non-parametric analysis of models and data

Non-parametric analysis of mass-action models and data

Heather Harrington

Theoretical Systems BiologyImperial 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 ThorneTheoretical Systems Biology, Imperial College London

(3) Multistationarity via spatial compartmentalization� Elisenda Feliu

Institute of Mathematical Sciences, University of Copenhagen

� Carsten WiufInstitute of Mathematical Sciences, University of Copenhagen

(4) Conclusions

Overview: Cell decisions

Cell decisionsCellular 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 ofresponse 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. Ina statistical framework, for example, we use the likelihood

L(θ) = P(D|θ).

Maximizing the likelihood gives us the value of the parameter θ thatmaximizes the probability of observing the data D.

Model Selection

If, however, we have a set of candidate models, M1,M2, . . . we haveto employ other criteria to choose which model is best.

The Akaike and Bayesian information criteria, for example, penalizemodels that are overly complex.

L(θ) = P(D|θ).

Model Selection

If, however, we have a set of candidate models, M1,M2, . . . we haveto employ other criteria to choose which model is best.

The Akaike and Bayesian information criteria, for example, penalizemodels that are overly complex.

L(θ) = P(D|θ).

Model Selection

If, however, we have a set of candidate models, M1,M2, . . . we haveto employ other criteria to choose which model is best.The Akaike and Bayesian information criteria, for example, penalizemodels that are overly complex.

Bayesian Inference

� In the Bayesian framework, parameter inference centers aroundfinding the posterior distribution

P(θ|D) =P(D|θ)π(θ)∫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 (marginallikelihood): ∫

P(D|θ)π(θ)dθ,

which is calculated by integrating the likelihood over the parameterspace.

� Given a set of models, we prefer the one for which the evidence isthe highest.

Bayesian Inference

� In the Bayesian framework, parameter inference centers aroundfinding the posterior distribution

P(θ|D) =P(D|θ)π(θ)∫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 (marginallikelihood): ∫

P(D|θ)π(θ)dθ,

which is calculated by integrating the likelihood over the parameterspace.

� Given a set of models, we prefer the one for which the evidence isthe highest.

The Problem of Model Selection

� In maximum likelihood estimation (or in optimization approachesmore generally) model selection needs to be addressed in an adhoc fashion.

� Bayesian approaches integrate out parameter dependencies alongthe way towards model selection.

� In a Bayesian framework, model selection is natural butcomputationally expensive: often prohibitively expensive.

� 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∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

� Dynamics from mass action kinetics:

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

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

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

� We eliminate these variables from the equations if possible.

N∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

N∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

N∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

These equations provide a quantitative description of the model.

N∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

These equations provide a quantitative description of the model.In principle, the equations can be used to test the model’s validity byassessing the degree to which they are satisfied by observed data.

N∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

N∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

N∑j=1

sijXjki−→

N∑j=1

s′ijXj , i = 1, . . . ,R

xi =R∑j=1

(s′ji − sji

) N∏k=1

xsjkj , i = 1, . . . ,N

Background: tools from algebraic geometry

For simple systems, this elimination can be done by hand. But ingeneral, a more systematic approach is often required.

� Grobner basis nonlinear generalization of Gaussian elimination.

� Elimination ideal allows us to perform elimination without havingto know the numerical values of the parameters a = (k1, . . . , kR)by treating them symbolically.

� Grobner bases automatically give equations that are fulfilled by anysteady-state solution and only involve a subset of variables.

Background: variable elimination and invariants

After variable elimination we are left with:

Ii (xobs; a) =

ni∑j=1

fij (a)

Nobs∏k=1

xtijkk , i = 1, . . . ,Ninv. (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 betweenobservable variables that hold a steady state for any givenrealization of parameter values, regardless of other factors (such asinitial conditions).

Ii (xobs; a) =

ni∑j=1

fij (a)

Nobs∏k=1

xtijkk , i = 1, . . . ,Ninv. (1)

Ii (xobs; a) =

ni∑j=1

fij (a)

Nobs∏k=1

xtijkk , i = 1, . . . ,Ninv. (1)

Ii (xobs; a) =

ni∑j=1

fij (a)

Nobs∏k=1

xtijkk , i = 1, . . . ,Ninv. (1)

Assessing coplanarity: overview

Data coplanar

Data not coplanar

Data coplanar

Data not coplanar

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

(Steady state measurements)

x1. . .

x2. . .

... . . .

xm. . .

Steady state invariants

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Calculate elimination ideal

Assess coplanarity

Reduce number of variables

to include only observables

Characterize steady states of models

Transform model variables,

parameters, and data

Assess coplanarity

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Transform model variables, parameters, and data

Data not coplanar

Model compatible

Model incompatible

Data not coplanar

Model compatible

Model incompatible

Models

Model 1 . . . Model L

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

� We are interested in how tocheck if models and data arecoplanar.

� Assess if the invariants anddata, when transformed, lie ona common plane.

� In a sense, we are checking thecoplanarity of transformedinvariants and data.

� Model rejection can then beperformed by assessing thedegree to which the transformeddata deviate from coplanarity.

