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Modeling and Simulation of Gene Regulatory

Network: A Comprehensive Survey

Surama Biswas, Tamal Pal, Sriyankar Acharyya

Abstract—The Gene Regulatory Network (GRN) specifies the series of regulatory interactions between different genes. A target gene is interacted by a signal which is originated from the expression of its regulator gene. A gene is known to be expressed when it synthesizes a protein and the degree of synthesizing the protein determines the level of its expression. The same gene can behave as a 'target' in one state of interaction and a 'regulator' in the next state. If there are many interacting genes in a biological system, a network can be formed out of them where the genes are treated as nodes and interaction between any two genes is treated as an edge. This network is known as Gene Regulatory Network. Simulation of GRN addresses the issue of reconstructing the network on the basis of the expression levels of the interacting genes. Various mathematical tools are used to design the system and different optimization techniques are used to find the optimal design. The process of designing starts with time-dependent (Time-series) and condition-dependent (Steady state) gene expression data, available from micro-array chips. The target gene is activated depending on the collective interactions made to it. The problem can be modeled using Neural Network and application of Fuzzy logic may improve the design. There are two issues to discuss. One is related to uncover the parameters involved in GRN called parameter estimation problem. The other is to predict the network structure step by step while learning the parameters. Applications of meta-heuristic algorithms are proved to be efficient in resolving both the issues.

Index Terms—Bayesian network, correlation, direct scale free network, fuzzy logic, gene expression, microarray, neural network, optimization.

1 INTRODUCTION

————————————————————
ENES, the functional and primarily protein coding parts of the DNA are the blue prints of life in an organism. Gene expression is the way of response of a gene to a
physiological phenomenon. The expression of a gene is com- posed of two main steps, viz., Transcription and Translation. Transcription is a bio-chemical change happening in the cod- ing region of a gene (exon) by some external or internal influ- ence which produces some intermediate products like primary RNA, messenger RNA or mRNA etc. [1], [2], [3]. Another bio- chemical reaction named translation when acted on the mRNA, the final product is formed that is known as protein. The extent of expression of a gene is a measurable quantity and named as expression level of the gene at the instance. Due to different cellular activities [4], [5], [12] the expression level of a gene may change and the protein thus produced may bind to the promoter site of any other gene causing it to be expressed. According to modern theory the sole cause of regu- lation is not the protein alone. In other words gene regulation does not occurs only in translational or post translational lev- els, it occurs also in transcriptional level as well by a short format RNA called micro RNA. The regulatory interaction may change the protein synthesis of the target gene in two ways, either to increase or to decrease [13], [15], [16]. They are

————————————————

Surama Biswas is currently working as a Research Assistant in Computer Science department in West Bengal University of Technology, WB, and In- dia, PH-+ 919433045471. E-mail: surama.biswas@gmail.com.

Tamal Pal is currently working as an Assistant Professor of Computer

Science and Engineering department in Dream Institute of Technology, WB, and India. PH-+ 9163692941. E-mail: tamalpal91@yahoo.co.in.

Dr. Sriyankar Acharyya is currently working as an Associate Professor in Computer Science department in West Bengal University of Technology, WB, and India, PH-+ 919903389062. E-mail: srikalpa8@gmail.com

named as enhancing or repressive regulatory interactions re- spectively.
The simulation of GRN can be done in two ways. One way is to construct the network from the microarray data by meas- uring similarity between expression levels at different time instances or conditions. This is called forward modeling. The other way of simulation is the Inverse modeling. In this pro- cess, network is constructed arbitrarily and expression level of the target gene is generated using certain mathematical formu- la. The error is calculated by comparing the simulated data with the original gene expression data. The structure of the gene regulatory network obtained from forward modeling provides a static road-map where the actual interest lies in the pattern of traffic, the cause of emerging such patterns, the way to control them so on and so forth. The structure of the net- work is similar to listing up the parts of a machine which does not give a clear picture of how the machine performs a partic- ular function and how a subset of machine parts interact to each other when performing the function[11]. If the network is simulated in steps, the dynamics of the system may be under- stood. At each time step, a new edge is attached to the net- work. This depicts the dynamics of the gene interactions in cellular activities.
All the genes of an organism are sparsely present in the ge- nome. The genome is embedded in each cell of the body of an organism [8], [9], [10]. But a gene may be expressed in one tissue and remain unexpressed in any other. In this context there are some questions to be answered. Which gene is ex- pressed? When will the gene be expressed, in which tissue of the organism and to what extent [14]. For example, Insulin, the best studied poly-peptide hormone in human being, is pro- duced by Human Insulin Gene (INS). It is produced as a result of increasing sugar level in blood and acts to lower the level of

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the sugar. The gene (INS) is expressed in the Beta Cells in Pan- creas only, though it is present in any other cell of the body.
The study of the regulatory system consists of the following components [11]:
i) Structure of the System: The gene regulatory network
(GRN) is represented by a graph G, where each gene i ∈ N (N
is the maximum number of genes in the system) represent a
node and regulatory interaction between i and j is an edge if
gene j has any influence to the expression of gene i.
ii) System Dynamics: Time dependent (producing time
series data) and condition dependent (producing steady state
data) behavior of a set of genes is represented in system dy- namics.
iii) Controlling Methods: The internal state of the cell may be controlled to minimize the malfunction as well as to facilitate drug finding applications.

