This document outlines a non-parametric approach to analyzing mass-action models and data using techniques from algebraic geometry. It discusses using Gröbner bases to perform variable elimination on dynamical equations describing chemical reaction networks, leaving invariants that must vanish at steady state. These invariants allow testing a model's validity by assessing how well they are satisfied by observed data, providing a way to perform parameter-free model selection and discrimination.

1. Non-parametric analysis of mass-action models and data
Heather Harrington
Theoretical Systems Biology
Imperial College London
May 8, 2012
Model checking, multistability, and spatial models Heather Harrington 1 / 40

2. Outline and collaborators
(1) Motivation
Michael Stumpf
Theoretical Systems Biology, Imperial College London
(2) Model checking using coplanarity
Kenneth Ho
Courant Institute of Mathematical Sciences, New York University
Thomas Thorne
Theoretical Systems Biology, Imperial College London
(3) Multistationarity via spatial compartmentalization
Elisenda Feliu
Institute of Mathematical Sciences, University of Copenhagen
Carsten Wiuf
Institute of Mathematical Sciences, University of Copenhagen
(4) Conclusions
Model checking, multistability, and spatial models Heather Harrington 2 / 40

3. Overview: Cell decisions
Cell decisions
Cellular decision-making is necessary for preservation of homeostasis in an
organism, e.g., apoptosis, proliferation, and differentiation.
Model checking, multistability, and spatial models Heather Harrington 3 / 40

4. Overview: Cell decisions
Cell decisions
Cellular decision-making is necessary for preservation of homeostasis in an
organism, e.g., apoptosis, proliferation, and differentiation.
Mechanisms that regulate these processes are often feedback loops.
Model checking, multistability, and spatial models Heather Harrington 3 / 40

5. Overview: Cell decisions
Cell decisions
Cellular decision-making is necessary for preservation of homeostasis in an
organism, e.g., apoptosis, proliferation, and differentiation.
Mechanisms that regulate these processes are often feedback loops.
Feedbacks can affect the behavior of the system (number of
response states).
Model checking, multistability, and spatial models Heather Harrington 3 / 40

6. Overview: Cell decisions
Cell decisions
Cellular decision-making is necessary for preservation of homeostasis in an
organism, e.g., apoptosis, proliferation, and differentiation.
Mechanisms that regulate these processes are often feedback loops.
Feedbacks can affect the behavior of the system (number of
response states).
Many models can be constructed to describe the same system.
Model checking, multistability, and spatial models Heather Harrington 3 / 40

7. Theoretical Systems Biology
Aims of the research group:
Reverse engineering
Inverse problems
Bayesian statistics
Model checking, multistability, and spatial models Heather Harrington 4 / 40

8. Statistical Inference
For any model, M(θ), we can infer the parameters in light of data. In
a statistical framework, for example, we use the likelihood
L(θ) = P(D|θ).
Maximizing the likelihood gives us the value of the parameter θ that
maximizes the probability of observing the data D.
Model checking, multistability, and spatial models Heather Harrington 5 / 40

9. Statistical Inference
For any model, M(θ), we can infer the parameters in light of data. In
a statistical framework, for example, we use the likelihood
L(θ) = P(D|θ).
Maximizing the likelihood gives us the value of the parameter θ that
maximizes the probability of observing the data D.
Model Selection
If, however, we have a set of candidate models, M1 , M2 , . . . we have
to employ other criteria to choose which model is best.
Model checking, multistability, and spatial models Heather Harrington 5 / 40

10. Statistical Inference
For any model, M(θ), we can infer the parameters in light of data. In
a statistical framework, for example, we use the likelihood
L(θ) = P(D|θ).
Maximizing the likelihood gives us the value of the parameter θ that
maximizes the probability of observing the data D.
Model Selection
If, however, we have a set of candidate models, M1 , M2 , . . . we have
to employ other criteria to choose which model is best.
The Akaike and Bayesian information criteria, for example, penalize
models that are overly complex.
Model checking, multistability, and spatial models Heather Harrington 5 / 40

11. Bayesian Inference
In the Bayesian framework, parameter inference centers around
finding the posterior distribution
P(D|θ)π(θ)
P(θ|D) = ,
P(D|θ)π(θ)dθ
where P(D|θ) is the likelihood and π(θ) is called the prior of θ.
Model checking, multistability, and spatial models Heather Harrington 6 / 40

12. Bayesian Inference
In the Bayesian framework, parameter inference centers around
finding the posterior distribution
P(D|θ)π(θ)
P(θ|D) = ,
P(D|θ)π(θ)dθ
where P(D|θ) is the likelihood and π(θ) is called the prior of θ.
For model selection, the key quantity is the Evidence (marginal
likelihood):
P(D|θ)π(θ)dθ,
which is calculated by integrating the likelihood over the parameter
space.
Given a set of models, we prefer the one for which the evidence is
the highest.
Model checking, multistability, and spatial models Heather Harrington 6 / 40

13. The Problem of Model Selection
In maximum likelihood estimation (or in optimization approaches
more generally) model selection needs to be addressed in an ad
hoc fashion.
Bayesian approaches integrate out parameter dependencies along
the way towards model selection.
In a Bayesian framework, model selection is natural but
computationally expensive: often prohibitively expensive.
Model checking, multistability, and spatial models Heather Harrington 7 / 40

14. The Problem of Model Selection
In maximum likelihood estimation (or in optimization approaches
more generally) model selection needs to be addressed in an ad
hoc fashion.
Bayesian approaches integrate out parameter dependencies along
the way towards model selection.
In a Bayesian framework, model selection is natural but
computationally expensive: often prohibitively expensive.
Can we do better? Can we do parameter-free model selection?
Model checking, multistability, and spatial models Heather Harrington 7 / 40

15. The Problem of Model Selection
In maximum likelihood estimation (or in optimization approaches
more generally) model selection needs to be addressed in an ad
hoc fashion.
Bayesian approaches integrate out parameter dependencies along
the way towards model selection.
In a Bayesian framework, model selection is natural but
computationally expensive: often prohibitively expensive.
Can we do better? Can we do parameter-free model selection?
We will try ...
Model checking, multistability, and spatial models Heather Harrington 7 / 40

16. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Model checking, multistability, and spatial models Heather Harrington 8 / 40

17. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Chemical reaction network:
N N
k
sij Xj −i
→ sij Xj , i = 1, . . . , R
j=1 j=1
Model checking, multistability, and spatial models Heather Harrington 8 / 40

18. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Chemical reaction network: Dynamics from mass action kinetics:
N N R N
k s
sij Xj −i
→ sij Xj , i = 1, . . . , R xi =
˙ kj sji − sji xj jk , i = 1, . . . , N
j=1 j=1 j=1 k=1
Model checking, multistability, and spatial models Heather Harrington 8 / 40

19. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Chemical reaction network: Dynamics from mass action kinetics:
N N R N
k s
sij Xj −i
→ sij Xj , i = 1, . . . , R xi =
˙ kj sji − sji xj jk , i = 1, . . . , N
j=1 j=1 j=1 k=1
These equations provide a quantitative description of the model.
Model checking, multistability, and spatial models Heather Harrington 8 / 40

20. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Chemical reaction network: Dynamics from mass action kinetics:
N N R N
k s
sij Xj −i
→ sij Xj , i = 1, . . . , R xi =
˙ kj sji − sji xj jk , i = 1, . . . , N
j=1 j=1 j=1 k=1
These equations provide a quantitative description of the model.
In principle, the equations can be used to test the model’s validity by
assessing the degree to which they are satisfied by observed data.
Model checking, multistability, and spatial models Heather Harrington 8 / 40

21. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Chemical reaction network: Dynamics from mass action kinetics:
N N R N
k s
sij Xj −i
→ sij Xj , i = 1, . . . , R xi =
˙ kj sji − sji xj jk , i = 1, . . . , N
j=1 j=1 j=1 k=1
These equations provide a quantitative description of the model.
In principle, the equations can be used to test the model’s validity by
assessing the degree to which they are satisfied by observed data.
However, in practice, the required variables are rarely available.
Model checking, multistability, and spatial models Heather Harrington 8 / 40

22. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Chemical reaction network: Dynamics from mass action kinetics:
N N R N
k s
sij Xj −i
→ sij Xj , i = 1, . . . , R xi =
˙ kj sji − sji xj jk , i = 1, . . . , N
j=1 j=1 j=1 k=1
These equations provide a quantitative description of the model.
In principle, the equations can be used to test the model’s validity by
assessing the degree to which they are satisfied by observed data.
However, in practice, the required variables are rarely available.
In particular the velocities x = (x1 , . . . , xN ) are difficult to measure, so we
˙ ˙ ˙
consider only the steady state x = 0.
˙
Model checking, multistability, and spatial models Heather Harrington 8 / 40

23. Background: Model selection using algebraic geometry
Techniques from algebraic geometry for model discrimination.
Using results from Manrai and Gunawardena (2008) Biophys J.
Chemical reaction network: Dynamics from mass action kinetics:
N N R N
k s
sij Xj −i
→ sij Xj , i = 1, . . . , R xi =
˙ kj sji − sji xj jk , i = 1, . . . , N
j=1 j=1 j=1 k=1
These equations provide a quantitative description of the model.
In principle, the equations can be used to test the model’s validity by
assessing the degree to which they are satisfied by observed data.
However, in practice, the required variables are rarely available.
In particular the velocities x = (x1 , . . . , xN ) are difficult to measure, so we
˙ ˙ ˙
consider only the steady state x = 0.
˙
We eliminate these variables from the equations if possible.
Model checking, multistability, and spatial models Heather Harrington 8 / 40

24. Background: tools from algebraic geometry
For simple systems, this elimination can be done by hand. But in
general, a more systematic approach is often required.
Model checking, multistability, and spatial models Heather Harrington 9 / 40

25. Background: tools from algebraic geometry
For simple systems, this elimination can be done by hand. But in
general, a more systematic approach is often required.
Gr¨bner basis nonlinear generalization of Gaussian elimination.
o
Model checking, multistability, and spatial models Heather Harrington 9 / 40

26. Background: tools from algebraic geometry
For simple systems, this elimination can be done by hand. But in
general, a more systematic approach is often required.
Gr¨bner basis nonlinear generalization of Gaussian elimination.
o
Elimination ideal allows us to perform elimination without having
to know the numerical values of the parameters a = (k1 , . . . , kR )
by treating them symbolically.
Model checking, multistability, and spatial models Heather Harrington 9 / 40

