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Introduction Model formulation Results Conclusion
Mathematical modeling of apoptosis: thedeath-inducing signaling complex
Kenneth L. Ho
Courant Institute, New York University
NYU Biomathematics Seminar (2009 Feb 3)
Introduction Model formulation Results Conclusion
What is apoptosis?
Apoptosis is a conserved, highly regulated cell death program inmulticellular organisms.
Involved in many physiological processes
Dysregulation associated with pathological conditions
Characteristic cell death morphology
Introduction Model formulation Results Conclusion
What is apoptosis?
Apoptosis is a conserved, highly regulated cell death program inmulticellular organisms.
Involved in many physiological processes
Dysregulation associated with pathological conditions
Characteristic cell death morphology
Introduction Model formulation Results Conclusion
What is apoptosis?
Apoptosis is a conserved, highly regulated cell death program inmulticellular organisms.
Involved in many physiological processes
Dysregulation associated with pathological conditions
Characteristic cell death morphology
Introduction Model formulation Results Conclusion
What is apoptosis?
Apoptosis is a conserved, highly regulated cell death program inmulticellular organisms.
Involved in many physiological processes
Dysregulation associated with pathological conditions
Characteristic cell death morphology
Introduction Model formulation Results Conclusion
Biochemistry of apoptosis
Hengartner (2000)
Introduction Model formulation Results Conclusion
Motivation for a detailed model
Previous models of apoptosis
Complete but unstable
Harrington et al. (2008)Hua et al. (2005)Okazaki et al. (2008)
Stable but incomplete
Bagci et al. (2006)Eissing et al. (2004)Legewie et al. (2006)
How to achieve a complete and stable model?
1 Mechanistic description of oligomerization
2 Inclusion of inhibitors (IAP, BAR, cFLIP)
3 Synthesis and degradation
Introduction Model formulation Results Conclusion
Motivation for a detailed model
Previous models of apoptosis
Complete but unstable
Harrington et al. (2008)Hua et al. (2005)Okazaki et al. (2008)
Stable but incomplete
Bagci et al. (2006)Eissing et al. (2004)Legewie et al. (2006)
How to achieve a complete and stable model?
1 Mechanistic description of oligomerization
2 Inclusion of inhibitors (IAP, BAR, cFLIP)
3 Synthesis and degradation
Introduction Model formulation Results Conclusion
Role of cFLIP: a toy model
Toy model:
Reactions:
DISC + cFLIPk1−−⇀↽−−k−1
DISC:cFLIP,
DISC + Casp8k2−→ DISC + Casp8∗,
∅ α−⇀↽−β
Casp8 , Casp8∗γ−→ ∅
Steady state:
[Casp8∗]ss =α/γ
1 + β/ (k2 [DISC]ss),
Introduction Model formulation Results Conclusion
Role of cFLIP: a toy model
10−2
100
102
0
10
20
30
[DISC]0 (nM)
[Cas
p8*]
ss (
nM)
[cFLIP]0 = 81 nM
0 50 100 150 2000
5
10
15
[cFLIP]0 (nM)
[Cas
p8*]
ss (
nM)
[DISC]0 = 1 nM
[DISC]0 (nM)
[cF
LIP
] 0 (nM
)
[Casp8*]ss
(nM)
10−2
10−1
100
101
102
0
50
100
150
200
5
10
15
20
cFLIP
no cFLIP
B
C
A
Introduction Model formulation Results Conclusion
Goals
Focus on the extrinsic pathway of apoptosis
DISC formation and its downstream interactions
Goals1 To construct a biologically relevant model
Bistability at physiological parametersMeaningful input-output map (FasL→ Casp3∗)
2 To study the function of cFLIP
How does it affect the apoptotic threshold?Does it confer robustness to the system?
Introduction Model formulation Results Conclusion
Goals
Focus on the extrinsic pathway of apoptosis
DISC formation and its downstream interactions
Goals1 To construct a biologically relevant model
Bistability at physiological parametersMeaningful input-output map (FasL→ Casp3∗)
2 To study the function of cFLIP
How does it affect the apoptotic threshold?Does it confer robustness to the system?
