Mathematical and computational analysis of intracelluar dynamics 3

33
Chapter 3
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 pro-
apoptotic 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 p53-
mediated pro-apoptotic cellular states within the first hour after oxidative stress

34
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).

35
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 MDM2-
MDM2a; AKTa and MDM2a denote biochemically active AKT and MDM2 proteins upon
phosphorylation. The v
r
’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

36
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):

=
dt
Pd
i
][
Rate of formation or generation – Rate of removal or degradation (3-1)

The left-hand side (LHS) denotes rate of change of the intracellular concentration
(denoted as []) of a part list P
i
, which equals to its rate of formation minus its rate of
removal as given by the respective rate terms, v
r
’s. As an example, the kinetic
equation of p53 is given by v
0
– v
2
– v

7
(refer to Figure 3-1 for the assignment of rate
term), where v
0
is rate of transcriptionally-active p53 synthesis and activation, v
2
is
rate of p53 degradation by MDM2a and v
7
is p53 self-degradation rate. Although p53
is also involved in the enzymatic reactions associated with the rate terms v
3
and v
5
,
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).

37
6116105
1266
983
44
11
720
]2[
]2[
][
]3[

][
]53[
vvvvv
dt
MDMd
vvv
dt
aMDMd
vvv
dt
PTENd
vv
dt
PIPd
vv
dt
AKTad
vvv
dt
pd
m
m
m
m
−−++=
−−=
−+=
−=
−=
−−=

(3-2)
where
v
0
: rate of transcriptionally-active p53 synthesis and activation;
v
1
: rate of AKTa formation by PIP3 phosphorylation of AKT;
v
m1
: rate of AKTa removal by basal dephosphorylation of AKTa;
v
2
: rate of transcriptionally-active p53 degradation by MDM2a;
v
3
: rate of PTEN formation by p53 transcription;
v
4
: rate of PIP3 formation by PI3K phosphorylation of PIP2;
v
m4
: rate of PIP3 removal by PTEN dephosphorylation of PIP3;
v
5
: rate of MDM2 formation by p53 transcription;
v
6
: rate of MDM2a formation by AKTa phosphorylation of MDM2;
v

m6
: rate of MDM2a removal by basal dephosphorylation of MDM2a;
v
7
: rate of active p53 inactivation (degradation or dephosphorylation);
v
8
: rate of PTEN formation by basal induction;
v
9
: rate of PTEN degradation;
v
10
: rate of MDM2 formation by basal induction;
v
11
: rate of MDM2 degradation;
v
12
: rate of MDM2a degradation


38
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] = [AKT
T
] – [AKTa] and [PIP2] = [PIP
T
] – [PIP3]. [AKT

T
]
and [PIP
T
] 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 [AKT
T
] 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:
1.

Enzymatic phosphorylation (v
1
, v
4
and v

6
), dephosphorylation (v
m1
, v
m4
and
v
m6
) and degradation (v
2
) reactions are assumed to have Michaelis-Menten
type expressions given by
[
]
[
]
[ ]
Sj
SEk
v
r
r
r
+
= , where k
r
and j
r
are kinetic
parameters (or constants) that denote the maximal rate of reaction and the

Michaelis constant, respectively. j
r
quantifies the binding affinity between

39
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, k
r
absorbs the variable [E], i.e.,
[
]
[ ]
Sj
Sk
v
r
r
r
+
= for v
m1
, v
4

and v
m6
.

2.

p53-dependent production of proteins (v
3
and v
5
) is assumed to have Hill-type
expressions given by
[
]
[ ]
n
n
r
n
r
r
pj
pk
v
53
53
+
= , where k
r
and j
r
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 = 1 while being positively cooperative when n > 1. In a positively
cooperative case, the binding of p53 to its promoter site increases the affinity
of the site for further p53 binding.

3. Self-degradation reactions (v
2
, v
9
, v
11
and v
12
) are assumed to have first-order
reaction kinetics type expression give by
[
]
irr
Pkv
=
, where k
r
is the rate
constant of the reaction and Pi represents the part list undergoing self-
degradation.

