Mathematical and computational analysis of intracelluar dynamics 3

Chapter Kinetic Models of the p53-AKT Network Several reported experimental observations suggest the possible existence of a cell survival-death switch involving p53 and AKT (for examples, see Gottlieb et al., 2002; Singh et al., 2002) For instance, the level of AKT in a cell population deprived of growth factors decreases rapidly and drastically upon irradiation, which is associated with p53 upregulation and p53-dependent induction of caspases However, in the presence of adequate growth factors, the upregulation of AKT can overcome the proapoptotic effects of p53 and ensures cell survival The goal of this chapter is to analyze p53-AKT kinetic models in order to gain insight on the control system of a cell’s decision to survive or die It will be shown that the models predict a bistable p53-AKT cell survival-death switch Bistability means the co-existence of two stable steady states with one unstable state in between (Tyson et al., 2003) Furthermore, model predictions such as network perturbations due to DNA damage and AKT inhibition are discussed The predictions of the models analyzed below are relevant at the single-cell level, and therefore experiments such those carried out by Nair et al (2004) - in which single-cell decisions between apoptosis and survival were shown - would be required to validate the models’ predictions Interestingly, Nair et al.’s results suggest a bistable behavior of the system in which individual cells commit to either ERK-mediated pro-survival or p53mediated pro-apoptotic cellular states within the first hour after oxidative stress 33 3.1 Overview of kinetic modeling of the p53-AKT network Kinetic models that describe the interaction among the part list of a studied system are used to study the governing systems dynamics The formulation of a kinetic model involves three major steps – encoding available mechanistic information about the p53-AKT network into abstract kinetic models (Section 3.2), deriving the kinetic equations (Section 3.2.1) and specifying the associated kinetic parameters (Sections 3.2.2) In a kinetic model, dynamics of interactions among the part lists in the modeled pathway are described by mathematical equations Generally, these mathematical equations are nonlinear and coupled (i.e., numerical values of the variables of an equation depends on other equations); the more complex the interactions, the more coupled the equations Therefore, except for the simplest mathematical model, they are analytically intractable and would need to be solved numerically on a computer, i.e., by computer simulations The solutions to these equations yield the time-courses and steady state behaviors of the system, which are then compared with existing experimental observations In the event that the simulation results differ considerably from experimental results, the model is modified until it can reproduce most if not all of the key experimental observations Subsequently, systems behaviors of the model to local perturbations of either kinetic parameters or quantity of part lists are analyzed Details of such analyses are described in Sections 3.3 and 3.4 Finally, novel systemic behaviors of the system are inferred and predicted from the simulation results (Sections 3.5 and 3.6) 34 3.2 Formulation of kinetic models Figure 3-1 Kinetic model of the p53-AKT network, Model M1 Model M1 encompasses the two feedback loops (p53-MDM2 and p53-AKT) and three phosphorylation-dephosphorylation cycles namely, PIP2-PIP3, AKT-AKTa and MDM2MDM2a; AKTa and MDM2a denote biochemically active AKT and MDM2 proteins upon phosphorylation The vr’s are the rate equations of each reaction step with units of concentration per unit time Broken edges denote enzymatic reactions whereas full edges denote mass action reactions All part lists are at the protein level In the model, p53 is transcriptionally-active where it transcribes target genes, MDM2 and PTEN The literature was reviewed to integrate experimental information available on p53 and AKT networks (see Chapter 2) to derive a kinetic model of the p53-AKT network (Figure 3-1), and shall be referred to as Model M1 hereafter It has been shown that regulatory cycles, which include feedback loops and phosphorylation- dephosphorylation cycles, affect the stability of a network (Aguda, 1999; Aguda and Algar, 2003) Hence, this is the motivation why the kinetic models formulated in this chapter are only those that contain cycles