iv) Design Methods: Simulation of GRN from gene ex- pression data uses different probabilistic models and employs various local search algorithms to optimize the construction procedure.
Some applications of GRN are described as under:
a) Biomarker detection: Biomarker detection is an im-
portant application of the study of gene regulatory network. The gene regulatory network depicts the cellular activity un- dergoing in the tissues. The regulatory interaction pattern changes from normal to diseased tissues. The gene expression
data from normal tissues and diseased tissues, tissues after therapeutic intervention and cure is obtained from microarray and regulatory interactions are modeled. It may be seen that a strongly connected component of the four different graphs corresponding to four different cases mentioned above, has been changed from one state to another. This sub-graph indi- cating the state of a disease is called bio-marker of the disease. Biomarker detection using Clustering algorithms has been done [49]. The study of gene regulatory network in the context of biomarker detection is being an interesting area of research [30].
b) Drug finding: The adjacency matrix of gene regulato- ry network is generally seen to be sparse in nature. Some nodes are strongly connected to the rest of the network. These genes are very few in number. They are called the hubs of the network. Drugs are designed concentrating on these hubs and aiming to reduce or increase the expression of the gene so as the disease can be taken under control. The hub genes detec- tion in the study of gene regulatory network is especially in- teresting for drug finding.
c) Detection of side effects of a drug: After the applica- tion of a drug it gradually starts taking part in cellular activi- ties. It may be considered as being a part of the regulatory network and sending regulatory interactions to the gene(s) targeted. But along with the targeted gene it may send undue
encompasses the data extraction and preprocessing, the co- expression matrix generation from processed data and finally probabilistic modeling. Section 3 defines the different soft- computing techniques including Recurrent Neural Network, Fuzzy Logic, Optimization Techniques and a software aiding the reconstruction, named as, GeneNetWeaver. In Section 4, the experimental issues including the source of data, the met- rics concerned in the performance evaluation of the simulation and a performance analysis from reviewed literature.

2. GRN: MATHEMATICAL FRAMEWORK

The microarray data, collected from image analysis are prone to errors. So the data is processed before being used in the simulation. The preprocessed data is used for co-expression matrix generation which gives the possible relation between genes. The graph generation process is based on either associ- ation computation between two samples of gene expressions or probabilistic reconstruction of the graph. Mathematical modeling is required to be applied in Data extraction and pre- processing as described in Section 2.1, Co-expression matrix generation as described in 2.2 and Probabilistic graph genera- tion as described in 2.3.

2.1 Data Extraction and Preprocessing

Gene expression level denotes a quantity which indicates the level of expression (protein generation process) for a particular gene in the cell and is represented in a floating point number obtained from microarray. The final input matrix contains N rows where each gene represents a row and T columns or samples where each column correspond either to a time point or to a condition. Here each entry represents the gene expres- sion level of a gene i at sample t. But the raw data collected from microarray is prone to contain various noise elements. So there should be methods of extraction [30] of original gene expression levels sampled earlier.
Consider, Yt denotes a set of microarray samples of all tar- get genes measured at sampling time t and Xt be the gene ex- pression levels at t, then Yt is mapped to Xt . The expression
level is sampled at certain time intervals to generate time se-
ries data. There are several models regarding extraction of
gene expression from microarray data.

2.1.1 Fully correlated model

This model is based on the assumption that microarray data is fully identical with the actual gene expression i.e. Yt = Xt .This model is too simplified and does not correspond to real micro- array experiments since it ignores noise factor.

2.1.2 Linear Gaussian model

White Gaussian noise is the linear combination of Gaussian noise (i.e. probability density function of the noise amplitude follows normal distribution) and white noise (noise with power carried by

regulatory interaction to other genes present in the network
and even to the tissues not having affected by the disease. So

wave is constant per unit frequency).Yt

is related to Xt
by a line-
side effect detection in the study of regulatory network is an important issue.
The rest of the paper is organized in following manner: Section 2 describes the GRN reconstruction framework which
ar combination considering white Gaussian noise to be mixed
with microarray data.
𝑌𝑡 = 𝑀 ⋅ 𝑋𝑡 +𝑎𝑜𝑏𝑠 +𝑣𝑡
M: Projection matrix is assumed to be identity matrix.

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aobs : A vector to adjust measurement error during observa- tion.
vt : White Gaussian noise vector
.

2.1.3 Gaussian model with discrete expression levels

In this model expression of each gene i at time t, xi (t) is related to corresponding micro-array data yi (t) with probability de- termined by Gaussian function. An individual probability for each gene indicates that this model accepts different noise lev- els for different genes of the network making the model non- linear and potentially better.
After extraction of the data it undergoes some pre- processing operation by calculating signal to noise ratio (SNR) and normalization [49]. Suppose two data sets of normal and tumor cells named as class1 and class2 are present in hand, then SNR is defined as

�𝑚𝑆𝑎𝑡(𝑆𝑙𝑎𝑠𝑠1) − 𝑚𝑆𝑎𝑡(𝑆𝑙𝑎𝑠𝑠2)�

xik is the expression level of gene i at kith sample mean(Xi ) is the mean of N expression levels of gene i
S.D(Xi ) is the Standard deviation of N expression levels of gene i
The Pearson Correlation Coefficient is compared to a threshold which may be hard (in the range of .6 - .8) in case of un-weighted network or as a function of different parameters in case of weighted network. In both the cases the coefficient is compared with the threshold σ as under.

