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BioSystems 92 (2008) 270–281

The role of dynamic stimulation pattern in the analysisof bistable intracellular networks

Thomas Millat a, Sree N. Sreenath b, Radina P. Soebiyanto b, Jayant Avva b,Kwang-Hyun Cho c, Olaf Wolkenhauer a,∗

a University of Rostock, 18051 Rostock, Germanyb Complex Systems Biology Center, Case Western Reserve University, Cleveland, OH 44106-7071, USA

c Department of Bio and Brain Engineering and KI for the BioCentury, Korea Advanced Institute of Science andTechnology (KAIST), Daejeon 305-701, Republic of Korea

Received 17 January 2007; received in revised form 3 January 2008; accepted 20 March 2008

Abstract

Bistable systems play an important role in the functioning of living cells. Depending on the strength of the necessary positive feedback onecan distinguish between (irreversible) “one-way switch” or (reversible) “toggle-switch” type behavior. Besides the well- established steady-stateproperties, some important characteristics of bistable systems arise from an analysis of their dynamics. We demonstrate that a supercritical stimulusamplitude is not sufficient to move the system from the lower (off-state) to the higher branch (on-state) for either a step or a pulse input. A switchingsurface is identified for the system as a function of the initial condition, input pulse amplitude and duration (a supercritical signal). We introducethe concept of bounded autonomy for single level systems with a pulse input. Towards this end, we investigate and characterize the role of theduration of the stimulus. Furthermore we show, that a minimal signal power is also necessary to change the steady state of the bistable system.This limiting signal power is independent of the applied stimulus and is determined only by systems parameters. These results are relevant forthe design of experiments, where it is often difficult to create a defined pattern for the stimulus. Furthermore, intracellular processes, like receptorinternalization, do manipulate the level of stimulus such that level and duration of the stimulus is conducive to characteristic behavior.© 2008 Elsevier Ireland Ltd. All rights reserved.

Keywords: Systems biology; Cell signalling; Multistability; Transient behavior; Bounded autonomy

1. Introduction

Living cells must continually sense their external and internalenvironment and induce changes on the basis of this informa-tion. In this way they are able to adapt to their environment,continue or stop their development, and form more complexstructures through intercellular communication (Wolkenhauer etal., 2005b). This processing of information in living cells is car-ried out by signalling networks (Downward, 2001; Wolkenhauerand Mesarovic, 2005). The character of information and thecorresponding responses include a wide range of physical and

∗ Corresponding author at: Systems Biology & Bioinformatics Group, Facultyof Computer Science and Electrical Engineering, Albert-Einstein-Str. 21, 18059Rostock, Germany. Tel.: +49 0381 498 7570/72; fax: +49 0381 498 7570/72.

E-mail addresses: [email protected] (T. Millat),[email protected] (O. Wolkenhauer).

URL: http://www.sbi.uni-rostock.de (O. Wolkenhauer).

chemical quantities, changes in temperature, pressure, waterbalance, concentration gradients, and pH-level.

Within these networks, information is transmitted bydynamic changes in protein concentrations. Besides con-tinuously varying signals, some cellular processes lead todiscontinuous, switch-like responses (Bhalla and Iyengar, 1999;Ferrell and Xiong, 2001; Huang and Ferrell, 1996; Melen et al.,2005). Such a bistable system toggles between two discrete,alternative stable steady states, in contrast to monostable sys-tems (Ferrell, 1998; Ferrell and Machleder, 1998; Markevich etal., 2004). Their separated branches of the steady-state responseallow the implementation of switches in biochemical networks.Other examples include cell cycle oscillations and mutuallyexclusive cell cycle phases (Pomerening et al., 2003; Tyson,1991; Tyson et al., 2002) as well as the generation of biochem-ical “memory” (Eißing et al., 2004; Lisman, 1985; Xiong andFerrel, 2003). Due to their properties, bistable systems also playan important role in development (Melen et al., 2005), cell dif-

0303-2647/$ – see front matter © 2008 Elsevier Ireland Ltd. All rights reserved.doi:10.1016/j.biosystems.2008.03.007


T. Millat et al. / BioSystems 92 (2008) 270–281 271

ferentiation, and evolution of biological systems (Laurent andKellershohn, 1999; Thomas and Kaufman, 2001).

Bistability may arise in signalling pathways that contain atleast one feedback loop or a combination of feedback loopswhose sum of signs is positive with respect to the consideredresponse component (Cinquin and Demongeot, 2002; Thomas,2004). The existence of positive feedback is a necessary butnot a sufficient condition for bistability (Angeli et al., 2004). Astandard graphical test in the phase plane can be used to analyzethese conditions, especially the parameter values, under whichthe system is bistable. Nevertheless, the analysis of complexpositive-feedback systems is difficult. In (Angeli, 2006; Angeliet al., 2004), a method to investigate systems with arbitrary orderwas presented within the framework of monotone systems. Inthis so-called ‘open-loop approach’, the feedback loop is cut andtreated as an additional input. The system can then be treated asa simple input/output system. A generalization to more complexfeedback structures is possible, if the feedback can be summedup in one single apparent feedback loop.

Positive feedback in signalling pathways has to be highly non-linear in order to create two asymptotically stable steady statesin the system. In biological signalling networks such behavioris often realized through ultrasensitive structures, such as cova-lent modification cycles (Goldbeter and Koshland, 1981; Tysonet al., 2003), protein cascades with multiple steps (Asthagiriand Lauffenburger, 2001; Bhalla and Iyengar, 1999; Heinrich etal., 2002; Huang and Ferrell, 1996), or inhibitor ultrasensitiv-ity (Ferrell, 1996; Thron, 1994). A simple autocatalytic reactioncan also bring about bistability (Schlogl, 1972).

In the present paper we do not focus on the investigation ofpossible mechanisms creating multistability but investigate thedynamics of a bistable system independent from the underlyingmechanism. Hence, we assume, that the considered system isbistable or in general multistable. Furthermore, we are not inter-ested in a general investigation of parameter dependencies butfocus on stimulation patterns that influence the bistable behavioras this is relevant for the design of cell signalling experiments.

The paper is organized as follows. In Section 2, we introducea minimal model to discuss dynamic properties of bistable sys-tems and provide necessary definitions. In Section 3, steady-stateproperties of bistable systems are reviewed. In the followingsection we discuss the dynamic behavior of bistable systems,using the introduced model of a bistable system. The conceptof “bounded autonomy” is defined for the system consideredin Section 5. From the example we derive characteristic timescales, which are important for the signal duration of the stim-ulus and the system’s low-pass filter characteristics. The resultsobtained in the previous sections are generalized to multistablesystems. Finally, we summarize our results in the last section.

