Chapter 1 Structural analysis of biological networks · 1 Structural analysis of biological...

Chapter 1

Structural analysis of biologicalnetworks

Franco Blanchini and Elisa Franco

Abstract We introduce the idea of structural analysis of biological networkmodels In general mathematical representations of molecular systems areaffected by parametric uncertainty experimental validation of models is al-ways affected by errors and intrinsic variability of biological samples Thususing uncertain models for predictions is a delicate task However given aplausible representation of a system it is often possible to reach general ana-lytical conclusions on the systemrsquos admissible dynamic behaviors regardlessof specific parameter values in other words we say that certain behaviorsare structural for a given model Here we describe a parameter-free qualita-tive modeling framework and we focus on several case studies showing howmany paradigmatic behaviors such as multistationarity or oscillations canhave a structural nature We highlight that classical control theory methodsare extremely helpful in investigating structural properties

11 Introduction

Structural analysis of a dynamical system aims at revealing behavioral pat-terns that occur regardless of the adopted parameters or at least for wideparameters ranges Due to their intrinsic parametric variability biologicalmodels are often subject to structural analysis which can be a very usefultool to reveal or rule out potential dynamic behaviors

Even for very simple networks simulations are the most common approachto structural investigation For instance three-node enzymatic networks areconsidered in [1] where numerical analysis shows that adaptability is mostlydetermined by interconnection topology rather than specific reaction param-

Franco BlanchiniDipartimento di Matematica ed Informatica Universita degli Studi di Udine Via delleScienze 206 33100 Udine Italy e-mail blanchiniuniudit

Elisa Franco

Department of Mechanical Engineering University of California at Riverside 900 Univer-

sity Avenue Riverside CA 92521 e-mail efrancoengrucredu

1

2 Franco Blanchini and Elisa Franco

eters In [2] through numerical exploration of the Jacobian eigenvalues fortwo three and four node gene networks the authors isolate a series of inter-connections which are stable robustly with respect to the specific parame-ters the isolated structures also turn out to be the most frequent topologiesin existing biological networks databases For other examples of numericalrobustness analysis see for instance [3 4 5 6 7 8]

Analytical approaches to the study of robustness have been proposed inspecific contexts A series of recent papers [9 10] focused on inputoutputrobustness of ODE models for phosphorylation cascades In particular thetheory of chemical reaction networks is used in [10] as a powerful tool todemonstrate the property of absolute concentration robustness Indeed thesondashcalled deficiency theorems are to date some of the most general results toestablish robust stability of a chemical reaction network [11] Monotonicityis also a structural property often useful to demonstrate certain dynamicbehaviors in biological models by imposing general interaction conditions [1213] Robustness has also been investigated in the context of compartmentalmodels common in biology and biochemistry [14] A survey on the problemof structural stability is proposed in [15]

Here we review and expand on the framework we proposed in [16] wherewe suggest a variety of tools for investigation of robust stability includingLyapunov setndashinvariance methods and conditions on the network graphWe will assume that certain standard properties or assumptions are verifiedby our model for example positivity monotonicity of key interactions andboundedness Based on such general assumptions we will show hows dynamicbehaviors can be structurally proved or ruled out for a range of examples Ourapproach does not require numerical simulation efforts and we believe thatour techniques are instrumental for biological robustness analysis [17][18]

The chapter begins with a motivating example and a brief summary of theanalysis framework in [16] Then we consider a certain number of ldquoparadig-matic behaviorsrdquo encountered in biochemical systems including multista-tionarity oscillations and adaptation through simple examples we showhow these behaviors can be deduced analytically without resorting to simu-lation As relevant case studies we consider a simplified model of the MAPKpathway and the lac Operon Finally we prove some general results on struc-tural stability and boundedness for qualitative models that satisfy certaingraphical conditions

111 Motivating example a qualitative model for transcriptionalrepression

Consider a molecular system where a protein x1 is translated at a certainsteady rate and represses the production of an RNA species x2 In turn x2is the binding target of another RNA species u2 (x2 and u2 bind and forman inactive complex to be degraded) unbound x2 is translated into proteinx3 A standard parametric model is for example in Equations (11) [19]

1 Structural analysis of biological networks 3

x1 = k1u1 minus k2x1

x2 = k31

Kn1 + xn1

minus k4x2 minus k5x2u2

x3 = k6x2 minus k7x3

(11)

One might ask what kind of dynamic behaviors can be expected by this sys-tem Since we cannot analytically solve these ODEs numerical simulationswould provide us with answers that depend on the parameters we believeare the most accurate in representing the physical system Parameters mighthave been derived by fitting noisy data so they are uncertain in practicallyall cases The purpose of this chapter is to highlight how we can achieve im-portant conclusions on the potential dynamic behavior of a molecular systemwithout knowing the value of each parameter

In this specific example we know that the system parameters are pos-itive and bounded scalars The Hill function H(x1) = k3 (Kn

1 + xn1 ) is adecreasing function sufficiently ldquoflatrdquo near the origin (ie with zero deriva-tive) with a single flexus (second derivative has a single zero) [19 20] Thenwe can say that for given u1 and u2 constant or varying on a slower timescalethan this system x1 will converge to its equilibrium x1 = k1u1k2 Similarlyx2 = H(x1)(k4 + k5u2) x3 = k6x2k7 Regardless of the specific parametervalues and therefore robustly the system is stable While the equilibriumvalue for the protein x1 could grow unbounded with u1 the RNA species x2is always bounded

12 Qualitative models for biological dynamical systems

The interactions of RNA species proteins and biochemical ligands are at thebasis of cellular development growth and motion Such interactions are oftencomplex and impossible to measure quantitatively Thus qualitative modelssuch as boolean networks and graph based methods are useful tools whentrying to make sense of very coarse measurements indicating a correlation orstatic relationship among different species When dynamic data are availableit is possible to build qualitative ordinary differential equation models Ratherthan choosing specific functional forms to model species interactions (such asHill functions or polynomial terms) one can just make general assumptionson the sign trend and boundedness of said interactions While such modelsare clearly not amenable to data fitting they still allow us to reach usefulanalytical conclusions on the potential dynamic behaviors of a systemThe general class of qualitative biological models we consider are ordinarydifferential equations whose terms belong to four different categories

xi(t) =sumjisinAi

aij(x)xj minussumhisinBi

bih(x)xh +sumsisinCi

cis(x) +sumlisinDi

dil(x) (12)

Variables xi i = 1 n are concentrations of species The different termsin equation (12) are associated with a specific biological and physical mean-


ing Terms aij(x)xj are associated with production rates of reagents typi-cally these functions are assumed to be polynomial in their arguments simi-larly terms bih(x)xh model degradation or conversion rates and are also likelyto be polynomial in practical cases Finally terms c(middot) and d(middot) are associ-ated with monotonic nonlinear terms respectively non-decreasing and non-increasing these terms are a qualitative representation of Michaelis-Mentenor Hill functions [20]

