dynamics and coding of a biologically-motivated … · carlos aguirre computer engineering...

International Journal of Bifurcation and Chaos, Vol. 16, No. 2 (2006) 383–394c© World Scientific Publishing Company

DYNAMICS AND CODING OFA BIOLOGICALLY-MOTIVATED NETWORK

CARLOS AGUIRREComputer Engineering Department,Univ. Autonoma de Madrid, Spain

JOAO MARTINSLabSEI, EST, IPS, Estefanilha 2914-508 Setubal, Portugal

R. VILELA MENDES∗CMAF, Complexo Interdisciplinar UL,

Av. Gama Pinto 2, 1649-003 Lisboa, [email protected]

Received August 2, 2004; Revised January 31, 2005

A four-node network consisting of a negative circuit controlling a positive one is studied. Itmodels some of the features of the p53 gene network. Using piecewise linear dynamics withthresholds, the allowed dynamical classes are fully characterized and coded. The biologicallyrelevant situations are identified and conclusions drawn concerning the effectiveness of the p53network as a tumor inhibitor mechanism.

Keywords : Biological networks; p53.

1. Introduction

In the modeling of biological processes, dynamicalnetworks play an important role. Metabolic pro-cesses of living beings are networks, the nodes beingthe substrates, which are linked together wheneverthey participate in the same biochemical reaction.Protein–protein as well as gene expression and reg-ulation are also dynamical networks.

A central role in the analysis of biologicalnetworks is played by the oriented circuits alsocalled feedback circuits or feedback loops [Snoussi &Thomas, 1993; Thomas et al., 1995]. Feedback cir-cuits are positive or negative according to whetherthey have an even or odd number of negative inter-actions. The general understanding is that positivecircuits generate multistability and negative circuitsgenerate homeostasis, that is, the variables in the

loop tend to middle range values, with or withoutoscillations [Thomas, 1981; Snoussi, 1998; Gouze,1998; Cinquin & Demongeot, 2002; Soule, 2003].

An interesting situation arises when positiveand negative circuits interact and control each other[Kaufman et al., 1985; Kaufman & Thomas, 2001a,2001b]. Here, one such network is studied. Besidesits interest as a dynamical system, it also tries tocapture some of the features of the p53 network.

The p53 gene was one of the first tumor-suppressor genes to be identified, its protein act-ing as an inhibitor of uncontrolled cell growth.The p53 protein has been found not to be act-ing properly in most human cancers, due either tomutations in the gene or inactivation by viral pro-teins or inhibiting interactions with other cell prod-ucts. (There are however a number of cases where

∗Author for correspondence.

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384 C. Aguirre et al.

apparently normal p53 does not achieve tumor con-trol.) Apparently not required for normal growthand development, p53 is critical in the preventionof tumor development, contributing to DNA repair,inhibiting angiogenesis and growth of abnormal orstressed cells [May & May, 1999; Vogelstein et al.,2000; Woods & Vousden, 2001; Taylor et al., 1999;Vousden, 2000]. In addition to its beneficial anti-cancer activities it may also have some detrimentaleffects in human aging [Sharpless & DePinho, 2002].

The p53 gene does not act by itself, but througha very complex network of interactions [Kohn,1999]. The simplified network that is discussed inthis paper, although not being accurate in biologi-cal detail, tends to capture the essential features ofthe p53 network as it is known today. In particu-lar, several different products and biological mech-anisms are lumped together into a single node whentheir function is identical. The network is depictedin Fig. 1. The arrows and signs denote the excita-tory or inhibitory action of each node on the othersand the letters b, c, p,m, g denote their intensities(or concentrations).

The p53 protein (p) is assumed to be producedat a fixed rate and to be degraded after ubiqui-tin labeling. The MDM2 protein being one of themain enzymes involved in ubiquitin labeling,1 theinhibitory node is denoted (m). There is a posi-tive feedback link from p53 to MDM2, because thep53 protein, binding to the regulatory region of theMDM2 gene, stimulates the transcription of thisgene into mRNA.

