Model Reduction for Biochemical Systems:
Computational Methods
Tom Snowden
T Snowden Model Reduction 1 / 42
Computational Reduction
For high dimensional, complex models many of the analyticalapproaches to model reduction (discussed in the previous presentation)will be di�cult to apply, as they often depend upon the researcherpossessing high degree of model intuition.
Instead it is common to seek computational algorithms for theapplication of model reduction in such settings.
In this presentaton we discuss a range of such methods anddemonstrate computational reduction via application to an example.
T Snowden Model Reduction 2 / 42
Presentation outline
How I de�ne model reduction
Review of existing methods
An example
Linking with pharmacokinetics
Conclusions
T Snowden Model Reduction 3 / 42
Chemical reaction network theory
Biochemical reaction networks are typically de�ned via systems ofinteracting chemical equations. Such networks can be expressed via threesets of information:
An n dimensional set S representingthe species in the network.
A p dimensional set C representingthe `complexes' in the network.
An m dimensional set R ⊂ C × Crepresenting the reations in thenetwork.
Example:
2A→ D
A + B � C → D + B
S = {A,B,C ,D} ,C = {2A,C ,D + B,A + B,D} ,R = {(2A,D), (A + B,C),
(C ,A + B), (C ,D + B)} .
T Snowden Model Reduction 4 / 42
Stoichiometric representation
Example:
2Ak1→ D
A + Bk2
�k3
Ck4→ D + B
N =
−2 −1 00 −1 10 1 −11 0 1
v =
k1x21 (t)
k2x1(t)x2(t)− k3x3(t)k4x3(t)
It is common to describe thedynamics of such networks enmasse via the Law of MassAction.
One common representation isvia the product of astoichiometry matrix N and avector of reaction rates v(x ,p),such that
x = Nv(x ,p)
where x gives the time-varyingmolecular concentration of eachof the species and p is a set ofparameters.
T Snowden Model Reduction 5 / 42
Control theoretic representation
However, it is also common for certain applications to seek to representsuch models in a control theoretic state-space representation, such that
x(t) = f (x(t),p) + g(x(t),p)u(t),
y(t) = h(x(t),p),
with:
u(t) ∈ Rl representing inputs which can be interpreted in some way ascontrolling the system.
y(t) ∈ Rv representing combinations of the species that can beconsidered outputs.
Within the context of QSP, the inputs may represent the dose of a drugwhilst the ouputs might represent the concentrations of species associatedwith some clinical response.
T Snowden Model Reduction 6 / 42
De�nition of model reduction
ε = ‖y(t)− y(t)‖
Hence, I de�ne a method of model reduction to be any method designed togive a system capable of satisfactorily reproducing the input-output
behaviour of the original model (under some given metric of error)whilst producing a reduction in the number of species S, reactions R,or complexes C.
T Snowden Model Reduction 7 / 42
Reducing systems biology models
Common disadvantages
1 Sti�ness:
K = λmax(Jf (x))λmin(Jf (x)) � 1
Presents issues for numericalmethods.
2 Nonlinearity: f (ax) 6= af (x)Presents issues for analyticalmethods.
3 Conservation relations:
∃Γ ∈ Rα×n : Γx(t) = xT , ∀tMust be handled carefully toavoid violation.
Common advantages
1 Asymptotic Stability:
limt→∞ ‖x(t)− x∗‖ = 0Enables a lot of theory.
2 Conservation relations:
xc = xT − Γcx i
Can be exploited to reducesystem for `free'.
Di�culty also arises from the wide range of aims associated with modelling in the �eld
of systems biology. The best available reduced model necessarily depends upon what it
will be used for.
T Snowden Model Reduction 8 / 42
Presentation outline
How I de�ne model reduction
Review of existing methods
An example
Linking with pharmacokinetics
Conclusions
T Snowden Model Reduction 9 / 42
Literature Review Introduction
The review limited itself to methods addressing deterministic
systems of ODEs and which had seen application to models of
biochemical reaction networks. Emphasis was placed on methods
with published use since 2000.
