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On the number of fixed points in regulatory

Boolean networks.

Julio Aracenaa,∗

aDepto. Ingenier´ ıa Matem´ atica, Universidad de Concepci´ on,

Av. Esteban Iturra s/n, Casilla 160-C, Concepci´ on, Chile.

Abstract

We study the number of fixed points in a particular class of Boolean networks called

regulatory Boolean networks (RBNs), where each interaction between the vertices

of the network are either a positive interaction or a negative interaction. We find

relationships between the positive and negative cycles of the graph of a RBN and

the existence of fixed points. As main result, we exhibit an upper bound for the

number of fixed points in terms of minimum cardinality of a set of vertices meeting

all positive cycles of the network (called positive feedback vertex set).

Key words: Boolean network, fixed point, positive and negative cycle, feedback

vertex set.

1Introduction

Boolean networks was first introduced by Kauffman as mathematical model to

study the dynamics of gene regulatory networks [6,10,11]. A gene expression

level is modeled by binary numbers, with 1 or 0, indicating either the active

and inactive state of a gene expression, respectively. Thus, a Boolean network

with n nodes has a total of 2npossible states. Each state maps to one state,

possibly itself. Hence, every network has at least one limit cycle or fixed point

(called attractors), and every trajectory will lead to such an attractor. Thus, a

general problem is finding the set of attractors. In Boolean networks with more

than a handful of nodes, the state space is too big to be searched exhaustively.

∗Corresponding author.

Email address: jaracena@ing-mat.udec.cl (Julio Aracena).

1Partially supported by Fondecyt project 1061008.

Preprint submitted to Elsevier Science23 February 2007

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Hence, the importance of having analytical results to efficiently determine this

set. At the present, There are only a few resuls on this matter [9].

In the modeling of gene regulatory networks, the attractors, in particular the

fixed points, are associated to distinct cell states defined by patterns of gene

activity. Thus, the number of attractors generated may correspond to the

number of possible cell types in an organism. Some numerical experimentals

carried out by Kauffman in [10,11] show that the number of attractors of

random Boolean networks increases as√n. Recently it has been shown that

the total number of attractors in Boolean networks grows faster than any

polynomial [15].

In many constructed models of real gene regulatory networks there are in-

teractions between gene and gene products which can be classified like acti-

vation action (positive interaction) or inhibition action (negative interaction)

[13,16,20]. In this paper we define a subclass of Boolean networks where each

link beetween the elements of the network represents a positive or negative

interaction, called regulatory Boolean networks (RBNs). In this family of net-

works we study the relationship between the positive and negative cycles of

the connection graph and the number of fixed points of the network. This

type of relation has been extensively studied in continous dynamical systems

[4,5,17,18] and in discrete neural networks [1,2]. We give necessary conditions

and sufficient condition for the existence on a given number of fixed points

in RBNs in terms of the structure of the positive and negative cycles of the

network.

The paper is organized as follows. Section II introduces the notation and the

main concepts of Regulatory Boolean Networks and Graph Theory. Necessary

conditions and sufficient conditions for the existence of fixed points in terms of

the positive and negative cycles of the connection graph are given in section III.

In section IV, we introduce the concept of set of vertices meeting all positive

cycles of a RBN (Positive Feedaback Vertex Set) and exhibit an upper bound

for the number of fixed points of a RBN in terms of the minimum cardinality

of a set of such vertices. Finally, in section VI we draws our conclusions and

future research.

2 Definitions and notation

A digraph is an ordered pair of sets G = (V,A) where V = {1,...,n} is a set

of elements called vertices (or nodes) and A is a set of ordered pairs (called

arcs) of vertices of V . The vertex set of G is referred to as V (G), its arc set

as A(G).

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We denote by V−(k) and V+(k) the sets of nodes j such that (j,k) ∈ A and

(k,j) ∈ A, respectively.

A path from a vertex v1to a vertex vmin a digraph G is a sequence of vertices

v1,v2,...,vm−1,vmof V (G) such that ∀k = 1,...,m−1, (vk,vk+1) ∈ A(G) or

(vk+1,vk) ∈ A(G). The vertices v1and vmare the initial and terminal vertex

of the path. A path is elementary if each vertex in the path appears only

once with the possible exception that the first and last vertex may coincide.

A path is closed if its initial and terminal vertices coincide. A circuit is a

closed elementary path. A path is called directed path if (vk,vk+1) for all

k = 1,...,m − 1. A cycle is a directed circuit, that is a closed elementary

directed path.

A digraph G is said to be connected if there is a path between every pair of its

vertices; and strongly connected if there is a directed path between every pair

of its vertices. More terminology about Graph Theory can be found in [7,19].

A Boolean network (BN) is defined by a 2-tuple N = (G,F), whith G = (V,A)

a digraph, where each node i has an associated state in {0,1}. This state is

updated by the application of a Boolean function fi: {0,1}n−→ {0,1} called

local activation function whose value depends on the values of its incident

nodes, i.e. fi(x) = fi(xj, j ∈ V−(i)). F = (f1,...,fn) : {0,1}n−→ {0,1}nis a

vector of local activation functions such that

∀x = (x1,...,xn) ∈ {0,1}n, F(x) = (f1(x),...,fn(x)).

