A simple model of NF�κB dynamics reproducesexperimental observations

Samuel Zambrano n, Marco E. Bianchi, Alessandra AgrestiSan Raffaele University and Scientific Institute, Division of Genetics and Cell Biology, Via Olgettina 58, 20132 Milan, Italy

H I G H L I G H T S

� We have built a simple model of thedynamics of the transcription factorNF�κB.

� The model reproduces the observedoscillations of NF�κB, both smoothand spiky.

� The model can also display non-oscillating dynamics of NF�κB.

� Different timings of gene expressioncan be reproduced by extendingour model.

� The model reproduces the effect ofLeptomycin B and Cycloheximide ingene expression.

G R A P H I C A L A B S T R A C T

a r t i c l e i n f o

Article history:Received 11 October 2013Received in revised form7 January 2014Accepted 8 January 2014Available online 18 January 2014

Keywords:Mathematical modelingSignalling pathwaysTranscription factor dynamicsOscillations

a b s t r a c t

The mathematical modeling of the NF�κB oscillations has attracted considerable attention in recenttimes, but there is a lack of simple models in the literature that can capture the main features of thedynamics of this important transcription factor. For this reason we propose a simple model thatsummarizes the key steps of the NF�κB pathway. We show that the resulting 5-dimensional dynamicalsystem can reproduce different phenomena observed in experiments. Our model can display smooth andspiky oscillations in the amount of nuclear NF�κB and can reproduce the variety of dynamics observedwhen different stimulations such as TNF�α and LPS are used. Furthermore we show that the model canbe easily extended to reproduce the expression of early, intermediate and late genes upon stimulation.As a final example we show that our simple model can mimic the different transcriptional outputsobserved when cells are treated with two different drugs leading to nuclear localization of NF�κB:Leptomycin B and Cycloheximide.

& 2014 Elsevier Ltd. All rights reserved.

1. Introduction

Transcription factors of the NF�κB family are responsible for thetranscription of a large number of genes that regulate a wide varietyof crucial processes in the human cells, which are known to have akey influence in inflammation and cancer. In resting cells, NF�κB

forms a cytoplasmic complex bound to inhibitor IκB proteins;upstream signals (e.g. TNF�α or LPS) activate the kinase IKK thatphosphorylates the inhibitors, which eventually are degraded. As aresult, NF�κB is free to translocate into the nucleus, where itactivates the expression of many genes (Hayden and Ghosh,2008). Among them, the genes encoding for the IκB inhibitors,which enact a negative feedback (since the NF�κB/IκB complexesrelocate to the cytoplasm, where they are transcriptionally inactive,Hoffmann et al., 2002). Due to this negative feedback, oscillationsarise in the system. These oscillations have been observed at a

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Journal of Theoretical Biology

0022-5193/$ - see front matter & 2014 Elsevier Ltd. All rights reserved.http://dx.doi.org/10.1016/j.jtbi.2014.01.015

n Corresponding author.E-mail address: samuel.zambrano@gmail.com (S. Zambrano).

Journal of Theoretical Biology 347 (2014) 44–53

single-cell level (Nelson et al., 2004; Paszek et al., 2009; Tay et al.,2010; Sung et al., 2009) and characterized with a high degree ofdetail. In particular it has been found that the dynamics is quiteheterogeneous among the cells (Paszek et al., 2010) and stronglydepends on the stimulus applied (Werner et al., 2005; Tay et al.,2010). As expected, this dynamics has an influence on geneexpression. The transcription of early, intermediate and late genes(Sung et al., 2009; Paszek et al., 2009; Tay et al., 2010) changeswhen the dynamics is modulated by varying the stimulus dose (Tayet al., 2010), applying pulsatile stimulations of different frequenciesand amplitudes (Paszek et al., 2009) or by interfering with differentdrugs (Sung et al., 2009).

The importance of NF�κB dynamics has motivated a number oftheoretical studies through mathematical modeling (Hoffmannet al., 2002; Nelson et al., 2004; Ihekwaba et al., 2005; Krishnaet al., 2005; Lipniacki et al., 2004; Tay et al., 2010; Wang et al.,2012; Turner et al., 2010; Paszek et al., 2010; Shih et al., 2010).NF�κB dynamics are modeled by systems of ordinary differentialequation (ODEs) typically derived from mass-action kinetics of thebiochemical reactions involved in the pathway, sometimes com-bined with stochastic processes for reactions involving specieswith low copy numbers (Tay et al., 2010; Paszek et al., 2010; Turneret al., 2010). All models reproduce at least some of the negativefeedbacks known to be present in the system, a necessary condi-tion for oscillatory dynamics (Snoussi, 1998). The signaling net-work of NF�κB is quite complex so these models typically have 20or more variables (Hoffmann et al., 2002; Nelson et al., 2004;Ihekwaba et al., 2005; Lipniacki et al., 2004; Tay et al., 2010; Wanget al., 2012; Turner et al., 2010; Shih et al., 2010) corresponding tobiochemical species present in the pathway, including the nuclearand cytoplasmic fractions of NF�κB, both free and bound to someof the known inhibitors IκBα,Iκ Bβ, Iκ Bε, Iκ Bδ, which can in turn bein different forms (phosphorylated and non-phosphorylated),among others. These models are able to reproduce differentfeatures observed experimentally. However, their complexitymakes them computationally expensive and hard to treat analy-tically, while the limited amount of information that is currentlyprovided by experiments would probably lead to over-fitting.In spite of these limitations, there is a lack of simple low-dimensional models of NF�κB dynamics that retain the ability toreproduce, at least qualitatively, many of the features observed inexperiments. An exception is the three-dimensional model of theNF�κB system that was proposed in Krishna et al. (2005) andTiana et al. (2007). This model is obtained from smart approxima-tions of the higher-dimensional model used in Nelson et al. (2004)that make use of the fast and slow timescales of the system. As aresult, this minimal model inherits the spiky oscillations of theoriginal model for a wide parameter range (Krishna et al., 2005).