27. Background: tools from algebraic geometry
For simple systems, this elimination can be done by hand. But in
general, a more systematic approach is often required.
Gr¨bner basis nonlinear generalization of Gaussian elimination.
o
Elimination ideal allows us to perform elimination without having
to know the numerical values of the parameters a = (k1 , . . . , kR )
by treating them symbolically.
Gr¨bner bases automatically give equations that are fulfilled by any
o
steady-state solution and only involve a subset of variables.
Model checking, multistability, and spatial models Heather Harrington 9 / 40

28. Background: variable elimination and invariants
After variable elimination we are left with:
ni Nobs
t
Ii (xobs ; a) = fij (a) xkijk , i = 1, . . . , Ninv . (1)
j=1 k=1
Model checking, multistability, and spatial models Heather Harrington 10 / 40

29. Background: variable elimination and invariants
After variable elimination we are left with:
ni Nobs
t
Ii (xobs ; a) = fij (a) xkijk , i = 1, . . . , Ninv . (1)
j=1 k=1
Ii is a polynomial in xobs that vanishes at steady state.
Model checking, multistability, and spatial models Heather Harrington 10 / 40

30. Background: variable elimination and invariants
After variable elimination we are left with:
ni Nobs
t
Ii (xobs ; a) = fij (a) xkijk , i = 1, . . . , Ninv . (1)
j=1 k=1
Ii is a polynomial in xobs that vanishes at steady state.
We call the Ii steady-state invariants.
Model checking, multistability, and spatial models Heather Harrington 10 / 40

31. Background: variable elimination and invariants
After variable elimination we are left with:
ni Nobs
t
Ii (xobs ; a) = fij (a) xkijk , i = 1, . . . , Ninv . (1)
j=1 k=1
Ii is a polynomial in xobs that vanishes at steady state.
We call the Ii steady-state invariants.
Invariants of a model (if they exist) describe relationships between
observable variables that hold a steady state for any given
realization of parameter values, regardless of other factors (such as
initial conditions).
Model checking, multistability, and spatial models Heather Harrington 10 / 40

32. Model Model Model1.1. .Model 22 L 2
Model
Model 1 Model Model Model
1 Model 2 Model 1Model
1 2
x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
Assessing coplanarity: overview .
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...
xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...
Models
Models Observed Data DataData Data
Observed
Observed Data Data
Observed Observed
Observed
Calculate elimination ideal
Models (Steadymeasurements)
Calculate elimination ideal elimination. .ideal (Steadymeasurements)
Calculate (Steady state state state measurements)
(Steady measurements)
Model 1 Model. 2 . . . . Model L (Steady state measurements)
Model 1 . . (Steady state state measurements)
Calculate elimination ideal ... ...
Calculate elimination. .ideal..Model xx1 Model 2 ...... . . . . . . .
. . . . . . . x1 . .. . x11 ˆˆ
ˆ ˆ 1 1 . .x1ˆ.
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
Assess coplanarity ˆ ˆ . ˆˆ . .x2 ........ . . . .. .. . . . .
.
Assess coplanarity1x2 . . . x.22 . xx2
Assess coplanarity . . . . . .
. .
x =
˙ x
ˆ 2 ˆ
. . ... ... ... ...
Assess coplanarity
..
. ... . . . . .. .. .. ........ . . . .. .. . . . .
. .. .
Assess coplanarity .
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
ˆ ˆ m ˆˆ
x
ˆ m x
Reduce number ˙of = variables ... ... ... ...
Reduce number xN variables
Reduce number of variables of
Reduce number of variables
to include only observables
observables only observables
to include
to include onlyReduce number of variables
to include only observablesSteady statestate invariants
Steady state invariants Data SteadySteady
Steady state invariants state invariants
Steady state state invariants
Steady invariants
invariants
Observed
to include only observablesstate invariants
Steady
(Steady state measurements) of models
Characterize steady states
Characterize steady states of. .models states of models
Characterize steady
.
x1Characterize steady
ˆ
Calculate elimination ideal states of models
Characterize steady states of elimination ideal
. . . Calculate models
x2
ˆ 1 1
1
11 1
. Transform model variables,
. Transform .model variables, parameters, and data
..
Transform model variables,
AssessTransform model variables,
coplanarity . . . and data
parameters,
xm parameters, and data
ˆ Transform model variables,
parameters, and data Assess coplanarity
parameters, and data
parameters, and data
Steady state invariants
Data coplanar Data not coplanar
1
1
1
2
2
Data not coplanar
2
2
2
Data not coplanar Model compatible
Model compatible Model incompatible
Data coplanar 2 Data not coplanar 3
Model checking, multistability, and spatial models Heather Harrington 11 / 40
Model incompatible

33. Model Model Model1.1. .Model 22 L 2
Model
Model 1 Model Model Model
1 Model 2 Model 1Model
1 2
x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
Assessing coplanarity: overview .
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...
xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...
Models
Models Observed Data DataData Data
Observed
Observed Data Data
Observed Observed
Observed
Calculate elimination ideal
Models (Steadymeasurements)
Calculate elimination ideal elimination. .ideal (Steadymeasurements)
Calculate (Steady state state state measurements)
(Steady measurements)
Model 1 Model. 2 . . . . Model L (Steady state measurements)
. . (Steady state state measurements)
Model 1
Calculate elimination ideal
Calculate elimination. .ideal..Model xx1 Model 2 ...... . . . . . . .
. . . . . . . x1 . .. . x11 ˆˆ
ˆ ˆ 1 1 . .x1ˆ. ... ... We are interested in how to
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
Assess coplanarity ˆ ˆ . ˆˆ . .x2 ........ . . . .. .. . . . .
Assess coplanarity1x2 . . . x.22 . xx2
Assess coplanarity . . . . . .
. .
. .
x =
˙
... ... ... ...
Assess coplanarity .
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
check if models and data are
Assess coplanarity ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
Reduce number ˙of =
Reduce number xN variables
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
Reduce number of variables
Reduce number of variables
to include only observables
observables only observables
to include
to include onlyReduce number of variables
to include only observablesSteady statestate invariants
Steady state invariants Data SteadySteady
Steady state invariants state invariants
Steady state state invariants
Steady invariants
invariants
Observed
to include only observablesstate invariants
Steady
(Steady state measurements) of models
Characterize steady states
Characterize steady states of. .models states of models
Characterize steady
.
x1Characterize steady
ˆ
Calculate elimination ideal states of models
Characterize steady states of elimination ideal
. . . Calculate models
x2
ˆ 1 1
1
11 1
. Transform model variables,
. Transform .model variables, parameters, and data
..
Transform model variables,
AssessTransform model variables,
coplanarity . . . and data
parameters,
xm parameters, and data
ˆ Transform model variables,
parameters, and data Assess coplanarity
parameters, and data
parameters, and data
Steady state invariants
Data coplanar Data not coplanar
1
1
1
2
2
Data not coplanar
2
2
2
Data not coplanar Model compatible
Model compatible Model incompatible
Data coplanar 2 Data not coplanar 3
Model checking, multistability, and spatial models Heather Harrington 11 / 40
Model incompatible

34. Model Model Model1.1. .Model 22 L 2
Model
Model 1 Model Model Model
1 Model 2 Model 1Model
1 2
x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
Assessing coplanarity: overview .
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...
xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...
Models
Models Observed Data DataData Data
Observed
Observed Data Data
Observed Observed
Observed
Calculate elimination ideal
Models (Steadymeasurements)
Calculate elimination ideal elimination. .ideal (Steadymeasurements)
Calculate (Steady state state state measurements)
(Steady measurements)
Model 1 Model. 2 . . . . Model L (Steady state measurements)
. . (Steady state state measurements)
Model 1
Calculate elimination ideal
Calculate elimination. .ideal..Model xx1 Model 2 ...... . . . . . . .
. . . . . . . x1 . .. . x11 ˆˆ
ˆ ˆ 1 1 . .x1ˆ. ... ... We are interested in how to
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
Assess coplanarity ˆ ˆ . ˆˆ . .x2 ........ . . . .. .. . . . .
Assess coplanarity1x2 . . . x.22 . xx2
Assess coplanarity . . . . . .
. .
. .
x =
˙
... ... ... ...
Assess coplanarity .
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
check if models and data are
Assess coplanarity ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
Reduce number ˙of =
Reduce number xN variables
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
Reduce number of variables
Reduce number of variables
to include only observables
observables only observables
to include
to include onlyReduce number of variables
to include only observablesSteady statestate invariants
Steady state invariants Data
Observed SteadySteady
Steady state invariants state invariants
Steady state state invariants
Steady invariants
invariants
Assess if the invariants and
to include only observablesstate invariants
Steady
(Steady state measurements) of models
Characterize steady states data, when transformed, lie on
Characterize steady states of. .models states of models
Characterize steady
.
x1Characterize steady
ˆ
Calculate elimination ideal states of models
x2
ˆ
Characterize steady states of elimination ideal
. . . Calculate models 1 1
1
11 1
a common plane.
. Transform model variables,
. Transform .model variables, parameters, and data
..
Transform model variables,
AssessTransform model variables,
coplanarity . . . and data
parameters,
xm parameters, and data
ˆ Transform model variables,
parameters, and data Assess coplanarity
parameters, and data
parameters, and data
Steady state invariants
Data coplanar Data not coplanar
1
1
1
2
2
Data not coplanar
2
2
2
Data not coplanar Model compatible
Model compatible Model incompatible
Data coplanar 2 Data not coplanar 3
Model checking, multistability, and spatial models Heather Harrington 11 / 40
Model incompatible