This section begins by reviewing computational approaches for theapplication of conservation analysis.
It then moves on to reviewing model reduction methods, these are dividedinto 4 categories:
1 Time-scale exploitation methods;
2 Optimisation approaches and sensitivity analysis;
3 Lumping; and
4 Singular value decomposition (SVD) based methods.
T Snowden Model Reduction 10 / 42
Conservation relations
α conservation relations imply that ∃Γ ∈ Rα×n : Γx(t) = xT , ∀t.
The conservation relations correspond to linear dependencies in therows of the stoichiometry matrix N.
It is possible to show1 that Γ = Null(NT ).
A numerically stable method for obtaining this null-space for largesystems is to employ QR factorisation via Householder re�ections 2.
1Reder, J. Theor. Biol., 1988.2Vallabhajosyula et al., Bioinformatics, 2006.
T Snowden Model Reduction 11 / 42
Time-Scale Exploitation Methods I
This refers to any method thatexploits the often largedi�erences in reaction rates thatcan occur within a biochemicalsystem.
Typically such methods partitionthe system into fast and slowcomponents - after some initialtransient period those fastportions are assumed to be inequilibrium with respect to theremainder of the network.
Such methods include singularperturbation approaches, ILDM,and CSP.
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Figure: An example of model reductionvia time-scale analysis
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Time-Scale Exploitation Methods II
Singular Perturbation
If a system of ODEs can beexpressed in the form
x(t) = f (x , z , t) ,
δz(t) = g (x , z , t) ,
then as δ → 0 this system can beapproximated by
x(t) = f (x , z , t) ,
z(t) = φ (x , t) ,
with φ (x , t) a root of theequations g (x , z , t) = 0.
Species Partitioning(xs
δx f
)=
(Ns
Nf
)v (xs , x f ,p)
Reaction Partitioning
x = (Ns Nf )
(v s (x ,p)
δ−1v f (x ,p)
).
x can then be decomposed intofast and slow contributions as asum, such that x = [x ]
s+ [x ]
f.
Hence
[x(t)]s
= Nsv s (x(t),p) ,
0 = Nf v f (x(t),p) .
T Snowden Model Reduction 13 / 42
Time-Scale Exploitation Methods III
PROS:
Species can maintain biologicalmeaning.
A large number of such methodsexist in the literature.
These methods are typicallyvalid in the reduction ofnonlinear systems.
CONS:
A system may not have a largeenough time-scale seperation tojustify reduction.
What happens during the initialtransient period may be ofinterest.
If a slow/fast partitioning is notknown a priori approaches fordetermining the mostappropriate one can becomputationally expensive.
T Snowden Model Reduction 14 / 42
Optimisation and Sensitivity Analysis Methods I
Reduction can be expressed as anoptimisation problem - i.e. obtain thelowest possible dimensional model(either in terms of species, reactions orcomplexes) for which a metric of error εremains within an acceptable bound,such that ε < εc .
Hence it is common to either:
1 Seek to measure how `sensitive' theconstraint variable ε is toperturbations and use this to guide areduction. Or;
2 Employ an iterative optimisationprocedure.
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Figure: An example of modelreduction via optimisation
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Optimisation and Sensitivity Analysis Methods II
A typical optimisationproceedure might involve`switching o�' ofreactions or species.
For example, kineticparameters can be givenswitch variables,
It is then an integerprogramming problemwith these switches todetermine a minimalreduced modelconstrained by an errorbound 3.
3Maurya et al., IET Syst Biol.,
2009.T Snowden Model Reduction 16 / 42
Optimisation and Sensitivity Analysis Methods III
PROS:
Species can maintain theirbiological meaning.
The application of such methodscan be highly algorithmic andcomputationally e�cient (e.g.heuristic approaches such asgenetic algorithms).
Common procedures areimplemented well in a number ofsoftware packages.
CONS:
For very large systemsperforming a su�ceint searchthrough the range of candidatesolutions may be highlycomputationally expensive.
Similarly, for sensitivity analysisconvincingly searching the entireparameter space may beimpossible.