Introduction Model formulation Results Conclusion
DISC model
Important features
1 Fas trimerization
2 Stepwise recruitment
3 Regulation by cFLIP
4 Pairwise Casp8 activation
Reactions [with (i , j , k, l) ≡FasL:Fasi :FADDj :cFLIPk :Casp8l ]:
(i , j , k, l) + Fas −−−⇀↽−−− (i + 1, j , k, l)
(i , j , k, l) + FADD −−−⇀↽−−− (i , j + 1, k, l)
(i , j , k, l) + cFLIP −−−⇀↽−−− (i , j , k + 1, l)
(i , j , k, l) + Casp8 −−−⇀↽−−− (i , j , k, l + 1)
(i , j , k, l) −−→ (i , j , k, l − 2) + 2Casp8∗
Model parameters(δ, σcFLIP, σCasp8):
1 Death domain clustering(δ = 1, 2, 3)
2 cFLIP stoichiometry(σcFLIP = 1, 2, 3)
3 Casp8 stoichiometry(σCasp8 = 2, 3)
Introduction Model formulation Results Conclusion
DISC model
Important features
1 Fas trimerization
2 Stepwise recruitment
3 Regulation by cFLIP
4 Pairwise Casp8 activation
Reactions [with (i , j , k, l) ≡FasL:Fasi :FADDj :cFLIPk :Casp8l ]:
(i , j , k, l) + Fas −−−⇀↽−−− (i + 1, j , k, l)
(i , j , k, l) + FADD −−−⇀↽−−− (i , j + 1, k, l)
(i , j , k, l) + cFLIP −−−⇀↽−−− (i , j , k + 1, l)
(i , j , k, l) + Casp8 −−−⇀↽−−− (i , j , k, l + 1)
(i , j , k, l) −−→ (i , j , k, l − 2) + 2Casp8∗
Model parameters(δ, σcFLIP, σCasp8):
1 Death domain clustering(δ = 1, 2, 3)
2 cFLIP stoichiometry(σcFLIP = 1, 2, 3)
3 Casp8 stoichiometry(σCasp8 = 2, 3)
Introduction Model formulation Results Conclusion
DISC model
Important features
1 Fas trimerization
2 Stepwise recruitment
3 Regulation by cFLIP
4 Pairwise Casp8 activation
Reactions [with (i , j , k, l) ≡FasL:Fasi :FADDj :cFLIPk :Casp8l ]:
(i , j , k, l) + Fas −−−⇀↽−−− (i + 1, j , k, l)
(i , j , k, l) + FADD −−−⇀↽−−− (i , j + 1, k, l)
(i , j , k, l) + cFLIP −−−⇀↽−−− (i , j , k + 1, l)
(i , j , k, l) + Casp8 −−−⇀↽−−− (i , j , k, l + 1)
(i , j , k, l) −−→ (i , j , k, l − 2) + 2Casp8∗
Model parameters(δ, σcFLIP, σCasp8):
1 Death domain clustering(δ = 1, 2, 3)
2 cFLIP stoichiometry(σcFLIP = 1, 2, 3)
3 Casp8 stoichiometry(σCasp8 = 2, 3)
Introduction Model formulation Results Conclusion
Extrinsic caspase model
Couple to DISC model, map from Casp8→ Casp3:
Reactions:
Casp8∗ + Casp3 −−→ Casp8∗ + Casp3∗
Casp3∗ + Casp8 −−→ Casp3∗ + Casp8∗
Casp8∗ + BAR −−⇀↽−− Casp8∗:BAR
Casp3∗ + IAP −−⇀↽−− Casp3∗:IAP
Introduction Model formulation Results Conclusion
DISC model (3,3,3)
10−2
100
2
4
6
8
10
12
[FasL]0 (nM)
[Cas
p8*]
ss (
nM)
[cFLIP]0 = 81 nM
0 50 100 150 2000.5
1
1.5
2
2.5
[cFLIP]0 (nM)
[Cas
p8*]
ss (
nM)
[FasL]0 = 0.1 nM
[FasL]0 (nM)
[cF
LIP
] 0 (nM
)
[Casp8*]ss
(nM)
10−2
10−1
100
101
0
50
100
150
200
2
4
6
8
10
12
cFLIP
no cFLIP
C
A B
Introduction Model formulation Results Conclusion
Comparison of DISC models
2 4 6 8 10 12 14 16 1810
−2
10−1
100
101
Model variant number
[Fas
L]0 (
nM)
[Casp8*]ss
(nM) at [cFLIP]0 = 81 nM
2
4
6
8
10
2 4 6 8 10 12 14 16 180
50
100
150
200
Model variant number
[cF
LIP
] 0 (nM
)
[Casp8*]ss
(nM) at [FasL]0 = 0.1 nM
1
1.5
2
B
A
Introduction Model formulation Results Conclusion
Coupled extrinsic caspase model (3,3,3)
10−4
10−3
10−2
0
20
40
60
80
αFasL
(nM/min)
[Cas
p3*]
ss (
nM)
[cFLIP]0 = 81 nM
10−4
10−2
0
100
2000
50
100