35. Model Model Model1.1. .Model 22 L 2
Model
Model 1 Model Model Model
1 Model 2 Model 1Model
1 2
x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
Assessing coplanarity: overview .
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...
xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...
Models
Models Observed Data DataData Data
Observed
Observed Data Data
Observed Observed
Observed
Calculate elimination ideal
Models (Steadymeasurements)
Calculate elimination ideal elimination. .ideal (Steadymeasurements)
Calculate (Steady state state state measurements)
(Steady measurements)
Model 1 Model. 2 . . . . Model L (Steady state measurements)
. . (Steady state state measurements)
Model 1
Calculate elimination ideal
Calculate elimination. .ideal..Model xx1 Model 2 ...... . . . . . . .
. . . . . . . x1 . .. . x11 ˆˆ
ˆ ˆ 1 1 . .x1ˆ. ... ... We are interested in how to
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
Assess coplanarity ˆ ˆ . ˆˆ . .x2 ........ . . . .. .. . . . .
Assess coplanarity1x2 . . . x.22 . xx2
Assess coplanarity . . . . . .
. .
. .
x =
˙
... ... ... ...
Assess coplanarity .
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
check if models and data are
Assess coplanarity ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
Reduce number ˙of =
Reduce number xN variables
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
Reduce number of variables
Reduce number of variables
to include only observables
observables only observables
to include
to include onlyReduce number of variables
to include only observablesSteady statestate invariants
Steady state invariants Data
Observed SteadySteady
Steady state invariants state invariants
Steady state state invariants
Steady invariants
invariants
Assess if the invariants and
to include only observablesstate invariants
Steady
(Steady state measurements) of models
Characterize steady states data, when transformed, lie on
Characterize steady states of. .models states of models
Characterize steady
.
x1Characterize steady
ˆ
Calculate elimination ideal states of models
x2
ˆ
Characterize steady states of elimination ideal
. . . Calculate models 1 1
1
11 1
a common plane.
. Transform model variables,
. Transform .model variables, parameters, and data
..
Transform model variables,
AssessTransform model variables,
coplanarity . . . and data
parameters,
xm parameters, and data
ˆ Transform model variables,
parameters, and data Assess coplanarity
In a sense, we are checking the
parameters, and data
parameters, and data coplanarity of transformed
Steady state invariants
Data coplanar Data not coplanar invariants and data.
1
1
1
2
2
Data not coplanar
2
2
2
Data not coplanar Model compatible
Model compatible Model incompatible
Data coplanar 2 Data not coplanar 3
Model checking, multistability, and spatial models Heather Harrington 11 / 40
Model incompatible

36. Model Model Model1.1. .Model 22 L 2
Model
Model 1 Model Model Model
1 Model 2 Model 1Model
1 2
x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
Assessing coplanarity: overview .
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...
xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...
Models
Models Observed Data DataData Data
Observed
Observed Data Data
Observed Observed
Observed
Calculate elimination ideal
Models (Steadymeasurements)
Calculate elimination ideal elimination. .ideal (Steadymeasurements)
Calculate (Steady state state state measurements)
(Steady measurements)
Model 1 Model. 2 . . . . Model L (Steady state measurements)
. . (Steady state state measurements)
Model 1
Calculate elimination ideal
Calculate elimination. .ideal..Model xx1 Model 2 ...... . . . . . . .
. . . . . . . x1 . .. . x11 ˆˆ
ˆ ˆ 1 1 . .x1ˆ. ... ... We are interested in how to
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
Assess coplanarity ˆ ˆ . ˆˆ . .x2 ........ . . . .. .. . . . .
Assess coplanarity1x2 . . . x.22 . xx2
Assess coplanarity . . . . . .
. .
. .
x =
˙
... ... ... ...
Assess coplanarity .
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
check if models and data are
Assess coplanarity ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
Reduce number ˙of =
Reduce number xN variables
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
Reduce number of variables
Reduce number of variables
to include only observables
observables only observables
to include
to include onlyReduce number of variables
to include only observablesSteady statestate invariants
Steady state invariants Data
Observed SteadySteady
Steady state invariants state invariants
Steady state state invariants
Steady invariants
invariants
Assess if the invariants and
to include only observablesstate invariants
Steady
(Steady state measurements) of models
Characterize steady states data, when transformed, lie on
Characterize steady states of. .models states of models
Characterize steady
.
x1Characterize steady
ˆ
Calculate elimination ideal states of models
x2
ˆ
Characterize steady states of elimination ideal
. . . Calculate models 1 1
1
11 1
a common plane.
. Transform model variables,
. Transform .model variables, parameters, and data
..
Transform model variables,
AssessTransform model variables,
coplanarity . . . and data
parameters,
xm parameters, and data
ˆ Transform model variables,
parameters, and data Assess coplanarity
In a sense, we are checking the
parameters, and data
parameters, and data coplanarity of transformed
Steady state invariants
Data coplanar Data not coplanar invariants and data.
1
1
1
Model rejection can then be
2
2
2
2
Data not coplanar
performed by assessing the
Data not coplanar
2
Model compatible
degree to which the transformed
data deviate from coplanarity.
Model compatible Model incompatible
Data coplanar 2 Data not coplanar 3
Model checking, multistability, and spatial models Heather Harrington 11 / 40
Model incompatible

37. Assess coplanarity: question
Data coplanarity
Given a set of steady-state measurements xobs,i for i = 1, . . . , m, and
ˆ
model with steady-state invariants I = {I1 , . . . , INinv }, we need a
procedure for deciding whether it is possible that the invariant is
compatible with the data, i.e.,
I (ˆobs,i ; a) = 0,
x i = 1, . . . , m, (2)
for some choice of a.
Model checking, multistability, and spatial models Heather Harrington 12 / 40

38. Assess coplanarity: transform variables and data
Consider an invariant I ∈ I, written in somewhat simplified form as
n Nobs
t
I (xobs ; a) = fj (a) xkjk (3)
j=1 k=1
To assess coplanarity (I (ˆobs,i ; a) = 0), we rewrite eq. 3 as:
x
n
I (ξ; α) = α i ξi .
i=1
Model checking, multistability, and spatial models Heather Harrington 13 / 40

39. Assess coplanarity: transform variables and data
Consider an invariant I ∈ I, written in somewhat simplified form as
n Nobs
t
I (xobs ; a) = fj (a) xkjk (3)
j=1 k=1
To assess coplanarity (I (ˆobs,i ; a) = 0), we rewrite eq. 3 as:
x
n
I (ξ; α) = α i ξi .
i=1
Let ϕ: xobs → ξ.
Model checking, multistability, and spatial models Heather Harrington 13 / 40

40. Assess coplanarity: transform variables and data
Consider an invariant I ∈ I, written in somewhat simplified form as
n Nobs
t
I (xobs ; a) = fj (a) xkjk (3)
j=1 k=1
To assess coplanarity (I (ˆobs,i ; a) = 0), we rewrite eq. 3 as:
x
n
I (ξ; α) = α i ξi .
i=1
ˆ
Compatibility implies that the transformed variable ξ = ϕ(ˆobs )x
x
corresponding to any observation ˆobs with coordinates
ˆ ˆ
(ξ1 , . . . , ξn ), lies on the plane defined by the coefficients α.
Model checking, multistability, and spatial models Heather Harrington 13 / 40

41. Assess coplanarity: transform variables and data
Consider an invariant I ∈ I, written in somewhat simplified form as
n Nobs
t
I (xobs ; a) = fj (a) xkjk (3)
j=1 k=1
To assess coplanarity (I (ˆobs,i ; a) = 0), we rewrite eq. 3 as:
x
n
I (ξ; α) = α i ξi .
i=1
ˆ
Compatibility implies that the transformed variable ξ = ϕ(ˆobs ) x
x
corresponding to any observation ˆobs with coordinates
ˆ ˆ
(ξ1 , . . . , ξn ), lies on the plane defined by the coefficients α.
In other words, compatibility with the data xobs,i implies that the
ˆ
corresponding transformed data ξ ˆi = ϕ(ˆobs,i ) are coplanar.
x
Model checking, multistability, and spatial models Heather Harrington 13 / 40

42. Assess coplanarity: SVD
ˆ
Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξ i .
Model checking, multistability, and spatial models Heather Harrington 14 / 40

43. Assess coplanarity: SVD
ˆ
Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξ i .
Then the data are coplanar if and only if Ξα = 0 for some column
vector α = 0.
Model checking, multistability, and spatial models Heather Harrington 14 / 40

44. Assess coplanarity: SVD
ˆ
Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξ i .
Then the data are coplanar if and only if Ξα = 0 for some column
vector α = 0.
Such a vector resides in the null space of Ξ, spanned by the right
singular vectors of Ξ corresponding to zero singular values.
Model checking, multistability, and spatial models Heather Harrington 14 / 40

45. Assess coplanarity: SVD
ˆ
Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξ i .
Then the data are coplanar if and only if Ξα = 0 for some column
vector α = 0.
Such a vector resides in the null space of Ξ, spanned by the right
singular vectors of Ξ corresponding to zero singular values.
Thus, assuming that m > n, if the smallest singular value σn of Ξ
is nonzero, then the data cannot be coplanar.
Model checking, multistability, and spatial models Heather Harrington 14 / 40

46. Assess coplanarity: SVD
ˆ
Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξ i .
Then the data are coplanar if and only if Ξα = 0 for some column
vector α = 0.
Such a vector resides in the null space of Ξ, spanned by the right
singular vectors of Ξ corresponding to zero singular values.
Thus, assuming that m > n, if the smallest singular value σn of Ξ
is nonzero, then the data cannot be coplanar.
More generally, σn = min α =1 Ξα gives the least squares
deviation of the data from coplanarity under the scaling constraint
α = 1.
Model checking, multistability, and spatial models Heather Harrington 14 / 40

47. Assess coplanarity: SVD
ˆ
Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξ i .
Then the data are coplanar if and only if Ξα = 0 for some column
vector α = 0.
Such a vector resides in the null space of Ξ, spanned by the right
singular vectors of Ξ corresponding to zero singular values.
Thus, assuming that m > n, if the smallest singular value σn of Ξ
is nonzero, then the data cannot be coplanar.
More generally, σn = min α =1 Ξα gives the least squares
deviation of the data from coplanarity under the scaling constraint
α = 1.
This measure depends only on the data and is therefore
parameter-free.
Model checking, multistability, and spatial models Heather Harrington 14 / 40

48. Assess coplanarity: remarks
(1) Note that this applies for any choice of α, regardless of whether
it can be realized by the original parameters a.
(2) In this sense, the condition of small σn provides a necessary but
not sufficient criterion for model compatibility.
Model checking, multistability, and spatial models Heather Harrington 15 / 40