that are destabilizing (i.e they could 35 generate unstable steady states); it is these destabilizing cycles that are taken as prime candidates for switching dynamics in the network Specifically, besides the two feedback loops (p53-MDM2 and p53-AKT), Model M1 encompasses three phosphorylation-dephosphorylation cycles namely, PIP2-PIP3, AKT-AKTa and MDM2-MDM2a (AKTa and MDM2a denote active AKT and MDM2 proteins respectively) 3.2.1 Deriving the kinetic equations In general, a kinetic equation is assigned to each part list of Model M1 (Figure 3-1), which has a mathematical form of an ordinary differential equation (ODE): d [ Pi ] = Rate of formation or generation – Rate of removal or degradation dt (3-1) The left-hand side (LHS) denotes rate of change of the intracellular concentration (denoted as []) of a part list Pi, which equals to its rate of formation minus its rate of removal as given by the respective rate terms, vr’s As an example, the kinetic equation of p53 is given by v0 – v2 – v7 (refer to Figure 3-1 for the assignment of rate term), where v0 is rate of transcriptionally-active p53 synthesis and activation, v2 is rate of p53 degradation by MDM2a and v7 is p53 self-degradation rate Although p53 is also involved in the enzymatic reactions associated with the rate terms v3 and v5, these rate terms are excluded from its kinetic equation because p53 is a catalyst in these reactions; the amount of catalyst is unaffected in a reaction The entire set of kinetic equations for Model M1 is listed in Eqn (3-2) 36 d [ p53] = v − v − v7 dt d [ AKTa ] = v1 − v m1 dt d [ PIP3] = v4 − v m dt d [ PTEN ] = v + v8 − v dt d [ MDM 2a ] = v − v m − v12 dt d [ MDM 2] = v5 + v10 + v m − v11 − v6 dt (3-2) where v0 : rate of transcriptionally-active p53 synthesis and activation; v1 : rate of AKTa formation by PIP3 phosphorylation of AKT; vm1 : rate of AKTa removal by basal dephosphorylation of AKTa; v2 : rate of transcriptionally-active p53 degradation by MDM2a; v3 : rate of PTEN formation by p53 transcription; v4 : rate of PIP3 formation by PI3K phosphorylation of PIP2; vm4 : rate of PIP3 removal by PTEN dephosphorylation of PIP3; v5 : rate of MDM2 formation by p53 transcription; v6 : rate of MDM2a formation by AKTa phosphorylation of MDM2; vm6 : rate of MDM2a removal by basal dephosphorylation of MDM2a; v7 : rate of active p53 inactivation (degradation or dephosphorylation); v8 : rate of PTEN formation by basal induction; v9 : rate of PTEN degradation; v10 : rate of MDM2 formation by basal induction; v11 : rate of MDM2 degradation; v12 : rate of MDM2a degradation 37 To minimize the number of kinetic equations in Model M1, kinetic equations are not assigned to AKT and PIP2 Instead, dynamics of AKT and PIP2 are obtained respectively by [AKT] = [AKTT] – [AKTa] and [PIP2] = [PIPT] – [PIP3] [AKTT] and [PIPT] denotes the respective intracellular concentrations of total AKT (AKT and AKTa) and PIP (PIP2 and PIP3), which can be assumed to be approximately constant within the time scale of the phosphorylation and dephosphorylation processes involved in the respective activations of AKT and PIP2 This is because these processes occur relatively faster than the transcriptional, translational and degradation processes in the model Furthermore, this assumption is supported by experimental observations in which [AKTT] remains relatively constant after irradiation or treatment with chemotherapeutic drugs even as [AKTa] decreases drastically (Gottlieb et al., 2002; Martelli et al., 2003) The next step is to derive mathematical expressions for each rate term As information about the biological mechanisms involved in the reaction steps, which is a requisite to derive the rate expressions, are often lacking, general rate expressions are used Hence, such a kinetic model is referred to as ‘abstract’ in the sense that the essential qualitative dynamics are captured by simple mathematical functions Biologically similar reaction steps are assumed to have similar rate expressions, as described below: Enzymatic phosphorylation (v1, v4 and v6), dephosphorylation (vm1, vm4 and vm6) and degradation (v2) reactions are assumed to have Michaelis-Menten type expressions given by v r = k r [E ][S ] , where kr and jr are kinetic j r + [S ] parameters (or constants) that denote the maximal rate of reaction and the Michaelis constant, respectively jr quantifies the binding affinity between 38 substrate (S) and enzyme (E) in which a large Michaelis constant indicates low binding affinity, and