Jonak et al. BMC Systems Biology (2016) 10:75 DOI 10.1186/s12918-016-0293-0

RESEARCH ARTICLE Open Access

A novel mathematical model ofATM/p53/NF-κB pathways points to theimportance of the DDR switch-off mechanismsKatarzyna Jonak1†, Monika Kurpas1†, Katarzyna Szoltysek2, Patryk Janus2, Agata Abramowicz2

and Krzysztof Puszynski1*†

Abstract

Background: Ataxia telangiectasia mutated (ATM) is a detector of double-strand breaks (DSBs) and a crucialcomponent of the DNA damage response (DDR) along with p53 and NF-κB transcription factors and Wip1phosphatase. Despite the recent advances in studying the DDR, the mechanisms of cell fate determination after DNAdamage induction is still poorly understood.

Results: To investigate the importance of various DDR elements with particular emphasis on Wip1, we developed anovel mathematical model of ATM/p53/NF-κB pathways. Our results from in silico and in vitro experiments performedon U2-OS cells with Wip1 silenced to 25 % (Wip1-RNAi) revealed a strong dependence of cellular response to DNAdamages on this phosphatase. Notably, Wip1-RNAi cells exhibited lower resistance to ionizing radiation (IR) resultingin smaller clonogenicity and higher apoptotic fraction.

Conclusions: In this article, we demonstrated that Wip1 plays a role as a gatekeeper of apoptosis and influences thepro-survival behaviour of cells – the level of Wip1 increases to block the apoptotic decision when DNA repair issuccessful. Moreover, we were able to verify the dynamics of proteins and transcripts, apoptotic fractions and cellsviability obtained from stochastic simulations using in vitro approaches. Taken together, we demonstrated that themodel can be successfully used in prediction of cellular behaviour after exposure to IR. Thus, our studies may providefurther insights into key elements involved in the underlying mechanisms of the DDR.

Keywords: Mathematical, Model, Deterministic, Stochastic, DNA damage, ATM, Wip1, p53

BackgroundDNA double-strand breaks (DSBs) appear in the non-dividing cells as a result of stress agents activity, whatleads to induction of the DNA damage response (DDR)and eventually DNA repair or cell apoptosis [1]. Signalabout DSBs is transmitted through ataxia telangiectasiamutated (ATM) – serine/threonine kinase – to p53 cellu-lar tumour antigen and nuclear factor NF-κB. These twotranscription factors are responsible for cell fate determi-nation; however, it is still not fully understood how the celldecides about its fate.

*Correspondence: krzysztof.puszynski@polsl.pl†Equal contributors1Faculty of Automatic Control, Electronics and Computer Science, SilesianUniversity of Technology, Akademicka 16, 44-100 Gliwice, PolandFull list of author information is available at the end of the article

Many studies confirmed that DNA lesions and incorrectmechanisms of the DDRmay lead to pathological changestransmitted to daughter cells, uncontrolled proliferationand tumour growth [2–7]. It has been reported that anessential role in the DDR is played by protein phosphates,among them Wip1 – a p53-induced protein phosphatase1D [8]. Wip1 is a crucial component of cancerogene-sis and is involved in the ATM/p53 pathways [9–11].Furthermore, it has been suggested that Wip1 regulatescell-autonomous decline in proliferation of self-renewingcells with advancing age [12].Investigating the DDR in general and connections

between ATM, p53, NF-κB andWip1 phosphatase in par-ticular is essential for understanding the cellular responseto DNA damages. As a result, these studies should help

© 2016 The Author(s). Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, andreproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to theCreative Commons license, and indicate if changes were made. The Creative Commons Public Domain Dedication waiver(http://creativecommons.org/publicdomain/zero/1.0/) applies to the data made available in this article, unless otherwise stated.

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to predict the behaviour of mammalian cells and deter-mine molecules that may become potential drug targets[13]. Although there have been many advances in the fieldof studying the DDR, the exact mechanism of cell fatedetermination is still largely unexplored.To answer the question whether Wip1 plays an impor-

tant role in cell fate determination and how sensitive thesystem is to various changes in the ATM/Wip1 mod-ules, we developed a novel mathematical model connect-ing ATM, Wip1, p53 and NF-κB. A new hybrid modelcombines stochastic and deterministic approaches and isbased on our previously constructed stochastic model ofp53/NF-κB pathways [14] and deterministic model of theATM/p53 pathways [15].Cell fate is determined by the accumulated levels of

p53 and its transcriptional targets, among them Cdk1-inhibitor p21, which initiates the cell cycle arrest [16],and Bax, which triggers the apoptotic events [17]. Over-experession of p21 and Bax has been found in manytypes of cancers, suggesting their high clinical importance[18–20].In this article, we describe a specific kinetics of Wip1

after the induction of DSBs by IR. According to in sil-ico and in vitro results, Wip1 plays a role as a gatekeeperin the ATM/p53 system. The phosphatase accumulates incells upon DSBs induction, in order to “turn off” the DDRsystem after the successful damage repair. However, if thelevel of Wip1 stays high, cells may become insensitive tofurther damages. Hence, the accurate regulation of Wip1is necessary to repair the system. It has been reported thatone of the main regulation factors important in the DDRsystem involves miRNAs [21, 22]. Therefore, we proposeto include miRNA-16 in the model as it plays essentialrole in restraining proliferation of tumour cells throughinhibition of Wip1 transcript [22, 23]. Based on the exper-imental data, we propose a model containing differentregulators of Wip1 transcript, among them miRNA-16and a stimulator of Wip1 transcription – CREB.

