tc method, the pcc clustering method and the lc method at two different cut-off p-values. the number of gene pairs extracted by procedure ii of our method is generally higher than that from procedure i, especially at the lower p-value. note that there is some overlapping between these two extraction procedures so that the total number of the gene pairs inferred is lower than the sum of the two procedures. for comparing the tc method with the other two methods the combined results of the two extraction procedures are considered in the following if not otherwise mentioned. p-values of e- and 1e- in the tc method are equivalent to pearson correlated coefficients of and in the pcc clustering method and scores of and in the local clustering method respectively. mathtype@mtef@5@5@+=feaafiart1ev1aaatcvaufkttlearuwrp9mdh5mbpbiqv92aaexatlxbi9gbaebbnrfifhhdyfgasaach8aky=wiffydh8gipec8eeeu0xxdbba9frfj0=oqffea0dxdd9vqai=hguq8kuc9pgc9s8qqaq=dirpe0xb9q8qilsfr0=vr0=vr0dc8meaabaqaciaacagaaeqabaqabegadaaakeaacqaiyagmcqgguaglcqai3awncqqglbqzdaahaawcbeqaaiabgkhitiabiodazaaakiabcqda6maalaaabagaegonayjaegimaajaegimaajaeginaqjaeginaqjaey4kasiaegynaujaegomaijaegynaujaegymaedabagaeginaqjaegyoakjaegynaujaegynaujaey4kasiaegymaejaegomaijaegonayjaegonayjaegymaejaegyoakdaaiabg2da9iabisda0iabimda5iabc6cauiabiada2iabcwcalaaa@4da3@. c and d, number of process-identity pairs among the detected pairs from the three methods; e and f, number of known protein interactions in collection dataset of protein interactions among the inferred gene pairs from the three methods. as can be clearly seen in figs. 2a and 2b, the number of inferred gene pairs depends much on the p-value in all the three methods. in general, the tc method infers a significantly lower number of gene pairs compared to the other two methods. % and % of the pairs inferred by the tc method are also found by the pcc and lc methods respectively. the common part of the three methods is somewhat lower and accounts only % for the lc method, % for the pcc method and % for the tc method. the number of gene pairs merely inferred by the tc method is considerable compared with those merely from the lc method and the pcc clustering method. by decreasing the p-value threshold to 1e- the number of inferred functional pairs decreases remarkably, especially for the tc and lc methods. however, the shared part of gene pairs predicted by the tc method increases up to % and % compared to the lc and pcc methods respectively, indicating an increased reliability of the prediction. the number of additional gene pairs merely inferred by the tc method amounts and is still significant. biological significance of the gene pairs inferred to assess the biological significance of the inferred functional associations, the gene pairs are compared to known biological processes and protein functions, known protein interactions and regulatory interactions in yeast respectively. the results from the three methods are also compared to each other. the results show that the tc method can significantly enhance the lc and pcc methods to infer functional linkages and biological network, and is well suited to explore temporal relationships of gene expression in time-series data as detailed below. biological process and protein function-similarity gene pairs in order to generally assess the biological significance of the gene pairs inferred, we first use two databases of biological processes and protein functions classification. the s. cerevisiae genome database mainly utilizes the gene ontology annotations. we use main biological processes. in this work, if two genes in the pair inferred are involved in the same biological process, we consider the gene pair as a process-identity one. % of the gene pairs inferred with the tc method at a p-value threshold of e- are found to be process-identity pairs. the detailed distribution of the process-identity pairs in each biological process is listed in table s a similar ratio of process homology gene pairs is found for the gene pairs inferred by the pcc method. only % of the gene pairs inferred by the lc method with the same p-value cutoff are process-identity pairs. if the results inferred by procedure i and procedure ii of the tc method are separately considered, % of the gene pairs detected by procedure i are process-identity pairs. the percentage of process-identity pairs among the gene pairs detected by procedure ii is slightly lower than that by procedure i but still somewhat higher than that of the lc method. the additional number of process-identity pairs inferred by the tc method is considerable compared to those resulted only from the lc method and the pcc method respectively. among the gene pairs predicted by the tc method with a p-value threshold of 1e- there are process-identity pairs. separately, only % of the gene pairs detected by procedure i are process-identity pairs, in contrast to as high as % for the gene pairs by procedure ii. considering the results from p-value ≤ e- mentioned above no general conclusion can be drawn with regard to the question which extraction procedure is more relevant. the percentage of the process-identity pairs among the genes pairs resulted from the lc method and the pcc method is 49% and 42% respectively. thus, the lower p-value threshold can significantly increase the portion of gene pairs involved in the same biological processes in all the three methods. at this low p-value the additional number of process-identity pairs merely inferred by the tc method amounts to % of those only resulted from the lc method. compared with those resulted by the pcc method this number declines to, suggesting that the gene pairs with a higher ranked