T Snowden Model Reduction 17 / 42
Lumping Based Methods I
Lumping is a classi�cationthat encompasses a range ofmethods.
In particular it pertains toany method that constructsa reduced system withstate-variables correspondingto subsets of the originalspecies.
These new states are referredto as `lumped' variables.
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Y1
Y2
Y3
(a) (b)
Figure: (a) Proper lumping - each of theoriginal species corresponds to, at most, oneof the lumped states. (b) Improper lumping- each of the original states can correspondto one or more of the lumped states.
T Snowden Model Reduction 18 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods II
Applying a lumping:
A set of species can be reduced via
some proper, linear lumping4
L ∈ {0, 1}r×n giving a reduced set of
species x ∈ Rr where x = Lx .
Via the Galerkin projection we can
obtain a reduced dynamical system of
the form:
˙x = Lf (Lx , p) + Lg(Lx , p)u
y = h(Lx , p).
Here L represents a generalised inverse
of L such that LL = Ir .
4Li & Rabitz, Chem. Eng. Sci., 1990.T Snowden Model Reduction 19 / 42
Lumping Based Methods III
PROS:
Lumping is a common methodin the reduction of chemicalkinetics - quite a large range ofliterature exists.
Algorithmic approaches that canbe implemented computationallyexist.
Lumped variables can be chosento be biological meaningful suchthat the reduced model maintinssome degree of biologicalintuitiveness.
CONS:
Many of the procedures in theliterature are highlycomputationally expensive forlarge systems.
Most methods in the literaturepertain to linear, proper lumping- better reduction is likely to beachieved by nonlinear and/orimproper lumping techniques,but this may lead to loss ofbiological meaning.
T Snowden Model Reduction 20 / 42
Singular Value Decomposition Based Approaches I
These methods are based upon thesignular value decomposition (SVD).
Crucially, via Eckart-Young-Mirskytheorem5 the SVD provides a way toapproximate a matrix via one oflower rank.
The most commonly applied suchmethod is balanced truncation.
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Z2
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u
u
y
y
~
Figure: Balanced truncation reducesa model whilst seeking to preservethe input-output relationship
5Eckart& Young, Psychometrika, 1936.T Snowden Model Reduction 21 / 42
Singular Value Decomposition Based Approaches II
Linear balanced truncation istypically applied to linearsystems of the form
x = Ax + Bu,
y = C x .
It requires the computation oftwo matrices P and Q:
1 The controllability GramianP provides information onhow the state-variables xrespond to perturbations ininputs u.
2 The observability Gramian Qprovides information on howthe outputs y respond toperturbations in thestate-variables x .
Balanced truncation done quick
1 Perform Cholesky factorisation ofboth gramians
P = LTL, Q = R
TR.
2 Take SVD of matrix LRT to obtain
LRT = (U1 U2)
(Σ1 00 Σ2
)(VT
1VT
2
)Where U1 is an n × r matrix, Σ1 is
an r × r diagonal matrix and VT
1 is a
r × n matrix.
3 Set
T1 = Σ− 1
2
1 VT
1 R, S1 = LTU1Σ
− 1
2
1 .
4 Finally
˙x = T1AS1x + T1Bu,
y = CS1x .
T Snowden Model Reduction 22 / 42
Singular Value Decomposition Based Approaches II
PROS:
Control theoretic description �tsneatly with the idea of systemspharmacology (i.e. the drugcontrolling subcellularprocesses).
They are highly algorithmicmethods - can potentially beautomated in a straightforwardmanner.
An a priori error bound can beobtained.
CONS:
Transformed/reduced states nolonger have biological meaning -only inputs and outputs preservetheir meaning.
Standard approach only existsfor linear models - butgeneralisations for nonlinearsystems do exist.
For large systems, empiricalbalanced truncation can behighly computational expensive.
T Snowden Model Reduction 23 / 42
Miscellaneous Methods
A number of other methods, with a limited publication record, do existincluding:
Motif replacment methods;
Methods for reduction of combinatorial complexity;
Complex reduction; and
Publications addressing general reduction heuristics.
