Mathematical and computational analysis of intracelluar dynamics 5

Limit cycle oscillations whose amplitudes are shown by the dots arise from the unstable steady states light gray curve in the lower A or upper B branch of the steady state curves.. As in

oscillations may be correlated with predisposition to cancer (Collister et al., 1998), as

suggested by studies with Bloom’s syndrome patients whose abnormal fibroblast cells display p53 oscillations distinct from wild-type cells upon DNA damage; Bloom’s syndrome patients have high tendency for tumorigenesis

Even so, biological functions of such oscillations remain elusive This might

be attributed to the fact that since p53 is involved in many regulatory feedback loops (Harris and Levine, 2005), any biological role of p53 oscillations is likely to be non-intuitive, which confounds the elucidation of these roles For this particular purpose, mathematical modeling is useful as it can elucidate counter-intuitive systems behaviors arising from complex non-linear interactions To study the roles of p53 oscillations however, mathematical models must encompass relevant pathways that crosstalk with the p53-MDM2 feedback loop One such pathway is the p53-AKT network studied previously, which embeds the p53-MDM2 feedback loop

In this chapter, the biological roles and consequences of p53 oscillations on the control between cell survival and death in the context of the p53-AKT network is investigated It will be shown that p53 oscillations markedly decrease the IR intensity threshold level at which switching to a pro-apoptotic state occurs Furthermore, several biological advantages conferred by p53 oscillations are demonstrated, including the increased ability of p53 as a transcription factor to induce expression of pro-apoptotic target genes at higher levels

5.1 Formulation of the kinetic model

The genesis of p53 oscillations requires a finite time lag between MDM2 protein expression and p53 degradation, which depends on cumulative time delays from transcriptional, translational and translocation processes of MDM2, as well as kinetics

of MDM2-mediated p53 degradation The cumulative delay is estimated as 40 min

(Ma et al., 2005) This is significantly shorter than the reported 90 to 150 min time delay of MDM2 oscillation peaks from p53 peaks (Geva-Zatorsky et al., 2006)

Furthermore, given that p53-MDM2 oscillations are observed only upon UV or gamma radiation in specific cell types, the time delay associated with MDM2 expression cannot explain for the presence of such oscillations as p53 transcription of

MDM2 is ubiquitous in a majority of cell types upon irradiation (Vogelstein et al., 2000; Momand et al., 2000; Bond et al., 2005; Piette et al., 1997; Momand and

Zambetti, 1997; Juven-Gershon and Oren, 1999; Moll and Petrenko, 2003; Iwakuma and Lozano, 2003; Alarcon-Vargas and Ronai, 2002; Horn and Vousden, 2007) For

these reasons, cell type and stress specific regulations of the p53-MDM2 feedback loop are likely to play a central role in the genesis of p53-MDM2 oscillations

Indeed, irradiation-specific post translational modifications of p53 and MDM2 lead to significant retardation of the kinetics of MDM2-mediated p53 degradation (Section 2.2.2 of Chapter 2), thereby extends the time lag between MDM2 expression and p53 degradation Interestingly, removal of ATM-mediated destabilization of MDM2 upon irradiation eradicates p53-MDM2 oscillations no matter how long the

MDM2 expression time delay is set (Ma et al., 2005; Wagner et al., 2005) Section

5.1.1 describes in detail as to how this particular regulatory mechanism is accounted

in the model Besides, p53 transcription of PTEN is cell type specific that is essential for p53 to antagonize AKT; p53 transcription of PTEN upon irradiation-induced DNA damage has been reported in cells and tissues where p53 oscillations are observed Notably, positive feedback loops, such as p53-AKT loop studied in this thesis, supplementing the p53-MDM2 negative feedback loop has been proposed to induce p53 oscillations (Section 4.2 of Chapter 4)

The mechanisms as described are implemented in the model shown in Figure 5-1 that shall be referred to from hereon as Model M4 It is an extension of Model M1 that is previously studied in Chapter 3 (see Figure 3-1 of Chapter 3), which exhibits bistability and predicts switching behavior between pro-survival and pro-apoptotic cellular states The extension included in Model M4 involves two additional part lists namely, the mRNA transcripts of the MDM2 and PTEN genes (these transcripts are symbolized by mdm2 and pten, respectively), as depicted in red

transcriptional and translational delays of p53-dependent MDM2 expression As explained above, an implicit inclusion of MDM2 expression delay in this context is adequate as it is not a likely reason for p53-MDM2 oscillations In fact, Model M4 combines key features of a relaxation oscillator and a delay oscillator