functional association inferred by the tc method is more similar to those resulted by the pcc method. with the percentage of process-identity pairs in the range of – at p-value ≤ e- the gene pairs inferred by the three methods seem to have a relatively low biological significance. if the common part of the gene pairs inferred by all the three methods is considered, the percentage of process-identity pairs increases to %, resulting in fairly good biological significance for an in silico method of biological function inference. the biological significance can be significantly increased by lowering the p-value. at p-value ≤ 1e- the percentage of process-identity pairs ranges from to 49% for the three methods. if the common part of the gene pairs inferred by the three methods at this p-value is considered, the percentage of process-identity pairs increases to %, resulting in a satisfactorily high biological significance. the second source used in this work for assessing biological relevance of the gene pairs is the munich information center for protein sequences functional catalogue database. for protein functional classification, the mips database contains up to different levels within the hierarchy. we use here the second level of mips as qian et al. did. altogether function classes are used. it should be mentioned that some functional classes merely belong to protein cellular functions of plants and animals. here if two genes in the pair inferred have the same protein cellular function, we term the pair as a function-similarity pair. the results of comparison among the three methods are similar to those of process-identity pairs, and detailed distribution of the function-similarity pairs in each protein cellular function by the trend correlation method is provided in table s. as in the case of function-similarity pairs, if the common part of the inferred gene pairs from the three methods is considered the percentage of function-similarity pairs increases to % and % at the two p-values respectively, resulting in a good basis for further experimental and functional studies of the gene pairs inferred. comparison of inferred gene pairs with known protein interactions to further assess the biological significance of the gene pairs inferred and especially for comparing the three methods we examine here the known protein-protein interactions in current databases of yeast among the gene pairs inferred from the yeast cell cycle data by the different methods. four databases and published high quality datasets are chosen for this purpose. the protein-protein interactions collection of yu et al integrates datasets from the databases of mips, the database of interacting proteins, the biomolecular interaction network database and the experimental datasets of yeast two-hybrid and high-throughput mass spectrometry measurements. many of the interactions are manually curated beyond the experimentally derived protein-protein interactions in the three databases mentioned above. with a p-value threshold of e- the tc method detects gene pairs among the known protein interactions. separately, extraction procedure i detects protein interaction pairs, compared to pairs by extraction procedure ii. with the same p-value threshold and such gene pairs can be detected by the lc method and the pcc method respectively. the relatively low number of protein interactions among the gene pairs inferred by the tc method is a consequence of the overall lower number of gene pairs inferred by the tc method. in fact, the percentage of protein interaction pairs in the overall gene pairs inferred by the tc method is comparable to that of the pcc method but higher than that of lc method. especially the extraction procedure ii of the tc method achieves a relatively higher percentage. it should be mentioned that the low percentage in all these cases is not surprising since only data from cell cycle is used in the inference and the cell cycle represents only a part of cellular functions involving protein-protein interactions in yeast. the results presented in this section are thus more suitable for comparing the different methods rather than for quantitatively assessing the biological significance of the gene pairs inferred. fig. 2e presents a comparison of the numbers of unique and common pairs of protein-protein interactions inferred by the three methods. the additional pairs merely detected by the tc method are significant compared with those resulted merely from the lc method and the pcc method. remarkably, protein-protein interactions among protein components of the large ribosomal subunit of yeast are only significantly detected by the tc method. the fact that the proteins connect with each other by the detected interactions is consistent with the assembling phenomena of components of the large ribosomal subunit. similarly, interactions among protein components of the small ribosomal subunit of yeast are also only significantly detected by the tc method. the proteins also connect with each other by the interactions. therefore, to have a more complete coverage of the protein interactions it is obvious that the tc method should be combined with the lc and pcc methods. by loosing the p-value threshold from e- to e- the tc method can detect significantly more pairs of known protein-protein interactions. a p-value of e- in the tc method is equivalent to a pearson correlated coefficient of in the pcc method and a score of in the lc method. with this p-value threshold, and pairs can be detected by the lc method and the pcc respectively. the number of protein-protein interactions additionally inferred by the tc method is also considerable. when the