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Physica A 384 (2007) 747–757

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Order or chaos in Boolean gene networks depends on themean fraction of canalizing functions

Fredrik Karlsson, Michael Hornquist�

Department of Science and Technology, Linkoping University, SE-601 74 Norrkoping, Sweden

Received 6 September 2006; received in revised form 12 May 2007

Available online 24 May 2007

Abstract

We explore the connection between order/chaos in Boolean networks and the naturally occurring fraction of canalizing

functions in such systems. This fraction turns out to give a very clear indication of whether the system possesses ordered or

chaotic dynamics, as measured by Derrida plots, and also the degree of order when we compare different networks with the

same number of vertices and edges. By studying also a wide distribution of indegrees in a network, we show that the mean

probability of canalizing functions is a more reliable indicator of the type of dynamics for a finite network than the classical

result on stability relating the bias to the mean indegree. Finally, we compare by direct simulations two biologically derived

networks with networks of similar sizes but with power-law and Poisson distributions of indegrees, respectively. The

biologically motivated networks are not more ordered than the latter, and in one case the biological network is even

chaotic while the others are not.

r 2007 Elsevier B.V. All rights reserved.

Keywords: Boolean; Network; Canalizing; Order; Chaos; Topology; Gene network; Noise

1. Introduction

The history of Boolean networks as models for gene networks started already in 1969 with the pioneeringwork by Kauffman [1]. Many interesting results within the realm of theoretical biology have come, see forexample the overview by Kauffman himself [2]. However, more experimentally oriented people have oftenviewed these studies as irrelevant to biology due to their less clear coupling to the experiments. The situationchanged somewhat with the advent in the late 1990s of new measurement technologies in molecular biology,such as microarrays, two-hybrid measurements on protein–protein interactions, genome-wide data for DNAbinding proteins, etc. These large-scale measurement have inspired a renewed interest in simple networkmodels, and many different approaches have been studied [3]. The Boolean networks belong to one extreme inthis spectrum, by being able to contain very large numbers of nodes but very crude in their approximation ofthe biology. Parts of the studies have been on the reconstruction of the Boolean nets from experimental data,and reverse engineering algorithms specially suited for inferring Boolean networks from microarray data havebeen developed (see Ref. [4] for an overview and Ref. [5] for a very recent example). Other studies have

e front matter r 2007 Elsevier B.V. All rights reserved.

ysa.2007.05.050

ing author.

esses: micho@itn.liu.se, Michael.Hornquist@itn.liu.se (M. Hornquist).

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concerned dynamical properties of the networks, and revealed an even more fascinating picture of Booleannetworks than provided by Kauffman [2]. For instance, the number of the attractors grows faster than anypolynomial with the number of nodes [6] and not as the square-root as earlier studies had indicated.

The use of a Boolean network as a model for a gene regulatory system contains many simplifyingassumptions. Most obvious is the approximation of the state of a gene to be either ‘‘on’’ or ‘‘off’’. Althoughthis is a questionable assumption [7], it has indeed been shown not to be without some biological relevance [8].Another commonly used approximation, which we also utilize here, is the synchronous updating. Dependingon which properties one consider, the result can be that the choice of updating schemes hardly matter [9], orthat they matter substantially [10]. An overview and introduction to random Boolean networks with respect to(artificial) life can be found in Ref. [11].

