A reduction method for noisy Boolean networks

(1)

arXiv:0912.3703v1 [physics.bio-ph] 18 Dec 2009

Frédéri Fourré

1∗

DenisBaurain

2†

De ember 18,2009

1

Aliation: MonteoreInstitute,DepartmentofAppliedMathemati s, Univer-sityofLiège,B28,B-4000Liège,Belgium

2

Aliation: Algology,My ologyandExperimentalSystemati s,Departmentof LifeS ien es, UniversityofLiège,B22,B-4000Liège,Belgium

Abstra t

This paper is on erned with the redu tion of a noisy syn hronous Booleannetworktoa oarse-grainedMarkov hainmodel. Consideran

n

-nodeBooleannetworkhavingatleasttwobasinsofattra tionandwhere ea hnodemaybeperturbedwithprobability

0 < p < 1

duringonetime step. Thispro essisadis rete-timehomogeneousMarkov hainwith

2 n

possiblestates. Nowunder ertain onditions,thetransitionsbetweenthe basins ofthe network may beapproximatedby a homogeneousMarkov hainwhereea hstateofthe hainrepresentsabasinofthenetwork,i.e. the size ofthe redu ed hainis the numberof attra torsof the original network.

Contents

1 NoisyBoolean networks(NBNs) 2

1.1 Denition . . . 2

1.2 Themeanspe i path. . . 5

1.3 Timeevolutionequation . . . 6

1.4 Stationaryprobabilitydistribution . . . 7

2 Methodfor the al ulation of the sojourn timedistribution in

a basin ofa NBN 9

3 The problem ofgeometri approximation 10

∗

PresentAddress:SystemsBiologyGroup,LifeS ien esResear hUnit,Universityof Lux-embourg,162a,avenuedelaFaïen erie,L-1511 Luxembourg,Grand-Du hyofLuxembourg. Corresponden eshouldbeaddressedtofrederi .fourreuni.lu

†

PresentAddress:UnitofAnimalGenomi s,GIGA-RandFa ultyofVeterinaryMedi ine, UniversityofLiège,B34,B-4000Liège,Belgium

(2)

4 The geometri approximation in

(n, C)

networks 20

4.1

(n, C)

networksand onden eintervals . . . 22

4.2 Simulationresultsanddis ussion . . . 25

4.3 Approximationformula forthe meantime spentin abasin of a NBN . . . 31

5 Methodforthe redu tion of a NBN 36 5.1 Dis rete-time redu tion . . . 36

5.1.1 Thelow

p

regime . . . 36

5.1.2 Case

p = 1/2

. . . 40

5.2 Continuous-timeredu tion. . . 40

6 Statisti al u tuationsin NBNs 42

7 Redu tion ofa NBN to a two-state Markov hain 46

8 Con lusion 46

9 Note 48

10A knowledgment 48

1 Noisy Boolean networks (NBNs)

1.1 Denition

Boolean networks (BNs) havebeenused for several de ades asmodels of bio- hemi al networks, mainly to predi t their qualitative properties and, in the aseofgeneti regulatorynetworks,to infertheinputsandintera tionrulesof theirnodesfrommi roarraydata(Kauman,1969;GlassandKauman,1973; Kauman,1993;Lietal.,2004;Martinet al.,2007; SamalandJain,2008).

An

n

-node BNmodel onsists of

n

intera ting nodeswhi h, in the ontext ofbio hemi alnetworks,generallyrepresentgenesormole ularspe iessu has proteins,RNAsormetabolites. Let

X

i

(k)

denotetheBooleanvariable des rib-ingthestateofnode

i

atdis retetime

k = 0, 1, . . .

Ifnode

i

representsaprotein, thenwesaythatifthevalueofthenodeis

1

,thentheproteinispresent,inits a tiveform and itstarget (or substrate)present. UsingDe Morgan'slaw, the negationofthis onjun tiongivestheinterpretation forvalue

0

: theproteinis absentorina tiveoritstarget(orsubstrate)absent.

Remark 1 A more on ise interpretation instead of present (resp. absent) is present and not being degraded (resp. absent orbeingdegraded)or evenpresent andprodu tion rate greater thandegradation rate (resp. absent ordegradationrate greaterthanprodu tion rate).

Foran

n

-nodesyn hronousBN,theintera tionsbetweenthenodesare mod-eledbyasetof

n

Booleanintera tionfun tions su h that:

(3)

X

i

(k + 1) = F

i

[X

1 (k), X

2 (k), . . . , X

n

(k)],

i = 1, 2, . . . , n,

(1)

with

F

i

intera tion fun tion of node

i

. If the state of node

i

at time

(k + 1)

depends onthe state of node

j

at previoustime

k

, then node

j

is saidto be aninputfornode

i

. Thenumberofinputsofanodeis alled the onne tivity of that node. Thenetwork is said to besyn hronousbe ause asexpressed in (1)the

n

nodesareupdatedsyn hronously. Alsofrom(1)thedynami sof the network deterministi .

Letustakeanexampletoillustratesomeessentialpropertiesof BNs. Con-siderthefollowingBN:

X

1 (k + 1) = ¬X

4 (k)

X

2 (k + 1) = ¬X

3 (k)

X

3 (k + 1) = X

1 (k) ∨ X

2 (k) ∨ X

4 (k)

X

4 (k + 1) = X

1 (k) ∧ X

2 (k) ∧ X

3 (k),

(2)

where symbols

¬

,

∨

and

∧

represent logi al operators NOT, OR and AND respe tively. The size of the state spa e of this network, i.e. the number of possiblestates, is

2 n

_{= 16}

sin e

n = 4

here. From(2), thenextstatefor ea h possiblestate an be omputedtoobtainthestatetableofthenetwork. Using this state table then, the state diagram an be built. This is shown in Fig 1 whereit anbe seenthat thestate spa ehasbeenpartioned intotwodisjoint sets. Thesesetsare alledbasinsofattra tionandaredenotedbyB1andB2in thegure. Ea h basin onsistsof anattra torand sometransientstates. The attra torof B1 is the xed point

1010

: whatever the initial state in B1, the network will onvergeto

1010

and stay there forever. Attra tors may also be periodi asisthe aseforbasinB2. Thesizeofabasinorofanattra toristhe numberofstatesthat onstituteit.

Now suppose that, due to random perturbations, ea h node of a BN has probability

0 < p < 1

to swit h its state (

0

to

1

or vi e versa) between any twotimes

k

and

(k + 1)

. Then we get anoisy BN(NBN)

1

. The introdu tion of disorder

p

is supported by the sto hasti nature of intra ellular pro esses oupled to the fa t that, thermodynami ally speaking, bio hemi al networks areopensystems.

One important dieren e between a BN and a NBN is that in the latter, transitionsbetweenthebasinsofthenetworkareallowed. Forexample,in the statediagramofFig.1,ifweperturbsimultaneouslynodes

2

and

3

ofattra tor state

1010

whi h isin B1,thenwegoto transientstate

1100

whi hbelongsto B2.

1

Here,forthesakeofsimpli ity,perturbationprobability

p

issupposedtobeindependent oftime

k

andofthestatesofthenodes.If,forinstan e,foraparti ularnode,

0

to

1

random swit hingprobabilityissetgreaterthan

1

to

0

one,thenitmeansthatrandomperturbations tendtoturnthenodeONratherthanOFF.

(4)

0110 1010 0001 1001 1000 1011 0010 0011 0111 1111 B1 B2

Figure1: Statediagramofthefour-node networkdened byintera tionrules(2). Intera tionsbetweenthenodesleadtothepartitioningofthestatespa eintotwo basinsB1 andB2 ofrespe tivesizes

6

and

10

.

Remark2 FlippingthevalueofsinglenodeshasbeenenvisagedbyKauman(1993 ) ingeneti Booleannetworkstostudytheirstabilitytowhathe alledminimal perturba-tions. Shmulevi h etal.(2002 )proposedamodelforrandomgeneperturbationsbased onprobabilisti Booleannetworks (a lassofBooleannetworks thaten losedthe lass ofsyn hronousBooleannetworks byassigning more than oneintera tion fun tion to ea hnode)inwhi hanygeneofthenetworkmayipwithprobability

p

independentlyof othergenesandinterpretedtheperturbationeventsastheinuen eofexternalstimuli onthea tivityof thegenes.