Sets Ai Bi Ci Di denote the subsets of variables affecting xi In generalmore than one species can participate in the same term affecting a givenvariable For instance one may have an interaction 2 rarr 1 influenced alsoby species x3 a12(x1 x3)x2 (The alternative notation choice a13(x1 x2)x3would be possible) To keep our notation simple we do not denote externalinputs with a different symbol Inputs can be easily included as dynamicvariables xu = wu(xu t) which are not affected by other states and have thedesired dynamics

121 General assumptions

We denote with xi = [x1 x2 ximinus1 xi+1 xn] the vector of nminus1 componentscomplementary to xi (eg in IR4 x2 = [x1 x3 x4]) Then f(x) = (xj xj) forall j In the remainder of this chapter we assume that system (12) satisfiesthe following assumptions

A1 (Smoothness) Functions aij(middot) bih(middot) cis(middot) and dil(middot) are nonnegativecontinuously differentiable functions

A2 Terms bij(x)xj = 0 for xi = 0 This means that either i = j orbij(xi 0) = 0

A3 Functions bij(x)xj and aih(x)xh are strictly increasing in xj and xhrespectively

A4 (Saturation) Functions cis(xs xs) are nonnegative and nonndashdecreasingin xs while dil(xl xl) are nonnegative and respectively nonndashdecreasing inxl Moreover cis(xsinfin) gt 0 and dil(xl 0) gt 0 Moreover they are globallybounded

In view of the nonnegativity assumptions and Assumption 2 our generalmodel (12) is a nonlinear positive system and its investigation will be re-stricted to the positive orthant We note that reducing dynamic interactionsto a form bij(x)xj and aih(x)xh is always possible under mild assumptionsfor instance if species j affects species i with a monotonic functional termfij(xj xj) if such term has a locally bounded derivative with f(xi 0) = 0 itcan always be rewritten as fij(x) = (fij(x)xj)xj = aij(x)xj (see [14] Sec-tion 21) Using the general class of models (12) and assumptions A1ndashA4 asa working template for analysis we will focus on a series of paradigmatic dy-namic behaviors which can be structurally identified or ruled out in examplesystems of interest


122 Glossary of properties

The structural analysis of system (12) can be greatly facilitated whenever itis legitimate to assume that functions a b c d have certain properties suchas positivity monotonicity boundedness and other functional characteristicsthat can be considered ldquoqualitative and structural propertiesrdquo [15] Throughsuch properties we can draw conclusions on the dynamic behaviors of theconsidered systems without requiring specific knowledge of parameters andwithout numerical simulations However it is clear that our approach requiresmore information than other methods such as boolean networks and othergraph-based frameworks

For the readerrsquos convenience a list of possible properties and their defini-tions is given below for functions of a scalar variable x

P1 f(x) = const ge 0 is nonnegativendashconstant

P2 f(x) = const gt 0 is positivendashconstant

P3 f(x) is sigmoidal it is nonndashdecreasing f(0) = f prime(0) = 0 if 0 lt f(infin) ltinfin and its derivative has a unique maximum point f prime(x) le f prime(x) for somex gt 0

P4 f(x) is complementary sigmoidal it is nonndashincreasing 0 lt f(0) f prime(0) =0 f(infin) = 0 and its derivative has a unique minimum point In simple wordsf is a CSM function iff f(0)minus f(x) is a sigmoidal function

P5 f(x) is constantndashsigmoidal the sum of a sigmoid and a positive constant

P6 f(x) is constantndashcomplementaryndashsigmoidal the sum of a complementarysigmoid and a constant

P7 f(x) is increasingndashasymptoticallyndashconstant f prime(x) gt 0 0 lt f(infin) lt infinand its derivative is decreasing

P8 f(x) is decreasingndashasymptoticallyndashnull f prime(x) lt 0 f(infin) = 0 and itsderivative is increasing

P9 f(x) is decreasingndashexactlyndashnull f prime(x) lt 0 for x lt x and f(x) = 0 forx ge x for some x gt 0

P10 f(x) is increasingndashasymptoticallyndashunbounded f prime(x) gt 0 f(infin) = +infin

As an example the terms d(middot) and c(middot) in general are associated with Hillfunctions which are sigmoidal and complementary sigmoidal functions Insome cases it will be extremely convenient to introduce assumptions whichare mild in a biological context but assure a strong simplification of themathematics One possible assumption is that a sigmoid or a complementarysigmoid is cropped (Fig 11) A cropped sigmoid is exactly constant above acertain threshold xminus and exactly null below another threshold x+ A cropped


x+xminus

f

f+LOW MEDIUM HIGH

x+xminus

f

f+LOW MEDIUM HIGH

Fig 11 Cropped sigmoids and complementary sigmoids

complementary sigmoid is exactly null above xminus and exactly constant belowx+

These assumptions extend obviously to multivariable functions just byconsidering one variable at the time For instance f(x1 x2) can be a sigmoidin x1 and decreasing in x2

123 Network graphs

Building a dynamical model for a biological system is often a long and chal-lenging process For instance to reveal dynamic interactions among a poolof genes of interest biologists may need to selectively knockout genes set upmicro RNA assays or integrate fluorescent reporters in the genome The dataderived from such experiments are often noisy and uncertain which impliesthat also the estimated model parameters will be uncertain However qualita-tive trends can be reliably assessed in the dynamic or steady state correlationof biological quantities Graphical representations of such qualitative trendsare often used by biologists to provide intuition regarding the network mainfeatures

Building on the general model (12) we can associate species to nodes ofa graph and different qualitative relationships between species with differenttypes of arcs terms a b c and d can be represented as arcs having differentendndasharrows as shown in Fig 12

a b

c d

1 3

2

b a31

Fig 12

These graphs can be immediately constructed by knowing the correlationtrends among the species of the network and serve as a support for theconstruction and analysis of a dynamical model For simple networks thesegraphs may facilitate structural robustness analysis


Our main objective is to show that at least for reasonably simple networksstructural robust properties can be investigated with simple analytical meth-ods without the need for extensive numerical analysis We suggest a twostage approach

bull Preliminary screening establish essential information on the networkstructure recognizing which properties (such as P1ndashP10) pertain to eachlink

bull Analytical investigation infer robustness properties based on dynamicalsystems tools such as Lyapunov theory set invariance and linearization

124 Example continued transcriptional repression

The model for the transcriptional repression system in Equations (11) [19]can be recast in the general class of models (12) and we can immediatelydraw the corresponding graph

x1 = u1 minus b11x1 (13)

x2 = d21(x1)minus b22x2 minus b2u2 x2 u2

x3 = a32x2 minus b33x3

1 2

3

u1

u2

b

b11

b22

b33

a32

a21d21

b2u2

Fig 13

Terms aij capture first order production rates bih capture first orderdegradation rates Term d21(x1) is our general substitute for the Hill func-tion [19 20] we assume it is a decreasing function with null derivative at theorigin whose second derivative has a single zero (flexus) and it is negativeon the left of the zero and positive on the right (such as 1(1 + xp1) n gt 1)

13 Robustness and structural properties

We now clarify the concepts of robustness and structural properties and theirrelations