49. Assess coplanarity: remarks
(1) Note that this applies for any choice of α, regardless of whether
it can be realized by the original parameters a.
(2) In this sense, the condition of small σn provides a necessary but
not sufficient criterion for model compatibility.
(3) This is in contrast to traditional approaches based on parameter
fitting, which provide a sufficient but not necessary condition,
since local minima may prevent a compatible model from being
fitted correctly.
(4) The additional degrees of freedom introduced by neglecting the
functional forms fj effectively linearizes the compatibility
condition (I (ˆobs,i ; a) = 0), allowing for a simple direct solution.
x
Model checking, multistability, and spatial models Heather Harrington 15 / 40

50. Assess coplanarity: noise in data
To account for the presence of noise, let x x
= ∆ˆobs / ˆobs be the relative error
x
in a measurement ˆobs .
Model checking, multistability, and spatial models Heather Harrington 16 / 40

51. Assess coplanarity: noise in data
To account for the presence of noise, let x x
= ∆ˆobs / ˆobs be the relative error
x
in a measurement ˆobs .
(1) Then from the perturbation equation ξ + ∆ξ = ϕ (x + ∆x) , this is
ˆ ˆ x
propagated to the transformed variables as ∆ξ / ξ ∼ β(ˆobs ) , where
ϕ(x) x
β (x) = ϕ(x)
is the noise amplification factor, and ϕ is the Jacobian of
ϕ, with elements ( ϕ)ij = ∂ξi /∂xj .
Model checking, multistability, and spatial models Heather Harrington 16 / 40

52. Assess coplanarity: noise in data
To account for the presence of noise, let x x
= ∆ˆobs / ˆobs be the relative error
x
in a measurement ˆobs .
(1) Then from the perturbation equation ξ + ∆ξ = ϕ (x + ∆x) , this is
ˆ ˆ x
propagated to the transformed variables as ∆ξ / ξ ∼ β(ˆobs ) , where
ϕ(x) x
β (x) = ϕ(x)
is the noise amplification factor, and ϕ is the Jacobian of
ϕ, with elements ( ϕ)ij = ∂ξi /∂xj .
(2) To quantify the overall level of noise across all measurements, we define
√
x x
β = β / m, where β = (β(ˆobs,1 ), . . . , β(ˆobs,m )) is a vector containing
each noise amplification factor, and the effective relative error as eff = β .
Model checking, multistability, and spatial models Heather Harrington 16 / 40

53. Assess coplanarity: noise in data
To account for the presence of noise, let x x
= ∆ˆobs / ˆobs be the relative error
x
in a measurement ˆobs .
(1) Then from the perturbation equation ξ + ∆ξ = ϕ (x + ∆x) , this is
ˆ ˆ x
propagated to the transformed variables as ∆ξ / ξ ∼ β(ˆobs ) , where
ϕ(x) x
β (x) = ϕ(x)
is the noise amplification factor, and ϕ is the Jacobian of
ϕ, with elements ( ϕ)ij = ∂ξi /∂xj .
(2) To quantify the overall level of noise across all measurements, we define
√
x x
β = β / m, where β = (β(ˆobs,1 ), . . . , β(ˆobs,m )) is a vector containing
each noise amplification factor, and the effective relative error as eff = β .
(3) Since the introduction of noise in Ξ of order eff in general gives a lower
√
bound of σn ∼ m eff ∼ β , we should reject the model only if σn β .
We therefore define the coplanarity error
σn
∆= ,
β
in terms of which the rejection criterion is simply ∆ 1. Observe that as
increases, ∆ decreases, so we lose rejection power, as expected.
Model checking, multistability, and spatial models Heather Harrington 16 / 40

54. Example application: multisite phosphorylation
Distributive Phosphorylation of MAPK
Disassociation
MAPKK MAPKK MAPKK
MAPKK MAPKK
P P P P P P
Model checking, multistability, and spatial models Heather Harrington 17 / 40

55. Example application: multisite phosphorylation
Distributive Phosphorylation of MAPK
Disassociation
MAPKK MAPKK MAPKK
MAPKK MAPKK
P P P P P P
Processive Phosphorylation of MAPK
MAPKK MAPKK
Slide
MAPKK MAPKK
P P P P P
Model checking, multistability, and spatial models Heather Harrington 17 / 40

56. Example application: multisite phosphorylation
Distributive Phosphorylation of MAPK
Disassociation
MAPKK MAPKK MAPKK
MAPKK MAPKK
P P P P P P
Processive Phosphorylation of MAPK
MAPKK MAPKK
Slide
MAPKK MAPKK
P P P P P
Dephosphorylation can also occur in a processive or a distributive
manner. We would like to know which mechanism operates in vivo.
Model checking, multistability, and spatial models Heather Harrington 17 / 40

57. Multisite phosphorylation: eliminate variables
Each enzyme can be either processive (P),
u cuv a
K + Su −− KSu −→ K + Sv ,
−− − where more than one phosphate modification
bu
may be achieved in a single step, or
vu αv γ
F + Sv −− FSv −→ F + Su ,
−− − distributive (D), where only one modification
βv
is allowed before the enzyme dissociates from
Phosphorylation the substrate.
E + S01 ES01
Models: PP, PD, DP and DD; where the first
letter designates the mechanisms of the
E + S00 ES00 E + S11
kinase, and the second, that of the
E + S10 ES10 phosphatase.
We considered only the concentrations
F S01 F + S01
xobs = (s00 , s01 , s10 , s11 ) as observable, and
were able to eliminate all other variables
F + S00 F S11 F + S11
except the concentration f of F from the
F S10 F + S10 dynamics of each model.
Dephosphorylation
Model checking, multistability, and spatial models Heather Harrington 18 / 40

58. Multisite phosphorylation: assess coplanarity
Each model has three steady-state invariants.
Model checking, multistability, and spatial models Heather Harrington 19 / 40

59. Multisite phosphorylation: assess coplanarity
Each model has three steady-state invariants.
Invariants share same transformed variables ξ = ϕ(xobs ) so only
the kinase is discriminative.
Model checking, multistability, and spatial models Heather Harrington 19 / 40

60. Multisite phosphorylation: assess coplanarity
Each model has three steady-state invariants.
Invariants share same transformed variables ξ = ϕ(xobs ) so only
the kinase is discriminative.
Data generated under this model: PP/PD DP/DD
Reject model PP/PD? No No
Reject model DP/DD? Yes No
ξ PP/PD = s00 s10 , s00 s11 , s01 s10 , s01 s11 , s10 , s10 s11 ,
2
ξ DP/DD = s00 s11 , s01 s10 , s01 s11 , s10 , s10 s11 .
2
Model checking, multistability, and spatial models Heather Harrington 19 / 40

61. Multisite phosphorylation: coplanarity results
Model checking, multistability, and spatial models Heather Harrington 20 / 40

62. Examples: apoptosis activation
Chapter 7. Fas trimerization model
s for each of the DISC, MAC, and apoptosome modules are described
145
tation is understood to apply only within each module.
Crosslinking model
!
rization kinetics are simplified from the crosslinking model (Delisi,
4, 1981) of Lai and Jackson, 2004 and follow the reactions
" 3kf !
FasL + FasR −− FasL-FasR,
−−
kr
2kf
FasL-FasR + FasR −− FasL-FasR2 ,
−−
2kr
kf
FasL-FasR2 + FasR −− FasL-FasR3 ,
−−
3kr
Lai & Jackson (2004) Math Biosci Eng
Figure 7.4: Comparison with the crosslinking model. (A) Process diagram (comply with the SBGN Pro-
cess Description language Level 1 (Le Nov` re et al., 2009)) of the crosslinking model. (B) Variation of
e
the steady-state signaling Fas fraction ζ∞ with respect to the model parameter κ. (C) Minimization errors
of the steady-state invariants ωH and ωC for the hysteron and crosslinking models, respectively (Ap-
pendix B.2), over data generated from each model (Datasets 3 and 4) using nonnegative least squares (see
Materials and methods for details).
Model checking, multistability, and spatial models Heather Harrington 21 / 40

63. [33], irreversible bistability is achieved, implementing a perma-
nent cell death decision. Thus, our model suggests a primary role
Examples: apoptosis activation
Chapter 7. Fas trimerization model
s for each of the DISC, MAC, and apoptosome modules are described
for death receptors in deciding cell fate. Moreover, our results offer
145 novel functional interpretations of ligand trimerism and receptor
pre-association and localization within the unified context of
The first reaction describes spo
closing; the second, constitutive
third, ligand-independent recept
bistability. fourth, ligand-dependent recepto
The orders of the cluster-stabiliz
tation is understood to apply only within each module.
Results parameters m and n, which captu
and Fas coordination by FasL, r
Model formulation stabilization (m~n~2) has been
Constructing a mathematical model of Fas dynamics is not higher-order analogues, for exam
entirely straightforward as receptors can form highly oligomeric interactions, are not unreasonabl
Crosslinking model Cluster model
!
rization kinetics are simplified from the crosslinking model (Delisi,
4, 1981) of Lai and Jackson, 2004 and follow the reactions
" 3kf !
FasL + FasR −− FasL-FasR,
−−
kr
Figure 1. Cartoon of model interactions. The transmembrane death receptor Fas natively adopts a closed co
2kf the binding of FADD, an adaptor molecule that facilitates apoptotic signal transduction. Open Fas can self-st
FasL-FasR + FasR −− FasL-FasR2 ,
−− interactions, which is enhanced by receptor clustering through association with the ligand FasL.
doi:10.1371/journal.pcbi.1000956.g001
2kr
PLoS Computational Biology | www.ploscompbiol.org 2 October 2010 |
kf
FasL-FasR2 + FasR −− FasL-FasR3 ,
−−
3kr
Lai & Jackson (2004) Math Biosci Eng Figure 2. Schematic of cluster-stabilization reactions. Examples
Ho & Harrington (2010) PLoS Comput Biol
of ligand-independent cluster-stabilization reactions involving unstable
(Y ) and stable (Z) open receptors of molecularities two (A), three (B),
Figure 7.4: Comparison with the crosslinking model. (A) Process diagram (comply with the SBGN Pro-
cess Description language Level 1 (Le Nov` re et al., 2009)) of the crosslinking model. (B) Variation of
e
and four (C). Higher-order reactions follow the same pattern. Ligand-
the steady-state signaling Fas fraction ζ∞ with respect to the model parameter κ. (C) Minimization errors dependent reactions are identical except that FasL (L) must be added
of the steady-state invariants ωH and ωC for the hysteron and crosslinking models, respectively (Ap- to each reacting state.
pendix B.2), over data generated from each model (Datasets 3 and 4) using nonnegative least squares (see doi:10.1371/journal.pcbi.1000956.g002
Materials and methods for details).
Formally, these reactions are to be interpreted as state transitions
Model checking, multistability, and spatial models Heather Harrington 21 / 40
on the space of cluster tuples. However, the reaction notation is