𝑆�𝑋𝑖 , 𝑋𝑗 � > 𝜎

If the condition satisfies an edge will be drawn between the
genes i and j.

2.2.2 Partial Correlation Coefficient

Correlation between two genes generally specifies either a direct relation between them or a common dependence on

(𝑆𝑁𝑅) = �

�𝑆. 𝐷(𝑆𝑙𝑎𝑠𝑠1) + 𝑆. 𝐷(𝑆𝑙𝑎𝑠𝑠2)�

some third gene. But the Partial Correlation only considers the direct relation between two genes and disobeys the theory of
Here S.D represents the standard deviation. After the calcu- lation of SNR, the dataset is arranged in the descending order of SNR and a subset of the data with high SNR is collected. Next task is to normalize each expression data to scale in a certain range (say 0 to 1). If the jth sample, named Xj contains xij, the expression level of ith gene, the formula of normaliza- tion of xij is given by:

�𝑥𝑖𝑗 − 𝑚𝑆𝑡𝑆𝑚𝑢𝑚�𝑋𝑗 ��

joint dependence of any third gene.

2.2.3 Information Theoretic Approach

In information theoretic approach Mutual Information(MI) is the measure to determine the statistical dependence between two variables. If there are M states of a system A, i.e. {a1, a2, a3, . . ., am } MI is computed using Shannon Entropy H(A) which is given as:

𝑁𝑜𝑟𝑚𝑎𝑙𝑆𝑧𝑆�𝑥𝑖𝑗 � =

�𝑚𝑎𝑥𝑆𝑚𝑢𝑚�𝑋𝑗 � − 𝑚𝑆𝑡𝑆𝑚𝑢𝑚�𝑋𝑗 ��

𝐻(𝐴) = − ∑𝑀𝐴 𝑆𝑏 (𝑎 ) ⋅ 𝑙𝑜𝑔�𝑆𝑏(𝑎 )� (i)

𝑖=1 𝑖 𝑖

This normalized data is arranged into tabular form which is suitable for processing.

2.2 Co-expression Matrix Generation

The two dimensional array of gene expression data xij ∈ X
is available after pre-processing where each row represents a
gene and each column (sample) represents either time points
(in time series data) or conditions (in steady state data). Now
the similarity or association between the gene expression lev-
els of two genes i and j is to be estimated. If the value satisfies
a predefined threshold then the two nodes may be connected
by an edge. The adjacency matrix formed in this context is
called co-expression matrix. The similarity is measured [31], [36], [44], [45], [46] by different means as under:

2.2.1 Pearson Correlation Co-efficient

The Pearson correlation coefficient S(Xi , Xj ) is estimated as:
pb(air ) is the probability for choosing air . Similarly joint en- tropy can be estimated for two systems A and B from joint probability pb(ai , bj). The Mutual Information is defined as:

𝑀𝐼(𝐴, 𝐵) = 𝐻(𝐴) + 𝐻(𝐵) − 𝐻(𝐴, 𝐵) ≥ 0 (ii)

The systems A and B are independent if MI(A, B) is 0 and
the statistical dependence increases with the increase in MI.
Two genes are taken as system A and B which have either
some continuous values in [0, 1] or some discrete value taken
in the interval [0, 1].

2.3 Probabilistic Modeling

The network may be constructed either using forward model- ing or inverse modeling. The basic graph representation tech- niques of GRN like Directed Graph and Directed Hyper-graph are discussed in subsections 2.3.1 and 2.3.2. Two Forward Modeling techniques, viz. Bayesian Network and Boolean Network and one Inverse Modeling technique, Directed Scale Free Graph are discussed in sections 2.3.3, 2.3.4 and 2.3.5 re-

1

𝑆�𝑋𝑖 , 𝑋𝑗 � = 𝑁 �

𝑘=1,𝑁

�𝑥𝑖𝑘 − 𝑚𝑆𝑎𝑡(𝑋𝑖 )� .