Here we focus on two recent results with respect to order in Boolean nets. The first is the result by Aldanafrom 2003 [12,13], which showed that networks with a power-law distribution of indegrees (sometimes called‘‘scale-free’’) provided a natural robustness and needed no fine-tuning, as networks with a Poissonian indegreedistribution (sometimes called Erd +os-Renyi networks) did. The other result was obtained by Kauffman et al.in 2004 [14] and concerned so-called canalizing functions. They proved that a network consisting only of suchfunctions always are stable, regardless of topology of the net. In the present article, we show numerically thatthe dynamical order for nets with a uniform distribution of Boolean functions can to a large extent beexplained by the fraction of canalizing functions. Further, we present simulations of two finite-size networksof different sizes (30 nodes and 56 edges, and 688 nodes and 1079 edges, respectively) inferred from theordinary baker’s yeast Saccharomyces cerevisiae. The results from these simulations are compared with resultsfrom networks of the same mean indegree, but with a Poissonian and power-law distribution, respectively. Themean probability for canalizing functions turns here out to be a better indicator of order than the classicalresult. Similar studies of finite-sized networks with various topologies have been performed in, e.g., [15,16].However, other measures for the order were utilized, and none of these studies considered any topology thatwere inferred from biological experiments.

The disposition of the present paper is as follows: First, in Section 2, we have some preliminaries where weintroduce concepts and measures. In Section 3, we show how the mean probability for canalizing functionsexplains most of the stability regions found by Aldana [12] and Aldana and Cluzel [13]. The simulations ofsome biologically motivated nets are presented in Section 4. Finally, in Section 5, we conclude with adiscussion of the results.

2. Preliminaries

2.1. Derrida plots

We will primarily utilize Derrida plots for measuring the degree of order the Boolean networks possess. Theideas behind the Derrida plot were introduced already in 1986 [17], and it has since then become a popular toolin this area (other measures used in the literature are, e.g., types of attractors, fraction of elements that areactive, and length of period, as discussed in Ref. [15]). In short, two different initial states with a Hammingdistance1 of DðTÞ are each evolved one time-step and the new Hamming distance DðT þ 1Þ is calculated. TheDerrida plot is obtained by plotting the updated distance versus the original distance, and a network isconsidered ordered if the updated distance is less than or equal to the original one, i.e., if DðT þ 1ÞpDðTÞ. Fora more detailed description of the Derrida plot see Ref. [18].

2.2. Canalizing functions

Canalizing functions are Boolean functions where at least one of the inputs has the possibility to determinethe outcome uniquely, regardless of the values of the other input variables.2 It has been shown that they have

1The Hamming distance between two states is the number of positions where the two states differ.2In Ref. [16], a canalizing function was defined as one where the canalizing input solely determines the output. We follow instead the

terminology from Ref. [2].

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Table 1

Fraction of canalizing functions and total number of Boolean functions when the bias is p ¼ 0:5 and there are k inputs

k Frac. of canal. func. Tot. num. of funct.

1 1.000 4

2 0.875 16

3 0.469 256

4 0.054 65 536

5 3.0e� 4 4.3e9

6 5.6e� 9 1.8e19

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the ability to suppress chaotic behaviour [14]. The probability for obtaining a Boolean function which iscanalizing from a uniform distribution of functions depends on the number of inputs, k, and the bias (thefraction of output values that are unity), p, as derived in [19]:

PrpðkÞ ¼ ð�1Þkðp2k

þ ð1� pÞ2k

Þ � 2kp2k�1

ð1� pÞ2k�1

þXk

i¼1

ð�1Þiþ1k

i

� �2iðp2k�2k�i

þ ð1� pÞ2k�2k�i

Þ

� �. ð1Þ

Since this expression is hard to grasp, we show in Table 1 how the fraction of canalizing functions rapidlydecreases when the number of input variables grows.

2.3. Network topologies

During recent years, there has been a great interest in various forms of networks and network properties.For short, there has been a shift in models utilizing network structures from the classical Erd +os–Renyinetwork with a Poissonian degree distribution to so-called scale-free networks with a power-law degreedistribution. Although there is much confusion about what it means for a network to be ‘‘scale-free’’ and whatit implies [20], there is at least much evidence that many biological networks (including gene networks) havedegree-distributions that sometimes are power-laws and almost always deviate strongly from being Poissonianby being much wider [21]. We will use the terms ‘‘scale-free’’ and ‘‘power-law distributed’’ as synonyms.

