Modeling metabolic networks: Advantages and limitations of approximated mathematical formalisms
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Modeling metabolic networks: Advantages and limitations of approximated mathematical formalisms Albert Sorribas Grup de Bioestadística i Biomatemàtica Departament de Ciències Mèdiques Bàsiques Institut de Recerca Biomèdica de Lleida (IRBLLEIDA) Universitat de Lleida
description
Modeling metabolic networks: Advantages and limitations of approximated mathematical formalisms. Albert Sorribas Grup de Bioestadística i Biomatemàtica Departament de Ciències Mèdiques Bàsiques Institut de Recerca Biomèdica de Lleida (IRBLLEIDA) Universitat de Lleida. Summary. - PowerPoint PPT Presentation
Transcript of Modeling metabolic networks: Advantages and limitations of approximated mathematical formalisms
Modeling metabolic networks: Advantages and limitations of approximatedmathematical formalisms
Albert Sorribas
Grup de Bioestadística i BiomatemàticaDepartament de Ciències Mèdiques BàsiquesInstitut de Recerca Biomèdica de Lleida (IRBLLEIDA)Universitat de Lleida
Summary Goals and strategies in analyzing complex
(metabolic) networks The need for a systemic perspective The role of mathematical models The need of an appropriate mathematical formalism
Alternative representations using approximate representations General ideas The power-law formalism as a modeling tool for
complex networks Definition and properties System analysis using power-law models
Model indentification from systemic data Conclusions and challenges
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The need for a systemic analysis of metabolic networks
Systems biology is an emerging field that enables us to achieve in-depth understanding at the system level. For this, we need to establish methodologies and techniques that enable us to understand biological systems as systems, which means to understand:(1) the structure of the system, such as gene/metabolic/signal transduction networks and physical structures,(2) the dynamics of such systems,(3) methods to control systems, and (4) methods to design and modify systems to generate desired properties. However, the meaning of ‘‘system-level understanding ’’is still ambiguous.
… Systems biology is both an old and new field in biology …… Concepts such as robustness and feedback control were already discussed at that time and extensively investigated.
Kitano, H. (2002) Curr.Gen.41:1-10
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System-level understanding of metabolic networks Understand complex metabolic networks
Gene regulatory networks, Signal transduction, Apoptosis, Cell cycle, Oxidative stress, etc. Adaptative response to changes in environmental
conditions Characterize pathological situations
Common questions (a mathematical formalism must be able of answering those questions)
Identify key features in complex systems (robustness, structural properties, …)
How are they regulated? Which are the key components?
Relate different levels of complexity: genomics, biochemistry, physiology. Which are the optimal expression profiles in a given
situation? Why a given expression profile leads to a pathological
situation? Make predictions (what happens if…) on future
observations
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Quantitative approaches:A Systems Biology perspective
Use abstraction to simplify the problem. Concentrate in class of systems.
Build-up mathematical models of complex systems. Use an appropriate formalism Use data to constraint the model. Use biological knowledge to challenge the model.
Concentrate in deriving general rules and in understanding design and operational principles. Design principles: Which are the evolutionary
advantages of a given regulatory network? Operational principles: Given a regulatory network,
which are the optimal expression profiles to adapt to environmental changes?
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Mathematical models of complex systems Mathematical models are a (simplified)
representation of the actual system. The challenge for system reconstruction should not
aim to have an exact picture of reality. If we concentrate in an exact picture of reality we
would came out with an object (model) as complex as reality.
Mathematical models are (incomplete) abstractions (and result from conceptual incomplete models of reality) We need a certain level of abstraction to be able of
understanding a complex network. It is possible to understand reality without knowing
every single piece of evidence in every imaginable situation.
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The need for appropriate mathematical formalisms Structural complex systems analysis
Graph theory, Boolean networks, Statistical and similar analysis (Identify connectivity properties, free-scale networks, etc.)
Quantitative approaches Stocihiometric analysis techniques Kinetic modeling
Detailed kinetic descriptions (lack of information, complicated descriptions, too many parameters, are they real in vivo? …)
Alternative strategies based on approximated representations (abstraction and simplification help in analyzing complex systems).
