Feedbacks, Bifurcations, and Cell Fate Decision-Making in the p53 System
[['The tumor suppressor p53 plays a pivotal role in cell growth control , DNA repair , cell cycle suppression and eventually in the initiation of apoptosis [1–4] .', 'It serves as a node of a complex and extensive gene regulatory network that integrates a variety of stress signals .', 'One of the most important ways of p53 activation is through DNA damage , which can be caused by , i . a . , ionizing radiation ( IR ) , UV radiation , hypoxia , heat shock , viral infection , or nutrient deprivation [1 , 5 , 6] .', 'Exposure to IR inflicts DNA double strand breaks ( DSBs ) , the most critical DNA lesions , which when unrepaired can lead to genomic instability resulting in either cell death or DNA mutations that can propagate to subsequent cell generations [7–9] .', 'The p53 regulatory network provides mechanisms that suppress cell cycle until DNA is repaired or trigger apoptosis when DNA damage is too extensive to be repaired [4 , 7 , 10] .', 'Unsurprisingly , mutations of the p53 gene ( TP53 ) turn out to be the most frequent genetic changes in human cancers [11] .', 'About half of all human cancer cells carry a mutation in TP53 .', 'In many other cancers , the genes which encode components of the p53 regulatory pathway are mutated [12 , 13] .', 'The lack of functional p53 protein decreases the efficiency of DNA repair and allows mutated cells to evade apoptosis leading to the propagation of mutations and eventually to cancer development and progression [13 , 14] .', 'Various malfunctions of the p53 regulatory pathway are determinants of tumor aggressiveness and p53 pathway components have become targets in anticancer therapies [14 , 15] .', 'The primary molecular function of p53 is the regulation of transcription [16 , 17] .', 'It controls expression of numerous genes which encode proteins of contradictory roles: pro-survival , cell cycle-suppressing , or pro-apoptotic [17] .', 'Its ability to regulate expression of distinct sets of genes is controlled by posttranslational modifications [5] .', 'In particular , p53 can assume two different phosphorylation states: p53ARRESTER ( when it is phosphorylated at Ser15 and Ser20 [5 , 18 , 19] ) and p53KILLER ( when it is additionally phosphorylated at Ser46 [20 , 21] ) .', 'In the arrester state , p53 triggers transcription of its main inhibitor , E3 ubiquitin ligase Mdm2 [22] , cell cycle-suppressing protein p21 [23] , and pro-survival phosphatase Wip1 [24 , 25] .', 'In the killer state , p53 triggers transcription of pro-apoptotic phosphatase PTEN [26] and another pro-apoptotic protein , Bax [27] .', 'In unstressed cells p53 remains inactive and is kept at a low level through constant proteasome-mediated degradation , regulated by Mdm2 [28–30] .', 'p53 is activated by kinase ATM , which is rapidly phosphorylated in response to DNA damage [31 , 32] .', 'The DNA damage detection system is sensitive; a handful of DSBs is sufficient for ATM and p53 activation; 1 Gy of irradiation induces about 35–40 DSBs [8 , 33] .', 'ATM phosphorylates p53 at several residues , including Ser15 and Ser20 , leading to its stabilization and activation as a transcription factor [19 , 34 , 35] .', 'ATM phosphorylates also p53 inhibitor , Mdm2 , promoting its degradation [36] .', 'As a result , p53 concentration increases 3–10-fold within an hour after DNA damage [34] .', 'Activity of p53 can be suppressed by growth factor stimulation leading to the activation of PI3 kinase ( PI3K ) , which phosphorylates phosphatidylinositol bisphosphate ( PIP2 ) into trisphosphate ( PIP3 ) [37–40] .', 'PIP3 allows for Akt recruitment to the plasma membrane where it can be activated via phosphorylation [41 , 42] by several kinases .', 'Activated Akt phosphorylates Mdm2 at Ser166 and Ser186 , allowing for its translocation to the nucleus where Mdm2 ubiquitinates p53 priming it for degradation [29 , 43] .', 'The p53-regulated proteins comprise a complex regulatory network governed by multiple intertwined feedback loops spanning diverse time scales [44] .', 'In addition to the primary p53 inhibitor , Mdm2 , other key regulatory players in the network are two phosphatases: pro-survival Wip1 [24] and pro-apoptotic PTEN [26] .', 'Wip1 attenuates signaling by dephosphorylating ATM [45] and inhibits apoptosis by facilitating dephosphorylation of p53KILLER to the p53ARRESTER form [46] .', 'When DNA repair is accomplished , Wip1 enables the return to the pre-stress state [24 , 47] .', 