𝑆. 𝐷(𝑋𝑖 )

�𝑥𝑗𝑘 − 𝑚𝑆𝑎𝑡�𝑋𝑗 ��

𝑆. 𝐷�𝑋𝑗

spectively. The reconstruction of regulatory network is based on a mathematical or probabilistic model. This is to avoid blind trial and error assumptions.
Xi is the set of expression levels of gene i in different sam- ples
N is the number of samples

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2.3.1 Directed graph

Here in the directed graph G = (V, E) each gene is represented
as a node i ∈ V (V maps to N, is the maximum number of
genes in the system) [14]. Each directed edge is represented by
a tuple {i, j, sn}, where i and j are the target and regulator
genes respectively and the symbol sn, (sn ∊ {+,-}) represents
the sign of the interaction. The interaction is either enhancive
(+) causing the rate of protein synthesis in target gene i to in-
crease or repressive (-) causing the rate of protein synthesis in

target gene i to decrease.
depending on some probability measures. The Bayesian net- work is a directed acyclic graph (DAG), G = (V, E) where each
node i ∈ V is associated with the expression level of gene i, 1 ≤
i ≤ N where N is the maximum number of genes in the system.
Each edge eij ∈ E is an interaction between gene i and j. The
two components of a Bayesian Network (BN) are the DAG and
a joint probability distribution of the expression levels in-
volved in the system. The conditional probability for each Xi is given by: pb(Xi | par(Xi )) where par(Xi ) represents the parent of Xi in the graph G, is a candidate from the set of regulators
in the system. Here actually we are computing the probability
of par(Xi) being the parent of Xi from different microarray dataset of same genes. And an edge is drawn from parent to child if the probability exceeds a certain threshold.

Figure 1: Directed Graph Representation of Gene Regulatory

Network.

The directed graph representation of gene regulatory network is shown in figure 1. The set of vertices is represented as V =
{1, 2, 3} and the set of edges E = {(2, 1, -), (1, 3, +), (3, 2, +), (2, 3,
-), (3, 3, -)} where each edge is (i, j, sn). Her i and j are node
numbers and sn is the sign of interaction shown in figure 1.

2.3.2 Directed Hyper-graph

The directed hyper-graph G' = (V', E') is composed of a node
set V' where each node i ∈ V (V maps to N, is the maximum
number of genes in the system) represents a gene as in 2.3.1
and each directed edge is represented by a tuple {i, J, Sn}. Here
i is the target gene which is expressed as a result of the regula-
tory interactions made on it and J is the set of regulators of i,
collective interactions of whom cause the expression of i and
Sn is the list of signs of the regulatory influences [14]. If the
graph of figure 1 is considered as the hyper-graph G', the set
of vertices V' = {1, 2, 3} and set of edges E' = {(2, [1,3], [-, -]), (1,
[3], [+]), (3, [2, 3], [+, -])}. For example the edge {(2, [1,3], [-, -])
in the edge set E represents that the target gene 2 in figure 1 is
interacted by the regulators 1 and 3 and both the regulatory
interactions are repressive i.e. having ‘-’ sign.

2.3.3 Bayesian Network

It is a probabilistic approach of graph construction [14] [30] which has become one of the efficient representations of re- construction of gene regulatory network. Bayesian network represents causal relationships between the nodes [37], [38], [39], [40], [41] rather than a flow of information. In gene regu- latory network this causal relation is drawn between the ex- pression levels Xi of the gene i involved in the system. In other words for a gene set the expression levels are obtained under different conditions. The causal relationship is reconstructed

Figure 2: Sample network of 5 nodes. The conditional proba- bility of each node of the network pb(Xi | par(Xi)) is determined.

The probability measures required for constructing such a graph is written as under:
The Conditional Probability measures of different variables as in figure 2 are described as under:
pb(X1), pb(X2), pb(X3 | X1, X2 ), pb(X4 | X3), pb(X5 | X4). The Joint Probability which represents the network as a
whole is given as in figure 2:
Pb(X1,X2, X3, X4, X5) = pb(X1)pb(X2)pb(X3 | X1, X2 )pb(X4
| X3)pb(X5 | X4).
An important term involved in this context, Conditional
Independence is given by the formula:
Ind(Xi : Non-descendants(Xi) | par(Xi)) which comes from
the famous Marcov assumptions.
Statement of Marcov Assumptions: Each variable (Xi) is in-
dependent of the Non-descendant of (Xi) given par(Xi).
So the set of Conditional Independence according to figure
2 is given by:
Ind(X1:X2, X5), Ind(X2: X1), Ind(X3: X5 | X1, X2), Ind(X4:
X1, X2, X5 | X3), Ind(X5: X1, X3, X4 | X2)
The probability distribution that obeys the Marcov assump-
tion and the chain rules of conditional probability can be writ-
ten as:

𝑃𝑏(𝑋) = � 𝑆𝑏 �𝑋𝑖 ∨ 𝑆𝑎𝑟(𝑋𝑖 )�

𝑖=1,𝑛

Each node of the Bayesian network carry a probability table
based on which expression level of a successor gene is deter-

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mined. Consider an instance where x is regulated by y and z in figure 3.

Figure 3: A gene regulating instance when genes y and z regulates x.


Expression levels are quantized into q=3 levels. The proba- bility table of x looks like Table1. For the instance when y=1, z=2 then probability of x=1 is 0.3, x=2 is 0.5 and x=3 is 0.2.

TABLE 1

PROBABILITY FOR GENEX

y z x=1 x=2 x=3

1 1 0.6 0.3 0.1

1 2 0.3 0.5 0.2

1 3 0.8 0.2 0

2 1 0.15 0.35 0.5

2 2 0.7 0.2 0.1

2 3 0.1 0.6 0.3

there with directed edge 4 to 6 or 6 to 4. Each member of the equivalence set of graphs is then evaluated for the score with respect to previous data D. The graph from the equivalence set with best score is the target network.