2. The Model

Throughout this study we use as an example the mutually-activated enzyme network, as described in Tyson et al. (2003)(Fig. 1). In this network, a linear system is coupled with asigmoidal system through a positive feedback loop. The cor-

Fig. 1. Graphical representation of the system used as a case study to dis-cuss multistability. The response component R is produced in a linear pathway,induced by an external stimulus S. The response component activates the enzymeE, which in turn facilitates the production of R through a positive feedback loop.This mechanism has been described as “mutual activation” (Tyson et al., 2003).The modification of the enzyme E follows a covalent reaction scheme (Goldbeterand Koshland, 1981; Tyson et al., 2003).

responding mathematical representation is given as

dR

dt= k0E

∗(R) + k1S − k2R, (1a)

E∗(R) = G(k3R, k4, J3, J4), (1b)

where the response component is R(t) and the external stim-ulus or input is S(t). For notational convenience, we do notshow the dependence on time for these two variables fromnow on. The kinetic constants ki and the Michaelis–Mentenconstants Ji (Segel, 1993; Cornish-Bowden, 2004) determinethe chemical properties of the involved biochemical species.Further, R(0) ≡ R0 denotes the initial condition, and G(·) theGoldbeter–Koshland function (Goldbeter and Koshland, 1981;Tyson et al., 2003) defined for system (1) as:

G(k3R, k4, J3, J4) = 2k3RJ4

X+√X 2 − 4(k4 − k3R)k3RJ4

, (2)

where

X = k4 − k3R + k4J3 + k3RJ4.

This function describes the concentration of the modified formE∗(R) as a steady-state equation. It is frequently used consid-ering (de)modification cycles, see Fig. 1 and Appendix A. TheGoldbeter–Koshland function (3) has a typical sigmoidal shape,as shown in Fig. 2. Due to its highly nonlinear behavior it cancause multiple steady states in the investigated system (1). Theuse of the Goldbeter–Koshland function assumes that the modifi-cation of enzyme E is much faster than the change of the responsecomponent R. This allows us to use a quasi-stationary approxi-mation (Millat et al., 2007). From Eq. (1), the production rate ofR is separated into a stimulus dependent and enzyme dependentcontribution. Due to the positive feedback loop the productionrate is regulated in an autocatalytic fashion. More on modelingof activation/deactivation cycles in signaling pathways can befound in Salazar and Hofer (2006).


272 T. Millat et al. / BioSystems 92 (2008) 270–281

Fig. 2. The ratio of modified enzyme and its inactive form as function of theresponse R considering the Goldbeter–Koshland function (3). The sigmoidalbehavior is highly nonlinear and provides a continuous switching behavior atk3R = k4. The parameters are chosen as specified in Appendix E.

We consider two types of input stimulus S. The first one is astep input defined as:

S = Smax · u(t), (3)

where Smax is the signal amplitude or strength, and u(t) rep-resents the step function (Bracewell, 1999) with u(t) = 1 fort ≥ 0, and u(t) = 0 for t < 0.

For a cellular environment, one would not expect a stimulusthat maintains a constant level over prolonged periods of times.If changes in the stimulus are much slower than the response ofthe system one can assume a quasi-stationary state (Millat et al.,2007) but in many cases one deals with various input profiles.We therefore also analyze the system using a pulse input:

S = Smax[u(t) − u(t − �τ)], (4)

where �τ > 0, is the pulse width or duration.In the following section, we review steady-state properties of

the bistable system (1). The nominal parameter values used fornumerical simulations can be found in Appendix E.

3. Steady-State Properties

In this section, we briefly summarize the steady-state prop-erties of the bistable system we are considering. We obtain thesteady state of system (1) with a step input (4) as a solution ofthe balance equation,

0 = rate of production − rate of degradation,

where production and degradation rates are balanced such that nomacroscopic net change is measurable. Specifically, for system(1) the balance equation can be written as,

k0E∗(RSS) + k1Smax − k2RSS = 0, (5)

where RSS refers to the steady states.The steady-state properties of a dynamic system provide us

with a wide range of information on bifurcation, stability andcontrol analysis (Fell, 1997; Heinrich and Schuster, 1996). For

Fig. 3. Steady-state step stimulus–response curves for different feedbackstrengths (k0/k1). The system behavior changes from a one-way switch to atoggle switch and finally to a monostable system. The second curve from theleft, separates the toggle-switch from a one-way switch. At Smax = 0 there is asecond critical point. The third curve is a typical example for hysteresis. If wefurther decrease the feedback strength, the critical points coincide and finallydisappear. For weak positive feedback the system is monostable and asymp-totically approaches the response curve for k0/k1 = 0, representing the stepstimulus–response of a linear system. Additionally, we plot the unstable regionin gray. In order to represent the complete course of all curves we extend theabscissa to negative stimuli, which is of course physically irrelevant.

complex reaction networks, the balance Eq. (6) can have mul-tiple solutions. The classification of the associated roots andtheir relevance cannot be determined analytically, but requiresknowledge of the particular biological system under considera-tion. Since we are dealing with measurable physical quantities,i.e., biochemical concentrations that are non-negative, the feasi-ble solutions should be real and non-negative. A mathematicalproof for the existence of steady states and their stability analy-sis is given in (Tyson and Othmer, 1978). The steady states, as afunction of a step input with stimulus strength Smax, can be com-puted by simply solving Eq. (6) numerically and selecting onlythe feasible solutions. This is visualized in the step stimulus–response curve (Fig. 3) for various feedback strengths. As seenfrom the figure there are either one or three steady states fordifferent parameter values.

On further investigation, we realize that for bistable behav-ior, the balance equation must have at least three bio-physicallyrealizable solutions. Of these, two stable branches are separatedby an unstable branch as shown in the bifurcation plot (Fig. 3).This unstable branch plays an important role in the dynamicsof bistable systems. From these considerations, the characteris-tic Eq. (6) has to be a polynomial of at least third degree, or ahighly nonlinear function. With a proper chosen set of parame-ters, the sigmoidal Goldbeter–Koshland function (3) fulfils thiscondition. It is clear from Fig. 3 that we can define

RSS ∈ {RSU, RUN, RSL}, (6)

where we drop the direct dependence on Smax for representa-tional convenience. RSU and RSL are the stable upper and lowersteady state respectively and RUN is the middle unstable state(indicated by the dashed line in Fig. 3).



What the bifurcation diagram does not reflect is the transientbehavior between the previous and the new steady state. Thetransient phase plays an important role in various biological pro-cesses, including cell differentiation (Laurent and Kellershohn,1999) and apoptosis (Aldridge et al., 2006).