'PTEN mediates a positive feedback loop by inhibiting Akt and Mdm2 , and by allowing p53KILLER to rise to a high level and initiate apoptosis [48] .', 'Dysregulation of the p53 pathway can occur as a consequence of gene amplification , gene loss , promoter methylation , or mutations which alter protein function .', 'Gene amplification or inhibition are the major mechanisms which enhance expression of genes involved in cancer development and progression [49] .', 'In particular , Wip1 amplification and overexpression have been found in multiple human cancers , predominantly in those that retain functional p53 [50–53] such as breast , lung , pancreas , bladder , and liver cancer , and in neuroblastomas [50 , 52 , 54 , 55] .', 'Conversely , Wip1-knockout mice are partially resistant to oncogene-induced cancer development , indicating that the inhibition of Wip1 activity may be potentially beneficial for cancer therapy [45 , 52 , 56 , 57] .', 'Decreased PTEN expression has been found in prostate , kidney , breast , bladder cancers , and in glioblastoma [58–60] .', 'Cell lines with defective PTEN have alterations in the cell cycle regulation and a defective apoptotic response , which places PTEN among the tumor suppressors most commonly lost in human cancers .', 'In contrast , overexpression of wild-type PTEN in cancer cells induces apoptosis and blocks cell-cycle progression [61 , 62] .', 'Another dysregulated protein observed in many cancers is PI3K .', 'The amplification of PI3K gene have been found in colorectal , gastric , hepatocellular , thyroid , breast , lung , and ovarian cancer , and in glioblastoma and acute leukemia [63–70] .', 'The clinical data shows that differences between cancer cells can be partially explained by different expression levels of PTEN , Wip1 , and PI3K .', 'The majority of breast cancers reveals amplification of Wip1 and reduced level of PTEN , which appear to correlate with poor prognosis as well as increased resistance to γ radiation-based therapy and apoptosis [55 , 58] .', 'A number of computational models have been constructed to investigate the regulatory mechanisms of the p53 pathway .', 'Earlier models were focused on explaining the origin of oscillations of p53 and Mdm2 levels in response to IR [71–76] , which were first observed in the cell population experiment on MCF-7 cells by Bar-Or et al . [71] and later in single-cell experiments by Lahav et al . [77] and Geva-Zatorsky et al . [73] .', 'Recently , the effort has been shifted to connecting p53 dynamics with cell fate decisions [75 , 78–85] .', 'For the sake of conciseness of the Introduction , an overview of p53 modeling results cited in this paragraph is provided in S1 Text .', 'In this study , we focus on the p53 regulatory core to demonstrate that the structure of the p53 regulatory network is such that for a broad range of parameters at persistent DNA damage the limit cycle oscillations of p53ARRESTER ( as well as p53KILLER ) coexist with the steady state characterized by a high level of p53KILLER .', 'The detailed bifurcation analysis shows that the direct switch between oscillations and high steady state is enabled by the existence of a Neimark—Sacker bifurcation .', 'These two recurrent solutions delineate temporal responses of the system , which can be interpreted unambiguously by two slaved modules which control cell cycle arrest and apoptosis .', 'This is confirmed by stochastic simulations of the model dynamics , showing clear separation of cells into surviving and apoptotic subpopulations about 30 hours after DNA damage .'], ['The proposed model consists of three modules: the core module , the cell cycle arrest module , and the apoptotic module .', 'In Fig 1 we show a simplified scheme which summarizes the key feedback loops present in the core module .', 'The detailed scheme with all model components is provided in S1 Fig . The regulatory pathway considered consists of 42 species ( see Tables A and B in S1 Text ) , coupled by 74 reactions parametrized by 97 reaction rate coefficients ( see Table C in S1 Text ) .', 'The stochastic model is equivalent to a Markov process while its deterministic approximation is represented by ODEs .', 'The levels of all substrates in stochastic as well as in deterministic simulations are expressed in the numbers of molecules per cell ( mlcs/cell ) .', 'In this section we focus on the discrimination of possible responses to DNA damage .', 'In the bifurcation analysis we assume that the number of DSBs remains