vice versa In the case whereby E is neither known nor modeled explicitly, kr absorbs the variable [E], i.e., v r = k r [S ] for vm1, v4 j r + [S ] and vm6 p53-dependent production of proteins (v3 and v5) is assumed to have Hill-type k r [ p53] n expressions given by v r = j rn + [ p53] n , where kr and jr are kinetic parameters that denote the maximal rate of production and the dissociation constant respectively, and n is the Hill coefficient A large dissociation constant indicates low p53 binding affinity to the target gene DNA promoter site, and vice versa The Hill coefficient, on the other hand, represents the degree of cooperativity of a reaction; a reaction is described as being noncooperative when n = while being positively cooperative when n > In a positively cooperative case, the binding of p53 to its promoter site increases the affinity of the site for further p53 binding Self-degradation reactions (v2, v9, v11 and v12) are assumed to have first-order reaction kinetics type expression give by v r = k r [Pi ] , where kr is the rate constant of the reaction and Pi represents the part list undergoing selfdegradation Basal production (v0, v8 and v10) of a part list is assumed to have a constant rate, i.e., vr = kr 39 Eqn (3-3) lists down the rate expressions for all the rate terms of Model M1 v0 = k o v1 = k1 [PIP3][AKT ] k1 [PIP3]{[ AKTT ] − [AKTa ]} = j1 + [ AKT ] j1 + [ AKTT ] − [ AKTa ] v m1 = v2 = k m1 [ AKTa ] j m1 + [ AKTa ] k [MDM 2a ][ p53] j + [ p53] n1 k [ p53] v3 = j3n1 + [ p53] n1 v4 = k [PIP 2] k {[PIPT ] − [PIP3]} = j + [PIP 2] j + [PIPT ] − [PIP3] vm = k m [PTEN ][PIP 3] j m + [PIP3] (3-3) k [ p53] n2 v5 = v6 = j5n + [ p53] n2 k [ AKTa ][MDM 2] j + [MDM 2] vm6 = k m [MDM 2a ] j m + [MDM 2a ] v = k [ p53] v8 = k v = k [PTEN ] 40 v10 = k10 (3-3) v11 = k11 [MDM ] v12 = k12 [MDM a ] 3.2.2 Specifying the kinetic parameters The final step in formulating the kinetic model is to specify numerical values for the kinetic parameters associated with the rate expressions listed in Eqn (3-3) Unfortunately, direct experimental measurements of kinetic parameters are difficult and they are therefore rarely determined Subsequently, they are determined from a range of values that are obtained from an extensive literature survey of similar reaction types whose kinetic parameters have been reported Table 3-1 tabulates the parameter values and the plausible biological ranges derived from the literature survey; a detail description of the determination of the parameter ranges is given in Appendix A-2 Table 3-1 The 28 parameters used in the model simulations for Model M1 KP (column 2) denotes kinetic parameter See Appendix A-2 for a detail description of the derivation of the values tabulated in columns and Item KP k0 k1 j1 Description Units Chosen Value Range Refs Production of active p53 PIP3-mediated phosphorylation of AKT Michaelis constant of PIP3-mediated phosphorylation of µM/ 0.1 0.002 to 0.2 Ma et al., 2005 /min 20 20 Giri et al., 2004 µM 0.1 0.1 Giri et al., 2004 41 Item KP km1 jm1 k2 j2 k3 j3 10 k4 11 j4 12 km4 13 jm4 14 k5 15 j5 Description AKT Dephosphorylation of AKTa Michaelis constant of dephosphorylation of AKTa MDM2-dependent degradation of p53 Michaelis constant of MDM2-dependent degradation of p53 p53-dependent production of PTEN Dissociation constant of p53-dependent production of PTEN Phosphorylation of PIP2 Michaelis constant of phosphorylation of PIP2 PTEN dephosphorylation of PIP3 Michaelis constant of PTEN dephosphorylation of PIP3 p53-dependent production of MDM2 Dissociation constant of p53-dependent production of MDM2 Units Chosen Value Range Refs µM/ 0.2 0.0000297 to 2.92 Qiu et al., 2004; Kholodenko, 2000 µM 0.1 0.1 Giri et al., 2004 /min 0.055 0.0184 to 0.092 Ma et al., 2005 µM 0.1 0.03 to 0.3 Ma et al., 2005 µM/ 0.006 0.006 Stambolic et al., 2001 µM >1 Stambolic et al., 2001 µM/ 0.15 0.15 Kholodenko, 2000 µM 0.1 0.1 Giri et al., 2004 /min 73 42.1, 73 ± 4.4 Giri et al., 2004; McConnachie et al., 2003 µM 0.5 0.1 to Giri et al., 2004; Georgescu et al., 1999; Vazquez et al., 2000 µM/ 0.024 0.024 Ma et al., 2005 µM ~1 Ma et al., 2005 k6 AKTa phosphorylation of MDM2 /min 10 0.42 to 64.8 17 j6 Michaelis constant of AKTa phosphorylation of MDM2 µM 0.3 0.00357 to 146 18 km6 Dephosphorylation of MDM2a µM/ 0.2 0.0000297 to 2.92 19 jm6 Michaelis constant of dephosphorylation of MDM2a µM 0.1 0.00238 to 2.23 20 k7 /min 0.05 0.02 to 0.2, 0.05 21 k8 µM/ 0.001 Hoffmann et al., 2002; Qiu et al., 2004; Schoeber et al 2002; Markevich