Mathematical and computational analysis of intracelluar dynamics 5

75
Chapter 5
Implications of p53 Oscillations in Cell Survival
and Death


Studies of global cellular regulatory networks have shown that p53 is a major network
hub whose perturbations are expected to affect many biological pathways regulating
cell cycle, apoptosis, ageing and development (Royds and Iacopetta, 2006). Thus, the
consequences of p53 oscillations may be wide-ranging. For example, these
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.

76
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

77
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
in Figure 5-1. The inclusion of mdm2 facilitates an implicit consideration of the

78
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
AKT

AKT
a

p53
MDM2

MDM2
a

mdm2
v
0
v
9
v
7
v
4
v
5
v
2
v
1
v
3
v
10
v
m8
v
6
v
m6
PIP2

PIP3

v
m16

v
16
v
8
pten
PTEN

v
11
v
12
v
13

v
15
v
14

79
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
proportional to IR intensity (
ρ
), i.e.,

80
ρ
ρ
*
*
,9,99
,0,00
IRbasal
IRbasal
kkk
kkk
+=
+
=
(5-1)
where
k
0,basal
and k
9,basal
are basal value of k
0
and k
9
under no DNA damage;

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.











81

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-
MDM2a pro-survival steady states due to the inclusion of time-delays via the
0 5 10 15 20 25
p (Gy)
A
[p53
ss
]
ρ

ρρ
ρ
(Gy)
0 5 10 15 20 25
p (Gy)
B
[MDM2a
ss
]
ρ
ρρ
ρ
(Gy)

82
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).








0
100
200
300
400
0 2 4 6 8 10 12 14 16
p (Gy)
Time (Min)
Limit cycle period
mdm2 time delay
MDM2 time delay
MDM2a time delay
ρ
ρρ
ρ
(Gy)

83
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 high-
amplitude oscillation that eventually settles down to the unique limit cycle at each
ρ
.
As an example, the time-courses of p53, mdm2, MDM2 and MDM2a are depicted in

84
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.


85

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


0 5 10 15 20
p (Gy)
[mdm2] (uM)
ρ
ρρ
ρ
*

ρ
ρ ρ
ρ
(Gy)

86

Figure 5-6. Schematic p53-MDM2a phase portrait diagrams illustrating the phase space
for the cases of
ρ
ρρ
ρ
<
ρ
ρρ
ρ
* and
ρ
ρρ
ρ
=
ρ
ρρ
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.








X

X

[MDM2a]
A B
[p53]

87















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.








0
4
0 5 10 15 20 25
p (Gy)
[ P5 3] (u M )
0
16
0 5 10 15 20 25
p (Gy)
[M D M 2] (u M )
ρ
ρρ
ρ

(Gy)



ρ
ρρ
ρ
*



0
1
0 5 10 15 20 25
p (Gy)
[MDM2a] (u M)

[p53]

[MDM2]












[MDM2a]


0
0.04
0 5 10 15 20 25
p (Gy)
[A KTa] (u M)
0
0.7
0 5 10 15 20 25
p (Gy)
[p ten ] (u M)
0
2.6

0 5 10 15 20 25
p (Gy)
[PTEN ] (u M)



[AKT
a
]













[pten]

[PTEN]



88
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.

89
0
10
20
30
40
50
60
70
80
90
100
6 8 10 12 14 16 18 20 20.8
p (Gy)

(Left knee)
(Right knee)
ρ
ρρ
ρ
(Gy)
Percentage of initial conditions
0
10
20
30
40
50
60
70
80
90
100
6 8 10 12 14 16 18 20 20.8
p (Gy)
(Left knee)
(Right knee)
ρ
ρρ
ρ
(Gy)
Percentage of initial conditions

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.



5.4.2 Cell-type variations

Measured basal concentrations of p53 in seven different cell lines range from about 1
x 10
4
to 22 x 10
4
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.


90
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;
k
0,basal

is set as 0.1 µM/min and j
5
is set as 0.5 µM in the computation of the p53
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.


91
3. Saddle-node switch (observed in 4% of the steady state curves), see Figure 5-
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 .

Figure 5-9. Schematic p53 steady state bifurcation diagrams obtained as kinetic
parameters k
0
and j
5
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.