constant and equal to 100 , which is considered as severe DNA damage .', 'Bifurcation analysis allows us to determine the recurrent solutions ( i . e . , steady states and limit cycles ) that delineate time-dependent responses to DNA damage .', 'Since the two positive feedback loops are controlled by phosphatases Wip1 and PTEN , expression levels of which are varied substantially among cancer cells , we choose Wip1 and PTEN synthesis rates as bifurcation parameters .', 'Additionally , PTEN has slow dynamics with respect to the other components and its gradual accumulation after DNA damage controls the behavior of the whole system .', 'As a result , the system behavior can be predicted by bifurcation analysis in which PTEN is a bifurcation parameter .', 'In Fig 4 we show the two-dimensional bifurcation diagram in the ( s1 , s2 ) -plane , where s1 and s2 are Wip1 and PTEN synthesis rates , respectively .', 'As shown , the system involves three types of bifurcations: Hopf , saddle-node , and Neimark—Sacker .', 'In the 2-D diagram , Neimark—Sacker bifurcations line arises at one zero—Hopf point ( s1 , s2 ) ≈ ( 0 . 1 , 0 . 012 ) and vanishes at another zero—Hopf point ( s1 , s2 ) ≈ ( 0 . 3 , 0 . 053 ) .', 'Between these points the Hopf bifurcations are supercritical , while outside them they are subcritical .', 'The supercritical Hopfs line lies between the saddle—node line and the Neimark—Sacker bifurcations line .', 'The saddle—node lines arise in a cusp point ( s1 , s2 ) ≈ ( 0 . 02 , 0 . 001 ) .', 'The bifurcation lines divide the ( s1 , s2 ) -plane into subregions in which the system exhibits different behaviors .', 'In short , the system is oscillatory in the parameter subdomain to the right of the Hopf bifurcations line , i . e . , oscillations can arise due either increase of Wip1 synthesis or decrease of PTEN synthesis .', 'The bistability ( understood here as either the coexistence of two stable steady states or coexistence of one stable steady state and one stable limit cycle ) can arise between two saddle—node lines .', 'The bifurcation lines divide the ( s1 , s2 ) -plane into seven subdomains characterized by different recurrent solutions: D1: monostability ( one stable steady state ) , D2: oscillatory ( stable limit cycle and one unstable steady state ) , D3: typical bistability ( two stable steady states and one unstable steady state ) , D4: typical bistability ( two stable steady states and one unstable steady state ) , D5: atypical bistability ( the stable limit cycle coexists with one stable steady state and two unstable steady states ) , D6: monostability ( one stable steady state , two unstable steady states and one unstable limit cycle ) , D7 ( the tiny region between supercritical Hopf line and upper saddle—node line ) : monostability ( one stable steady state and two unstable steady states ) .', 'We chose ( s1 , s2 ) = ( 0 . 1 , 0 . 03 ) as nominal model values for the Wip1 and PTEN mRNA synthesis rates .', 'According to Fig 4 , ( s1 , s2 ) = ( 0 . 1 , 0 . 03 ) lies in domain D1 , in which the system with persistent DNA damage has only one solution , the stable steady state .', 'As we will see , this state is characterized by the level of p53KILLER high enough to trigger apoptosis .', 'In Fig 5 we show bifurcation diagrams of p53KILLER as a function of the PTEN synthesis rate , s2 .', 'We consider two values of s1 , s1 = 0 . 2 and s1 = 0 . 4 .', 'The character of the bifurcation diagram for s1 = 0 . 2 is qualitatively similar to that for nominal s1 = 0 . 1 , however it is visually clearer since the distance between Neimark—Sacker , Hopf , and saddle—node bifurcations is larger .', 'For s1 = 0 . 4 ( Fig 5B ) the bifurcation diagram is qualitatively different from that for s1 = 0 . 1 and s1 = 0 . 2 ( Fig 5A ) .', 'As shown in Fig 4 , with s2 increasing ( for fixed s1 = 0 . 