Definition 1 Let C be a class of systems and P be a property pertainingsuch a class Given a family F sub C we say that P is robustly verified by F in short robust if it is satisfied by each element of F

Countless examples can be brought about families F and candidate prop-erties Stability of equilibria for instance is one of the most investigatedstructural properties [21 13 2]

When we say structural property we refer to the properties of a family Fwhose ldquostructurerdquo has been specified In our case the structure of a system


is the fact that it belongs to the general class (12) thus it satisfies assump-tions 1ndash3 and it enjoys a set of properties in the set P1ndashP8

A realization is any system with assumed structure and properties achievedby specific functions which satisfy these assumptions The set off all realiza-tion is a class For instance going back to the transcriptional repressionexample the dynamical system

x1 = u1 minus 2x1

x2 =1

1 + xn1minus x2 minus 2x2u2

x3 = 2x2 minus 2x3

is a realization of the class represented by system (13)

Definition 2 A property P is structural for a class C if any realizationsatisfies P

Note that demonstrating a structural property for a system is harder thanproving that it does not hold (the latter typically only requires to show theexistence of a system which exhibits the considered structure but does notsatisfy the property) For example consider matrices

A1 =

[minusa bminusc minusd

]A2 =

[minusa bc minusd

]with a b c and d positive real parameters To show that A1 is structurallystable one has to show that its eigenvalues have negative real part (in thiscase a simple proof) Conversely to show that A2 is not structurally stableit is sufficient to find a realization which is not stable such as a = 1 b = 1c = 2 and d = 1

14 Paradigmatic structural properties

We introduce an overview of properties particularly relevant in systems andsynthetic biology Through simple examples we highlight how our generalapproach can be used to determine analytically the structural nature of suchproperties

141 Mutistationarity

A multistationary system is characterized by the presence of several possibleequilibria Of particular interest are those systems in which there are threeequilibria of which two are stable and one unstableie the system is bistable

We consider a simple example of a multistationary system

x1 = x0 + c12(x2)minus b11x1 (14)



STABLEUNSTABLE

Fig 14 Sketch of a bistable system

with b11 b22 and a21 positive constants and with c12(x2) a (non-decreasing)sigmoidal function We assume x0 ge 0 The following proposition holds

Proposition 1 For x0 small enough and for b11b22a21 small enough sys-tem (14) has three equilibria two stable and one unstable Conversely forx0 large or b11b22a21 large the system admits a unique stable equilibrium

Explanation Setting x1 = 0 and x2 = 0 we find the equilibria as the rootsof the following equation

c12(x2) + x0 =b11b22a21

x2

From Fig 15 it is apparent that if x0 is small and the slope of the lineb11b22a21

x2 is small there must be three intersections Conversely there is a

single intersection for either x0 or b11b22a21

large

x2

C

BA

Fig 15 Sketch of the nullclines for system (14)

If three intersections (points A B C in Fig 15) are present there aretwo stable points A and B and one unstable This can be seen by inspectingthe Jacobian

J =

[minusb11 cprime12(x2)a21 minusb22

]

whose characteristic polynomial is

p(s) = s2 + (b11 + b22) + b11b22 minus a21cprime12(x2)


The stability of this second order polynomial is equivalent to b11b22 minusa21c

prime12(x2) gt 0 or

cprime12(x2) ltb11b22a21

x2

namely the slope of the sigmoidal function must be smaller that the slope ofthe line b11b22a21 This is the case of points A and C while the conditionis violated at point B

142 Oscillations

Oscillations in molecular and chemical networks are a well-studied phe-nomenon (see for instance [22]) Periodicity in molecular concentrations un-derlies cell division development and circadian rhythms One of the first ex-

Fig 16 Schematic representation of oscillatory behavior

amples considered in the literature is the well known Lotka Volterra predatorndashprey system whose biochemical implementation has been studied and at-tempted in the past [23 24] In our general setup the Lotka Volterra modelis

x1 = a11x1 minus b12(x2)x1

x2 = a21(x1)x2 minus b22x2

where all functions are strictly increasing and asymptotically unbounded inall arguments The system admits a single nonndashtrivial equilibrium the solu-tion of equations

0 = a11 minus b12(x2)

0 = a21(x1)minus b22

The Jacobian of this system at the unique equilibrium is

J =

[0 minusbprime12(x2)x1

aprime21(x1)x2 0

]

This matrix clearly admits pure imaginary eigenvalues for any realization ofthe functional terms Thus oscillations are a structural property

In second order systems the presence of sustained oscillations requires thepresence of a positive self loop (autocatalytic reactions) represented in thiscase by the a11 term


To achieve oscillations without a positive loop reaction the system mustbe of at least third order For instance the following model

x1 = x10d13(x3)minus b11x1x2 = a21x1 minus b22x2 (15)


where d13(x3) is a complementary sigmoid and the constant are positiveis a candidate oscillator Term x10 is an external input which catalyzes theproduction d13(x3)

Proposition 2 System (15) admits a unique equilibrium If the minimumvalue of the slope dprime13(x3) is sufficiently large there exists an interval (possiblyunbounded from above) of input values x10 inducing an oscillatory transitionto instability

Explanation The unique equilibrium point can be derived by the conditionsx1 = x2 = x3 = 0

x10d13(x3) =b11b22b33a21a32

x3

Fig 17) shows the qualitative trend of the nullclines above and clearly high-lights that they admit a single intersection

x2

A

x3

Fig 17 Qualitative trend of the nullclines for system (15)

Assume that the slope in the intersection point A is large The Jacobianof the system in this point is

J =

minusb11 0 minusmicroa21 minusb22 00 b32 minusb33

micro = minusx0dprime13(x3) gt 0

The corresponding characteristic polynomial is

p(s) = (s+ b11)(s+ b22)(s+ b33) + a21a32micro = s3 + p2s2 + p1s+ p0 + a21a32micro


This polynomial has a pair of complex conjugate positive roots as it can beinferred from the RuthndashHurwitz table

+ 1 p1+ p2 p0 + a21a32micro (p1p2 minus a21a32micro)p2+ a21a32micro

for large micro there are two sign changes which means that there are two unstableroots These roots cannot be real because the polynomial coefficients are allpositive so unstable roots must be complex conjugate

To see that in general the must exists ldquoan intervalrdquo for oscillations notethat for x0 small the intersection occurs in a region where the slope of micro =minusx0dprime13(x3) is small thus there are no changes in the Routh-Hurwitz tableand the system is stable

Note that it is not necessarily true that for large x0 the system is unsta-ble in addition the instability interval of x0 may be bounded In fact theequilibrium x3 increases for large x0 but it may transition to a region wheredprime13 is very small compensating for the increase of x0

143 Adaptation

A system is adaptive if when perturbed by a persistent input signal itsoutput always reverts to a neighborhood of its value prior to the perturbationin general after a transient [1 25 26] A sketch of this behavior is in Fig 18Adaptation is said to be perfect if the systemrsquos output reverts to its exactvalue prior to the perturbation