A different possible approach leading to simple models is justtrying to summarize the processes taking place into fewer ones insuch a way that they encapsulate the crucial phenomena takingplace in the system. A nice example of this modeling approach,related with NF�κB, is the simple telegraph model of geneexpression (Peccoud, 1995). This model summarizes the multiplesteps of the transcription and translation processes (Coulon, 2013)in a few ones: the gene activation and inactivation, the transcrip-tion and the degradation of the RNA and the translation anddegradation of the protein. In spite of its simplicity, it is able toreproduce the expression of a wide variety of genes in mammals(Suter et al., 2011).

Following this modeling philosophy, we propose here a simplemodel of NF�κB dynamics and we explore its ability to reproducedifferent phenomena that have been observed experimentally.Our model adds the minimum number of “layers of species” to afirst layer formed by a telegraph-like model of transcription andtranslation of the inhibitor IκBα in order to reproduce the basic

processes taking place in the NF�κB signaling pathway. Thedynamics of the resulting system are modeled using ordinarydifferential equations (ODEs) derived from mass-action kinetics ofthe processes considered. We show here that the resulting five-dimensional dynamical system can already reproduce a number offeatures of the NF�κB signaling that have been observed in theexperiments.

The paper is organized as follows: in Section 2 we present thebasic processes that we try to model, encapsulating the key phe-nomena occurring in the NF�κB signaling pathway. In Section 3 wedescribe how we perform the parameter selection and we show thatthe system displays the characteristic oscillations, whereas in Section4 we show that different types of oscillations can be observed for oursystem for different parameter combinations. In Section 5 we showthat the parameters governing the stimulus layer of the system havea capital influence in the dynamics, while in Section 6 we show thatour model can be used to reproduce the different timings of thetranscription of genes observed in this system. Furthermore, weshow that the model is able to reproduce the different effects intranscription obtained when the system is perturbed with two drugsleading to nuclear localization of NF�κB: Leptomycin B and Cyclo-heximide. Finally, in Section 7 we discuss the results and draw themain conclusions of this work.

2. Basic processes in the NF�κB system

As we said previously, our plan is to build the model by puttingtogether different “layers of species”. The first layer is where thetranscription and the translation of IκBα are modeled, the inhibitorsynthesis layer. The second is the regulatory layer, which includeshow NF�κB modulates the IκBα synthesis and the negative feed-back that arises from this regulation. Finally, the third layer is thestimulus layer, where the processes taking place in the cell uponstimulation are summarized. We describe these layers in moredetail below. We will designate in the same way the biochemicalspecies and their copy number.

2.1. The inhibitor synthesis layer

We first consider the layer with the elementary processesrelated with the transcription and translation of the protein IκBα,as shown in Fig. 1. We choose this particular inhibitor of NF�κBbecause it is the key ingredient in the negative feedback giving rise

Fig. 1. Scheme of the simple model of the NF�κB pathway that we propose. It consistsof three layers. The first layer summarizes the inhibitor synthesis, comprising activationof the gene Gα , transcription of IκBαRNA and translation into the inhibitor protein Iκ Bα.The second layer is the regulatory layer, where the free NF�κB and IκBα are inassociation–dissociation equilibriumwith the complex (NF�κB : IκBα). The free NF�κBactivates the gene expression, whereas free IκBα can induce gene inactivation. Thethird stimulus layer encloses the processes prompted by the stimulation: activation ofIKKa and its gradual inactivation. The appearance of IKKa leads to the degradation ofIκBα and thus frees NF�κB. The degradation of IκBα and IκBαRNA is also taken intoconsideration, as well as the inactivation of IKK.

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–53 45

to oscillations (Hoffmann et al., 2002; Nelson et al., 2004), asreproduced by mathematical models (Ihekwaba et al., 2005; Basaket al., 2011). The basic processes in this layer describe faithfully thegene expression in mammals (Suter et al., 2011).

The first step is the transcription of the i-th active copy of thegene encoding for IκBα, Gα;i;on:

Gα;i;on-KR Gα;i;onþ IκBαRNA: ð1Þ

Each copy of the gene can only be active or inactive, so∑iGα;i;onþGα;i;off ¼ G0 (and typically G0¼2). We describe the degra-dation of IκBαRNA as

IκBαRNA-dR ∅: ð2Þ

Translation of the RNA IκBαRNA into IκBα can be written as

IκBαRNA-KpIκBα: ð3Þ

Finally we have the spontaneous degradation of IκBα:

IκBα-dI ∅: ð4Þ

In this simple model we are not modeling explicitly importantprocesses involved in gene expression such as the splicingdynamics, which has been recently shown to play a key role inthe temporal order of early, intermediate and late genes (Hao andBaltimore, 2013). Our model of gene expression is similar to theone used in Tay et al. (2010) and Suter et al. (2011), so we followthem in interpreting IκBαRNA as the mature RNA. Thus, all the ratesabove implicitly regulate the dynamics of the processes leading tothe production of mature RNA.

2.2. The regulatory layer

In this layer we describe how the transcription and the transla-tion of IκBα are modulated. We consider that NF�κB can either befree or bound to Iκ Bα to form the complex (NF�κB : IκBα),summarized in Fig. 1. The free NF�κB can be considered to benuclear, and will modulate the activation of the gene Gα, whereasthe complex is essentially cytoplasmic and is thus inactive. In factwe have verified in a number of existing models that the freenuclear NF�κB and the cytoplasmic fraction of (NF�κB : IκBα) arethe two predominant species of NF�κB. Provided that we have onlytwo possible “states” for NF�κB and that NF�κB degradation istypically neglected in the timescales of the oscillatory dynamics(Hoffmann et al., 2002; Nelson et al., 2004; Ihekwaba et al., 2005;Lipniacki et al., 2004; Tay et al., 2010; Wang et al., 2012), we assumethat ðNF�κB : IκBαÞþNF�κB¼NF�κB0 where NF�κB0 is the totalamount of NF�κB.