Data coplanar

Data not coplanar

Data coplanar

Data not coplanar

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Data not coplanar

Model compatible

Model incompatible

Data not coplanar

Model compatible

Model incompatible

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Data coplanar

Data not coplanar

Data coplanar

Data not coplanar

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Data not coplanar

Model compatible

Model incompatible

Data not coplanar

Model compatible

Model incompatible

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Data coplanar

Data not coplanar

Data coplanar

Data not coplanar

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Data not coplanar

Model compatible

Model incompatible

Data not coplanar

Model compatible

Model incompatible

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Data coplanar

Data not coplanar

Data coplanar

Data not coplanar

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Data not coplanar

Model compatible

Model incompatible

Data not coplanar

Model compatible

Model incompatible

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Assess coplanarity: question

Data coplanarity

Given a set of steady-state measurements xobs,i for i = 1, . . . ,m, andmodel with steady-state invariants I = {I1, . . . , INinv

}, we need aprocedure for deciding whether it is possible that the invariant iscompatible with the data, i.e.,

I (xobs,i ; a) = 0, 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

I (xobs; a) =n∑

fj (a)

Nobs∏k=1

xtjkk (3)

To assess coplanarity (I (xobs,i ; a) = 0), we rewrite eq. 3 as:

I (ξ;α) =n∑

αiξi .

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

� In other words, compatibility with the data xobs,i implies that the

corresponding transformed data ξi = ϕ(xobs,i ) are coplanar.

I (xobs; a) =n∑

fj (a)

Nobs∏k=1

xtjkk (3)

I (ξ;α) =n∑

αiξi .

Let ϕ: xobs → ξ.

I (xobs; a) =n∑

fj (a)

Nobs∏k=1

xtjkk (3)

I (ξ;α) =n∑

αiξi .

I (xobs; a) =n∑

fj (a)

Nobs∏k=1

xtjkk (3)

I (ξ;α) =n∑

αiξi .

Assess coplanarity: SVD

� Let Ξ ∈ Rm×n be the matrix whose rows consist of the ξi .

� Then the data are coplanar if and only if Ξα = 0 for some columnvector α 6= 0.

� Such a vector resides in the null space of Ξ, spanned by the rightsingular vectors of Ξ corresponding to zero singular values.

� Thus, assuming that m > n, if the smallest singular value σn of Ξis nonzero, then the data cannot be coplanar.

� More generally, σn = min‖α‖=1 ‖Ξα‖ gives the least squaresdeviation of the data from coplanarity under the scaling constraint‖α‖ = 1.

� This measure depends only on the data and is thereforeparameter-free.

Assess coplanarity: remarks

(1) Note that this applies for any choice of α, regardless of whetherit can be realized by the original parameters a.

(2) In this sense, the condition of small σn provides a necessary butnot sufficient criterion for model compatibility.

(3) This is in contrast to traditional approaches based on parameterfitting, which provide a sufficient but not necessary condition,since local minima may prevent a compatible model from beingfitted correctly.

(4) The additional degrees of freedom introduced by neglecting thefunctional forms fj effectively linearizes the compatibilitycondition (I (xobs,i ; a) = 0), allowing for a simple direct solution.

Assess coplanarity: remarks

(1) Note that this applies for any choice of α, regardless of whetherit can be realized by the original parameters a.

(2) In this sense, the condition of small σn provides a necessary butnot sufficient criterion for model compatibility.

(3) This is in contrast to traditional approaches based on parameterfitting, which provide a sufficient but not necessary condition,since local minima may prevent a compatible model from beingfitted correctly.

(4) The additional degrees of freedom introduced by neglecting thefunctional forms fj effectively linearizes the compatibilitycondition (I (xobs,i ; a) = 0), allowing for a simple direct solution.

Assess coplanarity: noise in data

To account for the presence of noise, let ε = ‖∆xobs‖/‖xobs‖ be the relative errorin a measurement xobs.

(1) Then from the perturbation equation ξ + ∆ξ = ϕ (x + ∆x) , this ispropagated to the transformed variables as ‖∆ξ‖/‖ξ‖ ∼ β(xobs)ε, where

β (x) = ‖∇ϕ(x)‖‖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 defineβ = ‖β‖/

√m, where β = (β(xobs,1), . . . , β(xobs,m)) is a vector containing

each noise amplification factor, and the effective relative error as εeff = βε.

(3) Since the introduction of noise in Ξ of order εeff in general gives a lowerbound of σn ∼

√mεeff ∼ ‖β‖ε, we should reject the model only if σn � ‖β‖ε.

We therefore define the coplanarity error

∆ =σn

‖β‖ε ,

in terms of which the rejection criterion is simply ∆� 1. Observe that as εincreases, ∆ decreases, so we lose rejection power, as expected.

∆ =σn

‖β‖ε ,

∆ =σn

‖β‖ε ,

∆ =σn

‖β‖ε ,

Example application: multisite phosphorylation

Distributive Phosphorylation of MAPK

Disassociation

Processive Phosphorylation of MAPK

SlideMAPKK

Dephosphorylation can also occur in a processive or a distributivemanner. We would like to know which mechanism operates in vivo.

Disassociation

SlideMAPKK

Disassociation

SlideMAPKK

Multisite phosphorylation: eliminate variables

K + Suau−−⇀↽−−bu

KSucuv−−→ K + Sv ,

F + Svαv−−⇀↽−−βv

FSvγvu−−→ F + Su,

E + S00 ES00

E + S11

E + S01

E + S10

Phosphorylation

F + S11FS11

F + S01

F + S10

F + S00

Dephosphorylation

� Each enzyme can be either processive (P),where more than one phosphate modificationmay be achieved in a single step, ordistributive (D), where only one modificationis allowed before the enzyme dissociates fromthe substrate.

� Models: PP, PD, DP and DD; where the firstletter designates the mechanisms of thekinase, and the second, that of thephosphatase.