Let

L(x; n, p)

be the probability that exa tly

x

nodes will be perturbed duringonetimestep. Thenletting

q = 1 − p

:

L(x; n, p) =







n

x

p

x

_q

n−x

if

0 < x < n,

q

n

if

x = 0,

p

n

if

x = n.

(3)

This is the binomial probability distribution with parameters

n

and

p

. On average,

np

nodes are perturbed at ea h time step. The probability that at leastonenodewillbeperturbedduring onetimestepistherefore:

n

X

x=1

L(x; n, p) = 1 − q

n

= r.

(4)

Sin ewearemainlyinterestedinthebehaviourofthenetworkas

p

varies,inthe following

n

and

x

will be onsideredas parameterswhile

p

will be onsidered asavariable. Thusweshould write

L(p; n, x)

insteadof

L(x; n, p)

2 . 2

(5)

p

0.002 0.02 0.2 0.5

L(p; 4, 1)

0.0080 0.0753 0.4096 0.2500

L(p; 4, 2)

0.0000 0.0023 0.1536 0.3750

L(p; 4, 3)

0.0000 0.0000 0.0256 0.2500

L(p; 4, 4)

0.0000 0.0000 0.0016 0.0625

L(p; 8, 1)

0.0158 0.1389 0.3355 0.0312

L(p; 8, 2)

0.0001 0.0099 0.2936 0.1094

L(p; 8, 3)

0.0000 0.0004 0.1468 0.2188

L(p; 8, 4)

0.0000 0.0000 0.0459 0.2734

Table1: Some valuesofthefun tion

L(p; n, x)

.

For

0 < x < n

,

L(p; n, x)

has a maximum at

p = x/n

and

L(0; n, x) =

L(1; n, x) = 0

.

TheMa laurinseriesof

L(p; n, x)

uptoorder

2

is:

L(p; n, x) =











1 − np + n(n − 1)p

2 /2 + . . .

if

x = 0,

np − n(n − 1)p

2 + . . .

if

x = 1,

n(n − 1)p

2 _{/2 + . . .}

if

x = 2,

0 + . . .

if

2 < x ≤ n.

Thus onehas:

L(p; n, x) =







1 − np + o(p)

if

x = 0,

np + o(p)

if

x = 1,

o(p)

if

2 ≤ x ≤ n.

(5)

This means that,

n

being xed, for su iently small

p

the probability that

2 ≤ x ≤ n

nodes be perturbed during onetimestep isnegligible ompared to theprobabilitythat just onenodebeperturbed. Table1givessomevaluesof

L(p; n, x)

roundedtofourde imalpla es. Weseethatat

p = 0.002

,

L(p; 4, 1) =

0.0080 = 4p

and

L(p; 8, 1) = 0.0158 ≈ 8p

.

1.2 The mean spe i path

Consideraseries ofBernoullitrials whereea h timestepdenesonetrialand where asu ess means that at least onenodehas beenperturbed. From (4), theprobabilitythattherstsu esswillo urontrial

i

is:

p

i

= r(1 − r)

i−1

,

i = 1, 2, . . . ,

(6)

whi h is ageometri distribution with parameter

r

. The rst momentof this distribution(the meantimebetweentwosu esses) is:

(6)

τ =

1 r

=

1 1 − q

n

.

(7)

Weseethat

τ

isade reasingfun tionof

p

and

n

,thattendsto

1

as

p → 1

and to

∞

as

p → 0

. Sin etime stepisunity,

τ

equalsthemeannumber

d

of state transitions between two su esses. By analogy with the mean free path of a parti lein physi s

3

,

d

will be alled mean spe i path. Thetermspe i isusedtore allthatbetweentwosu esses,thetraje toryin thestatespa eis spe i tonodeintera tions,i.e.,itisentirelydeterminedbynodeintera tions.

Forsu ientlysmall

p

onehas

r = 1 − q

n

_{≈ np ≈ L(p; n, 1)}

,wherethelast approximation omesfrom (5). Hen e,forsu ientlysmall

p

wemaywrite:

p

i

≈ np(1 − np)

i−1

,

i = 1, 2, . . .

Dependingon thevalueof

p

, dierentdynami alregimes arepossible. For

p = 0

, the network is trapped by an attra tor where it stays forever. For

0 < p < 1

, transitions between the basins of the network o ur. The more

p

is lose to one, theshorter the times spent on the attra tors,the more the network suers from fun tional instability. In the low

p

regime, the network may both maintain a spe i a tivity for a long time period and hange its a tivity: fun tionalstabilityandexibility(ordiversity) oexist.

1.3 Time evolution equation

ANBNis infa t adis rete-timeMarkov hain

{X

k

, k = 0, 1, . . .}

,where

X

k

is therandomvariablerepresentingthestateofthenetworkattime

k

. Thestate spa eofan

n

-nodeBNwill bedenoted by

{1, 2, . . . , E}

,with

E = 2

n

andwith state

i

orrespondingto binaryrepresentationof

(i − 1)

.

Let

π

ij

= Pr{X

k+1

= j|X

k

= i} ≥ 0

bethe onditional probabilitythat the networkwill beinstate

j

at

(k + 1)

whileinstate

i

at

k

. Thematrixofsize

E

whoseelementsare the

π

ij

'swill bedenoted by

Π

andis alled thetransition probabilitymatrixin Markovtheory. Thesumofelementsin ea hrowof

Π

is unity. Let

z

(k)

i

betheprobabilitythat thenetwork will bein state

i

at time

k

and denote by

z

(k)

theve torwhose elementsare the stateprobabilities

z

(k)

i

. Givenaninitialstateprobabilityve tor

z

(0)

,theve tors

z

(1)

_{, z}

(2)

_{, . . .}

arefound from(Kleinro k,1975):

z

(k+1)

_{= z}

(k)

_Π,

_{k = 0, 1, . . .}

(8)

Matrix

Π

isthesumof twomatri es: 1. Theperturbationmatrix

Π

′

,whose

(i, j)

thelementisequalto

π

′

ij

=

p

h

ij

_q

n−h

ij

if

i 6= j,

0

if

i = j,

3

Inkineti theory of gases,the meanfree path ofa gas mole uleis the meandistan e traveledbythemole ulebetweentwosu essive ollisions.

(7)

where

h

ij

refers to

(i, j)

th element of Hamming distan e matrix

H

, a symmetri matrix that depends only on

n

. Element

h

ij

is equal to the numberof bits that dierin the Boolean representationsof states

i

and

j

. Diagonalelementsof

H

aretherefore

0

. Forexample,if

n = 2

,

H

takes theform:

H =







0 1

1 2

1 0

2 1

1 2

0 1

2 1

1 0







.

(9)

Note that in ea h row of

H

, integers

x = 1, 2, . . . , n

appear respe tively

n

x

times. Thusfrom(3) and(4), thesumofelementsin ea h rowof

Π

′

mustbe

r

.

2. Theintera tionmatrix

Π

′′

. Ifthenodeintera tionsaresu hthatstarting in state

i

thenetwork isfor ed totransition to state

j

in onetimestep, then

(i, j)

th element of

Π

′′

equals

q

n

_{= 1 − r}

, otherwise it is

0

. Noti e that if

i

th diagonalelementof

Π

′′

equals

q

n

thenstate

i

is axed point (attra torofsize

1

).

Therefore,forxed

n

,anyprobability

π

ij

willbeeither

0

orafun tionof

p

.

1.4 Stationary probability distribution

Sin e

p

does not depend on

k

, the

π

ij

's are independent of time. The hain isthushomogeneous. Asshown by Shmulevi hetal.(2002)forgeneti proba-bilisti Boolean networks,for

0 < p < 1

,the hainisirredu ibleandaperiodi . Consequently, forgiven

p

there is aunique stationary distribution

¯

z

whi h is independentof

z

(0)

(Kleinro k,1975). Thestationary distributionsatises the twoequations:

z

_{= zΠ}

and

X

i

z

i

= 1.

(10) Inaddition,wehave:

lim

k→∞

Π

k

_{= ¯}

_Π,

(11)

whereea h rowof

Π

¯

isequaltove tor

¯

z

.