With these two species we can reproduce the essential pro-cesses of gene regulation taking place.

The NF�κB modulation of gene activation (that has been shownto be noncooperative Giorgetti et al., 2010) can be written as

Gα;i;off þNF�κB-KonGα;i;onþNF�κB: ð5Þ

Note that cooperativity could be modeled by adding more copiesof NF�κB in the left and right sides of the above reaction.

It has also been suggested, both in mathematical models (Tayet al., 2010; Paszek et al., 2010) and by experiments (Arenzana-Seisdedos et al., 1995), that the presence of nuclear IκBα can inducedetaching of NF�κB from the promoter, and thus the inactivationof the gene. The fact that (a fraction of) IκBα can play this role canbe summarized by this process:

Gα;i;onþ IκBα-Koff

Gα;i;off þ IκBα: ð6ÞNotice that the activation/inactivation of the i-th copy of the

gene eventually affects the production of IκBαRNA. Hence Kon and

Koff implicitly modulate the dynamics of the processes leading tothe production of mature RNA.

The inhibition of the activity of NF�κB occurs through itsbinding to form the complex (NF�κB : IκBα) that is mostly cyto-plasmic and thus cannot induce gene activation. We summarize allthe processes by which the nuclear NF�κB is bound by Iκ Bα andforms a cytoplasmic complex (including the transport of all thereactants between the nucleus and the cytoplasm) simply by theseassociation–dissociation biochemical reactions:

NF�κBþ IκBα⇌A

dðNF�κB : IκBαÞ: ð7Þ

We also assume that there might be a spontaneous degradationof IκBα forming the (NF�κB : IκBα) complex to a rate proportionalby a factor γ to the degradation rate of free IκBα, hence

ðNF�κB : IκBαÞ-γdINF�κB: ð8Þ

2.3. The stimulus layer

Finally, we introduce the basic processes that take place whenan inflammatory stimulus hits the cell. Interaction of TNF�α or LPSwith the corresponding cell receptor triggers the appearance ofthe active kinase IKKa (Werner et al., 2005) that phosphorylatesthe IκB inhibitors (either free or bound to (NF�κB : IκBα)) inducingtheir degradation. In our simple system of just one inhibitor wecan summarize these processes by the two following simplereactions, for bound IκBα:

ðNF�κB : IκBαÞþ IKKa-PNF�κBþ IKKa; ð9Þ

while for free IκBα we expect that the IKK-driven degradation ratewill be proportional to the former one by a factor κ, so

IκBαþ IKKa-κPIKKa: ð10Þ

In the first models of NF�κB dynamics (see i.e. Hoffmann et al.,2002; Nelson et al., 2004) a TNF�α stimulation in t¼0 h wasmodeled by setting the amount of IKKa equal to a certain value,and this amount would decrease to zero following certain inacti-vation rates. In other models, in which the NF�κB�driven A20negative feedback (Lipniacki et al., 2004) on a detailed IKKactivation module is taken into account (Werner et al., 2005),the IKKa reaches a steady state (Wang et al., 2012). Simulationsshow that in this model there are sustained oscillations, as it isobserved in other simple mathematical models in which IKKa isconsidered constant (Krishna et al., 2005), due to a Hopf bifurca-tion (Tiana et al., 2007; Wang et al., 2012). However most long-term experimental observations of NF�κB dynamics post-stimulusat a single cell level (Sung et al., 2009; Tay et al., 2010) showdamped oscillations before the return to the pre-stimulus state ofthe system, which means that the active kinase IKKa levelsdecrease to zero. Thus, we will assume that the behavior of theIKKa post-stimulus is a simple inactivation of the form

IKKa-dK ∅: ð11Þ

The stimulus will be simply modeled by setting at time t¼0 theamount of IKKa equal to certain IKK0. As we shall see now, the five-dimensional dynamical system that can be derived from this set ofreactions displays a rich variety of dynamical behaviors.

3. The normalized model

Using mass-action kinetics equations the described system ofsix biochemical species interacting through 11 biochemical reac-tions (and thus 14 parameters, if we consider NF�κB0, IKK0 andG0¼2 as free parameters) can be modeled by the following set of

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–5346

differential equations:

dKdt

¼ �dK � K ð12Þ

dNdt

¼ d � ð1�NÞþγ � dIð1�NÞþp � K � ð1�NÞ�a � N � I ð13Þ

dIdt

¼ d � ð1�NÞ�κ � p � K � I�a � N � Iþkp � R�dI � I ð14Þ

dRdt

¼ dR � ðG�RÞ ð15Þ

dGdt

¼ kon � N � ð1�GÞ�koff � I � G ð16Þ

This model describes the evolution in time of the followingnormalized variables: K is proportional to the amount of IKKa, N isthe nuclear (free) to total ratio of NF�κB, I is proportional to theamount of inhibitor, R is proportional to the amount of RNA of theinhibitor, and G is proportional to the gene activity. The preciserelation between the variables of the normalized model and thenumber of copies of each species (represented, as usual, by thespecies name) is K ¼ IKKa=IKK0, N¼NF�κBnuclear=NF�κB0,I ¼ IκBα=NF�κB0, R¼ IκBαRNA � dR=ðKm � G0Þ, G¼∑iGα;i;on=G0

Our normalization is chosen in such a way that all the variableswill be Oð1Þ variables. Being this a set of ordinary differentialequations derived from mass-action kinetics, it is possible to showthat it presents nonnegative solutions (Chellaboina et al., 2009).If we set the stimulation time as t¼0 h, the initial conditions thathave to be used in this normalized system are Kð0Þ ¼ 1, Nð0Þ ¼Nn,Ið0Þ ¼ In, Rð0Þ ¼ Rn and Gð0Þ ¼ Gn. The values Nn, In, Rn and Gn

correspond to the values of the only stable equilibrium arising inthe system when K is set equal to zero, which is equivalent tomaking IKKa equal to zero: the equilibrium situation that might beexpected for unstimulated cells.

