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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.

1 Introduction

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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On the other hand, given C?a cycle in G and x a vector satisfying (2), then

?

(vi,vj)∈C?

wF(vi,vj) =

?

j:vj∈C?(2xj− 1) ·

?

i:vi∈C?(2xi− 1) =

?

j:vj∈C?(2xj− 1)2≥ 0.

Thus, C?is a positive cycle.

2

The following theorem is a necessary condition for the existence of fixed points

in a RBN.

Theorem 2 Given a RBN N =(G,F), if N has at least one fixed point, then

G has at least one positive cycle.

PROOF. Let x ∈ {0,1}nbe a fixed point of N. Fix i ∈ V (G) and suppose

that xi= 0. Then by Property P2 of the unate functions there exists either

j ∈ I+(fi), xj= 0 or j ∈ I−(fi), xj= 1. Analogously, if xi= 1, then there

exists either j ∈ I+(fi), xj= 1 or j ∈ V−(fi), xj= 0. Thus, in both cases,

there exists j ∈ I−(fi) such that wF(j,i) = 1 ⇐⇒ xi= xj. Applying now this

argument for j ∈ V (G), we can find k ∈ V−(j) such that wF(k,j) = 1 ⇐⇒

xj= xkand so on. Thus we can construct a sequence of nodes: i1,i2,... such

that

∀j ∈ N, wF(ij+1,ij) = 1 ⇐⇒ xij= xij+1.

Since set of nodes is finite, there exist j,k ∈ N such that ij = ij+k in the

sequence. Thus, by Lemma 1 the cycle

ij,ij+1,...,ij+k

is positive, which completes the proof.

2

The following corollary is a stronger necessary condition for the existence of

fixed points in a RBN.

Corollary 1 Given a RBN N = (G,F), if there exists a strongly connected

component H of G such that all cycles of H are negative and for each arc

(vi,vj) ∈ A(G), vj∈ V (H) =⇒ vi∈ V (H) (see example in Fig. 2), then N

has no fixed points.

PROOF.

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−

+

−

−

+

+

+

H

G:

+

+

Fig. 2. A RBN without fixed points.

The proof is an immediate consequence of Theorem 2 and from the fact that

if x is a fixed point of N, then (xj, j ∈ H) is a fixed point of the sub-network

induced by H, NH, with digraph H.

2

The condition of Theorem 2 is not a sufficient condition as shown in the

following example:

Example 3 Let N = (G,F) be a RBN where G is given in the Fig. 3 and

f1(x) = (x3∨ x5) ∧ x7and ∀j = 2,...,7, fj(x) = xi, i ∈ V−(j). It is easy to

check that N does not have any fixed point.

+

−

3

+

+

2

+

6

+

7

1

+

−

5

+

4

Fig. 3. A RBN without fixed points, but with a positive cycle C=1,2,3,1.

Lemma 4 Let N = (G,F) be a RBN with G strongly connected and such all

the cycles of the signed digraph (G,wF) are positive. Then, for all i,j ∈ V (G)

the directed paths from i to j have equal sign.

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PROOF. Let Pijand Qijbe two directed paths from i to j in G, then since

G is strongly connected, there exists Rjia directed path from j to i. Thus,

C = Pij,Rjiand C?= Qij,Rjiare two circuits in G which can be decomposed

into cycles. Since by hypothesis all cycles of (G,wF) are positive, C and C?

are both positive circuits, hence the number of negative arcs of Pij and Qij

are equal and therefore they have the same sign.

2

Theorem 5 Let N = (G,F) be a RBN. If G is a strongly connected digraph

and the cycles of (G,wF) are all positive, then N has at least two fixed points.

PROOF. Let x ∈ {0,1}nbe a vector defined as follows.

x1= 1, and

∀i = 2,...,n, xi=

1 if there exists a positive directed path from1toi

0 otherwise

First, let us prove that for all (i,j) ∈ A(G), wF(i,j) = 1 ⇐⇒ xi= xj.

Given an arc (i,j) ∈ A(G) we have the following cases.

- Case i = 1. By definition of x and Lemma 4, xj= 1 if and only if the arc

(1,j) is positive. Thus, wF(1,j) = 1 ⇔ x1= xj.

- Case j = 1. By hypothesis there exists P1i a directed path from 1 to i.

Thus, P1i,1 is a cycle of G, which by hypothesis is positive. Therefore, (i,1)

is positive if and only if P1iis positive, that is to say, wF(i,1) = 1 ⇔ xi= x1.

- Case i ?= 1 and j ?= 1. Let us suppose that there exists Q1,ia directed path

from 1 to j which does not contain j. Thus, Q1,i,j is a directed path from

1 to j, and therefore xi= xj if and only if wF(i,j) = 1. Otherwise, there

exist P1,jand Qj,ia directed path from 1 to j and a directed path from j to

i respectively. Thus, Qj,i,j is a cycle in G which by hypothesis is positive.

Therefore, wF(i,j) = 1 ⇔ xi= xj.