Model developmentTo investigate the importance of various components ofthe DDR system on cell fate determination, we devel-oped a novel mathematical model of the ATM/p53/NF-κBpathways.Proposed model is based on our previous p53/NF-

κB crosstalk model [14] and preliminary ATM pathwaymodel presented in [15]. To cover the experimentallyobservedWip1 behaviour, we added regulators of its tran-scription in form of miRNA-16 and CREB. Our modelis one of a few that includes the spatio-temporal regu-lation of the studied ATM/p53 pathways (we distinguishthe nucleus, the cytoplasm and the extracellular matrix)and according to our knowledge the only one that com-bines that with stochasticity. This extension together with

the introduction of p21 and Bax molecules responsible forcell cycle blockade and initiation of apoptosis allow us tosimulate a single cell response to DNA damage from thedetection system to the mechanisms directly involved incell fate determination.The interactions between molecules are presented in

Fig. 1. All of the processes occurring in the system weredescribed with ordinary differential equations (ODEs) dis-closed in Additional file 1. Numerical implementation ofthe model was based on the Haseltine-Rawlings postu-late – a hybrid approach used to combine stochastic anddeterministic methods [24].Stochasticity in our model is present as a random gene

switching (main source), DNA damage and repair events(as an appearance/disappearance of a particular break-age), and TNF receptors activation/deactivation events.The model equations were built using basic biochem-

istry laws: law of mass action and Michaelis-Mentenkinetics. Detailed information about the numerical imple-mentation procedure is available in Additional file 2.Detailed information about the model variables andparameters is available in Additional files 3 and 4.

Activation of the DNA damage responseIn this section, we briefly describe the activation pro-cesses of the studied DDR pathways included in the newmodel. A detailed biological interactions are presented inAdditional file 5 and in the references within the arti-cles [14] and [25].We assume that detection of DSBs is triggered by ATM

kinase and Mre11-Rad50-Nbs1 (MRN) complex in thenucleus, as proposed in the previous deterministic model[15]. ATM triggers the activation of p53, Chk2 and indi-rectly a cytoplasmic form of Mdm2 – a natural inhibitorof p53. Active p53 and Chk2 stimulate the DNA dam-age signal increasing the activity of proteins involved inATM and p53 pathways, among them Wip1 phosphatase.Wip1 dephosphorylates main system kinases and tran-scription factors leading to their inactivation or, as forMdm2, activation.We extended our existing p53 model [25] and

introduced an additional form of Mdm2 – a multi-phosphorylated inactive nuclear Mdm2. We assumed thattwo Mdm family members, Mdm2 and MdmX, can betreated as one (Mdm2), what simplify the model descrip-tion [14, 26].In the model, we included a complex ATM-dependent

regulation of Wip1: through mRNA inhibitor miR-16(miRNA-16) and its activator KSRP, and Wip1 synthesisactivator CREB. Furthermore, Wip1 transcription is up-regulated by p53 and NF-κB. This link betweenWip1 andNF-κB leads to the transcriptional inhibition of NF-κB-dependent genes, like these encoding A20, IκB, p53 andWip1 itself. In contrast, ATM activates NF-κB pathway via

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Fig. 1 Detailed schematic of the ATM/p53/NFκB pathway model. Active and inactive states of proteins and transcripts (mRNA) are presented. Themodel distinguishes three compartments: the extracellular matrix, the cytoplasm and the nucleus. There are four main modules of the model: p53(black), NF-κB (gray), ATM (green) and Wip1 (blue). Dotted lines with arrow-heads stand for positive regulation between components, solid lines fortransitions between states of the components, crossed circles for degradation of the protein or transcript, and “P” for phosphorylated form of theprotein; “a” for ATM stands for fully active form of this kinase

the cytoplasmic Iκ B complex. Finally, the DNA damagesignal is transmitted to decision-making proteins: p21 andBax.

Gene switching stochasticityStochasticity in the model occurs mainly due to thestochastic gene switching. Genes encoding Wip1 andChk2 are activated with the probabilities proportional tothe amount of p53 transcription factor. Additionally, geneencoding ATM depends on CREB activity. Gene for Wip1is activated by three factors: p53, CREB and NF-κB. How-ever, Wip1 inhibits its own NF-κB-dependent activationthrough its negative regulation. Notice that p53 inhibitsChk2 transcription. Deactivation of genes included in themodel is assumed as background/spontaneous.We assumed that each gene has two copies (alleles) with

three possible states: both alleles can be active, only oneor none of them. In stochastic approach, these states aregiven by values 2, 1 and 0, respectively, while in determin-istic model they are within the range <0, 2> that is themean state for cells population.