4. Basal production (v
0
, v

8
and v
10
) of a part list is assumed to have a constant
rate, i.e., v
r
= k
r
.

40
Eqn (3-3) lists down the rate expressions for all the rate terms of Model M1.













(3-3)

[ ][ ]
[ ]
[ ] [ ] [ ]
{ }
[ ] [ ]
[ ]
[ ]
[ ][ ]
[ ]
[ ]
[ ]
[ ]
[ ]
[ ] [ ]
{ }
[ ] [ ]
[ ][ ]
[ ]
[ ]
[ ]
[ ][ ]
[ ]
[ ]
[ ]

[ ]
[ ]
PTENkv
kv
pkv
aMDMj
aMDMk
v
MDMj
MDMAKTak
v
pj
pk
v
PIPj
PIPPTENk
v
PIPPIPj
PIPPIPk
PIPj
PIPk
v
pj
pk
v
pj
paMDMk
v
AKTaj
AKTak

v
AKTaAKTj
AKTaAKTPIPk
AKTj
AKTPIPk
v
kv
m
m
m
n
n
n
m
m
m
T
T
n
n
n
m
m
m
T
T
o
99
88
77

6
6
6
6
6
6
2
2
5
2
5
5
4
4
4
4
4
4
4
4
1
1
3
1
3
3
2
2
2
1

1
1
1
1
1
1
1
0
53
2
2
2
2
53
53
3
3
3
3
2
2
53
53
53
532
33
=
=
=
+

=
+
=
+
=
+
=
−+
−
=
+
=
+
=
+
=
+
=
−+
−
=
+
=
=

41

(3-3)

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 5 and 6.

Item KP Description Units
Chosen
Value
Range Refs
1
k
0

Production of active
p53
µM/

min
0.1
0.002 to
0.2
Ma et al., 2005
2
k
1

PIP3-mediated
phosphorylation of
AKT
/min 20 20 Giri et al., 2004
3
j
1

Michaelis constant of
PIP3-mediated
phosphorylation of
µM 0.1 0.1 Giri et al., 2004
[ ]
[ ]
aMDMkv
MDMkv
kv
2
2
1212
1111

1010
=
=
=

42
Item KP Description Units
Chosen
Value
Range Refs
AKT
4
k
m1

Dephosphorylation of
AKTa
µM/
min
0.2
0.0000297
to 2.92
Qiu et al., 2004;
Kholodenko, 2000
5
j
m1

Michaelis constant of
dephosphorylation of

AKTa
µM 0.1 0.1 Giri et al., 2004
6
k
2

MDM2-dependent
degradation of p53
/min 0.055
0.0184 to
0.092
Ma et al., 2005
7
j
2

Michaelis constant of
MDM2-dependent
degradation of p53
µM 0.1 0.03 to 0.3 Ma et al., 2005
8
k
3

p53-dependent
production of PTEN
µM/
min
0.006 0.006 Stambolic et al., 2001
9

j
3

Dissociation constant
of p53-dependent
production of PTEN
µM 2 > 1 Stambolic et al., 2001
10
k
4

Phosphorylation of
PIP2
µM/
min
0.15 0.15 Kholodenko, 2000
11
j
4

Michaelis constant of
phosphorylation of
PIP2
µM 0.1 0.1 Giri et al., 2004
12
k
m4

PTEN
dephosphorylation of

PIP3
/min 73
42.1, 73 ±
4.4
Giri et al., 2004;
McConnachie et al.,
2003
13
j
m4

Michaelis constant of
PTEN
dephosphorylation of
PIP3
µM 0.5 0.1 to 1
Giri et al., 2004;
Georgescu et al., 1999;
Vazquez et al., 2000
14
k
5

p53-dependent
production of MDM2
µM/
min
0.024 0.024 Ma et al., 2005
15
j

5

Dissociation constant
of p53-dependent
production of MDM2
µM 1 ~1 Ma et al., 2005
16
k
6

AKTa
phosphorylation of
MDM2
/min 10
0.42 to
64.8
Hoffmann et al., 2002;
Qiu et al., 2004;
Schoeber et al. 2002;
Markevich et al., 2005;
Kholodenko, 2000;
Giri et al., 2004
17
j
6