et al., 2005; Kholodenko, 2000; Giri et al., 2004 Hoffmann et al., 2002; Qiu et al., 2004; Schoeber et al 2002; Markevich et al., 2005; Kholodenko, 2000; Giri et al., 2004 Qiu et al., 2004; Kholodenko, 2000 Qiu et al., 2004; Schoeber et al 2002; Markevich et al., 2005; Kholodenko, 2000; Giri et al., 2004 Ma et al., 2005; Zhou et al., 2001; Bar-Or et al., 2000 Unknown 16 Inactivation of active p53 (degradation or dephosphorylation) Basal induction of PTEN 42 3.4.1 Model M1 The range at which kinetic parameters of Model M1 are varied in the sensitivity analysis is tabulated in Table 3-2 Values of kinetic parameters that have not been measured directly in experiments are varied according to their uncertainty, i.e., parameters whose values are set arbitrarily are varied over a wider range than those estimated or inferred indirectly from experimental data or modeling papers (see Appendix A-2) In the sensitivity analysis, values of these kinetic parameters are varied simultaneously This approach is more comprehensive and accurate than varying one parameter at a time because, firstly, it is able to cover significantly more combinations of parameter values and, secondly, cellular noise/fluctuations affect all biochemical and biophysical kinetics concurrently On the other hand, kinetic parameters that were measured directly in reported experiments are not varied Table 3-2 Model M1: variation ranges of the kinetic parameters for sensitivity analysis For kinetic parameters (KP) marked as “Arbitrary”, they are varied –50% and at least +50% from their respective values used in Model M1 (Table 3-1) For kinetic parameters marked as “Estimated” (last column), they are varied ±20% from the values used in Model M1 For kinetic parameters denoted as “Direct experimental data”, they are fixed throughout the sensitivity analysis Altogether, 22 kinetic parameters are varied Item 10 11 12 13 KP k0 k1 j1 km1 jm1 k2 j2 k3 j3 k4 j4 km4 jm4 Base value 0.1 20 0.1 0.2 0.1 0.055 0.1 0.006 0.15 0.1 73 0.5 Variation range 0.08 - 0.12 16 - 24 0.08 - 0.12 0.16 - 0.24 0.08 - 0.12 0.044 - 0.066 0.08 - 0.12 0.25*k5 1.6 - 2.4 0.12 - 0.18 0.08 - 0.12 0.4 - 0.6 Remarks Estimated Estimated Estimated Estimated Estimated Estimated Estimated Direct experimental data Estimated Estimated Estimated Direct experimental data Estimated 53 Item 14 15 16 17 18 19 20 21 22 23 24 25 26 KP k5 j5 k6 j6 km6 jm6 k7 k8 k9 k10 k11 k12 [PIPT] Base value 0.024 10 0.3 0.2 0.1 0.05 0.001 0.0054 0.018 0.015 0.015 Variation range 0.0192 - 0.0288 0.8 - 1.2 – 12 0.24 - 0.36 0.16 - 0.24 0.08 - 0.12 0.0005 - 0.0015 0.0144 - 0.0216 0.012 - 0.018 0.012 - 0.018 0.5, Remarks Estimated Estimated Estimated Estimated Estimated Estimated Direct experimental data Arbitrary (varied +/- 50%) Direct experimental data Estimated Estimated Estimated Arbitrary (varied -50% & +100%) Computer simulations of all the possible permutations of the 22 kinetic parameters to be varied are not only unrealistic but also computationally intractable Therefore, Latin hypercube sampling (as described briefly in Appendix A-8) is used to generate a sample of N unique sets of kinetic parameters values With the exception of [PIPT], a random sample of 10,001 (i.e., N = 10,001) Latin sets is sampled from the 21-dimensional Latin hypercube A total of four independent random samples are generated, which total 40,004 Latin sets For each Latin set, the conservation of bistability is determined The above steps are repeated for [PIPT] = 0.5, and µM The samplings from the Latin hypercube and the determination of bistability are implemented in MATLAB The number of Latin sets of kinetic parameters that exhibit bistability is shown in Figure 3-9 54 Number of Latin sets of kinetic parameters that exhibit bistability Figure 3-9 Model M1: number of Latin sets of kinetic parameters that exhibit bistability The number of Latin sets (vertical axis) that exhibit a bistable switch between p53 and AKTa in the four independent random samples, each consisting of 10,001 Latin sets of kinetic parameters, is plotted [PIPT] is fixed for each sample run For the three different values (0.5, and µM) of [PIPT], the simulations are repeated using the same Latin sets For each [PIPT], the average percentage of the number of Latin sets that exhibits bistability among the four samples is shown above the bars Bistability is determined using identical methods as described in Section 3.3.1 and Appendix A-3 The variance of the total Latin sets of kinetic parameters that exhibit