Discovering time-lagged rules from microarray data using gene profile classifiers.

Abstract

Gene regulatory networks have an essential role in every process of life. In this regard, the amount of genome-wide time series data is becoming increasingly available, providing the opportunity to discover the time-delayed gene regulatory networks that govern the majority of these molecular processes.
This paper aims at reconstructing gene regulatory networks from multiple genome-wide microarray time series datasets. In this sense, a new model-free algorithm called GRNCOP2 (Gene Regulatory Network inference by Combinatorial OPtimization 2), which is a significant evolution of the GRNCOP algorithm, was developed using combinatorial optimization of gene profile classifiers. The method is capable of inferring potential time-delay relationships with any span of time between genes from various time series datasets given as input. The proposed algorithm was applied to time series data composed of twenty yeast genes that are highly relevant for the cell-cycle study, and the results were compared against several related approaches. The outcomes have shown that GRNCOP2 outperforms the contrasted methods in terms of the proposed metrics, and that the results are consistent with previous biological knowledge. Additionally, a genome-wide study on multiple publicly available time series data was performed. In this case, the experimentation has exhibited the soundness and scalability of the new method which inferred highly-related statistically-significant gene associations.
A novel method for inferring time-delayed gene regulatory networks from genome-wide time series datasets is proposed in this paper. The method was carefully validated with several publicly available data sets. The results have demonstrated that the algorithm constitutes a usable model-free approach capable of predicting meaningful relationships between genes, revealing the time-trends of gene regulation.

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METHODOLOGY ARTICLEOpen Access
Discovering time-lagged rules from microarray
data using gene profile classifiers
Cristian A Gallo1, Jessica A Carballido1and Ignacio Ponzoni1,2*
Abstract
Background: Gene regulatory networks have an essential role in every process of life. In this regard, the amount of
genome-wide time series data is becoming increasingly available, providing the opportunity to discover the time-
delayed gene regulatory networks that govern the majority of these molecular processes.
Results: This paper aims at reconstructing gene regulatory networks from multiple genome-wide microarray time
series datasets. In this sense, a new model-free algorithm called GRNCOP2 (Gene Regulatory Network inference by
Combinatorial OPtimization 2), which is a significant evolution of the GRNCOP algorithm, was developed using
combinatorial optimization of gene profile classifiers. The method is capable of inferring potential time-delay
relationships with any span of time between genes from various time series datasets given as input. The proposed
algorithm was applied to time series data composed of twenty yeast genes that are highly relevant for the cell-
cycle study, and the results were compared against several related approaches. The outcomes have shown that
GRNCOP2 outperforms the contrasted methods in terms of the proposed metrics, and that the results are
consistent with previous biological knowledge. Additionally, a genome-wide study on multiple publicly available
time series data was performed. In this case, the experimentation has exhibited the soundness and scalability of
the new method which inferred highly-related statistically-significant gene associations.
Conclusions: A novel method for inferring time-delayed gene regulatory networks from genome-wide time series
datasets is proposed in this paper. The method was carefully validated with several publicly available data sets. The
results have demonstrated that the algorithm constitutes a usable model-free approach capable of predicting
meaningful relationships between genes, revealing the time-trends of gene regulation.
Background
The genome encodes thousands of genes whose products
enable cell survival and numerous cellular functions. The
amount and the temporal pattern in which these
products appear in the cell are crucial to the processes of
life. Gene Regulatory Networks (GRNs) govern the levels
of these gene products. A GRN is the collection of
molecular species and their interactions, which together
control gene product abundance [1,2]. Numerous cellular
processes are affected by regulatory networks.
Innovations in experimental methods have enabled
large scale studies that allow parallel genome-wide gene
expression measurements of the products of thousands
of genes at a given time, under a given set of conditions
and for several cells/tissues of interest. This technology,
called DNA microarray, introduces a variety of data ana-
lysis issues (due to the large amount of information to
analyze) that are not present in traditional molecular
biology [3].
Over the past few years, several statistical and artificial
intelligence techniques have been proposed to carry out
the reverse engineering of GRNs from monitoring and
analyzing gene expression data [1-6]. These techniques
vary from the simplest Boolean models to Continuous
and Single Molecule Level models [2]. In this regard,
model-free approaches are decidedly attractive because
of the complexities of dynamic molecular networks [7].
