Maximum number of fixed points in regulatory Boolean networks.

Abstract

Boolean networks (BNs) have been extensively used as mathematical models of genetic regulatory networks. The number of fixed points of a BN is a key feature of its dynamical behavior. Here, we study the maximum number of fixed points in a particular class of BNs called regulatory Boolean networks, where each interaction between the elements of the network is either an activation or an inhibition. We find relationships between the positive and negative cycles of the interaction graph and the number of fixed points of the network. As our 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, which can be applied in the design of genetic regulatory networks.

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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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