T Snowden Model Reduction 24 / 42
Conclusions of Literature Review I
This literature review enabled several speci�c conclusions:
There is no `one-size-�ts-all' method of model reduction.
Whilst many of these methods can be highly automated, the onus ison the modeller to choose the correct tool for the task.
Consider what the reduced model will be used for to judge whichmethod is most appropriate.
T Snowden Model Reduction 25 / 42
Conclusions of Literature Review II
T Snowden Model Reduction 26 / 42
Presentation outline
How I de�ne model reduction
Review of existing methods
An example
Linking with pharmacokinetics
Conclusions
T Snowden Model Reduction 27 / 42
Aims
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Figure: An example of aproper lumping
This section introduces acomputational modelreduction algorithm developedduring my PhD.
Three existing methods arebrought together in thisapproach:
Conservation analysis.
Proper lumping.
Empirical balancedtruncation.
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Z2
Z3
u
u
y
y
~
Figure: Schematic outline
of Balanced Truncation -
the method focuses on
preserving the
input-output relationship
of the system.
T Snowden Model Reduction 28 / 42
Combined model reduction algorithm
Given this context, we have
developed the an algorithm for
model reduction which combines
previously existing methods in a
novel way. The following
schematic outlines its operation:
T Snowden Model Reduction 29 / 42
Combined method justi�cation
The core justi�cation of the combined reduction algorithm is the
use of proper lumping as a preconditioner for the application of
empirical balanced truncation.
Empirical balanced truncation (EBT) should, in theory, produce moreaccurate reduced networks than proper lumping.
In practice, EBT often fails for highly sti� systems.
Proper lumping, however, will tend to sum together thosestate-variables that interact on faster timescales than their neighbours.
Hence the reduced model will often contain a smaller range oftimescales and be less sti� with each additional dimension eliminated.
T Snowden Model Reduction 30 / 42
ERK Activation Model
Figure: Block schematic overview of EGF
and NGF dependent ERK signalling
network7. Model consists of 150 reactions
and 99 species. There are 23 conservation
relations in this system enabling the model
to be reduced to 76 states.
c1.SOS
c1.pSOS
c1.SOS_Grb2
c1.Grb2
c1.Dok
c1.pDok
c1.Crk
c1.FRS2
c1.Shc
c1.pSOS_Grb2
c1.Rap1_GDP
c1.MEK
c1.MKP3
c1.pShc_dpEGFR
c1.dpEGFR_c_Cbl
c1.B_Raf_Rap1_GTP
c1.pShc_dpEGFR_c_Cbl
c1.pFRS2_dpEGFR_c_Cbl