Under normal circumstances the network is“off” or operates at a low level. The main activation

g

+

+

+

− −

b c

mp+

g−

−Fig. 1. A network with a controlled negative circuit control-ling a positive one: a simplified p53 network.

pathways are the detection of cell anomalies, likeDNA damage, or abnormal growth signals, such asthose resulting from the expression of several onco-genes. They inhibit the degradation of the p53 pro-tein, which may then reach a high level. There areseveral distinct activation pathways. For example,in some cases phosphorylation of the p53 proteinblocks its interaction with MDM2 and in others itis a protein that binds to MDM2 and inhibits itsaction. However, the end result being a decrease inthe MDM2 efficiency, they may both be described asan inhibitory input control (g) to the MDM2 node.

The p53 protein controls cell growth and prolif-eration (c), either by blocking the cell division cycle,or activating apoptosis or inhibiting the blood-vessel formation (b).

Blood-vessel formation (b) is stimulated by sev-eral tumors and is essential for anomalous cellgrowth (c). That is, c and b form a positive circuit.Expression of oncogenes or abnormalities is codedas an external control (g) that operates both on thep–m negative circuit and the c–b positive circuit.These two (controlled) circuits will first be studiedseparately and later their interaction is taken intoaccount.

The dynamical evolution of the network is mod-eled by a piecewise linear discrete-time system sim-ilar to the one used in [Volchenkov & Lima, 2003]for other gene expression networks, namely

p(t + 1) = app(t) + WpmH(Tm − m(t))m(t + 1) = amm(t) + WmpH(p(t) − Tp)

+ WmgH(Tg − g)c(t + 1) = acc(t) + WcbH(b(t) − Tb)

+ WcpH(Tp − p(t)) + WcgH(g − Tg)b(t + 1) = abb(t) + WbcH(c(t) − Tc)

+ WbpH(Tp − p(t))

(1)

H is the Heaviside function, Ti’s are the thresholdsand in all cases

ai +∑

k

Wik = 1 (2)

Wik > 0. This condition insures that all intensitiesremain in interval [0, 1].

A similar network was studied in [VilelaMendes, 2005] modeled by a differential equa-tion system. Both differential and discrete-timesystems with thresholds, as opposed to purely logi-cal schemes, automatically include in the dynamics

1MDM2 binding to p53 can also inhibit p53’s transcription.

Dynamics and Coding of a Biologically-Motivated Network 385

the action delays characteristic of biological pro-cesses. The actual biological process is, of course,a continuous time process. In this sense our modelrepresents a stroboscopic image of the physicalprocess. The main justification to use a piece-wise linear discrete-time model is that it allows fora complete mathematical characterization of thedynamical solutions and how they depend on theparameter values.

The study that we perform analyses the twobasic network modules, namely the positive andnegative circuits. This type of modular analysis hasbeen proposed before [Thieffry & Romero, 1999] asa tool to study biological networks. However this isan adequate methodology only if, in each case, theeffect of the other modules is taken into accountas an external control. This changes the usual con-ditions for multistationarity and stable periodicity.We will follow this extended modular approach, byconsidering the p–m circuit as controlled by the gsignal and the c–b circuit as doubly controlled by gand p.

A characterization is obtained of the dynamicalregimes of the controlled p–m and c–b circuits andof the coupled system in biologically relevant condi-tions. The main biological significance of the resultsmight be the identification of the dynamical roleof the thresholds as a guide for therapeutic actionand a possible dynamical reason for the eventualfailure of genetically normal p53 (see Remarks andConclusions).

