Non-parametric analysis of models and data

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

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


Overview: Cell decisions

Cell decisions
Cellular decision-making is necessary for preservation of homeostasis in an
organism, e.g., apoptosis, proliferation, and diﬀerentiation.



Cell decisions

Mechanisms that regulate these processes are often feedback loops.



Cell decisions

Feedbacks can aﬀect the behavior of the system (number of
response states).



Cell decisions

Feedbacks can aﬀect the behavior of the system (number of
response states).
Many models can be constructed to describe the same system.


Theoretical Systems Biology

Aims of the research group:

Reverse engineering
Inverse problems
Bayesian statistics


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.




L(θ) = P(D|θ).

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.




L(θ) = P(D|θ).

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.


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.


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.

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.



hoc fashion.
Can we do better? Can we do parameter-free model selection?



hoc fashion.
Can we do better? Can we do parameter-free model selection?
We will try ...


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




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





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.





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

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.





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

However, in practice, the required variables are rarely available.





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

In particular the velocities x = (x1 , . . . , xN ) are diﬃcult to measure, so we
˙ ˙ ˙
consider only the steady state x = 0.
˙





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

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.

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



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.



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.


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



ni Nobs
t
j=1 k=1

Ii is a polynomial in xobs that vanishes at steady state.



ni Nobs
t
j=1 k=1

We call the Ii steady-state invariants.



ni Nobs
t
j=1 k=1

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 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

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
1 2

x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...

xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...

Models
Observed
Observed Data Data
Observed Observed
Observed
. . (Steady state state measurements)
Model 1
. . . . . . . x1 . .. . x11 ˆˆ
ˆ ˆ 1 1 . .x1ˆ. ... ... We are interested in how to
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
. .
. .
x =
˙
... ... ... ...
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
check if models and data are
Assess coplanarity ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
Reduce number ˙of =
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
to include
Steady state invariants Data SteadySteady
Steady invariants
invariants
Observed
Steady
Characterize steady states
Characterize steady
.
ˆ
. . . Calculate models
x2
ˆ 1 1
1
11 1

..
parameters,

1

1

1

2
2
Data not coplanar
2
2

2



Model incompatible

Model
1 2

x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...

xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...

Models
Observed
Observed Data Data
Observed Observed
Observed
Model 1
. . . . . . . x1 . .. . x11 ˆˆ
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
. .
. .
x =
˙
... ... ... ...
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
to include
Steady state invariants Data
Observed SteadySteady
Steady invariants
invariants
Assess if the invariants and
Steady
Characterize steady states data, when transformed, lie on
Characterize steady
.
ˆ
x2
ˆ
. . . Calculate models 1 1
1
11 1
a common plane.
..
parameters,

1

1

1

2
2
Data not coplanar
2
2

2



Model incompatible

Model
1 2

x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...

xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...

Models
Observed
Observed Data Data
Observed Observed
Observed
Model 1
. . . . . . . x1 . .. . x11 ˆˆ
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
. .
. .
x =
˙
... ... ... ...
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
to include
Steady invariants
invariants
Steady
Characterize steady
.
ˆ
x2
ˆ
1
11 1
a common plane.
..
parameters,
In a sense, we are checking the
parameters, and data coplanarity of transformed
Data coplanar Data not coplanar invariants and data.
1

1

1

2
2
Data not coplanar
2
2

2



Model incompatible

Model
1 2

x1 = x˙11 =. xx1=......x.1..=. . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ . .˙ ˙ = ˙ .. ...
.
x = 1
. . . . . . ...... . .... . . .. ...... . . . ...... .. . ....... . . . . . .
..
. .. . ...

xN = x˙N =xxN=.....x. ...= . .. ...... . . . ...... .. . ....... . . . . . .
˙ ˙ N .=. ˙ = ˙ N . .
x
. ˙N . ...

Models
Observed
Observed Data Data
Observed Observed
Observed
Model 1
. . . . . . . x1 . .. . x11 ˆˆ
x1 =˙ 1 = . . . . . .
˙ x x
ˆ
. .
. .
x =
˙
... ... ... ...
x
ˆ 2 ˆ.
... . . . . .. .. .. ........ . . . .. .. . . . .
. ..
.
.
xN =N = . . . . . .
˙ x ˙ . . . . . . . . .xm . .. .. .xm xxm . .ˆ.m ...... . . . . . . .
of
ˆ
variables
ˆ m ˆˆ
x
ˆ
...
m x
... ... ...
coplanar.
to include
Steady invariants
invariants
Steady
Characterize steady
.
ˆ
x2
ˆ
1
11 1
a common plane.
..
parameters,
In a sense, we are checking the
parameters, and data coplanarity of transformed
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 incompatible

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.


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



n Nobs
t
j=1 k=1

x
n
I (ξ; α) = α i ξi .
i=1

Let ϕ: xobs → ξ.



n Nobs
t
j=1 k=1

x
n
I (ξ; α) = α i ξi .
i=1

ˆ
Compatibility implies that the transformed variable ξ = ϕ(ôbs )x
x
corresponding to any observation ôbs with coordinates
ˆ ˆ
(ξ1 , . . . , ξn ), lies on the plane defined by the coefficients α.



n Nobs
t
j=1 k=1

x
n
I (ξ; α) = α i ξi .
i=1

ˆ
Compatibility implies that the transformed variable ξ = ϕ(ôbs ) x
x
corresponding to any observation ôbs 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 î = ϕ(ôbs,i ) are coplanar.
x

Assess coplanarity: SVD

ˆ
Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξ i .



ˆ
Then the data are coplanar if and only if Ξα = 0 for some column
vector α = 0.



ˆ
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.



ˆ
vector α = 0.
Thus, assuming that m > n, if the smallest singular value σn of Ξ
is nonzero, then the data cannot be coplanar.



ˆ
vector α = 0.
More generally, σn = min α =1 Ξα gives the least squares
deviation of the data from coplanarity under the scaling constraint
α = 1.



ˆ
vector α = 0.
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.


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.


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 (ôbs,i ; a) = 0), allowing for a simple direct solution.
x


Assess coplanarity: noise in data

To account for the presence of noise, let x x
= ∆ôbs / ôbs be the relative error
x
in a measurement ôbs .



x
(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 .



x
ˆ ˆ x
ϕ(x) x
β (x) = ϕ(x)
(2) To quantify the overall level of noise across all measurements, we define
√
x x
β = β / m, where β = (β(ôbs,1 ), . . . , β(ôbs,m )) is a vector containing
each noise amplification factor, and the effective relative error as eff = β .



x
ˆ ˆ x
ϕ(x) x
β (x) = ϕ(x)
(2) To quantify the overall level of noise across all measurements, we define
√
x x
β = β / m, where β = (β(ôbs,1 ), . . . , β(ôbs,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.


Example application: multisite phosphorylation

Distributive Phosphorylation of MAPK
Disassociation

MAPKK MAPKK MAPKK

MAPKK MAPKK
P P P P P P



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



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.


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


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.



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


Multisite phosphorylation: coplanarity results


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).


[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
on the space of cluster tuples. However, the reaction notation is

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