Fig 18 System capable of adaptation

For small perturbations linearization analysis suggests that adaptationrequires the presence of a zero in the systemrsquos transfer function If the systemincludes a feedback loop then the presence of a pole at the origin (integrator)is required[26 25] Establishing criteria to detect a systemrsquos capability foradaptation is thus simple Consider the system

x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)


We assume all the constants are positive and that function b21(x1) is acropped sigmoid namely it is strictly increasing and exactly positive constantabove a certain threshold Term x0 is a constant and u ge 0 is a perturbinginput

Proposition 3 If x0 is sufficiently large and u = 0 then system (16) has astable equilibrium point Takin y = x2 as the systemrsquos output perfect adap-tation is achieved with respect to constant perturbations on u gt 0

Explanation For u = 0 the equilibrium conditions are b21(x1)x2 = x0 anda12x1 minus b22x2 Therefore the equilibrium x1 can be expressed as the solutionof

b21(x1)a12b22

x1 = x0 (18)

For x0 suitably large x1 increases until it falls in the range where b21 (acropped sigmoid) is constant thus b21(x1) = b21(infin) and bprime21(x1) = 0

In this range the linearized system is[x1x2

] [0 minusb21(x1)a21 minusb22

] [x1x2

]+

[01

]u y =

[0 1

] [x1x2

]with output y(t) = x2(t) The state matrix is a stable matrix with charac-teristic polynomial p(s) = s2 + b22s + b21(x1)a21 The transfer function isw(s) = sp(s) has a zero at the origin and thus the system locally exhibitsperfect adaptation

If u gt 0 increases as a step input after a transient the output x2 returnsto its original value x2 prior to the perturbation However the equilibriumof x1 increases to a new value such that a12x1 = b22x2 + u

144 Spiking and Persistency the MAPK network as a case study

Spiking is a phenomenon observed in several molecular networks in which asystem subject to a step input grows rapidly and subsequently undergoes arelaxation as sketched in Fig 19 The relaxation bring the system to a newequilibrium distinct from the equilibrium prior to the input stimulation

Fig 19 System presenting a spiking behavior

Persistency is closely related to bistability it occurs when a transient inputvariation causes the system to switch its output to a new value which persistsupon removal of the input as shown in Fig 110


Fig 110 System presenting a persistent response

1441 A qualitative model of the MAPK pathway

Experiments show that the mitogen-activated protein kinase (MAPK) path-way in PC12 rat neural cells exhibits dynamic behaviors that depend onthe growth factor they are exposed to as an input The response to Epi-dermal Growth Factor (EGF) is a spike followed by a relaxation while theresponse to Nerve Growth Factor (NGF) is persistent In the latter casethe system can be driven to a new state which persists after the stimulushas vanished Ultimately these dynamic behaviors correspond to differentcell fates EFG stimulation induces proliferation while NGF stimulation in-duces differentiation The biochemical mechanisms responsible for the differ-ent input-dependent dynamic response are still unclear One hypothesis isthat each input generates a specific interaction topology among the kinasesStarting from experimental results that support this hypothesis [27] in ourprevous work we considered the two network topologies and we derived andanalyzed qualitative models which exhibit structural properties [28] Herewe use a simplified third order model for the pathway We refer the readerto [28] for a more detailed model and its derivation In our reduced ordermodel we neglect double-phosphorylation dynamics and model the activeconcentration of each MAPK protein with a single state variable We alsoneglect mass conservation assumptions regarding the total amount of MAPKprotein [13 16]

MAP3K x1 = u(x3 x0)minus b11x1 (19)

MAP2K x2 = c21(x1)minus b22x2 (110)


Output y = x3 (112)

We assume c21 and c32 are strictly increasing asymptotically constant iec21(infin) = c21 lt infin c32(infin) = c32 lt infin and null at the origin c21(0) =c32(0) = 0 Terms bii are positive constants In essence this model capturesthe fact that each protein in the cascade is activated by its predecessor in thechain in the absence of term u(x3 x0) the system would be an open loopmonotonic cascade [12] Term u(x3 x0) is a feedback term modulated by anexternal input x0 and we consider two cases


EGF u = a10(x3)x0 where a10(x3) is a complementary sigmoid exactlyconstant below a threshold η and exactly null over a threshold ξ Thisconfiguration is characterized by the presence of a negative feedback loop

NGF u = a10(x3) + x0 where a10(x3) is a sigmoid exactly null below athreshold η and exactly constant over a threshold ξ This configuration ischaracterized by the presence of a positive feedback loop

Under these assumptions we show that in the EFG configuration the outputexhibits a spike while in the NGF configuration the output is persistent

145 The EGF-induced patwhay and its spiking behavior

The system in this configuration admits a single equilibrium this can beshown as for the third order oscillator model (15)

Consider c21(infin) = c21 c32(infin) = c32 the saturation value Let x2 =c21b22 be the corresponding ldquosaturationrdquo limit value of x2 Let in turn

x3 = c32(x2)b33

be the limit value of x3 For large increasing values of the input x0 thevariable x1 increases and the equilibrium values of x2 and x3 approach x2and x3 The following proposition holds

Proposition 4 Assume that the limit value for x3 is x3 gt ξ Then for x0constant sufficiently large and for xi(0) = 0 we have a) First x3 growsarbitrarily close to x3 b) Subsequently x3 relaxes below ξ

Proof Since a10(x3) is constant for a small values of x3 if x0 is large thenby continuity x1 can grow arbitrarily large in an arbitrarily small amount oftime τ gt 0 Then considering the time interval [τ T ] where T is arbitrarilylarge and given an arbitrary micro gt 0 by picking x1(τ) sufficiently large we canguarantee

x1(t) ge micro for t isin [τ T ] (113)

In fact we have x1 ge minusb11x1 thus x1(t) ge x1(τ)eminusb11t on [τ T ] thereforepicking a large initial value x1(τ) equation (113) is verified Thus we canguarantee that variables x2 and x3 have values arbitrarily close to the upperlimit x2 and x3 being micro and T arbitrarily large

If x3 increases at some point in time the condition a10(x3) = 0 is metThis ldquoswitches offrdquo the first variable whose dynamics become x1 = minusb11x1thus x1 starts decreasing variables x2 and x3 follow the same pattern Theseconcentrations decrease until x3 le ξ

146 The NGF-induced pathway is an example ofpersistent network

Let us now define a10(infin) = a10 as a saturation value If x3 is greater than thethreshold ξ then a10(x3) = a10 then for x0 = 0 we can find the equilibriafrom the following conditions


0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)

which yield x1 = a10b11 x2 = c21(x1)b22 x3 = c32(x2)b33 The assump-tion x3 gt ξ means that the positive feedback given by the term a10 is ableto sustain this positive equilibrium

Now consider the case where the input x0 becomes arbitrarily large Thusx1 becomes arbitrarily large Defining c21 = c21(infin) we find the correspond-ing limit values for the steady states x2 = c21b22 and x3 = c32(x2)b33 Itis immediate that x1 ge x1 x2 ge x2 x3 ge x3 because the ldquohatrdquo equilibriumvalues are achieve by means of an arbitrarily large input x0 while the ldquobarrdquovalues are achieved by the bounded input a10