3. Order and canalizing functions

A recent result on the stability of Boolean networks was derived by Aldana [12] and Aldana and Cluzel [13].There the original annealed computation on order from 1986 [22] was generalized to encompass generalindegree distributions (with a finite first moment). A primary result was that the classical result, about theborder between the ordered and the chaotic regimes, still holds.3 This border is given by

2pð1� pÞK ¼ 1, (2)

where p is the bias as before and K is the mean indegree K ¼P

kkPinðkÞ. Although the result should be validfor any distribution of indegrees, [12,13] discuss its consequences only for networks possessing a Poisson and apower-law degree distribution, respectively.

Because of the result that Booleans networks with only canalizing functions always are stable [23], we find itnatural to see whether we can relate the fraction of canalizing functions with the stability border given by (2).In Fig. 1 we have depicted Eq. (2) (solid curve) together with the level curvesX

k

PinðkÞPrpðkÞ ¼ 0:69, (3)

3The same result was obtained already two years earlier in [15], but that derivation was for a linearized model.

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Fig. 1. Solid curve: border between chaotic and ordered regime, as given by Eq. (2), in the pK-plane. Dashed curve: level curve (3) for a

network with a Poisson distribution of indegrees (4). Triangles: same as dashed curve, but for a delta distribution (hence it takes values

only for integer mean indegrees). The ordered region is situated to the left of the curve. The resemblance of these curves is striking.

F. Karlsson, M. Hornquist / Physica A 384 (2007) 747–757750

where Pin is some distribution of indegrees and Prp is the distribution of canalizing functions from (1). Thenumber ‘‘0.69’’ is obtained from the fraction of canalizing functions when p ¼ 0:5 and K ¼ 2 for a networkwith a Poisson distribution of indegrees.

For the Poisson distributed networks of indegrees, where Pin takes the form

PinðkÞ ¼ e�K Kk

k!, (4)

we obtain the dashed curve in Fig. 1. The triangles correspond to networks where all nodes have the sameindegree, i.e., the degree distribution is a delta function (this was the kind of network originally studied,termed the NK-model). The resemblance of these curves is striking, especially in the interval 0:2opo0:8,although not perfect. The similar results for the two distributions are not unexpected, though, since thePoisson distribution is narrow around the mean, and hence it resembles the delta distribution well.

In order to see how well the mean probability for canalizing functions explains the properties of (2) whenthe distribution of indegrees is wide, we turn to power-law distributed networks with an indegree distributionof the form

PinðkÞ ¼1

zðgÞ � kg . (5)

Here z is the Riemann zeta function [24] and is just a normalization constant. The Eq. (2) can now be written as

2pð1� pÞzðg� 1Þ

zðgÞ¼ 1. (6)

In accordance with Refs. [12,13], we depict in Fig. 2 the two regions of order and chaos in the gp-plane. Thistime, the deviation between the curve given by (6) (which is nothing but (2) for a power-law distribution) (solidcurve) and the level curve given by (3) (dashed curve) is more pronounced. Although the functional forms aresimilar to each other, there is a clear difference in the form of a translation between the curves.

Because of this observation, we explore by Derrida plots the region in Fig. 2 between the solid curve from(2) [or, equivalently, from (6)] and the dashed curve from (3) [with PinðkÞ from (5)]. We concentrate on the caseof equal bias, i.e., when p ¼ 0:5, and let g ¼ 2:1 for the scale-free networks (we utilize 50 different networksand present the mean values). These networks are composed by 10 000 nodes and 31 189 edges, resulting in amean indegree K ¼ 3:1 and a mean probability for canalizing functions Pr0:5 ¼ 0:8035.4 The out-degreedistribution is the same as the in-degree distribution. According to the classical result, Eq. (2), the dynamics of

4The rather low number of K comes from a cut-off effect. An infinite scale-free network with the same g should have K ¼ 6:78. However,

the present size is enough for showing our statements.