2.3.4 Boolean Network

Boolean network is a suitable medium of the reconstruction of gene regulatory network [14]. It is represented as graph G (V, E, F) where V is the set of nodes represented as genes. Each
node i ∈ V is associated with the expression level of gene i, 1 ≤
i ≤ N where N is the maximum number of genes in the system.
Each edge eij ∈ E is an interaction between gene i and j. At time instance t the state of gene i is denoted by £i ∈ {0, 1} which is determined by a Boolean function fi ∈ F involving k genes. So
there are 2K possible values of £i . £i = 1 means gene i is ex-
pressed and £i = 0 means gene i is not expressed. An edge is extracted if gene i is expressed or £i flips from 0 to 1. Another function fj , taking this output as an input along with (k-1) oth-
er inputs evaluates the state of some other gene. The genes i ∊
{1,2,....,n} at time point t constitute an n element vector X(t) {£1 ,
£2 , ….,£n } which represent the state of the system at time point t. At time point t+1 system changes its state to X(t+1) by the

set of functions F.

Figure 4: W iring Diagram of a Sample Boolean Network

3

1

0.5

0.2

0.3

3

2

0.2

0.4

0.6

Figure 4 and Figure 5 shows the change of state from time point t, X(t) to the state at time point t+1, X(t+1). The four

functions in the set F are some standard Boolean gates. In real

3 3 0.17 0.23 0.6

practice they may be some complex Boolean functions.

Two Bayesian Network (BN) are said to be equivalent if they contain the same set of conditional independences, i.e. If Ind(G) = Ind(G') two BNs are equivalent. In this context the statement of Perl and Verma is important.
Statement of Perl and Verma [37]: Two DAGs are equiva- lent if and only if they have the same underlying undirected graph and same v-structures (i.e. same set of converging edges into one node. e.g. Xj →Xi ←Xk ).
This implies the equivalence graphs with some fixed di- rected edges like Xj →Xi and some undirected edges like Xj — Xi which takes the form of Xj→Xi or Xi→Xj . In the graph of figure 2 to keep the v structure X1 →X3 ←X2 intact, all the nodes should have the same direction as in the graph. But if a
6th node be connected from 4 then an equivalence set might be

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Figure 5: State Change from t to t+1 in the Circuit Diagram of a

Sample Boolean Network

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Each node of the Boolean network carry a data table based on which expression level of a successor gene is determined. Consider a regulatory interaction where x is regulated by y and z in figure 1.

Expression levels are quantized into 0 and 1 where zero in- dicates gene is not expressed and one indicates gene is ex- pressed. The probabilities of x taking different values are pre- sented in Table2. For instance at time t if y is not expressed and z is expressed then gene x is expressed at t+1.

TABLE 2

ACTIVATION OF GENE X INA BOOLE-

AN NETWORK

y z x

0 0 0

0 1 1

1 0 1

1 1 0

2.3.5 Direct Scale Free Network (DSF)

Scale free nature of directed graph obeying Power law has well been implemented in many types of graph representation [14]. Novel Graph construction applications like web graph, where websites or web pages are the vertices and the direct link between them represent the edges of the graph, the movie actor graph, scientific collaboration graph are examples where the scale free nature of directed graph are implemented. In gene regulatory network construction direct scale free graph is implemented in recent past [26]. Graph construction process of gene regulatory network is based on degree based preferen- tial attachments. Initially we have a set of genes named Nnew with which the GRN is to be constructed. The graph construc- tion starts with an initial graph G(t0 ) at time point t0 . It has a single node randomly chosen from Nnew with the initial de- gree ≥1 (to make the calculation meaningful). In each step a new edge (regulatory interaction between two genes) will be attached to the graph and after t time units the graph will ex- actly have t number of edges. Whenever a node is fetched from Nnew and being used as the source (regulator) or target node of the edge to be attached it will be deleted from Nnew and being placed in a set named Nold , the set of existing nodes which have already been used in the graph. To attach an edge to the graph G(t) there are three steps:
i) Choosing a regulator, ii) Choosing a target,
iii) Placing a directed edge from regulator to target.
Here a new node (regulator/target) i ∈Nnew having degree(i) =
0 and an existing node (regulator/target) j ∈Nold having de-
gree(j) ≥1.
The phrase 'degree based preferential attachment' indicates
that edge connection and the choice of regulator and target is
entirely based on degree [42], [26]. In case that a new node is
chosen the question of degree does not arises and therefore the
selection of new node may be random. But in case that an ex-
isting node if chosen as the target the selection is based on its
in-degree of the node and if chosen as the regulator the selec-
tion is based on the out-degree of the node.
Selection of regulator and target node (gene) may be done in three possible phases:
Case A: New regulator and existing target, Case B: Existing regulator and existing target Case C: Exiting regulator and new target
There are some probability constants like α, ß, γ, δinp and δout involved in this computation. They are non-negative real numbers provided α+ß+γ = 1 and δinp , δout ≥ 0. Each of the
three cases A, B and C are executed with the probability α, ß
and γ respectively. The above mentioned cases are described
as under:
Case A: The case is chosen with probability α. New regula- tor v, is chosen from the set Nnew randomly. In case of choos- ing existing node as target w from the set Nold a probability measure is to be estimated. The probability that a node wi tak- en from the set Nold is the required target node w is given by:
Pb(w = wi ) = (Indeg(wi ) + δinp ) / (t + δinp .n(t))
Here, Indeg(wi ) is the indegree of the node wi , t represents the time instance and n(t) represents number of nodes the graph contains in the current time point t.
Case B: The case is chosen with probability ß. The regulator v, being an existing node is chosen from the set Nold with some probability. The probability that a node vi taken from the set Nold is the target node v of our interest is given by:
Pb(v = vi ) = (Outdeg(vi ) + δout ) / (t + δout .n(t)) Outdeg(vi ) is the out-degree of the node vi in the current
graph.
In case of choosing existing node as target w from the set
Nold a probability is to be estimated. The probability that a
node wi taken from the set Nold is the required target nod w is given by:
Pb(w = wi ) = (Indeg(wi ) + δinp ) / (t + δinp .n(t))
Case C: The case is chosen with probability Ƴ. The regula- tor v, being an existing node is chosen from the set Nold with some probability. The probability that a node vi taken from the set Nold is the target node v of our interest is given by:
Pb(v = vi ) = (Outdeg(vi ) + δout ) / (t + δout .n(t))