� We considered only the concentrationsxobs = (s00, s01, s10, s11) as observable, andwere able to eliminate all other variablesexcept the concentration f of F from thedynamics of each model.

Multisite phosphorylation: assess coplanarity

� Each model has three steady-state invariants.

� Invariants share same transformed variables ξ = ϕ(xobs) so onlythe kinase is discriminative.

Data generated under this model: PP/PD DP/DD

Reject model PP/PD? No No

Reject model DP/DD? Yes No

ξPP/PD =(s00s10, s00s11, s01s10, s01s11, s

210, s10s11

ξDP/DD =(s00s11, s01s10, s01s11, s

210, s10s11

Multisite phosphorylation: coplanarity results

Fig. 2. Selection of multisite phosphorylation models. (A) Coplanarity error � of thesteady-state invariants of the PP/PD (left) and DP/DD (right) models along time coursetrajectories simulated from the PP model, corrupted by various levels of noise (lined,✏ = 10�9; dashed, ✏ = 10�6; dotted, ✏ = 10�3). At each noise level, the errors forthree invariants are shown (blue, I1; green, I2; red, I3). (B) Coplanarity error � of DP/DDinvariants on PP data at steady state as a function of the noise level ✏; invariants coloredas in (A). The shaded region indicates the regime over which the DP/DD models can bereliably rejected (� & 100). (C) Invariant error ✓ for each model (blue, PP; green, PD;red, DP; cyan, DD) on data generated from the PP (top left), PD (top right), DP (bottomleft), and DD (bottom right) models.

Footline Author PNAS Issue Date Volume Issue Number 7

Examples: apoptosis activation

Crosslinking model

Chapter 7. Fas trimerization model 145

Figure 7.4: Comparison with the crosslinking model. (A) Process diagram (comply with the SBGN Pro-

cess Description language Level 1 (Le Novere et al., 2009)) of the crosslinking model. (B) Variation of

the steady-state signaling Fas fraction !! with respect to the model parameter ". (C) Minimization errors# of the steady-state invariants $H and $C 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).

6.3 Methods 108

6.3.2 Steady-state abstraction of oligomerization kinetics

The oligomerization kinetics of the DISC, MAC, and the apoptosome are abstracted us-

ing steady-state results; this abstraction is a demonstration of a simple technique for

modularization and model reduction. For an oligomer X with intermediate structures

X1, . . . , Xn and dynamics

dt= f ([X] , [X]1 , . . . , [X]n) ! µ [X] ,

where f is the oligomerization rate function and µ the degradation rate, use the steady-

state approximation f " fss # [X]ss. This allows the modeling of only the final complex

and hence significant simplification of the dynamical equations. Although the time de-

pendence of the oligomerization rate is neglected, information regarding the long-term

behavior is retained. For the present application, f = [X]ss with proportionality constant

The abstractions for each of the DISC, MAC, and apoptosome modules are described

below, where the notation is understood to apply only within each module.

DISC module

The DISC oligomerization kinetics are simplified from the crosslinking model (Delisi,

1980; Perelson, 1984, 1981) of Lai and Jackson, 2004 and follow the reactions

FasL + FasR3kf!!!"!!kr

FasL-FasR,

FasL-FasR + FasR2kf!!!"!!2kr

FasL-FasR2,

FasL-FasR2 + FasRkf!!!"!!3kr

FasL-FasR3,

Lai & Jackson (2004) Math Biosci Eng

� The activation signal is defined for each model.

� Each model has one steady-state invariant.

Crosslinking model

6.3 Methods 108

dt= f ([X] , [X]1 , . . . , [X]n) ! µ [X] ,

DISC module

FasL-FasR,

FasL-FasR2,

FasL-FasR3,

Cluster model

that the number of receptors that each ligand can coordinate is atleast three. This hence gives a theory for the trimeric character ofFasL. Furthermore, at high concentrations, for example, throughreceptor pre-association [30–32] or localization onto lipid rafts[33], irreversible bistability is achieved, implementing a perma-nent cell death decision. Thus, our model suggests a primary rolefor death receptors in deciding cell fate. Moreover, our results offernovel functional interpretations of ligand trimerism and receptorpre-association and localization within the unified context ofbistability.

Results

Model formulationConstructing a mathematical model of Fas dynamics is not

entirely straightforward as receptors can form highly oligomeric

clusters [27,33]. A standard dynamical systems description wouldtherefore require an exponentially large number of state variablesto account for all combinatorial configurations. To circumventthis, we considered the problem at the level of individual clusters.Each cluster can be represented by a tuple denoting the numbersof its molecular constituents, the cluster association being implicit,so only these molecule numbers need be tracked.In our model, a cluster is indexed by a tuple (L, X , Y , Z),

where L represents FasL and X , Y , and Z are three posited formsof Fas, denoting closed, open and unstable, and open and stable,i.e., active and signaling, receptors, respectively. Within a cluster,we assumed a complete interaction graph and defined thereactions

kcY , !1a"

Z DAku Y , !1b"

jYz i{j! "Z DAki! "s

j{k! "Yz i{jzk! "Z,i~2, . . . ,m,

j~1, . . . ,i,

k~1, . . . ,j,

>:!1c"

LzjYz i{j! "Z DAki! "l

Lz j{k! "Yz i{jzk! "Z,i~2, . . . ,n,

j~1, . . . ,i,

k~1, . . . ,j:

>:!1d"

The first reaction describes spontaneous receptor opening andclosing; the second, constitutive destabilization of open Fas; thethird, ligand-independent receptor cluster-stabilization; and thefourth, ligand-dependent receptor cluster-stabilization (Figure 2).The orders of the cluster-stabilization events are limited by theparameters m and n, which capture the effects of receptor densityand Fas coordination by FasL, respectively. Although only pair-stabilization (m~n~2) has been observed experimentally [27],higher-order analogues, for example, as facilitated by globularinteractions, are not unreasonable.