αFasL
(nM/min)[cFLIP]0 (nM)
[Cas
p3*]
ss (
nM)
αFasL
(nM/min)
[Cas
p3*]
(nM
)
[cFLIP]0 = 81 nM
0 1 2 3 4
x 10−4
0
20
40
60
80
100
apoptosis
stable
unstable
αFasL
(nM/min)
[Cas
p3*]
(nM
)
[cFLIP]0 = 0 nM
0 1 2 3 4
x 10−4
0
20
40
60
80
100
apoptosis
stable
unstable
cFLIP
no cFLIP
A B
C D
Introduction Model formulation Results Conclusion
Apoptotic threshold
[cFLIP]0 (nM)
α Fas
L* (n
M/m
in)
0 20 40 60 80 100 120 140 160 180 2001
1.5
2
2.5
3
3.5
4
4.5x 10
−4
(<3,1,⋅)
(3,1,⋅)
(<3,2,⋅)
(3,2,⋅)
(<3,3,⋅)
(3,3,⋅)
2 4 6 8 10 12 14 16 180
2
4x 10
−4
Model variant number
α Fas
L* (n
M/m
in)
cFLIP
no cFLIP
B
A
Introduction Model formulation Results Conclusion
DISC model time courses
Time (min)
[Cas
p8*]
(nM
)
0 50 100 150 200 250 300 350 400 450 5000
2
4
6
8
10
12
σcFLIP
= 1
σcFLIP
> 1
[FasL]0 = 0.1 nM
[FasL]0 = 2 nM
Introduction Model formulation Results Conclusion
Robustness with FasL
β: minimum fractional inhibitor reduction for apoptosis
[IAP]0 7→ (1− β) [IAP]0 and [BAR]0 7→ (1− β) [BAR]0β is a measure of robustness
Introduction Model formulation Results Conclusion
Robustness with FasL
β: minimum fractional inhibitor reduction for apoptosis
[IAP]0 7→ (1− β) [IAP]0 and [BAR]0 7→ (1− β) [BAR]0β is a measure of robustness
αFasL
(nM/min)
β
10−6
10−5
10−4
10−3
0
0.05
0.1
0.15
0.2
0.25
0.3
0.35
0.4
σcFLIP
= 1
σcFLIP
> 1
no cFLIP
Introduction Model formulation Results Conclusion
Robustness with cFLIP
[cFLIP]0 (nM)
β
αFasL
= 5 × 10−5 nM/min
0 20 40 60 80 100 120 140 160 180 2000.12
0.14
0.16
0.18
0.2
0.22
0.24
0.26
(<3,1,⋅)
(3,1,⋅)
(<3,2,⋅)
(3,2,⋅)
(<3,3,⋅)
(3,3,⋅)
2 4 6 8 10 12 14 16 180
0.1
0.2
0.3
Model variant number
β
cFLIP
no cFLIP
A
B
Introduction Model formulation Results Conclusion
Conclusion
1 Model exhibits irreversible bistability
Mechanistic descriptionSynthesis and degradation
2 Only significant parameter is σcFLIP; otherwise, degenerate3 cFLIP protects cell survival and robustness (σcFLIP > 1)
Increases apoptotic thresholdMaintains robustness of bistability
But the only truly significant result is bistability; the rest is justminor details if we’re interested in bigger questions...
Introduction Model formulation Results Conclusion
Conclusion
1 Model exhibits irreversible bistability
Mechanistic descriptionSynthesis and degradation
2 Only significant parameter is σcFLIP; otherwise, degenerate
3 cFLIP protects cell survival and robustness (σcFLIP > 1)
Increases apoptotic thresholdMaintains robustness of bistability
But the only truly significant result is bistability; the rest is justminor details if we’re interested in bigger questions...
Introduction Model formulation Results Conclusion
Conclusion
1 Model exhibits irreversible bistability
Mechanistic descriptionSynthesis and degradation
2 Only significant parameter is σcFLIP; otherwise, degenerate3 cFLIP protects cell survival and robustness (σcFLIP > 1)
Increases apoptotic thresholdMaintains robustness of bistability
But the only truly significant result is bistability; the rest is justminor details if we’re interested in bigger questions...