2. The p–m System: A ControlledNegative Circuit

p(t + 1) = app(t) + WpmH(Tm − m(t))m(t + 1) = amm(t) + WmpH(p(t) − Tp)

+ WmgH(Tg − g)(3)

The thresholds Tp and Tm divide the unitsquare [0, 1]2 into four distinct regions with differ-ent dynamical laws. This is illustrated in Fig. 3(for the case g > Tg), where close to the cornerof each region we have drawn the next-time map

g

+

p m

−

−

Fig. 2. The p–m system. A controlled negative circuit.

corresponding to the evolution of p and m. Becauseai’s are smaller than one, the dynamics tends todrive the system to a fixed point. However if thefixed point of the dynamics is outside that region,the dynamical law will change whenever the systemmoves to another region. Therefore, the system maynever reach a fixed point and evolve forever. Thisis illustrated in the Fig. 4, where the regions andthe corresponding fixed points are color coded forthe two cases (g = 0 or g = 1). Notice also that, forfuture reference, the four regions are coded as 0, 1,2 and 3.

Define the quantities

fmp =Wmp

1 − am

fmg =Wmg

1 − am

(4)

m p

m p

m p

m p

Tp

Tm

0 2

1

g>Tg

Wmp

3

Fig. 3. The p–m system for g > Tg . The dynamical lawsoperating in each region are depicted near their corners.

Fig. 4. Color coded dynamical regions and correspondingfixed points for the p–m system.


Notice that from (2) it follows

fmp + fmg = 1 (5)

Examining Fig. 4 and considering all qualitatively different positions of the thresholds one obtains acomplete classification of all possible dynamical behaviors of this system, which are listed in the followingtable:

Table 1. Asymptotic behavior and coding of the controlled p–msystem.

g Asympt. behavior Coding

Tm < fmgTm > fmp

g < Tg

———

g > Tg

p → 0, m → fmg

————————

p → 1, m → fmp

0 → 3 → 1↑2

—————1 → 0 → 2

↑3

—————————————————————————————

Tm < fmgTm < fmp

g < Tg

———

g > Tg

p → 0, m → fmg

————————

oscillation

0 → 3 → 1↑2

—————0 → 2↑1←

↓3

—————————————————————————————

Tm > fmgTm > fmp

g < Tg

———

g > Tg

oscillation

————————

p → 1, m → fmp

0 → 2↑1←

↓3

—————1 → 0 → 2

↑3

—————————————————————————————

Tm > fmgTm < fmp

g < Tg

———

g > Tg

oscillation

————————

oscillation

0 → 2↑1←

↓3

—————0 → 2↑1←

↓3

(6)

The period of the oscillations will depend onthe specific values of the parameters, but the qual-itative nature of the dynamics only depends onthe relative positions of the thresholds and thequantities fmg and fmp. The coding representsthe transitions between the regions defined bythe thresholds. Depending on the numerical val-ues of the parameters, the system may stay fora number of time steps in each region but thenit is forced to follow the arrows in the codingscheme.

Of the above four working regimens, the firsttwo (Tm < fmg) are the biologically relevantones, because in the absence of oncogenes (g),normal p53 is expected to be at a low level.The other two regimens correspond to biologically

detrimental action of this gene, provoking prema-ture aging.

Oscillations in the p53-MDM2 in response to astress condition had been considered before [Bar-Oret al., 2000; Brewer, 2002]. Our conclusion is that, inthe biological normal region (Tm < fmg), a steadyresponse is also possible if Tm > fmp.

3. The c–b System: A DoublyControlled Positive Circuit

b(t + 1) = abb(t) + WbcH(c(t) − Tc)+ WbpH(Tp − p(t))

c(t + 1) = acc(t) + WcbH(b(t) − Tb)+ WcpH(Tp − p(t)) + WcgH(g − Tg)

(7)


Here we consider this system as a positive cir-cuit, with dynamical variables b and c and two con-trols p and g. The relevant quantities are

fcp =Wcp

1 − ac

fcb =Wcb

1 − ac(8)

fcg =Wcg

1 − ac

with

fcp + fcb + fcg = 1 (9)

and

fbc =Wbc

1 − ab

fbp =Wbp

1 − ab

(10)

with

fbc + fbp = 1 (11)

As before, the dynamics is characterized by ana-lyzing the position of the fixed points for the fourpossible situations (p ≶ Tp, g ≶ Tg). This is illus-trated in Fig. 6, for particular values of the thresh-olds. The regions defined by the thresholds are, inthis case, coded as a, b, c, d. We recall that thecoding, listed in the tables, represents the requireddynamical transitions between the regions. Depend-ing on the numerical values of the parameters, thesystem may remain in each region a certain numberof time steps before each transition. By adjustingthe parameters we may therefore model differentaction delays in the network. The most importantparameters controlling the delays are the dissipa-tion parameters ai.