Moreover, most of gene networks are hard to be
mapped precisely by any parsimonious mathematical
model. Then, data mining approaches offer a way to
identify regulatory mechanisms directly from the input/
* Correspondence: ip@cs.uns.edu.ar
1Laboratorio de Investigación y Desarrollo en Computación Científica
(LIDeCC), Departamento de Ciencias e Ingeniería de la Computación,
Universidad Nacional del Sur, Av. Alem 1253, 8000, Bahía Blanca, Argentina
Full list of author information is available at the end of the article
Gallo et al. BMC Bioinformatics 2011, 12:123
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© 2011 Gallo et al; licensee BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons
Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in
any medium, provided the original work is properly cited.

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output data without any underlying model. In particular,
rule-based approaches offer several advantages when
data-driven analysis is performed. They are highly
abstract model-free techniques and hence, they require
the least amount of data, with an important ability to
perform inferences [2]. Additionally, the simplicity of
these approaches allows the inference of large size mod-
els with a higher speed of analysis. On the other hand,
they can merely display qualitative dynamic behavior [2].
Another important aspect to be considered, when deal-
ing with this biological problem, is constituted by the
manner in which the temporal patterns of a GRN are cap-
tured. As it was mentioned in some other studies [8,9], the
time-delayed gene regulation is a common phenomenon.
Thereby, multiple time-delayed gene regulations can be
considered the norm, while single time-delayed associa-
tions the exception [7]. This issue of the time-delayed
gene regulations is well recognized by several authors
[7,10-13], although, in most cases, they merely deal with
the gene networks delayed for one unit of time due to the
inherent complexity and computational cost involved.
In this paper, a new machine-learning approach for the
inference of time-lagged rules from time series gene
expression data is assessed. The discovered relationships,
that represent potential interactions between genes, may
be used to predict the gene expression states of a gene in
terms of the gene expression values of other genes and, in
this way, a putative GRN may then be reconstructed by
applying and combining these rules. The approach offers
several relevant and distinguishing features in relation to
most of the existing methods. First of all, the gene expres-
sion value discretization criterion performed in this work
is neither arbitrary nor uniform. Secondly, it can infer
rules with multiple time-delays. Also, the results can be
easily interpreted since the rules are derived from schemes
that classify the different regulation states. As well, the
algorithm can infer the relationships between genes auto-
matically from multiple microarray time series data.
Finally, the new method is capable of processing large
scale datasets in order to perform genome-wide studies.
The rest of the paper is organized as follows: in the
next subsection, several machine learning techniques
available in the literature for GRN inference are over-
viewed. Following, the underlying methodology and the
main characteristics of the new algorithm are presented.
Next, two experimental phases are described. The first
one is constituted by a detailed comparison with several
related methods; the second one contains a performance
analysis of the method in a genome-wide scale. Finally,
some conclusions are put forward.
Related work
As it was aforementioned, several statistical and artificial
intelligence techniques have been proposed in order to
reconstruct a GRN from gene expression data. In this
section, some of the approaches from the area of
machine learning will be summarized. For a more
detailed review please refer to [2,3,6].
Clustering techniques are one of the most used com-
putational strategies for analyzing microarrays [14-16].
These approaches approximate regulatory networks by
identifying groups of co-expressed genes and by analyz-
ing relationships between their regulatory regions and
DNA binding motifs targeted by known transcription
factors. However, determining the interactions that can
exist between different genes is not easily achieved by
direct clustering, particularly because genes can partici-
pate in more than one gene network. Another limitation
of these approaches is that they assume co-expression is
always equivalent to regulation. Moreover, these meth-
ods imply symmetric relationships between the genes,
which might not always correspond to biological phe-
nomena [10].
Bayesian Networks also constitute the basis of several
approaches for GRN inference [11,17,18]. These meth-
ods employ conditional probabilistic distributions for
gene interaction modeling. Particularly, Friedman et al.
[17] proposed a heuristic algorithm to produce networks
which appeared biologically plausible for the yeast cell
cycling array data. As another example, in Zou and
Conzen [18] a model for genetic regulatory interactions
that combines the simple Boolean logic semantics of
Boolean Networks and the uncertainty offered by Baye-
sian Networks was proposed. Despite the strong theore-
tical rationale behind these approaches, the exponential
explosion of the parameter space required for these
models, together with the large quantity of data needed
to make reliable inferences, reduces their capacity to
infer complex GRNs by only using gene expression data.