Cell fate decisionThe output of our model is the levels of molecules ata certain time after irradiation and cell fate decision.Following Kracikova et al. findings [27], we implementeda novel p53-dependent threshold mechanism to deter-mine behaviour of cells after introducing DNA damages.For simplification, we considered only two of p53 prod-ucts: Bax and p21. The lower first threshold determinesthe cell cycle arrest and is based on the p21 protein level.Due to the fact that p21 is p53-dependent protein andstrictly follows its level, the first threshold is also indirectlydependent on active p53.The higher second threshold is responsible for driving

cells to apoptosis and it combines the levels of Bax andactive p53. In the model, we used both proteins to reflectthe fact that p53 may have also a non-transcriptionalinfluence on the apoptotic decision [28–30].Briefly, if during the simulation time the levels of active

p53 and Bax cross the given thresholds simultaneously,the studied cell is considered as apoptotic. If the p21 levelis above the threshold, the cell is considered as one with

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arrested cycle. If this cell dies before the first division orits cycle is arrested for more than 63.9 h (which preventslarge-enough single cell origin colony formation), it is con-sidered as not viable. Detailed description of determiningthe thresholds is available in Additional file 6.

Model fitting and parameters justificationThe information about the half-lives of majority of thetranscripts and proteins was obtained from the litera-ture or our experiments performed on U2-OS cell line(immunoblots are presented in Additional file 7). Theparameters involved in the system activation by IR wereobtained by fitting the model to the number of DSBsacquired from our experiments and data from the Kohnand Bohr studies [1]. Transcription and translation ratesfollow the Levin upper limits [31]. Detailed informationabout the parameters are available in Additional file 4. Theparameters indicated as “fitted” were guessed by a trial-and-error method in such way that they allow the modelresponse to be in agreement with the experimental datapresented in Figs. 2, 3, 4b and 4c (training set).It is known that the parameters estimation is com-

plicated for systems with large number of parametersand mostly for stochastic and hybrid models [32]. Onehas to remember that complex models with many freeparameters suffer from so called “model sloppiness”. Ithas been demonstrated [33] that even when some indi-vidual parameters are poorly constrained (sloppy), collec-tive fitting could result with well-constrained predictions.Moreover, in such case sensitivities spectrum could havethe eigenvalues distributed over many decades. In [33], itis postulated that models with constitute rule of sloppyparameters are not exception.The second common problem of the complex biologi-

cal network models is a practical non-identifiability – theexistence of various parameters sets that have more orless equivalent fitting capability [34]. If the practical non-identifiability exists, the minimum of the performance

index in the automatic fitting algorithms is surroundedby a large flat region or multiple local minima of compa-rable “depth”. Therefore, from the identifiability point ofview, time consuming automatic algorithms do not bringany advantage over the trial-and-error method. Even ifsuch an algorithm would be able to find the true min-imum, remembering that usually user has to determinethe parameters box in which the search take place, the“uniqueness” of such parameter set would be questionableon the ground of biological meaning.Taking into account this knowledge, we decided to use

the trial-and-error method to fit the model to the experi-mental data. However, we do not claim the uniqueness ofour parameters set.To test the sensitivity of the model to the fitted param-

eters, we performed sensitivity analysis according to theprocedure described in [35] and in Additional file 8. Theresults show that for the time periods of measurementof apoptotic fractions and viability the model outputs areinsensitive to the change of parameters values. This makesour predictions reliable. The results of sensitivity analysisare presented in Additional file 8.

ResultsWip1 exhibits a non-oscillatory behaviour after high dosesof IRTo investigate the role of Wip1 in cell fate determination,we performed stochastic simulations for 1000 cells and invitro experiments on U2-OS cells with wild-type expres-sion of Wip1 (Ctr-RNAi) and expression reduced to ca.25 % of initial value (Wip1-RNAi). Due to the fact thatimmunoblot experiments were performed on human can-cer cell line, we were detecting Mdm2 human analogue –Hdm2.To investigate the kinetics of Wip1 after exposure

to IR, we performed immunoblot assay and stochas-tic simulations for Ctr-RNAi cells treated with 10 Gyof IR. We compared the levels of Wip1 in irradiated

Fig. 2 Kinetics of Wip1. a Immunoblots of Ctr-RNAi cells show Wip1 kinetic for control cells (“Ctr”) and for cells irradiated with dose of 10 Gy.b Results of stochastic simulations on 1000 irradiated cells. Solid line stays for median values, dashed for 1st and 3rd quartile, while dashed black linesfor time points when experimental data were gathered

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Fig. 3 Influence of Wip1 gene silencing on cellular response to DNA damage. a Immunoblots of Wip1, active p53, Chk2 and Hdm2 (Mdm2 formouse) for control and time points 2, 8 and 24 h after exposure to 10 Gy of IR. Experiments were performed for Ctr-RNAi and Wip-RNAi cells.b–e: Results of stochastic simulations on 1000 cells irradiated with dose of 10 Gy show the levels of various proteins. Solid line stays for medianvalues, dashed for 1st and 3rd quartile, while dashed black lines for time points when experimental data were gathered

cells to the levels in untreated cells (Fig. 2). Here, ourmain studies were focused on verification whether Wip1exhibits an oscillatory response after strong irradiationof cancer cells and if it follows the oscillations (levels)of p53.In the literature, it has been reported that the transcript

of Wip1 follows the levels of p53 and the highest level isobserved two hours after irradiation [36, 37]. In themodel,we fitted the parameters in a way that the response ofWip1 mRNA is comparable.We performed immunoblot experiments to verify the