64. [33], irreversible bistability is achieved, implementing a perma-
nent cell death decision. Thus, our model suggests a primary role
Examples: apoptosis activation
Chapter 7. Fas trimerization model
s for each of the DISC, MAC, and apoptosome modules are described
for death receptors in deciding cell fate. Moreover, our results offer
145 novel functional interpretations of ligand trimerism and receptor
pre-association and localization within the unified context of
The first reaction describes spo
closing; the second, constitutive
third, ligand-independent recept
bistability. fourth, ligand-dependent recepto
The orders of the cluster-stabiliz
tation is understood to apply only within each module.
Results parameters m and n, which captu
and Fas coordination by FasL, r
Model formulation stabilization (m~n~2) has been
Constructing a mathematical model of Fas dynamics is not higher-order analogues, for exam
entirely straightforward as receptors can form highly oligomeric interactions, are not unreasonabl
Crosslinking model Cluster model
!
rization kinetics are simplified from the crosslinking model (Delisi,
4, 1981) of Lai and Jackson, 2004 and follow the reactions
" 3kf !
FasL + FasR −− FasL-FasR,
−−
kr
Figure 1. Cartoon of model interactions. The transmembrane death receptor Fas natively adopts a closed co
2kf the binding of FADD, an adaptor molecule that facilitates apoptotic signal transduction. Open Fas can self-st
FasL-FasR + FasR −− FasL-FasR2 ,
−− interactions, which is enhanced by receptor clustering through association with the ligand FasL.
doi:10.1371/journal.pcbi.1000956.g001
2kr
PLoS Computational Biology | www.ploscompbiol.org 2 October 2010 |
kf
FasL-FasR2 + FasR −− FasL-FasR3 ,
−−
3kr
Lai & Jackson (2004) Math Biosci Eng Figure 2. Schematic of cluster-stabilization reactions. Examples
Ho & Harrington (2010) PLoS Comput Biol
of ligand-independent cluster-stabilization reactions involving unstable
(Y ) and stable (Z) open receptors of molecularities two (A), three (B),
The activation signal is defined for each model.
Figure 7.4: Comparison with the crosslinking model. (A) Process diagram (comply with the SBGN Pro-
cess Description language Level 1 (Le Nov` re et al., 2009)) of the crosslinking model. (B) Variation of
e
and four (C). Higher-order reactions follow the same pattern. Ligand-
dependent reactions are identical except that FasL (L) must be added
the steady-state signaling Fas fraction ζ∞ with respect to the model parameter κ. (C) Minimization errors
of the steady-state invariants ωH and ωC for the hysteron and crosslinking models, respectively (Ap- to each reacting state.
pendix B.2), over data generated from each model (Datasets 3 and 4) using nonnegative least squares (see doi:10.1371/journal.pcbi.1000956.g002
Materials and methods for details).
Formally, these reactions are to be interpreted as state transitions
Model checking, multistability, and spatial models Heather Harrington 21 / 40
on the space of cluster tuples. However, the reaction notation is

65. [33], irreversible bistability is achieved, implementing a perma-
nent cell death decision. Thus, our model suggests a primary role
Examples: apoptosis activation
Chapter 7. Fas trimerization model
s for each of the DISC, MAC, and apoptosome modules are described
for death receptors in deciding cell fate. Moreover, our results offer
145 novel functional interpretations of ligand trimerism and receptor
pre-association and localization within the unified context of
The first reaction describes spo
closing; the second, constitutive
third, ligand-independent recept
bistability. fourth, ligand-dependent recepto
The orders of the cluster-stabiliz
tation is understood to apply only within each module.
Results parameters m and n, which captu
and Fas coordination by FasL, r
Model formulation stabilization (m~n~2) has been
Constructing a mathematical model of Fas dynamics is not higher-order analogues, for exam
entirely straightforward as receptors can form highly oligomeric interactions, are not unreasonabl
Crosslinking model Cluster model
!
rization kinetics are simplified from the crosslinking model (Delisi,
4, 1981) of Lai and Jackson, 2004 and follow the reactions
" 3kf !
FasL + FasR −− FasL-FasR,
−−
kr
Figure 1. Cartoon of model interactions. The transmembrane death receptor Fas natively adopts a closed co
2kf the binding of FADD, an adaptor molecule that facilitates apoptotic signal transduction. Open Fas can self-st
FasL-FasR + FasR −− FasL-FasR2 ,
−− interactions, which is enhanced by receptor clustering through association with the ligand FasL.
doi:10.1371/journal.pcbi.1000956.g001
2kr
PLoS Computational Biology | www.ploscompbiol.org 2 October 2010 |
kf
FasL-FasR2 + FasR −− FasL-FasR3 ,
−−
3kr
Lai & Jackson (2004) Math Biosci Eng Figure 2. Schematic of cluster-stabilization reactions. Examples
Ho & Harrington (2010) PLoS Comput Biol
of ligand-independent cluster-stabilization reactions involving unstable
(Y ) and stable (Z) open receptors of molecularities two (A), three (B),
The activation signal is defined for each model.
Figure 7.4: Comparison with the crosslinking model. (A) Process diagram (comply with the SBGN Pro-
cess Description language Level 1 (Le Nov` re et al., 2009)) of the crosslinking model. (B) Variation of
e
and four (C). Higher-order reactions follow the same pattern. Ligand-
dependent reactions are identical except that FasL (L) must be added
the steady-state signaling Fas fraction ζ∞ with respect to the model parameter κ. (C) Minimization errors
of the steady-state invariants ωH and ωC for the hysteron and crosslinking models, respectively (Ap- to each reacting state.
Each model has one steady-state invariant.
pendix B.2), over data generated from each model (Datasets 3 and 4) using nonnegative least squares (see
Materials and methods for details).
doi:10.1371/journal.pcbi.1000956.g002
Formally, these reactions are to be interpreted as state transitions
Model checking, multistability, and spatial models Heather Harrington 21 / 40
on the space of cluster tuples. However, the reaction notation is

66. Apoptosis activation: coplanarity results
Model checking, multistability, and spatial models Heather Harrington 22 / 40

67. x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ ..
x = 1
...
Asessing coplanarity: overall findings . .. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
. . ...
xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...
Models
Models Observed Data DataData Data
Observed
Observed Data Data
Observed Observed
Observed
Calculate elimination ideal
Models (Steadymeasurements)
Calculate elimination ideal elimination. .ideal (Steadymeasurements)
Calculate . . . . .
Model Model 2
1 Model (Steady state state state measurements)
Model 1 elimination. ideal L (Steady state measurements)
. (Steady measurements)
(Steady state state measurements)
Calculate
Calculate elimination. .ideal.. . x11 ˆ1 Model xx1 Model 2 ........ . . ..... . . . .
1ˆ . .x1 .
x1 =˙ 1 = . . . . . .
˙ x . . . . . . . x1 . .. ˆ
ˆ x
ˆ ˆ
Assess coplanarityx2 x1 ˆ
˙ = x.2. . xx2 . .x2 ........ . . . .. .. . . . .
ˆ ˆˆ ˆ.
Assess coplanarity Assess coplanarity. . .. .. . x2 ˆ 2
. . ... ...
. . ... ... ..
Assess coplanarity
..
. .. . . . . .. ...... . . . . . . .
.. . . . . . .
Assess coplanarity . ... ...
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
ˆ ˆ ˆˆ
m x
Reduce numberx = x.m . ˙of variables
ˆ
. ... ... ...
Reduce number ofReduce number of variables
N
variables
Reduce number of variables
to include only observables
to include only observables
Reduce number of variables
to include only observables
to include only observablesSteady statestate invariants
Steady state invariants Data SteadySteady
Steady state invariants state invariants
Steady state state invariants
Steady invariants
invariants
Observed
to include only observablesstate invariants
Steady
(Steady state measurements) of models
Characterize steady states
Characterize steady states of models
Characterize steady states of. .models
x1Characterize steady
ˆ .
Calculate elimination ideal states of models
Characterize steady states of elimination ideal
. . . Calculate models
x2
ˆ 1 1
1
11 1
Transform model variables,
Transform . . .
Transform model variables, model variables, parameters, and data
.
Transform model variables,
Assess coplanarity . . . and data
parameters,
xm parameters, and data
ˆ Transform model variables,
parameters, and data Assess coplanarity
parameters, and data
parameters, and data
Steady state invariants
Data coplanar Data not coplanar
1
1
1
2
2
Data not coplanar
2
2
2
Data not coplanar Model compatible
Model compatible Model incompatible
Data coplanar 2 Data not coplanar 3
Model incompatible
Model checking, multistability, and spatial models Heather Harrington 23 / 40

68. Asessing coplanarity: overall findings
Novel model selection scheme based on steady-state coplanarity
that does not require parameter estimation.
Model checking, multistability, and spatial models Heather Harrington 23 / 40

69. Asessing coplanarity: overall findings
Novel model selection scheme based on steady-state coplanarity
that does not require parameter estimation.
Method is not always effective– steady-state invariants may not
exist, or there may be additional degrees of freedom.
Model checking, multistability, and spatial models Heather Harrington 23 / 40

70. Asessing coplanarity: overall findings
Novel model selection scheme based on steady-state coplanarity
that does not require parameter estimation.
Method is not always effective– steady-state invariants may not
exist, or there may be additional degrees of freedom.
Coplanarity adds to the spectrum of model selection methods,
especially when no knowledge of parameter is known.
Model checking, multistability, and spatial models Heather Harrington 23 / 40

71. Asessing coplanarity: overall findings
Novel model selection scheme based on steady-state coplanarity
that does not require parameter estimation.
Method is not always effective– steady-state invariants may not
exist, or there may be additional degrees of freedom.
Coplanarity adds to the spectrum of model selection methods,
especially when no knowledge of parameter is known.
This model selection is computationally much quicker than
optimization methods.
Model checking, multistability, and spatial models Heather Harrington 23 / 40