When possible, the parameters in the above equations arecalculated from the reaction rates used in Tay et al. (2010), thevalues are provided in Table 1. It is important to notice that due tothe normalization chosen, some of the parameters of the normal-ized model are not the biochemical reaction rates, that is the caseof parameters p, a, kp, kon and koff (lowercase parameters arerelated to uppercase reaction rates). The relation between theseparameters and the biochemical reaction rates used to obtainthem are given in Appendix A.

On the other hand, in some cases finding a straightforwardcorrespondence between the reaction rates considered in bothmodels is not possible, because the model from Tay et al. (2010) isremarkably more complex than ours. Still, in some cases it is possible

to establish a correspondence, as for example with parameters γ or κor IKK0 (computed as the ratios of the rates of degradation andphosphorylation for free and bound fractions of IκBα, and from theapproximate IKKa values obtained when the system is stimulatedwith 10 ng/ml of TNF�α). Those cases and those where the reactionrates have been only slightly varied from those given in Tay et al.(2010) are noted as “Approx. from Tay et al. (2010)” in Table 1. Therest of the parameters are chosen to fit qualitatively the experimentalobservations of oscillatory dynamics. The resulting parameters of thenormalized model are dK¼0.15 h�1, d¼3 h�1, γ ¼ 0:2, dI¼0.24 h�1,p¼9 h�1, a¼200 h�1, κ¼ 0:2, kp¼16 h�1, dR¼2.7 h�1,kon ¼ 7:5 h�1, koff ¼ 15 h�1. Note that the normalization used allowsone to reduce the number of parameters of the model.

The results of our simulations for this parameter set can beobserved in Fig. 2(a). We can observe oscillations in nuclear tototal ratio of NF�κB (N), which is one of the basic features ofNF�κB dynamics. These oscillations are followed by oscillations inthe rest of the variables that represent the gene state G, theamount of mature RNA R and the inhibitor amount I, with theexpected sequentially increasing delay. The trajectories obtainedreproduce the damped oscillations observed in the experiments;however, we have to notice that we used parameters that werechosen to fit experimental observations with a particular cell type,together with other parameters that we fitted ourselves. For thisreason, it is mandatory to explore in further depth the dynamicsfor different parameters in order to have a clearer picture of thedescriptive capacities of the simple model.

Before doing so, we note that in the above equations we havemodeled the gene state (G) using a differential equation. Another

Table 1Reaction rates.

Rate Value Units From

KR 0.2 s�1 copies�1 Tay et al. (2010)dR 7.5�10�4 s� Tay et al. (2010)KP 0.25 s�1 copies�1 Approx. from Tay et al. (2010)dI 6.7�10�5 s�1 Approx. from Tay et al. (2010)Kon 6.9�10�8 s�1 copies�1 FitKoff 1.4�10�8 s�1 copies�1 FitA 1.9�10�6 s�1copies�2 Fitd 8.4�10�4 s�1copies�1 New (not in Tay et al., 2010)γ 0.2 adimensional Tay et al. (2010)P 2.5�10�8 s�1copies�2 FitdK 4.2�10�5 s�1 FitNF�κB 3�104 copies Our experimental measurementsIKK0 105 copies Approx. from Tay et al. (2010)G0 2 copies Tay et al. (2010)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

KNI

RG

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

G, G

appr

ox, N

t (hours)

Fig. 2. (a) Evolution of the variables of the model, which are proportional to theactive kinase IKKa (K), the nuclear to total ratio of NF�κB (N), the amount ofinhibitor IκBα(I), the amount of RNA IκBαRNA (R) and the gene state (G). Delayedoscillations of N and I are observed, as expected for this system. (b) Comparison ofthe evolution of the gene state G (solid line) with the values Gapprox that would beobtained using a Michaelis–Menten-like approximation (diamonds). We also plothere the evolution of N (thick green line). The delayed gene activation resultingfrom the explicit modeling of the activation dynamics can be exploited to modelthe different timing in gene expression observed in the experiments.

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–53 47

modeling approach (that would reduce the dimensions of ourmodel to four) would have been to use a Michaelis–Menten likeequation for the gene activity (Alon, 2007), provided that NF�κB isan activator while IκBα prevents the binding of NF�κB to thepromoter (Arenzana-Seisdedos et al., 1995), which in practicalterms is similar to the role of a repressor. The Michaelis–Mentenapproximation assumes that the free and bound fraction of theactivators and repressors are in equilibrium, and this implies that

G� Gapprox ¼konN

konNþkoff I: ð17Þ

We can see in Fig. 2(b) the difference between the value of Gprovided by this approximation (Gapprox) and the one obtained forthe model. Both are reasonably similar, but the modeling of Gusing a differential equation introduces a delay, also with respectto the oscillations of N, which makes sense from a biological pointof view provided that NF�κB triggers the recruitment of themachinery that activates the gene, which might take some time.Furthermore, as we shall show later this delay can be exploited tomodel qualitatively the gene expression of early, intermediate andlate genes taking place upon different TNF�α stimulations (Sunget al., 2009; Tay et al., 2010).

4. Exploration of the oscillatory dynamics

We have shown that our mathematical model of NF�κB, bycombining parameters from Tay et al. (2010) with others that wefitted manually, can reproduce oscillating dynamics. However,many of the parameters from Ref. Tay et al. (2010) are manual fitsthemselves. For this reason, we have decided to explore theparameter space of the system by varying the parameters indifferent ranges. The ranges chosen account for the fact that forcertain parameters we have a bigger uncertainty than for others,either because the parameters have not been experimentallymeasured or because the parameters considered derive from ratesthat summarize a number of biochemical process (as for examplethe parameter a that derives from A). For this reason, in ourparameter exploration we vary the parameters by multiplyingthem by random numbers in the ½10�n;10n� interval, taking n¼1for parameters dk, a, d; n¼0.5 for parameters kp, p, kon and koff andn¼0.25 for κ, γ, dR and dI. In other words, we allow a variation of upto two orders of magnitude in some of the parameters of ourmodel. We generated thus 2000 trajectories and decided toclassify them according to the evolution of the variable N, whichis the one that mirrors the translocations between the nucleus andthe cytoplasm of NF�κB typically observed in the single-cellexperiments.