kinetics of Wip1 in 24-h time course after irradiation. Thedata from our biological experiments and simulations areconsistent with the reported data about the kinetics ofWip1 in U2-OS cell line [38].For the control wild-type cells we found the presence

of Wip1 protein only 15 h after the initiation of theexperiment. The levels of the phosphatase elevated andreached the maximum at 18 h. These clear bands of Wip1

might be an effect of the activation of the DDR in responseto DNA damages occurring spontaneously in cells. Forthe irradiated cells, we observed that the level of Wip1increased to reach the maximum at around 18 h. Thesehigh levels persisted longer than in untreated cells. At24 h, the levels of Wip1 started to decrease, what suggeststhat when DSBs are finally repaired by the systemWip1 isbeing reset in the subsequent cells.Although some groups reported oscillatory behaviour

of Wip1 activated upon DSBs induction [38–40], weobserved its non-oscillatory behaviour in U2-OS cellsafter treatment with 10 Gy of IR during a long 24-htime course. Similar results were obtained by [41] forshorter duration of the experiments. We observed onlysmall rises and falls of Wip1 during the general eleva-tion after IR. Because other research groups reportingoscillatory behaviour of this phosphatase used differentdamaging agents and/or cell lines, we think that Wip1response pattern may differ depending on the cell line and

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Fig. 4 Impact of IR and TNFα on cells viability. a Percentage of cells viability for Ctr-RNAi and Wip1-RNAi cells after treatment with various doses ofIR. b Percentage of Ctr-RNAi and Wip1-RNAi apoptotic cells 24 and 48 h after irradiation with various doses of IR. c Impact of TNFα on fraction size ofapoptotic cells. Note: Because all p-values > 0.05 (for α = 0.05), most of them > 0.85, there is no reason to reject the null hypothesis thatexperimental and simulation values are no different

damaging agent, maybe even IR doses. This will be thescope of our future research.It is equivocal if Wip1 phosphatase exhibits oscillatory

or non-oscillatory behaviour in general. Here, we decidedto use term “rise and fall” of Wip1 levels instead of “oscil-lations”, which we use rather to describe clear and robustp53/Mdm2 oscillations.

Reduction ofWip1 affects the DDR systemTo further investigate the influence of IR on Wip1behaviour, we performed biological experiments onWip1-RNAi cells treated with 4 Gy, what gave us ananswer to the question about the influence of Wip1 onvarious molecules from ATM/p53/NF-κB pathways. Oursimulations and biological experiments (Fig. 3a) clearlydemonstrate that the levels of Wip1 decrease aroundfour-fold in Wip1-RNAi cells comparing to Ctr-RNAi.For p53 (Fig. 3a and d), the highest change between

Wip1-RNAi and Ctr-RNAi cells was observed around 2 hafter irradiation.The kinetics of p53 and Mdm2 in Wip1-RNAi did

not change drastically comparing to Ctr-RNAi: in bothcases these proteins demonstrated high peak of activityafter irradiation and then extinguishing oscillations with

smaller amplitudes (Fig. 3a, d and e). The response ofboth proteins in their active forms was stronger in Wip1-RNAi and both stayed longer at these high levels. Incontrast, we observed that Chk2 did not oscillate as p53(Fig. 3a, b and c). The response of Chk2 was stronger inWip-RNAi than in Ctr-RNAi cells. However, the biggestchange was detected after 8 h for irradiated cells, wherein Wip1-RNAi cells Chk2 levels were decreasing slowercomparing to Ctr-RNAi cells that stabilised at around18 h. To summarise, we found that silencing Wip1 genewith RNAi enhances the levels of proteins essential in theDDR system.

Wip1maintains cell viabilityTo analyse cells viability after irradiation and silencingWip1, we performed biological experiments. Clonogenicabilities (Fig. 4a) of the irradiated cells were measuredin vitro and in silico. The clonogenic cell survival assayallows to determinate how many cells are able to pro-liferate by meaning of at least 30 cells large colony for-mation during 10 days period. We used received resultsas training set for stochastic model for Ctr-RNAi andWip1-RNAi cells. In our study, the inability to prolifer-ate is equivalent to exceeding at least the first threshold

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described by Kracikova et al. [27]. In the clonogenic cellsurvival assay, the apoptotic cells and arrested cells are notdistinguishable.Simulated cells were considered as not viable if they

died before the end first division or their p21 level waselevated above a given threshold for ∼64 h – these cellswere not able to form colonies large enough. After irradia-tion, reduction of clonogenic potential was observed withincreased dose of IR in both Ctr-RNAi and Wip1-RNAi.In cells with silenced Wip1, the clonogenic potential wastwo-fold lower than in the control for doses of 2 and 4 Gy.After the treatment with doses over 6 Gy, cells lost theirclonogenic potential entirely. We observed that dosesabove 4 Gy successfully stopped cellular division. Thesedata confirmed that Wip1 has a pro-survival effect andregulates apoptosis in mammalian cells, what is consistentwith the previous reports [42].