72. Asessing coplanarity: overall findings
Novel model selection scheme based on steady-state coplanarity
that does not require parameter estimation.
Method is not always effective– steady-state invariants may not
exist, or there may be additional degrees of freedom.
Coplanarity adds to the spectrum of model selection methods,
especially when no knowledge of parameter is known.
This model selection is computationally much quicker than
optimization methods.
Potential new class of model selection methods based on geometric
structure.
Model checking, multistability, and spatial models Heather Harrington 23 / 40

73. De
Phos
Phos
Cellular states 10-4
0 1
Stimulus (Etot)
D
Substrate (S, S*)
10 25 40
Stimulus (Etot)
Model checking, multistability, and spatial models Heather Harrington 24 / 40

74. Cellular information processing
Information processing
One central aspect of biological information processing is the mapping of
environments onto intra-cellular states given by the abundances of the molecular
species (proteins, mRNAs, metabolites etc.) under consideration. To process
information, one or more environmental variables need to be represented in a way
that facilitates the appropriate response (discrete, continuous).
Model checking, multistability, and spatial models Heather Harrington 25 / 40

75. Cellular information processing
Information processing
One central aspect of biological information processing is the mapping of
environments onto intra-cellular states given by the abundances of the molecular
species (proteins, mRNAs, metabolites etc.) under consideration. To process
information, one or more environmental variables need to be represented in a way
that facilitates the appropriate response (discrete, continuous).
The number of response states is of particular interest if there is a
regime of conditions where a system can occupy more than one
state.
Model checking, multistability, and spatial models Heather Harrington 25 / 40

76. Cellular information processing
Information processing
One central aspect of biological information processing is the mapping of
environments onto intra-cellular states given by the abundances of the molecular
species (proteins, mRNAs, metabolites etc.) under consideration. To process
information, one or more environmental variables need to be represented in a way
that facilitates the appropriate response (discrete, continuous).
The number of response states is of particular interest if there is a
regime of conditions where a system can occupy more than one
state.
If more than one state exists (e.g., switch-like systems), this is
called multistationarity.
Model checking, multistability, and spatial models Heather Harrington 25 / 40

77. Cellular information processing
Information processing
One central aspect of biological information processing is the mapping of
environments onto intra-cellular states given by the abundances of the molecular
species (proteins, mRNAs, metabolites etc.) under consideration. To process
information, one or more environmental variables need to be represented in a way
that facilitates the appropriate response (discrete, continuous).
The number of response states is of particular interest if there is a
regime of conditions where a system can occupy more than one
state.
If more than one state exists (e.g., switch-like systems), this is
called multistationarity.
The number of states is linked to the flexibility in the decision
making of a cell.
Model checking, multistability, and spatial models Heather Harrington 25 / 40

78. Enzyme sharing as a cause of multistationarity
Downloaded from rsif.royalsocietypublishing.org on May 1, 2012
Enzyme sharing and multi-stationarity E. Feliu and C. Wiuf 1225
one-site modification
(a) (b) (c) (d)
E E E1, E2 E1, E2
S0 S1 S0 S1 S0 S1 S0 S1
F E F F1 ,F2
two-site modification
(e) E1 E2 (f) ( g)
E E E E
S0 S1 S2 S0 S1 S2 S0 S1 S2
F1 F2 F1 F2 F F
modification of two substrates
(h) E (i) E
S0 S1 P0 P1 S0 S1 P0 P1
F1 F2
F
two-layer cascade
( j) (k) (l)
E E E
S0 S1 S0 S1 S0 S1
F1 F F1
P0 P1 P0 P1 P0 P1
F2 F F2
Figure 1. Motifs composed of one or two one-site cycles. Motifs with purple label, and only these, admit multiple biologically
Feliu & Wiuf (2012) J R Soc Interface
meaningful steady states. Si and Pi are substrates with i ¼ 0,1,2 phosphorylated sites. E, E1, E2 denote kinases, and F, F1, F2
phosphatases. In Motif (b), the kinase and the phosphatase are the same enzyme.
Model the motifs from a one-site phosphorylation cycle
build checking, multistability, and spatial models Heather Harrington 26 / 40
phosphatases. This motif represents by symmetry also a

79. Protein kinase cascades
Protein kinase cascades
A canonical system for investigating multistationarity are protein kinase cascades,
e.g., mitogen activated protein kinase (MAPK).
Cytoplasm
F
Y
Plasma
membrane S S*
X
E
Cytoplasm
E
X
S S*
Y
F
Nucleus Nucleus
Figure 1: Spatial signaling schematic.
Model checking, multistability, and spatial models Heather Harrington 27 / 40

80. Protein kinase cascades
Protein kinase cascades
A canonical system for investigating multistationarity are protein kinase cascades,
e.g., mitogen activated protein kinase (MAPK).
The ultimate function of MAPK is to
Cytoplasm
initiate transcriptional responses.
F
Y
Plasma
membrane S S*
X
E
Cytoplasm
E
X
S S*
Y
F
Nucleus Nucleus
Figure 1: Spatial signaling schematic.
Model checking, multistability, and spatial models Heather Harrington 27 / 40

81. Protein kinase cascades
Protein kinase cascades
A canonical system for investigating multistationarity are protein kinase cascades,
e.g., mitogen activated protein kinase (MAPK).
The ultimate function of MAPK is to
Cytoplasm
initiate transcriptional responses.
F
Y
Plasma
membrane S Spatial organization plays a
S*
pronounced role to increasing the
E
X
Cytoplasm biological information processing.
E
X
S S*
Y
F
Nucleus Nucleus
Figure 1: Spatial signaling schematic.
Model checking, multistability, and spatial models Heather Harrington 27 / 40

82. Protein kinase cascades
Protein kinase cascades
A canonical system for investigating multistationarity are protein kinase cascades,
e.g., mitogen activated protein kinase (MAPK).
The ultimate function of MAPK is to
Cytoplasm
initiate transcriptional responses.
F
Y
Plasma
membrane S Spatial organization plays a
S*
pronounced role to increasing the
E
X
Cytoplasm biological information processing.
E
We Xfind that compartmentalization
S S*
increases the number of states that
Y
F
Nucleus Nucleus can become simultaneously accessible
to the cell.
Figure 1: Spatial signaling schematic.
Model checking, multistability, and spatial models Heather Harrington 27 / 40

83. E
X
One-site model
E X
S S⇤
F
Y
Nucleus Cytoplasm
F
Y
Plasma
membrane S S*
Figure 1: Shuttling of a one-site phosphorylation cycle between the nucleus and the cytoplasm.
X
2 Shuttling in a one-site phosphorylation cycle E
2.1 Reactions
Cytoplasm
E
We consider a one-site phosphorylation cycle with species: S, S ⇤ (the unphosphorylated and phospho-
rylated substrates), E (kinase), F (phosphatase), and X, Y (intermediate complexes). Phosphorylation
X
and dephosphorylation are assumed to follow a Michaelis-Menten mechanism (see below and main
S
text). This motif cannot admit multiple steady states, and it is monostable [4]. S*
To study the effect of compartmentalization, we asume that the species S, S ⇤ , E, X can shuttle
Y
between the cytoplasm and the nucleus (see Figure 1). We let Z c denote species Z in the Fcytoplasm.
Nucleus
Then, the reactions in play are as follows: Nucleus
• Reactions in the nucleus:
k1 k3 k4 k6
E+S o / / ⇤ ⇤ / /
Figure X Spatial signaling schematic. + S
1: E + S F +S Y F o
k 2 k 5
• Reactions in the cytoplasm:
k7 k9 k10 k12
Ec + Sc o / / Ec + Sc F c + S c⇤ o / / F c + S c⇤
Xc Yc
k8 k11
• Shuttling reactions:
k13 k14 k15 k16
/ / / /
Eo Ec Xo Xc So Sc S⇤ o S c⇤
k17 k18 k19 k20
To ease the notation below, we have changed the notation of the reaction constants k⇤ in the main
Model checking, multistability, and spatial models Heather Harrington 28 / 40

84. Determining whether a system is capable of
multistationarity
Jacobian conjecture for quadratic polynomials (Bass, Connel,
Wright 1982). The Jacobian conjecture, which is true for polynomial
functions whose components are up to degree two, guarantees that if the
Jacobian of a function f never vanishes in a convex domain, then the function
is injective in that domain.
Model checking, multistability, and spatial models Heather Harrington 29 / 40

85. Determining whether a system is capable of
multistationarity
Jacobian conjecture for quadratic polynomials (Bass, Connel,
Wright 1982). The Jacobian conjecture, which is true for polynomial
functions whose components are up to degree two, guarantees that if the
Jacobian of a function f never vanishes in a convex domain, then the function
is injective in that domain.
If injective, we’re done and the system, for no parameters/total
amounts can elicit multistationarity.
Model checking, multistability, and spatial models Heather Harrington 29 / 40

86. Determining whether a system is capable of
multistationarity
Jacobian conjecture for quadratic polynomials (Bass, Connel,
Wright 1982). The Jacobian conjecture, which is true for polynomial
functions whose components are up to degree two, guarantees that if the
Jacobian of a function f never vanishes in a convex domain, then the function
is injective in that domain.
If injective, we’re done and the system, for no parameters/total
amounts can elicit multistationarity.
Failure of the Jacobian injectivity criterion is not sufficient to
conclude that multistationarity occurs. Use other methods,
e.g.,CRNT toolbox (Ellison, Feinberg, Ji, 2011) software which
implements algorithms to determine when a network can have multiple positive
steady states for fixed conserved amounts (using mass-action kinetics).
Model checking, multistability, and spatial models Heather Harrington 29 / 40