Moreover, since they are acyclic directed graphs, they
cannot represent an auto regulation or a time-course
regulation in a straightforward way [19].
As well, the Apriori Algorithm is also a classic
method, designed to operate on databases for learning
association rules [20]. In Baralis et al. [21], this data
mining technique was used for the extraction of time-
delayed association rules in gene expression data. They
mine the rules by means of the application of the algo-
rithm on matrices of time-lagged gene expression pro-
files, similar to those used in [7]. Following the same
basis, in Nam et al. [22] a modified version of the
Apriori Algorithm was proposed. In this work, they
extended the original method in order to consider tem-
poral item sets, allowing the extraction of temporal
association rules. Since the performance of this method
highly depends on the parameters set being selected,
they employ a parameter fitting phase that uses known
regulation information in order to find the best setup
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for a given dataset. However, Apriori based methods
also scale poorly and are sometimes impractical with
high dense datasets such as microarrays [3], due to the
high-computational cost of the evaluation of candidate
and test sets.
Decision trees are also among the most popular classi-
fication algorithms in current use within data mining
and machine learning research areas. In this sense, Soi-
nov et al. [12] approached the task of reconstructing
GRNs as a classification problem, proposing the applica-
tion of decision trees to infer classifiers that may repre-
sent regulatory rules (relationships) between genes. In
this work the authors have considered at most one unit
of time-delay and have applied the C4.5 algorithm to
infer the decision trees [23]. In the same regard, Li et al.
[7] proposed the application of decision trees to infer
relationships with one or more units of time-delay, as a
generalization of Soinov’s method. For each target gene,
they constructed its time-delayed gene expression profile
and then used a decision tree to discover the time-
delayed regulations that modulate the activities of the
target gene. Although these are sound and conceptually
interesting approaches, working directly on large data-
sets of thousands of genes, they can be computationally
highly demanding.
Boolean Networks were one of the first models to be
employed in GRNs inference [24,25] and new variations
of this approach have been recently published [26].
These models basically aim at inferring logical rules
from a discretization of gene expression time series.
Even though these models can be easily applied, they
depend on arbitrary discretizations of the gene expres-
sion values [12], which impose strong assumptions and
restrictions about the biological system under study. In
order to overcome this limitation, Ponzoni et al. [10]
proposed a machine-learning algorithm called GRNCOP
based on combinatorial optimization that does not
assume arbitrary nor uniform gene expression value dis-
cretizations. The thresholds are calculated dynamically
by applying the same continuous-valued attribute discre-
tization techniques as those used for classification algo-
rithms based on decision trees. However, this
discretization is performed merely for regulatory genes,
since the thresholds for the target genes are calculated
by using the mean expression value. Another limitation
is that it is only able to infer rules of one unit of time-
delay at most. This method is in fact the antecessor of
the approach proposed in this work, called GRNCOP2,
in which all these limitations are overcome.
With the exception of clustering approaches, all of the
aforementioned rule-based mining algorithms have been
assessed only for highly reduced datasets. Although per-
forming over a small amount of data can give an idea of
the performance of a method, in any real scenario the
large size and the amount of datasets available impose
another challenge on the reconstruction of GRNs that
few authors have considered: the scalability problem [3].
This issue represents one of the most important weak-
nesses of the previously cited studies for rule-based
inference methods, due to the lack of evidence that they
can actually perform over large datasets, thus preventing
their applicability in any complex study. In this context,
the algorithm presented in this work exhibits most of
the desirable features mentioned before, and in addition
it successfully deals with the main drawbacks detected
in the existing methods.
Results and Discussion
For this work, the time series encoded in the gene expres-
sion dataset are represented by means of a gene expression
data matrix, X, where the rows and columns represent
genes and time-points, respectively. In this way, each ele-
ment xijof X contains the expression value of geneiin the
time-point (sample or experimental condition) j. Although
the gene expression values belong to a continuous range
of the real numbers, it is possible to define a finite expres-
sion state set for each gene by means of a discretization
procedure. Such a procedure is required in order to
encode the inputs for any combinatorial optimization pro-
cess or machine-learning method. In this paper we work
with two states for each gene: upregulated (when the gene
is expressed with a value greater than a specific discretiza-
tion threshold) and downregulated (when the gene is
expressed with a value lower or equal to a specific discreti-
zation threshold).