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Some uses of mathematical modelsThe user perspective
Fit experimental data to derive parameter values that characterize the processes of interest
Reconstruct and identify the topology of reactions and regulation in biological pathways and circuits
Analyze design principles Optimize specific properties of the system Integrate different levels of the cellular response
and create a network that accounts for the dynamic behavior of genes, proteins and metabolites
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Basic mathematical models
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Basic mathematical modelsNode equations
Node equations
Aggregated node equations
nivdt
dX p
rrir
i ,..,11
niVVvvdt
dXii
p
rrir
p
rrir
i ,..,111
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Network structure: ij
Regulatory structure: vr
Dynamic model: select a mathematical representation for vr
ExampleNode equations
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1 2
3 4
56
53222 vvvX
dt
dX
6
5
4
3
2
1
6
5
4
3
2
1
110000
001100
010110
000011
v
v
v
v
v
v
X
X
X
X
X
X
X
ExampleAggregatednode equations
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1 2
3 4
56
2253253222 VVvvvvvvX
dt
dX
6
5
4
3
2
1
6
5
4
3
2
1
100000
001000
010100
000010
010000
000100
000010
000001
v
v
v
v
v
v
v
v
v
v
v
v
X
Steady-state equations
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1 2
3 4
56
0
0
0
0
0
0
110000
001100
010110
000011
6
5
4
3
2
1
v
v
v
v
v
v
X
0VS
Methods based on the stoichiometric matrix
Flux balance analysis Find the optimal flux distribution constrained to some goal
(maximum growth, maximum flux,..) Predict the effect of knocking-out a given gene
As the steady-state equation must be fulfilled, fluxes must be changed to match the effect of knocking-out a gene
Advantages Genome-wide models Ready to go from a simple conceptual description Independent of detailed kinetic information
Problems Does not include regulatory information Does not include metabolite levels Does not include dynamic changes
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Flux balance analysis
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1 2
3 4
56
1 2
3 4
56
110000
001100
010110
000011
S
110000
001100
010110
000011
S
Different systemsSame stoichiometry
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Basic mathematical modelsNode equations
Node equations
Include regulatory structure Which are the metabolites that affect each reaction? Which is the influence of a change in a metabolite on
the properties of each reaction?
nivdt
dX p
rrir
i ,..,11
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Network structure: ij
Regulatory structure: vr
Regulatory effects
Use kinetic equations? Lack of information, complicated
representations Use approximated representations
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1 2
3 4
56
110000
001100
010110
000011
S
231222 ,,, XXEvv
Regulatory effects
Use kinetic equations Which is the available information? Is the in vitro information relevant? How many parameters are required? Can they be indentified?
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1 2
3 4
56
110000
001100
010110
000011
S
231222 ,,, XXEvv
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Mathematical formalisms based on approximate representations
Consider a process that depends on different metabolites and parameters
Homogeneous function of the enzyme
θX,,ii Ev
Parameters:
..effectors, s,Metabolite :
Enzyme:
θ
XiE
θX, ii Ev
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The power-law formalism
),..,()ln(),....,..,,( 121 mniimnni yyvXXXXv