NF-κB pathway activated before irradiation leads toincreased cell survivalOne can notice on Fig. 1 that ATM not only activatesthe p53 pathway but also the NF-κB module through thephosphorylation of IKK. NF-κB itself is also responsiblefor transcriptional activation of p53. These dependen-cies increase the number of p53 molecules after irradi-ation and thus increase the following apoptotic fractionthrough p53 activation with simultaneous Mdm2 degra-dation and increase of p53 synthesis. To investigate whichof these two interactions is dominant, we performed invitro and in silico experiments with IR-based activation,only TNFα-based NF-κB activation and simultaneous IR-and TNFα-based activation.Here, we focused on a difference in apoptotic fraction

in wild-type cells treated with pro-survival agent TNFα.Again, we performed stochastic simulations for 1000 cellsand biological experiments on U2-OS cell line. For bothcases, we irradiated cells 24 h after initiation of the exper-iment with various doses of IR. For TNFα, we used onlyone dose (10 ng/ml), because testing the effects of variousdoses of this cytokine was not a subject of our study.The apoptotic fraction for Ctr-RNAi cells was deter-

mined experimentally by flow cytometry assay and thenused as a training set for stochastic simulation of 1000cells per dose (Fig. 4b). Following Kracikova et al. [27],simultaneously elevated levels of p53 and Bax above thegiven threshold was equivalent to simulated apoptosisof the cells population. In our studies, cells labelled asapoptotic 24 h after irradiation were excluded from theapoptotic fraction at 48 h. Therefore, the total number ofcells that died after DSBs induction was a sum of apop-totic cells that did not survive at various time points. Weobserved that for cells counted 48 h after treatment withIR, the highest apoptotic fraction was observed for 10 Gy,while the lowest was detected for 6 Gy (Fig. 4b). Thus,

we concluded that when the irradiation dose is too highcausing irreparable damages of DNA chains, Wip1 has nobigger impact on cells viability.To analyse the impact of NF-κB pathway activation pre-

ceding radiation exposure, we performed simulations forCtr-RNAi cells for three cases: 1) without exposure to IRbut with TNFα cytokine added in dose of 10 ng/ml for1 h; 2) without TNFα stimulation, but irradiated with 4Gy; 3) with TNFα cytokine stimulation (10 ng/ml for 1 h)and irradiated with dose of 4 Gy two hours after finish-ing administration of TNFα. We counted the apoptoticfraction using flow cytometry technique 24 and 48 h aftertreatment with IR. We observed (Fig. 4c) small apoptoticfraction for untreated cells with TNFα added to the U2-OS cell culture. For irradiated cells with TNFα counted48 h after irradiation, we detected more apoptotic cellsbut three-fold less than for cells treated with 4 Gy ofIR. Our results about the radio-protective effect of TNFαadded before the irradiation are consistent with the pre-vious reports [43, 44]. TNF administration 6 h after IRincreased the number of apoptotic cells in the studiedpopulation. Our model predicted 1.3 % of apoptotic cellsafter 24 h and 4.5 % after 48 h. We investigated depen-dency between TNF-IR shift and apoptotic fraction indetails in [14].In summary, the results of in silico and in vitro experi-

ments confirmed that the number of apoptotic cells in thepopulation treated with cytokine before irradiation wassmaller than in the population exposed only to IR (Fig. 4c).Moreover, we noticed that for TNFα-treated cells, theyneed at least 24 h for the effect of the treatment to besignificantly visible.

DiscussionIn this study we constructed a new stochastic modelof ATM/p53/NF-κB pathways with Wip1 phosphatasein the main core and additional components responsi-ble for its regulation. Our model is one of a few thatincludes the spatio-temporal regulation of the studiedATM/p53 pathways [45]. Most of the models of ATM/p53pathways do not specify the localisation of molecules[39, 46–50]. Contrary, our model includes three compart-ments contributing to a time delay in the system.

Wip1 kineticsIn our model, an essential role in cell fate determinationafter DSBs induction is played by Wip1 protein phos-phatase. Similarly, several previously developed modelsconsidered the effects of this phosphatase on ATM mod-ule alone [47, 50, 51] or on ATM and Mdm2 [52] or Chk2[48]. In contrast, there are still models that do not includeWip1 as a component of the DDR system [46, 49, 53–56].Mostly, models that include Wip1 show its oscillatory

behavious similarly as for p53s [39, 47, 48, 51, 57]. The

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only model that does not show oscillatory behaviour ofWip1 is presented in [50]. Biological experiments showdifferent patterns of Wip1 for different cell lines, such asMCF-7 [41], HCT116 [58], RPE [59], BJ fibroblasts [41]and U2-OS [38, 41]. For example, it has been reportedthat in MCF7 cell line small oscillations of Wip1 arepresent up to 12 hours after irradiation [39, 51]. Our stud-ies on U2-OS are consistent with studies performed onthe same line by other groups [38, 41]. Worth noting isthat in our study samples are collected for longer timeperiod: 24 and 48 hours. Other groups often limit theduration of the experiment to 12 hours. We think thatit does not fully reflect the kinetics of Wip1. Moreover,we noticed that treatment with different factors, like UV[60], cisplatin [61], 4-hydroxytamoxifen [62] or IR (i.e. inour studies), may have a crucial effect on the kinetics ofWip1 protein. Futhermore, Wip1 regulation may be dose-dependent [40], but we did not perform verification ofthese observations.Recent studies [63] indicate that Wip1 activation

depends also on a phase of cell cycle. It is another possiblereason why sometimes oscillatory behaviour of Wip1 isobserved, especially when we take into account the influ-ence of cells synchronisation. These differences of Wip1behaviour will be a scope of our future work. Using ourmodel we observed dependency which is known from lit-erature – Wip1 is a main deactivation agent for manyproteins involved in the DDR pathway. We confirmed thatWip1 is also important in maintaining cells viability. Itsaccumulation is essential for resumption of cell cycle pro-gression and for deactivation of pro-apoptotic proteinsafter successful DNA repair [42].