Proposition 5 Assume that x3 gt ξ and that the previous inequalities arestrict x1 gt x1 x2 gt x2 x3 gt x3 Then for xi(0) = 0 the following happensa) If x0 is constant and sufficiently large and it is applied for a sufficientlylong time interval [0 T ] then x3 grows arbitrary close to x3b) If after time T the input signal x0 is eliminated (x0 = 0) then x3 remainsabove ξc) Finally x3 converges to x3 from above

Proof We have seen that when x0 = 0 x1 x2 x3 are admissible equilibriaof the system Exactly as done in the EGF-driven network example we canshow that for a sufficiently large input x0 variables x1 x2 and x3 can growarbitrarily close to x1 x2 and x3 above x1 x2 and x3

We only need to show that if all xi(t) grow above the corresponding xithen they will not reach values below xi afterx0 is removed

We begin by defining the new variables zi = ximinus xi then zi = xi given byequations (19)ndash(111) After x0 is removed the input is a10(x3) in additionsince we assume x3 ge x3 ge ξ (so z3 ge 0) we have a10(x3) = a10 If weconsider also the steady state equations (114)ndash(116) we get

z1 = minusb11z1 (117)

z2 = c21(z1 + x1)minus c21(x1)minus b22z2 (118)

z3 = c32(z2 + z2)minus c32(z2)minus b33z3 (119)

This is a positive system in the z variables So because we assumed that atsome point zi(τ) gt 0 (prior to the removal of x0) we can immediately seethat this situation is permanent

To prove convergence note that z1 goes to zero in view of equation (117)Then c21(z1 + x1) minus c21(x1) goes to 0 so z2 converges to 0 For the samereason z3 converges to 0


15 Structural boundedness and stability

Our qualitative modeling framework is generally described by equation (12)

xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)

The general assumptions we made on functions a b c and d guarantee non-negativity of the states which is a required feature to meaningfully modelconcentrations of molecules Another important feature of most biochemicalsystem models is boundedness of their states (possibly with the exception ofpathological cases) In the following we outline additional assumptions andconsequent results regarding structural boundedness of the solutions to ourgeneral model (12)

151 Structural boundedness

Consider the case in which states in model (12) are dissipative ie the dy-namics of each variable include a degradation term minusbii(x)xi We also assumethat

bii(x) gt βi gt 0

Obviously this property alone does not assure the global boundedness of thesolution However if no unbounded andashterms were present it would be simpleto show that the solutions are globally bounded

Let us assume that each aij(x) term is bounded by a positive constant 0 leaij(x) lt aij Then we ask under what conditions we can assure structuralboundedness of the solutions We build a graph G(A) associated with the aijterms where there is a directed arc from node j to node i for every term aij Then the following theorem holds

Theorem 1 The system solution is structurally globally bounded for anyinitial condition x(0) ge 0 if and only if G(A) has no cycles (including selfndashcycles) including aii terms

In other words structural boundedness is guaranteed if and only if there isno autocatalysis in the system

Proof We first show that the condition is structurally necessary Assumeab absurdo that there is a cycle which includes a term aij Without restrictionassume that the cycle if formed by the first r nodes 12 r forming asequence a12 a23 ar1 also assume that each term aij is lower boundedby a constant κ We finally assume that the sum of all bik terms appearingin the first r equations is upper bounded by η

rsumi=1

sumkisinBi

bik le η

Consider the Lyapunovndashlike function


V (x1 x2 xr) = x1 + x2 + middot middot middot+ xr

and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1

aii+1xi+1 minus ηrsumi=1

xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V

Then if η lt rκ V increases and the equilibrium is not stable Thus struc-tural boundedness cannot hold

Let us now consider the sufficiency part If there are no cycles in G(A)then there exists necessarily a node which is a root ie its dynamics do notinclude aij terms Let us assume without loss of generality that node x1does not have any a1j term Then

x1 = minussumhisinB1

bih(x)xh +sumsisinC1

c1s(x) +sumlisinD1

d1l(x)

le minusβ1x1 +sumsisinC1

c1s(x) +sumlisinD1

d1l(x)

Since the c and d terms are bounded then the solution x1 is bounded withoutloss of generality assume x1 le ξ1 ξ1 gt 0

If x1 is bounded then all terms (if any) of type ak1(x)x1 in other equationsremain bounded ak1(x)x1 le aj1ξ1

Let us consider the other nodes x2 x3 xn Since there are no cyclesincluding aij terms there is at least one variable whose equation has eitherno a terms or has only ak1(x)x1 terms from x1 which are bounded Let usassume node x2 fulfills this statement Then

x2 = ai1(x)x1 minussumhisinB2

bih(x)xh minussumhisinB2


c2s(x) +sumlisinD2

d2l(x)

le minusβ2x2 + aj1ξ1 +sumsisinC2

c2s(x) +sumlisinD2

d2l(x)

The above inequality implies boundedness of the solution x2The proof can be concluded recursively by noticing that there must exists

a new variable say x3 whose equation includes either no aij terms or onlybounded a3j terms coming from x1 and x2 and so on

The following corollary holds

Corollary 1 The solution to the general model (12) is bounded if and onlythere are no aij terms and all bii terms are lower bounded by a positiveconstant bii gt βi


This corollary highligths boundedness is structurally assured in systemswhere each species is degraded by terms of at least first order and all theinteraction terms are bounded

Example 1 As an example we consider the well known lac Operon geneticnetwork We will propose and analyze a qualitative model or class the classi-cal model proposed in [29] is a realization whitin this class The state variablesof our model are the concentration of nonfunctional permease protein x1 theconcentration of functional permease protein x2 the concentration of inducer(allolactose) inside the cell x3 and the concentration of β-galactosidase x4 aquantity that can be experimentally measured The concentration of inducerexternal to the cell is here denoted as an input function u A model for thissystem can be written in the following form (see [16] for details)

x1 = c13(x3)minus b11x1x2 = a21x1 minus b22x2 (120)

x3 = a32(u)x2 minus b32(x3)x2 + c3uuminus b33x3x4 = c43(x3)minus b44x4

1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111

where c13(x3) = f1(x3) b11 = δ1 a21 = β1 b22 = δ2 a32(u) = f2(u) =b32(x3) = f3(x3) c3u = β2 b33 = δ3 c43(x3) = γf1(x3) and b44 = δ4 Thiscorresponds to the network in Fig 111We assume that c13 is constantndashsigmoidal a32(u) and b32(x3) are increasingndashasymptoticallyndashconstant and the remaining functions a21 b11 b22 and b33 arepositivendashconstant

The arcs associated to aij terms in Fig 111 do not form any cycles Eachnode is dissipative therefore the solution is structurally bounded

The requirement of having no aij cycles can be strong especially in chemicalreaction networks [11] However the conditions in Theorem 1 are necessaryand sufficient we believe it is unlikely that stronger results can be foundwithout assuming bounds on the dynamic temrs