2 ) the system proceeds sequentially through five subdomains in the ( s1 , s2 ) -plane: D2 , D5 , D6 , D7 , and D1 .', 'Fig 5A visualizes details of recurrent solutions along this line .', 'In the description below we focus on stable steady states and limit cycles because these solutions shape time-dependent responses .', 'For s2 < sSN1 , i . e . , in subdomain D2 , the system has a single stable limit cycle .', 'This solution is characterized by oscillations of the levels of p53KILLER and p53ARRESTER .', 'Amplitude of p53ARRESTER oscillations is high enough to suppress the cell cycle , but the level of p53KILLER remains below the apoptotic threshold .', 'At s2 = sSN1 the first saddle—node bifurcation arises , and for s2 ∈ ( sSN1; sN−S ) , i . e . , in D5 subdomain , the system exhibits atypical bistability in which the stable limit cycle coexists with the stable steady state .', 'This steady state is characterized by the p53KILLER level high enough to trigger apoptosis .', 'Therefore , in this parameter range the apoptotic state and cell cycle arrest state coexist .', 'At s2 = sN−S the subcritical Neimark—Sacker bifurcation arises , in which the stable limit cycle loses its stability merging with an unstable invariant torus , and for s2 > sN−S the only stable recurrent solution is the stable steady state associated with apoptosis .', 'Therefore , as parameter s2 increases ( which is associated with the increase of the PTEN level ) , the cell with damaged DNA at s2 = sN−S switches abruptly to apoptosis .', 'This biologically plausible behavior , allowing for unambiguous cell fate decisions , is associated with the specific bifurcation structure involving the subcritical Neimark—Sacker bifurcation .', 'Then the cycle vanishes at s2=sHsuper in the supercritical Hopf bifurcation ( see the Fig 5A insert ) .', 'Subsequently , two remaining unstable steady states ( one with 3 eigenvalues of positive real parts ) , and the other ( with 2 eigenvalues of positive real parts ) “annihilate” in s2 = sSN2 .', 'Let us notice that unstable steady states in the analyzed diagram have at most 3 eigenvalues with positive real parts and the unstable cycle has 2 Floquet multipliers of moduli larger than 1 .', 'This implies that despite the system has complex structure which involves numerous feedback loops , its dynamics is essentially 3-dimensional , meaning that locally there exists a 3-dimensional attracting manifold .', 'A qualitatively different is the bifurcation diagram depicting the recurrent solutions along line s1 = 0 . 4 with increasing s2 ( Fig 5B ) , i . e . , when the system proceeds sequentially through D2 , D5 , D4 , and D1 subdomains ( shown in Fig 4 ) .', 'Similarly to the previous case for s2 < sSN1 , the system has the stable limit cycle ( cell cycle arrest state ) .', 'Then , at s2 = sSN1 , the first saddle—node bifurcation arises , and for s2\xa0ϵ ( sSN1;sHsub ) the stable limit cycle coexists with the stable steady state ( apoptotic state ) .', 'However , in contrast to bifurcation diagram shown in Fig 5A , the stable limit cycle does not lose its stability , but it is replaced by a steady state at s2=sHsub through the subcritical Hopf bifurcation .', 'In this steady state , p53KILLER does not exceed the apoptotic threshold .', 'This steady state annihilates with an unstable steady state at s2 = sSN2 in the second saddle—node bifurcation .', 'As in the previous case , for s2 > sSN2 the system has a single recurrent solution , the stable steady state associated with apoptosis .', 'We think that the bifurcation structure in Fig 5A reflects the experimental observations better than that of Fig 5B , suggesting that upon DNA damage the level of p53 either oscillates at a relatively low amplitude , or builds up to the high level in which apoptosis is triggered .', 'Therefore , for further analysis we assume s1 = 0 . 1 , resulting in a bifurcation diagram as in Fig 5A .', 'The bifurcation diagrams corresponding to s1 = 0 . 1 ( or qualitatively equivalent s1 = 0 . 2 ) calculated with respect to Wip1 , PI3K , and the number of DSBs are shown in S3 Fig . Qualitatively , diagrams with respect to Wip1 and PI3K are mirror images of the bifurcation diagram with respect to PTEN ( Fig 5A ) , confirming the antagonistic action of Wip1 and PTEN .', 'The antagonism of PI3K and PTEN is straightforward because PI3K phosphorylates PIP2 to PIP3 , while PTEN dephosphorylated PIP3 to PIP2 .', 'The bifurcation diagram with respect to DNA damage ( assumed , for this diagram , to be persistent ) is similar to the bifurcation diagram with respect to PTEN , but the limit cycle oscillations start at a non-zero value of the bifurcation parameter .', 'In this section we analyze time-dependent dynamics in response to DNA damage .', 'First , we consider the case in which DNA repair is suppressed; then , we consider the nominal model in which DNA repair mechanism is active; finally , we analyze how cell fate depends on the expression levels of Wip1 and PTEN .', 'In order to analyze the cell-to-cell variability we performed stochastic simulations according to the Gillespie algorithm [90] implemented in BioNetGen [91] .', 'In Fig 9 we compare