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Fig. 2. Solid curve: border between chaotic and ordered regime, as given by Eq. (6), in the pg-plane. Dashed curve: level curve (3) for a

network with a power-law distribution of indegrees (5). The ordered region is situated to the right of the curves. Although the same

functional form, there is a translation between the curves.

Fig. 3. Derrida plot for scale-free networks with 10 000 nodes and mean indegree K ¼ 3:12 (triangles) and for a Poisson distributed

network of the same size and mean degree (squares). The solid line DðT þ 1Þ ¼ DðTÞ is the border between order and chaos, and any curve

above this line indicates a chaotic behaviour of the corresponding system. According to Eq. (2) both networks should posses chaotic

dynamics, but obviously the scale-free networks have ordered dynamics. The standard deviations are negligible and not visible in the

figure.

F. Karlsson, M. Hornquist / Physica A 384 (2007) 747–757 751

these networks should be chaotic, but according to fraction of canalizing functions, it should be ordered. InFig. 3 we show the Derrida plots for these scale-free networks and, as comparison, for ER-networks withPoisson distributions of in- and outdegrees and the same number of nodes and edges, and a mean probabilityfor canalizing functions Pr0:5 ¼ 0:4854 (i.e., these networks should possess chaotic dynamics according to bothmeasures). We see how the Poisson distributed networks have a chaotic dynamics, as expected by Eq. (2), butthat the scale-free networks have an ordered dynamics. Hence, Eq. (2) cannot be valid for a wide distributionof the kind we explore here. Instead, the fraction of canalizing functions provides a more accurateclassification. This apparent contradiction, however, is resolved by noting that the result in Ref. [12,13] isclaimed to be valid only in the thermodynamic limit when the number of nodes tends to infinity. We note thatreal regulatory networks always are finite, which motivates our study although there already are analyticalresults.

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To summarize this section, we noticed there is a clear connection between the fraction of canalizingfunctions in a Boolean network and the stability properties of the system. The general result in (2) does nothold for finite networks with general distributions of indegrees, which we numerically showed for networkswith power-law distributions of indegrees.

4. Simulations

In order to see the effects of different topologies with respect to order and chaos, we now explore twonetworks of various sizes, both derived from ordinary baker’s yeast, Saccharomyces cerevisiae. Ascomparisons, we also explore networks of the same size and mean indegree, but with Poisson and power-law degree distributions, respectively.

4.1. A small yeast network, the Lee-network

The network studied in this section is fetched from Ref. [23], which in their turn have obtained it from directmeasurements in Ref. [25] with a certain cut-off. This is a proposed transcriptional network for yeast, and weutilize a version with self-feedback, but where the vertices with no outputs have been removed. The networkhas then 30 vertices and 56 edges, and we will refer to it as the Lee-network. Its distribution of indegrees isshown in Fig. 4. We study the order with Derrida plots.

As comparison, we also study ER-networks and scale-free networks with the same number of vertices andedges as the Lee-network. Fifty Poisson distributed and 50 power-law distributed networks are generated.They are denoted ERLee and SFLee, respectively. The maximum number of indegrees to a vertex is limited to 10for the power-law distributed networks. In order to obtain reasonable statistics, 50 sets of independentlyuniformly random Boolean functions are used for the Lee-network. For all cases, both for the Lee topology aswell as for the scale-free and Poisson distributed networks, we use the bias p ¼ 0:5. The curves in the Derridaplot, Fig. 5, are based on 1000 initial states per network. Somewhat in contrast to the general belief thatrobustness should have been a favoured property by evolution, we notice from the figure that both the ER-networks and the scale-free networks have dynamics which are more ordered than the biologically derivednetwork. On the other hand, it has also been argued that biological networks have to be on the ‘‘edge ofchaos’’ [2] for life to proceed.