Figure 5-1 Kinetic model of the oscillatory p53-AKT network, Model M4

Model M4 is an extension of Model M1 (Figure 3-3 of Chapter 3) in which mRNAs of PTEN

and MDM2 genes are included (depicted in red) The v r’s are the rate equations of each reaction step Broken edges denote enzymatic reactions whereas full edges denote mass

action reactions All part lists are at the protein level except for those depicted in red AKTa and MDM2a denote biochemically active AKT and MDM2 proteins upon phosphorylation

In the model, p53 is transcriptionally-active where it transcribes target genes, MDM2 and PTEN

The ODEs, rate expressions and the associated kinetic parameters values of Model M4 are given in Appendix A-10 Some of the kinetic parameter values, which are associated with rate equations that describe the p53-MDM2 feedback loop, are modified from Model M1 to fit experimental observations in single cells (reviewed in Section 4.1 of Chapter 4) namely, periods of sustained oscillations of p53 and MDM2

range from 4 to 7 hours, peaks of MDM2 oscillations lag behind p53 peak for 1.5 to 2.5 hrs and periods of oscillations decrease with increasing IR intensity The modified values are within the order of magnitudes that were obtained from literature surveys as described in Appendix A-2

5.1.1 Simulating the DNA damage signal transduction

pathways

As p53-MDM2 oscillations are exhibited upon DNA damage, kinetics of the DNA damage signaling to the p53-AKT network has to be considered As reviewed in Section 2.2.2 of Chapter 2, post-translational modifications of p53 and MDM2 upon DNA damage translate to an increase in the kinetic rate parameter for p53 synthesis

and activation (k 0 ) as well as the degradation rate parameter of active MDM2 (k 9) in Model M4 The trend of these two kinetic parameters as a function of the extent of DNA damage depends chiefly on the kinetics of the DNA damage signal transduction pathways However, as reviewed in Section 2.3 of Chapter 2, several key issues confound the simulation of these pathways First and foremost, current knowledge about the part lists and their interaction mechanisms are still incomplete Despite that, existing network topologies of these pathways are surprisingly complex – besides the many feedback loops, several redundant sub-networks are present (see Figure 2-3 of Chapter 2) Particularly, the relative contribution of each of the sub-networks on the

regulation of the kinetic parameters k 0 and k 9 might be cell type specific, which has

not been determined so far As such, both k 0 and k 9 are assumed to be directly

ρ

*

*, 9 ,

9 9

, 0 ,

0 0

IR basal

IR basal

k k

k

k k

k 0,IR and k 9,IR are proportional constants

(values of k 0,basal , k 9,basal , k 0,IR and k 9,IR are given in Appendix A-10)

Nonetheless, simulations are repeated for all other biologically plausible variations of

k 0 and k 9 with ρ (see Section 5.9)

5.2 Steady states and oscillations

Steady states of Model M4 were determined using the method as described in Section

3.3.1 of Chapter 3 The steady states of p53 and active MDM2 (MDM2a) as

functions of ρ (ionizing radiation intensity) are shown in Figure 5-2

Figure 5-2 Model M4: steady-state bifurcation diagrams of p53 and MDM2a

Steady states of (A) p53 and (B) active MDM2 (MDM2a) as a function of ρ (the abscissa) Local stability of these states is indicated as either stable (black curve: stable nodes; dark gray curve: stable spirals) or unstable (broken black curve: saddle nodes; light gray curve: limit cycles) Limit cycle oscillations whose amplitudes are shown by the dots arise from the unstable steady states (light gray curve) in the lower (A) or upper (B) branch of the steady state curves Units of vertical axes are in µM Units of abscissa are in Gy

Model M4 also exhibits multiplicity of steady states within a range of ρ (6 to 20.8 Gy – in Figure 5-2) As in Model M1, within this range of ρ, the high-p53 and

low-MDM2a (and low-AKTa) pro-apoptotic steady states are all locally stable nodes,

and the middle branch of steady states are all unstable saddle points (local stability is determined by the method as described in Appendix A-4) In contrast to Model M1, however, Model M4 exhibits oscillatory dynamics in all the low-p53 and high-

additional of mdm2; no oscillations are obtained in the pro-apoptotic states (local stability analysis is used to determine steady states exhibiting oscillatory dynamics as described in Appendix A-11) The steady states enveloped by the gray dotted curves

in Figure 5-2 exhibit unstable spirals that lead to sustained oscillations or limit cycles The peaks and troughs of these limit cycle oscillations are indicated by the dots above and below the steady states, respectively The remaining pro-survival steady states, which flank the steady states exhibiting limit cycles, exhibit stable spirals that lead to damped oscillations It is interesting to note that the steady state at ρ = 0 Gy is a stable spiral; this result could explain why some cells have been observed to show

oscillatory behavior despite the absence of DNA damage (Geva-Zatorsky et al.,

2006)