trajectories obtained in single-cell stochastic simulations with the deterministic trajectory as well as the population trajectory , i . e . , trajectory obtained by averaging over 1000 single-cell stochastic trajectories .', 'In the figure we show trajectories of p53ARRESTER and p53KILLER in response to three irradiation doses: small of 2 Gy , intermediate of 4 Gy ( which is just below the critical dose for the deterministic approximation ) and large of 8 Gy .', 'As one can observe , the stochastic trajectories follow closely the deterministic trajectory for the low and high doses , and in these two cases the population trajectory almost matches the deterministic one .', 'This is in contrast to the intermediate dose of 4 Gy , for which the deterministic trajectory after a few oscillations ( during which DNA is repaired ) returns to the initial state characterized by low levels of p53ARRESTER and p53KILLER , while the stochastic trajectories separate into two groups corresponding to the apoptotic cells ( of high p53 levels ) and surviving cells ( of low p53 levels ) .', 'As a result , for 4 Gy the deterministic trajectory is much different from the population-averaged trajectory .', 'Such discrepancy between deterministic and stochastic solutions has been demonstrated earlier for various , even very simple , nonlinear or multistable systems [92–97] .', 'It is especially pronounced in the cases of bistable ( or multistable ) systems when the deterministic trajectory converges to one of steady states , while the stochastic trajectories split into two ( or more ) groups .', 'In this case the population average ( observed in population experiments by Western blotting or PCR ) does not correspond to any single-cell trajectory .', 'The fraction of apoptotic cells increases with the irradiation dose ( see Fig 10 , where 5000 single-cell stochastic simulations were performed for each of 14 different doses from the 0–8 Gy range ) .', 'For the critical ( deterministic ) dose equal 4 . 05 Gy , the fraction of apoptotic cells was found to be close to 50% .', 'The fraction of apoptotic cells increases from about 10% to about 90% as the irradiation dose increases from 2 . 5 Gy to 6 Gy .', 'In order to determine how gene switching noise contributes to the heterogeneity in cell fate decisions we performed simulations in a model variant in which gene activity was assumed to be a deterministic function of the level of p53 ( in a gene-specific phosphoform ) .', 'As expected , after removing one source of stochasticity the individual cell responses has become less heterogeneous ( see also S4 Fig ) , which is manifested by the steeper sigmoid of the fraction of apoptotic cells ( violet vs . pink line ) .', 'Surprisingly , the difference between these two cases is not very large , which indicates that transcriptional and translational noises significantly contribute to the overall stochasticity .', 'As showed above for the intermediate doses , the cell population splits into apoptotic and surviving subpopulations .', 'In Fig 11 we analyze how the cell fate decision after 4-Gy irradiation is reached in time .', 'In Fig 11A we show trajectories of PTEN , which is a key protein mediating slow positive feedback loop and thus its level was found to be a good predictor of the ultimate cell fate decision .', 'As shown , the trajectories of surviving cells ( blue ) are almost fully separated from trajectories of apoptotic cells ( red ) after about 36 hours .', 'The separation between apoptotic and surviving trajectories progresses in time , as shown in Fig 11B where temporal evolution of the Kolmogorov—Smirnov statistic between 6 distributions of different variables characterizing apoptotic and surviving cells is calculated .', 'This figure confirms that the cell fate decision is reached after about 36 hours , as for this time Kolmogorov—Smirnov statistics ( for all variables ) reaches 1 , which indicates that distributions corresponding to apoptotic and surviving cells fully separate .', 'The number of DSBs that arise during the irradiation phase ( which , due to the stochasticity , can be different for cells irradiated with the same dose ) is the fastest predictor .', 'PTEN and Aktu ( levels of which can be different prior to the irradiation due to stochastic fluctuations in resting cells ) are also fast predictors and , moreover , the Kolmogorov—Smirnov statistics corresponding to their distributions grow monotonically .', 'This is in contrast to p53KILLER , p53ARRESTER , and Wip1 levels , which exhibit pronounced oscillations after irradiation , and for which the Kolmogorov—Smirnov statistics reach 1 non-monotonously .'], ['p53 is thoroughly studied as a pivotal signaling node that integrates and transforms diverse stress signals into downstream responses including cell cycle arrest and apoptosis to limit proliferation of cells with the damaged genetic material .', 'Based on the experimental knowledge about feedbacks and interactions within the p53 network , we proposed a modular model of p53 signaling pathway , in which the dynamics of the core p53 module controls downstream modules that govern cell cycle arrest and apoptosis .', 'The model has the purposeful bifurcation structure that delineates plausible temporal responses and allows for , we think , the most unambiguous cell death , cycle arrest , or survival decisions .', 'The stochastic , yet unambiguous , processing of analog irradiation signals to reach a digital cell fate outcome is assured by the bistability of cell cycle arrest and apoptotic modules , but primarily by a specific-type bistability of the core p53 module .', 'As demonstrated by the two-parameter bifurcation analysis of the core module , in a relatively broad range of parameters the limit cycle oscillations ( of p53ARRESTER and p53KILLER levels ) coexist with the stable steady state in which the p53KILLER level is very high , while the level of phosphorylated Akt is very low .', 'The limit cycle oscillations of p53ARRESTER induce cell cycle arrest , while the high p53KILLER level and low level of phosphorylated Akt induce apoptosis .', 'The key difference between dynamics of the cycle arrest module and the apoptotic module is that the former is based on reversible bistability and involves one input signal , p53ARRESTER , which suppresses cell cycle when above the threshold and releases the cell cycle arrest when below the lower threshold .', 'In turn , apoptosis induction requires two signals ( increase of p53KILLER and decrease of phosphorylated Akt ) simultaneously , and is an irreversible process .', 'The bifurcation structure delineates temporal responses to DNA damage .', 'Small DNA damage results in temporal cell cycle arrest , followed by DNA repair and return to the resting cell state , while high DNA damage results in the cell cycle arrest and subsequent apoptosis ( without returning to cell cycle ) .', 'The direct jump between limit cycle oscillations and the stable steady state is enabled by the presence of a subcritical Neimark—Sacker bifurcation in which the limit cycle loses its stability merging with an unstable invariant torus .', 'Interestingly , this type of bifurcation , rarely analyzed in the context of molecular regulatory networks , can exist in at least three-dimensional systems .', 'Although the core module is 18-dimensional , the stability analysis of the steady state showed that at most three eigenvalues have positive real parts which implies that locally the system could be reduced to a three-dimensional one .', 'It remains an open question whether such reduction is possible globally .', 'We propose that in the process of cell fate decision-making the p53 system can exhibit the most robust transitions when the only stable recurrent solutions are a stable limit cycle and a stable steady state ( out of this limit cycle ) , provided there exists a parameter region for which these recurrent solutions coexist .', 'Such structure allows cells to switch to the cell cycle arrest ( associated in the model with limit-cycle oscillations of the levels of p53 phosphoforms ) and then to direct transition to either apoptosis or recovery .', 'Interestingly , there exists an even simpler bifurcation structure , shown in S5 Fig , that allows for such transitions .', 'In this hypothetical case , as the bifurcation parameter s2 increases the limit cycle simply vanishes in the saddle—loop bifurcation .', 'The most pronounced dynamical difference between the modeled and the hypothetical scenario is that the period of oscillations diverges to infinity as the bifurcation parameter approaches the saddle—loop bifurcation point .', 'This would imply that in a heterogeneous cellular population some cells would exhibit much longer oscillation periods , which is not observed in experiments [73] .', 'As discussed by us earlier [76] , we thus expect that oscillations in the p53 system ( as well as in the NF-κB system ) do not arise or vanish through a saddle—loop bifurcation or a saddle—node