If we consider the stability region given by (2), all the present networks should be stable (since K ¼ 5630� 1:87

and p ¼ 0:5). Meanwhile, the mean probabilities for canalizing functions for the Lee-network, the ERLee

network and the SFLee network are 0.592, 0.723, and 0.823, respectively. The order is in accordance with thedegree of order, and the value for the Lee-network is below 0.69, which indicates a chaotic behaviour. Indeed,

Fig. 4. Cumulated distribution of indegrees for the Lee-network, withP1

k0¼kPinðk

0Þ depicted as function of indegree k.

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Fig. 5. Derrida plots for the Lee-network (squares), the ERLee networks (triangles) and the SFLee networks (circles). The straight line is the

diagonal DðT þ 1Þ ¼ DðTÞ and corresponds to the border between order and chaos. The standard deviations are calculated for each

Hamming distance separately, but the smoothness of the curves indicate they are considerably less. The curves for ERLee and SFLee

networks are below the diagonal, and hence their dynamics are ordered according to this graph. The curve for the Lee-network is slightly

above the diagonal for some values of DðTÞ, which shows that the network possesses chaotic dynamics.

Fig. 6. Time-evolution of Hamming distance, showing the effect of noise for the Lee-network (squares), the ERLee networks (triangles)

and the SFLee networks (circles).

F. Karlsson, M. Hornquist / Physica A 384 (2007) 747–757 753

a closer inspection of Fig. 5 shows that the curve exceeds the diagonal for DðTÞt6, and hence the dynamics ischaotic.

Considering the response to noise, we let two identical initial states be updated in identical networks underthe same set of Boolean functions. One of the trajectories is exposed to noise by our flipping of the node valueswith a certain probability per update and node. For each time-step, we calculate the Hamming distance, anddepict it as a function of time. In Fig. 6 we see the results for the Lee-network and the corresponding Poissonand power-law distributed networks. Every curve is based on 50 initial pair of states per network5 and theprobability for noise is 0.01. The variation is considerable, and for clarity not shown in the figure. However,

5For the Lee-network, we do not change the topology, but only the initial states and the Boolean functions.

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the order of the curves is robust. An interesting feature is that a higher mean probability for canalizingfunctions still corresponds to a higher degree of order, confirming the results from the Derrida plot.

4.2. A large yeast network, the Milo-network

The second network of interactions in yeast we consider is compiled by Milo et al. [26] and we will refer to itas the Milo-network. This network has 688 vertices and 1079 edges, where the former represent genes (orcomplexes of genes) and the latter direct interactions between these. Its distribution of indegrees is shown inFig. 7. Although not an exact power-law, it is broad and deviates strongly from the narrow peak characteristicfor a Poisson distribution.

As before, 50 Poisson distributed and 50 power-law distributed networks with the same number of nodesand edges serve as comparison, and they are denoted ERMilo and SFMilo, respectively. For the power-lawdistributed networks the maximum number of indegrees is 10. Fifty sets of Boolean functions of the same kindas for the Lee-network are randomly assigned to the Milo-network. All the results presented under this topic

Fig. 7. Cumulated distribution of indegrees for the Milo-network, withP1

k0¼kPinðk

0Þ depicted as function of indegree k.

Fig. 8. Derrida plots for the Milo-network (squares), the ERMilo networks (triangles) and the SFMilo networks (circles). The solid line

DðTÞ ¼ DðT þ 1Þ is the border between order and chaos. The standard deviations are calculated for each Hamming distance separately,

but the smoothness of the curves indicate they are considerably less. All networks are stable according to this graph, since all of them are

below the diagonal.

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Fig. 9. Time-evolution of Hamming distance, showing the effect of noise for the Milo-network (squares), the ERMilo networks (triangles)

and the SFMilo networks (circles). The clear deviation of the power-law distributed networks cannot be explained by the mean probability

for canalizing functions in the systems.