Michaelis constant of
AKTa
phosphorylation of
MDM2

µM 0.3
0.00357 to
146
Hoffmann et al., 2002;
Qiu et al., 2004;
Schoeber et al. 2002;
Markevich et al., 2005;
Kholodenko, 2000;
Giri et al., 2004
18
k
m6

Dephosphorylation of
MDM2a
µM/
min
0.2
0.0000297
to 2.92
Qiu et al., 2004;
Kholodenko, 2000
19
j
m6

Michaelis constant of
dephosphorylation of
MDM2a
µM 0.1

0.00238 to
2.23
Qiu et al., 2004;
Schoeber et al. 2002;
Markevich et al., 2005;
Kholodenko, 2000;
Giri et al., 2004
20
k
7

Inactivation of active
p53 (degradation or
dephosphorylation)
/min 0.05
0.02 to
0.2, 0.05
Ma et al., 2005;
Zhou et al., 2001;
Bar-Or et al., 2000
21
k
8

Basal induction of
PTEN
µM/
min
0.001 Unknown

43
Item KP Description Units
Chosen
Value
Range Refs
22
k
9
Degradation of PTEN /min 0.0054
0.0025 to
0.0083
Georgescu et al., 1999
23
k
10

Basal induction of
MDM2
µM/
min
0.018 ~0.018 Ma et al., 2005
24
k
11

Degradation of
MDM2
/min 0.015
0.0028,
0.0347

Ma et al., 2005;
Bar-Or et al., 2000
25
k
12

Degradation of
MDM2a
/min 0.015
0.0028,
0.0347
Ma et al., 2005;
Bar-Or et al., 2000
26
n1
Hill coefficient of
p53-dependent
production of PTEN
- 3 3 Ciliberto et al., 2005
27
n2
Hill coefficient of
p53-dependent
production of MDM2
- 3 3 Ciliberto et al., 2005
28
[PIP
T
]
Sum of [PIP2] and

[PIP3]
µM 1 Arbitrary


3.3 Analyses of kinetic models

The biological plausible steady states and the stability of each steady state manifested
by Model M1 are determined (Section 3.3.1). Subsequently, Model M1 is simplified
sequentially to kinetic models with decreasing mechanistic details between p53 and
AKT interaction to determine whether the bistability phenomenon predicted by Model
M1 is still conserved (Section 3.3.2).


3.3.1 Steady states of Model M1

A system achieves steady state when the rate of change of every component is zero.
Therefore, to determine steady states, the LHS of the kinetic equations in Eqn (3-2)
are all set to zero. For each value of [AKT
T
] specified, steady state concentrations of
every part list is determined by solving the corresponding systems of nonlinear

44
algebraic equations numerically using Maple (version 7.0). Steady states that do not
meet the following biological conditions are rejected: [P
i
] > 0, [AKTa] < [AKT
T
] and
[PIP3] < [PIP

T
]. The resulting biological plausible steady states as functions of
control parameters such as [AKT
T
] are referred to as steady-state bifurcation
diagrams. For examples, steady-state bifurcation diagrams of p53 and AKTa are
depicted in Figure 3-2; bifurcation diagrams of PTEN and MDM2 are qualitatively
similar to p53 whereas those of PIP3 and MDM2a are qualitatively similar to AKTa
(data not shown).









Figure 3-2. Model M1: steady-state bifurcation diagrams of p53 and AKTa.
The steady state values of p53 ([p53
ss
], gray curve) and AKTa ([AKTa
ss
], black curve) are
plotted as functions of [AKT
T
] (the abscissa). There is a range of [AKT
T
] where three steady
states of p53 and AKTa coexist, and is referred to as the bistable range or hysteresis. Units

are in µM.