Note that Theorem 1 only requires that bounds on the functional termsexist while their specifc values need not be known If such bounds are known


we obtain less restrictive conditions Note that model (12) can be writtencompactly as

x(t) = A(x(t))x(t)minusB(x(t))x(t) + C(x(t)) +D(x(t)) (121)

or asx(t) = M(x(t))x(t) + C(x(t)) +D(x(t)) (122)

where M(x(t)) = A(x(t)) minus B(x(t)) If the elements of matrix M(x(t))are constrained in a closed (even better if compact) set M(middot) isin M and ifand if we can demonstrate exponential stability of the associated differentialinclusion [30]

x isinMx

then we can show the overall boundedness of the systemsrsquo solution To proveboundedness it is convenient to exclude a neighborhood of the origin Nν =x xi ge ν

Theorem 2 Assume that M(x) isin M for x isin N and assume that the dif-ferential inclusion is bounded and admits a positively homogeneous functionV (x) as Lyapunov function

V (x) = nablaV (x)Mx le minusγV (x)

for all M isinM Then the system solution is bounded

Proof The proof is an immediate consequence of the fact that the trajec-tories of the original linear systems are a subset of the possible trajectoriesof the linear differential inclusions

An exponentially stable differential inclusion has bounded solutions if per-turbed by bounded terms

x isinMx+ C +D

as in our case

Example 2 Consider a biological network composed by two proteins x1 andx2

x1 = +c10 + a12(x1)x2 minus b11x1x2 = +c20 minus b21(x2)x1 minus b22x2

In this model we suppose that both x1 and x2 are produced in active format some constant rates (terms c10 and c20) but they are inactivated or de-graded at some speed proportional to their concentration (terms b11 andb22) However suppose protein x1 is activated by binding to x2 this inter-action in turn inactivates x2 this pathway is modeled by terms a12(x1) and


b21(x2) which we assume are sigmoidal functions asymptotically constantconsistently with a cooperative Hill function-type protein interaction

We can rewrite the above equations as[x1x2

]=

[minusb11 a12 + δ12

minusb21 minus δ21 minusb22

] [x1x2

]+

[c10c20

]

where δ12 = a12(x1)minus a12 and δ21 = a21(x1)minus a21 and where a12 = a12(infin)and b21 = b21(infin)

If the region near the origin is delimited by a ldquoradiusrdquo ν sufficiently largethe bounds on δ12 and δ21 can be taken arbitrarily tight

So inside N for large ν gt 0 we may assume |δ12| le ε and |δ21| le ε withsmall Since the nominal systems for δ12 = δ21 = 0 is quadratically stablenamely it admits a quadratic Lyapunov function inside Nν the large ν thisis a Lyapunov function for the actual systems since the contribution of theterms δ12x2 and δ21x1 is negligible

This technique allows us to prove boundedness but not stability of theoriginal system Boundedness does imply the existence of equilibria but theirstability may be or may be not verified

152 Structural stability of equilibria

If we can establish boundedness of a system the existence of equilibria isautomatically assured Then we can ask two main questions

bull How many equilibria are presentbull Which equilibria are stable

Several results from the so-called degree theory help us find answers seefor instance [31 32 33 34] Here we recall one particularly useful theorem

Theorem 3 Assume that all the systemrsquos equilibria x(i) are strictly positiveand assume that none of them is degenerate ie the Jacobian evaluated ateach equilibrium has nonndashzero determinant Thensum

i

sign det[minusJ(x(i))] = 1

How does this theorem help us answering our questions We describe in-formally three cases that we can immediately discriminate as a consequenceof this theorem Suppose analytical expressions for the Jacobian are availableas a function of a generic equilibrium point

1) If we can establish that the determinant of minusJ is always positive re-gardless of specific values for parameters or equilibria then there is a uniqueequilibrium

2) If at an equilibrium point we have sign det[minusJ ] lt 0 then such equilib-rium must be unstable (because the characteristic polynomial has a negativeconstant term p0 = det[minusJ ]) A consequence of Theorem 3 is that other equi-libria must exist if they are not degenerate then there must be at least twoequilibria


3) If there are two stable equilibria then necessarily another unstablestable equilibrium must exist

In a qualitativeparameter-free context general statements about stabil-ity of equilibria are difficult to demonstrate If we restrict our attention tospecific classes of systems however we can find structural stability resultsWe mention a few well known examples

bull Chemical reaction networks modeled with mass action kinetics the zerondashdeficiency theorem [11] guarantees uniqueness of the equilibrium andasymptotic stability of networks satisfying specific structural conditionsthat do not depend on the reaction rate parameters

bull Monotone systems if a system is monotone [35] then its Jacobian hasnonnegative non-diagonal entries in other words it is a Metzler matrixFor a Metzler matrix stability is equivalent to having a characteristicpolynomial with all positive coefficients This property is easy to checkanalytically in systems of small dimension

bull Planar systems Plenty of straightforward methods are available to findstructural stability conditions

We conclude this section with a paradox

Difficulty Structural stability investigation is generally speaking an un-solved problem which typically requires a casendashbyndashcase study

Interest Stability is generally of little interest to biologists because manynatural behaviors in biology are known to be (obviously) stable In otherwords formal proofs of stability are not very informative However lackof stability of an equilibrium can be a hallmark for other interesting be-haviors such as multistationarity and periodicity

Conclusions

A property is structurally robust if it is satisfied by a class of models re-gardless of the specific expressions adopted or of the parameter values in themodel This chapter highlighted how qualitative parameter-free models ofmolecular networks can be formulated by making general assumptions on thesign trend and boundedness of the species interactions Linearization Lya-punov methods invariant sets and graphical tests are examples of classicalcontrol theoretic tools that can be successfully employed to analize such qual-itative models often reaching strong conclusions on their admissible dynamicbehavior

Robustness is often tested through simulations at the price of exhaustivecampaigns of numerical trials and more importantly with no theoreticalguarantee of robustness We are far from claiming that numerical simulationsare useless they are useful for instance to falsify ldquorobustness conjecturesrdquoby finding suitable numerical counterexamples In addition for very complexsystems in which analytical tools cannot be employed simulations are the


only viable method for analysis A limit of our qualitative modeling andanalysis approach is its lack of systematic scalability to complex modelsHowever the techniques we employed can be successfully used to study alarge class of low dimension systems and are an important complementarytool to simulations and experiments

References1 Ma W Trusina A El-Samad H Lim WA Tang C Defining Network Topologies

that Can Achieve Biochemical Adaptation Cell 2009 138(4)760ndash7732 Prill RJ Iglesias PA Levchenko A Dynamic Properties of Network Motifs Con-

tribute to Biological Network Organization Public Library of Science Biology

2005 3(11)e3433 Kwon YK Cho KH Quantitative analysis of robustness and fragility in biolog-

ical networks based on feedback dynamics Bioinformatics 2008 24(7)987ndash9944 Gomez-Gardenes J Y M Florıa LM On the robustness of complex heteroge-

neous gene expression networks Biophysical Chemistry 2005 115225ndash229

5 Gorban A Radulescu O Dynamical robustness of biological networks withhierarchical distribution of time scales IET Systems Biology 2007 1(4)238ndash