Fig. 2showsthestationary stateprobabilities

z

¯

i

forthenetwork ofFig. 1 andtwo

p

values. Bluepoints orrespondto

p = 0.04

and redonesto

p = 0.4

. As anbeseenin thegure,when

p = 0.04

,thestationary probabilitiesof the transientstatesare small ompared to those of theattra torstates(attra tor statesarerepresentedinboldtypeinthegure). Thereforeinthelow

p

regime, thea tivityofthenetworkinthelongtermisgovernedmostlybyitsattra tors

4 . 4

Equivalently,inthelow

p

regime, itismostly attra torsthatdeterminethe fateof the ells.

(8)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

Boolean states

S

ta

ti

on

ar

y

pr

ob

ab

ili

ti

es

¯z

i

Figure 2: Stationary state probabilitiesfor the network shownin Fig. 1and two

p

values. Blue points:

p = 0.04

; red points:

p = 0.4

. Attra tor states are in boldtype. The stationary probability ofB1 (obtained by summingthe stationary probabilitiesofthestatesofB1)is

0.3270

when

p = 0.04

and

0.3708

when

p = 0.4

. Sin ethesizeof B1is

6

, thisprobability onvergesto

6/16 = 0.375

as

p

tendsto

1

.

Infa t,thestationaryprobabilitiesofthetransientstatestendto

0

as

p

tends to

0

. Alsonoti efor

p = 0.04

thestationaryprobabilitiesofthese ondattra tor almostequal.

As

p

tendsto unity,thestationaryprobabilitiestend tobeuniform,i.e. as

p → 1

onehas

z

¯

i

→ 1/E = 1/16 = 0.0625 ∀i

(seethe ase

p = 0.4

). Thusfor

p

su iently lose tounity, thestationary probabilitiesof thetransient states arenotnegligible omparedtothoseoftheattra torstates.

Remark 3 Forsu iently small

p

, the stationary probabilityof an attra tor state maybeapproximatedby

¯

Z

∗

_/A

where

¯

Z

∗

isthelimitas

p → 0

ofthestationary o upa-tionprobability

Z

¯

ofthebasin ontainingtheattra torand

A

thesizeoftheattra tor (see further 4.3). Thus in the ase of the network of Fig. 1, we get for attra tor state

11

(xed point) that

z

¯

11 ≈ 1/3 = lim

p→0

Z/1

¯

, and forthe states of these ond attra tor,thatea hstationary stateprobability

≈ 1/6 = lim

p→0

¯

Z/4

.

(9)

distribution in a basin of a NBN

ConsideraBNhavingatleasttwobasins. ConsiderabasinBofsize

B

ofthis network andaxedperturbationprobability

p

. Let

Π

b

=

Q

a

0 ₁

.

(12)

Thesquare

B × B

sub-matrix

Q

ontainstheone-step transition probabilities

π

ij

between the states of B. For example, in the ase of basin B1 of Fig. 1, element

(1, 4)

of

Q

isequaltotheone-steptransitionprobabilitybetweenstate

2

(

0001

) and state

10

(

1001

). Element

i

of ve tor

a

represents the one-step transitionprobabilitybetweenstate

i ∈

B andan absorbingstate

5 regrouping states

j /

∈

B ,i.e.

a

i

=

X

j /

∈

B

π

ij

=

X

j /

∈

B

π

′

ij

=

X

j /

∈

B

p

h

ij

_q

n−h

ij

_.

0

isarowve toroflength

B

withallelementszeroand

1

isas alar. Frommatrix

Q

,we an al ulatetheprobability

W

(k)

thatthenetworkwill beinbasinBattime

k

givenaninitialprobabilityve tor

b

(0)

oflength

B

with sumofelements

1

:

b

(k+1)

_{= b}

(k)

_Q,

_{k = 0, 1, . . . ,}

(13) andbydenition:

W

(k)

=

B

X

i=1

b

(k)

_i

,

k = 0, 1, . . . ,

with

W

(0)

_{= 1}

.

Nowwewantthe sojourntime distributionin B. Weshall denoteby

S

the dis reterandomvariablerepresentingthesojourntimeinB,withsamplespa e

s = 1, 2, . . .

, andby

ψ

s

theprobabilitydistribution of

S

, i.e.

ψ

s

= Pr{S = s}

. Foragivenbasin,

ψ

s

dependson

p

andinitial ve tor

b

(0)

. Itisgivenby:

ψ

s

= W

(s−1)

− W

(s)

,

s = 1, 2, . . . ,

(14)

whilethe umulativedistributionfun tionof

S

is:

ˆ

ψ

s

= Pr{S ≤ s} = 1 − W

(s)

,

s = 1, 2, . . .

(15)

Matrix

Π

b

in(12)representsanabsorbingMarkov hain(Snell ,1959),i.e. it hasatleastoneabsorbingstateandfromeverynon-absorbingstate(everystate

∈

B )one anrea hanabsorbingstate(anystate

∈

/

B ). Ifthe hain isinitially in state

i ∈

B , then the mean time spent in state

j ∈

B before absorption is

5

(10)

the

(i, j)

th elementof fundamental matrix

(I − Q)

−1

(Snell, 1959). Thus if

µ

denotesthemeanof

ψ

s

then:

µ = b

(0)

(I − Q)

−1

1 _.

(16)

Here

1

is a olumnve toroflength

B

withea helement

1

.

Alsonoti e from the denitions ofthe meanof

S

and of

ψ

s

that

µ

anbe expressedasfollows:

µ =

∞

X

k=0

W

(k)

= 1 + W

(1)

+ W

(2)

+ . . .

3 The problem of geometri approximation

Let

g

s

be a geometri distribution with parameter

p

µ

= 1/µ

, i.e.

ψ

s

and

g

s

have same mean. Consider the maximum deviation between the umulative distributionfun tions ( dfs)of thesetwodistributions,namely:

δ

∗

_{= max}

s≥1

δ

(s)

_,

(17) with

δ

(s)

_{= | ˆ}

_ψ

s

− ˆ

g

s

|

and

g

ˆ

s

the df of

g

s

,i.e.

g

ˆ

s

= 1 − (1 − p

µ

)

s

,

s = 1, 2, . . .

In probabilitytheory,

δ

∗

is alledtheKolmogorovmetri . As

ψ

s

,

δ

∗

isnotdened for

p = 0

and, for a given basin, depends on

p

and

b

(0)

. It omes from (15) that:

δ

(s)

= |(1 − p

µ

)

s

− W

(s)

|,

s = 1, 2, . . .

(18) Thus

δ

(s)

representstheabsoluteerrorbetween

W

(s)

anditsgeometri approx-imation

(1 − p

µ

)

s

. Forbasinsof size

B = 1

(one xed pointand notransient states),

ψ

s

is given by(6) whi h is geometri . Hen e for su h basins

δ

(s)

_{= 0}

∀s, p

andthus

δ

∗

_{= 0 ∀p}

. Thisisaparti ular aseandinthefollowingweshall studythebehaviourof

δ

∗

as

p

tendsto

0

withoutanyassumptionson

ψ

s

. Theprobabilitytoexit Bduring

(k, k + 1)

is:

p

e

(k, k + 1) = Pr{S = k + 1|S > k} = 1 −

W

(k+1)

W

(k)

,

k = 0, 1, . . .

(19)

Thisprobabilitydependson

k

,

p

and

b

(0)

. Inparti ular,for

B = 1

,

p

e

(k, k + 1)

is onstant and equalto

r

. When

p > 0

, any state

i ∈

B ommuni ates with any state

j /

∈

B, thus

p

e

(k, k + 1) > 0 ∀k = 0, 1, . . .

and therefore probability

W

(k)

isstri tlyde reasing. Inversely,sin e

W

(0)

_{= 1}

,wehave:

W

(s)

₌

s−1

Y

k=0

[1 − p

e

(k, k + 1)],

s = 1, 2, . . .

(20)

(11)

δ

(s)

= |(1 − p

µ

)

s

−

s−1

Y

k=0

[1 − p

e

(k, k + 1)]|,

s = 1, 2, . . .

