Fuzzy Logical on Boolean Networks as Model of Gene Regulatory Networks

Honglin Xu, Shitong Wang

School of Information

Jiangnan University

Wuxi, China

xuhonglin2005@yahoo.com.cn

Abstract

A novel gene regulatory network model via the

fuzzy logic is proposed. Fuzzy logic can effectively

model gene regulation and interaction to accurately

reflect the underlying biology. By judging genes

expression level on the fuzzy rule, Fuzzy Boolean

network (FBN) makes it possible to handle

simultaneously the randomness and fuzziness of

biological phenomena.

1. Introduction

Recently, the study of gene regulation network is

receiving more and more attention of researchers,

which is an effective way to study the mutual regulation

and influence between genes from an overall and

dynamical situation. The main purpose of research on

genetic regulatory network (GRN)[1] is to analyze the

mechanism of the birth, grow function, death of genes.

In order to simulate the biology adjustment of most

reality, choosing an appropriate model is a quite critical.

So far, several kinds of genes regulatory network

models have been proposed, including linear model,

Bayesian network, neural network, differential

equations, Boolean network, etc.

According to many biology experiments and cancer

research, studies have gain remarkable achievements

on building the models of gene regulatory networks.

However, the major difficulty is, from gene expression

data to theoretical models, there be a large number of

model parameters, but sparse sample data (usually a

few score gene sample observations, but thousands of

parameters). As an example of continuous model——

Differential Equations Model, a large number of

parameters need to be estimated, and it is really

difficult under the low sample rate. On the other hand,

Boolean network, as a discrete model, simplifies the

data structure by desecrating expression data on time

points, mapping gene expression data into two states:

on and off. Obviously, the structure is simplified with a

significant reduction in the parameters, but at the same

time a lot of valuable regulatory information is lost.[2]

Boolean network provides a conceptual framework

to describe the gene regulatory by simulating a discrete

dynamical process. However, the binary assumption is

too over-idealizing to express the gene mutual

information [2]. Aiming at the proposed shortage of

Boolean network, this paper improves the model with

Fuzzy Logic. By definition of the different levels of

gene regulatory in some fuzzy rules, we build a new

model of gene regulatory network lies between

continuous and discrete, which is provided to be

stricter in stimulating the uncertainty and complexity of

biology system[3][4].

2. Boolean Network

The model system that has received, perhaps, the

most attention, not only from the biology community,

but also in physics, is the Boolean Network model,

originally introduced by Kauffman .In this model, gene

expression is quantized to only two levels: ON and

OFF. A Boolean network ),( FVG is defined by a set of

nodes (genes) },{ 1 nxxV L= and a list of Boolean

functions },{ 1 nffF L= . Each }1,0{∈ix ,

ni ,,1L= is a binary variable and its value at time

1+t is completely determined by the values of some

other genes )()(2)(1 ,, ikjikik ixxx L at time t by means

of a Boolean function Ff i ∈ . That is, there are ij

genes assigned to gene ix and the

mapping },,1{},,1{: nnk j LL → , ijj ,,1L=

determines the wiring of gene ix . Thus we can write:

( ))(),(),()1( )()(2)(1 tXtXtXftX ikjikiki L=+

2009 International Joint Conference on Artificial Intelligence

978-0-7695-3615-6/09 $25.00 © 2009 IEEE

DOI 10.1109/JCAI.2009.78

501

Each ix represents the expression of gene i , where

1=ix represents the fact that gene i is expressed and

0x i = means it is not expressed. The list of Boolean

functions F represents the rules of regulatory

interactions between genes. That is, any given gene

transforms its inputs (regulatory factors that bind to it)

into an output, which is the state or expression of the

gene itself. The maximum connectivity of a Boolean

networks is defined beii jmaxJ = .

Figure 1.Topology structure and logical rule of

Boolean network

We give a simple Boolean network with 3 nodes

(genes) as an example. In Fig 1 regulatory relationships

can be found. With 3 nodes means there are 823=

system states. In the dynamical process of system

evolution, each state is represented by a circle and the

arrows between states show the transitions of the

network in Fig.2. In this way, all states of time

sequence converge to the cycle of state-space

dynamically.