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The target w, being a new node is chosen from the set Nnew
either randomly or with some estimated probability.
work is the proper model applicable in this context. The varia- tion of the expression level represented as the dynamics [43] of the system is given as under:

3 SIMULATION

Reconstruction of Gene Regulatory network involves a step by

𝑇𝑖

𝑑𝑥𝑖

𝑑𝑡

= 𝑆�∑𝑁 𝑤 ⋅ 𝑥 (𝑡) − 𝑏 � − 𝑘 ⋅ 𝑥 (𝑡) (1)

step procedure of graph construction where at each instance a new weighted edge is added to the network. This may require a new node to be attached in the network. The final outcome is the simulated network with a set of parameters. The different

𝑑𝑥𝑖 =

𝑑𝑡

𝑙𝑖𝑚𝛥𝑡→0 𝑥𝑖 (𝑡+𝛥𝑡)−𝑥𝑖 (𝑡) (2)

𝛥𝑡

parameters involved are i) weight parameters wij ∊ W, where
W is an NxN matrix expressing the weight of each edge in the
Substituting (2) to (1) we get

𝛥𝑡

network, ii) Basal expression parameter bi , expression level of

𝑥𝑖 (𝑡 + 𝛥𝑡) = 𝑆�∑𝑁

𝑤𝑖𝑗

gene i in non-excited condition which is an 1xN vector and iii)
timing parameter ti , the delay factor of gene i which is also an

𝑗=1 ⋅ 𝑥𝑗 (𝑡) − 𝑏𝑖 � − 𝑘𝑖 ⋅ 𝑥𝑖 (𝑡) �1 −

𝑇𝑖

⋅ 𝑘𝑖 (3)

1xN vector. Here N is the number of genes involved in the network. So total number of parameters are N2+2N or N(N+2). The problem domain may be decomposed into two parts, network reconstruction and parameter estimation. Different soft computing tools and techniques like Artificial Neural Network, Fuzzy logic and different advanced search and op- timization techniques are implemented in this area. There are some softwares which aid the work by supplying ready to use data and platforms for comparing original and simulated net- work. The following subsections depict the different soft com- puting tools, algorithms and software which aids the recon- struction:

3.1 Useful Software

Since construction of Gene Regulatory Network is a well- known problem domain over one and half decades. There are softwares like GeneNetWeaver, GNA (Genetic Network Ana- lyzer), GINsim, GeNESis etc. which have been developed to aid the GRN reconstruction. One such well known open source software is GeneNetWeaver [28], [29] available in http://sourceforge.net. The software supplies time series gene expression data of organisms like Yeast and E.coli for their known regulatory network. It extracts strongly connected subgraphs of the known genetic graphs of the aforesaid organ- isms. The number of nodes of the subgraphs may be specified by the user. It visualizes the original network and which can be used for future reference. It provides the facility for knock- out tests (removing a node or gene from the network to see change in behavior of other nodes) and knockdown tests (low- ering the expression level for a specific gene of interest and to note the corresponding change of other gene expressions). It facilitates some of the reconstruction algorithm including ARACNE [15] for simulation. It also provides the comparison platform for original to simulated network and publishes the result as pdf.

3.2 Artificial Neural Network(ANN)

Here each gene is considered as an artificial neuron [12], [18], [19], [20] which is activated and send interaction to another gene by the additive effects of interactions of other genes. Here the genes of a certain layer of ANN not only can send
xi (t) is the expression level of gene i at time instance t,
f(.) is a nonlinear sigmoid function f(x) =1/ (1+e-x), ki is the decay constant of ith gene,
T i is the time constant of gene i,
bi is the basal expression of gene i,
wij is the sigmoid weight from regulator j to target i, N is the maximum number of genes in the system.
In each time step according to mathematical model a regu- lator and target is probabilistically chosen, parameters are im- posed (initially randomly), the expression level of the target is determined by expression (3) specified above. Error is estimat- ed by comparing the original gene expression value to the es- timated gene expression value. The objective of optimization is to reduce the error. The expression level of the target gene and the RNN parameters of current iteration are making tar- get gene capable to determine regulatory interaction.