on invariant cycle bifurcation ( also called SNIPER ) in which limit cycle arises from orbit homoclinic to the saddle .', 'Biochemical signal processing depends on the topology and time scales of feedback loops .', 'Regulatory networks with negative feedback ( s ) can produce oscillatory or pulsed responses to tonic stimuli , and allow for adaptation to the change in the level of stimulation .', 'The existence of positive feedbacks , in turn , can give rise to bistability [98] ( or multistability [99] ) that can be harnessed for cell fate decisions .', 'Combinations of positive and negative feedbacks allow for more complex behaviors .', 'At least three qualitatively different type of responses leading to distinct regulatory outcomes are possible .', '( 1 ) In the case when a positive feedback loop acts on a short time scale and is embedded in a longer negative feedback loop , the system can switch periodically between two predefined states ( e . g . , the autophagy/translation switch [100] ) .', '( 2 ) In the case when a positive feedback loops encompasses the negative one , the system can exhibit autonomous oscillations ( e . g . , oscillations in the TNF—NF-κB autocrine system [101] ) .', '( 3 ) In the case when a slow positive feedback loop can turn off a negative feedback , the system can exhibit biphasic responses in which after an oscillatory phase it either returns to the initial state ( more likely when the signal is short-lasting ) or it switches to another steady state ( more likely when the signal is persistent or lasts sufficiently long [102] ) .', 'The last case ( which is a simplification of the p53 regulatory core organization discussed in this study ) seems to be a good candidate for the unbiased and proportionate process of cell fate decisions .', 'The oscillatory phase provides a time interval to collect and integrate various signals , and estimate the severity of stress , before a ( potentially irreversible ) switch to another committed state is done .', 'Stochastic simulations of the model allowed to analyze how the death-or-life decision is achieved in time , and what are the best or earliest indicators of apoptosis .', 'By calculating time-dependent Kolmogorov—Smirnov statistic between distributions of protein levels corresponding to two populations of cells with apoptotic and surviving fates , we found that the clear separation between these two populations is reached at about 30 hours after irradiation .', 'PTEN , the p53-induced phosphatase that mediates slow positive feedback loop , was found to be the most robust and fast apoptosis predictor .', 'Interestingly , the Kolmogorov—Smirnov statistics for Wip1 , p53ARRESTER and p53KILLER are non-monotonous in time , what possibly follows from the pronounced oscillations of these components and indicates that at some time-points of the response the elevated or decreased levels of pro- or anti-apoptotic proteins may not correlate with the ultimate cell fate decision .', 'Most of experimental data were gathered for cancer ( or immortalized ) cell lines , which became cancerous as a result of malfunctions in the p53 network .', 'Much less is known about regulation of normal , i . e . , non-cancer , non-immortalized cells .', 'Nevertheless , we expect that the plausible responses of normal cells should consist of the cell cycle suppression phase ( which can be associated with p53 oscillations ) , followed by either apoptosis or recovery , depending on the extent of the DNA damage .', 'Accordingly , in contrast to the majority of previous studies focused on modeling of cancer cell lines , we aimed at constructing the model that reflects behavior of normal cells after γ irradiation .', 'Within this model , the responses of different cancer cell lines can be analyzed by adjusting the appropriate parameters such as mRNA synthesis rates or steady-state levels of proteins that are assumed to be not regulated in the model .', 'In this way one can simulate cancer cell lines that are known to have elevated expression of Wip1 such as MCF-7 , BT474 ( breast cancers ) , OVCAR4 ( ovarian cancer ) , MOLT4 ( lymphoma ) , U2OS ( osteosarcoma ) [50–52 , 54 , 103] , lines of decreased PTEN expression such as MCF-7 ( breast cancer ) , H1299 , H322 , Calu1 ( lung cancers ) [58–60] , or with amplification of PI3K gene , such as MKN1 , SNU1 ( gastric cancers ) , OVCAR4 , A2780 ( ovarian cancers ) [63–70] .', 'The life-or-death decision is reached in the interplay of two antagonistic phosphatases Wip1 