A
C
B
A
C
B
[p53
ss

]
ρ
ρρ
ρ
(Gy)

92





0.2
1.2
0.01 0.2
k0
j5
No limit cycle
Early switch
Saddle-node switch
Monostable
k
0
,
b
a
s
a
l

93
Figure 5-10. Types of p53 steady state bifurcation curve obtained at each combination
of kinetic parameters values of k
0,basal
and j
5
.
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
[ ]
( )












+= t
P
AMtp
yoscillator
π
2
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.



v
15
p53
X
m

X
p

v
11
v
13
v
14

94
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 X
p
(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 X
p

expression for both oscillatory and non-oscillatory p53. As clearly shown, the rate of

X
p
expression is increased by p53 oscillations. Figure 5-13B shows that the resulting
steady state mean level of X
p
(and X
m
, data not shown) is increased due to p53
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
[TP53] (uM)
ρ
ρρ
ρ



(Gy)



[p53] uM

95
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 non-
oscillatory p53.
(A) Representative time-courses of X
p

(protein of p53-target gene X) expression induced by
both non-oscillatory (gray) and oscillatory p53 (black). The oscillation profile of oscillatory
p53 is identical to the limit cycle of Model M4 at
ρ
= 12Gy. (B) Steady state amount of X
p

induced by both non-oscillating (gray curve) and oscillating (black curve) p53 for the entire
0
4
8
12
0 500 1000 1500
Tim e (min)
[PTEN] (nM)
A
0
0.2
0.4
0.6
0.8
0 500 1000 1500
Time (m in)
[TP53] (uM)
0
2
4
6
8
10

12
14
0 4 8 12 16
p (Gy)
[PTEN] (nM)
B
ρ
ρρ
ρ
(Gy)
0
4
8
12
0 500 1000 1500
Tim e (min)
[PTEN] (nM)
A
0
0.2
0.4
0.6
0.8
0 500 1000 1500
Time (m in)
[TP53] (uM)
0
2
4
6

8
10
12
14
0 4 8 12 16
p (Gy)
[PTEN] (nM)
B
ρ
ρρ
ρ
(Gy)
[p53] uM
0
4
8
12
0 500 1000 1500
Tim e (min)
[PTEN] (nM)
A
0
0.2
0.4
0.6
0.8
0 500 1000 1500
Time (m in)
[TP53] (uM)
0

2
4
6
8
10
12
14
0 4 8 12 16
p (Gy)
[PTEN] (nM)
B
ρ
ρρ
ρ
(Gy)
0
4
8
12
0 500 1000 1500
Tim e (min)
[PTEN] (nM)
A
0
0.2
0.4
0.6
0.8
0 500 1000 1500
Time (m in)

[TP53] (uM)
0
2
4
6
8
10
12
14
0 4 8 12 16
p (Gy)
[PTEN] (nM)
B
ρ
ρρ
ρ
(Gy)
[p53] uM
[X
p
] (nM)
[X
p
] (nM)

96
range of
ρ
where limit cycle exists in Model M4. The black curve is obtained by taking the
average of the crest and trough of X

p
at steady state. In (A) and (B), the level of non-
oscillatory p53 is equal to the mean of the oscillatory p53.


Next, the effects of p53 oscillation amplitudes and periods on X
p
steady state
mean level are studied using Model M5. Interestingly, the steady state mean level of
X
p
does not depend on the period P of p53 oscillations (see Figure 5-14A for two
representative plots). It does depend, as expected, on amplitude A of p53 oscillations
in which the steady state mean levels of X
p
increase with A (Figure 5-14B). The
trends depicted in Figures 5-14A and 5-14B holds for other values of p53 oscillation
means, amplitudes and periods varied as well; they are varied over a significantly
wider range than those inferred from the limit cycles generated in Model M4.
Specifically, M is varied from 0.1 to 2 µM, A is varied from 0 to 0.5 µM and P is
varied from 4 to 7 hrs as measured experimentally (Geva-Zatorsky et al., 2006).

4 5 6 7
TP53 Oscillation Period (hr)
[PTEN] (uM)
A
[X
p
] (nM)
p53 oscillation period (hr)
B
[X
p
] (nM)
p53 oscillation amplitude (µM)

97
Figure 5-14. Consequences of p53 oscillations on gene expression.
(A) Steady state mean level of X
p
(protein of p53-target gene X) induced as p53 oscillation
period is varied. For illustration, the amplitudes of oscillatory p53 are fixed at 0.5 µM for
both of the cases. The oscillation means of oscillatory p53 are fixed at 2 µM (top line) and
0.6 µM (bottom line). (B) Steady state mean level of X
p
induced as p53 oscillation
amplitudes is varied. For illustration, the mean and period of oscillatory p53 are fixed at 0.5
µM and 4 hr, respectively. The trends shown in the two figures hold for other values of p53
oscillation amplitudes, means and periods varied as well.