01
000
01
0 )ln()ln(
)ln()ln( jj
mn
jijijj
mn
j j
iii yyfvyy
y
vvv
)( ii XLogy
0
0
0
00
)ln(ij
i
ji
j
i fv
X
X
v
y
v
j
mn
j
fj
mn
j
fjiij
mn
jijj
mn
jijii
ijij XXvvyfyfvv11
001
00
1
00
00
)ln()ln(
mn
j
fjiiijXv
1
0
mn
j
fjiiijXv
100
0
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The power-law formalism
mn
i
fjimnniijXXXXXv
121
0
),....,..,,(
order Kinetic :
constant Rate :
ij
i
f
0
0
0
0
0
0
ij
ij
ij
f
f
f
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The power-law formalism
mn
i
fjimnniijXXXXXv
121 ),....,..,,(
X1 X2
X3 X4v2
Example
X12
(-)
X5 is the enzyme in reaction v2
212252321125312125312 ),,,( ffff XXXXXXXXv
2524232117141311
948682
736662
3225242321
5148
54312743115
496284
376263
23543122
15841
ffffffff
fff
fff
fffff
ff
XXXXXXXXX
XXXX
XXXX
XXXXXX
XXX
X2
X5
X1X7
X8
X3
X6
X4
X6
1
3
4 5
6
7
8
9
(+) (+)
(+)
(+)(-)
(-) (-)
(-)
(-) (+)
(-)(+)
(-) (-)
Building-up models:Building-up models:Generalized Mass Generalized Mass Action (GMA model)Action (GMA model)
2
Thyroid hormone metabolismSorribas & Gonzalez (1999) J.Theor.Med. 2:19-38
Automatic model generation from
the scheme
Matrix representation of a GMA model
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i
ij
iii
r
j
j
rrjI
r
j
j
rrjD
irir
r
j,iV
vVrrV
mnnjr
X
X
vfmrF
njr
X
X
vfnrF
NrnN
)1(
0)(
,..,1)(
,..,1)(
)(
0
0
0
0
0
0
0
From a given scheme, we can automatically generate these matrices. The GMA analysis is straightforward using these matrices
Steady-state characterization in GMA models
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0
0
0
),(i
j
j
iji X
p
dp
dXpXS
0
0
0
),(i
j
j
iji X
p
dp
dXpXS
•GMA analysis is based on sensitivity theory.
•It provides a complete steady-state characterization through parameter sensitivity (robustness) and log-gains (response to environmental changes).
•Dynamic responses are analyzed through numerical simulations.
I1
DID
IIDD
II
1I
I
D1DD
D
1
II
D
DI
N·V·F)(N.V.F)X,S(X
N·V·F)X,·S(XN.V.F
XX
VN·V·VX
X
XXX
X
VN·V·V
X
VN
X
X
X
VNVN·
X
N·V
0
0
0··)(
0
d
d
d
d
d
d
I1
DID
IIDD
II
1I
I
D1DD
D
1
II
D
DI
N·V·F)(N.V.F)X,S(X
N·V·F)X,·S(XN.V.F
XX
VN·V·VX
X
XXX
X
VN·V·V
X
VN
X
X
X
VNVN·
X
N·V
0
0
0··)(
0
d
d
d
d
d
d
Logarithmic gains
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A simple example on design principles
X1 X2
X4
X3
(+)
(-)
X5
• Which are the requirements (design principles) for havingan increase of X3 as a response to an increase of X5?• Can we design a system in which an increase in X5 willproduce a decrease of X3?
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GMA modelX1 X2
X4
X3
(+)
(-)
X5 X1 X2
X4
X3
(+)
(-)
X5
645251
4332
5251322321
232115
462154
34233
215233122
312511
fff
ff
fffff
fff
XγXXγX
XγXγX
XXγXγXXγX
XXγXγX
64
5251
43
32
2321
000
00
000
000
00
0000
f
ff
f
f
ff
FD
0
0
0
015f
FI
110000
001100
010110
000011
N
1 2
34
5
6
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GMA modelX1 X2
X4
X3
(+)
(-)
X5 X1 X2
X4
X3
(+)
(-)
X5
64
5251
43
32
2321
000
00
000
000
00
0000
f
ff
f
f
ff
FD
0
0
0
015f
FI
110000
001100
010110
000011
N
1 2
34
5
6
IDID FVNFVNXXS ·····),( 1
Logarithmic gains
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X1 X2
X4
X3
(+)
(-)
X5 X1 X2
X4
X3
(+)
(-)
X51 2
34
5
6 IDID FVNFVNXXS ·····),( 1
513223525
3324321
515
1213215
53 ),(
ffffv
vfff
fv
vfff
XXS
21
51
5
1
515
12153 00),(
f
f
v
v
fv
vfXXS
Systemic response
Design principle
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Design of specific systemsResponse to a 10% increase of X5 over the reference state
0 2 4 6 8 10
0.95
1
1.05
1.1
875.0/
2/
2151
51
ff
vv21
51
5
1
f
f
v
v
6/
2/
2151
51
ff
vv
Steady-state characterization in GMA systems
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0