3.3 Fuzzy Logic

Fuzzy logic has been successfully implemented [48] for pa- rameter estimation problem and gene expression level estima- tion problem related to the present context [50]. It enhances the probability of choosing a value in the domain to be true. In other word it improves the Probability Mass Function. Sup- pose a parameter value is to be chosen from a range -30 to +30. Let us take discrete values in the range at 0.1 interval. The dis- crete set is {-30, -29.9, -29.8, . . . -0.1, 0, 0.1, . . . 29.8, 29.9, 30}. There are almost600 elements in the set. If we arbitrarily choose a value, the probability of right choice is 1/600. If we fuzzify the set in [0, 1] taking values of the interval 0.1, i.e. the typical membership values will lie in the range {0, 0.1, 0.2, 0.3,
…., 0.9, 1}, the probability of choosing a value to be right value is almost 1/11 because of the presence of 11 elements in the fuzzy set. This probability improvement is the main cause of fuzzification. The work implements the parameter estimation using fuzzy set where the crisp set is taken as {-30, -15, 0.001,
15, 30}. 0.001 is taken instead of 0 to make the defuzzification
meaningful. Fuzzification is done dividing the crisp values by
random numbers. Defuzzification is done using centroidal
defuzzification using the formula written below:

𝑚 𝑚

interaction to the layer next, rather it can send the interaction
even to a gene of a previous layer. So Recurrent Neural Net-

𝑤�𝚤𝚥 (𝑡) =

𝐿

𝑚=1

𝑤𝑖𝑗 × 𝜇 �𝑤𝑖𝑗

𝑚

� (𝑡)

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http://www.ijser.org

𝐿

𝑚=1

𝜇 �𝑤𝑖𝑗

� (𝑡)

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𝑚

𝑤𝑖𝑗is the crisp value,

𝑚

𝜇 �𝑤𝑖𝑗 � (𝑡)is the fuzzy values generated in tth iteration
𝑤�𝚤𝚥 (𝑡)is the defuzzified value of tth iteration.
In each iteration a set of five fuzzified values are generated
for each of all N(N+2) parameters. After defuzzification it is
been used for the gene expression value estimation and there-
by error calculation.

3.4 Optimization Techniques

In both parameter estimation and graph construction different meta-heuristic algorithms [50] like Genetic Algorithm (GA), Simulated Annealing (SA), Differential Evolution (DE) [48], Particle Swarm Optimization (PSO) [21], [22], Ant Colony Op- timization (ACO) [23] are successfully implemented. ACO has shown better result in graph reconstruction problem [27] whereas PSO has proven it's excellence in parameter estima- tion problem [26]. The optimization algorithms may be ob- tained in different publications [26], [27], [48], [50] etc. The optimization problem may be modeled either for step by step graph construction problem learning the parameter in each iteration or it may be presented as parameter estimation prob- lem where whole set of parameters are initialized arbitrarily and iteratively trying for better solution minimizing the error comparing the simulated value of equation (3) to the corre- sponding micro-array gene expression data. The cost functions of the two types of problem specified above are written as mean squared error in equation (4) and (5) respectively.

4.1 Data Source

The starting point of the study lies in the biological data avail- able from Micro-array. Micro-array is a suitable technology [6], [7] from where the gene expression data for large number of genes are available. Microarray is a glass slide where single stranded DNA molecules are attached by biological methods like spectrometry in fixed locations of the slide. These fixed locations are called spots. Each spot generally represent a gene. There may be thousands of genes represented in a mi- croarray. The microarray data is obtained from different pub- lic repositories [32], [33], [34], [35] in different data formats like .cel, .tab etc. The data is then represented in some popular tabular format like spreadsheet by using different software like R, bio-conductor, orange etc. The data may be of two types: time series data and steady state data. Both are repre- sented as 2D arrays [49] where rows are represented by N Number of genes and columns are represented either by M time points (in case of time series where gene expressions are plotted in different time points) or by M conditions (in case of steady state where gene expression for different conditions like normal tissues, diseased tissues, tissues after therapeutic intervention are plotted). Apart from genes there may be one extra column present in the raw data which correspond to the class label. The expression level of ith gene at jth time point is situated at the intersection point of ith row and jth column. There are two sub-categories of time series data: short time series (containing 3 – 8 time points) and long time series (more than 8 time points). But two important difficulties are involved here are: firstly, many biological and experimental noises are involved in the data and secondly, compared to the number of genes the number of time steps in which their expression lev- els measured are few. This phenomenon is called the “curse of

𝑆𝑖 =

1 ⋅ ∑𝑇 �𝑥 (𝑡) − 𝑥 (𝑡)�2

𝑇

(4)
dimensionality” [27]. The data thus collected from gene chip or microarray by using image analysis tools are needed to be
E = 1 ⋅ ∑𝑇
𝑁

2

�𝑥 (𝑡) − 𝑥 (𝑡)�
(5)
properly extracted to get actual expression levels of the genes.