Influence of complex Wip1 regulation on cell viabilityOur previously developed preliminary model of ATM[15] allowed us to predict the overall behaviour of cellstreated with IR. However, that model was not fitted toin vitro data or the fitting was not complete. Moreover,observed by us long Wip1 half-lives and elevating lev-els of mRNA and protein under DSBs induction, resultedin inability of cells to enter apoptosis pathway even ifthe damages were not repaired. On the other hand, weobserved that elimination of Wip1 from the model resultsin much higher percentage of cells deaths. These cellsactivated their DNA damage repair processes much latercomparing to the wild-type situation. This fact resultedin high difficulties in fitting the model in its previousversions. We found that the solution to this problemmay be to introduce miR-16 and a complex Wip1 reg-ulatory mechanisms. miRNAs are found to be impor-tant for regulation of the p53 pathway, however thereare not many models that include them [64]. Thus, weexpanded the preliminary model to obtain results com-parable to the biological experiments more accurately

[23, 65, 66]. In case of deletion of miR-16 from Wip1module, we observed increased cells viability (Additionalfile 9), which is consistent with literature reports[22, 66]. miR-16 is not the only miRNA down-regulatingWip1 [22, 67, 68]. Probably the cumulative activity ofmore of these molecules cause stronger response of thephophatase.

Model activation and final responseThe DDR system is usually described by the activity ofp53 protein and related molecules. Our previous modelsof p53 pathway [14, 25] and number of studies on dynam-ics of this module [69–71], describe DNA damages in asimplified way – as a direct input to the model causingactivation of p53. In some of the previously describedmodels [51, 55], repair process is also simplified. In [51],p21 was included as a component responsible for cellcycle arrest, allowing DNA repair. Also, in [72] p21-basedsignalling pathway was added without including detailsabout damage repair. For the apoptosis pathway, Bax wasintroduced in one of the p53 models [70].In [73], the authors focused on the order of the

first events in the senescence pathway and in [74] onthe detailed non-homologous-end-joining (NHEJ) repair.However, we did not intend to built a model of repair andsenescence processes. Therefore, similarly as in [51], weincluded simplified decision making pathways: p21 andBax. Our assumptions about cell fate are consistent withthe experiments from [75], where the higher dosages ofIR resulted in increased formation of DSBs and decreasedrate if DNA damage repair.A simplified model described in [39] includes the main

feedback loops between Mdm2, Chk2 and Wip1 proteins.In [76], it is suggested that cell fate is not fully dependenton the concentration of active p53 forms, but rather on thedynamics of its oscillations. The results from our experi-ments and simulations of the stochastic model agree withthat suggestion and are true for these cells with wild-typelevels of Wip1 and reduced levels to 25 %. The modelsimulations matched with the dynamics of the main com-ponents, like p53, Mdm2 and ATM, obtained from ourbiological experiments on U2-OS cells. Our model, sim-ilarly as in [74], matches the cell-to-cell variability and isable to explain the specific behaviour of the main compo-nents of the system.Some models were developed with simplified ATM/p53

pathways without including their complex regulation[48–50, 56]. In [50], ATM is considered together witha kinase responsible for detection of a single-strandedDNA: ataxia telangiectasia and Rad3-related protein(ATR). In one of the recent models [46], ATM is treated asa parameter that does not change over time. Furthermore,to our knowledge there is not many models includinginteractions of p53 pathway with other transcription

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factor responsible for cell fate determination – NF-κB[49, 77].It has been shown that the pulses occurring in p53

system are dependent on intrinsic DSBs and small oscil-lations in ATM detector module [39]. In [78], the authorsproposed a model with the pulses of p53 maintained bythe auto-regulatory positive feedback loop allowing thethreshold activation of p53. The activation of the ATMmodule was shown to exhibit different pattern what possi-bly depend on the amount of DNA damages per cell [79],similarly as we observed for Wip1.To date majority of the models on ATM and p53

pathways do not include stochasticity. We have alreadyobserved in our previously developed models of p53/NF-κB pathways [14] and ATR/p53 pathways [26] thatintroducing stochastic effects is crucial for modelling andpredicting the behaviour of a single cell and populationof cells. The stochasticity for random damage inductionand DNA repair was included in the model presented in[80]; however, this model could not reproduce the resultsconcerning the cell-to-cell variability in the pulses of p53and cell fate. On the other hand, the results of the simula-tions of stochastic model of NHEJ repair [74] was able tomatch the observed variability. Our stochastic model hasan advantage as the results of simulations on 1000 cellswere consistent with the experimental data regarding vari-ability after exposure to IR [76] and the results from ourin vitro experiments on U2-OS cell line with the wild-typeand reduced levels of Wip1.