The periods of the sustained oscillations in p53, mdm2, MDM2 and MDM2a

are identical, which decrease with increasing ρ (Figure 5-3), in accord with

experimental observations (Geva-Zatorsky et al., 2006) The model predicts

sustained oscillation periods of 3.5 to 5.2 hours that falls within reported experimental

values (Geva-Zatorsky et al., 2006)

Figure 5-3 Model M4: limit cycle oscillation periods and time-delays of mdm2, MDM2 and MDM2a

Oscillation properties of p53, mdm2, MDM2 and MDM2a are depicted for the entire range of

ρ where the system exhibits limit cycle The gray curve depicts the oscillation periods of p53,

mdm2, MDM2 and MDM2a The remaining curves depict the time-delays of the peaks of mdm2, MDM2 and MDM2a pulses relative to the peaks of p53 pulses (see inset)

Furthermore, the model reproduces experimentally measured time-delays

between the peaks of MDM2a and p53 oscillations (1.2 to 1.9 hours) As expected, both MDM2 and MDM2a have longer time-delay than mdm2 due to transcription and

translation processes (Figure 5-3) Generally, these time-delays are not sensitive to ρ

On the other hand, the amplitudes of the oscillations in AKTa, PIP3, pten and PTEN

are insignificant; i.e., the concentration difference between crest and trough is less than 4% of the mean concentration (data not shown) that could be difficult to detect experimentally

5.3 Cells exposed to increasing IR intensities

Computer experiments using Model M4 are performed to simulate the behavior of a cell that is exposed to a pulse of IR with fixed intensity The experiment is repeated

as ρ is increased in the range where limit cycle oscillations are exhibited For each simulated experiment, the initial cellular levels of the proteins and transcripts are those of the steady states of a cell unexposed to IR (i.e., the steady state of Model M4

at ρ = 0 Gy) Interestingly, simulations show that the system initially displays a amplitude oscillation that eventually settles down to the unique limit cycle at each ρ

high-Figure 5-4 at ρ = 6 Gy in which their first pulse amplitudes are larger than their respective limit cycle Notably, larger initial p53 oscillation amplitudes are also

observed in single cells (Geva-Zatorsky et al., 2006)

Figure 5-4 Time-courses of Model M4 at ρ = 6 Gy

Time-courses of p53 (red), mdm2 (blue, broken line) and MDM2 (blue) and MDM2a (black)

are depicted

The amplitudes of these initial oscillations are shown in Figure 5-5 superimposed with the steady-state curve of mdm2 This figure shows that the initial amplitudes increase with ρ and, interestingly, that there is a particular ρ = 13.8 Gy (symbolized by ρ*) where the system crosses a boundary surface associated with the unstable steady states (the dotted middle branch of steady states) and gets attracted to the upper branch of stable steady states These steady states correspond to high-p53 pro-apoptotic states Since ρ* is less than the value of ρ corresponding to the right-knee of the steady state curve, ρ* is defined as an early-switching point

Figure 5-5 First pulse amplitudes of mdm2 oscillations plotted along with its state curve

steady-The first pulse amplitudes are indicated as black square boxes that are determined from the time-courses of mdm2 at various ρ For each ρ considered, the initial concentrations of the species is set to the steady state concentration under no DNA damage, i.e., at ρ = 0 Gy Simulations show that the system can switch from an oscillatory low-p53 state to the high-p53 state before the right knee of the steady state curve at ρ* = 13.8 Gy (as indicated by the vertical line) ρ* is termed as the early-switching point.

Figure 5-6 explains (schematically) what occurs in phase space as ρ* is crossed One of the stable manifolds of the saddle point (o) shifts position (from Figure 5-6A to 5-6B) so that the initial state (indicated by X) is now located in the basin of attraction of the high p53-stable node (●) In general, it is the stable manifolds of the saddle point that define the boundary between the basins of attraction

of the oscillatory low-p53 limit cycle and the high-p53-stable node Depending on the initial conditions of the molecular species in the model, the boundary manifolds may

or may not be crossed as the cells are exposed to increasing ρ

Figure 5-6 Schematic p53-MDM2a phase portrait diagrams illustrating the phase space

for the cases of ρ < ρ* and ρ = ρ*

(A) For ρ < ρ*, the initial state corresponding to ρ = 0 Gy (as indicated by X) is located

within the basin of attraction of the limit cycle (B) At ρ = ρ*, one of the stable manifolds (bold curve) of the saddle point (o) shifts position so that X is now located in the basin of attraction of the high-p53 stable node (●) Note: the actual phase portrait is high-dimensional, however, for the purpose of illustrating the change in phase spaces of ρ before and at ρ*, a two-dimensional diagram is given