𝑁𝑇

𝑡=1

𝑖=1 𝑖 𝚤

ei : The predicted error of the temporal expression pattern
of gene i,
E: Error associated with the predicted time series,

xi (t) and xi (t): The original and estimated expression level of gene i at time point t respectively,
T: The number of time-points in the time-series micro array data,
N: is the number of genes in the system.

4 REVIEW OF EXPERIMENTAL RESULTS

This section provides the details of the source of microarray gene expression data. The measurement metrics for compari- son of the simulated network to the original known network are discussed. Here original network may be obtained from the software like GeneNetWeaver. Lastly the listing of the ap- plications of soft computing tools in different reviewed prob- lems is presented.

4.2 Metric

The performance of simulation is estimated by comparing the edges obtained from co-expression matrix to the edges of the inverse modeled network. Comparing the original adjacency matrix to the simulated one an edge eij may be named as true positive (tp), false positive (fp), true negative (tn) or false neg- ative (fn). The Sensitivity of the simulation is defined by the True Positive Rate which indicates the share of correctly in- ferred edges over total number of edges that should be in- ferred (tp and fn) edges [27], [49]. On the other hand Specifici- ty of the simulation is defined by the False Positive Rate of the predicted network which indicates the share of falsely inferred edges over total number of edges that should not be inferred (tp and tn). Finally the Precision of the simulated network, also referred as the Positive Predictive Value, indicates the share of correctly inferred edges over all inferred edges. The Sensitivity, Specificity and the Precision are defined as:

𝑡𝑆

𝑆𝑆𝑡𝑠𝑆𝑡𝑆𝑣𝑆𝑡𝑆 =

𝑡𝑆 + 𝑆𝑡

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𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑆𝑡𝑆 =

𝑃𝑟𝑆𝑆𝑆𝑠𝑆𝑜𝑡 =


𝑆𝑆
𝑆𝑆 + 𝑡𝑡

𝑡𝑆

of different genes and the resulting regulatory network of genes that causes the variation in their shapes, activities and other characteristics. The applications in detecting biomarker of a disease and finding drug for the disease have extensive
tp: true positive edge, fp: false positive edge, tn: true negative edge, fn: false negative edge.

4.3 Performance Analysis

𝑡𝑆 + 𝑆𝑆

importance in bio-informatics. But at the time of simulating
the network or estimating the parameters for gene expression
profiling we have only time series or steady state gene expres-
sion data at our hand which is again prone to errors. The prior
knowledge of the structure of the graph is also not available.
This situation makes the network prediction job a tedious one.
In this article focus is made to each issue of modeling and
simulation of the network and certain measures are suggested
In this paper the relative performances of soft computing tools
applied in different works are presented. Different soft com- puting tools presented here are Recurrent Neural Network (RNN), Fuzzy Logic, Particle Swarm Optimization (PSO), Ant Colony Optimization (ACO) and Differential Evolution

(DE).The soft computing tools are applied in two types of problems, Parameter-set Generation problem and Dynamic Graph Generation Problem. Performances of different re- viewed works are presented with the paper reference number in the table.

TABLE 3

PERFORMANCE ANALYSIS

to reduce the complexity. Different mathematical models re- quired for the simulation of GRN like data extraction proce- dure, co-expression network generation, different forward and reverse probabilistic models are discussed. The optimization techniques to reduce the state space search during graph re- construction and parameter estimation, different soft compu- ting techniques to facilitate the reconstruction are stated. The microarray technology, the public data sources of Time Series and Steady State gene expression data, different metrics to measure the performance of the simulation are presented and the application area is explained here. The software named GeneNetWeaver, discussed here, is not only useful as a plat- form for producing time series gene expression data needed

Sl.

Soft-computing

Parameter-set

Dynamic Graph Generation

for experiments but also provides a platform for different per-
turbation experiments like Knock Out and Knock Down tests.

No. tools

Generation

problem

problem

In this paper, the focus has been made on the optimal struc- ture of GRN, but in future there is an ample scope for research

Graph

construction

Parameter

learning

in a particular disease domain, such as, to detect the bi- omarker of the disease, to find pathway for the disease, to find drug based on the simulation of GRN.

1 RNN √, [48,50] √, [26,27,43]

2 Fuzzy Logic √, [48, 50]

3 PSO √, [50] √, [26] √, [26,27]

4 ACO √, [27]

5 DE √, [48]

√ specifies good result,
[i] Denote the paper number in reference.

5 CONCLUSION

In this article the effort is made to present a complete rele- vance and procedure of gene regulatory network and its re- construction. A clear explanation of what GRN is, how does the formation of GRN take place as a result of certain cellular activities, what will be its actual structure, and why the recon- struction process of such networks are so important is given. It is seen that the infant of an organism has the same orientation of genes in its genome as in the adult of that organism. The only thing differs in these two cases, is the way of expression

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