and PTEN .', 'Our analysis shows that sensitivity to apoptosis increases with PTEN expression and decreases with Wip1 or PI3K expression , which confirms the pro-survival functions of Wip1 and PI3K , and pro-apoptotic functions of PTEN .', 'The other key regulatory pro-apoptotic protein present in network is kinase HIPK2 that converts p53ARRESTER to the p53KILLER phosphoform [104] .', 'The model shows that undamped oscillations of p53ARRESTER and p53KILLER levels observed in MCF-7 cells can result from the low expression of PTEN .', 'Analogously , our analysis indicates that cell lines with Wip1 overexpression ( resulting from , e . g . , gene amplification ) can exhibit persistent oscillations and resistance to apoptosis .', 'Strong dependence of the critical irradiation dose on PTEN , Wip1 , and PI3K expression may suggest that pretreatment of cancer cells ( with intact p53 ) with drugs that act temporally to elevate the expression of PTEN or to repress the expression of Wip1 or PI3K can enhance the effectiveness of radiotherapy .', 'Very recently , Puszyński et al . [105] have modified his earlier model [80]
['Introduction', 'Results', 'Discussion', 'Materials and Methods']
['The p53 transcription factor is a regulator of key cellular processes including DNA repair , cell cycle arrest , and apoptosis .', 'In this theoretical study , we investigate how the complex circuitry of the p53 network allows for stochastic yet unambiguous cell fate decision-making .', 'The proposed Markov chain model consists of the regulatory core and two subordinated bistable modules responsible for cell cycle arrest and apoptosis .', 'The regulatory core is controlled by two negative feedback loops ( regulated by Mdm2 and Wip1 ) responsible for oscillations , and two antagonistic positive feedback loops ( regulated by phosphatases Wip1 and PTEN ) responsible for bistability .', 'By means of bifurcation analysis of the deterministic approximation we capture the recurrent solutions ( i . e . , steady states and limit cycles ) that delineate temporal responses of the stochastic system .', 'Direct switching from the limit-cycle oscillations to the “apoptotic” steady state is enabled by the existence of a subcritical Neimark—Sacker bifurcation in which the limit cycle loses its stability by merging with an unstable invariant torus .', 'Our analysis provides an explanation why cancer cell lines known to have vastly diverse expression levels of Wip1 and PTEN exhibit a broad spectrum of responses to DNA damage: from a fast transition to a high level of p53 killer ( a p53 phosphoform which promotes commitment to apoptosis ) in cells characterized by high PTEN and low Wip1 levels to long-lasting p53 level oscillations in cells having PTEN promoter methylated ( as in , e . g . , MCF-7 cell line ) .']
['Cancers are diseases of signaling networks .', 'Transcription factor p53 is a pivotal node of a network that integrates a variety of stress signals and governs critical processes of DNA repair , cell cycle arrest , and apoptosis .', 'Somewhat paradoxically , despite the fact that carcinogenesis is prevalently caused by p53 network malfunction , most of our knowledge about p53 signaling is based on cancer or immortalized cell lines .', 'In this paper , we construct a mathematical model of intact p53 network to understand dynamics of non-cancerous cells and then dynamics of cancerous cells by introducing perturbations to the regulatory system .', 'Cell fate decisions are enabled by the presence of interlinked feedback loops which give rise to a rich repertoire of behaviors .', 'We explain and analyze by means of numerical simulations how the dynamical structure of the regulatory system allows for generating unambiguous single-cell fate decisions , also in the case when the cell population splits into an apoptotic and a surviving subpopulation .', 'Perturbation analysis provides an explanation why cancer cell lines known to have vastly diverse expression levels of p53 regulators can exhibit a broad spectrum of responses to DNA damage .']
['phosphorylation', 'cell death', 'cell cycle inhibitors', 'gene regulation', 'cell cycle and cell division', 'regulatory proteins', 'cell processes', 'dna-binding proteins', 'dna damage', 'transcription factors', 'dna', 'proteins', 'gene expression', 'biochemistry', 'biochemical simulations', 'cell biology', 'nucleic acids', 'post-translational modification', 'apoptosis', 'genetics', 'biology and life sciences', 'dna repair', 'computational biology']