mips, dip and bind updated databases are used individually for comparison similar results are obtained and summarized in table though the majority of the three databases is consistent some differences exist among them. we chose them to cover more known protein interactions and could therefore obtain more reliable statistics results. comparison with known regulatory interactions to examine the biological significance of functional associations inferred by the different methods it is also interesting to know if the gene pairs inferred cover some of the known regulatory interactions, especially those involved in the regulation of cell cycle. for this purpose, we use two regulatory datasets as a comparison basis. we first use the dataset including regulatory interactions which are confirmed with a p-value threshold of 1e- by genome wide location analysis. with a p-value threshold of e- only a relative low number of the regulatory interactions is detected by the lc method, the pcc method and the tc. regulatory interactions detected by the tc method are significantly less than those by the other two methods. nevertheless, the additional number of regulatory interactions detected by the tc method is considerable compared with those only resulted from the lc method and the pcc method. similar results are obtained when the regulatory interaction collection dataset of luscombe et al. is used. the additional number of interactions predicted by the tc method is and compared with those resulted only from the lc method and the pcc with a p-value threshold of e- with a p-value threshold of e- the results of comparison are similar to those at a p-value cutoff of e- some of the typical interactions between transcriptional regulators and target genes which are detected by the tc method but cannot be significantly detected by the lc and/or the pcc method are summarized in table s a detailed investigation of the results from a p-value threshold of e- shows that the transcriptional factors and/or the targeted genes of among known regulatory interactions inferred by the tc method are involved in cell cycle regulation according to luscombe et al.. among the interactions only six interactions have both the activated regulator and the differentially expressed targeted gene according to the gene set of cell cycle regulation given in luscombe et al.. the fact that of the known interactions have either an inactivated regulator or a non-differentially expressed targeted gene in the cell cycle condition as detected by the tc method with a high p-value cutoff seems to indicate controversies in the designation of activated regulators and differentially expressed genes in the cell cycle condition. for example, the gene fkh is annotated in hollenhorst et al. as a transcriptional factor of the forkhead family that regulates the cell cycle according to sgd. furthermore, many interactions known to involve fkh are detected by at least one of the three methods. however, fkh is not regarded as a regulator in the cell cycle condition by luscombe et al.. we further present two examples of predicted functional linkages which have well-documented biological relationships but cannot be significantly detected by the pcc method and the lc method. the regulatory relationship between rcs and gcn was confirmed by gwla with a p-value of e- the pearson correlation coefficient for this gene pair is as low as and the score of the lc is only. we detect this gene pair with a maximal matched inverted change trend sc of after rcs is shifted forward by one time point and obtain a cc value of between the matched change levels by the tc method. another example is the proteins of the genes sac and hrp that form a protein complex. though the score of the lc method is only and the pearson correlation coefficient is, we could detect this gene pair with a maximal matched similar change trend sc of and obtained a cc value of by the tc method. the inference of regulatory circuit and network as shown above, functional linkages inferred by the individual method do not give a complete picture. it is conceivable that this would be especially the case for inferring regulatory circuits and further for reconstructing regulatory network. fig. shows comparisons of inferred functional relationships by the individual methods and their combination which are mapped into regulatory circuits and networks based on the datasets. in consistence with the results of detected regulatory interactions the lc method is superior to the other two methods but still gives a very much incomplete picture. the tc and pcc methods can augment the lc method significantly. to infer more complete regulatory circuits and network the three methods should be therefore combined. this is exemplified with two relatively simple regulatory circuits. for the regulatory chain from the regulator tos to the target gene cct the negative relationship between tos and swi can only be detected by the lc method with a shift of two-time-points. the interaction between swi and abf can be detected by all the three methods. however, the interaction between abf and cct can only be detected by the tc method. the three interactions can form one regulatory chain among the complex combination of several motifs. another example about a single input motif is explained in fig. s. discussion from a practical point of view there are two key general issues in the analysis of gene expression data. first, we ought to infer functional relationships with high statistic significance and as completely as possible. second, the functional relationships inferred should have a high biological significance. we proposed a new method in this work and