Therefore, the inference process requires the definition
of discretization thresholds in order to infer putative reg-
ulatory relationships between genes. These “discretization
thresholds” have traditionally been estimated as unique
static values for all of the genes under study. For exam-
ple, ad hoc methods based on mean expression values
have been applied. However, a more biologically mean-
ingful scheme should model the fact that a gene may
actually have distinct discretization thresholds in relation
to different genes in the GRN [10]. For example, regard-
ing the regulatory network under study in this work that
corresponds to the Saccharomyces cerevisiae organism,
the CLB2 and SWI5 genes are shown to be potentially
activated by CLB1 gene, but their respective upregulation
thresholds are different. Therefore, a fundamental pro-
blem consists in the estimation of the regulation thresh-
olds for each gene in relation to every potential target
gene, which can reflect significant interactions between
them with a higher level of accuracy.
In this regard, two different types of discretizations are
defined in this paper. Broadly, the first one is to set the
state of each target gene, and it is called Target Discreti-
zation Threshold (TDT). The second one is to evaluate
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the potential interaction between each pair of genes and
it is calculated in an adaptive gene-pair-specific way.
This last discretization is called Relative Regulation
Threshold (RRT).
At this point, our hypothesis is stated as follows: rules
- potential regulatory relationships - may be accurately
inferred from gene time series data to reveal how the
present and future state of a gene may be affected by
the gene expression values of the other genes, taking
into account their RRT. In this paper, we consider time-
lagged rules that represent the situation in which the
state of a geneiin a time-point j depends on the gene
expression values of other genes in the previous time-
point (that is to say, previous experimental condition) j
- w, where w is a non-negative integer value represent-
ing the time-delay in the relation. The syntax of the
rules is: < symbol >< gener>w®< symbol >< genei>
where generand geneistand for gene regulator and
gene target respectively. The symbol + (-) on the left
side of the rule indicates above (below) some RRT for
the generw.r.t. genei, whereas the symbol + (-) on the
right side of the rule indicates upregulated (downregu-
lated) state, depending on the TDT for the genei. For
example, the rule +/- CLB1 3® +/- CLB5 denotes that,
if CLB1 is above its RRT in relation to CLB5, tCLB1,CLB5,
in a time-point j, then CLB5 will be upregulated in the
time-point j+3 and, if CLB1 is below or equal to tCLB1,
CLB5in a sample j, then CLB5 will be downregulated in
the sample j+3. The types of rules obtained through this
scheme are similar to those studied in [7,10-12]. The
main difference is that this scheme allows the represen-
tation of both simultaneous and time-lagged rules
spanned in any unit of time-interval, which constitutes
the kind of rules that GNRCOP2 is capable of inferring.
GRNCOP2 infers the association rules described above
by exploring the possible combinations of interactions
between each pair of genes. In this sense, six particular
cases are assumed, which are represented by the non null
integer numbers between -3 and 3, and a special case that
indicates the absence of any relation represented by the
number 0. All of these cases are described in Table 1.
In mathematical terms, the inference of the rules to
reconstruct a GRN can be expressed as the following
combinatorial optimization problem:
n?
subject to:
i=1
max
πi∈Pσ ∗ (πi,δ (X,i)),
(1)
• n is the number of genes in the microarray dataset.
• m is the number of time-points in the microarray
dataset.
• X Î ℜnxmis the matrix with the expression data.
• P is the space of all vectors v of dimension n such
that v(r) Î {-3, -2, -1, 0, 1, 2, 3} ∀ r, r = 1..n.
• δ (X, i) is the discretization function such that δ
(X, i) = Diand DiÎ {-1,1}nxm.
• πiÎ P is a classifier for Di.
• s*(πi, Di) is a general performance function of πi
as a classifier of Di.
From now on, the symbol ∏windicates the set of
optimal classifiers, ∏w= {π1, π2, ..., πn}, for a given
time-delay w. It is important to note that the general
optimization problem is the same for all the time-lagged
rules. The only difference lies in the definition of the
discretization function δ (X, i), because the delayed rules
are based on expression value discretizations of X that
consider the required temporal shift.