Figure 1. Cartoon of model interactions. The transmembrane death receptor Fas natively adopts a closed conformation, but can open to allowthe binding of FADD, an adaptor molecule that facilitates apoptotic signal transduction. Open Fas can self-stabilize via stem helix and globularinteractions, which is enhanced by receptor clustering through association with the ligand FasL.doi:10.1371/journal.pcbi.1000956.g001

Author Summary

Many prominent diseases, most notably cancer, arise froman imbalance between the rates of cell growth and deathin the body. This is often due to mutations that disrupt acell death program called apoptosis. Here, we focus on theextrinsic pathway of apoptotic activation which is initiatedupon detection of an external death signal, encoded bya death ligand, by its corresponding death receptor.Through the tools of mathematical analysis, we find that anovel model of death ligand-receptor interactions basedon recent experimental data possesses the capacity forbistability. Consequently, the model supports threshold-like switching between unambiguous life and death states;intuitively, the defining characteristic of an effective celldeath mechanism. We thus highlight the role of deathreceptors, the first component along the apoptoticpathway, in deciding cell fate. Furthermore, the modelsuggests an explanation for various biologically observedphenomena, including the trimeric character of the deathligand and the tendency for death receptors to colocalize,in terms of bistability. Our work hence informs themolecular basis of the apoptotic point-of-no-return, andmay influence future drug therapies against cancer andother diseases.

Bistability in Apoptosis by Receptor Clustering

PLoS Computational Biology | www.ploscompbiol.org 2 October 2010 | Volume 6 | Issue 10 | e1000956

Formally, these reactions are to be interpreted as state transitionson the space of cluster tuples. However, the reaction notation issuggestive, highlighting the contribution of each elementary event,which we modeled using constant reaction rates (for simplicity, weset uniform rate constants k(i)s and k(i)l for all ligand-independentand -dependent cluster-stabilization reactions of molecularity i,respectively). Then on making a continuum approximation, wereinterpreted the molecule numbers as local concentrations andapplied the law of mass action to produce a dynamical system foreach cluster in the concentrations (l, x, y, z) of (L, X , Y , Z).Validity of the model requires that the molecular concentrations arenot too low and that the timescale of receptor conformationalchange is short compared to that of cluster dissociation.To study the long-term behavior of the model, we solved the

system at steady state (denoted by the subscript ?). Introducingthe nondimensionalizations

s, !2a"

s, !2b"

s, !2c"

s, !2d"

t~kct, !2e"

where s is a characteristic concentration and t is time, and

ko~kokc

, !3a"

ku~kukc

, !3b"

k i! "s ~

ki! "s si{1

kc, i~2, . . . ,m, !3c"

k i! "l ~

ki! "l si

kc, i~2, . . . ,n, !3d"

this is

j?~s{f?1zko

, !4a"

g?~koj?, !4b"

where s~jzgzf is the nondimensional total receptor density,and f? is given by considering

k i! "s

gifi{jXj

k i! "l

gifi{jXj

k{kuf !5"

and solving df=dt~0 with (j, g, f).(j?, g?, f?), a polynomial

in f? of degree maxfm, ng. Clearly, the model is bistable only if

maxfm, ng§3 (two stable nodes must be separated by an unstable

node as the model is effectively one-dimensional in f).We used f as a measure of the apoptotic activation of a cluster.

In principle, all open receptors contribute to apoptotic signaling,but g is small, at least at steady state (since ko%1 due to theassumed prevalence of the closed form [29]), and so can beneglected.

Bistability and receptor clusteringWhile nmeasures the coordination capacity of FasL and hence

may be equated with its oligomeric order (e.g., n~3 in thebiological context), an appropriate value for m, relating to thetotal receptor concentration, is somewhat more elusive. There-fore, we began our analysis by performing a simple receptordensity estimate. Approximating the cell as a cube of lineardimension*10 mm, the associated volume of*1 pL implies thecorrespondence 1 nM *600 molecules *10{6 molecules/nm2

on restricting to the membrane, i.e., by averaging over thesurface area of *600 mm2. Thus, for a conservative receptorconcentration estimate of 100 nM [7,9,12,13], the number of Fasmolecules in the neighborhood of each receptor is only *1,assuming a charateristic size of 100 nm. We hence found thatreceptors may be very sparsely distributed. In this low densitymode, high-order Fas interactions in the absence of ligand can beneglected (m~2). Therefore, in this context, bistability is possibleonly if n§3, and the trimerism of FasL thus demonstrates thelowest-order complexity required for bistability.From the form of df=dt, this bistability is reversible as a

function of the FasL concentration l since the governingpolynomial for f? is of degree only m~2 at l~0. This suggeststhat at the cluster level, the cell death decision can be reversed,which may have adverse effects on cellular and genomic integrity.However, irreversible bistability at higher receptor densities mayalso be achieved. Researchers have observed tendencies for deathreceptors both to pre-associate as dimers or trimers [30–32] and toselectively localize onto membrane lipid rafts [33]. The result ofeither of these processes may be to increase the local receptorconcentration. In this high density mode, we set m§3, as thepreceeding approximation is no longer valid. Irreversible bist-ability then becomes attainable, representing a committed celldeath decision.

