Mathematical and computational analysis of intracelluar dynamics 3

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 th

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 mediated pro-apoptotic cellular states within the first hour after oxidative stress

p53-3.1 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)

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 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

MDM2-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

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)

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

P

d[ 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 Pi, 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 v0 – v 2 – v 7 (refer to Figure 3-1 for the assignment of rate term), where v0 is rate of transcriptionally-active p53 synthesis and activation, v2 is rate of p53 degradation by MDM2a and 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)

6 11 6 10 5

12 6 6

9 8 3

4 4

1 1

7 2 0

]2[

]2[

][

]3[

][

]53[

v v v v v dt

MDM d

v v v dt

a MDM d

v v v dt

PTEN d

v v dt

PIP d

v v dt

AKTa d

v v v dt

p d

m m

m m

−

−++

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

To minimize the number of kinetic equations in Model M1, kinetic equations are not assigned to AKT and PIP2 Instead, dynamics of AKT and PIP2 are obtained respectively by [AKT] = [AKTT] – [AKTa] and [PIP2] = [PIPT] – [PIP3] [AKTT] and [PIPT] denotes the respective intracellular concentrations of total AKT (AKT and

AKTa) and PIP (PIP2 and PIP3), which can be assumed to be approximately constant

within the time scale of the phosphorylation and dephosphorylation processes involved in the respective activations of AKT and PIP2 This is because these processes occur relatively faster than the transcriptional, translational and degradation processes in the model Furthermore, this assumption is supported by experimental observations in which [AKTT] remains relatively constant after irradiation or

treatment with chemotherapeutic drugs even as [AKTa] decreases drastically (Gottlieb et al., 2002; Martelli et al., 2003)

The next step is to derive mathematical expressions for each rate term As information about the biological mechanisms involved in the reaction steps, which is

a requisite to derive the rate expressions, are often lacking, general rate expressions are used Hence, such a kinetic model is referred to as ‘abstract’ in the sense that the essential qualitative dynamics are captured by simple mathematical functions Biologically similar reaction steps are assumed to have similar rate expressions, as described below:

1 Enzymatic phosphorylation (v1, v4 and v6), dephosphorylation (vm1, vm4 and

v m6) and degradation (v2) reactions are assumed to have Michaelis-Menten

type expressions given by [ ][ ]

[ ]S j

S E k v

r

r r

+

= , where kr and jr are kinetic

parameters (or constants) that denote the maximal rate of reaction and the

Michaelis constant, respectively jr quantifies the binding affinity between

substrate (S) and enzyme (E) in which a large Michaelis constant indicates low binding affinity, and vice versa In the case whereby E is neither known nor

modeled explicitly, kr absorbs the variable [E], i.e., [ ]

[ ]S j

S k v

r

r r

r

n r r

p j

p k v

53

53+

= , where kr and jr are kinetic parameters

that denote the maximal rate of production and the dissociation constant

respectively, and n is the Hill coefficient A large dissociation constant

indicates low p53 binding affinity to the target gene DNA promoter site, and vice versa The Hill coefficient, on the other hand, represents the degree of cooperativity of a reaction; a reaction is described as being noncooperative

when n = 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 (v2, v9, v11 and v12) are assumed to have first-order

reaction kinetics type expression give by v = r k r[ ]P i , where kr is the rate

constant of the reaction and Pi represents the part list undergoing degradation

self-4 Basal production (v0, v8 and v10) of a part list is assumed to have a constant rate, i.e., vr = k r

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

k v

p k v

a MDM j

a MDM k

v

MDM j

MDM AKTa

k v

p j

p k v

PIP j

PIP PTEN k

v

PIP PIP

j

PIP PIP

k PIP j

PIP k v

p j

p k v

p j

p a MDM k

v

AKTa j

AKTa k

v

AKTa AKT

j

AKTa AKT

PIP k AKT j

AKT PIP k v

k v

m

m m

n n

n m

m m

T T

n n

n

m

m m

T T o

9 9

8 8

7 7

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

22

22

5353

33

3

32

2

5353

53

532

33

−

=+

−

=+

=

=

(3-3)

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

[MDM a]

k v

MDM k

v

k v

2

2

12 12

11 11

10 10

=

=

=

Item KP Description Units Chosen

Value Range Refs

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 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 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;

µM 0.3 0.00357 to

146

Hoffmann et al., 2002; Qiu et al., 2004; Schoeber et al 2002; Markevich et al., 2005;

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

Item KP Description Units Chosen

Value Range Refs

22 k 9 Degradation of PTEN /min 0.0054 0.0025 to

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 [PIPT] 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)

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 [AKTT] specified, steady state concentrations of every part list is determined by solving the corresponding systems of nonlinear

algebraic equations numerically using Maple (version 7.0) Steady states that do not meet the following biological conditions are rejected: [Pi] > 0, [AKTa] < [AKTT] and

[PIP3] < [PIPT] The resulting biological plausible steady states as functions of control parameters such as [AKTT] 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 ([p53ss], gray curve) and AKTa ([AKTass], black curve) are plotted as functions of [AKTT] (the abscissa) There is a range of [AKTT] 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 [AKTT ] 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

[AKTT ]

[p53ss] [AKTa ss ]

[p53ss]

[AKTa ss ]

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

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 [AKTT] 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 survival to

pro-[AKTa]

Time (min)

[p53]

Time (min)

apoptotic state; in other words, increasing [AKTT] 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

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

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 M2 is derived from Model M1 by removing PTEN and the PIP2-PIP3

phosphorylation-dephosphorylation cycle; accordingly, rate equations v3, v4, vm4, v8 and v9 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 ([p53ss], gray curve) and AKTa ([AKTass], black curve) are plotted as functions of [AKTT] (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

[AKTT ] [p53ss] [AKTa ss ]