Finally, let j ∈ V (G). If xj= 0, by the above result for all i ∈ V ((G) such that

wF(i,j) = 1, xi= 0, and for all i ∈ V (G) such that wF(i,j) = −1, xi= 1.

Thus by Property P2 of the unate functions fj(x) = 0. Analogously, if xj= 1,

then fj(x) = 1, that is to say, x is a fixed point of N. A simliar conclusion

can be drawn in the case of ¬x starting with x1= 0. Therefore, x and ¬x are

both fixed points of N.

2

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4 An upper bound of the number of fixed points.

Theorem 2 show us a relationship between the positive cycles and the existence

of fixed points in a RBN. In this section we will show the existence of a

monotone increasing function involving both ones.

The following theorem gives us an upper bound on the number of fixed points

of a RBN N = (G,F) in terms of the positive cycles of (G,wF). A similar

bound has also been exhibited in networks with continuous local transition

functions in terms of disjoint cycles [17]. We first give some definitions.

Definition 6 A feedback vertex set (FVS) of a digraph G = (V,A) is defined

to be a subset of vertices that contains at least one vertex of each cycle of G.

The minimum number of vertices of a FVS is denoted by τ(G). Analogously, a

positive vertex set (PFVS) of the signed digraph (G,wF) of a RBN N = (G,F)

is defined as a subset of vertices that contains at least one vertex of each

positive cycle of (G,wF). The cardinality of the minimum number of a PFVS

is denoted by τp(G).

The name PFVS is referred to the classical problem in graphs and digraphs

Feedback Vertex Set.

Example 7 For the signed digraph G in Figure 4, {1,7} and {7} are examples

of FVS and PFVS respectively. Thus, τ(G) = 2, and τp(G) = 1.

−

+

+

G:

1

2

3

6

5

4

7

+

−

+

−

−

+

+

−

Fig. 4. Example of signed digraph G with τ(G) = 2, and τp(G) = 1.

Theorem 8 Given a RBN N = (G,F) with at least a positive cycle, the

number of fixed points of N is less than or equal to 2τp(G). And this upper

bound can be reached.

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PROOF. Fix T = {vt1,vt2,...,vtτp(G)} ⊆ V (G) a PFVS of cardinality equal

to τp(G). Let

g : {x|xis a fixed point of N} −→ {0,1}τp(G),

x −→ g(x) = (xt1,xt2,...,xtτp(G)).

Let us prove that the function g is injective. Let x1,x2∈ {0,1}nbe two fixed

points of N such that x1?= x2. Let us define the partition V1,V2of V (G) as

follows:

V1= {j ∈ V (G)|x1

j= x2

j},V2= {j ∈ V (G)|x1

j?= x2

j}

By hypothesis, V2?= ∅.

Fix i1∈ V2and assume without loss of generality that x1

x2

exists a positive cycle i1,i2,...,ik= i1, k ∈ N such that

∀j ∈ {1,...,k}, wF(ij+1,ij) = 1 ⇐⇒ x1

and

∀j ∈ {1,...,k}, wF(vij+1,vij) = 1 ⇐⇒ x2

i1= 1, and therefore

i1= 0. Hence, by using the same argument from proof of Theorem 2, there

ij= x1

ij+1

ij= x2

ij+1.

Hence, x1

such that ij∈ T and x1

the function g is injective and

ij?= x2

ij, ∀j ∈ {1,...,k}. It follows that there exists j ∈ {1,...,k}

ij?= x2

ij, which implies that g(x1) ?= g(x2). Therefore,

Card({x|x is fixed point of N}) ≤ Card({0,1})τP(G)= 2τ(G)

Let us now see one family of RBNs where the bound is reached. Given n ∈ N

odd, let Nn= (Gn,Fn) be a RBN defined by:

• Gn= (Vn,An), where

- Vn= {1,...,n},

- An=

?

k=1

• Fn= (f1,...,fn),

(n−1)/2

{(v2k−1,v2k)} ∪ {(v2k,v2k−1)} ∪ {(v2k,vn)} (see Fig. 5).

fi(x) =

x2k−1

ifi = 2k, k ∈ N,

i = 2k − 1, k ∈ N, and i ?= n,x2k

if

x2+ x4+ ··· + xn−1otherwise.

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It is easy to check that x ∈ {0,1}nis a fixed point of Nnif and only if x2k=

x2k−1, ∀k = 1,...,(n−1)/2 and xn= 1 if there exists k ∈ {1,...,(n−1)/2},

x2k= 1. Hence the total number of fixed points is 2(n−1)/2. On the other hand,

τP(G) = (n − 1)/2, thus the bound is reached.

2

1

+

+

+

+

+

+

+

+

n−2

n−1

4

3

2

n

+

Fig. 5. Example of RBN reaching the upper bound on the number of fixed points.

By using the central idea of the above Theorem we can drawn a similiar result

in the case of Boolean networks not necessarily regulatory ones.