0
0
),(i
j
j
iji X
p
dp
dXpXS
0
0
0
),(i
j
j
iji X
p
dp
dXpXS
ijij ff ·)X(LogΓ)·,S(X),S(X
Γ)·F,S(X·N·V·FN·V·F)X,S(X
·N·VN·V·FΓ),S(X
ijDD
IDI1
DID
1DD
ijij ff ·)X(LogΓ)·,S(X),S(X
Γ)·F,S(X·N·V·FN·V·F)X,S(X
·N·VN·V·FΓ),S(X
ijDD
IDI1
DID
1DD
Metabolite sensitivities
Flux sensitivities
ijij ff ·
)
)
)X(LogΓ)·S(V,)S(V,
F)X,S(XFXS(V,
IΓ),S(XFΓS(V,
ij
IIDDI
DD
ijij ff ·
)
)
)X(LogΓ)·S(V,)S(V,
F)X,S(XFXS(V,
IΓ),S(XFΓS(V,
ij
IIDDI
DD
Use of sensitivity analysis
Check model consistency High sensitivity may indicate a ill-defined
model. Compare design performance
If design (a) has a lower sensitivity than design (b), then design (a) can be a better choice for the considered function
Relate local and global properties Sensitivity is a global property that depends on
the underlying processes on the network
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Purine metabolism in man
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Curto et al. (1998) Math.Biosc. 151:1-49
Kinetic-order sensitivities
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Analytical methods provided by the power-law formalismBiochemical Systems Theory (BST)
Automatic model generation from a conceptual scheme
Algebraic methods for analyzing model characteristics and design Mathematical controlled comparisons (design
principles) Quantitative modeling and analysis
Sensitivity analysis (assess model robustness) Parameter scanning (operational principles) Simulation
Optimization Canonical modeling strategies (recasting non-linear
models into power-law models) http://www.udl.es/Biomath/PowerLaw/
Alternative kinetic formalisms based on approximated representations
Mathematical formalisms based on approximate representations
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01
0
0
0
0, jj
n
j j
ii
i
iiii XX
X
vEE
E
vvEv
θX,
Linear (1)
1101 0
0
0
000
0
0
00j
jn
j i
j
j
ii
ii
i
i
iiii X
X
v
X
X
vv
E
E
v
E
E
vvvv
1101
00
0
000
j
jn
jiji
i
iiEiii X
Xfv
E
Efvvv
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11101
0
0
0
0 j
jn
jij
i
iiE
i
i
X
Xf
E
Ef
v
v
0
0
0
0
i
j
j
iij v
X
X
vf
Kinetic-order
j
n
jijiiiEiii XfvEfvbv
1
00
000
Collect constant terms
Mathematical formalisms based on approximate representations
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11101
0
0
0
0 j
jn
jij
i
iiE
i
i
X
Xf
E
Ef
v
v
Linear (1)
00
ln1i
i
i
i
E
E
E
E
01
0
0
0
0
lnln1j
jn
jij
i
iiE
i
i
X
Xf
E
Ef
v
v
(log)linear
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HOTE
E
E
E
yE
Ey
HOTyyy
HOTyyy
yy
HOTyyy
yyy
i
i
i
i
i
i
1ln
1
)1()ln(1
)(1
)ln()ln(
)()ln(
)ln()ln(
00
00
0
00
0
0
0
0
Taylor series Reversing the result
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Mathematical formalisms based on approximate representations
Linear Taylor’s series on Linear(1)
Linear Taylor’s series on Linear(2)
Linear(1) and (y-1)≈Ln(y) (log)linear
Linear(2) and (y-1) ≈Ln(y) Lin-log
Linear Taylor’s series on (log-log coordinates)
Power-law
Linear Taylor’s series on(generalized inverse coordinates)
Saturable and cooperative (SC)
θX,,iE
θX,
θX,,iE
θX,,iE
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The Saturable and Cooperative formalism
Consider a transformation:
Aproximate by Taylor series and return to cartesian coordinates
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mn
j
n
j
j
ij
ij
mn
j
n
j
ji
iij
ij
X
X
p
p
X
XV
v
1 0
1 0
1
0 if lim
0 if lim
1
0
0
0
0
0
ijiX
jm
ijiX
jm
jm
iij
ij
ijij
fvV
fvV
V
vp
p
fn
j
j
ijnjjii Xzvw 1
Sorribas A, Hernández-Bermejo B, Vilaprinyo E, Alves R. Cooperativity and saturation in biochemical networks: a saturable formalism using Taylor series approximations. Biotechnol Bioeng. (2007) 97(5):1259-1277.