As the feedback strength in system (1), represented by theratio k0/k1, is varied, the steady-state step response has differentcharacteristics (Fig. 3). If the ratio is k0/k1 ≥ 29.4, the systemacts as a one-way switch. Once activated it remains in this state.In an intermediate range of the feedback strength, the system isa reversible switch and shows hysteresis-like behavior (Angeliet al., 2004; Ferrell, 1998; Tyson et al., 2003). In both cases,the solution of balance Eq. (6) consists of three branches (Tysonand Othmer, 1978), two stable ones (solid lines for RSU andRSL), separated by an unstable branch (dashed lines for RUN), asdescribed before. From a biochemical point of view the stabilityof the response component is determined by the partial derivativeor the Jacobian of system (1) evaluated for a given parameterset (for instance nominal parameter set in Appendix E) and aspecific Smax (Gray and Scott, 1994),

J = ∂

∂R

dR

dt= k0

∂E∗(R)

∂R− k2,

={

negative for : RSU and RSL (stable),

positive for : RUN (unstable),(7)

where the stable steady states are minima of this function andunstable states are maxima (see Fig. 4). The Jacobian J is animplicit function of Smax since system (1) is affine. A smallperturbation in response component R will relax back to theformer steady state if Eq. (8) is negative. The correspondingsteady state and its neighborhood are stable (Schlogl, 1971).Otherwise, the system is unstable and an infinitesimal fluctuationmoves the system to one of the neighboring stable states.

For a one-dimensional system as in (1), at steady state (RSS),the Jacobian represents the eigenvalue of the system linearizedaround this state. Using MAPLE (Maplesoft, Waterloo, Ontario)we computed the Jacobian (8) at the steady states with nominalparameters (Appendix E) and for stimulus Smax ∈ [0, 40]. Whilethe real part of the eigenvalue is negative for RSU and RSL,indicating stability, the real part of the eigenvalue is positive forRUN, indicating instability.

The three branches are separated by critical or bifurcationpoints, where the system’s behavior changes abruptly. The crit-

Fig. 4. Schematic representation of the stability criterion (8). The stable statescorresponds to minima of the Jacobian and unstable states to a maximum. Smallperturbations of a stable steady return the system back to the same state, whereasthe system moves from the unstable state into one of the stable states.

ical points define a range where no stable steady state exists(presented by the gray area in Fig. 3).

Note how varying the feedback strength impacts on the sys-tem’s behavior. For instance, when we weaken the feedbackstrength, the system eventually loses its bistable nature. In thelimit of vanishing feedback k0/k1 → 0, the system (1) is of first-order and linear. This limit also defines the asymptotic behaviorfor small external stimuli, where only a small amount of enzymeE is modified (corresponding to weak feedback). Note that wedid not change the properties of the enzymatic modification cyclebut only modified k0 to vary the feedback strength. The modi-fication cycle of the system is considered ultrasensitive for allparameter sets used.

The following section introduces dynamic characteristics ofbistable systems.

4. Dynamic Behavior

The response of biological systems to changing environmen-tal or cellular conditions is a dynamic process (Wolkenhauer etal., 2005a). The system’s transient relaxation into a new steadystate depends on the previous state of the system, the inputstrength and duration, as well as, on the structure and the kineticparameters of the biological system under consideration.

In multistable systems the transition between the differentstable branches is of special interest. Extending the discussionfrom Section 3, we use the terms “lower branch” and “upperbranch” for the bistable system to distinguish between bothstable solutions of the balance Eq. (6). Note, the assignmentof an “off” and an “on”-state depends on the biological sys-tem, not the mathematical solutions. In the following sectionwe discuss the transient response to an external stimulus S. Thekinetic parameters and total concentrations remain unchanged(see Appendix E). Initially we investigate the transient responseof a step input, followed by that of a pulse input. In the lattercase, the change of stimulus allows us to separate the dynamicsinto periods with different but constant values of the stimulus S(Bracewell, 1999).

4.1. Step Response of the Bistable System

Following Tyson et al. (2003), for a given set of parameters,initial condition R0, we define the critical stimulus S = Scritu(t)as the smallest step stimulus amplitude that changes the cellularresponse “abruptly and irreversibly”. Any Smax > Scrit is labeledsupercritical stimulus. In contrast, a subcritical stimulus refersto an amplitude that does not affect such change in the cellularresponse, i.e., Smax < Scrit.

To begin with, we investigate the dynamic behavior of system(1) for a specific subcritical stimulus Smax < Scrit with differentinitial conditions (Fig. 5). Depending on the initial concentra-tion R0, the transient response signal tends to either the loweror the upper stable branch. The considered system therefore hassome form of memory. The unstable solution RUN of balanceEq. (6) acts as a separatrix between both behaviors. If the ini-tial concentration is lower than the unstable state, the responseapproaches the lower stable branch. In the other case, the new



Fig. 5. Steady-state response for a subcritical signal Smax < Scrit for differentinitial conditions. The unstable state is a separatrix. The gray area denotes theforbidden region.

steady state is located on the upper branch. The stable states actas attractors and the unstable state as a repeller. This propertyof an unstable state will become important for our subsequentdiscussion. Of course, for a supercritical stimulus Smax > Scrit,the concentration of the response component always tends tothe upper stable state. Additionally, we plot in the figure theforbidden region, where no stable solution of balance Eq. (6)exists. Because system (1) acts as an one-way switch for theused nominal parameters, this region is defined through the sumof the unstable area and the range of irrelevant solutions, seealso Fig. 3.

Fig. 6 shows us how we can use a given initial condition R0,to determine the critical stimulus Scrit from the middle segmentof the curve (corresponding to the unstable steady state). Conse-quently, for R0 < RSL, or for RSL < R0 < RUN, the trajectoryof R(t) converges to RSU asymptotically. In essence, for a givenR0 there exists a corresponding Scrit.

Fig. 6. Relationship between initial condition R0 (rectangles), stable steadystates (RSU, RSL), unstable steady state (RUN) and the corresponding criticalvalue of step amplitude Scrit. Arrows indicate the asymptotic convergence oftrajectories over time, starting from a given initial condition. Nominal parametershave been assumed (Appendix E).

Fig. 7. Transient pulse response of R to a supercritical stimulus S > Scrit asa function of the signal duration �τ from the origin. After �τ the externalstimulus returns to its initial value. The grey region denotes the forbidden areawhere no stable solution exists. The critical pulse duration for a critical pulsewith amplitude Smax = 14 is �τcrit = 3.1.