246

6 Kartal O Ebenhoh O Ground State Robustness as an Evolutionary DesignPrinciple in Signaling Networks Public Library of Science One 2009 4(12)e8001

7 Aldana M Cluzel P A natural class of robust networks Proceedings of the

National Academy of Sciences of the USA 2003 100(15)8710ndash87148 Tian T Robustness of mathematical models for biological systems ANZIAM

J 2004 45C565ndashC577

9 Shinar G Milo R Rodrıguez Martınez M Alon U Input-output robustness insimple bacterial signaling systems Proceedings of the National Academy of Sci-

ences of the USA 2007 10419931ndash19993510 Shinar G Feinberg M Structural Sources of Robustness in Biochemical Re-

action Networks Science 2010 327(5971)1389ndash1391

11 Feinberg M Chemical reaction network structure and the stability of com-plex isothermal reactors I The deficiency zero and deficiency one theorems

Chemical Engineering Science 1987 422229ndash2268

12 Sontag E Monotone and near-monotone biochemical networks Systems andSynthetic Biology 2007 159ndash87

13 Angeli D Ferrell JE Sontag ED Detection of multistability bifurcations and

hysteresis in a large class of biological positive-feedback systems Proceedingsof the National Academy of Sciences of the USA 2004 101(7)1822ndash1827

14 Jacquez J Simon C Qualitative Theory of Compartmental Systems Society

for Industrial and Applied Mathematics Review 1993 3543ndash7915 Nikolov S Yankulova E Wolkenhauer O Petrov V Principal difference between

stability and structural stability (robustness) as used in systems biologyNonlinear Dynamics Psychology and Life Sciences 2007 11(4)413ndash33

16 Blanchini F Franco E Structurally robust biological networks Bio Med Central

Systems Biology 2011 57417 Abate A Tiwari A Sastry S Box Invariance for biologically-inspired dynamical

systems In Proceedings of the IEEE Conference on Decision and Control 20075162ndash5167

18 El-Samad H Prajna S Papachristodoulou A Doyle J Khammash M Advanced

Methods and Algorithms for Biological Networks Analysis Proceedings of

the IEEE 2006 94(4)832ndash85319 De Jong H Modeling and simulation of genetic regulatory systems a liter-

ature review Journal of Computational Biology 2002 967ndash103


20 Alon U An Introduction to Systems Biology Design Principles of Biological Circuits

Chapman amp HallCRC 200621 Kitano H Systems biology A brief overview Science 2002 295(5560)1662ndash1664

22 Goldbeter A Gerard C Gonze D Leloup JC Dupont G Systems biology of cel-

lular rhythms FEBS Letters 2012 5862955ndash296523 Ackermann J Wlotzka B McCaskill JS In vitro DNA-based Predator-Prey Sys-

tem with Oscillatory Kinetics Bulletin of Mathematical Biology 1998 60(2)329ndash

35424 Balagadde FK Song H Ozaki J Collins CH Barnet M Arnold FH Quake SR You

L A synthetic Escherichia coli predator-prey ecosystem Molecular SystemsBiologyl 2008 4ndash [httpdxdoiorg101038msb200824]

25 Yi TM Huang Y Simon MI Doyle J Robust perfect adaptation in bacte-

rial chemotaxis through integral feedback control Proceedings of the NationalAcademy of Sciences of the USA 2000 97(9)4649ndash4653

26 Drengstig T Ueda HR Ruoff P Predicting Perfect Adaptation Motifs in Reac-

tion Kinetic Networks The Journal of Physical Chemistry B 2008 112(51)16752ndash16758

27 Santos SDM Verveer PJ Bastiaens PIH Growth factor-induced MAPK net-

work topology shapes Erk response determining PC-12 cell fate Nature CellBiology 2007 9(3)324ndash330

28 Franco E Blanchini F Structural properties of the MAPK pathway topologies

in PC12 cells Journal Mathematical Biology 2012 [To appear]29 Vilar JMG Guet C Leibler S Modeling network dynamics the lac operon a

case study Journal of Cell Biology 2003 161(3)471ndash47630 Blanchini F Miani S Set-theoretic methods in control Volume 22 of Systems amp

Control Foundations amp Applications Boston Birkhauser 2008

31 Ortega R Campos J Some Applications of the Topological Degree to StabilityTheory In Topological Methods in Differential Equations and Inclusions Kluwer

Academic Publishing Dordrecht 1995377ndash409

32 Hofbauer J An index theorem for dissipative semiflows Rocky Mountain Jour-nal of Math 1990 20(4)1017ndash1031

33 Meiss J Differential Dynamical Systems SIAM 2007

34 Ambrosetti A Prodi G A Primer of Nonlinear Analysis SIAM 199535 Smith HL Monotone Dynamical Systems An Introduction to the Theory of Compet-

itive and Cooperative Systems American Mathematical Society 2008

Structural analysis of biological networks


Introduction

Motivating example a qualitative model for transcriptional repression

Qualitative models for biological dynamical systems

General assumptions

Glossary of properties

Network graphs

Example continued transcriptional repression

Robustness and structural properties

Paradigmatic structural properties

Mutistationarity

Oscillations

Adaptation

Spiking and Persistency the MAPK network as a case study

The EGF-induced patwhay and its spiking behavior

The NGF-induced pathway is an example of persistent network

Structural boundedness and stability

Structural boundedness

Structural stability of equilibria

References


eters In [2] through numerical exploration of the Jacobian eigenvalues fortwo three and four node gene networks the authors isolate a series of inter-connections which are stable robustly with respect to the specific parame-ters the isolated structures also turn out to be the most frequent topologiesin existing biological networks databases For other examples of numericalrobustness analysis see for instance [3 4 5 6 7 8]

Analytical approaches to the study of robustness have been proposed inspecific contexts A series of recent papers [9 10] focused on inputoutputrobustness of ODE models for phosphorylation cascades In particular thetheory of chemical reaction networks is used in [10] as a powerful tool todemonstrate the property of absolute concentration robustness Indeed thesondashcalled deficiency theorems are to date some of the most general results toestablish robust stability of a chemical reaction network [11] Monotonicityis also a structural property often useful to demonstrate certain dynamicbehaviors in biological models by imposing general interaction conditions [1213] Robustness has also been investigated in the context of compartmentalmodels common in biology and biochemistry [14] A survey on the problemof structural stability is proposed in [15]

Here we review and expand on the framework we proposed in [16] wherewe suggest a variety of tools for investigation of robust stability includingLyapunov setndashinvariance methods and conditions on the network graphWe will assume that certain standard properties or assumptions are verifiedby our model for example positivity monotonicity of key interactions andboundedness Based on such general assumptions we will show hows dynamicbehaviors can be structurally proved or ruled out for a range of examples Ourapproach does not require numerical simulation efforts and we believe thatour techniques are instrumental for biological robustness analysis [17][18]