5.6 p53-transcriptional regulation of apoptosis

As regulation of transcription occurs in a cell nucleus, oscillations of nuclear p53
protein levels could have implications on p53-transcriptional regulation of apoptosis
upon DNA damage. Mechanisms of non-transcriptional p53 regulation of apoptosis
on the other hand, are not adequately characterized thus far for model construction. In
view of these facts, only p53-transcriptional regulation of apoptosis is studied.

As reviewed in Section 2.5 of Chapter 2, both p53 and AKT impinge
extensively on the pro-apoptotic and pro-survival members of the BCL2 protein
family, the key regulator of the release of cytochrome c from the mitochondria that is
required to activate the intrinsic caspase-dependent apoptotic pathways. Among
them, p53 transcribes pro-apoptotic members BAX (Miyashita and Reed, 1995) and
BAD (Jiang et al., 2006), which binds and inhibits pro-survival members BCL-2
(Hemann and Lowe, 2006) and BCL-XL (Yang et al., 1995; Adams and Cory, 2001)
respectively. AKTa however, phosphorylates BAD after which it is sequestrated by
14-3-3 proteins and thereby cannot bind BCL-XL (Datta et al., 1997; Datta et al.,
2000). Indeed, the intrinsic apoptosis program is triggered in the event when the
protein levels of the pro-survival members BCL-XL and BCL-2 are substantially

98
downregulated (Adams and Cory, 2001; Cory and Adams, 2002). Furthermore, the
ratio of BAX to BCL-2 protein levels as a determinant of apoptotic threshold has been
illustrated in both experimental and modeling studies (Danial and Korsmeyer, 2004;
Bagci et al., 2006). Therefore, analysis of a kinetic model that includes these specific
and critical BCL2 family members could aid in the understanding of the regulation of
the intrinsic apoptotic pathways by the p53-AKT network. And since BAD is a
common target of p53 and AKT, the implications of their opposing regulations on

BAD can be studied.


5.6.1 Formulation of a model of apoptosis








Figure 5-15. Kinetic model depicting the regulation of pro-apoptotic and pro-survival
genes by p53 and AKT.
Kinetic model in which the regulation of pro-apoptotic BAX and BAD proteins and pro-
survival BCL-2 and BCL-XL proteins by p53 and AKT are considered. 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 bax (mRNA of
BAX) and bad (mRNA of BAD). BADp denotes phosphorylated form of BAD.



The “Apoptotic Model” that overly simplifies the actual apoptotic pathway is
formulated as a prototype to preliminary explores the consequences of p53
AKT
a

v

17
p53
BAX*BCL-2

bax

BAX

BCL-2

bad
BAD

BCL-XL

BAD*BCL-XL
BAD
p

v
18
v
19
v
20
v
21
v
22
v

23
v
24
v
25
v
26
v
27
v
28
v
29
v
30
v
31
v
m31

v
32

99
oscillations on the regulation of apoptosis upon DNA damage. The model is
constructed by appending the kinetic model depicted in Figure 5-15 to Model M4
(Figure 5-1). The Apoptotic Model includes all the ODEs, rate equations and the
associated kinetic rate parameters used in Model M4 (see Appendix A-10). The
additional ODEs, rate equations and the associated kinetic rate parameters describing
the kinetic model in Figure 5-15 are given in Appendix A-12; refer to Appendix A-13

for a description of the derivation of these kinetic parameter values. Note that the rate
equations and values of the kinetic parameters associated with the reactions of bax,
BAX and BCL-2 are set to be identical to those of bad, BAD and BCL-XL,
respectively.

To account for DNA damage repair, the DNA damage signal is assumed to
decay exponentially with timescale
λ
(min), which is based on the observation that
DNA damage repair reduces the number of DSBs exponentially for at least 24 hrs
after exposure to
ρ
ranging from 0.2 to 80 Gy (Rothkamm and Lobrich, 2003). Note:
although p53 could influence DNA damage repair by inducing DNA repair genes, it is
not included in the model due to paucity of data for this mechanism. Subsequently,
Eqn (5-1) becomes
t
IRbasal
t
IRbasal
ekkk
ekkk
λ
λ
ρ
ρ
−
−
+=
+=

**
**
,9,99
,0,00
(5-2)
p53-dependent transcription of bax and bad is also assumed to decay exponentially
with the same timescale
λ
as DNA damage is progressively repaired. This is because
phosphorylation of p53 especially at serine 46 by DNA damage signal transducers
such as ATM and DNA-PK is needed to express pro-apoptotic proteins, as reviewed
in Section 2.3 of Chapter 2.