Figure 2. Schematic of cluster-stabilization reactions. Examplesof ligand-independent cluster-stabilization reactions involving unstable(Y ) and stable (Z) open receptors of molecularities two (A), three (B),and four (C). Higher-order reactions follow the same pattern. Ligand-dependent reactions are identical except that FasL (L) must be addedto each reacting state.doi:10.1371/journal.pcbi.1000956.g002

Ho & Harrington (2010) PLoS Comput Biol

Crosslinking model

6.3 Methods 108

dt= f ([X] , [X]1 , . . . , [X]n) ! µ [X] ,

DISC module

FasL-FasR,

FasL-FasR2,

FasL-FasR3,

Cluster model

Results

kcY , !1a"

Z DAku Y , !1b"

j{k! "Yz i{jzk! "Z,i~2, . . . ,m,

j~1, . . . ,i,

k~1, . . . ,j,

>:!1c"

Lz j{k! "Yz i{jzk! "Z,i~2, . . . ,n,

j~1, . . . ,i,

k~1, . . . ,j:

>:!1d"

Author Summary

s, !2a"

s, !2b"

s, !2c"

s, !2d"

t~kct, !2e"

ko~kokc

, !3a"

ku~kukc

, !3b"

k i! "s ~

ki! "s si{1

kc, i~2, . . . ,m, !3c"

k i! "l ~

ki! "l si

kc, i~2, . . . ,n, !3d"

this is

j?~s{f?1zko

, !4a"

g?~koj?, !4b"

k i! "s

gifi{jXj

k i! "l

gifi{jXj

k{kuf !5"

Crosslinking model

6.3 Methods 108

dt= f ([X] , [X]1 , . . . , [X]n) ! µ [X] ,

DISC module

FasL-FasR,

FasL-FasR2,

FasL-FasR3,

Cluster model

Results

kcY , !1a"

Z DAku Y , !1b"

j{k! "Yz i{jzk! "Z,i~2, . . . ,m,

j~1, . . . ,i,

k~1, . . . ,j,

>:!1c"

Lz j{k! "Yz i{jzk! "Z,i~2, . . . ,n,

j~1, . . . ,i,

k~1, . . . ,j:

>:!1d"

Author Summary

s, !2a"

s, !2b"

s, !2c"

s, !2d"

t~kct, !2e"

ko~kokc

, !3a"

ku~kukc

, !3b"

k i! "s ~

ki! "s si{1

kc, i~2, . . . ,m, !3c"

k i! "l ~

ki! "l si

kc, i~2, . . . ,n, !3d"

this is

j?~s{f?1zko

, !4a"

g?~koj?, !4b"

k i! "s

gifi{jXj

k i! "l

gifi{jXj

k{kuf !5"

Apoptosis activation: coplanarity results

Fig. 3. Selection of cell death signaling models. (A) Coplanarity error� of the steady-stateinvariants of the crosslinking (left) and cluster (right) models along time course trajectoriessimulated from the cluster model, corrupted by various levels of noise (blue, ✏ = 10�9;green, ✏ = 10�6; red, ✏ = 10�3). (B) Coplanarity error � of model invariants (blue,crosslinking; green, cluster) on cluster data at steady state as a function of the noiselevel ✏. The shaded region indicates the regime over which the crosslinking model canbe reliably rejected (� & 100). (C) Invariant error ✓ for each model (blue, crosslinking;green, cluster) on data generated from the crosslinking (left) and cluster (right) models.

8 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author

Asessing coplanarity: overall findings

Data coplanar

Data not coplanar

Data coplanar

Data not coplanar

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Models

Model 1 Model 2 . . . . . .

x1 = . . . . . . . . . . . .

... . . . . . . . . . . . .

xN = . . . . . . . . . . . .

Assess coplanarity

Data not coplanar

Model compatible

Model incompatible

Data not coplanar

Model compatible

Model incompatible

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

Models

x1 = . . . . . . . . .

... . . . . . . . . .

xN = . . . . . . . . .

Observed Data

x1. . .

x2. . .

... . . .

xm. . .

� Novel model selection scheme based on steady-state coplanaritythat does not require parameter estimation.

� Method is not always effective– steady-state invariants may notexist, 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 thanoptimization methods.

� Potential new class of model selection methods based on geometricstructure.

Cellular states

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 4010-4

Stimulus (Etot)5010 20 30 40

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

Stimulus (Etot)25 4010

Cellular information processing

Information processingOne 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 aregime of conditions where a system can occupy more than onestate.

� If more than one state exists (e.g., switch-like systems), this iscalled multistationarity.

� The number of states is linked to the flexibility in the decisionmaking of a cell.

Enzyme sharing as a cause of multistationarity

build the motifs from a one-site phosphorylation cyclewhich is monostable [16–19] and shown in Motif (a).A specific kinase (phosphatase) catalyses phosphoryla-tion (dephosphorylation) and all modifications can bereversed. In general, protein phosphoforms are denotedby S andP (figure 1). If one phosphoform is converted intoanother, an arrow is drawn and the enzyme (E or F)catalysing the reaction is indicated.

Motifs (a)–(d) cover different possibilities for a one-site modification process. In Motif (b), the same enzymecatalyses phosphorylation and dephosphorylation.Motifs (c) and (d) account for competition betweenkinases and/or phosphatases to catalyse the samemodification(s).