4.2. Pulse Response of the Bistable System

Apart from stepwise constant stimuli, biological systemsoften have to respond to transient signals. Thereby, the responseto pulse-like signals is of special interest. The reason for such asignal may be a perturbation in the environment, degradation ofligands (Lauffenburger and Linderman, 1993), receptor internal-ization (Madshus, 2004) amongst others. The system responsefor a pulse input (5) is shown in Fig. 7 for the system with thenominal parameters. Switching depends on R0, Smax and �τ.Starting with initial condition R0 = 0, we apply a supercriticalpulse stimulus Smax = 14 for varying pulse widths �τ. For sig-nal durations �τ < 3.1 the response asymptotically convergesto zero. If the signal duration exceeds this limit, the responseapproaches a steady state on the upper branch and remains onthis branch. This pulse duration is defined as the critical pulseduration �τcrit, and the corresponding pulse amplitude is definedas Scrit. We therefore note that it is the combination of sig-nal strength and signal duration which defines the steady state.Pulses which do not fulfill this criterion cannot change the stateof the system. The bistable system cannot resolve these pulsesand acts as a low-pass filter. Given an initial state R0, we can finda combination of Scrit and �τcrit, such that the system switchesto the upper branch (and asymptotically converges to the uppersteady state). For the nominal parameters, the system asymptot-ically converges to RSU = 0.38, which corresponds to Smax = 0(see Fig. 6).

In the next section, we discuss the characteristics of theswitching surface obtained for the pulse response of the sys-tem for varying {R0, Scrit, �τcrit}, in terms of the concept ofbounded autonomy for a single level system.

5. Bounded Autonomy in Single Level Systems

Bounded autonomy has been described in Mesarovic et al.(2004) for multi-level systems where the higher level does notintervene to modify the behavior of the lower level system as



Fig. 8. Switching surface of the system’s pulse response to an initial condition R0, critical stimulus Scrit and critical pulse duration �τcrit. Regions A–F representcharacteristics of the switching surface. Heatmap color refers to the critical power Pcrit. A contour plot included in the inset indicates further details about the surface.

long as the system is ‘performing’ at a ‘satisfactory’ level, andintervenes to modify the behavior of the lower level only in orderto bring the complex system performance to a satisfactory level.We will adapt this concept to a single-level system using theconcept of a switching surface.

To this end, for a pulse input (5) we plot in three dimen-sions the critical stimulus Scrit, pulse width or duration �τcrit,and initial condition R0 as represented in Fig. 8. The plot wasgenerated by fixing Smax and �τ and gradually increasing R0in the dynamic system simulation till the system pulse responseswitches from the lower branch to the upper branch in steadystate. The value of the pulse strength is then the critical stimulusScrit and the pulse duration is the critical duration �τcrit for thegiven initial condition R0. The simulation period was 15 units oftime with ranges for input and initial condition was as follows:�τcrit ∈ [0, 15] with resolution 0.01; Scrit ∈ [0, 50] with reso-lution 0.01; and for R0 ∈ [0, 0.2] with resolution 0.001 whichcomes to maximum of (15/0.01) × (50/0.01) × (0.2/0.001) =1.5 billion dynamic simulations. These were done using MAT-LAB (The MathWorks, Natick, MA) with ‘ode45’ equationsolver on a 2.81 GHz, 2 GB RAM Windows based machine withAMD Athlon processor and numerical accelerator card. In real-ity, the number of simulations conducted were 2.073 millionpoints on the switching surface, since only simulations for val-ues of R0 below the switching surface were considered. Lowerresolution simulations (less than 100,000 points) resulted in arti-facts that disappeared when high resolution simulations wereconducted.

We now define the input signal power as the integral of thestimulus over time as:

P =∫ ∞

0S(t) dt. (8)

Particularizing this for the pulse input with Pcrit as the power,

Pcrit = Scrit · �τcrit.

Then the critical power represents the “dosage” or input neededfor the system to switch assuming that the objective is to inducethe system to switch. The color “heat-map” in Fig. 8 indicatesthis critical power as a function of R0, Scrit, and �τcrit. Belowwe describe regions of the switching surface in Fig. 8. There is adeadband for �τcrit ∈ [0, 0.1] since the system does not switchfor pulse width equal to zero, and the pulse amplitude is not high.

Region A: Low values of R0 and �τcrit, require a high value ofpulse amplitude or Scrit for switching.

Region B: With ever so slight increase in �τcrit and for thesame low values of R0, a much lower value of Scritis needed for switching.

Region C: At a fixed, but high level of R0 (for example R0slightly below 0.2) the switching is independentof critical pulse duration (�τcrit = [0.1, 15]), and,only a low level of stimulation amplitude Scrit < 8is needed for switching. Also for the Scrit axis, for8 > Scrit, the switching surface falls at a nearly con-stant rate of 0.04 along the initial condition range0.2 > R0 > 0.18.

Region D: As we move along the Scrit axis, for 10 > Scrit > 8,the switching surface slopes faster along the initialcondition range R0 = [0.15, 0.18).

Region E: Between 10.4 > Scrit > 10.1, the switching surfacerapidly falls indicating that the system can switchfor smaller values of R0 for given �τcrit and ever sosmall increase in Scrit, finally having the ability toswitch at R0 = 0.

Region F: This flat region indicates that with initial conditionR0 = 0, the system can switch for a wide range ofa rectangular area spanned by Scrit ∈ [10.4, 50], and�τcrit ∈ [1, 15], and bounded by the curve formed bythe intersection of the switching surfaces and the Scritand �τcrit plane.



Fig. 9. Plot of the minimum critical power (Pcrit) required for the system toswitch from a specific initial condition R0 and pulse input.

Further, a contour plot given in the inset in Fig. 8 indicatesa lip that is formed at Region C, and also illuminates the shapeof the switching surface. If the system state and the stimulusis below the surface indicated in Fig. 8 (Region F), then thesystem response stays within bounds and reverts back to thezero state eventually. If however, the system state and the stim-ulus are at or above the switching surface then the system stateswitches to a higher value than the initial state. This phenomenonis interpreted as bounded autonomy for single level systems, i.e.,the system is autonomous below the switching surface, and isdependent on the system state and input stimulus combinationto switch to a different state.

One philosophical approach to understanding biology is toconsider the biological system purposive or teleological. Perfor-mance is then related to the objective of the system. For instanceconstitutive activation (such as switching) of a key system statecould be implicated in development of cancer, e.g., JAK2 consti-tutive activation is implicated in chronic myelogenous leukemia(CML) in murine models. To consider a system to be performingwith the objective of not becoming cancerous, then constitutiveactivation resulting from bistability needs to be avoided. If thevalue of a parameter (say a rate constant) is bounded and nottriggering switching then the system can be defined to be per-forming at a satisfactory level. Thus within these bounds of theparameter the system behavior remains non-cancerous and thusfollows the organizing principle of bounded autonomy.