bistability among the four independent random samples is very small relative to the mean, indicating that a sample size of N = 10,001 sets is sufficiently large Bistability is conserved about 86.4% of the time based on [PIPT] = (Figure 3-9) Conservation of bistability is reduced by 5.9% when [PIPT] is decreased by 50% whereas it is increased by 2.5% when [PIPT] is increased by 100% The quantity of PIPT correlates positively albeit weakly with conservation of bistability in Model M1 Therefore, the bistability phenomenon is not sensitive to variations in kinetic parameters whose values are not measured directly in experiments 55 3.4.2 Model M2 Because kinetic parameter values of Model M1 that give rise to bistability are handed down to identical steps found in Model M2, only parameters in the latter model that correspond to a group of steps in Model M1 are varied; since conservation of bistability is insensitive to variation of parameter values in Model M1 The only kinetic parameter whose value cannot be inferred from Model M1 is k*1 (refer to Appendix A-6 for more details), which is associated with the reaction step that summarizes p53 inhibition of AKT via PTEN-dependent dephosphorylation of PIP3 Thus, parameter k*1 is swept to determine the range where bistability is conserved (if any) Simulation results show that bistability is conserved when k*1 > 3.2 min-1 This minimum value is significantly lower than the corresponding experimental values for PTEN-dependent dephosphorylation of PIP3 (73 min-1 [McConnachie et al., 2003, from experimental measurement] and 42.1 min-1 [Giri et al., 2004, from parameter fitting]) Hence, it is very likely that bistability is also conserved in Model M2 3.4.3 Model M3 Similarly, only kinetic parameter of Model M3 whose value cannot be inferred from its parent model (Model M2) is varied The only such kinetic parameter is k*2 (refer to Appendix A-7 for more details), which is associated with step that summarizes MDM2a-mediated degradation of p53 Thus, parameter k*2 is swept to determine the range where bistability is conserved (if any) Simulation results show that there is no bistability for any range of k*2 In fact, p53 is always at the “on” state while AKTa is 56 at the “off” state, which indicates that the strength of the antagonism of p53 on AKT is too strong This is not surprising given that Model M3 does not have the p53MDM2 negative feedback loop to attenuate the strength of p53 To counter the strength of p53 antagonism on AKT, k1 (phosphorylation rate constant for AKT) is varied simultaneous with k*2 To show the extent of bistable regions in the k1-k*2 parameter plane, phase diagrams are plotted in Figure 3-10 in which different regions in these diagrams represent different numbers of steady states (3 states imply bistability; state implies monostability) As shown in Figure 3-10B, there exists a biologically reasonable bistable region in k1-k*2 plane Since k*1 is an arbitrary parameter in Model M2, the effect of halving (3.53 min-1) and doubling (14.1 min-1) the k*1 on the bistable region is investigated Figure 3-10A shows that reducing k*1 will diminish the bistable region whereas increasing k*1 will enlarge it (Figure 3-10C) 57 Figure 3-10 Model M3: phase diagrams mapping parameter plane to bistable regions The plots delineate, on the k1-k*2 parameter plane, regions of bistability (3 steady states, circled No 3) from regions of monostability (1 steady state, circled No 1), for three values of k*1 For each plot, the curves represent the boundaries between bistable and monostable regions The parameters used are k1 (phosphorylation rate of AKT), k*2 (MDM2-dependent degradation rate of p53) and k*1 (dephosphorylation rate of AKTa induced by p53) 3.5 Biological significance of bistability The most significant among the predictions of the models is the existence of the robust phenomenon of bistability in the p53-AKT network As discussed in the following, the existence of a bistable range for [AKTT] confers three biological significance Furthermore, the advantages of a bistable over a monostable cellular switch are discussed 3.5.1 Setting definite threshold levels The presence of a bistable range defines threshold points for [AKTT] where irrevocable decisions are made As depicted in Figure 3-11, the threshold points correspond to the left (Lk) and right (Rk) knees of the steady-state bifurcation 58 diagrams When [AKTT] < Lk, the system can only be at the high-p53 and low-AKTa pro-apoptotic cellular state and