In eukaryotes, phosphorylation of most proteinstakes place in more than one site [20], potentially withdifferent biological effects [21]. Combination of twoone-site cycles into a two-site sequential cycle yieldsthree motifs: (e) all enzymes are different, (f) onlyone kinase but two phosphatases, and (g) one kinaseand one phosphatase. By symmetry, Motif (f) rep-resents as well a motif with one phosphatase but twokinases. We assume for simplicity that both phos-phorylation and dephosphorylation proceed in asequential and distributive manner [22]—that is, onesite is (de)phosphorylated at a time in a specific order.

Motif (h) represents one-site modification of twosubstrates that share the same kinase but use different

phosphatases. This motif represents by symmetry also asystem with a shared phosphatase. If both the kinaseand the phosphatase are shared, we obtain Motif (i).

Finally, two one-site modification cycles can be com-bined in a cascade motif, where the activated substrateof the first cycle acts as the kinase of the next. Theinterplay between enzymes is represented by three cas-cades: ( j) dephosphorylation at each layer uses differentphosphatases, (k) the phosphatase is not layer specific,and (l) the kinase of the first layer catalyses the modifi-cation in the second layer as well.

2.2. Mathematical modelling

We assume that any modification S! S* follows theclassical Michaelis–Menten mechanism in which anintermediate complex Z is formed reversibly but dis-sociates into product and enzyme G irreversibly:

S !G !a

bZ "c! S# !G

The phosphate donor, generally ATP, is assumed tobe in large constant concentration and hence embeddedinto the rate constants. Imposing mass action kinetics,the species concentrations over time can be modelledby a system of polynomial differential equations. Forexample, in Motif (a) the equations are (here E alsorefers to the concentration of the kinase E, and similarly

one-site modification

S0 S1 S0 S1 S0 S1

S2 S0 S1 S2 S0 S1 S2

S0 S1 S0 S1 S0 S1

(b) (c) (d)E

(c) E1, E2 E1, E2

F1 ,F2

two-site modification

(e) E1 E2

F1 F2 F1 F2

( f )E E ( g)

modification of two substrates

P0 P1 P0 P1 P0 P1

E (i) E

two-layer cascade

Figure 1. Motifs composed of one or two one-site cycles. Motifs with purple label, and only these, admit multiple biologicallymeaningful steady states. Si and Pi are substrates with i ! 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.

Enzyme sharing and multi-stationarity E. Feliu and C. Wiuf 1225

J. R. Soc. Interface (2012)

on May 1, 2012rsif.royalsocietypublishing.orgDownloaded from

Feliu & Wiuf (2012) J R Soc Interface

Protein kinase cascades

Protein kinase cascadesA canonical system for investigating multistationarity are protein kinase cascades,

e.g., mitogen activated protein kinase (MAPK).

Plasma membrane

Cytoplasm

Nucleus Nucleus

Cytoplasm

Figure 1: Spatial signaling schematic.

0 5 10 15 20 25 30 35 40 45 500

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 40

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

Figure 2: One cycle shuttling toy model.

� The ultimate function of MAPK is toinitiate transcriptional responses.

� Spatial organization plays apronounced role to increasing thebiological information processing.

� We find that compartmentalizationincreases the number of states thatcan become simultaneously accessibleto the cell.

Plasma membrane

Cytoplasm

Nucleus Nucleus

Cytoplasm

0 5 10 15 20 25 30 35 40 45 500

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 40

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

Plasma membrane

Cytoplasm

Nucleus Nucleus

Cytoplasm

0 5 10 15 20 25 30 35 40 45 500

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 40

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

Plasma membrane

Cytoplasm

Nucleus Nucleus

Cytoplasm

0 5 10 15 20 25 30 35 40 45 500

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 40

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

One-site model

Plasma membrane

Cytoplasm

Nucleus Nucleus

Cytoplasm

0 5 10 15 20 25 30 35 40 45 500

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 40

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

2 Molecular localization causes multistability in biochemical reaction networks

Cytoplasm

Nucleus

S S⇤

Figure 1: Shuttling of a one-site phosphorylation cycle between the nucleus and the cytoplasm.

2 Shuttling in a one-site phosphorylation cycle

2.1 Reactions

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). Phosphorylationand dephosphorylation are assumed to follow a Michaelis-Menten mechanism (see below and maintext). This motif cannot admit multiple steady states, and it is monostable [4].

To study the effect of compartmentalization, we asume that the species S, S⇤, E, X can shuttlebetween the cytoplasm and the nucleus (see Figure 1). We let Zc denote species Z in the cytoplasm.Then, the reactions in play are as follows:

• Reactions in the nucleus:

E + Sk1

// Xk2

k3// E + S⇤ F + S⇤ k4

// Yk5

k6// F + S

• Reactions in the cytoplasm:

Ec + Sck7

k9// Ec + Sc F c + Sc⇤ k10

// Y c

k12// F c + Sc⇤

• Shuttling reactions:

oo Xk14

oo Sk15

oo S⇤ k16// Sc⇤

To ease the notation below, we have changed the notation of the reaction constants k⇤ in the mainModel checking, multistability, and spatial models Heather Harrington 28 / 40

Determining whether a system is capable ofmultistationarity

� 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/totalamounts can elicit multistationarity.

� Failure of the Jacobian injectivity criterion is not sufficient toconclude 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).

Multistationarity by localization

Species shuttling conditions7 Compartment shuttling

# 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}{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}

{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 }{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.