It is easy to see that certain states will be revisited

infinitely, depending on the initial starting state, the

network happens to transition into them. Such states are

called attractors and the states that lead into them

comprise their basins of attraction. In Figure1. the

states (0, 0, 0) and (1, 1, 1) are both attractors, and the

other states leading into them are their basins of

attraction.

Figure 2. The state-transition diagram for the

Boolean network

3. Fuzzy logical on Boolean Networks

3.1 Why Fuzzy? Traditional Boolean network is essentially a certain

model. Actually, the relationship between genes is not

only a simple discussion of "on and off" or interaction

and non-interaction; it's varied and uncertain as a

system process of biology body. The level of gene

expression inferred by two major conditions: the

external and the internal. The external condition is

complex, we may consider it in three aspects: normal

environment, abnormal environment, experimental

environment; the internal conditions may be the healthy

state and the life stage of the body. Considering all

these mentioned above, it’s important to improve the

uncertainty, time-dependence and continuity of the

models.

Thinking of the complicated biology phenomena,

the easy description on gene regulation behavior （be

expressed or not）is not so strict. In this paper, in

order to simulate the deference of gene regulation in

deferent life term, we give a fuzzy division on the level

of gene expression. Combined with the theory of

Boolean Network, Fuzzy Boolean networks as a new

model of gene regulatory network are preceded.

3.2 FBNs A number of additional justifications for introducing

fuzzy logic to modeling gene regulatory have been

research. Here, we briefly give a definition of fuzzy

logic rules on Boolean Networks in a new way.

According to the experimental data of gene expression,

a fuzzy set division is proposed; the membership of

fuzzy subsets is given by membership functions. That is

to say, referring to the giving biology knowledge and

practical experiments, for each parameter of the fuzzy

membership function from gene regulatory data, gene

expression can be divided into the following five levels:

C B A

”’

C B A

1. A activate B

2. B activate A & C

3. C inhibit A

0,0,0

0,1,0 0,0,1

1,0,0

1,1,1

1,1,0

1,0,

1

502

Weak, Middle weak, Middle, Middle strong, Strong,

described in a set of },,,,{ SMSMMWW ,as the

TABLE 1.

Table 1．．．． Rank Levels rank

level W MW M MS S

Fold

change

[0,0.25] (0.25,0.5) 0.5 (0.5,0.75) [0.75,1]

Considering revising, a fuzzy Boolean network is

defined in a set of genes },{ 1 nxxV L= and a list of

fuzzy membership functions },,{ 21 nuuuU L= .

Each }1,0{∈ix ni ,,1L= is a binary variable and

its value at time 1+t is completely determined by the

values of some other genes regulated at time t under

the corresponding fuzzy rules, with the regulatory

function ( )ixf ., where },{ 1 nffF L= .

Basing on the character of natural distribution, we

use the Gauss Function as the membership function of

fuzzy subset:

)2/)/)((exp()( 2σuxxU −−=

This function is determined by two

parameters },{ σu . Here, u is the center of the

function and σ means the width. Inutility, quantized

expression level of genes as following (in this paper

initial 5.0,1 == σu ):

Table 2. Quantized expression level of genes

0 0.25 0.5 0.75 1

W 1 0.8

MW 0.2 1 0.2

M 0.2 1 0.2

MS 0.2 1 0.2

S 0.8 1

Gene regulatory means the process of symbolization

and regulation of the rules between genes. In building

gene network models, the most important point is how

to attract and mine further knowledge in an effective

and suitable way. In fuzzy logic, fuzzy division and

production rules “if…, then…” can be used for

building the reasonable model.

Define the fuzzy regulatory rules (between gene X

and gene Y regulatory on the gene Z) as following:

• Rule1: if the expression variable of gene X is W and Y is W, Then Z can be expressed

）（）（）（）（）（ 5

11

4

11

3

11

2

11

1

11 θθθθθ

SMSMMWW

• Rule2: if the expression variable of gene X is W and Y is MW, Then Z can be expressed

）（）（）（）（）（ 5

12

4

12

3

12

2

12

1

12 θθθθθ

SMSMMWW

M

• Rul24: if the expression variable of gene X is MS, and Y is S, Then Z can be expressed

( ) ( )

）（）（）（ 5

54

4

54

3

54

2

54

1

54 θθθθθ

SMSMMWW.