Algorithm
Although the basic ideas behind GRNCOP, more spe-
cifically the adaptive regulation thresholds and the
combinatorial optimization of rules classifiers, remain
in GRNCOP2, the new method constitutes a signifi-
cant evolution of the previous algorithm due to the
challenges that impose the improvements being pro-
posed. Figure 1 shows an abstract representation of
the approach. The machine-learning process used to
obtain the rules iteratively performs the search
through all the datasets for all the required time-
delays. The algorithm receives as input a set of K
microarray time series datasets and returns an array П
of dimension W that contains, in each position w, the set
of rules obtained with a time-delay w. The following sub-
section will explain in detail the main characteristics of the
procedure.
An improved discretization technique
for the target genes
In order to obtain the TDTs of the target genes,
GRNCOP2 employs a technique that is able to
infer the gene states in a more precise way, compared
to the mean expression value used in GRNCOP. In
Table 1 Types of rules inferred by GRNCOP2
Rule typeTime-lagged rule associated
+ generw® - genei
- generw® + genei
+/- generw® -/+ genei
generdoes not interact with genei
+/- generw® +/- genei
+ generw® + genei
- generw® - genei
-3
-2
-1
0
1
2
3
Summary of the different types of rules inferred by GRNCOP2, where w
denotes the time-delay in the regulation.
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mathematical terms, the procedure for the discretiza-
tion of a geneican be defined as follows:
min
S1,S2⊂S(var(S1) + var(S2)),
subject to:
(2)
• S is the set of sample values for the genei.
• S1∩ S2= ∅, S1∪ S2= S, |S1|>1 and |S2|>1.
• var(S1) and var(S2) are the variance of S1and S2
respectively.
• S1and S2represent the two expression states for
the genei.
Basically, the procedure divides the samples of the genei
in the two sets that have the minimum sum of its var-
iances. The cardinality of S1and S2is required to be
greater than one in order to avoid the effects of a possible
outlier in the samples, since it is improbable that a gene is
clearly expressed or inhibited in only one time-point.
Thus, when the samples of a geneiare separated in a parti-
tion that violates this restriction, the geneiis no longer
considered in the inference process for the actual datasets.
Another approach could have been to exclude the conflic-
tive time-point in the search of the partition. However,
this can lead to the same situation described before, thus
reintroducing the issue that was supposed to be fixed.
This technique is in essence a clustering procedure
similar to a k-means with k = 2. However, since the num-
ber of clusters is 2 and the elements of S have a total
ordering, the problem can be optimally and efficiently
solved through the following deterministic procedure:
• Sort the elements of S on an array L.
• Search for the element e such that var(L[1..e]) +
var(L[e+1 ..|S|]) be the minimum.
• Return (L[e]+L[e+1])/2 as the TDT of the genei.
It is important to state that, according to Figure 1,
TDT values are calculated for each dataset separately.
Rule Consensus Process
In essence, the main loop of the algorithm applies the
same inference method to the K microarray time series
datasets given as input, and then returns the intersection
of the results for all of the datasets. The objective of this
procedure, incorporated by GRNCOP2, is to automati-
cally assess the rules obtained by the algorithm through
different datasets, thus increasing the degree of evidence
required for the potential regulatory relationships to be
returned. The intersection of the rules obtained from
two datasets k1and k2is defined as follows:
?k1∩ ?k2= (?1
k1∩ ?1
k2,?2
k1∩ ?2
k2,...,?W
k1∩ ?W
k2), (3)
where:
• W is the maximum time-delay established by the
researcher.
• ?k1and ?k2are the rules obtained from datasets k1
and k2respectively in all time-delays W.
• ?w
and k2respectively with a time-delay w.
k1and ?w
k2are the rules obtained from datasets k1
Basically, the intersection of the results is the intersec-
tion of each component, i.e., time-delay w, of the set of
rules obtained from the K datasets. In an ideal scenario,
all the time series microarray datasets should have the
same sampling rate and then, a direct mapping between
the time-delay w and the real time-points would be
achieved. However, this is far from any real scenario
Figure 1 General schema of the GRNCOP2 Algorithm.
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