2.4 Sets of shuttling species

We next inspected what the sets of shuttling species that provide multistationarity are. Theresults are summarized in Table 1. The systematic way employed to classify each motif is thefollowing. First, we check if the systems fulfill the conditions of Jacobian conjecture and decideif the system is injective. If the coefficients of the polynomial in x given by the determinant ofthe Jacobian (as above) are all positive, then the system cannot exhibit multistationarity, forany set of total amounts. If this criterion fails, then we have use the CRNT toolbox.

We have obtained that if only one species shuttles, then multistationarity cannot occur.That is, at least two species, e.g. {S1, X} or {S0, Y }, are required to obtain multistationarityfor certain total amounts and rate constants. Adding more shuttling species keeps multista-tionarity.

3 Shuttling in a two-site phosphorylation cycle

In eukaryotes, most protein phosphorylation events take place in more than one site. It is wellknown that multisite phosphorylation can cause multistationarity by itself [5, 6]. However,multistationarity does not occur for all choice of rate constants.

We next investigate if shuttling of species can induce multistationarity in two-site (sequen-tial) phosphorylation systems that independently cannot exhibit multistationarity. For that, we

One-site phosphorylation system. For all possible sets of shuttling species it is

indicated if the system has the capacity for multiple steady states or not.

Necessary conditions for monostability

There are two conditions suffice to guarantee monostationarity,namely:

C2 =k9k12(k15 − k16)(k18 − k17) + k9k14k15k16 + k12k14k15k16

+ k12k14k16k17 + k9k15k16k18 + k12k15k16k18 + k12k16k17k18 > 0,

+ k9k14k19k20 + k12k14k19k20 + k9k18k19k20 + k12k18k19k20 > 0.

There are two conditions suffice to guarantee monostationarity,namely:

+ k12k14k16k17 + k9k15k16k18 + k12k15k16k18 + k12k16k17k18 > 0,

+ k9k14k19k20 + k12k14k19k20 + k9k18k19k20 + k12k18k19k20 > 0.

By inspection of these two expressions, we conclude thatmultistationarity 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.

(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.

Cytoplasm

Nucleus

S S⇤

2.1 Reactions

E + Sk1

// Xk2

k3// E + S⇤ F + S⇤ k4

// Yk5

k6// F + S

Ec + Sck7

k9// Ec + Sc F c + Sc⇤ k10

// Y c

k12// F c + Sc⇤

oo Xk14

oo Sk15

oo S⇤ k16// Sc⇤

To ease the notation below, we have changed the notation of the reaction constants k⇤ in the main

(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.

Note that these only involve the rate constants for the shuttlingreactions.

Cytoplasm

Nucleus

S S⇤

2.1 Reactions

E + Sk1

// Xk2

k3// E + S⇤ F + S⇤ k4

// Yk5

k6// F + S

Ec + Sck7

k9// Ec + Sc F c + Sc⇤ k10

// Y c

k12// F c + Sc⇤

oo Xk14

oo Sk15

oo S⇤ k16// Sc⇤

To ease the notation below, we have changed the notation of the reaction constants k⇤ in the main

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 formultistationarity:

(1) If X shuttles into the nucleus slower than E then S shuttles intothe cytoplasm slower than S∗ and vice versa.

(2) If X shuttles into the cytoplasm slower than E then S shuttlesinto the nucleus slower than S∗ and vice versa.

Bistability by changing total amounts

Plasma membrane

Cytoplasm

Nucleus Nucleus

Cytoplasm

0 5 10 15 20 25 30 35 40 45 500

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 40

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

Bistability by changing total amounts

BistableRegime

MonostableHigh

MonostableLow

Bistable

Stimulus (Etot)

0 5010 20 30 4010-4

Stot=15

Stot=25Stot=35

Stot =45

Stot=150

Bistability by changing shuttling rates

MonostableLow

Bistable

MonostableHigh

0 0.1 0.50.2 0.40.3

eus (k in

Bifurcation parameter (shuttling rate) Rate of S* entering the nucleus (kin,S*)

kin,E kin,X

kin,S*

Rate of S* exiting the nucleus (kout,S*)

0 0.60.1 0.50.2 0.40.30 0.60.1 0.50.2 0.40.30

4 Etot=30

Etot=28

Etot=26

Etot=24

Etot=22

� 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.

Bistability by changing shuttling rates

MonostableLow

Bistable

MonostableHigh

0 0.1 0.50.2 0.40.3

eus (k in

Bifurcation parameter (shuttling rate) Rate of S* entering the nucleus (kin,S*)

kin,E kin,X

kin,S*

Rate of S* exiting the nucleus (kout,S*)

0 0.60.1 0.50.2 0.40.30 0.60.1 0.50.2 0.40.30

4 Etot=30

Etot=28

Etot=26

Etot=24

Etot=22

Rate constant Rate-response curvek13, k14, k19, k20 For large rate constant, only a low stable steady state is obtainedk15, 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.

Example system

Two-site modificationWe consider a two-site modification system, such as MAPK, at parameter values

which cannot permit multistationarity.

! "! #!! #"! $!! $"! %!! %"! &!! &"! "!!#!

40 45 50 55 600

0 100 200 300 400 500 6000

0 10 20 30 40 50 60 70 800

Nucleus

Cytoplasm

Zoomed-in (linear scale)

100 200 300 400

Stimulus (Etot)

Cytoplasmic Phosphatase (Fctot)Rate of S1 leaving the nucleus (k28)

0 8020 6040

0 200 600400

40 45 50 55 60

Multistable Regime

Figure 3: Two site phosporylation example.