Arbitrary switching to high-p53 pro-apoptotic state due to random spike in the oscillation amplitudes of any one species is unlikely As depicted in Figure 5-7, at the early-switching point, ρ*, the initial oscillation amplitudes of all the species must cross their respective saddle nodes simultaneously for the system to switch to high-p53 state

[MDM2a]

[p53]

Figure 5-7 First pulse oscillations amplitudes of Model M4 plotted along with the steady-state curves

First pulse amplitudes of the species in Model M4 are superimposed with their respective steady state curves as functions of ρ (the abscissas) In each plot: the first pulse amplitudes are indicated as gray square boxes; the early-switching point is indicated by the green vertical line at ρ* = 13.8 Gy; limit cycles are indicated in blue; the peaks and troughs of these limit cycles are indicated respectively by the blue dots above and below the limit cycles; stable spirals are shown as gray lines; unstable saddle nodes are indicated in red; stable nodes are indicated in black

5.4 Effects of cell-cell and cell-type variations

5.4.1 Cell-cell variations

Variation of the intracellular levels of part lists among individual cells is expected due

to random fluctuations To study how cell-cell variation affects the systems behavior

at steady state, initial concentrations of all the part lists are varied simultaneously using Latin hypercube sampling (see Section 3.4.1 of Chapter 3) For each part list, its initial concentration is varied from 0 µM up to 200% of its steady state concentration at ρ = 0 Gy The variation range is then divided equally into 100 intervals and then randomly permutated to generate 101 Latin sets of initial conditions

in each Latin hypercube sample A total of 10 independent Latin hypercube samplings are performed

For each Latin set of initial concentrations, time-courses are computed for each ρ within the bistable range (6 to 20.8 Gy in Figure 5-2) The percentages of Latin sets of initial concentrations in a Latin hypercube sample that switches the system to high-p53 steady states are given in Figure 5-8 Note that between 10 to 20% of initial conditions can lead to an early switching point at ρ = 8 Gy At 18 Gy about 90% of the cells are predicted by the model to make this transition Thus, early switch to the high-p53 state can be induced by cell-cell variations in their initial concentrations at ρ that are significantly lower than that corresponding to the right-knee of the steady-state curve (see Figure 5-2) Generally, the propensity for early switch to high-p53 state becomes more deterministic or less dependent on cell-cell variations as ρ increases

Figure 5-8 Percentage of Latin sets of initial concentrations leading to high-p53 state

Percentage of Latin sets of initial conditions in each Latin hypercube sample that leads to high-p53 state at various selected ρ is computed A total of 10 Latin hypercube samplings are performed and the simulation result of each hypercube sample is shown in the figure as a bar

Measured basal concentrations of p53 in seven different cell lines range from about 1

x 104 to 22 x 104 molecules per cell (Ma et al., 2005), suggesting a cell

type-dependent rate of p53 synthesis and therefore a significant variation in the value of the

kinetic parameter k 0,basal of Model M4 In addition, several reports (Toledo and Wahl, 2006; Appella and Anderson, 2001; Bode and Dong, 2004) have documented cell-type specific post-translational modifications of various DNA-binding domains of p53, which will affect the binding affinity towards the promoter sequences of its

target genes Hence, the kinetic parameter j 5 (associated with p53’s affinity to the promoters of target genes) is also subject to wide variations depending on cell types;

high j 5 means low dissociation rate of p53 from DNA, and vice versa

To investigate how these specific cell-type differences affect the conservation

of the early switch phenomenon, kinetic parameters k 0,basal and j 5 are simultaneously

varied k 0,basal is varied from 0.01 to 0.2 µM/min at an interval of 0.002 µM/min

whereas j 5 is varied from 0.2 to 1.2 µM at an interval of 0.02 µM For each

combination of k 0 and j 5, a p53 steady-state curve as a function of ρ is computed;

steady state curve in Figure 5-2 above A total of 4896 computed steady state curves are characterized Expectedly, all steady-state curves with multiple steady states exhibit either limit cycles or damped oscillations, or both, at the low-p53 steady state branch Three typical types of bifurcation curves are obtained within the varied parameter space (Figures 5-9 and 5-10):