For a time series N(t) we consider a peak as a sequence of alocal minimum, a local maximum and a local minimum that areobserved at times tmin;L, tmax and tmin;R respectively; oscillations canbe described as the appearance of more than one of such peaks inN(t). Oscillatory NF�κB dynamics have been observed in a numberof experiments, although the shape of such oscillations appears tovary depending on the experimental conditions and the cell type.In some experiments spiky oscillations are observed (Nelson et al.,2004; Turner et al., 2010) while in others oscillations have asmoother appearance (Sung et al., 2009; Tay et al., 2010). Toquantify the spikyness of the oscillations observed for differentparameter sets we propose the following quantifier S of thespikyness of each peak, defined as

S¼NðtmaxÞ�0:5 � ðNðtmin;LÞþNðtmin;RÞÞ⟨NðtÞ⟩�minðNðtmin;LÞ;Nðtmin;RÞÞ

ð18Þ

(the t¼0 point, when the stimulation is applied, is considered as alocal minimum) where ⟨NðtÞ⟩ is the average value of N(t) in the

interval ½tmin;L; tmin;R�. With this parameter we measure the ratiobetween the “height” of each peak and the average value of N(t)above the minimum, an average that should be small for peaks ofvery spiky oscillations. The spikyness quantifier provided inKrishna et al. (2005) is based on a similar idea, but S has theadvantage of being invariant by changes of scale and to shifts bothin N and in time t.

In Fig. 3 we show time series obtained integrating our modelfor which three peaks are detected in 5 h (which is compatiblewith the period of the oscillations of approximately 1.5 h observedin experiments, Tay et al., 2010; Turner et al., 2010). We define asspiky oscillations those having an average value of SZ3 in thethree first peaks, while the rest will be considered smoothoscillations. Similarly, we consider time series with a highresponse value those for which the maximum value of N(t) isbigger than 0.6, while the rest are considered low response timeseries. Following this simple classification, we show in Fig. 3(a)non-spiky large amplitude oscillations, in Fig. 3(b) spiky largeamplitude oscillations, in Fig. 3(c) smooth low amplitude oscilla-tions and in Fig. 3(d) spiky low amplitude oscillations. Thus, ourmodel is able to reproduce oscillations that have been observed ina number of experimental situations. In particular, spiky oscilla-tions of large amplitude were observed in Nelson et al. (2004),whereas spiky oscillations of low amplitude were observed for lowdoses of TNF�α in Turner et al. (2010). On the other hand, the highand low amplitude smooth oscillations observed in Tay et al.(2010) for high and low doses of TNF�α are reminiscent of thoseshown in this figure. The fact that a simplified model of NF�κB likeours is able to reproduce different types of dynamics indicates thatoscillations are strongly dependent on the system parameters; forthis reason discrepancies are not surprising when consideringexperiments in which different cell lines were used and thus therates of the biochemical reactions involved should differ.

Up to here we have described oscillating dynamics of NF�κB,which has been predominantly observed experimentally and that inour model is summarized by the evolution of the variable N.However we also observed that other types of dynamics can alsoarise. For example, for certain combinations of the parametersoscillations are not present and the variable N decays slowly fromits maximum value. As we shall show now, the parameters related tothe stimulus layer play a key role in the type of dynamics observed.

5. Role of the stimulus layer

In different experimental works the response of the NF�κBsystem to different stimuli, such as different doses of TNF�α (Tayet al., 2010; Turner et al., 2010) or LPS (Lee et al., 2009), has beencharacterized in detail. In practice in our model any stimulus ismodeled by two parameters: p (proportional to IKK0, the activeIKK right after the stimulus application) and dK (inactivation rateof IKKa). In order to evaluate the role of these parameters in thesystem we have varied them by two orders of magnitude aroundthe values used in Section 3, that we denote here as dk;0 and p0,and calculated the values of certain quantities that characterizethe dynamics of the resulting time series. In Fig. 4(a) we show thevalue of the maximum of N, Nmax, sometimes also referred to asthe response. We can see that the key parameter governing theresponse value is p, that is related with IKK0 and ultimately withthe dose of TNF�α applied to the system: higher doses of TNF�αshould give higher activation of IKK and thus higher values of IKK0

and p (Werner et al., 2005). The increase of the response with pthen mirrors the increase of the response of the NF�κB systemwith increasing doses of TNF�α that has been observed experi-mentally in different works (Tay et al., 2010; Turner et al., 2010). InFig. 4(b) we plot the value of the timing of the response, Tresponse,

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–5348

that is the position in time of the first local maximum of N. We seethat lower values of p induce later responses of the system.This again corresponds with what is observed when the cells arestimulated with lower and lower doses of TNF�α (Tay et al., 2010;Turner et al., 2010).

Fig. 4(c) shows the value of T, the period of the oscillationsarising after t¼0. Such period T is estimated by calculating thetime difference between the first and the second relative max-imum of N(t). The black zones displayed correspond to thosecombinations of the parameters for which the system does notpresent two of such maxima. Thus, the existence of dampedoscillations in this nonlinear system strongly depends on thevalues of the parameters p and dK. However, it is trivial to showthat these two parameters do not affect the stability of the fixedpoint of the system because they do not appear in the Jacobiancomputed in the equilibrium point for which we know thatK ¼ Kn ¼ 0. Hence we can infer that this behavior is due to thenonlinear nature of the system that is specially relevant when weare far from the equilibrium.