Introduction Model formulation Results Conclusion
Conclusion
1 Model exhibits irreversible bistability
Mechanistic descriptionSynthesis and degradation
2 Only significant parameter is σcFLIP; otherwise, degenerate3 cFLIP protects cell survival and robustness (σcFLIP > 1)
Increases apoptotic thresholdMaintains robustness of bistability
But the only truly significant result is bistability; the rest is justminor details if we’re interested in bigger questions...
Introduction Model formulation Results Conclusion
Ideas!
Use the DISC model as a component in a larger model to achieve asystems-level description that is complete and stable
1 What are the determinants of cell survival or death?
Linear classifier: y = θ(∑
i wixi )SVD (principal components)GSVD as a comparison tool (Alter et al., 2003)
2 Similarly, study how bistability is maintained3 What controls the apoptotic threshold?
Linear regression
Any suggestions?
Introduction Model formulation Results Conclusion
Ideas!
Use the DISC model as a component in a larger model to achieve asystems-level description that is complete and stable
1 What are the determinants of cell survival or death?
Linear classifier: y = θ(∑
i wixi )SVD (principal components)GSVD as a comparison tool (Alter et al., 2003)
2 Similarly, study how bistability is maintained3 What controls the apoptotic threshold?
Linear regression
Any suggestions?
Introduction Model formulation Results Conclusion
Ideas!
Use the DISC model as a component in a larger model to achieve asystems-level description that is complete and stable
1 What are the determinants of cell survival or death?
Linear classifier: y = θ(∑
i wixi )SVD (principal components)GSVD as a comparison tool (Alter et al., 2003)
2 Similarly, study how bistability is maintained
3 What controls the apoptotic threshold?
Linear regression
Any suggestions?
Introduction Model formulation Results Conclusion
Ideas!
Use the DISC model as a component in a larger model to achieve asystems-level description that is complete and stable
1 What are the determinants of cell survival or death?
Linear classifier: y = θ(∑
i wixi )SVD (principal components)GSVD as a comparison tool (Alter et al., 2003)
2 Similarly, study how bistability is maintained3 What controls the apoptotic threshold?
Linear regression
Any suggestions?
Introduction Model formulation Results Conclusion
Ideas!
Use the DISC model as a component in a larger model to achieve asystems-level description that is complete and stable
1 What are the determinants of cell survival or death?
Linear classifier: y = θ(∑
i wixi )SVD (principal components)GSVD as a comparison tool (Alter et al., 2003)
2 Similarly, study how bistability is maintained3 What controls the apoptotic threshold?
Linear regression
Any suggestions?
Introduction Model formulation Results Conclusion
References
Alter O, Brown PO, Botstein D (2003) Generalized singular value decomposition for comparative analysis ofgenome-scale expression data sets of two different organisms. Proc Natl Acad Sci USA 100(6):3351–3356.
Bagci EZ, Vodovotz Y, Billiar TR, Ermentrout GB, Bahar I (2006) Bistability in Apoptosis: Roles of Bax,Bcl-2, and Mitochondrial Permeability Transition Pores. Biophys J 90(5):1546–1559.
Eissing T, Conzelmann H, Gilles ED, Allgower F, Bullinger E, Scheurich P (2004) Bistability Analyses of aCaspase Activation Model for Receptor-induced Apoptosis. J Biol Chem 279(35):36892–36897.
Harrington HA, Ho KL, Ghosh S, Tung KC (2008) Construction and analysis of a modular model ofcaspase activation in apoptosis. Theor Biol Med Model 5(1):26.
Hengartner MO (2000) The biochemistry of apoptosis. Nature 407(6805):770–776.
Hua F, Cornejo MG, Cardone MH, Stokes CL, Lauffenburger DA (2005) Effects of Bcl-2 Levels on FasSignaling-Induced Caspase-3 Activation: Molecular Genetic Tests of Computational Model Predictions. JImmunol 175(2):985–995.
Lai R, Jackson TL (2004) A Mathematical Model of Receptor-Mediated Apoptosis: Dying to Know WhyFasL is a Trimer. Math Biosci Eng 11(2):325–338.
Legewie S, Bluthgen N, Herzel H (2006) Mathematical Modeling Identifies Inhibitors of Apoptosis asMediators of Positive Feedback and Bistability. PLoS Comput Biol 2(9):e120.
Okazaki N, Asano R, Kinoshita T, Chuman H (2008) Simple computational models of type I/type II cellsin Fas signaling-induced apoptosis. J Theor Biol 250(4):621–633.
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