5.1. Critical Power for the Bistable System with a PulseInput

Fig. 9 depicts the minimum critical power required to switchfrom a given initial condition R0, i.e.,

Pcrit, min(R0) = minScrit,�τcrit

Pcrit(R0).

The trend in Fig. 9 indicates that as the initial conditionincreases the amount of minimum critical power needed toswitch to the upper branch decreases. This follows from thefact that an increase in the initial response component decreasesthe displacement between R0 and the unstable state. As one can

see in Fig. 7, the chemical activity of the considered system ishigh in the forbidden region. Hence, it takes less time to bridgethe gap between the initial value and the unstable state.

Interestingly, for low values of R0 < 0.1, the curve of thePcrit, min fluctuates (Fig. 9). The exact reason for such fluctua-tions is hard to pinpoint. We recognize that analytically findingthe minimum critical power is not possible. Further we note thatthe simulated range of the input amplitude and duration limitsthe numerically computed minimum critical power, which wehave verified numerically. Hence Fig. 8 is indicative. Also, givena Pcrit, min value, the open circles indicates the appropriate R0to reach this critical minimum power. One interpretation is thatgiven the state of the biological system (indicated by the initialcondition) the minimum dosage (indicated by the power) neededfor the system to switch can be read off from the curve.

6. Estimation of Critical Signal Duration

As the numerical analysis of system (1) with a step inputshows, not only signal amplitude Smax defines the final state(see Eq. (6)). In this section we estimate the critical durationof stimulus �τmin, which is the minimal time a step stimulushas to be applied to reach the unstable state given that the ini-tial condition of the system R0 < RUN. Unfortunately, system(1) cannot be solved analytically due to the complexity of theGoldbeter–Koshland function (3).

For the sake of simplicity but without loss of generality, werestrict our discussion to the one-way mutual activation switch(1) and investigate the transient changes between the lower to theupper branches for an supercritical stimulus. We assume a smallconcentration of the response component to begin with. We canthen expand the Goldbeter–Koshland function (3) with respect tothe response component R. The derivation is given in AppendixA. This is then used to replace the complex Goldbeter–Koshlandfunction G(·) with the leading first-order term of the expansion,yielding the linear differential equation

dR

dt= k1Smax − (k2 − k0C1)R. (9)

Due to the positive feedback loop, there is now an additionalcontributing term, which depends on the concentration of R.This term reduces the rate of degradation, where the constantC1 follows from the Taylor expansion (17). The resulting lineardifferential equation can be solved analytically and we obtainfor the critical signal duration

�τmin = 1

k′2

lnRSS − R0

RSS − RUN, (10)

where we introduce the new effective rate constant

k′2 = k2 − k0C1 (11)

and the steady state concentration RSS = k1Smax/k′2.

In Fig. 10, the approximation (11) is compared to a numer-ical analysis of system (1). Due to an underestimation of thepositive feedback contribution, the approximation deviates forweak stimuli from the intrinsic critical duration time �τmin.



Fig. 10. The critical signal duration �τmin as a function of the step stimulusamplitude Smax. We compare the results of a numerical calculation (solid line)with the linear approximation (11). For weak supercritical stimuli the approxi-mation differs from the numerical result because of the underestimated positivefeedback contribution in the production rate of the response component. Forhigher levels of stimulus, the approximation agrees with the numerical solution.

Here, the nonlinear character of the Goldbeter–Koshland func-tion (3), describing the feedback dependence, is important. Theinclusion of higher orders of the series expansion (A.2) wouldprovide a better description for weak supercritical signals, butthe resulting differential equation becomes nonlinear and can-not be solved with respect to the critical duration. For strongsignals both representations tend to a common asymptote (seeAppendix C)

�τmin ≈ RUN − R0

k1Smax

which is inversely proportional to the production rate k1Smax anddirect proportional to the displacement between unstable stateand the initial state.

7. Signal Power

In Section 4, we demonstrated that for system (1) with aninitial condition lower than the unstable equilibrium R0 < RUN,and a step input, there exists a minimum input amplitude Scrit, forwhich the system switches to the upper stable equilibrium. Nextwe define �τmin as the minimal necessary signal duration of asupercritical signal Scrit such that response reaches the unstableequilibrium R(�τmin) = RUN. From our discussion in Section4, it is clear that

limt→∞R(t) = RSU.

It is now appropriate to modify the signal power definition in(9) and extend this analysis for a step input for the above caseby defining the signal power needed Pmin as,

Pmin = Smax · �τmin. (12)

A comparison of the numerical analysis for the nonlinear sys-tem (1) and its linear approximation (10) is shown in Fig. 11.The signal power increases for weak supercritical signals andtends to infinity in the limit Smax → Scrit. On the other hand, for

Fig. 11. Semilogarithmic plot of the signal power as a function of the externalstimulus for the nonlinear system (1) (solid line) and for the linear approxima-tion (10)(dashed line). The dash-dotted line is the minimal signal power (14)following from the strong stimuli approximation. The line defines the minimalsignal power required to switch the system between the stable states.

strong stimuli the signal power approaches a constant limitingvalue Pmin. We are now going to calculate this limiting value.Towards this end, we again use the logarithmic expansion of thecritical duration time (11) (Appendix C) and neglect all termshigher than first-order with respect to the stimulus S. Due to thestimulus-dependent prefactor of Eq. (11) the remaining depen-dencies on the signal cancel out so that we obtain the constantlimiting signal power (see Appendix D for a derivation)

Pmin = RUN − R0

k1. (13)

This defines the minimal signal power which is necessary topush the system from the lower to the upper state. It dependsonly on the difference between the unstable and the initial state,and the kinetic parameter k1. The critical power decreases withdecreasing displacement of initial and unstable state. This isqualitatively equivalent to the critical power for pulse input inFig. 9, where we investigated the critical power as function ofthe initial state for a fixed stimulus. It is interesting that thereis no contribution of the feedback loop and the degradation rateto the derived minimal signal power. For weaker supercriticalstimuli as in Fig. 9, where the positive feedback is dominant,this leads to an increase of the necessary signal power. On theother hand, this will decrease the sensitivity of the system in theneighborhood of the critical point, ensuring a proper decision onthe basis of stable stimulus and not based on small fluctuations.