87. Multistationarity by localization
7 Compartment shuttling
Species shuttling conditions
# shuttling species No multistationarity Multistationarity
1 All None
{S0 , S1 } {E, Y } {F, X} {E, F } {X, Y } {S1 , X} {S0 , Y }
{S1 , E} {S0 , F } {S0 , X}
2
{S1 , Y } {E, X} {F, Y }
{S0 , E} {S1 , F }
{X, E, F } {Y, E, F } {S0 , E, X} {S1 , F, Y } {S0 , E, Y }
{X, Y, E} {X, Y, F } {S1 , F, X} {S0 , E, S1 } {S1 , F, S0 }
3
{S0 , F, X} {S1 , E, Y } {S0 , X, Y } {S1 , Y, X} {S0 , F, Y }
{S0 , E, F } {S1 , F, E} {S1 , E, X} {S0 , S1 , X} {S0 , S1 , Y }
{Y, X, E, F } {S0 , S1 , X, F } {S0 , S1 , Y, E} {S0 , E, X, Y }
{S1 , F, X, Y } {S0 , F, X, Y } {S1 , E, X, Y }
4
{S0 , E, F, X} {S1 , E, F, Y } {S0 , E, F, Y }
{S1 , E, F, X} {S0 , S1 , X, E} {S0 , S1 , Y, F }
{S0 , S1 , X, Y } {S0 , S1 , E, F }
5, 6 None All
Table 1: Sets of shuttling species that add or not multistationarity to the system.
One-site phosphorylation system. For all possible sets of shuttling species it is
indicated2.4 the system has the capacity for multiple steady states or not.
if Sets of shuttling species
We next inspected what the sets of shuttling species that provide multistationarity are. The
results are summarized in Table 1. The systematic way employed to classify each motif is the
following. First, we check if the systems fulfill the conditions of Jacobian conjecture and decide
if the system is injective. If the coefficients of the polynomial in x given by the determinant of
the Jacobian (as above) are all positive, then the system cannot exhibit multistationarity, for
any set of total amounts. If this criterion fails, then we have use the CRNT toolbox.
Model checking, multistability,that ifspatial models shuttles, then multistationarity cannot occur. / 40
We have obtained and only one species Heather Harrington 30
That is, at least two species, e.g. {S1 , X} or {S0 , Y }, are required to obtain multistationarity

88. Necessary conditions for monostability
There are two conditions suffice to guarantee monostationarity,
namely:
C2 =k9 k12 (k15 − k16 )(k18 − k17 ) + k9 k14 k15 k16 + k12 k14 k15 k16
+ k12 k14 k16 k17 + k9 k15 k16 k18 + k12 k15 k16 k18 + k12 k16 k17 k18 > 0,
C8 =k9 k12 (k14 − k13 )(k19 − k20 ) + k12 k13 k14 k20 + k12 k13 k18 k20
+ k9 k14 k19 k20 + k12 k14 k19 k20 + k9 k18 k19 k20 + k12 k18 k19 k20 > 0.
Model checking, multistability, and spatial models Heather Harrington 31 / 40

89. Necessary conditions for monostability
There are two conditions suffice to guarantee monostationarity,
namely:
C2 =k9 k12 (k15 − k16 )(k18 − k17 ) + k9 k14 k15 k16 + k12 k14 k15 k16
+ k12 k14 k16 k17 + k9 k15 k16 k18 + k12 k15 k16 k18 + k12 k16 k17 k18 > 0,
C8 =k9 k12 (k14 − k13 )(k19 − k20 ) + k12 k13 k14 k20 + k12 k13 k18 k20
+ k9 k14 k19 k20 + k12 k14 k19 k20 + k9 k18 k19 k20 + k12 k18 k19 k20 > 0.
By inspection of these two expressions, we conclude that
multistationarity cannot occur in any of the following cases:
(i) k20 ≤ k19 , k18 ≥ k17 , k16 ≤ k15 , k14 ≥ k13 ,
(ii) k20 ≥ k19 , k18 ≥ k17 , k16 ≤ k15 , k14 ≤ k13 ,
(iii) k20 ≤ k19 , k18 ≤ k17 , k16 ≥ k15 , k14 ≥ k13 ,
(iv) k20 ≥ k19 , k18 ≤ k17 , k16 ≥ k15 , k14 ≤ k13 .
Model checking, multistability, and spatial models Heather Harrington 31 / 40

90. Necessary conditions for monostability
X E
S S⇤
F
Y
Nucleus
By inspection of these two expressions, we conclude that
multistationarity1:cannot aoccur in any cycle the followingthe cytoplasm.
Figure Shuttling of one-site phosphorylation of between the nucleus and cases:
2 Shuttling in ak19 , kphosphorylation cycle 15 ,
(i) k20 ≤ one-site 18 ≥ k17 , k16 ≤ k k14 ≥ k13 ,
2.1 (ii) k
Reactions
20 ≥ k19 , k18 ≥ k17 , k16 ≤ k15 , k14 ≤ k13 ,
We consider a one-site phosphorylation cycle with species: S, S ⇤ (the unphosphorylated and phospho-
(iii) k20 ≤(kinase), F (phosphatase), and X, Y16 ≥ k15 , complexes). Phosphorylation
rylated substrates), E k19 , k18 ≤ k17 , k (intermediate k14 ≥ k13 ,
and dephosphorylation are assumed to follow a Michaelis-Menten mechanism (see below and main
text). This motif cannotk19 , multiple steady states, and 16 ≥ k15 , [4]. 14 ≤ k13 .
(iv) k20 ≥ admit k18 ≤ k17 , k it is monostable k
To study the effect of compartmentalization, we asume that the species S, S ⇤ , E, X can shuttle
between the cytoplasm and the nucleus (see Figure 1). We let Z c denote species Z in the cytoplasm.
Then, the reactions in play are as follows:
• Reactions in the nucleus:
k1 k3 k4 k6
E+S o / / E + S⇤ F + S⇤ o / /F +S
X Y
k2 k5
• Reactions in the cytoplasm:
k7 k9 k10 k12
Ec + Sc o / / Ec + Sc F c + S c⇤ o / / F c + S c⇤
Xc Yc
k8 k11
• Shuttling reactions:
k13 k14 k15 k16
/ / / /
Eo Ec Xo Xc So Sc S⇤ o S c⇤
k17 k18 k19 k20
To ease the notation below, we have changed the notation of the reaction constants k⇤ in the main
Model checking, multistability, and spatial models Heather Harrington 31 / 40

91. E
X
Necessary conditions for monostability
E X
S S⇤
F
Y
By inspection of these two expressions, we conclude that
Nucleus
multistationarity cannot occur in any of the following cases:
Figure 1: Shuttling of a one-site phosphorylation cycle between the nucleus and the cytoplasm.
(i) k20 ≤ k19 , k18 ≥ k17 , k16 ≤ k15 , k14 ≥ k13 ,
2 (ii) k20 in ak19 , kphosphorylation cycle 15 ,
Shuttling ≥ one-site 18 ≥ k17 , k16 ≤ k k14 ≤ k13 ,
2.1 Reactions
(iii) k20 ≤ k19 , k18 ≤ k17 , k16 ≥ k15 , k14 ≥ k13 ,
We consider a one-site phosphorylation cycle with species: S, S ⇤ (the unphosphorylated and phospho-
rylated substrates), ≥(kinase), F (phosphatase), and X, Y (intermediate complexes). Phosphorylation
(iv) k20 E k19 , k18 ≤ k17 , k16 ≥ k15 , k14 ≤ k13 .
and dephosphorylation are assumed to follow a Michaelis-Menten mechanism (see below and main
text). This motif cannot admit multiple steady states, and it is monostable [4].
Note that these only involve the rate constants for the shuttling
To study the effect of compartmentalization, we asume that the species S, S ⇤ , E, X can shuttle
between the cytoplasm and the nucleus (see Figure 1). We let Z c denote species Z in the cytoplasm.
reactions. Then, the reactions in play are as follows:
• Reactions in the nucleus:
k1 k3 k4 k6
E+S o / / E + S⇤ F + S⇤ o / /F +S
X Y
k2 k5
• Reactions in the cytoplasm:
k7 k9 k10 k12
Ec + Sc o / / Ec + Sc F c + S c⇤ o / / F c + S c⇤
Xc Yc
k8 k11
• Shuttling reactions:
k13 k14 k15 k16
/ / / /
Eo Ec Xo Xc So Sc S⇤ o S c⇤
k17 k18 k19 k20
To ease the notation below, we have changed the notation of the reaction constants k⇤ in the main
Model checking, multistability, and spatial models Heather Harrington 31 / 40

92. Necessary conditions for multistability
We notice that the rate constants go in pairs:
the shuttling rate constants of S relate to those of S ∗ , and
the shuttling rate constants of E to those of X .
In particular, the following conditions are necessary for
multistationarity:
(1) If X shuttles into the nucleus slower than E then S shuttles into
the cytoplasm slower than S ∗ and vice versa.
(2) If X shuttles into the cytoplasm slower than E then S shuttles
into the nucleus slower than S ∗ and vice versa.
Model checking, multistability, and spatial models Heather Harrington 32 / 40

93. Bistability by changing total amounts
4
102
A Bistable B
3.5 Regime
Phosphorylated Substrate (S*)
Phosphorylated Substrate (S*)
3
100
2.5
Activation
2
De-activation
10-2
1.5
1
0.5
10-4
0
0 5 10 15 20 25 30 35 40 45 50
0 1
Stimulus (Etot)
Model checking, multistability, and spatial models Heather Harrington 33 / 40

94. Bistability by changing total amounts
A Bistable B 102 Stot=150 C 100 Monostable
Regime High
Phosphorylated Substrate (S*)
Phosphorylated Substrate (S*)
Total Substrate (Stot)
St St 80
100 ot = ot =35 Sto =
45 t 25
Activation
Stot=15 60
De-activation
Bi
10-2
st
ab
40
le
Monostable
10-4 20 Low
0 10 20 30 40 50 10 20 30 40 50
Stimulus (Etot) Stimulus (Etot) Stimulus (Etot)
D
Substrate (S, S*)
Model checking, multistability, and spatial models
10 25 40 Heather Harrington 33 / 40

95. Bistability by changing shuttling rates
A 4 B 5 Etot=30 C 0.004
kin,E kin,X
Phosphorylated Substrate (S*)
Phosphorylated Substrate (S*)
Rate of S exiting the nucleus (kin,S)
Etot=28
3 4
Etot=26 0.003
kin,S* 3 Etot=24
2 0.002
Etot=22
2
1 0.001
1
Monostable
0 Low
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1
Bifurcation parameter (shuttling rate) Rate of S* entering the nucleus (kin,S*) Rate of S
kout,E is the rate at which E exits the nucleus.
kin,X is the rate at which X enters the nucleus.
kin,S∗ is the rate at which S enters the nucleus.
Model checking, multistability, and spatial models Heather Harrington 34 / 40