c1.Shc_dpEGFR
c1.c_Cbl
c1.RasGAP c1.c_Raf
c1.B_Raf
c1.ERK
c1.PP2A
c1.Ras_GDP
c1.Rap1GAP
c1.C3G
c1.pShc
c1.pFRS2_dpEGFR
c1.pTrkA_endo
c1.MEK_ERK
c1.pMEK_ERK
c1.FRS2_dpEGFR_c_Cbl_ubiq
c1.Crk_C3G_pFRS2_dpEGFR_c_Cbl
c1.pShc_dpEGFR_c_Cbl_ubiq
c1.Crk_C3G_pFRS2_dpEGFR
c1.Grb2_SOS_pShc_dpEGFR_c_Cbl_ubiq
c1.Grb2_SOS_pShc_dpEGFR_c_Cbl
c1.Shc_dpEGFR_c_Cbl_ubiq
c1.dpEGFR_c_Cbl_ubiq
c1.proteasome
c1.Grb2_SOS_pShc
c1.Shc_dpEGFR_c_Cbl
c1.Grb2_SOS_pShc_dpEGFR
c1.pFRS2
c1.FRS2_dpEGFR
c1.pDok_RasGAP
c1.pMEK
c1.FRS2_dpEGFR_c_Cbl
c1.pFRS2_dpEGFR_c_Cbl_ubiq
c1.Ras_GTP
c1.Crk_C3G_pFRS2_dpEGFR_c_Cbl_ubiq
c1.c_Raf_Ras_GTP
c1.B_Raf_Ras_GTP
c1.ppMEK
c1.ppERK
c1.Crk_C3G
c1.Rap1_GTP
c1.ppMEK_ERK
c1.dppERK
c1.Shc_pTrkA
c1.Shc_pTrkA_endo
c1.pShc_pTrkA
c1.pFRS2_pTrkA
c1.FRS2_pTrkA
c1.pShc_pTrkA_endo
c1.FRS2_pTrkA_endo
c1.pFRS2_pTrkA_endoc1.Crk_C3G_pFRS2_pTrkA_endo
c1.Grb2_SOS_pShc_pTrkA
c1.Crk_C3G_pFRS2_pTrkA
c1.Grb2_SOS_pShc_pTrkA_endo
c1.c_Raf_Ras_GTP_MEK
c1.c_Raf_Ras_GTP_pMEK
c1.c_Raf_Ras_GTP_MEK_ERKc1.c_Raf_Ras_GTP_pMEK_ERK
c1.B_Raf_Ras_GTP_MEK
c1.B_Raf_Ras_GTP_pMEK
c1.B_Raf_Ras_GTP_MEK_ERK
c1.B_Raf_Ras_GTP_pMEK_ERK
c1.B_Raf_Rap1_GTP_MEK
c1.B_Raf_Rap1_GTP_pMEK
c1.B_Raf_Rap1_GTP_MEK_ERK
c1.B_Raf_Rap1_GTP_pMEK_ERK
c1.ppERK_MKP3
c1.dppERK_MKP3
c1.pro_TrkA
c1.pro_EGFR
c1.degradation
compartment.EGFR
compartment.L_EGFR
compartment.L_EGFR_dimer
compartment.L_dpEGFR
compartment.NGFRcompartment.pTrkA
compartment.L_NGFR
compartment.NGF
compartment.EGFform_EGFreceptor
EGFbinding
dimerization
binding_SOS_Grb2
binding_pSOS_Grb2
EGFRphosphorylation
binding_cCbI_dpEGFR
binding_pShc_LdpEGFR
pDOKdephosphorylation
binding_cCbl_pShc_dpEGFR
SOSdephosphorylation
pSOS_Grb2_dephosphorylation
binding_Shc_LdpEGFR
Shc_dpEGFR_phosphorylation
dpEGFR_c_Cbl_ubiquitination
dpEGFR_cCbl_degrad
binding_cCbl_Shc_dpEGFR
Shc_dpEGFR_c_CBl_Ubiquitination
Shc_dpEGFR_c_Cbl_ubiq_DegradationpShc_dpEGFR_c_Cbl_ubiquitination
pShc_dpEGFR_c_Cbl_ubiq_degradation
Shc_dpEGFR_c_Cblphosphorylation
binding_Grb2_SOS_pShc
binding_Grb2_SOS_pShc_dpEGFRbinding_Grb2_SOS_pShc_dpEGFR_1
binding_c_Cbl_Grb2_SOS_pShc_dpEGFR
binding_Grb2_SOS_pShc_to_dpEGFR_c_Cbl
Grb2_SOS_pShc_dpEGFR_c_Cbl_ubiquitination
Grb2_SOS_pShc_dpEGFR_c_Cbl_ubiq_degradation
Grb2_SOS_pShc_Dissociation
Unnamed Reaction
pShc_dephosphorylation
pFRS2_dephosphorylation
binding_Crk_to_C3G
binding_L_dpEGFR_to_FRS2
binding_pFRS2_to_L_dpEGFR
FRS2_dpEGFRphsphorylation
binding_Crk_C3G_to_pFRS2_pRTK
binding_c_Cbl_to_FRS2_dpEGFR
binding_c_Cbl_to_pFRS2_dpEGFRpFRS2_dpEGFR_c_Cbl_ubiquitiation
FRS2_dpEGFR_c_Cbl_ubiquitination
FRS2_dpEGFR_c_Cbl_phosphorylation