8. Generalization to Multistable Systems

In the last section we investigated and estimated the proper-ties of bistable systems. In this section, we extend our findings tomultistable systems which have more than two stable branches.Some general features of the dynamical behavior of multi-stable system can be directly inferred from bistable systemsinvestigated in the previous sections. To this end, we considermultistable systems as a combination of neighboring bistablesystems. Then the transition between neighboring stable states



depends on the combination of external stimulus and its duration.As in the case of bistable systems, each pair of of stable branchesis separated by an unstable branch. This unstable branch has tobe exceeded also in the case of multistable systems. Then thesystem approaches the neighbored stable branch, even if thestimulus switch back to a level corresponding to a steady stateat the former branch. It should be noted, that a signal is requiredwhich has a stable solution on both branches. Furthermore, in ourdiscussion about bistable systems we use a “toggle switch” asan example which switches irreversibly. Reversible bistable andmultistable can also switch back to lower stable branch. Also, theseparating unstable branch determines the characteristic signalduration.

Even if the analytical analysis of multistable systems ismore complex than our example of a mutually activated system(1), from a mathematical point of view and from their diver-sity of initial conditions, we are able to transfer directly somedynamical properties of bistable systems. The transient switch-ing between neighboring stable branches underlies the samequalitative restrictions as in the bistable system.

9. Discussion

Multistable systems play an important role in the functionand regulation of biological systems. Depending on the valuesof kinetic parameters the cellular system can have multiple sta-ble steady states, which are separated through unstable ones.Whereas the final state of the system is determined by the sta-ble state, the dynamic response depends on transient propertiesof the corresponding stimulus and is strongly affected by theunstable state between the two neighboring stable branches. Thetransition between the stable branches is triggered by changingthe signal strength of the stimuli in case of a step input. A super-critical signal with respect to the current stable branch leads toa jump to the next stable branch. If the system is a reversibleswitch, a subcritical signal forces the system back to a lowerbranch until the lowest one is approached.

Note that the widely used stimulus–response analysis doesnot provide us with information about the dynamics of the inves-tigated system. The presented steady state, as a function of a stepinput stimulus, is formally reached after an infinite time. From adynamic perspective this means, the system is much faster thanthe variation of the stimulus. In biological systems, say in sig-nal transduction, we deal with the reverse situation, where theenvironmental changes are faster or in the same time scale as theintrinsic time the system requires to adapt to the new context.The importance of studying dynamical properties of bistablesystems has recently been shown in the context of apoptosis(Aldridge et al., 2006), where crucial decisions are determinedby the transient activation of active caspase-3. How the choice oftime-delays incorporated into the mathematical models changethe dynamic behavior without influencing the steady state ofthe considered system was demonstrated in Veflingstad et al.(2005). As a consequence, the consideration of the dynamicbehavior is important as well as the consideration of steady statesto understand the function and the structure of a biochemicalnetwork. Different biochemical systems may have qualitatively

equal steady states, but their transient behavior can discriminatebetween biologically favorable models from unfavorable ones.

In bistable systems mainly two questions arise: What are theconditions for a significant change of system’s properties andsecondly, how long does it take? A necessary condition for achange of the systems’ properties is a supercritical signal, push-ing the system from one stable state to a neighborhood stablestate. In this context, supercritical means that the signal lies out-side the current stable branch. However, as we have shown, thesignal has to be applied for a certain critical time �τcrit (forpulse input) and �τmin (for step input). The reason is that theunstable branch acts as a separatrix. Only, if it is exceeded, thesystem approaches the new stable branch, otherwise it returnsto its previous branch. Using the mutually activated system asan example, we showed that the critical duration time dependson the displacement of the initial and the unstable state fromthe corresponding steady state for a step input. Of course, thekinetic parameters do also determine this duration. In general,we can conclude that the critical duration will decrease withincreasing external stimuli. As shown in Sections 6 and 7, thesignal power has to fulfil a limiting criterion. It follows from thedemonstrated strong stimulus approximation, that there is a min-imal signal power. The applied signal power has to exceed thisminimum to push the system from one stable state to the other.It should be mentioned, that the necessary power increases ifthe stimulus decreases, even if the positive feedback loop inour example becomes more important for the dynamics. Bothdynamic properties have consequences for the signal processingand the decision making of biological systems.

The concept of bounded autonomy was introduced for sin-gle level systems, with the autonomy being characterized by aswitching surface (see Section 5). These concepts, combinedwith the minimum critical power required for switching, are ofgreat practical value. For instance, given the system state, wecan use them to evaluate the minimum dose that has a sufficienttherapeutic index. This concept needs further elaboration andwill constitute future work.

Biological signals are often noisy and one reason for theappearance of bistability in systems may be the fact that theyhave an intrinsic mechanism to suppress noise. We say they areacting as a “low-pass” filter. This is particularly important ifthe switching is crucial for the further development of the bio-logical system, such as in cell differentiation or apoptosis. Theintrinsic noise suppression can also have a stabilizing effect onoscillating biochemical systems, e.g., the circadian rhythm. Ina stochastic simulation (Barkai and Leibler, 2000), the authorscompared the stability of a three-step oscillating system with anegative feedback loop and a combination of negative and posi-tive feedback. They found, that the combination of negative andpositive feedback makes the system much more stable againstits own intrinsic noise; especially, the period that was stabilized.This finding can be explained with the properties of the bistablesub-system discussed in this paper.

As shown elsewhere, the dynamic properties of biochemicalsystems depends on the chosen level of approximation in themathematical representation of the considered model (Millat etal., 2007). The numerical study of different mathematical rep-



resentations of the same model system shows strong deviationsfor weak supercritical signals. In particular, the critical signalduration was much longer for a more detailed representation.Furthermore, the steady states of the system may be also affectedby the approximations used. As recently showed in Bluthgen etal. (2006), the neglect of complexes in the derivation of theGoldbeter–Koshland function may lead to dramatic changes inthe properties of a system. The oscillations in a MAPK-cascadewith negative feedback, predicted in Kholodenko (2000), wereshown to vanish if one considers more complex representations.The reason is the disappearance of the ultrasensitive behavior inthe cascade (Goldbeter and Koshland, 1981; Millat et al., 2007).

Naturally, the values of kinetic parameters play an importantrole for the dynamic properties of a bistable system. However, aswe have shown, an appropriate design of experiments requiresthe experimentalist to consider well-defined patterns of stimula-tion. Whereas bistable systems are often discussed only on thelevel of their steady states, the response to time-varying stimulistrongly influences their dynamical properties.