It is possible though to have a deeper insight on this nonlinearphenomenon. To do so, we can study what happens if we fix thevalue of K¼1 for tZ0 and set dk¼0 while varying the parameter p(this is equivalent to analyze the dynamics if the stimulus signalwas not degraded). The types of dynamics around the fixed pointof the system can be inferred by computing the eigenvalues fλig ofthe Jacobian. We find that all of them have real parts smaller thanor equal to zero. However, their imaginary parts become zero asthe value of p is increased, as we can see in Fig. 4(e). If we observethe trajectories obtained for each value of the parameters con-sidered, we can see that this implies that no damped oscillationstowards the fixed point are observed for sufficiently high values of

p, as shown in Fig. 4(f). This behavior is thus mirrored by ournonlinear system when dK≳0 and the value of p is sufficiently big.On the other hand, it is interesting to note that this situation withdk¼0 gives also hints of the behavior observed in our systemwhenp is increased: the response value Nmax (the value of the firstmaximum) and the response time Tresponse (its timing) increase anddecrease respectively with p, as we saw in Fig. 4(a) and (b) for oursystem.

Finally, Fig. 4(d) shows the relative decay of N from its maximumvalue after a total time of 6 h, a number in the ½0;1� intervalindicating the fraction that N(t) decreases with respect to Nmax:0 corresponds to no decay at all, while 1 corresponds to a decay fromNmax to the initial value. Interestingly enough we can see that for acombination of sufficiently low dk and sufficiently high value of psuch decay is very slow (note that eventually all trajectories shouldreturn asymptotically to the basal state, provided that K tends to zeroas t increases). Thus, there is somehow a saturation of the system ifthe stimulus applied is very high, leading to a large amount of IKKa,which in turn induces a sustained nuclear localization of NF�κB. Thiscan also be related with the profiles of N observed for high values ofp in Fig. 4(f). For more complex models of NF�κB dynamics it wasalso shown that the profiles of IKK activation upon stimulation canlead to this kind of slowly decaying and non-oscillating dynamics(Werner et al., 2005), a phenomenon that has also been observed at asingle cell level when cells are stimulated with LPS (Lee et al., 2009)and with doses of TNF�α higher than the “canonical” dose of 10 ng/ml (Kalita et al., 2012). Hence, our simple model of IKK activationgives rise to qualitatively different types of dynamics. Being oursystem essentially a nonlinear oscillator (Guckenheimer and Holmes,1983) we consider that a wider variety of dynamics (even chaos)could be obtained by adding a time-dependent term in Eq. (12) of

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

N

t (hours)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

N

t (hours)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

N

t (hours)

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

N

t (hours)

Fig. 3. When varying the parameters of our model, different types of dynamics arise. We classify the trajectories according to the value of the first peak and the spikyness ofthe peaks of N. Here we show examples of trajectories with (a) smooth oscillations and high response values, (b) spiky oscillations with high response values, (c) smoothoscillations with low response values and (d) spiky oscillations with low response values. Thus, this simple model gives a variety of behaviors of NF�κB consistent withdifferent experimental observations.

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–53 49

our model, that would account for more temporally complexstimulations of the system.

6. A minimal model of transcription with an example ofapplication

As a final application of our model, we propose to extend it tomodel the transcription of genes that are not involved in thenegative feedback but are necessary to provide an adequate cellresponse. In particular we aim to mimic the timing of early,intermediate and late transcription of genes that has beenobserved in a number of experiments (Tay et al., 2010; Sung

et al., 2009). The model that we propose is

dR0

dt¼ d0R � ðG0 �R0Þ ð19Þ

dG0

dt¼ k0on � NðtÞ � ð1�G0Þ�k0off � IðtÞ � G0: ð20Þ

These equations describe the activation of a given gene G0, thatis modulated (as for the IκBα gene) by N(t) and I(t), and the timeevolution of its mature RNA, proportional to R0. The parameters k0onand k0off are proportional to the activation and inactivation rates ofthe gene considered respectively, while d0R is the degradation rateof the RNA. In order to model the transcription profile of a gene in

0

0.2

0.4

0.6

0.8

1

d K/d

K,0

p/p0

Nmax 10

1

0.10.1 1 10

0

0.2

0.4

0.6

0.8

1

d K/d

K,0

p/p0

Tresponse (hours) 10

1

0.10.1 1 10

0.511.522.533.544.55

d K/d

K,0

p/p0

T (hours) 10

1

0.10.1 1 10

0

0.2

0.4

0.6

0.8

1

d K/d

K,0

p/p0

Relative decay 10

1

0.10.1 1 10

-6

-4

-2

0

2

4

6

10-1 100 101

Im(λ

i)

p/p0

0

0.2

0.4

0.6

0.8

1

0 1 2 3 4 5

N(t)

t (hours)

e

Fig. 4. For our simple model the parameters governing the external stimulus are two: p (proportional to IKK0, the active IKK right after the stimulus application) and dK(inactivation rate of IKKa). Here we show the effect of varying these parameters around the values used originally, p0 and dk;0 in (a) the response value Nmax, (b) the timing ofthe response Tresponse, (c) the period of the oscillations (black means that it is not defined) and (d) decay of N relative to Nmax in 6 h post-stimulation. (e) Imaginary part of theeigenvalues of the system when dK¼0 h and K¼1, as p is increased they become zero, which explains why for high p values and dK≳0 no damped oscillations are observed.(f) Time series of N(t) obtained when dk¼0 using the p values of the previous figure. The disappearance of the second (and subsequent) peaks as p increases (and all theeigenvalues of the Jacobian of the system become real) is evident.

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–5350

a population of cells, we generate N¼50 trajectories from oursystem varying the parameters 50 % around the values given inSection 3 (simulating then the extrinsic noise of the system), inorder to mimic the wide dynamical heterogeneity observed in thedynamics among a population of cells (Tay et al., 2010; Paszeket al., 2010; Sung et al., 2009) upon stimulation. The resulting N(t)and I(t) are then used in Eqs. (19) and (20). In other words, in orderto simulate the different timings of gene expression we will keepthe parameters k0on, k

0off and d0R fixed to mirror the behavior of

different classes of genes, while we will allow a variability in thetrajectories N(t) and I(t) used as inputs for the model to account forthe cell-to-cell variability. This way “single-cell” traces of R0ðtÞ andG0ðtÞ are obtained.