A non-intuitive feature of the steady-state bifurcation diagrams is the existence
of a range of [AKT
T
] where three steady states of AKTa and of p53 coexist. This
range is referred to as the bistable range. The local stability of the steady states in the
bistable range is determined using standard linear stability analysis (as described in
[AKT
T
]
[p53
ss
] [AKTa
ss
]
[p53
ss
]
[AKTa
ss
]

45
Appendix A-3), which involves determining the sign of the eigenvalue of each steady
state, as illustrated in Appendix A-4. Results from the linear stability analysis
indicate that all the steady states are stable (i.e., all eigenvalues have negative real
parts) except the middle steady states (at least one eigenvalue has positive real part).
The stable states are termed as stable nodes whereas the unstable states are termed as

saddle nodes. The system could nevertheless rest at the saddle node perpetually
provided that the concentration of all the part lists remains fixed at their respective
saddle node values. However, random fluctuations in the concentration of any part
list will cause the system to deflect away from the saddle node, as illustrated in Figure
3-3. To demonstrate this, a time-course analysis is performed.








Figure 3-3. Model M1: initial state of the system determines its steady state behaviors.
The arrows in the diagram indicate the state at which the system will rest at steady state. For
instance, in the bistable region, an initial state above the saddle node (middle branch of the
curve) will cause the system to get attracted to the upper stable node, and vice versa.



Two initial states within the bistable range are selected for time-course
analysis (as described in Appendix A-5), indicated as A and B in Figure 3-3. As
expected, Figure 3-4 shows that the system is attracted to the upper p53 and lower
AKTa state when it starts from state A (red curves) whereas it is attracted to the lower
[AKT
T
]
[AKTa
ss
]

A

B

[AKT
T
]
[p53
ss
]
A

B


46
p53 and upper AKTa state when it starts from state B (blue curves). In fact, the
model predicts a situation at steady state where either p53 is “on” and AKTa is “off”
or vice versa, depending on which protein happens to have the upper hand. In other
words, high-p53 steady state is associated with low-AKTa steady state, and vice
versa.







Figure 3-4. Model M1: time-courses of [p53] and [AKTa].
Time-courses of [p53] (left) and [AKTa] (right) starting from two initial states A and B,

which lie in the bistable region of the steady-state bifurcation curves shown in Figure 3-3, are
shown. The red curves corresponds to initial state A that attracts the system to the high-p53
and low-AKTa state while the blue curves corresponds to initial state B that attracts the
system to the low-p53 and high-AKTa state.


As p53 promotes but AKT inhibits apoptosis, the high-p53 and low-AKTa
state is deduced as a pro-apoptotic state whereas the low-p53 and high-AKTa state is
deduced as a pro-survival state. This is supported by experimental observations in
which a high level of p53 is predisposed to trigger apoptosis (Laptenko and Prives,
2006). Therefore, the system could either be at the pro-apoptotic or pro-survival state
in the bistable region. Interestingly, as [AKT
T
] increases, the middle steady state
branch of p53 increases whereas those of AKTa decreases (Figure 3-2). This
indicates that more amounts of p53 and AKTa must be simultaneously upregulated
and downregulated respectively to switch the system from pro-survival to pro-
[AKT
a
]

Time
(min)

[p53]
Time
(min)


47

apoptotic state; in other words, increasing [AKT
T
] in the bistable region augments the
cell’s resistance to death. On the other hand, the system can only be at the pro-
apoptotic state before the bistable region or at the pro-survival state after the bistable
region.


3.3.2 Hierarchy of kinetic models

One of the key questions asked in this work is how robust the bistability predicted by
Model M1 is. The approach to answer this question is to determine whether the
bistability phenomenon is conserved firstly, as Model M1 is simplified sequentially to
kinetic models with decreasing mechanistic details between p53 and AKT interaction
and secondly, as the kinetic parameters used in the model are varied simultaneously
(see Section 3.4).

A hierarchy of kinetic models with decreasing degree of mechanistic details
from Model M1 is analyzed, as shown in Figure 3-5. Model M2 is obtained by
simplifying Model M1. Likewise, Model M3 is obtained by simplifying Model M2.
The formulations and analyses of Models M2 and M3 are described as follows.







48

Figure 3-5. A hierarchy of kinetic models analyzed.
A hierarchy of kinetic models with decreasing degree of mechanistic details from Model M1
is analyzed to illustrate the conservation of bistability of the p53-AKT network. First column
shows the qualitative network and second column gives the corresponding kinetic models.