Role of NF-κB pathway in the DDROur model shows a cytoprotective effect of TNFα onirradiated cells. Number of the apoptotic cells in pop-ulation treated with this cytokine before irradiation issmaller than in cells irradiated with the same IR dosewithout TNFα treatment. Such influence of TNF, whichnormally increases apoptotic fractions [81], is observedonly in case of treatment with TNF preceding irra-diation [43, 44] and was described in our previouswork [14].The results regarding cell fate determination after expo-

sure to IR obtained from the stochastic simulations fordifferent timing of added TNFα are in agreement withthe previous works. As noted in [77], we observed thatthe functional consequences of NF-κB activation by addi-tion of TNFα probably depend on the number, period andamplitude of the p53 oscillations. The results of simu-lations and our biological experiments demonstrate thatwhile NF-κB is being activated 3 hours before the induc-tion of DSBs, the levels of proteins responsible for signaldetection and transduction decrease comparing to thecase where NF-κB pathway was not activated by TNFαbefore irradiation.While analysing in silico results, we sawincreased period of p53, Mdm2 and p21 oscillations and

decreased amplitude of the pulses. Thus, the treatmentresulted in lowered apoptotic fraction.It has been reported that the effect of the activation of

NF-κB pathway by the cytokine may be different depend-ing on the timing of adding TNFα [81]. The higher apop-totic fraction was observed for TNFα added few hoursafter treatment with DNA damaging factor.We confirmedit by simulating 1000 cells when TNFawas added six hoursafter irradiation. When the DDR is already switched onbefore TNFα treatment, the activation of the NF-κB path-way enhances the effect of p53 and results is increasedapoptosis of population of cells.

Limitations of the modelDespite the accuracy of the model predictions, one shouldremember that our model is based on U2-OS cell lineand therefore the results it provides may be inaccu-rate for other cell lines, DNA damaging agents and/ordoses. Especially the number of DSBs after treatmentwith IR, proteins and mRNAs half-lives, genes activa-tion/inactivation rates and DSBs repair rate may differbetween cell lines. Moreover, some cell line-specific mod-ifications, like 25-fold Mdm2 gene amplification in SJSA-1cells or PTEN blockade in MCF-7 or H157 cells, shouldbe included when other cell line is considered. The abovemutations in the system can be compensate to some pointby changing the model parameters, i.e. degradation ratesor total number of active gene alleles. The model doesnot consider the influence of a phases of cell cycle oncells response to the damage. In our opinion, including thechanges in i.e. number of active molecules in various cellcycle phases may be important for cell fate determination.

ConclusionsThe presented stochastic model captures relevant biolog-ical aspects of the mechanism of the DDR system basedon the ATM/p53/NF-κB network. Stochastic modellingallows to observe differences in a single cell response aftertreatment with various doses of stress agents and manip-ulation of the components of the DDR system. Thesedisparities among single cells reacting to the same stimuliunder the same conditions are reflected by the apoptoticfractions size and viabilities of the surviving cells.Thus, the model presented in this study allows investi-

gating the process of determining the cell fate in responseto DSBs and provides enhanced explanatory and predic-tive power. In comparison to our previous works, the cur-rentmodel describes the studied pathways inmore details.As we presented in this article, some of the intermediatemolecules and interactions omitted in previously devel-oped models provide important delays to the dynamics ofthe studied system. Moreover, after conducting sensitivityanalysis, we noticed that our model is characterised by arelatively large robustness, what allows to use the model

Jonak et al. BMC Systems Biology (2016) 10:75 Page 10 of 12

for different cell lines and parameters obtained for them.This greatly increases the cognitive and predictive value ofour model.Collectively, we provide clear evidences that the model

can be successfully used in the studies on cellularbehaviour after introducing DSBs in the system. Thus, webelieve that our studies may provide important insightsinto mechanisms of the complex DDR pathways activatedby DSBs induction.

MethodsCell lines The human osteosarcoma U2-OS cells (ATCCHTB96) were grown in DMEM supplemented with 10 %foetal bovine serum and gentamicin at 37 °C in 5 % CO2.For Wip1 down-regulation cells were transduced with alentivirus containing shRNA sequence specific to PPM1Dgene (Wip-RNAi) or a scrambled sequence shRNA (Ctr-RNAi) (sequences from Santa Cruz Biotechnology); trans-duced cells were selected with puromycin.

Cell treatments Treatments started 48 h after cells inoc-ulation; all steps, with exception of irradiation, wereperformed in a humidified incubator at 37 °C and 5 %CO2. Cells were incubated for 60 min in medium con-taining 10 ng/ml of TNFα cytokine (Sigma), irradi-ated with ionizing radiation or subjected to combinedtreatment of both cytokine stimulation and irradiation.After cytokine stimulation TNFα-containingmediumwasreplaced with fresh TNFα-freemedium and cells collectedat different time points after stimulation; during experi-ments with combined cytokine and irradiation treatment,after cytokine stimulation TNFα-containing medium wasreplaced with normal one for an additional 3 h beforefurther treatment with radiation. The ionizing radiation(IR) was generated by a linear accelerator (Clinac 600;Varian); cells were exposed to dose range between 2 and10 Gy (at a dose rate of 1 Gy/min). Irradiated cells wereincubated for an additional time (differ during differentanalyses: 60 min to 24 h for Western blot analyses or 24and 48 h for cytometry-based analyses) and then collectedand analysed. Appropriate controls, either untreated orTNFα-treated only, were processed in parallel. To deter-mine the half-life of proteins and transcripts (mRNA),cells were treated with cycloheximide (100 μg/ml) andactinomycin D (5 μg/ml) respectively, and collected atspecific time points (15 min to 8 h from the start oftreatment).