thus, Lk sets the pro-apoptotic threshold for [AKTT] Likewise, Rk sets the pro-survival threshold for [AKTT] as the system can only be at the pro-survival cellular state when [AKTT] > Rk For this reason, switching between the two cellular states can be achieved by just varying [AKTT] from one threshold point to the other, without the need to regulate other part lists In fact, as [AKTT] is the control parameter here, one could either vary AKT or AKTa, or both L [p53ss] R Lk [AKTT ] Rk R [AKTass] L Lk [AKTT ] Rk Figure 3-11 Threshold points for [AKTT] set by the bistable region Schematic steady-state bifurcation diagrams of p53 (top) and AKTa (bottom) where the left (Lk) and right (Rk) knees of the bistable region define threshold points for [AKTT] (abscissa) The steady states corresponding to pro-apoptotic (high-p53 and low-AKTa) cellular state are demarcated in red while steady states corresponding to pro-survival (low-p53 and highAKTa) cellular states are demarcated in blue The arrows indicate paths of the system when [AKTT] is varied (see main text for details) 59 3.5.2 Resistance to random switching between cellular states With clearly defined pro-apoptotic and pro-survival threshold points, random switching between the two cellular states due to random fluctuations of intracellular concentrations is curtailed For instance, suppose the system is initially at point L of the steady-state bifurcation curves whose [AKTT] is less than Lk (Figure 3-11), i.e., at pro-apoptotic state As [AKTT] is gradually increased beyond the pro-survival threshold, Rk, the system switches to pro-survival state, as indicated by the red arrows In the event that random fluctuation causes [AKTT] to fall below Rk, the system would still be however at pro-survival state so long as [AKTT] does not fall below the proapoptotic threshold, Lk Therefore, the bistable region buffers against random fluctuation of [AKTT] and thereby prevents random switching between pro-survival and pro-apoptotic states induced by cellular noise As such, a wider bistable region confers a larger buffering capacity, and vice versa This buffering effect also applies when the system switches from pro-survival state at point R to pro-apoptotic state as [AKTT] is lowered below Lk, as indicated by the blue arrows in Figure 3-11 3.5.3 Switching by finite perturbations In the bistable region, finite perturbations of the concentration of the part lists could switch the system from one cellular state to the other, while keeping the control parameter [AKTT] fixed As an illustration, AKTa is perturbed in Model M3 by introducing InhibitorX, which binds and inhibits AKTa; examples of commercially 60 available InhibitorX are described in Hu et al (2000), Kozikowski et al (2003) and Yang et al (2004) The modified kinetic model of Model M3 is given in Appendix A-9 The time-courses of [p53] in response to different amounts of InhibitorX are simulated; in each simulation, the initial state of the system is set to be at pro-survival steady state in the bistable region As depicted in Figure 3-12, as a consequence of the bistability, there is a threshold quantity of InhibitorX that switches the system to pro-apoptotic steady state 1.6 1.4 1.2 [InhibitorX] = [InhibitorX] = 1.17 [InhibitorX] = 1.18 [InhibitorX] = 1.19 [InhibitorX] = 25 [P53] [p53] 0.8 0.6 0.4 0.2 0 50 100 150 200 Time (min) 250 300 350 400 Figure 3-12 Model M3: Time-courses of [p53] in response to different levels of perturbations Perturbation of Model M3 is introduced by InhibitorX, which binds and inhibits AKTa Various levels of InhibitorX are used (see inset) to illustrate a discontinuous response of the system from a low-p53 pro-survival to a high-p53 pro-survival steady state, as a consequence of the bistability Initial state of the system in each simulation is set as: [p53] = 0.248, [AKTa] = 0.0973 and [AKTT] = 0.5; this initial state corresponds to the low-p53 and highAKTa steady state of the system in the bistable region All concentrations are in µM This could explain observations that treatment of human leukemia cells (HL60) with inhibitors of AKT restored their sensitivity towards apoptosis-inducing chemotherapeutic drugs only when the level of AKTa is decreased more than 3.5 61 times (Martelli et al., 2003), thus indicating a possible apoptotic threshold level for AKTa Furthermore, they reported that the effectiveness of AKTa inhibition depends on [AKTa] rather than [AKTT]; this suggests a switching mechanism involving finite perturbation