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m
j j
jij
i
i
i
i
X
Xf
E
E
v
v
1 0
0
00
ln1
m
j j
jij
i
iiE
i
i
X
Xf
E
Ef
v
v
1 0
0
0
0
0
111
m
j j
jij
i
i
i
i
X
Xf
E
E
v
v
1 0
0
00
11
m
j j
jij
i
iij
i
i
X
Xf
E
Ef
v
v
1 0
0
0
0
0
lnlnLn
mn
j
n
j
j
ij
ij
mn
j
n
j
ji
iij
ij
X
X
p
p
X
XV
v
1 0
1 0
1
0 if lim
0 if lim
1
0
0
0
0
0
ijiX
jm
ijiX
jm
jm
iij
ij
ijij
fvV
fvV
V
vp
p
fn
j
j
m
j j
jij
i
iij
i
i
X
Xf
E
Ef
v
v
1 0
0
0
0
0
lnln1
Linear (1) Linear (2)
(log)linear Lin-log
Power-law
Saturable and cooperative
θX,,ii Ev θX, ii Ev
A family picture of the different formalisms based on Taylor series
►40
41
Using the SC formalism(unknown rate-laws)
0
2
4
11.25
1.51.75
2
0
2
4
6
0
2
4
11.25
1.51.75
2
Mixed AC inhibiton with fractal kinetics
E + S ES E + Pk1
k-1
k2
+
I
k3 k-3
EIk1
k-1
ESI
+
I
k3 k-3
02.119.2
02.119.2
34.202.1
19.23
IS
ISv
225
5.112.1
322
33
211
SET ggE
kk
kkk
SE gg SEkv 11
S
I
v1
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Mathematical formalisms based on approximate representations
Common characteristics of all these representations Exact representations at a given operational point Same operational point values for fluxes and
metabolites
Local sensitivity at the operational point (kinetic-order, elasticity)
00 ,Xiv
0
0
i
j
j
iij v
X
X
vf
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Are those important limitations for analyzing systemic design?
Characterization of systemic properties at a given steady-state can be made with approximated representations as they are exact at that point.
If we can prove that a given design is better (based on criteria of functional effectiveness) independently of the parameter values, then this conclusion holds for any steady state. Issues of accuracy are not relevant in this case.
Accuracy issues become relevant when “goodness” of design depends on parameter values because calculations of systemic behavior away from the operational point become less acurate.
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Characterization from systemic measurements Collect information of the integrated behavior of
the target system Steady-state values at different conditions Dynamic response to perturbations
Fitting a model to dynamic data Non-linear regression, Neural networks, Alternative
regression, etc. In any of those cases, the resulting representation is
no longer local. It smoothes the available data within a given range (for instance by using a least-squares criteria)
►44
• Hernández-Bermejo B, Fairén V, Sorribas A. Power-law modeling based on least-squares criteria: consequences for system analysis and simulation. Math Biosci. 2000,167(2):87-107. • Hernández-Bermejo B, Fairén V, Sorribas A. Power-law modeling based on least-squares minimization criteria. Math Biosci. 1999, 161(1-2):83-94. 2008 Oeiras
Parameter estimation from dynamic data
Which are the limitations of approximated formalisms? Which information is contained in dynamic
data? Can we identify a model from a single
experiment? Experimental design for model identification
from dynamic data Which are the minimum requirements in
collecting dynamic data? Simulated examples
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Which kind of information is contained in dynamic data?