Corollary 2 A Boolean network N = (G,F) with at least a cycle has at most

2τ(G)fixed points. And this upper bound can be reached.

PROOF.

Notice that if x1and x2are two distinct fixed points of N with x1

there exists j ∈ V−(i) such that x1

C = i1,...,ik,i1such that x1

is a vertex beloning to a FVS of G, then x1

a bijection simliar to whose in previous Theorem. The rest of the proof is

analogous.

2

i?= x2

i, then

j?= x2

ij, ∀j = 1...,k. Hence, if il, l ∈ {1,...,k}

il?= x2

jand so on. Thus, there exists a cycle

ij?= x2

il. Thus, we can construct

Thus, from above Theorem the structure of a connection digraph G can

strongly determine, in some cases, the number of fixed points of a RBN, spe-

cially when τp(G) is little.

5Conclusions

A first conclusion from this paper is that the structure of the graph of a RBN

can strongly determine its dynamical behavior, in particular the maximum

number of fixed points of the network. More precisely, if all cycle of a RBN

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are negative, there are no fixed points and if there is at least a positive cycle in

the graph, then the maximum number of fixed points of the network is at most

2τp(G). This bound is of interest wether τp(G) is little, for example O(logn), as

observed in many constructed gene regulatory networks. The bad new is that

the problem of finding a PFVS of minimum cardinality and thus determining

τp(G) has been proved to be a NP-Complete problem [3].

Other important conclusion is that the negative cycles of a RBN can determine

the existence of limit cycles. In fact, the Theorem 2 gives us a necesarry

condition for having at least a limit cycle in the dynmaics of a RBN. Hence,

a next challenge is determing a bound for the number of limit cycles of RBNs

in terms of its positive and negative cycles.

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References

[1] J. Aracena, Modelos matem´ aticos discretos asociados a los sistemas biol´ ogicos.

Aplicaci´ on a las redes de regulaci´ on g´ enica, PhD thesis, U. Chile & UJF,

Santiago, Chile, & Grenoble, France (2001).

[2] J. Aracena, J. Demongeot, E. Goles, Positive and negative circuits in discrete

neural networks, IEEE Transactions on Neural Networks 15 (1) (2004) 77-83.

[3] J. Aracena, A. Gajardo and M. Montalva, On the complexity of feedback vertex

set problems in signed digraphs, 2007, submitted.

[4] O. Cinquin and J. Demongeot, Positive and negative feedback: striking a

balance between necessary antagonists, J. Theor. Biol., 216 (2002), 229-241.

[5] J. Demongeot, M. Kaufmann and R. Thomas, Positive regulatory circuits and

memory, C.R. Acad. Sci. 323 (2000), 69-80.

[6] L. Glass and S.A. Kauffman,J. Theor. Biol. 39 (1973), 103.

[7] F. Harary, Graph Theory, Addison-Wesley, Reading, MA, 1969.

[8] S. Huang. Gene expression profiling, genetic networks and cellular states: an

integrating concept for tumorigenesis and drug discovery.J. Mol. Med. 77 (1999),

469-480.

[9] D. J. Irons, Physica D: Nonlinear Phenomena 217 (2006), 7-21.

[10] S.A. Kauffman, Metabolic stability and epigenesis in randomly constructed

genetics nets, J. Theor. Biol. 22 (1969), 437.

[11] S.A. Kauffman,The Origins of Order, Self-Organization and Selection in

Evolution, Oxford University Press, 1993.

[12] A. Mochizuki, An analytical study of the number of steady states in gene

regulatory networks.J. Theor. Biol. 236 (2005), 291-310.

[13] L. Mendoza and E. Alvarez-Buylla, Dynamics of the genetic regulatory network

for Arabidopsis Thaliana flower morphogenesis, J. Theor. Biol. 193 (1998), 307-

319.

[14] S. Nikolajewa, M.Friedel, T. Wilhelm, Boolean networks with biologically

relevant rules show ordered behavior, BioSystems (2006), in press.

[15] B. Samuelsson and C. Troein, Phys. Rev. Lett. 90 (2003), Art. No. 098701.

[16] L. S´ anchez and D. Thieffry, A Logical Analysis of the Drosophila Gap-gene

System,J. Theor. Biol. 211 (2001), 115-141.

[17] R. Thomas and J. Richelle, Positive feedback loops and multistationarity,

Discrete. Appl. Math. 19 (1988), 381-396.

13

Page 14

[18] R. Thomas, The role of feedback circuits: positive feedback circuits are a

necessary condition for positive real eigenvalues of the Jacobian matrix, Ber.

Bunsenges. Phys. Chem. 98 (9) (1994), 1148-1151.

[19] D. West, Introduction to Graph Theory, Prentice Hall, 1996.

[20] Z. Xie and D. Kulasiri, Modelling of circadian rhythms in Drosophila

incorporating the interlocked PER/TIM and VRI/PDP1 feedback loops. J.

Theor. Biol. 245 (2007), 290-304.

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