(21)

Wesee thatif

p

e

(k, k + 1) = p

µ

∀k

then

δ

∗

_{= 0}

. Alsonotethefollowing: 1.

lim

p→0

µ = ∞ ∀b

(0)

_.

(22)

Whatevertheinitial onditions,as

p

be omessmaller,ittakesonaverage moreandmoretimetoleavethebasin. Alsothismeansthat

∀b

(0)

:

p

µ

→ 0

as

p → 0

. 2.

lim

p→0

p

e

(k, k + 1) = 0

∀k, b

(0)

_.

(23)

Whatever

k

andthe initial onditions,the probability toleavethebasin during

(k, k + 1)

goesto

0

as

p → 0

. Indeed,fromequations(13)and(19), it omesthat

p

e

(k, k + 1) =

X

i∈

B

a

i

ˆb

(k)

i

=

X

i∈

B

n

X

x=1

Γ

x

i

p

x

q

n−x

ˆb

(k)

i

,

k = 0, 1, . . . ,

(24) with

Γ

x

i

≥ 0

thenumberofwaysofleavingBbyperturbing

x

bitsofstate

i ∈

B 6 and

ˆb

(k)

i

=

b

(k)

i

W

(k)

,

i.e.

X

i∈

B

ˆb

(k)

i

= 1,

k = 0, 1, . . .

Inordertoshowthat

∀b

(0)

any

δ

(s)

tendsto

0

as

p

tendsto

0

,weintrodu e twopropositions.

Proposition1 ForanybasinofanoisyBooleannetworkwithxed

0 < p < 1

, thefundamental matrix

(I − Q)

−1

has areal simpleeigenvalue

λ

∗

_{> 1}

whi h is greaterin modulusthanany other eigenvaluemodulus, thatis

λ

∗

_{> |λ|}

for any othereigenvalue

λ

of

(I − Q)

−1

. Wehave:

lim

k→∞

p

e

(k, k + 1) =

1 λ

∗

.

6

Thisnumberisequaltothenumberofelementsinrow

i

oftheHammingdistan ematrix

(12)

Foragiven basin,the rate of onvergen edependson

p

and

b

(0)

whilethelimit

1/λ

∗

dependsonly on

p

.

From that proposition,weseethat if

ψ

s

is geometri then

p

µ

= 1/λ

∗

. For example,if

B = 1

then

p

µ

= r = 1/λ

∗

. Demonstration.

Wehavetoshowthat:

lim

k→∞

W

(k+1)

W

(k)

= λ

b

,

with

λ

b

= 1 − 1/λ

∗

arealsimpleeigenvalueof

Q

greaterin modulusthanany othereigenvaluemodulusand

0 < λ

b

< 1

.

Matrix

Q

isnonnegative. Itisirredu ibleandaperiodi thusprimitive. From thePerron-Frobeniustheorem,wehavethat: (1)

Q

hasarealeigenvalue

λ

b

> 0

whi h is greater in modulus than any other eigenvalue modulus. Sin e

Q

is substo hasti , wehave

0 < λ

b

< 1

. (2)

λ

b

isasimplerootofthe hara teristi equationof

Q

. Moreover(DouglasandBrian,1999):

lim

k→∞

Q

k

λ

k

b

= vu

T

,

with

v

and

u

rightandlefteigenve torsasso iatedwith

λ

b

hoseninsu haway that

u > 0

,

v > 0

and

u

T

_{v = 1}

. Thegreater

k

, thebetterthe approximation

Q

k

_{≈ λ}

k

b

vu

T

. Thus:

W

(k+1)

_{= b}

(0)

_Q

k+1

₁

_{≈ b}

(0)

_λ

k+1

b

vu

T

1 ≈ b

(0)

λ

b

Q

k

1 = λ

b

W

(k)

.

Therefore:

W

(k+1)

W

(k)

→ λ

b

as

k → ∞.

Proposition 2

µ

isasymptoti ally equivalentto

λ

∗

:

lim

p→0

µ

λ

∗

= 1.

Sin e

λ

∗

doesnot depend on

b

(0)

, the last statement is equivalent to say thatas

p

tendsto

0

,

µ

islessandlessdependenton

b

(0)

. Inotherwords,from (16),theelementsofve tor

(I − Q)

−1

₁

tendstobeequalas

p → 0

. Demonstration.

If

ψ

s

isgeometri then

µ

doesnotdependon

b

(0)

. Thusfrom(16)it omes that:

(I − Q)

−1

₁

_{= µ1,}

orequivalently

Q1 = (1 −

1

(13)

This means that

(1 − 1/µ)

is an eigenvalue of

Q

with asso iated eigenve tor

1

. Sin e

Q

isnonnegativeprimitive,bythePerron-Frobeniustheorem,

(1 −

1 µ

)

mustequal

λ

b

,whi himplies

µ > 1

. Now

λ

b

= 1 − 1/λ

∗

. Thus

µ = λ

∗

. From (25) we see that the sum of elements in any row of

Q

is a onstant. This is be ausewhen

ψ

s

isgeometri ,theprobabilitytoleavethebasinfromanystate is onstant.

When

Q

isnotgeometri ,thePerron-Frobeniustheoremgives:

Qv = λ

b

v,

0 < λ

b

< 1.

Nowas

p

tendsto

0

,

Q

tendstoasto hasti matrix(be auseve tor

a

tendsto ve tornull). Therefore as

p

tends to

0

,

λ

b

musttend to

1

and

v

must tendto

1

. Thusforsu ientlysmall

p

wemaywrite:

Q1 ≈ λ

b

1 ,

(26)

orequivalently

(I − Q)

−1

1 _{≈ λ}

∗

1 _.

Therefore

µ = b

(0)

_{(I − Q)}

−1

₁

_{≈ b}

(0)

_λ

∗

₁

_{= λ}

∗

_.

Thus as

p → 0

,

µ

willbelessand lessdependentoninitial onditionsand the error in the aboveapproximationwill tend to

0

. Note that sin e

λ

b

→ 1

as

p → 0

wemusthave

µ → ∞

as

p → 0

whi hisresult(22).

Remark 4 Aswillbe dis ussed later, approximation

µ ≈ λ

∗

may be good in some neighborhood ofsome

p

(typi allyintheneighborhoodof

p = 0.5

, seeFig.13in4.3).

From Proposition 2now,

(1 − p

µ

)

s

will tend to

(1 − 1/µ)

s

_{= λ}

s

b

as

p → 0

. Ontheotherhand,from(26),weget

Q

s

1 _{≈ λ}

s

_b

1 _,

andthus:

W

(s)

= b

(s)

1 _{= b}

(0)

_Q

s

1 _{≈ λ}

s

_b

_.

Hen efrom(18)any

δ

(s)

willtendto

0

as

p

tendsto

0

,whatever

b

(0)

. Thusthe maximumwill doso:

lim

p→0

δ

∗

_{= 0}

_∀b

(0)

_.

Sin e the maximum deviation between the two dfs of

ψ

s

and

g

s

vanishes as

p → 0

, these two dfs tend to oin ide asp tends to

0

. Alsothis means that

(1 − p

µ

)

s

isagoodapproximationtowithin

±δ

∗

of

W

(s)

atanyandevery

s

and thatthe error an bemade arbitrarilysmall bytaking

p

su iently loseto

0

(butnotequalto

0

).

(14)

Inaddition,from(22)wemaywriteforsu ientlysmall

p

:

ˆ

g

s

= Pr{S ≤ s} = 1 − e

s ln(1−p

µ

)

≈ 1 − e

−p

µ

s

.

Yetthe dfofanexponentialdistribution withparameter

γ

is:

Pr{T ≤ t} = 1 − e

−γt

.

For

t = s

and

γ = p

µ

,thetwoprobabilitiesareequal. Therefore,as

p → 0

and

∀b

(0)

,the dfof

ψ

s

tendsto oin idewiththe dfofanexponentialdistribution. Sin e

µ

is lessand less dependent on

b

(0)

as

p → 0

(see Proposition 2), one has for su iently small

p

that an exponential distribution an be used to approximatethetimespentinabasinofaNBN.

Remark5 Dependingon

b

(0)

,

ψ

s

maybevery losetoageometri distributionwhile

p

isnot small. See Examples1and2below.