The vertices are updated simultaneously at every time step according to the

local activation functions. Hence, the dynamics of the network is given by the

equations:

x(0) ∈ {0,1}n

∧

x(t + 1) = F(x(t)), ∀t ∈ N.

We say that a vector x ∈ {0,1}nis a fixed point of N if x is invariant under

the application of the complete sequence of updates, that is,

F(x(t)) = x(t)∀t ∈ N.

A Boolean function f : {0,1}m−→ {0,1} is increasing monotone on input i if

∀x ∈ {0,1}mxi= 0, f(x) ≤ f(x + ei),

and decreasing monotone on input i if

∀x ∈ {0,1}m, xi= 0, f(x) ≥ f(x + ei),

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where ei∈ {0,1}mdenotes the binary vector with all entries equal to 0, except

for entry i, which equals 1. A Boolean function f : {0,1}m−→ {0,1} is said

to be a unate function if for each i = 1,...,m, is either increasing monotone

or decreasing monotone on input i. Equivalently, a Boolean function is unate

if it can be represented by a formula in disjuntive normal form in which all

ocurrences of any given literal are either negated or nonnegated. Example

of unate Boolean functions are the hierarchically canalyzing functions [14].

And a well-known example of non-unate Boolean function is XOR, that is,

XOR(x1,x2) = x1x2+ x1x2.

Given a unate function f : {0,1}m−→ {0,1} we denote by I+(f) and I−(f)

the set of indices where f is increasing monotone and decreasing monotone

respectively.

From definition, every unate function f : {0,1}m−→ {0,1} satisfies the fol-

lowing properties:

P1: For all vectors x,y ∈ {0,1}m, with xi≤ yifor all i ∈ I+(f) and xi≥ yi

for all i ∈ I−(f), f(x) ≤ f(y).

P2: For all vectors x,y ∈ {0,1}m, with xi= 0∀i ∈ I+(f) and xi= 1∀i ∈

I−(f), f(x) = 0. Analogously, if xi= 1∀i ∈ I+(f) and xi=,∀i ∈ I−(f),

then f(x) = 1.

A BN where each local activation function is a unate function will be called a

Regulatory Boolean Network (RBN).

In this paper we will suppose w.l.o.g. that a RBN has not constant local

activation functions and each local activation function fidepends really on

the values of its incident nodes, that is to say, j ∈ V−(i) if and only if

∃x ∈ {0,1}m, fi(x1,...,xj= 0,...,xm) ?= fi(x1,...,xj= 1,...,xm),

where m = |V−(i)|. Notice that if fiis not a constant function, then |V−(i)| ≥

1. It follows that there exists at least one cycle in G (you can even have a cycle

of the form (j,j) called a loop).

Thus, for all i ∈ V , the set {I+(fi),I−(fi)} is a partition of the set V−(i).

Hence, for every RBN N = (G,F) we can define a weight function

wF: A(G) → {−1,1} with

wF(i,j) = −1 if i ∈ I−(fj) and wF(i,j) = 1 if i ∈ I+(fj).

An arc (i,j) ∈ A(G) will be called positive if wF(i,j) = 1 and negative

otherwise. We will say that a path is positive if the number of its negative

arcs is even, and negative otherwise. (G,wF) will be called signed digraph of

N. In Fig. 1, an example of the RBN is depicted.

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f = x2

f2= x1x3

f3= x x3+x2

1

1

.

,

,

+

+

+

3

1

2

−

−

−

Fig. 1. An example of RBN. Here, C1= 1,3,2,1 and C2= 1,2,1 are negative and

positive cycles, respectively.

3Relationships between the fixed points and the positive and neg-

ative cycles of a RBN.

Here and subsequently, we will assume G to be a connected digraph; otherwise,

one may apply the results to each of the connected components of G.

The following lemma is a characterization of the positive and negative cycles

in a RBN N.

Lemma 1 Given a RBN N = (G,F), a cycle C = v1,...,vm,v1 in G is

positive if and only if there exists a vector x = (x1,...,xm) ∈ {0,1}msuch

that

wF(vi,vj) = 1 ⇐⇒ xi= xj,

∀(vi,vj) ∈ C, (1)

or equivalently

wF(vi,vj) = (2xi− 1)(2xj− 1)

∀(vi,vj) ∈ C.(2)

Notice that if the vector x verifies (1), then the vector ¬x = (x1,...,xm) as

well, where xi= 1 ⇔ xi= 0,∀i = 1,...,m. Besides, there is no other vector

distinct from x and ¬x verifying (1).

PROOF. Let C = v1,...,vm,v1be a positive cycle in G. Let us define the

vector x ∈ {0,1}mas follows:

• x1= 1,

• ∀j = 2...,m,xj=

xj−1if wF(vj−1,vj) = 1,

xj−1if wF(vj−1,vj) = −1.

Hence, it is clear that x verifies (1).

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