The chapter begins with a motivating example and a brief summary of theanalysis framework in [16] Then we consider a certain number of ldquoparadig-matic behaviorsrdquo encountered in biochemical systems including multista-tionarity oscillations and adaptation through simple examples we showhow these behaviors can be deduced analytically without resorting to simu-lation As relevant case studies we consider a simplified model of the MAPKpathway and the lac Operon Finally we prove some general results on struc-tural stability and boundedness for qualitative models that satisfy certaingraphical conditions

111 Motivating example a qualitative model for transcriptionalrepression

Consider a molecular system where a protein x1 is translated at a certainsteady rate and represses the production of an RNA species x2 In turn x2is the binding target of another RNA species u2 (x2 and u2 bind and forman inactive complex to be degraded) unbound x2 is translated into proteinx3 A standard parametric model is for example in Equations (11) [19]



x2 = k31

Kn1 + xn1



(11)






xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x) (12)




























x+xminus

f

f+LOW MEDIUM HIGH

x+xminus

f

f+LOW MEDIUM HIGH




123 Network graphs



a b

c d

1 3

2

b a31

Fig 12








x1 = u1 minus b11x1 (13)



1 2

3

u1

u2

b

b11

b22

b33

a32

a21d21

b2u2

Fig 13










x1 = u1 minus 2x1

x2 =1


x3 = 2x2 minus 2x3




A1 =


]A2 =

[minusa bc minusd







x1 = x0 + c12(x2)minus b11x1 (14)



STABLEUNSTABLE





c12(x2) + x0 =b11b22a21

x2




large

x2

C

BA



J =


]





prime12(x2) gt 0 or


x2


142 Oscillations







0 = a11 minus b12(x2)



J =


aprime21(x1)x2 0

]










x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References



x2 = k31

Kn1 + xn1



(11)






xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x) (12)




























x+xminus

f

f+LOW MEDIUM HIGH

x+xminus

f

f+LOW MEDIUM HIGH




123 Network graphs



a b

c d

1 3

2

b a31

Fig 12








x1 = u1 minus b11x1 (13)



1 2

3

u1

u2

b

b11

b22

b33

a32

a21d21

b2u2

Fig 13










x1 = u1 minus 2x1

x2 =1


x3 = 2x2 minus 2x3




A1 =


]A2 =

[minusa bc minusd







x1 = x0 + c12(x2)minus b11x1 (14)



STABLEUNSTABLE





c12(x2) + x0 =b11b22a21

x2




large

x2

C

BA



J =


]





prime12(x2) gt 0 or


x2


142 Oscillations







0 = a11 minus b12(x2)



J =


aprime21(x1)x2 0

]










x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References



























x+xminus

f

f+LOW MEDIUM HIGH

x+xminus

f

f+LOW MEDIUM HIGH




123 Network graphs



a b

c d

1 3

2

b a31

Fig 12








x1 = u1 minus b11x1 (13)



1 2

3

u1

u2

b

b11

b22

b33

a32

a21d21

b2u2

Fig 13










x1 = u1 minus 2x1

x2 =1


x3 = 2x2 minus 2x3




A1 =


]A2 =

[minusa bc minusd







x1 = x0 + c12(x2)minus b11x1 (14)



STABLEUNSTABLE





c12(x2) + x0 =b11b22a21

x2




large

x2

C

BA



J =


]





prime12(x2) gt 0 or


x2


142 Oscillations







0 = a11 minus b12(x2)



J =


aprime21(x1)x2 0

]










x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References







x1 = u1 minus b11x1 (13)



1 2

3

u1

u2

b

b11

b22

b33

a32

a21d21

b2u2

Fig 13










x1 = u1 minus 2x1

x2 =1


x3 = 2x2 minus 2x3




A1 =


]A2 =

[minusa bc minusd







x1 = x0 + c12(x2)minus b11x1 (14)



STABLEUNSTABLE





c12(x2) + x0 =b11b22a21

x2




large

x2

C

BA



J =


]





prime12(x2) gt 0 or


x2


142 Oscillations







0 = a11 minus b12(x2)



J =


aprime21(x1)x2 0

]










x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References




x1 = u1 minus 2x1

x2 =1


x3 = 2x2 minus 2x3




A1 =


]A2 =

[minusa bc minusd







x1 = x0 + c12(x2)minus b11x1 (14)



STABLEUNSTABLE





c12(x2) + x0 =b11b22a21

x2




large

x2

C

BA



J =


]





prime12(x2) gt 0 or


x2


142 Oscillations







0 = a11 minus b12(x2)



J =


aprime21(x1)x2 0

]










x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References


STABLEUNSTABLE





c12(x2) + x0 =b11b22a21

x2




large

x2

C

BA



J =


]





prime12(x2) gt 0 or


x2


142 Oscillations







0 = a11 minus b12(x2)



J =


aprime21(x1)x2 0

]










x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References



prime12(x2) gt 0 or


x2


142 Oscillations







0 = a11 minus b12(x2)



J =


aprime21(x1)x2 0

]










x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References








x10d13(x3) =b11b22b33a21a32

x3


x2

A

x3



J =











143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References







143 Adaptation




x1 = minusb21(x1)x2 + x0 (16)

x2 = a12x1 minus b22x2 + u (17)





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References





b21(x1)a12b22

x1 = x0 (18)




] [x1x2

]+

[01

]u y =

[0 1

] [x1x2














Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References








Output y = x3 (112)









x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References








x3 = c32(x2)b33










0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References


0 = a10 minus b11x1 (114)

0 = c21(x1)minus b22x2 (115)

0 = c32(x2)minus b33x3 (116)







z1 = minusb11z1 (117)








xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References




xi(t) =sumjisinAi



cis(x) +sumlisinDi

dil(x)




bii(x) gt βi gt 0






rsumi=1

sumkisinBi

bik le η




and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References



and its derivative

V =

rsumi=1

xi gersumi=1

[aii+1xi+1 minus

sumkisinBi

bikxk

]ge

rsumi=1


xi

ge (rκminus η)

rsumi=1

xi = (rκminus η)V





c1s(x) +sumlisinD1

d1l(x)


c1s(x) +sumlisinD1

d1l(x)







c2s(x) +sumlisinD2

d2l(x)


c2s(x) +sumlisinD2

d2l(x)










1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References






1 2

3 4u

b11

b22

b33

b44

a32

b32

a21

c13

c43

c3u

Fig 111










x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References






x isinMx







x isinMx+ C +D

as in our case







]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References




]=



] [x1x2

]+

[c10c20

]










i














Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References










Conclusions












246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References










246




J 2004 45C565ndashC577













































Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References



























Introduction



General assumptions


Network graphs




Mutistationarity

Oscillations

Adaptation







References

Chapter 1 Structural analysis of biological networks · 1 Structural analysis of biological...

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Transcript of Chapter 1 Structural analysis of biological networks · 1 Structural analysis of biological...