Suppose that for ea h basin of a NBN the sojourn time

S

is su iently memoryless. Then, under someadditional assumptions that will bedis ussed in se tion5,onemaydeneanother dis rete-timehomogeneousMarkov hain

˜

Y

_k

whosestatesarethebasinsofthenetwork. (1)Thisnewsto hasti pro ess is oarserthan

X

k

sin eonlytransitionsbetweenthebasinsofthenetworkare des ribed. (2) The size of its state spa e is in general mu h smaller than

E

, thesize ofthestatespa eoftheoriginalpro ess. (3)Ithastobeviewedasan approximation. Thetildesymbolin

Y

˜

k

isusedtore allthatingeneral,abasin doesnotretaintheMarkovproperty(i.e.,

ψ

s

is notgeometri ).

Example 1 Cal ulation of sojourn time distributions and illustration of the geo-metri approximation problem. Let us take basin B1 of Fig. 1 with

p = 0.02

and

b

(0)

_i

= 1/6 ∀i = 1, 2, . . . , 6

, andlet us ompute

ψ

s

and

g

s

. Thetwo distributionsare plottedinFig.3a. The ir lesstandfortheprobabilities

ψ

s

(

µ = 13.3530

),thepoints forthe probabilities

g

s

(

p

µ

= 1/µ = 0.0749

). Fig. 3bshows

W

(k)

and itsgeometri approximation

(1 − p

µ

)

k

versus time

k

. Maximum deviation

δ

∗

between the two is

2.3007%

.

Table2gives

µ

and

δ

∗

fordierent

p

valuesanddierentinitial onditionsforthe twobasinsofFig. 1 . Ea hof thefour olumnsbelow

µ

and

δ

∗

orrespondsto one

p

value,fromlefttoright:

p = 0.002

,

0.02

,

0.2

and

0.8

. Themeanvaluesofthe olumns arealsogiven(seetherowswithMean). Forthe al ulationof

µ

,we onsideredtwo typesofinitial onditions: (1)thenetworkstartsinonestateofthebasin(su essively ea hstateofthebasinwastakenasinitialstate)and(2)

b

(0)

i

= 1/B ∀i = 1, 2, . . . , B

(seeUnif. intheTable). Weseethatfor therstthreevaluesof

p

themeansojour timesinB1aresmallerthanthoseinB2(seetherstthree olumnsbelow

µ

). ThusB1 islessstablethanB2. Also al ulatedforea hbasinandthefour

p

valuesthevariation oe ientofthesojourntimesobtainedwiththersttypeofinitial onditions. The resultsare( olumns

1

,

2

,

3

and

4

):

0.1876

,

1.7880

,

9.3454

and

15.9039%

forB1;

0.1566

,

1.3030

,

2.5152

and

17.0044%

forB2. Therefore

µ

islessandlesssensitivetotheinitial stateas

p

de reases,whi hisina ordan ewithProposition2.

Anotherimportantobservationinthistableisthat,dependingontheinitial ondi-tions,

ψ

s

maybe losetoageometri distributionwhile

p

isnotsmall(seeinthetable thevaluesof

δ

∗

when

p = 0.2

). Fig. 4showsthefun tions

δ

∗

_(p)

forthetwobasinsof Fig. 1andthe uniforminitial ondition. Weseethat inboth ases

δ

∗

_{(p) → 0}

(15)

p → 0

,

δ

∗

_{(0.5) = 0}

(

ψ

s

isgeometri at

p = 0.5

) 7 and

δ

∗

_(p)

tendstoaglobalmaximum as

p → 1

. ForbasinB1(blue urve),thereisonelo almaximumat

p = 0.112

,while for basinB2 (green urve) thereare two lo al maxima, one at

p = 0.0530

and the otherat

p = 0.3410

,and onelo alminimumat

p = 0.2600

(

0.0677%

). Asmentioned above,fun tion

δ

∗

_(p)

dependsoninitial onditions. Forexample,ifthenetworkstarts instate

0000

,thenB2hasstillonelo alminimumbutthisnowo ursat

p = 0.3640

(

0.0830%

).

Example2 Inthisexample,weillustratethefa tthatwhile

p

isnotsmall,

ψ

s

may be lose toageometri distribution,dependingon

b

(0)

. Wetake basinB1 ofFig. 1 and al ulate

ψ

s

wheninitiallythenetworkis(a)instate

0101

and(b)instate

0110

. Fig. 5showstheresults. As anbeseeninthegure,whenthenetworkstartsin

0110

,

ψ

s

is lose toageometri distributionwhi hisnotthe ase whentheinitialstate is

0101

.

To omparethetwoapproximations,we omputedthevarian es

σ

2

and

σ

2 g

of

ψ

s

and

g

s

respe tivelyandthetotalvariationdistan e

d

T V

between

ψ

s

and

g

s

(another probabilitymetri )whi hisgivenby:

d

T V

=

1

2 X

s≥1

|ψ

s

− g

s

|.

ThevaluesoftheseparametersaregiveninTable 3 . Weseethatthetotal variation distan eisabout

10

timesgreaterwhentheinitialstateis

0101

thanwhenitis

0110

. Themoregeometri

ψ

s

,thesmaller

d

T V

,thebettertheapproximations

σ

2 _{≈ σ}

2 g

and

µ ≈ λ

∗

.

Tillnow,wehaveassumedea hnodeofthenetworkhasthesame probabil-ityofbeingperturbed. InExample

3

below,welookathowthenetworkofFig. 1behaveswhenoneof itsnodes isperturbed withaprobabilitywhi his high ompared to the other nodes. Suppose nodes represent proteins. Within the ellinterior, some proteinsmay be moresubje tto ompeting rea tionsthan others. Theserea tionsmaybeassumedtoa trandomly

8

,eithernegativelyor positively,onthestateofthetargetproteins. Forexample,somerea tionsmay leadto protein unfolding while others, like those involving mole ular haper-ones,may res ue unfoldedproteins(Dobson,2003). Onthe other hand,some proteinsmaybemoresensitivetophysi o- hemi alfa tors,liketemperatureor pH,in reasingtheirprobabilitytobeperturbed.

Example3 Inthisexample,itisassumedthatonenodeisperturbedwitha proba-bilitywhi hishigh omparedtotheothernodes. Thebehaviourofthenetworkwhen

p → 1

is omplexand will notbedis ussed here so we take

p

1 = p

2 = p

3 = 10

−3

(therstthreenodesare rarelyperturbed)and

0 < p

4 ≤ 0.9

. Fig. 6shows

¯

z

for two valuesof

p

4

. For

p

4 = 0.04

(thebluepoints)the networkbehavesasifall thenodes hadthe sameprobability

p = 0.04

ofbeing perturbed(seeFig. 2 , theblue points). Simulationshave shown this to be true whateverthe three rarelyperturbednodes. When

p

4

in reasesin

]0, 0.9]

,thestationarystateprobabilitiesdonottendtobeequal (seeFig. 6 , the ase

p

4 = 0.9

), rather,sometransient states, typi allystate

0011

of B2,tendtobemorepopulatedattheexpenseofattra torstates(seeattra torstates

7

Whatevertheinitial onditions,at

p = 0.5

,

µ = 8/5

forB1and

8/3

forB2. See5.1.2for analyti alexpressionof

µ

when

p = 0.5

.

8

(16)

0

10

20

30

40

50

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08 Sojourn time (s)

P

ro

ba

bi

lit

ie

s

(a)

0

10

20

30

40

50

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Time (k)

W

(k

)

(b)

Figure 3: Sojourn time distribution and geometri approximation for basin B1 of Fig. 1 with

p = 0.02

and

b

(0)

i

= 1/6 ∀i = 1, 2, . . . , 6

. (a) Cir les: sojourn timedistribution

ψ

s

(with mean

µ = 13.3530

); points: approximatinggeometri distribution

g

s

(with parameter

p

µ

= 1/µ = 0.0749

). (b) Cir les:

W

(k)

; points: geometri approximation

(1 − p

µ

)

k

. Maximum deviation

δ

∗

is

2.3007%

.