References and Notes

1. W. Xiong, J. E. Ferrell Jr, Nature 426, 460 (2003).

2. U. S. Bhalla, R. Iyengar, Science 283, 381 (1999).

3. F. Ortega, J. Garces, F. Mas, B. N. Kholodenko, M. Cascante, FEBS J 273, 3915 (2006).

4. E. Feliu, C. Wiuf, J R Soc Interface (2011).

5. A. C. Ventura, J.-A. Sepulchre, S. D. Merajver, PLoS Comput Biol 4, e1000041 (2008).

6. S. Legewie, B. Schoeberl, N. Bluthgen, H. Herzel, Biophys J 93, 2279 (2007).

7. N. Bluthgen, et al., FEBS J 273, 895 (2006).

8. B. N. Kholodenko, J. F. Hancock, W. Kolch, Nature Reviews Molecular Cell Biology 11,

414 (2010).

Multistability in two-site modification

Nucleus

Cytoplasm

100 200 300 40010-6

Stimulus (Etot)

Cytoplasmic Phosphatase (Fctot)Rate of S1 leaving the nucleus (kout,S1)

0 8020 6040

0 200 600400

40 45 50 55 60

Multistable Regime

� Steady state analysis onparameter shuttling rateconstants.

� Analysis indicates the two-sitephosphorylation cycle canundergo hysteresis.

� Large region of multistability(32 ≤ Etot ≤ 445), most ofwhich is bistable.

Versatility of MAPK

Bifurcations of shuttling rate (kout,S1) and total amount (F ctot)

Nucleus

Cytoplasm

100 200 300 40010-6

Stimulus (Etot)

Cytoplasmic Phosphatase (Fctot)Rate of S1 leaving the nucleus (kout,S1)

0 8020 6040Ph

2)0 200 600400

40 45 50 55 60

Multistable Regime

Steady states of the system can be regulated through reversibleswitches governed by shuttling of parameters and other total

amounts.

Spatial localization: overall findings

� Species localization serves as a mechanism for multistationarity:the number of states may be higher for spatially structured systemscompared to homogenous systems.

� Thereby cellular computational capacity and informationprocessing capacity is driven by spatial organization.

� Provide a method for precluding whether a system is capable ofhaving multistationarity, irrespective of parameter values.

� Identify necessary conditions for multistationarity, which dependonly on the shuttling rate constants.

Conclusions

� Many methods for analyzing models are limited (e.g., simulationtime, nonlinear objective functions, among others).

� Here, we proposed non-parametric methods for analyzingmass-action models with data.

(1) We presented a novel method for rejecting models based onsteady-state coplanarity.

(2) We argued that compartmentalization serves as a mechanism formultistationarity and increases information processing capacity.

� We look forward to combining these parameter-free approacheswith the other spectrum of existing methods.

Conclusions

Acknowledgements

I would like to thank and acknowledge:

� Kenneth Ho

� Tom Thorne

� Carsten Wiuf

� Elisenda Feliu

� Michael Stumpf

� Leverhulme Trust

� Theoretical Systems Biology Group

� Mathematical Biosciences Institute

� Thank you for your attention!

Acknowledgements

� Kenneth Ho

� Tom Thorne

� Carsten Wiuf

� Elisenda Feliu

� Michael Stumpf

Acknowledgements

� Kenneth Ho

� Tom Thorne

� Carsten Wiuf

� Elisenda Feliu

� Michael Stumpf

Acknowledgements

� Kenneth Ho

� Tom Thorne

� Carsten Wiuf

� Elisenda Feliu

� Michael Stumpf

Acknowledgements

� Kenneth Ho

� Tom Thorne

� Carsten Wiuf

� Elisenda Feliu

� Michael Stumpf

Future work

Is there a way to be more precise with rejecting a model usingcoplanarity error?

� In the absence of rigorous criteria we have to rely on heuristics,and those heuristics essentially are the cost of getting our modelselection wrong. We ourselves don’t find this an entirelysatisfaction.

� We have a naıve hope that combining this with non-Bayesianparametric statistics will help us solve this issue.

Is there a way to precisely determine what parameters will yieldmulti-stationarity in a system with spatial localization?

� We hope that combining optimization techniques with thenecessary conditions for multistationarity would improve ourunderstanding of how large of a parameter space is capable ofmultistationarity.

Grobner Bases

Manrai & Gunawardena procedure:

� Let Q[a] be the polynomial ring consisting of all polynomials in theparameters a = (k1, . . . , kR) with coefficients from the rationalnumbers 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 inx = (x1, . . . , xN) with coefficients in K.

� Note that the parameters a have been absorbed into the coefficientfield K.

� By performing all operations over K, we can treat a symbolically,i.e., without specifying any particular parameter values.

Characterize Steady State

To characterize the steady state (x = 0):� Construct the ideal J = 〈x〉 generated by x, consisting of all

polynomials∑N

i=1 fi xi , where each fi ∈ K[x].� 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 idealJi = J ∩ K[xobs], where xobs = (xi+1, . . . , xN) denotes the“observable” variables.

� Use Grobner bases, which are special sets of generators with theso-called elimination property that if g = (g1, . . . , gM) is a Grobnerbasis for J under the lexicographic ordering x1 > · · · > xN , thenJi = 〈gobs〉, where gobs = g ∩ K[xobs] are precisely those elementsof g containing only the variables xobs.

� The polynomials gobs generate all elements of K[xobs] that vanishat steady state and so characterize the projection of the steadystate onto the variables xobs.

non-parametric analysis of models and data

Technology