1 Monostable (observed in 9% of the steady state curves), see Figure 5-9A The only state of the system is a high-p53 stable node and no oscillation is manifested This occurs at relatively high p53 production rate and fast dissociation rate of p53 from promoter site, demarcated as in Figure 5-10

2 Early switch (observed in 87% of the steady state curves), see Figure 5-9B This is the predominant type observed, indicated as black circles (O) in Figure 5-10 The early-switching point is searched using the method as described in Section 5.3 Interestingly, the initial oscillation amplitudes of the stable spirals causes the early switch to high-p53 state for steady state curves that do not exhibit limit cycles, indicated as in Figure 5-10

3 Saddle-node switch (observed in 4% of the steady state curves), see Figure 9C For this type, the system switches to high-p53 state exactly at the right-knee of the curve although some of them exhibit limit cycles (demarcated as O in Figure 5-10); those that do not exhibit limit cycles are indicated as

5-Figure 5-9 Schematic p53 steady state bifurcation diagrams obtained as kinetic parameters k0 and j5 are simultaneously varied

Three types of p53 steady state bifurcation diagrams as functions of ρ are obtained when

k 0.basal and j 5 are simultaneously varied namely (A) monostable, (B) bistable with early switch and (C) bistable with saddle-node switch.

[p53ss]

ρ(Gy)

k 0,basal

Figure 5-10 Types of p53 steady state bifurcation curve obtained at each combination

of kinetic parameters values of k0,basal and j5

Each point represents a p53 steady state bifurcation curve computed for a particular value of

k 0,basal and j 5; a total of 4896 bifurcation curves are computed Note that all bifurcation curves exhibit bistability except for those denoted as ‘Monostable’ See text for details

5.5 Consequences of p53 oscillations on expression of

target genes

Since p53 is a transcription factor with many target genes (Qian et al., 2002), it is of

interest to determine what the effects of nuclear p53 oscillations are on the expression levels of these genes A simplistic model for the expression of a representative p53-target gene, X, is considered (Model M5), as shown in Figure 5-11 The model

includes the rate equations v 11 , v 13 , v 14 and v 15, whose expressions and associated kinetic parameters are identical to those used in Model M4 (Figure 5-1) The time-

course of an oscillatory p53 is estimated by [ ] ( ) 

P A M t

sin153

where M is oscillation mean (µM), A is oscillation amplitude (µM) and P is oscillation

period (hr)

Figure 5-11 Kinetic model of Model M5: p53-dependent expression of a representative target gene

Model M5 is extracted from Model M4 (Figure 5-1) in which the rate equations v 11 , v 13 , v 14

and v 15 are identical to Model M4 X m and X p denote mRNA and protein of a representative target gene of p53 Broken edges denote enzymatic reactions whereas full edges denote mass action reactions

Values of M, A and P are inferred from the limit cycles generated in Model

M4 Figure 5-12 depicts the crest (blue curve) and trough (red curve) of p53 limit cycle oscillations for the entire range of ρ where Model M4 exhibits limit cycles For each ρ, the oscillation mean, M, is obtained by taking the average of p53 crest and trough levels (black curve) The oscillation amplitude, A, is then the difference between the crest and the oscillation mean, M, at each ρ The oscillation period, P,

equals to the limit cycle period at each ρ, as given in Figure 5-3 above

Figure 5-12 p53 limit cycle profiles as functions of ρ generated by Model M4

The crest (blue) and trough (red) levels of p53 oscillations are plotted as functions of ρ for the entire range of ρ where Model M4 exhibits limit cycle The black curve represents the average between the crest and trough

At each ρ within the range of ρ where Model M4 exhibits limit cycles, the resulting steady state level of Xp (protein of p53-target gene X) induced by an oscillatory and non-oscillatory p53 are compared; the level of non-oscillatory p53

equals to M Figure 5-13A shows an example of the temporal profiles of Xpexpression for both oscillatory and non-oscillatory p53 As clearly shown, the rate of

Xp expression is increased by p53 oscillations Figure 5-13B shows that the resulting steady state mean level of Xp (and Xm, data not shown) is increased due to p53

oscillations for the entire range of ρ where limit cycles exists in Model M4, despite the fact that the mean level of these oscillations is equal to that of the non-oscillatory level

Figure 5-13 Oscillatory p53 expresses more amounts of target gene than a oscillatory p53

non-(A) Representative time-courses of Xp (protein of p53-target gene X) expression induced by both non-oscillatory (gray) and oscillatory p53 (black) The oscillation profile of oscillatory

ρ

0 4 8 12