(17)

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

5

10

15

20

25

30

35 p

δ

∗

(p

)

Figure 4: Maximumdeviation

δ

∗

(in

%

)as a fun tion of

p

for the twobasinsof Fig. 1 andinitial onditions

b

(0)

i

= 1/B ∀i = 1, 2, . . . , B

. Blue urve: basin B1. Green urve: basinB2. One has

δ

∗

_{(0.5) = 0}

forbothbasins(

ψ

s

is geometri at

p = 0.5

).

(18)

Ini. ond.

µ

δ

∗

B1

0001

126.00 13.52 2.37 1.54 0.39 3.44 7.47 10.39

0101

126.00 13.52 2.39 2.07 0.39 3.47 8.49 4.09

0110

125.50 13.02 1.94 2.28 0.00 0.00 0.92 10.66

1001

126.00 13.52 2.39 2.07 0.39 3.47 8.49 4.09

1010

125.50 13.02 1.94 2.28 0.00 0.00 0.92 10.66

1101

126.00 13.52 2.37 1.54 0.39 3.44 7.47 10.39 Mean 125.83 13.35 2.23 1.96 0.26 2.30 5.63 8.38 Unif. 125.83 13.35 2.23 1.96 0.26 2.30 4.59 2.72 B2

0000

252.85 27.68 4.27 3.77 0.39 2.96 2.31 5.40

0010

251.86 26.81 4.08 2.25 0.00 0.11 0.50 20.87

0011

252.36 27.26 4.22 3.77 0.20 1.63 2.59 5.42

0100

251.86 26.80 4.02 3.60 0.00 0.12 1.99 0.93

0111

251.86 26.80 4.02 3.60 0.00 0.12 1.99 0.93

1000

251.86 26.80 4.02 3.60 0.00 0.12 1.99 0.93

1011

251.86 26.80 4.02 3.60 0.00 0.12 1.99 0.93

1100

252.36 27.26 4.22 3.77 0.20 1.63 2.59 5.42

1110

251.86 26.81 4.08 2.25 0.00 0.11 0.50 20.87

1111

252.85 27.68 4.27 3.77 0.39 2.96 2.31 5.40 Mean 252.16 27.07 4.12 3.40 0.12 0.99 1.88 6.71 Unif. 252.16 27.07 4.12 3.40 0.12 0.81 0.24 2.81

Table2: Meansojourntime

µ

andmaximumdeviation

δ

∗

(in

%

)forthetwobasins of Fig. 1. Twotypes of initial onditions have been onsidered: (1) the network startsinonestateofthebasin(su essivelyea hstateofthebasinistakenasinitial state)and(2)ea helementof

b

(0)

isonedividedbythesizeofthebasin(seeUnif. intheTable). The four olumnsbelowaparameter(

µ

or

δ

∗

) orrespondfromleft toright to

p = 0.002

,

0.02

,

0.2

and

0.8

. The mean valueforea h olumn isalso indi ated(seeMean). Attra torstatesareinboldtype.

0101

0110

δ

∗

(%) 8.49 0.92

d

T V

(%) 10.75 1.18

σ

2

2.38 1.91

σ

2 g

3.34 1.82

λ

∗

2.00 2.00

µ

2.39 1.94

(19)

0

2

4

6

8

10

12

14

16

18

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45 s

P

ro

ba

bi

lit

ie

s

(a)

0

2

4

6

8

10

12

14

16

18

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7 s

P

ro

ba

bi

lit

ie

s

(b)

Figure5: Comparisonbetweentwogeometri approximations. Twodierentinitial statesaretakenin basinB1 ofFig. 1with

p = 0.2

(seetable2). Cir les: probabil-ities

ψ

s

. Points: geometri approximation

g

s

. (a)Initial state

0101

.

δ

∗

_{= 8.49%}

,

µ = 2.39

. (b)Initialstate

0110

.

δ

∗

_{= 0.92%}

(20)

0

0.05

0.1

0.15

0.2

0.25

0.3

0.35 0000

0001

0010

0011

0100

0101

0110

0111

1000

1001

1010

1011

1100

1101

1110

1111

Boolean states

S

ta

ti

on

ar

y

pr

ob

ab

ili

ti

es

¯z

i

Figure6: Stationarystate probabilitiesforthenetwork shown inFig. 1and non-equiprobablenodeperturbations. The rstthreenodesarerarely perturbed:

p

1 =

p

2 = p

3 = 10

−3

. Blue points:

p

4 = 0.04

; red points:

p

4 = 0.9

. Attra torstates areinboldtype.

1000

,

1010

,

1011

and

1110

inFig. 6 ) 9

.

Fig. 7 ashows distribution

ψ

s

with approximating distribution

g

s

for basinB2,

p

4 = 0.1

andstate

0011

as initialstate. Maximumdeviation

δ

∗

is

8.2011%

. Notethe splittingof

ψ

s

into twosubdistributions, one for odd frequen ies and the other for evenfrequen ies. Probabilities

W

(k)

andapproximations

(1 − p

µ

)

k

areplottedinFig. 7b.

4 The geometri approximationin

(n, C)

networks In Example 1, we illustrated, using the network of Fig. 1, the fa t that as

p

tendsto

0

,the dfofthetimespentinabasinofattra tionapproa hesthe df of ageometri (resp. exponential) distribution and that theexpe ted sojourn timetends tobeindependentof initial onditions(see thede reasesof

δ

∗

and 9

Atthe ellpopulationlevel,thismeansthata ellpopulationwouldexhibitphenotypes that ouldnothavebeenobserved(ornot easilyobserved) inthe low

p

regime. Notethat thesephenotypesarenotne essarilysurvivableforthe ells.

(21)

0

10

20

30

40

50

60

70

0

0.01

0.02

0.03

0.04

0.05

0.06

0.07

0.08

0.09

0.1 Sojourn time (s)

P

ro

ba

bi

lit

ie

s

(a)

0

10

20

30

40

50

60

70

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Time (k)

W

(k

)

(b)

Figure 7: Sojourntimedistribution andgeometri approximation for basin B2 of Fig.1. Therstthreenodesarerarelyperturbed:

p

1 = p

2 = p

3 = 10

−3

;

p

4 = 0.1

. Initially, the network is in state

0011

. (a) Cir les: distribution

ψ

s

(with mean

µ = 22.9653

); points: approximating geometri distribution

g

s

(with parameter

p

µ

= 0.0435

). Forthesakeof larity,sojourntimes

s > 70

havebeenomitted. (b) Cir les:

W

(k)

;points: geometri approximation

(1 − p

µ

)

k

. Maximumdeviation

δ

∗

(22)

variation oe ient as

p

be omes smaller, respe tively in Table 2 and in the text). As will be seen later, for this network,a two-state dis rete-time (resp. ontinuous-time) homogeneousMarkov hain may be used for approximating basin transitions in the low

p

regime,i.e., a oarse-graineddes riptionof this network existsinthelow

p

regime.

Nowweaddresstheproblem ofgeometri approximationin randomly on-stru ted

(n, C)

networks. Arandom

(n, C)

networkisbuiltbyrandomly hoos-ingforea hnode

C

inputsandoneintera tionfun tion (Kauman,1993).

4.1

(n, C)

networks and onden e intervals

Six

(n, C)

ensembleswereexaminedtaking

n = 6

,

8

or

10

(

E = 64

,

256

or

1024

) and

C = 2

or

5

. Ea hbasinofarandomly onstru tednetwork wasperturbed with

p = 0.002

,

0.01

,

0.02

,

0.1

,

0.3

,

0.5

and

0.8

.

A ordingtothe lassi ationofKauman(1993),

(n, 2)

networksare om-plexwhile

(n, 5)

onesare haoti . Onedieren ebetweenthesetwoensembles of networks,whi h is of parti ular interest here, is that in the former If the stabilityof ea h state y leattra tor isprobed by transient reversing ofthe a -tivity of ea h element in ea h state of the state y le, then for about

80

to

90

per ent ofall su hperturbations,the systemowsba ktothe samestate y le. Thusstate y lesareinherentlystabletomostminimaltransientperturbations. (Kauman, 1993, p. 201). In haoti networks, the stability of attra tors to minimalperturbationsisat best modest(Kauman,1993,p. 198).