Notice that the oscillations coded as b � c canonly be realized if the dynamics jumps directly

g

p

+

+

+

− −

b c

Fig. 5. The c–b system: a doubly controlled positive circuit.

Fig. 6. Color coded dynamical regions and correspondingfixed points for the c–b system.

between these two regions. Otherwise, if in thecourse of the dynamical evolution, the systemcrosses the regions a or d, it will be attracted totheir fixed points. Therefore this oscillation mightonly be implemented for exceptional parameter val-ues. The emergence of these oscillations resultsfrom the simultaneous updating of the variablesat discrete times. In the corresponding continuous-time system, of which the discrete-time one isa stroboscopic image, only the two fixed pointsare attractors because the oscillations are unstablesolutions.

4. Biological Implications andthe Coupled System

The analysis performed in Secs. 2 and 3 is quite gen-eral and applies to any parameter region. However,not all parameter values will model biologically nor-mal conditions. For example, in the absence of cellabnormalities or oncogenes (g < Tg), normal p53will be at a low level. Therefore, as follows fromTable 1,

Tm < fmg (12)

On the other hand, in the absence of oncogenes andlow p53, that is, under nonpathological conditions,cell growth should not be explosive. Therefore thiscondition requires for the model

Tc > fcp + fcb

Tb > fbp(13)

as follows from Table 2(a).With the above (normal condition) require-

ments we are now ready to analyze the behavior forthe other regimens. First, it follows from (13) that


Tc > fcb also. Therefore one sees, from Table 2(b)that if g < Tg and p > Tp, then b → 0 and c → 0.That is, in the absence of oncogenes, expression ofp53 completely blocks cell growth.

On the other hand, one notices, from Table2(c), that in all cases, when g > Tg (presenceof oncogenes) and p < Tp, there is explosive cellgrowth (c → 1). Only in the special case Tc >fcg + fcp and Tb > fbp there are other solu-tions, but even in this case c → 1 is a possiblesolution.

Finally, the most interesting situation is to ana-lyze the effect of expressed p53 (p > Tp) in the pres-ence of oncogenes (g > Tg) — Table 2(d):

• For Tc > fcg + fcb (and any Tb) p53 effectivelycontrols cell growth, keeping it at a low level(c → fcg). In this network model, a large valueof Tc is related to the effectiveness of cell growthto stimulate the required increased level of bloodsupply. The larger Tc the less effective is the stim-ulation signal.

Table 2(a). Case g < Tg and p < Tp.

g < Tg , p < Tp Asympt. behavior Coding

Tc < fcpTb < fbp

b → 1c → fcp + fcb

a → d ← b↑c

—————————————————————————————

Tc < fcpTb > fbp


a → b → d↑c

—————————————————————————————

fcp + fcb > Tc > fcpTb < fbp


a → c → d↑b

—————————————————————————————

fcp + fcb > Tc > fcpTb > fbp

b → 1, c → fcp + fcbor

b → fbp, c → fcpor

oscillation

d �

a �

b � c—————————————————————————————

Tc > fcp + fcbTb < fbp

b → fbpc → fcp + fcb

a → c ← b↑d

—————————————————————————————

Tc > fcp + fcbTb > fbp

b → fbpc → fcp

b → c → a↑d

Table 2(b). Case g < Tg and p > Tp.

g < Tg, p > Tp Asympt. behavior Coding

Tc < fcbTb < fbc

b → 0, c → 0or

b → fbc, c → fcbor

oscillation

a �

d �

b � c——————————————————————————

Tc < fcbTb > fbc

b → 0c → 0

c → b → a↑d

——————————————————————————

Tc > fcbTb < fbc

b → 0c → 0

b → c → a↑d

——————————————————————————

Tc > fcbTb > fbc

b → 0c → 0

c → a ← b↑d


Table 2(c). Case g > Tg and p < Tp.