Acknowledgments

O.W.’s contributions were supported by the European Com-munity as part of the FP6 project AMPKIN and by the GermanResearch Foundation (DFG) as part of the Research TrainingGroup 1387 dIEM oSiRiS. T.M. and O.W. acknowledge sup-port by the German Federal Ministry for Education & Research(BMBF) as part of the European Transnational Network - Sys-tems Biology of Microorganisms (SysMo). S.N.S., R.P.S., andJ.A. gratefully acknowledge support of US NIH grant K25 CA113133, and US NIH grant P20 CA 112963. K.-H. C.’s workwas supported by the Korea Science and Engineering Founda-tion (KOSEF) grant funded by the Korea government (MOST)(M10503010001-07N030100112) and also supported from theKorea Ministry of Science and Technology through the NuclearResearch Grant (M20708000001-07B0800-00110) and the 21CFrontier Microbial Genomics and Application Center Program(Grant MG05-0204-3-0).

Appendix A. Expansion of the Goldbeter–KoshlandFunction

The modification/demodification of a protein can bedescribed as a system of two Michaelis–Menten like steps(Goldbeter and Koshland, 1981)

E + Rk−1�k1

C1k2−→R + E∗, (A.1a)

E∗ + Pk−3�k3

C2k4−→P + E, (A.1b)

where R modifies the enzyme E and P catalyzes the reversereaction. For example, in case of phosphorylation R is a kinaseand P a phosphatase. The steady-state solution of this reactionscheme is the Goldbeter–Koshland function (3) (Goldbeter andKoshland, 1981; Tyson et al., 2003) where we assume that thephosphatase P is constant over time (Millat et al., 2007).

If we assume a small amount of the response component R,we can expand Eq. (3) into a Taylor series (Abramowitz andStegun, 1972) with respect to R

E∗(R) = J4

1 + J3

k3

k4R + 1 + J3 + J3J4

(1 + J3)3

k23

k24

R2 +O(R3)

= C1R + C2 R2 +O(R3), (A.2)

where we introduced the constant prefactors Ci for each expan-sion order i. These prefactors are determined by the ratio of thekinetic coefficients k3 and k4 and a combination of the MichaelisconstantsJ3 andJ4. We can now insert the series expansion (A.2)into the differential equations, which describe the temporal evo-lution of the system under consideration (1). This is done in thefollowing sections.

Appendix B. Linear Approximation

The bistable system (1) cannot be solved analyti-cally. We therefore use the series expansion (A.2) of theGoldbeter–Koshland function introduced in Appendix A witha step input, S = Smax · u(t). In this linear approximation weneglect all terms higher than first-order. Inserting this approxi-mation into the differential equation for the response component,leads us to the linear differential equation

dR

dt= k1Smaxu(t) − (k2 − k0C1)R. (B.1)

Due to the positive feedback, the degradation rate is reducedby the first-order term of the expansion. For a constant exter-nal stimulus, this approximate differential equation can thenbe solved analytically. The temporal evolution of the responsecomponent is

R(t) = k1Smax

k′2

(1 − exp{−k′2t}) + R0 exp{−k′

2t}, (B.2)

where the first term describes the approach to the steady stateand the second the ‘degradation’ of the initial state R0. Theeffective rate constant k′

2 = k2 − k0C1 describes the reduceddegradation due to the positive feedback. The steady state forthe linear approximation is

RSS = k1Smax

k′2

. (B.3)

Note, because we neglect higher feedback terms, this steadystate is slightly different from the steady state of the nonlinearsystem (1).

Consider the case such that R0 < RUN (see Fig. 6). Definethe minimum duration �τmin as the duration for which the stepstimulus should be on such that R(�τmin) = RUN.

After a sequence of transformations, we obtain for the criticalduration time

�τmin = 1

k′2

lnRSS − R0

RSS − RUN. (B.4)

The minimum duration �τmin depends on the displacementof initial state R0 and steady state RSS, on the displacement of



unstable state RUN and steady state RSS, and on kinetic parame-ters, especially on the effective rate constant of the degradationk′

2. The effect of the external stimulus is hidden in the steady stateRSS, whereas the new signal strength determines the unstablestate RUN.

Appendix C. Asymptotic Behavior for Strong Signals

The critical signal duration, derived in Appendix B, dependson the external stimulus, which determines the steady state of theconsidered system. For strong pulse-like signals we are able tofind an asymptotic expression for the duration time. We rewriteEq. (21) in a slightly different form

�τ = 1

k′2

lnx0 − 1

x∗ − 1, (C.1)

where we transform the arguments of the logarithm into a(x − 1)-form. For a simpler notation we introduced the abbrevi-ations

x0 = k′2R0

k1Smaxand x∗ = k′

2RUN

k1Smax.

For strong stimuli Smax k′2RUN/k1 we have x � 1 such that

we can expand the logarithm

ln(x − 1) ≈ iπ −∞∑j

xj

j,

where i = √−1 is the imaginary unit. Finally we obtain thelogarithmic ratio

lnx0 − 1

x∗ − 1≈ x∗ − x0 + x∗2 − x2

0

2+O(x∗3 − x3

0). (C.2)

This depends on the difference of the contributions for the unsta-ble state and the initial state. Insertion this into Eq. (C.1) leadsto the asymptote

�τ = RUN − R0

k1Smax. (C.3)

The asymptote for strong signals is inversely proportional tothe signal-dependent production rate only. It is scaled by thedifference between the unstable state and the initial state. Inter-estingly, there is no dependence on the degradation rate and thefeedback strength.

Appendix D. Signal Power

The signal power is a further interesting property of the con-sidered dynamic switching process. It is defined as the integral

P =∫

S(t) dt

for the time dependent external stimulus. Here we are interestedin the minimum signal power needed for the system state to reachthe unstable state. In other words, we simplify the minimumsignal power to the product of the magnitude of the step stimulus

and the minimum time �τmin needed for the state to reach theunstable state RUN from an initial condition R0 < RUN, i.e.,

Pmin =∫ �τmin

0Smax dt = Smax · �τmin.

Insertion of the derived expressions (B.4) for the critical time�τ gives

Pmin = Smax

k′2

lnRSS − R0

RSS − RUN, (D.1)

where the steady state is given by Eq. (20). If we assume onlystrong stimuli, we can use the logarithmic expansion (C.2) andsimplify the expression. Because of the signal-dependent pref-actor in Eq. (D.1) the contribution of the external stimulus iscanceled in the first-order term of the expansion and we obtaina signal power of

Pmin = RUN − R0

k1. (D.2)

The required signal power for strong stimuli is independentof the signal strength, if we assume the unstable steady stateas a constant parameter. It is determined by the displacementbetween the initial and the unstable states and increases if thisdisplacement increases. Surprisingly, it is inversely proportionalto rate coefficient k1 only. The effective rate constant k′

2 whichdescribes mainly the dynamic relaxation, has no influence onthe signal power.