evaluated it mainly regarding these two aspects with the microarray data of yeast cell cycle. the new method is also compared with the local clustering method and the pearson correlation coefficient based clustering method. the number of functional gene pairs inferred depends very much on the p-value threshold in all the three methods. with the two p-value cutoffs applied the tc method detects general significantly lower number of gene pairs. nevertheless, a considerable number of gene pairs is only detected by the tc method, ranging from % of the number merely detected by the lc method to as high as % of that merely detected by the pcc method at p-value ≤ e- the ratio of additional gene pairs inferred by our method remains similar at p-value ≤ 1e- compared with the lc method but decreases to a relatively lower value compared with the pcc method. this can be explained by the fact that the statistically higher ranked correlations have mostly simultaneous relationships that can be well detected by all the three methods. since p-value ≤ e- represents a relatively high statistic significance it can be concluded that the new method detects a significantly high portion of additional functional relationships compared to the other two methods. since the shared part of gene pairs of the three methods at p-value ≤ e- is less than 50% it is also obvious that these methods should be combined to have a more complete exploitation of functional associations of genes buried in time-series expression data. concerning the biological significance % and % of the gene pairs inferred by our method are process-identity ones by comparing with the known biological processes in the s. cerevisiae genome database. this is compared with a process-identity ratio in the range of –49% for the lc method and – % for the pcc method. similar results are obtained when the mips protein functional catalogue database is used to assess the biological significance. these results suggest that the gene pairs detected by the three methods achieve fairly good and comparable biological significance. as in the case of the total gene pairs the additional gene pairs with biological significance inferred merely by the trend correlation method is considerable compared to the other two methods. we also examined the gene pairs inferred by the three methods with regard to known protein interactions and known regulatory interactions in yeast. in general, the percentage of protein interaction pairs in the overall gene pairs is low and comparable for all the methods, % for the lc method and % for the pcc method). this can be understood in view of the fact that only cell cycle data under very specific conditions are used here and the cell cycle represents only a small portion of the cellular activities. furthermore, only protein interaction pairs with significantly changed expression levels of all the involved partner proteins can be theoretically detected. this applies also to the known regulatory interactions. it is shown that most of the detected regulatory interactions are indeed involved in the cell cycle regulation. nevertheless, the number of protein and regulatory interactions additionally inferred merely by the trend correlation method is significant in all the cases. it is obvious that more molecular interactions can be obtained if more microarray datasets under different conditions are considered. given the large number of functionally associated gene pairs inferred by the different methods an important question arises as to how we can find gene pairs which have really a high biological significance and thus would be best candidates for further experimental and functional studies. to this end, the shared part of gene pairs inferred by all the three methods is of particular interest, especially at low p-values. it is found for example that the percentage of process and function-similarity gene pairs of the shared pairs at p-value ≤ 1e- can be as high as % and %, building a very good basis for experimental study. the common part of the tc and lc methods would be also of particular interest for finding time-delayed and/or inverted functional relationships which have received so far less attention. it should be mentioned that all the functional associated pairs predicted have a high probability to be true and can thus serve as hypotheses for further study in view of the high statistic significance criteria applied. we would also like to emphasize that the combined use of the different methods is not only useful for finding more and highly possible potential candidates of functional association but also very important to infer more complete regulatory circuits and network as exemplified in this study.  the major difference between our method and the other current methods is that the trend correlation method is based on the main change trend and comprehensively considers correlation coefficient between the main change trend of two genes, whereas the other methods are mainly based on the correlation of point-to-point expression levels of two genes. hence the trend correlation method can reveal additional gene pairs with same function or in the same biological process but yet not significantly co-expressed. it therefore also can infer additional protein-protein and regulatory network as demonstrated. as mentioned above, a combined use of different methods is presently necessary for the analysis of time-series microarray data. as clearly demonstrated in this work the proposed new method can significantly augment the currently major methods and is well suitable for exploring temporal relationships of gene expression in time-series data. 