g > Tg , p < Tp Asympt. behavior Coding

Tc < fcg + fcpTb < fbp

b → 1c → 1

c → d ← b↑a

——————————————————————————

Tc < fcg + fcpTb > fbp

b → 1c → 1

a → b → d↑c

——————————————————————————

Tc > fcg + fcpTb < fbp

b → 1c → 1

a → c → d↑b

——————————————————————————

Tc > fcg + fcpTb > fbp

b → 1, c → 1or

b → fbp, c → fcg + fcpor

oscillation

d �

a �

b � c

Table 2(d). Case g > Tg and p > Tp.

g > Tg , p > Tp Asympt. behavior Coding

Tc < fcgTb < fbc

b → fbcc → fcg + fcb

a → b → d↑c

————————————————————————————

Tc < fcgTb > fbc

b → fbcc → fcg

a → b ← d↑c

————————————————————————————

fcg + fcb > Tc > fcgTb < fbc

b → fbc, c → fcg + fcbor

b → 0, c → fcgor

oscillation

d �

a �

b � c————————————————————————————

fcg + fcb > Tc > fcgTb > fbc

b → 0c → fcg

c → b → a↑d

————————————————————————————

Tc > fcg + fcbTb < fbc

b → 0c → fcg

b → c → a↑d

————————————————————————————

Tc > fcg + fcbTb > fbc

b → 0c → fcg

c → a ← b↑d

• For Tc values smaller than fcg + fcb the inhibitoryeffect of p53 may not be so effective. For example,for fcg + fcb > Tc > fcg and Tb < fbc there areseveral solutions and the outcome will depend onthe initial conditions. For Tc < fcg and Tb < fbcno control is possible.

For parameter values of Tm and Tp for whichthe p–m system is driven towards a fixed point,the behavior of the coupled system may be readdirectly from the c–b tables of the previous sec-tion and the conclusions are as listed above. For

oscillatory conditions on the p–m system, thebehavior is more complex and several different reg-imens may be accessed. Here we will illustrate, bydirect numerical simulation, the behavior of thecoupled system, for regions of parameter valueswhich, as discussed above, are biologically relevant(Tm < fmg, Tc > fcp + fcb and Tb > fbp).

Figures 7 and 8 show the dynamical evolutionof the variables p, m, b and c for g > Tg and theparameter values

fmg = 0.55, fbp = 0.4, fcb = 0.35, fcp = 0.2


5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

p

5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

m

5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

b

5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

c

Fig. 7. Dynamical evolution of p, m, b, and c for fmg = 0.55, fbp = 0.4, fcb = 0.35, fcp = 0.2, ai = 0.8, Tp = 0.5,Tm = 0.5, Tc = 0.6, Tb = 0.5, g > Tg.

5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

p

5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

m

5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

b

5 10 15 20 250

0.2

0.4

0.6

0.8

1

t

c

Fig. 8. Same as Fig. 7 with different initial conditions.


10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

p

10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

m

10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

b

10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

c

Fig. 9. Same as Fig. 7 with Tm = 0.4 instead of Tm = 0.5.

10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

p

10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

m

10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

b

10 20 30 40 500

0.2

0.4

0.6

0.8

1

t

c

Fig. 10. Same as Fig. 9 with different initial conditions.


and

Tp = 0.5, Tm = 0.5, Tc = 0.6, Tb = 0.5

This corresponds to the situation

Tm < fmg, Tm > fmp

in the p–m system and

fcg + fcb > Tc > fcg, Tb < fbc

in the c–b system.The parameter values are the same for both fig-

ures. One sees that depending on the initial condi-tions, in one case the p53 action keeps c at a lowlevel and in the other it does not.