• Rule25: if the expression variable of gene X is S, and Y is S, Then Z can be expressed

( ) ( )

）（）（）（ 5

55

4

55

3

55

2

55

1

55 θθθθθ

SMSMMWW.

In the rules above, gene X and Y regulate gene Z in

fuzzy logic, ransacks the 5 fuzzy

sets },,,,{ SMSMMWW , and

)5,2,1,(, L=jil

jiθ constructs the fuzzy membership

distribution of gene Z. Easily the fuzzy membership

function of Z can be calculated.

Basing on the revising fuzzy rules, we define the

regulatory functions between genes in a successive way.

In the end, conferring the character of Boolean

Network, the regulatory gene can be judged if it is

expressed by function (3) as following:

≤≤

<≤

==

∑

∑

∑

∑

=

=

1

),(),(

),(),(

5.01

5.0

),(),(

),(),(

0 0

),(Z

5

,

5

1,

5

,

5

1,

ji

YX

ji

YXlij

ji

YX

ji

YXlij

yuxu

yuxu

when

yuxu

yuxu

when

YXf

σσ

σσθ

σσ

σσθ

(2)

As the following Figure 3, a basic building block

of a FBN can be directly described. Clearly, when the

fuzzy membership function be simplified into 1=iu for

503

Figure 3. A basic building block of a FBN.

all genes, then the FBN is simply reduces to a standard

Boolean network.

4. Discussion Usually, a theoretical and mathematical derivation is

very difficult. Nevertheless, a simplified analysis based

upon a simple model would definitely provide a much

needed visualization of the biological behavior and

associated phenomenon. It is only through

mathematical analysis would get a better chance in

understanding the complex phenomenon. This is

partially true when the simplified model can be viewed

as the basic building blocks of a complicated situation.

As we just discussed, most genetic networks are

stable in the sense that they typically operate in sets of

states that are stable to perturbations. In Boolean

networks, this corresponds to a likely return to the

attractor; in FBNs, it corresponds to a low sensitivity of

the steady-state probabilities. The ideas are

fundamentally the same.

A preliminary study of 3-gene regulation networks

using fuzzy sets was carried out. This example can be

evaluated by hand. We observed that different logic

operations (Logic1, 2, 3 ,4 or 5), fuzzy membership

functions, and initial membership values led to

different attractor and limit cycles for the 3-gene

regulation network.

We list our observations as below:

CBA ∧←'

BB ←'

CC ←'

If AND gene holds

highest value: （111）is

attractor

If un-AND gene holds

middle value: （000）

attractor.

CBA ∨←'

BB ←'

CC ←'

Two steps to pick the

maximum of 3 genes: （111）is attractor.

5. Conclusion We have introduced a new class of models for

genetic regulatory networks. This new class constitutes

a Fuzzy Logic generalization of the well-know Boolean

network models and offers a more flexible and

powerful modeling framework. FBNs also present

many interesting and challenging problems.

A fascinating aspect of the research on FBNs is that

it involves and spans so many fields and topics, such as

the reasoning of logical structure of FBN, the robust

under interference, etc. We will do further research on

the model.

Acknowledgment This work is partially supported by the Hong Kong

Poly-technic University Grant (Grant no. Z-08R),

National 973 Key Project (Grant no. 2006CB705700),

National Science Foundation of China (Grant nos.

60773206/F010106 and 60704047/F030304),

New_century Outstanding Young Scholar Grant pf

Ministry of Education of China (Grant no. NCET-04-

04960, National KeySoft Laboratory at Nanjing

University, the Key Laboratory of Computer Science at

Institute of Software, CAS, China.

nx 3x 2x 1x

nx 2x

1f 2f 3f

Rule1 Rule 2 Rule 3 Rule 4 Rule 5

nf

Fuzzy rules

Input nodes

Functions

Output nodes

1x

504

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