For ea h

(n, C)

ensemble, we generated about

2500

basins (whi h orre-sponds to about

700

to

800

networks depending on the ensemble). We es-tablished onden e intervals for three statisti al variables of whi h two are probabilities:

1. Consideran

n

-node network. Forea h of its basins, one an dene two onditional probabilities: (1)the onditional probability

α

of leavingthe basingiventhatoneattra tor bitoutof

nA

hasbeenipped,with

A

the size oftheattra tor,and (2)the onditional probability

β

ofleaving the basingiventhatonebasinbitoutof

nB

hasbeenipped.

In ea h basin sample, a small proportion of basins having

α = 0

were found. The

25

thper entileofthe relativesize

B/E

of

α = 0

basinswas morethan

0.8

for

C = 2

networks and

0.9

for

C = 5

ones. Thus

α = 0

basinsaremostoftenbigbasins. We al ulated

2

ratios: ratio

κ

¯

between the median mean sojourn times of

α = 0

and

α > 0

basins and ratio

κ

∗

betweenthemaximummeansojourntimesofthetwotypesofbasins. TheseratiosaregiveninTable4for

(n, 2)

networksandfour

p

values. In the aseof

α = 0

basins,the omputationof

ψ

s

for

p = 0.01

andstopping riterion

ψ

ˆ

s

> 0.9999

variesbetweenafewminutestoseveraldaysusinga PowerEdge2950serverwith

2

Quad-CoreIntelXeonpro essorsrunningat 3.0GHz. As anbeseenfromthistable,with

p

de reasing,themaximum meansojourntimeof

α = 0

basinsin reasesdrasti ally omparedtothat

(23)

p

0.03 0.05 0.1 0.5

¯

κ

n = 6

30.1612 19.9012 12.1390 6.8534

n = 8

44.3541 26.0081 15.1438 11.9929

n = 10

35.5351 21.6376 14.5090 10.3930

κ

∗

_{n = 6}

122.3414 19.3446 3.6270 1.0000

n = 8

300.3482 32.4353 3.1643 1.0000

n = 10

542.8109 67.3100 4.9887 1.0000

Table 4: Comparison between medianmean sojourn times of

α = 0

and

α > 0

basins(ratio

κ

¯

)andbetweenmaximummeansojourntimesofbothtypesofbasins (ratio

κ

∗

)for

(n, 2)

networksandfour

p

values. Cal ulationof

ψ

s

: attime

0

,the statesof thebasinareequiprobable. The proportionsof

α = 0

basinsfor samples

n = 6

,

8

and

10

werefoundtobe

4.0759

,

3.1989

and

2.5971%

respe tively.

of

α > 0

basins 10

. Theproportion of

α = 0

basins variesin oursamples from

1.7

to

4.1%

dependingontheensemble. Forxed

C

,itisade reasing fun tionof

n

andforxed

n

,itissmallerinthe haoti regimethaninthe omplexone. Although we dobelievethat

α = 0

basins havenot to be reje tedfromabiologi alinterpretationperspe tive,networkswithatleast one

α = 0

basinwereomittedduringthesamplingpro edure(essentially be ause of the high omputational time that is needed to ompute

ψ

s

when

α = 0

and

p

issmall).

Fig. 8 shows the onditional probabilities

α

for sample

(8, 2)

. For an

n

-nodenetwork,one andene

n

mainprobabilitylevels

1, 2, . . . , n

orre-spondingto probabilities

1/n, 2/n, . . . , 1

. The per entage ofdata points lo atedinmainprobabilitylevelsaregiveninTable5forthesixsamples and thetwo onditional probabilities

α

and

β

. It anbe seenfrom this tablethat whateverthe onne tivityandthe onditionalprobability,this per entagede reaseswhen

n

in reases. Foragiven onditionalprobability andany

n

,itisgreaterinthe omplexregimethaninthe haoti one. Thestatisti al analysis of thesix data sampleshas shownthe following. For

C = 2

,there is a learquantizationof

α

andaweakoneof

β

. Also thehistogramsof

α

and

β

are symmetri (skewnessequalto

∼ 0.2

). For

C = 5

,thequantizationof

α

isweakandthereisnoobviousquantization of

β

. Thehistogramsof

α

and

β

arenegativelyskewed(skewnessequalto

∼ −0.9

)andthelasttwomainprobabilitylevels

(n−1)

and

n

arestrongly populated omparedto theotherones.

10

For

α = 0

basins,the onditionalprobability

α

x

ofleavingthebasingiventhat

2 ≤ x ≤ n

bitsofanattra torstatehavebeenperturbedmaybenonnull.However

L(p; n, x) = o(p)

for

2 ≤ x ≤ n

. Ontheotherhand,fortransientstatesandsu ientlylarge

k

onehas

ˆ

b

(k)

i

→

0

(24)

0

500 1000

1500

2000

2500

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1 Basins

α

Figure8: Conditionalprobabilities

α > 0

sortedin as endingorder,sample

(8, 2)

. About

74%

ofthedatapointsarein main probabilitylevels. These orrespondto probabilities

1/8, 2/8, . . . , 1

.

C

n

α

β

2

6

78.6 52.0

8

73.9 45.9

10

70.2 35.3

5

6

56.6 31.5

8

50.8 23.5

10

43.6 15.4

Table 5: Per entages of datapoints lo ated in main probability levels for the six ensembles

(n, C)

andthetwo onditionalprobabilities

α > 0

and

β > 0

.

(25)

α

β

C

n

Mean Median Mean Median

2

6

0.5057 0.5000 0.4817 0.5000

8

0.4523 0.4375 0.4294 0.4144

10

0.4149 0.4000 0.3907 0.4000

5

6

0.6869 0.7917 0.6852 0.7778

8

0.7032 0.8125 0.7038 0.8219

10

0.7050 0.8125 0.7106 0.8375

Table6: Meanandmedianof onditionalprobabilities

α > 0

and

β > 0

forthesix ensembles

(n, C)

.

Table6 givesthe mean and median of probabilities

α

and

β

for thesix ensembles

(n, C)

. Forxed

n

, themeanand median ofboth onditional probabilities arehigher for haoti basins than for omplex ones. Addi-tionally,for

C = 2

basins,themeanandmediande reasewith

n

whilefor

C = 5

,bothin reasewith

n

(ex eptforthemedian of

α

).

2. The third statisti al variable is the mean time spent in a basin,

µ

. For

C = 2

andany

n

,thereisa learquantizationat

p = 0.002

whi hrapidly disappearsas

p

in reases. For

C = 5

andany

n

,thereareonlytwolevels ofquantizationat

p = 0.002

andtheserapidlyvanishas

p

in reases. Also noti e that the minimum of

µ

is equal to the mean spe i path

d = τ

whi h, a ordingto(7), doesnotdependon

C

.

Most ofthe limitsof the

95%

onden e intervalsranged between

1.5

and

2.5%

(for

α

and

β

: onden e intervalforthe mean if

C = 2

, forthe median andforthetrimmedmeanif

C = 5

;for

µ

: onden e intervalforthetrimmed mean)

11 .

4.2 Simulation results and dis ussion Mostofthebasinvariables(su has

µ

or

δ

∗

)werepositivelyskewed. Therefore forea hofthesevariableswe al ulatedquartiles

Q

1

,

Q

2

(themedian)and

Q

3

.

Forthe al ulationof

µ

,twotypesofinitial onditionswere onsidered:

1. Ea helementof

b

(0)

isequalto

1/B

. We allthis onditiontheuniform initial ondition.

11

Twomethodswereused:anonparametri methodbasedonthebinomialdistribution(for the medianonly) andthe bootstrap method (usedforthe medianand the trimmedmean). Forthetrimmedmean,weaveragedthesampledatathatwere(1)betweenthe

25

thand

75

th per entiles,(2)lessthanthe

75

thper entileand(3)lessthanthe

90

thper entile.

(26)

in that state(the elementsof

b

(0)

areall

0

ex ept onewhi his equalto

1

). We allthis onditiontherandominitial ondition.