of AKTa rather than via the control parameter [AKTT] Similarly, since PI3K inhibitors indirectly inhibit AKT activation, various cell lines such as acute myeloid leukaemia cells (Grandage et al., 2005), HTLV-1-transformed cells (Jeong et al., 2005) and Ewing’s Sarcoma family of tumors (Toretsky et al., 1999) have been shown to restore their sensitivity towards apoptosis-inducing chemotherapeutic drugs after incubation with specific threshold amounts of such inhibitors Conversely, sufficient upregulation of AKTa could perturb the system from a pro-apoptotic to a pro-survival state Indeed, AKTa upregulation in Ewing’s sarcoma family of tumors cell line leads to a gain of resistance to doxorubicin, an apoptosisinducing chemotherapeutic drug (Toretsky et al., 1999) Other ways to perturb the system includes for example, modifying p53 degradation, varying p53-dependent inhibition of AKTa and etc Or all of the aforementioned ways can be used in various combinations The important idea is that within the bistable region, switching between the two stable cellular states can be implemented by perturbing various components of the network model 3.5.4 Bistable switch versus monostable switch Alternatively, a monostable system can also function as a cellular switch Monostable switch can be either graded (Figure 3-13A) or discrete (Figure 3-13B) In the latter 62 case, the system is predisposed to random switching between pro-survival and proapoptotic states upon random fluctuation of [AKTT] around the sole threshold point, LRk (Figure 3-13B) On the other hand, while a graded monostable switch is less susceptible to random switching between cellular states, an apparent distinction between the two cellular states cannot be discerned for [AKTT] between Lk and Rk (Figure 3-13A); in a bistable switch, this is not a problem as the system undergoes a discontinuous change in its steady state concentrations when [AKTT] is increased from Lk to Rk, or vice versa (Figure 3-11) Remarkably, these shortcomings could be eradicated in the presence of a bistable region, as manifested in a bistable switch Moreover, bistability also offers an additional means to regulate the cellular switch In particular, finite perturbations can be applied to a bistable system (Section 3.5.3) to induce switching from one cellular state to the other, while keeping the control parameter [AKTT] fixed In contrast, the control parameter [AKTT] is the only mean to regulate a monostable switch A [TP53 ] [p53ssss] B [TP53 ] [p53ssss] Lk [AKTT] Rk LRk [AKTT] Figure 3-13 Schematic monostable p53 steady-state bifurcation diagrams Steady states corresponding to pro-apoptotic state are demarcated in red whereas steady states corresponding to pro-survival states are demarcated in blue (A) A graded monostable switch (B) A discrete monostable switch 63 3.6 Effect of DNA damage The effects of DNA damage have been investigated using Models M1, M2 and M3 and found their responses to be qualitatively similar Hence, the following discussion only refers to results using Model M3 3.6.1 Simulation of DNA damage To simulate the effect of DNA damage in Model M3, the kinetic equation of p53 becomes v7 v2 d [ p53] , where [DNADam] denotes the = v0 − − dt + [DNADam ] + [DNADam ] extent of DNA damage (arbitrary units) The functional form of the second and third terms on the right-hand side of this equation captures the essence that DNA damage stabilizes active p53 (v7) and attenuates MDM2-mediated degradation of p53 (v2) p53 steady-state bifurcation diagrams as functions of [AKTT] for an increasing levels of DNA damage is shown as curves to in Figure 3-14 On the other hand, increasing the rate of p53 synthesis and activation (v0) with [DNADam] can also simulate DNA damage stabilization of p53; simulations results are nevertheless qualitatively equivalent to those of Figure 3-14 As the extent of DNA damage increases, the middle curve of unstable steady states shifts down and lengthens in range (Figure 3-14) This downshift may be interpreted as the tendency to favor apoptosis over survival because of a lowered threshold for switching to the upper p53 stable states The significant increase in the 64 range of the bistable region has important implication on the plausible mechanism to switch to pro-apoptotic state upon sustaining substantial DNA damage, as dicussed in the following section [P53ss] 1.5 [p53ss] 0.5 0 10 [AKTT ] Figure 3-14 p53 steady-state bifurcation diagrams for various extents of DNA damage Steady states