Figures 9 and 10 use the same parameters forthe c–b system, the only change being the valueof Tm that is now 0.4, that is, Tm < fmg andTm < fmp. In this case the p–m system oscillatesinstead of converging to a fixed point, but the actionon the c–b system is similar and it again dependson the initial conditions.

5. Remarks and Conclusions

(i) A simplified four-node model displays bio-logically reasonable features which might, atleast, serve as a toy model for the p53network. The model allows for a completemathematical characterization of its dynam-ical solutions and how they depend on theparameter values (thresholds and couplings).In biology, to change the threshold positionscorresponds to stimulate or inhibit the mech-anisms of gene expression. Therefore thedynamical identification of their role on thedynamics of the network might be a guide fortherapeutic action.

(ii) For very large values of the Tc threshold, ourmodel p53 (p) achieves control of cell growthin the presence of oncogenes. However, as illus-trated before, there is a biologically reasonableregion of parameters for which the success-ful action of p, as an inhibitor of cell growth,strongly depends on the initial conditions. Inpractice it means that it is only effective if thep level is already large enough before the pos-itive feedback effect of c on b starts to play arole.

It is known that in about half of humancancers p53 action is lost through mutationin the p53 gene. In many of the remaining

tumors, the p53 gene is intact but it doesnot achieve the proper response [Woods &Vousden, 2001]. Several mechanisms have beenproposed to explain the failure of geneti-cally normal p53. They include abnormalconformation, targeting for degradation byviral proteins, defective localization to thenucleus, amplification of MDM2, loss of abil-ity to inhibit MDM2 through proteins suchas p14ARF or loss of kinases that phosphory-late MDM2 and p53. What our simple modelsuggests is that there is a range of biolog-ical parameters for which success or failureof the p53 action is very much a questionof chance, that is, it depends on the initialconditions of the process. This would be apurely dynamic explanation for the failure ofnormal p53.

(iii) The actual biological p53 network is highlycomplex and contains very many nodes andpathways [Kohn, 1999]. Our toy model has con-centrated on extracting the abstract functionalactions rather than reproducing the biologi-cal detail. However, there are still many otherdynamical mechanisms in the organic controlof DNA abnormalities or pathological cell pro-liferation. A number of other genes either playa similar role or interact with p53. On theother hand some of the “defense” mechanismsof cancer cells are directly related to the oper-ation of the basic network. For example, oxy-gen starving (hypoxia) of tumor cells by lackof blood supply (the b node intensity) activatesthe CXCR4 gene and this activation causes thetumor cells to migrate to other organs (metas-tasis) [Staller et al., 2003]. Inclusion of this andother qualitatively different effects seems aninteresting possibility.

(iv) Sections 2 and 3 contain a complete classi-fication of the fixed versus periodic natureof the orbits. However, as mentioned before,the actual period of the oscillations dependsstrongly on the values of the parameters. InFig. 11 we illustrate this dependence by colorcoding the logarithm of the period s of theorbits, in the negative circuit, as a functionof the parameters am and ap. The parame-ters are chosen such that Wpm = 1 − am,Wmp = 0.5(1 − am) and Wmg = 0.5(1 − am).These conditions satisfy the oscillation condi-tion and maintain normalization.


Fig. 11. Color-coded log-period of the closed orbits in the negative circuit. g > Tg, Tp = 0.5, Tm = 0.4, fmg = 0.5.

Fig. 12. Color-coded rotation number of the periodic orbits in the negative circuit. The parameters are the same as in Fig. 11.


To visualize the orbits corresponding toany parameter value a tool for Windows isavailable for download at http://www.ii.uam.es/aguirre/p53 emu.zip.

The behavior of the periods is rather discontin-uous, switching from small to large values for smallparameter changes. This occurs, in particular, whensome points of the orbit lie close to the thresholds.

In Fig. 12 the rotation number i (the num-ber of cycles divided by the period of the orbit,is depicted. This quantity displays a more conti-nous behavior that the period of the orbits. High-est values of i correspond to high values of both am

and ap. However, there are still some discontinuitiesdue mainly to orbits close to the intersection of thethresholds.

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