Let's start with the basin variable

µ

. First the uniform initial ondition. Coe ients of skewness for

µ

distributions mostly ranged from

5

to

15

with meanof

10.0568

(strongskewness). Forexample,forsample

(8, 5)

with

p = 0.01

, weobtaineda oe ient ofskewnessof

10.2415

, ameanof

44.7471

,amedian of

15.7922

, a

75

th per entile of

26.0523

, a

99

th per entile of

608.4743

and a maximumof

3.0117 × 10

3

. A small proportionof the meansojourntimesare thereforeveryfarfrom the median (takenhereasthe entral tenden y). The threequartilesof

µ

versus

ln p

areshowninFig.9forthesixensembles

(n, C)

. The blue squares orrespond to

C = 2

and the red triangles to

C = 5

. The size of a symbol is proportional to

n

. Forthe sakeof larity, the abs issæ of thepoints orresponding toensembles

(6, C)

and

(10, C)

havebeentranslated (respe tivelytotheleftandtotherightofthe

p

values).

It an beseenfromFig. 9that: (1)forxed

p

andany

C

,themedianof

µ

is ade reasingfun tion of the size

n

of the network(not trueat

p = 0.5

and

0.8

),although

α

de reaseswith

n

(seeTable6). (2)Forxed

C

andany

n

,the medianof

µ

isade reasingfun tionof

p

. (3)Forxed

p

andany

n

,median

µ

of haoti basins(

C = 5

)islessthanthatof omplexones(

C = 2

). At

p = 0.002

, themediansof

µ

for

C = 5

ensemblesarerespe tively

1.58

,

1.86

and

2.02

times smallerthan those for

C = 2

ones. Similarbut smallervalueswere foundfor

p = 0.01

and

p = 0.02

. Wetherefore onrmtheresultsofKauman (1993, p. 198,201and488-491)that haoti basinsarelessstabletonodeperturbations than omplexones.

We looked forthe relationshipbetweenthe median of

µ

and

p

. Wefound thatforsu ientlysmall

p

:

Q

(2;µ)

≈

c

2 p

,

(27)

i.e. the median of the mean sojourntime is inversely proportional to

p

. The proportionality onstant

c

2

has been estimated by the least squares method forthesixensembles

(n, C)

and wasfoundto de reasewhen

n

in reases(only therstthree data pointswere tted,i.e. thepoints withabs issa

p = 0.002

,

0.01

and

0.02

). For

C = 2

, we obtained

c

2 = 0.3342

,

0.2871

and

0.2510

; for

C = 5

,

c

2 = 0.2117

,

0.1547

and

0.1242

. Thusforxed

n

, haoti basinsareless sensitivetoavariationin

p

than omplexones. Theresultoftheleastsquares tispresentedinFig. 10. Graph(a)showsthehyperboli relationshipbetween themedian of

µ

and

p

forthesixensembles

(n, C)

. Forthesakeof larity, the fun tions were drawnup to

p = 0.05

. Whenalogarithmi s ale forea h axis is used, oneobtains the graph(b) whi h showsfor the three ensembles

(n, 2)

a linear relationship between

ln Q

(2;µ)

and

ln p

at low

p

(up to

p = 0.02

on thegraph)

12

. Thesameruleappliesto thethreeensembles

(n, 5)

. Wewillsee furtherhowtoexpress

c

2

infun tion of

n

andmedian

α

probability.

12

(27)

0.002

0.01

0.02

0.1

0.3 0.5 0.8

0

50

100

150

200

250

300 p

Q

ua

rt

ile

s

of

µ

Figure9: Medianandinterquartilerangeofmeansojourntime

µ

versus

ln p

forthe sixensembles

(n, C)

. Cal ulationof

ψ

s

withtheuniforminitial ondition. Quartiles

Q

1

and

Q

3

areindi atedbyhorizontalbars. Bluesquares: omplexregime(

C = 2

); redtriangles: haoti regime(

C = 5

). Thesize ofasymbol(squareortriangle)is proportionalto

n

.

(28)

0

0.01

0.02

0.03

0.04

0.05

0

20

40

60

80

100

120

140

160

180 p

M

ed

ia

n

µ

(a)

0.002

0.01

0.02

0.1

0.3 0.5 0.8

1

5

10

50

150 p

M

ed

ia

n

µ

(b)

Figure10: Fitof(27)bytheleastsquareste hnique. (a)Linears aleforbothaxis, tsfor thesixensembles

(n, C)

. Forthesakeof larity, thepointswith

p > 0.05

have been omitted. Blue squares:

C = 2

; red triangles:

C = 5

. The size of a symbol is proportional to

n

. (b) Logarithmi s ale for both axis (ln-ln), ts for ensembles

(n, 2)

only. Thesizeofasymbolisproportionalto

n

.

(29)

With therandominitial ondition,thequartilesof

µ

arevery losetothose obtainedwiththeuniforminitial ondition.

We then al ulatedfor ea h basin the relative errorbetween themean so-journtimeobtainedfrom theuniform initial onditionandthatobtainedfrom therandominitial ondition. This erroris nullat

p = 0.5

(as forthenetwork ofFig. 1,thisisbe ause

ψ

s

isgeometri at

p = 0.5

). Thequartilesoftheerror tendto

0

as

p → 0

andtheyrea hamaximumat

p = 0.1

whateverthenetwork ensemble. Thusthesmaller

p

,themore

µ

isindependentofinitial onditions.

To end, we turn to the maximum deviation

δ

∗

. First the uniform initial ondition. Theresultsforthesix

(n, C)

ensemblesarepresentedinFig. 11. At

p = 0.002

andforxed

C

,thequartilesin reaselinearlywith

n

,ex epttherst quartileof

C = 2

basinswhi his onstantandapproximatelyequalto

0.0005%

(whatever

n

,

25%

of

C = 2

basins have a sojourn time that is geometri or loselyfollowsageometri distribution).

Forboth onne tivities,thefun tions

Q

(2;δ

∗

)

(p)

and

Q

(3;δ

∗

)

(p)

have similar-itieswiththefun tions

δ

∗

_(p)

ofFig. 4: theytend to

0

when

p → 0

,theyhave atleastonelo almaximumin

0 < p < 0.5

andarenullat

p = 0.5

.

With therandom initial ondition, these ond andthird quartiles of

δ

∗

in- reaseformostofthesixensembles

(n, C)

omparedtotheuniform ase. Qual-itatively, thebehaviourof thequartiles with respe tto

p

is similar tothe one foundwiththeuniforminitial ondition.

Tosummarizese tions3and4,wehavethefollowingproposition:

Proposition 3 Consider abasinof anoisy Boolean network. As

p

tends to

0

then:

1. whateverthe initial onditions,

µ → ∞

.

2.

µ

tends to be independent of the initial onditions:

µ → λ

∗

with

λ

∗

_{> 1}

the Perron-Frobeniuseigenvalue offundamentalmatrix

(I − Q)

−1

. 3. whatever the initial onditions,

δ

∗

_{→ 0}

,where

δ

∗

is the Kolmogorov dis-tan e between the umulative distributionfun tion of the sojourn time

S

inthebasinandthatofageometri distribution

g

s

havingthesamemean as

S

.

4.

g

s

onvergestoan exponentialdistribution.

Proposition 3 doesnot guaranteethe existen e ofa oarser representation ofaNBNin thelow

p

regime. What anbesaidfrom thispropositionis that if su h a representation exists, then it is in general an approximation of the originalpro essandit analwaysbeexpressedinadis reteor ontinuoustime framework. Thusweareleadtothepropositionbelowthatwillbedis ussedin moredetailsin thenextse tion:

thelinearizedproblem:

Y

2 = a

2 −

X

where

Y

2 = ln Q

(2;µ)

,

a

2 = ln c

2

and

X = ln p

.Onlythe rstthreedatapointsweretted.

(30)

0.002

0.01

0.02

0.1

0.3 0.5 0.8

0

0.5

1

1.5

2

2.5 p

Q

ua

rt

ile

s

of

δ

∗

Figure11: Geometri approximationfor

(n, C)

ensembles. Medianandinterquartile rangeof

δ

∗

(in

%

)versus

ln p

. Cal ulationof

ψ

s

withtheuniforminitial ondition. Bluesquares:

C = 2

; red triangles:

C = 5

. As

p → 0

, the threequartilesof

δ

∗

(31)

oarser time-homogeneous representation of this network in the low

p

regime. Then the dynami s between the basins of the network may be approximated by a system of linear ordinary dierential equations with size being equal to the numberofattra torsof the network.

Re all that Proposition 3 does not mean that