96. Bistability by changing shuttling rates
A 4 B 5 Etot=30 C 0.004
kin,E kin,X
Phosphorylated Substrate (S*)
Phosphorylated Substrate (S*)
Rate of S exiting the nucleus (kin,S)
Etot=28
3 4
Etot=26 0.003
kin,S* 3 Etot=24
2 0.002
Etot=22
2
1 0.001
1
Monostable
0 Low
0 0
0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1 0.2 0.3 0.4 0.5 0.6 0 0.1
Bifurcation parameter (shuttling rate) Rate of S* entering the nucleus (kin,S*) Rate of S
Rate constant Rate-response curve
k13 , k14 , k19 , k20 For large rate constant, only a low stable steady state is obtained
k15 , k17 , k18 For a small rate constant, only a high stable steady state is obtained
k16 Similar to the previous case, but the high branch decreases.
Model checking, multistability, and spatial models Heather Harrington 34 / 40

97. Example system
Two-site modification
We consider a two-site modification system, such as MAPK, at parameter values
which cannot permit multistationarity.
2
A Cytoplasm B 10 Multistable Regime
Fc Fc
De-activation
c c
Y1 Y2
100
Sc Sc Sc
Phosphorylated Substrate (S2)
0 1 2
c c
X1 X2
Ec Ec 10-2
E E 10-4
X1 X2
S0 S1 S2
10-6
Y1 Y2
F F
Nucleus 0 100 200 300
Stimulus (Etot)
70 45
C D
Zoomed-in (linear scale)
6060 40
40
45
ubstrate (S2)
ubstrate (S2)
ubstrate (S2)
35
40
40
50
3030 35
Model checking, multistability, and spatial models
40 40
25
Heather Harrington 35 / 40
3030

98. Multistability in two-site modification
2
B 10 Multistable Regime
Fc
De-activation
Activation
Sc 100
Phosphorylated Substrate (S2)
2
Steady state analysis on
10-2
parameter shuttling rate
constants.
10-4
S2 Analysis indicates the two-site
F 10-6
phosphorylation cycle can
0 100 200 300 400 500
Stimulus (Etot)
Zoomed-in (linear scale)
undergo hysteresis.
Large region of multistability
Phosphorylated Substrate (S2)
40
30 (32 ≤ Etot ≤ 445), most of
20 which is bistable.
10
0
0 60 80 40 45 50 55 60
osphatase (Fctot) Stimulus (Etot)
Model checking, multistability, and spatial models Heather Harrington 36 / 40

99. E E
Phosph
Versatility of X1
MAPK X2
10-4
S0 S1 S2
Y1 Y2
F F 10-6
Nucleus c 0 1
Bifurcations of shuttling rate (kout,S1 ) and total amount (Ftot )
2 2
C 10 D 10 Zoomed
101
Phosphorylated Substrate (S2)
101
Phosphorylated Substrate (S2)
Phosphorylated Substrate (S2)
40
100 100 30
10-1 10-1 20
10
10-2 10-2
10-3 0
10-3
0 200 400 600 0 20 40 60 80 40
Rate of S1 leaving the nucleus (kout,S1) Cytoplasmic Phosphatase (Fctot)
Steady states of the system can be regulated through reversible
switches governed by shuttling of parameters and other total
amounts.
Model checking, multistability, and spatial models Heather Harrington 37 / 40

100. Spatial localization: overall findings
Species localization serves as a mechanism for multistationarity:
the number of states may be higher for spatially structured systems
compared to homogenous systems.
Model checking, multistability, and spatial models Heather Harrington 38 / 40

101. Spatial localization: overall findings
Species localization serves as a mechanism for multistationarity:
the number of states may be higher for spatially structured systems
compared to homogenous systems.
Thereby cellular computational capacity and information
processing capacity is driven by spatial organization.
Model checking, multistability, and spatial models Heather Harrington 38 / 40

102. Spatial localization: overall findings
Species localization serves as a mechanism for multistationarity:
the number of states may be higher for spatially structured systems
compared to homogenous systems.
Thereby cellular computational capacity and information
processing capacity is driven by spatial organization.
Provide a method for precluding whether a system is capable of
having multistationarity, irrespective of parameter values.
Model checking, multistability, and spatial models Heather Harrington 38 / 40

103. Spatial localization: overall findings
Species localization serves as a mechanism for multistationarity:
the number of states may be higher for spatially structured systems
compared to homogenous systems.
Thereby cellular computational capacity and information
processing capacity is driven by spatial organization.
Provide a method for precluding whether a system is capable of
having multistationarity, irrespective of parameter values.
Identify necessary conditions for multistationarity, which depend
only on the shuttling rate constants.
Model checking, multistability, and spatial models Heather Harrington 38 / 40

104. Conclusions
Many methods for analyzing models are limited (e.g., simulation
time, nonlinear objective functions, among others).
Model checking, multistability, and spatial models Heather Harrington 39 / 40

105. Conclusions
Many methods for analyzing models are limited (e.g., simulation
time, nonlinear objective functions, among others).
Here, we proposed non-parametric methods for analyzing
mass-action models with data.
Model checking, multistability, and spatial models Heather Harrington 39 / 40

106. Conclusions
Many methods for analyzing models are limited (e.g., simulation
time, nonlinear objective functions, among others).
Here, we proposed non-parametric methods for analyzing
mass-action models with data.
(1) We presented a novel method for rejecting models based on
steady-state coplanarity.
Model checking, multistability, and spatial models Heather Harrington 39 / 40

107. Conclusions
Many methods for analyzing models are limited (e.g., simulation
time, nonlinear objective functions, among others).
Here, we proposed non-parametric methods for analyzing
mass-action models with data.
(1) We presented a novel method for rejecting models based on
steady-state coplanarity.
(2) We argued that compartmentalization serves as a mechanism for
multistationarity and increases information processing capacity.
Model checking, multistability, and spatial models Heather Harrington 39 / 40

108. Conclusions
Many methods for analyzing models are limited (e.g., simulation
time, nonlinear objective functions, among others).
Here, we proposed non-parametric methods for analyzing
mass-action models with data.
(1) We presented a novel method for rejecting models based on
steady-state coplanarity.
(2) We argued that compartmentalization serves as a mechanism for
multistationarity and increases information processing capacity.
We look forward to combining these parameter-free approaches
with the other spectrum of existing methods.
Model checking, multistability, and spatial models Heather Harrington 39 / 40

109. Acknowledgements
I would like to thank and acknowledge:
Kenneth Ho
Tom Thorne
Carsten Wiuf
Elisenda Feliu
Michael Stumpf
Model checking, multistability, and spatial models Heather Harrington 40 / 40

110. Acknowledgements
I would like to thank and acknowledge:
Kenneth Ho
Leverhulme Trust
Tom Thorne
Carsten Wiuf
Elisenda Feliu
Michael Stumpf
Model checking, multistability, and spatial models Heather Harrington 40 / 40

111. Acknowledgements
I would like to thank and acknowledge:
Kenneth Ho
Leverhulme Trust
Tom Thorne
Theoretical Systems Biology Group
Carsten Wiuf
Elisenda Feliu
Michael Stumpf
Model checking, multistability, and spatial models Heather Harrington 40 / 40

112. Acknowledgements
I would like to thank and acknowledge:
Kenneth Ho
Leverhulme Trust
Tom Thorne
Theoretical Systems Biology Group
Carsten Wiuf
Mathematical Biosciences Institute
Elisenda Feliu
Michael Stumpf
Model checking, multistability, and spatial models Heather Harrington 40 / 40

113. Acknowledgements
I would like to thank and acknowledge:
Kenneth Ho
Leverhulme Trust
Tom Thorne
Theoretical Systems Biology Group
Carsten Wiuf
Mathematical Biosciences Institute
Elisenda Feliu
Thank you for your attention!
Michael Stumpf
Model checking, multistability, and spatial models Heather Harrington 40 / 40

114. Future work
Is there a way to be more precise with rejecting a model using
coplanarity error?
In the absence of rigorous criteria we have to rely on heuristics,
and those heuristics essentially are the cost of getting our model
selection wrong. We ourselves don’t find this an entirely
satisfaction.
We have a na¨ hope that combining this with non-Bayesian
ıve
parametric statistics will help us solve this issue.
Is there a way to precisely determine what parameters will yield
multi-stationarity in a system with spatial localization?
We hope that combining optimization techniques with the
necessary conditions for multistationarity would improve our
understanding of how large of a parameter space is capable of
multistationarity.
Model checking, multistability, and spatial models Heather Harrington 41 / 40

115. Gr¨bner Bases
o
Manrai & Gunawardena procedure:
Let Q[a] be the polynomial ring consisting of all polynomials in the
parameters a = (k1 , . . . , kR ) with coefficients from the rational
numbers Q.
Let K be its fraction field, comprising all elements of the form f /g ,
where f , g ∈ Q[a].
Clearly, each xi ∈ K[x], the ring of all polynomials in
˙
x = (x1 , . . . , xN ) with coefficients in K.
Note that the parameters a have been absorbed into the coefficient
field K.
By performing all operations over K, we can treat a symbolically,
i.e., without specifying any particular parameter values.
Model checking, multistability, and spatial models Heather Harrington 40 / 40

116. Characterize Steady State
To characterize the steady state (x = 0):
˙
Construct the ideal J = x generated by x, consisting of all
˙ ˙
polynomials N fi xi , where each fi ∈ K[x].
i=1 ˙
Clearly, J contains all elements of K[x] that vanish at steady state.
To obtain only those elements of J that do not depend on the
variables x1 , . . . , xi , we consider the ith elimination ideal
Ji = J ∩ K[xobs ], where xobs = (xi+1 , . . . , xN ) denotes the
“observable” variables.
Use Gr¨bner bases, which are special sets of generators with the
o
so-called elimination property that if g = (g1 , . . . , gM ) is a Gr¨bner
o
basis for J under the lexicographic ordering x1 > · · · > xN , then
Ji = gobs , where gobs = g ∩ K[xobs ] are precisely those elements
of g containing only the variables xobs .
The polynomials gobs generate all elements of K[xobs ] that vanish
at steady state and so characterize the projection of the steady
state onto the variables xobs .
Model checking, multistability, and spatial models Heather Harrington 40 / 40