binding_Crk_C3G_to_pFRS2_pFRS2_dpEGFR_c_Cbl
Crk_C3G_pFRS2_dpEGFR_c_Cbl_ubiquitination
FRS2_dpEGFR_c_Cbl_ubiq_dissociation
pFRS2_dpEGFR_c_Cbl_ubiq_dissociation
binding_RasGAP_to_pDOK
SOS_Grb2_phosphorylationSOS_phosphorylation
binding_c_Raf_to_Ras_GTP
binding_B_Raf_to_Rap1_GTP
binding_B_Raf_to_Ras_GTP
ppMEK_dephosphorylation
pMEK_dephosphorylation
ppMEK_ERK
pMEK_ERK_dephosphorylation
ppERK_dimerization
Ras_GTP_dephosphorylation
Rap1_GTP_dephosphorylation
Rap1_GTP_phosphorylation
Ras_GDP_phosphorylation
binding_NGF_to_NGFRTrkA_phosphorylation
pTrkA_intermalization
pTrkA_endo_degradation
pTrkA_degradation
binding_Shc_to_pTrkA
binding_pShc_to_pTrkA
binding_FRS2_to_pTrkA
binding_pFRS2_to_pTrkA
binding_Shc_to_pTrkA_endo
binding_pShc_to_pTrkA_endo
Shc_pTrkA_endo_phosphorylationShc_pTrkA_phosphorylation
pFRS2_pTrkA_phosphorylation
binding_FRS2_to_pTrkA_endo
binding_pFRS2_to_pTrkA_endo
FRS2_pTrkA_endo_phosphorylation
FRS2_pTrkA_degradation
pFRS2_pTrkA_degradation
Shc_pTrkA_degradation
pShc_pTrkA_degradation
FRS2_pTrkA_endo_degradation
Shc_pTrkA_endo_degradation
pShc_pTrkA_endo_degradation binding_Grb2_SOS_to_pShc_pTrkA_endo
binding_Grb2_SOS_to_pShc_pTrkAGrb2_SOS_pShc_pTrkA_ubiquitination
Crk_C3G_pFRS2_pTrkA_ubiquitinationpFRS2_pTrkA_ubiquitination
FRS2_pTrkA_ubiquitination
pShc_pTrkA_ubiquitination
Shc_pTrkA_ubiquitination
binding_Crk_C3G_to_pFRS2_pTrkA
binding_Crk_C3G_to_pFRS2_pTrkA_endo
binding_Grb2_SOS_pShc_to_pTrkA
binding_Grb2_SOS_pShc_to_pTrkA_endo
Crk_C3G_pFRS2_pTrkA_degradation
Crk_C3G_pFRS2_pTrkA_endo_degradation
Grb2_SOS_pShc_pTrkA_degradation
Grb2_SOS_pShc_pTrkA_endo_degradation
pFRS2_pTrkA_endo_degradation
form_NGFreceptor
binding_Shc_to_dpEGFR_c_Cbl
binding_pShc_to_dpEGFR_c_Cbl
binding_SOS_Grb2_to_pShc_dpEGFR_c_Cbl
binding_c_Cbl_to_Crk_C3G_pFRS2_dpEGFR
binding_FRS2_to_dpEGFR_c_Cbl
binding_pFRS2_to_dpEGFR_c_Cbl
Ras_GTP_dephosphorylation_1
RAP1_GTP_dephosphorylation
Dok_phosphorylation
Grb1_SOS_pShc_dissociation
binding_MEK_to_ERK
binding_ERK_to_pMEK
binding_ERK_to_ppMEKppMEK_ERK_dissociationc_Raf_Ras_GTP_dissociationB_Raf_Ras_GTP_dissociation
B_Raf_Rap1_GTP_dissociation
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Figure: Full ERK activation pathway model
in petri-net form
6Sasagawa et al., Nat. Cell Biol., 2005.7Sasagawa et al., Nat. Cell Biol., 2005.T Snowden Model Reduction 31 / 42
ERK Activation Reduction Results I
Results for thereduction of the 99dimensionalErk-activation model.`#' implies Matlabcould not simulatethis reduction usingode15s due tonumerical error. `-'implies the error atthis point was equalto the lumping error.