of p53 as functions of [AKTT] for various extents of DNA damage Model M3 was used in the computer simulations (see text for details) Extent of DNA damage (arbitrary units) increases from curves to 4, i.e., 0.01, 0.02, and 0.1, respectively For curve 1, DNA damage is absence, which is identical to the curve depicted in Figure 3-7 3.6.2 Switching to pro-apoptotic state upon substantial DNA damage Upon substantial DNA damage, the system could switch from the pro-survival state at the lower branch of the p53 bifurcation diagram (Figure 3-14) to the pro-apoptotic state at the upper branch of the figure by either downregulating the control parameter [AKTT] below the pro-apoptotic threshold (Section 3.5.1) or finite perturbations of the 65 components of the network model in the bistable region (Section 3.5.3) However, [AKTT] is relatively constant after cells are irradiated (Gottlieb et al., 2002; Martelli et al., 2003); [AKTT] is regulated by growth factor signalling rather than by DNA damage Furthermore, the pro-apoptotic threshold of [AKTT] is insensitive to extent of DNA damage (Figure 3-14) For these reasons, plausibility of the first mechanism is diminished To utilize the second mechanism however, the system must be in the bistable region This requirement is met with ease given that the range of the bistable region lengthens considerably and thereby shifts the pro-survival threshold Rk to the right when extent of DNA damage is increased slightly from curves to (Figure 3-14) In particular, Curve demonstrates that there is a specific extent of DNA damage at which Rk extends to practically infinity Thus, an unstressed system, whose [AKTT] lies outside of the bistable region, could find itself in the bistable region upon substantial DNA damage Nevertheless, it has to be noted that confining the system in the bistable region only confers potentiality for (but does not guarantee) switching to pro-apoptotic state Whether or not the switch occurs is dependent on the magnitudes at which components of the p53-AKT network are perturbed, which are influenced by the specific sources that drive the perturbations In this particular context, signals emanating from DNA damage and repair pathways are possible sources of perturbation, whereby the perturbation magnitudes could possibly correlate with extent of DNA damage It will be shown in Chapter that, in response to DNA damage, perturbation in the form of oscillations of intracellular concentration of p53 could switch the system to pro-apoptotic state 66 3.7 Summary Kinetic models corresponding to the p53-AKT networks analyzed in the previous chapter are formulated and studied in detail The most significant among the predictions of the kinetic models is the existence of the robust phenomenon of bistability in the p53-AKT network This phenomenon is conserved as the kinetic models are simplified sequentially to kinetic models with decreasing mechanistic details between p53 and AKT interaction, and when the values of the kinetic parameters used in the models are varied simultaneously Furthermore, several biological advantages of bistability in the regulation of the cell survival and death switch are contrasted with a monostable switch A bistable region can provide a buffer against random fluctuation of [AKTT], which curtails random switching between pro-apoptotic and pro-survival cellular states In addition, a bistable region allows finite perturbations to the components of the network model as a means to regulate the switching of the system from one cellular state to the other Lastly, the effect of DNA damage on bistability is studied Interestingly, the range of the bistable region is lengthened considerably as the extent of DNA damage is slightly increased that extends to practically infinity upon a particular extent of damage, which primes the cell for switching to pro-apoptotic state 67 ... Sections 3. 3 and 3. 4 Finally, novel systemic behaviors of the system are inferred and predicted from the simulation results (Sections 3. 5 and 3. 6) 34 3. 2 Formulation of kinetic models Figure 3- 1 Kinetic... p 53] n1 k [ p 53] v3 = j3n1 + [ p 53] n1 v4 = k [PIP 2] k {[PIPT ] − [PIP3]} = j + [PIP 2] j + [PIPT ] − [PIP3] vm = k m [PTEN ][PIP 3] j m + [PIP3] (3- 3) k [ p 53] n2 v5 = v6 = j5n + [ p 53] n2 k [... conserved (Figure 3- 7) p53ss AKT*ss [p53ss] [AKTass ] [AKTT ] Figure 3- 7 Model M3: steady-state bifurcation diagrams of p 53 and AKTa The steady state values of p 53 ([p53ss], gray curve) and AKTa ([AKTass],