The signal power derived in the used approximation is a min-imal estimation of the necessary power to push the system fromthe lower branch to the upper. The inclusion of higher orderswith respect to the signal strength is especially important forstimuli near the critical point. As one can see from the logarith-mic expansion (C.2) this will lead to an increase of the criticalsignal power.

Appendix E. Parameters of the Nominal System

For numerical simulations and analytical calculation of themutually activated system (1) we used the kinetic coefficients

k0 = 0.4; k1 = 0.01; k2 = 1;

k3 = 1; k4 = 0.2; J3 = J4 = 0.05;

except in Fig. 3, where we vary the feedback strength. For thisinvestigation we choose the above value of k0 as starting pointand increase it. The corresponding k0 values can be calculatedfrom the ratio k0/k1, given in Fig. 3.

For the above set of parameters the system is an irreversible“one-way switch”. The only relevant critical point is located at

[Scrit, Rcrit] = [10.127, 0.147].

For a vanishing stimulus the system has three steady states,where the stable response values are

RSL(0) = 0 and RSU(0) = 0.38.

The corresponding unstable solution is

RUN(0) = 0.2.



If not stated otherwise, the transient response starts with aninitial value of R0 = 0.

The subcritical stimulus in Fig. 5 is Smax = 6 and Smax = 14in Fig. 7.

References

Abramowitz, M., Stegun, I., 1972. Handbook of Mathematical Functions,number 55 in Applied Mathematics Series, 10th ed. National Bureau ofStandards.

Aldridge, B., Haller, G., Sorger, P., Lauffenburger, D., 2006. IEE Proc. Syst.Biol. 153 (6), 425.

Angeli, D., 2006. IEE Proc. Syst. Biol. 153 (2), 61.Angeli, D., Ferrell, J.E., Sontag, E.D., 2004. Proc. Natl. Acad. Sci. U.S.A. 101

(7), 1822.Asthagiri, A., Lauffenburger, D., 2001. Biotechnol. Prog. 17, 227.Barkai, N., Leibler, S., 2000. Nature 403 (6767), 267.Bhalla, U., Iyengar, R., 1999. Science 283 (5400), 381.Bluthgen, N., Bruggemann, F., Legewie, S., Herzel, H., Westerhoff, H., Kholo-

denko, B., 2006. FEBS J. 273 (5), 895.Bracewell, R., 1999. The Fourier Transform and Its Applications, third ed.

McGraw Hill, New York.Cinquin, O., Demongeot, J., 2002. J. Theor. Biol. 216, 229.Cornish-Bowden, A., 2004. Fundamentals of Enzyme Kinetics, third ed. Port-

land Press.Downward, J., 2001. Nature 411 (6839), 759.Eißing, T.G.E., Conzelmann, H., Allgower, F., Bullinger, E., Scheurich, P., 2004.

J. Biol. Chem. 279 (35), 36892.Fell, D., 1997. Understanding the Control of Metabolism. Portland Press.Ferrell, J.E.E., Xiong, W., 2001. Chaos 11 (1), 227.Ferrell Jr., J., 1996. Trends Biochem. Sci. 21 (12), 460.Ferrell Jr., J., 1998. Trends Biochem. Sci. 23 (12), 461.Ferrell, J., Machleder Jr., E., 1998. Science 280, 895.Goldbeter, A., Koshland Jr., D., 1981. Proc. Natl. Acad. Sci. U.S.A. 78 (11),

6840.Gray, P., Scott, S., 1994. Chemical Oscillations and Instabilities: Nonlinear

Chemical Kinetics. Oxford University Press.

Heinrich, R., Neel, B.G., Rapoport, T.A., 2002. Mol. Cell 9 (5), 957.Heinrich, R., Schuster, S., 1996. The Regulation of Cellular Systems. Chapmann

and Hall.Huang, C.-Y., Ferrell Jr., J., 1996. Proc. Natl. Acad. Sci. U.S.A. 93 (19), 10078.Kholodenko, B.N., 2000. Eur. J. Biochem. 267 (6), 1583.Lauffenburger, D., Linderman, J., 1993. Receptors: Models for Binding, Traf-

ficing, and Signalling. Oxford University Press.Laurent, M., Kellershohn, 1999. Trends Biochem. Sci. 24 (11), 418.Lisman, J., 1985. Proc. Natl. Acad. Sci. U.S.A. 82 (9), 3055.Madshus, I., 2004. Signalling from Internalized Growth Factor Receptors. In:

Current Topics in Microbiology and Immunology. vol. 286. Springer.Markevich, N.I., Hoek, J.B., Kholodenko, B.N., 2004. J. Cell Biol. 164 (3),

353.Melen, G., Levy, S., Barkai, N., Shilo, B., 2005. Mol. Syst. Biol. 1, E1.Mesarovic, M., Sreenath, S., Keene, J., 2004. IEE Proc. Syst. Biol. 1 (1), 19.Millat, T., Bullinger, E., Rohwer, J., Wolkenhauer, O., 2007. Math. Biosci. 207

(1), 40.Pomerening, J., Sontag, E., Ferrell Jr., J., 2003. Nat. Cell Biol. 5 (4), 346.Salazar, C., Hofer, T., 2006. Biosystems 83 (2–3), 1195.Schlogl, F., 1971. Z. Physik 243 (4), 303.Schlogl, F., 1972. Z. Physik 253 (2), 147.Segel, I., 1993. Enzyme Kinetics. John Wiley and Sons.Thomas, R., 2004. Quantum Noise, Springer Series in Synergetics, third ed.

Springer, Berlin, pp. 180–193.Thomas, R., Kaufman, M., 2001. Chaos 11 (1), 170.Thron, C., 1994. BioSystems 32 (2), 97.Tyson, J., 1991. Proc. Natl. Acad. Sci. U.S.A. 88, 7328.Tyson, J., Csikasz-Nagy, A., Novak, B., 2002. BioEssays 24, 1095.Tyson, J., Othmer, H., 1978. Prog. Theor. Biol. 5, 1.Tyson, J.J., Chen, K.C., Novak, B., 2003. Curr. Opin. Cell Biol. 15 (2), 221.Veflingstad, S., Plahte, E., Monk, N., 2005. Physica D 207, 254.Wolkenhauer, O., Mesarovic, M., 2005. Mol. BioSyst. 1 (1), 14.Wolkenhauer, O., Sreenath, S., Wellstead, P., Ullah, M., Cho, K.-H., 2005a.

Biochem. Soc. Trans. 579 (3), 507.Wolkenhauer, O., Ullah, M., Wellstead, P., Cho, K., 2005b. FEBS Lett. 579 (8),

1846.Xiong, W., Ferrel Jr., J., 2003. Nature 426, 460.

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