Determining the values of the model parameters Theanalysis of half-lives of transcripts and proteins was per-formed using qPCR method and immunoblot, respec-tively. Degradation rates for themodel were counted usingthe regression analysis with expotential approximation.The formation of DSBs through IR was measured using

γ -H2AX detection. The detailed information about theprocedures of defining the values of the parameters areavailable in Additional file 10.

Analysis of cell fate The analysis of apoptotic fraction(Sub-G1 fraction) was performed with flow cytometryassay. Clonogenic cell survival assay was performed toverify the viability of cells. Details about experimentalprocedures are available in Additional file 10.

Computational simulations Hybrid deterministic-stochastic simulations were performed according toHaseltine and Rawlings [24]. We used Runge-Kutta 4thorder algorithm for deterministic and direct Gillespie forstochastic part of the simulations. The simulation timewas set to 951000 s, simulation step for 0.1 s, and savingdata for every 10 s. We performed 1000 stochastic simula-tion for each of the experiment. Numerical simulations ofthe presented model were performed using Solvary (tooldeveloped by Roman Jaksik and Krzysztof Puszynski,Silesian University of Technology).

χ2 test Differences between results from biologicalexperiments and simulations were determined by p-valuesobtained from χ2 test for two sets of samples. Nullhypothesis: experimental and simulation values are nodifferent. Significance level assumed as α = 0.05.

Additional files

Additional file 1: Model equations. Transcribed model equations. For thestochastic variables we present reaction propensities, while fordeterministic proper ordinary differential equations (ODEs). (PDF 204 kb)

Additional file 2: Numerical implementation. Description and details ofnumerical implementation. (PDF 132 kb)

Additional file 3: Definition of the variables. Values of the model variablesare presented separately for Ctr-RNAi and Wip1-RNAi cells. (PDF 35.2 kb)

Additional file 4: Model parameters. Values and description of modelparameters. (PDF 202 kb)

Additional file 5: Biological description. Description of themain biologicalinteractions between the components included in the model. (PDF 96.3 kb)

Additional file 6: Cell fate decision. Description of the procedure ofdetermining the proper thresholds for the decision about the cell fate inthe model. (PDF 103 kb)

Additional file 7: Determining the half-life time of proteins included inthe model. Immunoblots of total ATM, unphosphorylated p53, p53phosphorylated on Ser15, Wip1, Hdm2, Hdm2 phosphorylated on Ser166,Chk2. (PNG 738 kb)

Additional file 8: Sensitivity analysis. Description and results of thesensitivity analysis of the fitted parameters. (PDF 1413 kb)

Additional file 9: Impact of miR-16. Impact of miR-16 deletion onirradiated cells. (PNG 2089 kb)

Additional file 10: Methods. Detailed description about the experimentalmethods described in the article. (PDF 132 kb)

Jonak et al. BMC Systems Biology (2016) 10:75 Page 11 of 12

AbbreviationsATM, ataxia telangiectasia mutated; ATR, ataxia telangiectasia mutated andRad3-related; Cdk1, cyclin-dependent kinase 1; Chk2, checkpoint kinase 2;DDR, DNA damage response; DNA, deoxyribonucleic acid; DSBs, double-strandbreaks; Hdm2, human Mdm2 homologue; IR, ionizing radiation; Mdm2, Mousedouble minute 2 homolog; MRN, Mre11-Rad50-Nbs1; NF-kB, nuclear factorkappa-light-chain-enhancer of activated B cells; NHEJ, non-homologous endjoining; ODE, ordinary differential equation; shRNA, small hairpin RNA; TNFa,tumor necrosis factor alpha; Wip1/PPM1D, protein phosphatase 1D

AcknowledgementsThe authors want to thank Roman Jaksik for the help in the development ofthe software Solvary for the purpose of performing deterministic andstochastic simulations of the described mathematical model. The authors wishalso to thank the anonymous referees for their useful suggestions.

FundingThis work was financially supported by the Polish National Science Center(NCN) grants no. DEC-2012/05/D/ST7/02072 (KJ, MK, KP) and N N518 287540(KS, PJ, AA).

Availability of supporting dataAll data sets supporting the results presented in the article are included withinthe article and additional files.

Authors’ contributionsKP suggested the problem and designed research. KJ and MK performedliterature review. KJ, MK and KP developed the model and performedsimulations. AA, KS and PJ performed biological experiments. KJ, MK and KPanalysed the data. KJ and MK prepared the figures. KJ, MK and KP prepared themanuscript. All authors read and approved the final manuscript.

Competing interestsThe authors declare that they have no competing interests.

Consent for publicationNot applicable.

Ethics approval and consent to participateNot applicable.

Author details1Faculty of Automatic Control, Electronics and Computer Science, SilesianUniversity of Technology, Akademicka 16, 44-100 Gliwice, Poland. 2MariaSklodowska-Curie Memorial Cancer Center and Institute of Oncology,Wybrzeze Armii Krajowej 15, 44-400 Gliwice, Poland.

Received: 17 January 2016 Accepted: 27 June 2016

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