F. Karlsson, M. Hornquist / Physica A 384 (2007) 747–757 755

are means based on 50 networks, or in the case of the Milo-network 50 sets of Boolean functions. The curvesin the Derrida plot, Fig. 8, are based on 50 initial states per network/sets of Boolean functions. All of thesenetworks should be stable according to Eq. (2), since K ¼ 1079

688� 1:57 and p ¼ 0:5. The fractions of canalizing

functions are 0.828, 0.780, and 0.884 for the Milo-, ERMilo and SFMilo network, respectively. That is, allprobabilities are above 0.69, and from Fig. 8 we see that they all correspond to stable systems. Indeed, alsohere the degree of order correspond to the ordering of the mean probability for canalizing functions.

In Fig. 9, where the results from a noise analysis is shown, every curve is based on 50 initial states pernetwork/sets of Boolean function. When noise is applied the probability for a node to be inverted is 0.001 pertime-step. The variation is considerable, and for clarity not shown in the figure. However, the order of thecurves is robust. It is interesting to note that a higher mean probability for canalizing functions corresponds toa higher degree of robustness. This time, however, there seems to be a more qualitative difference between theMilo- and ERMilo networks one side, and the SFMilo network on the other. The large variation makes this,however, somewhat speculative. The Hamming distance for the SFMilo networks almost saturates quiterapidly, and remain on a low level, while the Hamming distance for the other networks increase more or lesslinearly. Considering the fact that the Milo-network only has a slightly less mean fraction of canalizingfunctions than the SFMilo networks, there has to be more relevant features than only this mean number. Theexploration of what these features can be is beyond the scope of the present paper.

5. Discussion and conclusions

Canalizing functions are known for their ability to suppress chaotic behaviour. Here we show that the meanfraction of canalizing functions in a Boolean network is enough for giving a clear indication on whether thedynamics of the network is chaotic or ordered. For the delta-distributed network, where all nodes have thesame indegree, and for the Poisson distributed networks, the classical result on stability regions in the bias-indegree plan is more or less recaptured for biases in the interval 0:2opo0:8. When the distribution ofindegrees gets wider, there will be a stronger deviation from this classical result. The power-law distributednetworks show quite a strong deviation. In order to test which of the two measures which provide the mostreliable estimate on the dichotomy between order and chaos when there is equal bias (p ¼ 0:5), we explored byDerrida plots the nature of the dynamics, and found the fraction of canalizing functions to yield a morereliable prediction of the behaviour of the network. The appearent contradiction might be resolved by notingthat the classical border 2pð1� pÞK ¼ 1 strictly is valid only for infinite networks, while the Derrida plots weutilize are based on simulations with networks consisting of 10 000 nodes.

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When we study the time evolution of a Hamming distance in certain biologically inspired networks, we findthat they are not more robust than power-law or Poisson distributed nets. This fact is opposite to the view thatevolution has made its networks insensitive to external perturbation, and shows that they are rather on the‘‘edge of chaos’’. The mean fraction of canalizing functions gives a better description of their stabilityproperties than does the classical result. The latter predicts all networks of the same size studied here to beordered, while the fraction of canalizing functions predicts their intrinsic order of stability, and even indicatethat one of them should be chaotic. Derrida plots and time-evolution graphs show that these predictions oforder and instability are correct.

In summa, with the support of the results from the calculations and simulations performed in this article, weclaim that the mean fraction of canalizing functions is a dominating factor affecting whether the dynamics of anetwork with randomly assigned Boolean functions is ordered or chaotic. The connection between the meanfraction of canalizing functions and order provides us with a new tool to determine the degree of order/chaosfor such networks.

Acknowledgement

Thoughtful and useful comments from Mika Gustafsson, who read a draft of the manuscript, areappreciated. Financial support from CENIIT (Centre for Industrial IT at Linkoping University) and from theCarl Trygger’s foundation is acknowledged.

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