We want to emphasize here that, as with the variable R of ourmodel, the average R0ðtÞ among the cells ⟨R0ðtÞ⟩ should be repre-sentative of the amount of mature RNA of the considered geneamong the population. For this reason all the parameters involvedin the dynamics of R0 (k0on, k

0off and d0R) implicitly are regulating the

dynamics of other processes important for gene expression thatwe are not modeling explicitly, such as the RNA splicing (Hao andBaltimore, 2013). In Fig. 5(a) we show the single-cell traces of thecells R0ðtÞ considered using k0on ¼ kon, k

0off ¼ koff and d0R ¼ dR and

normalized by R0;max, the maximum value of ⟨R0ðtÞ⟩ in the intervalconsidered.

As we said previously, transcription profiles have been obtainedexperimentally in a number of works with different degrees of

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4t (hours)

⟨ R(t) ⟩

R(t)

/R0,

max

’

’

0

0.2

0.4

0.6

0.8

1

1.2

0 0.5 1 1.5 2 2.5 3 3.5 4t (hours)

⟨ R(t)

⟩ /R

0,m

ax (

RN

A a

mou

nt)

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4

N (g

reen

), I (

red)

, G (b

lue)

t (hours)

0

0.5

1

1.5

2

0 0.5 1 1.5 2 2.5 3 3.5 4t (hours)

⟨ R(t)

⟩ /R

0,m

ax (

RN

A a

mou

nt)

’ + LMB

+ CHX

0

0.5

1

1.5

2

2.5

3

3.5

4

0 0.5 1 1.5 2 2.5 3 3.5 4t (hours)

⟨ R(t)

⟩ /R

0,m

ax (

RN

A a

mou

nt)

’ + LMB

+ CHX

0

0.5

1

1.5

2

2.5

0 0.5 1 1.5 2 2.5 3 3.5 4t (hours)

⟨ R(t)

⟩ /R

0,m

ax (

RN

A a

mou

nt)

’

+ CHX

+ LMB

Fig. 5. (a) Single-cell time series of R0 (cyan) and average ⟨R0ðtÞ⟩ (‘þ ’) obtained by using time series of N(t) and I(t) from our normalized model, where we have allowed a variationof 50% in the parameters of the model while using k0on ¼ kon, k

0off ¼ koff and d0R ¼ dR . (b) Average ⟨R0ðtÞ⟩ obtained using parameters for early (blue ‘þ ’), intermediate (red diamonds)

and late (green squares) genes, obtained varying adequately the parameters k0on , k0off and d0R . (c) Time series of N(t) (green), I(t) (red) and G(t) (blue) obtained when the system is

perturbed with LMB (circles) and CHX (lines). Levels of nuclear localization are similar in both cases but gene activity is quite different due to the differences in I(t). (d) Values of⟨R0ðtÞ⟩ normalized to the maximumvalue in the absence of co-treatment R0;max (blue ‘þ ’) and when the system is co-treated with CHX (blue diamonds) or with LMB (blue squares).(e) Same for intermediate genes (red ‘þ ’) for co-treatment with CHX (red diamonds) or with LMB (red squares). (f) Same for late genes (green ‘þ ’) when the system is perturbedwith CHX (green diamonds) or with LMB (green squares). Overall, co-treatment with LMB leads to a decrease in gene expressionwhile co-treatment with CHX to a global increase,as observed in the experiments.

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–53 51

detail (Tay et al., 2010; Sung et al., 2009; Tian et al., 2005; Giorgettiet al., 2010) and different modeling and bioinformatics approacheshave been used to characterize them (Giorgetti et al., 2010;Sivriver et al., 2011). The shape of our numerical transcriptionprofiles of ⟨R0ðtÞ⟩ will depend strongly on the parameters of themodel and on the number of trajectories considered, as long asasynchronous oscillations tend to be blurred when the number ofoscillators considered is large (Sung et al., 2009). However,mimicking precisely the observed profiles is beyond the scope ofthe present work. On the other hand, the fold change uponstimulation can be easily tuned by modifying the ratio of k0on andk0off (which determines the steady state value of G0 and R0). For thisreason, here we simply aim to model the timing of the transcrip-tion. In Tay et al. (2010) and Sung et al. (2009) a classificationbetween early, intermediate and late genes was done for genes forwhich the RNA amount upon stimulation with TNF�α peaked att � 1 h, t � 2–3 h, and tZ4 h respectively. The example shown inFig. 5(a) for k0on ¼ kon, k

0off ¼ koff and d0R ¼ dR would then correspond

to an early gene, which is not surprising because these parametersare those used to simulate the expression of the gene of IκBα,which is an early gene (Sung et al., 2009; Tay et al., 2010).

Using adequate parameters it is also possible to obtain traces ofR0ðtÞ and the average ⟨R0ðtÞ⟩ corresponding to intermediate and lategenes using our model and the above procedure: keeping fixed thevalues of k0on, k

0off and d0R and simulating cell-to-cell variability by

using trajectories of N(t) and I(t) obtained from parameter rando-mization of our model. In Fig. 5(b), together with the ⟨R0ðtÞ⟩previously obtained corresponding to an early gene (blue ‘þ ’),we show the ⟨R0ðtÞ⟩ obtained for k0on ¼ 0:2kon, k0off ¼ 0:3koff andd0R ¼ 0:6dR (red ‘þ ’) and for k0on ¼ kon=8, k

0off ¼ koff =8 and d0R ¼ dR=8

(blue ‘þ ’) that can be considered to correspond to intermediateand late genes respectively. Thus, our model can reproduce thedifferent timings of gene expression observed in the experimentsupon stimulation of the cells.