change trends between consecutive time points in time-series data have been so far not well explored. in this work we present a new method based on extracting main features of the change trend and level of gene expression between consecutive time points. the tc method is demonstrated with data from yeast cell cycle and compared with the lc method and the widely used pearson correlation coefficient based clustering method. % compared with the lc method and % with the pcc method for a p-value threshold of e- moreover, the percentage of the inferred gene pairs consistent with databases by our method is generally higher than the lc method and similar to the pcc method. it should be emphasized that the overlapping of gene pairs detected by the three methods is normally not very high, indicating a necessity of combining the different methods in search of functional association of genes from time-series data. the tc method can significantly augment the current major methods to infer functional linkages and biological network and is well suitable for exploring temporal relationships of gene expression in time-series data. although the tc method predicts an overall lower number of gene pairs than the other two methods at a same p-value threshold, the additional number of gene pairs inferred by the tc method is considerable: e.g. for a p-value threshold of 1e- the percentage of process-identity and function-similarity gene pairs among the shared part of the three methods reaches % and % respectively, building a good basis for further experimental and functional study. a significant number of the gene pairs only inferred by the tc method are process-identity or function-similarity pairs or have well-documented biological interactions, including known protein interactions and some known cell cycle related regulatory interactions. the biological significance of the gene pairs is examined with several large-scale yeast databases. the increasing availability of time-series expression data opens up new possibilities to study functional linkages of genes. present methods used to infer functional linkages between genes from expression data are mainly based on a point-to-point comparison. the new method considers time shifts and inverted relationships in a similar way as the local clustering method but the latter is merely based on a point-to-point comparison. the method, termed as trend correlation, includes two major steps: calculating a maximal local alignment of change trend score by dynamic programming and a change trend correlation coefficient between the maximal matched change levels of each gene pair; inferring relationships of gene pairs based on two statistical extraction procedures. furthermore, the combined use of methods is important to infer more complete regulatory circuits and network as exemplified in this study.
 the increasing availability of time-series expression data opens up new possibilities to study functional linkages of genes. present methods used to infer functional linkages between genes from expression data are mainly based on a point-to-point comparison. change trends between consecutive time points in time-series data have been so far not well explored.  in this work we present a new method based on extracting main features of the change trend and level of gene expression between consecutive time points. the method, termed as trend correlation, includes two major steps: calculating a maximal local alignment of change trend score by dynamic programming and a change trend correlation coefficient between the maximal matched change levels of each gene pair; inferring relationships of gene pairs based on two statistical extraction procedures. the new method considers time shifts and inverted relationships in a similar way as the local clustering method but the latter is merely based on a point-to-point comparison. the tc method is demonstrated with data from yeast cell cycle and compared with the lc method and the widely used pearson correlation coefficient based clustering method. the biological significance of the gene pairs is examined with several large-scale yeast databases. although the tc method predicts an overall lower number of gene pairs than the other two methods at a same p-value threshold, the additional number of gene pairs inferred by the tc method is considerable: e.g. % compared with the lc method and % with the pcc method for a p-value threshold of e- moreover, the percentage of the inferred gene pairs consistent with databases by our method is generally higher than the lc method and similar to the pcc method. a significant number of the gene pairs only inferred by the tc method are process-identity or function-similarity pairs or have well-documented biological interactions, including known protein interactions and some known cell cycle related regulatory interactions. it should be emphasized that the overlapping of gene pairs detected by the three methods is normally not very high, indicating a necessity of combining the different methods in search of functional association of genes from time-series data. for a p-value threshold of 1e- the percentage of process-identity and function-similarity gene pairs among the shared part of the three methods reaches % and % respectively, building a good basis for further experimental and functional study. furthermore, the combined use of methods is important to infer more complete regulatory circuits and network as exemplified in this study.  the tc method can significantly augment the current major methods to infer functional linkages and biological network and is well suitable for exploring temporal relationships of gene expression in time-series data. 