v2
X1
X3 X1 X2
(-)
v2 v3
►Define a reference model using classical kinetics►At t=0 increase X3 and run the simulation►From the simulations compute Xi(t) and vi(t)
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X2
X1
0 2 4 6 80
2
4
6
8
10
Time
Xt
0 2 4 6 8 1 00
2
4
6
8
1 0
1 2
Time
v20 5 1 0 1 5 2 0
0
2
4
6
8
1 0
1 2
1 4
Time
v3
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X1
X2
X2
X1
Out[204]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
A single S-system can fit the experiment
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X3 X1 X2
(-)
v2 v3
11.12
44.012
44.01
97.13
42.121
71.035.2X
35.238.0X
XX
XXX
Several models can fit a single experiment
The information contained in a single experiment allows fitting approximated models as the variation of metabolites and fluxes is limited
In that case, approximated models are a good representation Different formalisms can provide a good model
However, since the information of a single experiment is limited, in most cases various models would be able of fitting the same data
In general, more information would be required to uniquely identify the best model for the actual system
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Which kind of information is contained in dynamic data?
v2
X1
X3 X1 X2
(-)
v2 v3
►Define a reference model using classical kinetics►At t=0 decrease X3 and run the simulation►From the simulations compute Xi(t) and vi(t)
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X2
X1
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0 2 4 6 80
2
4
6
8
10
Time
Xt
0 2 4 6 8 100
2
4
6
8
10
12
Time
v20 5 10 15 20
0
2
4
6
8
10
12
14
Time
v3
X2
X1
X1
X2
Out[204]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
A different S-system can fit each experiment
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X3 X1 X2
(-)
v2 v3
11.12
44.012
44.01
97.13
42.121
71.035.2X
35.238.0X
XX
XXX
Out[222]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
33.12
69.112
69.11
38.13
29.021
49.199.0X
99.035.3X
XX
XXX
X1
X2
X1
X2
Limitations of approximated models in fitting different experiments
No single S-system (or (log)linear/lin-log) model can fit both experiments
The range of variation demands a more flexible representation
The SC formalism can provide a solution
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Out[161]=
0 2 4 6 8 100
2
4
6
8
10
12
Time
v2
0 5 10 15 20
0
2
4
6
8
10
12
14
Time
v3
0 2 4 6 8 100
2
4
6
8
10
12
Time
v2
0 5 10 15 20
0
2
4
6
8
10
12
14
Time
v3
Exp.1
Exp.2
Exp.1
Exp.2
X2
X1
The SC formalism as a tool for fitting dynamic dataFitting single SC model to both experiments
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Out[323]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
Out[325]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
Out[327]=
0 2 4 6 80
2
4
6
8
10
Time
Xt The SC fromalism can fit the
experiments 1-2, but further information is required to fully identify the model and predict the outcome of a new experiment.
Experiment 1 Experiment 2
New experiment
Reference
Predicted
The SC formalism as a tool for fitting dynamic dataFitting single SC model to both experiments
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Out[323]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
Out[325]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
Three experiments were required in that case to fully recover the reference model. The fitted SC model can now reproduce other experiments
Experiment 1 Experiment 2
Out[332]=
0 2 4 6 80
2
4
6
8
10
Time
Xt
Experiment 3
Out[352]=
0 2 4 6 80
5
10
15
20
Time
Xt
New Experiment
The importance of collecting data from different perturbations
If possible, collect metabolite levels and fluxes. Plot flux versus metabolites would provide a good
estimate of the range of dynamic variation Data from different perturbations allows for
characterizing the systemic behavior at a wider range The predictive value of the fitted models would be
better Single experiments can be, in many cases, easily
reproduced. However, the resulting model (of any kind) would have a limited predictive value Full characterization would require complementary
information from several experiments
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ConclusionsRelevance of models based in approximated representations for Systems Biology applications
Approximated representations are required for practical reasons. Systematic representation. Models can be produced
automatically from schemes. Qualitative information can be incorporated into models. Models can be easily updated and shared.
Any of the non-linear approximations discussed can be used, although each one has its own range of application. Parameter estimation is an issue. So far, power-law and SC
formalisms appear as more indicated, specially with dynamic data.
The power-law formalism has a whole set of tools and strategies that facilitates the investigation of design and operational principles. (log)linear and lin-log approximations, at best, can produced
results similar to those obtained using a power-law formalism.
The SC formalism can be used to complement the results of the power-law formalism, particularly in the dynamic range.
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