Model M3
T

p53

p53
Model M1
Model M2
T
p53
p53
p53
p53

49
Model M2 is derived from Model M1 by removing PTEN and the PIP2-PIP3
phosphorylation-dephosphorylation cycle; accordingly, rate equations v
3
, v
4
, v
m4
, v
8

and v
9
are not used. In fact, the qualitative network of Model M2 is identical to the
model suggested by Gottlieb et al., 2002. The regulatory steps of PTEN and PIP3 are
then substituted with a direct p53 inhibition of AKT phosphorylation, represented by
the rate equation v
*
1
(Figure 3-5). The kinetic equations and the associated parameter
values in Model M1 are handed down to identical steps found in Model M2. The

kinetic and rate equations of Model M2 and the kinetic parameters used are given in
Appendix A-6. The steady-state bifurcation diagrams of p53 and AKTa are shown in
Figure 3-6, which is qualitatively similar to those of Model M1 (Figure 3-2). Thus,
the bistable property of the death-survival switch is conserved when one of the
phosphorylation-dephosphorylation cycles is removed.








Figure 3-6. Model M2: steady-state bifurcation diagrams of p53 and AKTa.
The steady state values of p53 ([p53
ss
], gray curve) and AKTa ([AKTa
ss
], black curve) are
plotted as functions of [AKT
T
] (the abscissa). Units are in µM.


In Model M3, the phosphorylation-dephosphorylation cycle of MDM2-
MDM2a is removed. This simplest model however, retains the AKT-AKTa cycle in
[AKT
T
]
[p53

ss
]
[AKTa
ss
]

50
addition to the mutual antagonism between p53 and AKT. The regulatory step of
MDM2a is subsequently substituted with a direct AKTa degradation of p53,
represented by the rate equation v
*
2
(Figure 3-5). Similarly, the kinetic equations and
the associated parameter values in Model M2 are handed down to identical steps
found in Model M3. The kinetic and rate equations of Model M3 and the kinetic
parameters used are given in Appendix A-7. Again, the bistable property of the
death-survival switch is conserved (Figure 3-7).








Figure 3-7. Model M3: steady-state bifurcation diagrams of p53 and AKTa.
The steady state values of p53 ([p53
ss
], gray curve) and AKTa ([AKTa
ss

], black curve) are
plotted as functions of [AKT
T
] (the abscissa). Units are in µM.



3.3.3 Phase portrait analysis of Model M3

As Model M3 is a 2-dimensional system (i.e., only two kinetic equations), its 2-
dimensional phase portrait, which depicts the system’s trajectories from different
initial states (Strogatz, 1994), can be shown clearly; phase portrait is indecipherable
for systems with higher than three dimensions. Trajectories from four representative
AKT*ss

p53ss

[AKT
T
]
[p53
ss
]
[AKTa
ss
]

51
initial states (blue diamonds) are depicted in the phase portrait shown in Figure 3-8;
[AKT

T
] is set as 1 µM, in the bistable region.












Figure 3-8. Model M3: phase portrait at [AKT
T
] = 1 µ
µµ
µM.
Trajectories from four representative initial states (blue diamonds), each lying in four regions
demarcated as basins of attraction 1, 2, 3 and 4, are depicted. The stable nodes are indicated
as black circles while the saddle node is shown as a red circle. The stable manifold separates
the plane into left (basins 1 and 2) and right (basins 3 and 4) regions whereas the unstable
manifold separates the plane into upper (basins 1 and 3) and lower (basins 2 and 4) regions.
The system is in the bistable region at [AKT
T
] = 1 µM.