Our model is also able to account for other interesting phe-nomena observed in experiments. In Sung et al. (2009) it wasshown that by co-treating cells stimulated with TNF�α with theinhibitor of nuclear protein export Leptomycin B (LMB) or theinhibitor of protein synthesis Cycloheximide (CHX) led to apersistent nuclear localization of NF�κB. This is more or lessexpected, provided that CHX should block the inhibitor synthesis,keeping NF�κB in the nucleus as well. However, it was found(Sung et al., 2009) that although the dynamics of NF�κB were verysimilar, the gene expression output was very different: co-treatment with CHX induces a high increase in the transcriptionof early, intermediate genes and moderate in the late genes (Sunget al., 2009), while co-treatment with LMB leads to a globaldecrease in the transcription.

A hypothesis to explain this experimental observation can beprovided using our model. The natural way to mimic the action ofCHX would be to set kp¼0, provided that kp is proportional to theprotein synthesis rate KP, see Appendix A. On the other hand, in orderto mimic the block of nuclear export due to LMB, provided that weconsider only two species of NF�κB, the free (and thus predomi-nantly nuclear) and the bound to IκBα (predominantly cytoplasmic),we can set a¼0. The reason for this is that this parameter isproportional to the rate A of formation of the complex (NF�κB:IκBα) and implicitly of its export to the cytoplasm, where it is nottranscriptionally active. In that case, according to our model therewould only be free and mostly nuclear NF�κB, which is a simplifica-tion because in principle a fraction of nuclear NF�κB should bebounded to IκBα. As we will show below, though, this does notchange the ability of the model to reproduce the observed decrease.In Fig. 5(c) we show that indeed both situations induce a persistentnuclear localization of NF�κB, giving a value of N close to one.However we can see that the values of I differ in the two cases, which

in turn affects the gene activity G, which is definitely lower when thesystem is co-treated with LMB than when CMX is used.

In order to simulate the transcriptional output in these twosituations, we proceed as before: we obtain a number of trajec-tories from our model by setting either a or kp equal to zero andvarying the rest of the parameters randomly up to 50% all theparameters to account for co-treatment and cell-to-cell variability.We use the resulting N(t) and I(t) in Eqs. (19) and (20) to obtain theaverage transcriptional output ⟨R0ðtÞ⟩. This is done for the threeparameter sets that we identified previously as able to reproducethe expression of early, intermediate and late genes. The results ofthese computations are shown in Fig. 5(d) for early genes, in Fig. 5(e) for intermediate genes, and in Fig. 5(f) for late genes. In oursimple model we observe that the co-treatment with LMB leads tolower transcription levels, while co-treatment with CHX leads tohigher transcription with respect to what is obtained when onlyTNF�α is applied to the cells. This is possible even if our modelsomehow overestimates the amount of nuclear free NF�κB. Thus,our simple model is also able to reproduce the effects of co-treatment of both CHX and LMB with TNF�α.

While the difference in result using two different inhibitors isstriking, it is not difficult to suggest the reason of this difference.As we anticipated previously, although both perturbations leadto similar N(t) profiles, they give different gene activities G(t).CHX interrupts the synthesis of IκBα, and hence the probability ofinactivation (process 6) is zero, so it is permanently active.However with LMB the inhibitor is continuously synthesized andhence the negative feedback is still present because IκBα increasesthe inactivation rate of the gene (while still binding the freeNF�κB), leading to its partial inactivation. This simple exampleshows that a simple model like ours can help us to formulateinteresting hypothesis on the basic mechanisms involved in theNF�κB signaling pathway.

7. Conclusions and discussion

In this paper we have shown that a simple model built usingfirst principles, just summarizing the basic processes that takeplace in the NF�κB signaling pathway, displays already a variety ofbehaviors that can be linked to many different experimentalobservations. The fact that this simple model is able to reproducefeatures like the spiky and smooth oscillations observed indifferent experiments, or the persistent nuclear localization ofNF�κB that might take place upon certain stimuli, is a clearevidence of the versatility of the NF�κB signaling system and itsability to provide different dynamical response in different con-texts, also for different cell types.

We believe that following our approach, simple models canalso be built for other transcription factors with negative feedbackand for which oscillations have been observed, such as p53 (Geva-Zatorsky et al., 2006). On the other hand, these simple models canprovide valuable insights on the dynamics of the system consid-ered, they are more amenable to analytical treatment and are lessprone to over-fitting. It is necessary to acknowledge, though, thatsimplified models have inherent limitations and there might besituations in which data from an experimental assay are difficult tomatch to the mathematical models. This can be the case forexample when trying to understand the effect of specific pertur-bations in one of the many processes that are summarized as justone process in the simplified model: in that case it might bedifficult to isolate precisely the effect of such perturbation. Thisneeds to be taken into account when considering what is the levelof mathematical complexity needed to gain insight on the phe-nomena that we are studying.

S. Zambrano et al. / Journal of Theoretical Biology 347 (2014) 44–5352

From a broader perspective we think that models like the oneproposed here with low number of species involved can be used toevaluate in a more precise way the role of stochasticity in thebiochemical reactions compared to more complex models wherethe number of uncertain parameters is much higher: this explorationwill be performed elsewhere. Finally, we suggest that future tissue-level models or whole-organism models in which many processesare modeled simultaneously (Karr et al., 2012) will require models ofsignaling with an adequate trade-off between complexity and abilityto capture single-cell dynamics, similar to the model that wepropose here.

Acknowledgments

S.Z. is supported by the Intra-European Fellowships for careerdevelopment-2011-298447 NonLin-kB and M.E.B. is supported bythe Italian AIRC 2010/2013 Project no. R0444.

Appendix A. Rates of the biochemical reactions and relationwith parameters of the normalized model

Some of the parameters of the normalized model are obtained byadequately normalizing the reaction rates. Here we give the relationbetween those parameters and the reaction rates considered (noticethan lowercase parameters are related with the same uppercasereaction rate). p¼ P � IKK0, a¼ A �NF�κB0, kp ¼ KP � KRG0=

ðdR � NF�κB0Þ, kon ¼ Kon � NF�κB0, koff ¼ Koff � NF�κB0.

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