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ERK Activation Reduction Results II
Figure: Simulated results for the output of the original 99-dimensional ERK activation modelvs the reduced 8 dimensional model. This plot emphasises the fact that the reduced model isdesigned to remain valid for any reasonable change in input. The system starts by being a�ectedby an agonist that increases the rate of EGF binding by 25% for over an hour (4000 seconds), atthis point the input �ips to an antagonist decreasing the rate of EGF binding by 25% and runsfor the same time period (an additional 4000 seconds). At any given time point the errorbetween the original and reduced model exceeds no more than 5%.
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Presentation outline
How I de�ne model reduction
Review of existing methods
An example
Linking with pharmacokinetics
Conclusions
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Reducing PBPK models I
In this section we explore the application of model reduction methodsto models of pharmacokinetics.
Pharmacokinetic models are typically linear which enables moreaccurate reduction as compared with, typically nonlinear, models ofbiochemical reaction networks.
A brief study of applying model reduction methods to physilogicallybased pharmacokinetic models was underaken.
The PBPK system we chose to employ was a deterministic, linear,16-dimensional, compartmental model.
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Reducing PBPK models II
Analysis was made of bothlumping and standardbalanced truncation as ameans for the reduction thissystem. Balanced truncationwas found to give the bestresults
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Linking I
Questions include:
Should spatial inhomogeneityin di�usion be explicitlyaccounted for?
What is the cumulativee�ect of the cellularresponse?
Should di�erent cell types(e.g. diseased and healthy)and their di�erences in druga�nity be accounted for?
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Linking II
We made the simplifyingassumption that the tissuee�ects were accounted for bythe PBPK model and thatthe cells/receptors werehomogeneously distributed inthe relevant tissuecompartment.
Hence they are partiallydecoupled and can bereduced separately as in theschematic given on the right.
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Linking Results: ERK activation
Figure: Linking the 10 dimensional reduced version of the ERK activation modelobtained under the combined model reduction algorithm with a 3 dimensionalreduced version of the PBPK model obtained via balanced truncation yields theresults above. In comparison to a linked version of the original model, the reducedversion maintains a 3% error bound.
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Presentation outline
How I de�ne model reduction
Review of existing methods
An example
Linking with pharmacokinetics
Conclusions
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Conclusions
We have hopefully demonstrated that model reduction methods canproduce signi�cant simpli�cations in a system whilst retaining a highdegree of accuracy.
The literature review shows that a wide range of such methods currentlyexist.
The aims of such reduction might include seeking to speed up simulationtime, obtaining a model of an appropriate scope relative to the avilabledata, or trying to analyse which components of a model are mostresponsible for driving the dynamical behaviour of interest.
Crucially, the optimal reduction method is deeply dependent upon yourresearch question!
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Thank you for listening.
Acknowledgments
Thank you to P�zer and EPSRC for their �nancial support throughoutthe PhD.
Thank you to Marcus Tindall and Piet van der Graaf for theirsupervision throughout the project.
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APPENDIX
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Petrov-Galerkin Projection
System trajectories can often be well approximated in a lower dimensional
subspace S : dim (S) = r .
Select a test basis B ∈ Rn×r of S, such that x(t) ≈ B x(t) with x(t) ∈ Rr
represents our reduced state vector.
Hence, B ˙x(t) = f (B x(t), p, u(t)) + r(t) where r(t) represents the residual
incurred via our approximation.
Constrain the residual to be orthogonal to a subspace C with an associated test
basis C ∈ Rn×r such that CT r(t) ≈ 0.
Therefore we left multiply by CT to obtain CTB ˙x(t) = C
T f (B x(t), p, u(t))
Assuming CTB is non-singular we can obtain
˙x(t) =(CTB
)−1
CTf (B x(t), p, u(t))
y = g (B x(t), p)
If B = C this is a special case known as a Galerkin projection.
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