The initial state plane can be divided into four basins of attraction demarcated

as 1, 2, 3 and 4 (Figure 3-8). The system gets attracted to the low-p53 and high-
AKTa stable node (black circle at upper left corner) when it starts from either basin 1
or 2, or to the high-p53 and low-AKTa stable node (black circle at lower right corner)
when it starts from either basin 3 or 4. Bifurcation of a saddle-node type is
distinguished by the stable and unstable manifolds, which intersect at the saddle node
(red circle). The stable manifold bifurcates the plane into left (basins 1 and 2) and
right regions (basins 3 and 4). Because each region is a sink for the stable node it
1

2

3

4

[p53]
[AKTa]

52
encompasses, a trajectory has to cross the stable manifold in order to switch the
system from one stable node to the other. If the system starts from a state along the
stable manifold, it will traverse along the manifold to the saddle node. However, any
perturbation to its trajectory will veer the system to either one of the two stable nodes.
The unstable manifold on the other hand, represents the shortest path to the stable
nodes, which bifurcates the plane into upper (basins 1 and 3) and lower regions
(basins 2 and 4).


3.4 Parameter sensitivity analysis

Here, sensitivity of the bistability phenomenon predicted by Models M1 to M3
against variations in the values of kinetic parameters used in the kinetic models is
investigated, firstly, by a brute-force method of varying parameters over biologically
reasonable ranges and, secondly, by mapping phase diagrams in parameter space.
Results of the parameter sensitivity analyses are described in the order of Models M1,
M2 and M3 below. Generally, the bistability phenomenon is conserved when values
of kinetic parameters are varied in each of the kinetic model. In fact, this result agrees
with the report by Gutenkunst et al. (2007) that most dynamic models are insensitive
to parameter values, and that efforts should be focused more on making predictions
rather than on estimating the parameter values.




53
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 KP Base value Variation range Remarks
1
k
0
0.1 0.08 - 0.12 Estimated
2
k
1
20 16 - 24 Estimated
3
j
1
0.1 0.08 - 0.12 Estimated
4
k
m1
0.2 0.16 - 0.24 Estimated
5
j
m1
0.1 0.08 - 0.12 Estimated
6
k

2
0.055 0.044 - 0.066 Estimated
7
j
2
0.1 0.08 - 0.12 Estimated
8
k
3
0.006 0.25*k
5
Direct experimental data
9
j
3
2 1.6 - 2.4 Estimated
10
k
4
0.15 0.12 - 0.18 Estimated
11
j
4
0.1 0.08 - 0.12 Estimated
12
k
m4
73 - Direct experimental data
13
j

m4
0.5 0.4 - 0.6 Estimated

54
Item KP Base value Variation range Remarks
14
k
5
0.024 0.0192 - 0.0288
Estimated
15
j
5
1 0.8 - 1.2
Estimated
16
k
6
10 8 – 12
Estimated
17
j
6
0.3 0.24 - 0.36
Estimated
18
k
m6
0.2 0.16 - 0.24
Estimated

19
j
m6
0.1 0.08 - 0.12
Estimated
20
k
7
0.05 - Direct experimental data
21
k
8
0.001 0.0005 - 0.0015 Arbitrary (varied +/- 50%)
22
k
9
0.0054 - Direct experimental data
23
k
10
0.018 0.0144 - 0.0216
Estimated
24
k
11
0.015 0.012 - 0.018
Estimated
25
k
12

0.015 0.012 - 0.018
Estimated
26
[PIP
T
] 1 0.5, 2 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 [PIP
T
], 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 [PIP
T
] =
0.5, 1 and 2 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.






55

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. [PIP
T
] is fixed for each sample run. For the three different values (0.5,
1 and 2 µM) of [PIP
T
], the simulations are repeated using the same Latin sets. For each
[PIP
T
], 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 [PIP
T
] = 1 (Figure 3-9). Conservation of
bistability is reduced by 5.9% when [PIP
T
] is decreased by 50% whereas it is
increased by 2.5% when [PIP
T
] is increased by 100%. The quantity of PIP
T
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.

Number of Latin sets of
kinetic parameters that exhibit bistability

56
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

57
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 p53-
MDM2 negative feedback loop to attenuate the strength of p53.

To counter the strength of p53 antagonism on AKT, k
1
(phosphorylation rate
constant for AKT) is varied simultaneous with k
*
2
. To show the extent of bistable
regions in the k
1
-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; 1 state implies monostability). As shown in Figure 3-10B,
there exists a biologically reasonable bistable region in k
1
-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).