what? why? how? an introduction to modeling biological systems eberhard o.voit department of...
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What? Why? How?What? Why? How?An Introduction to ModelingAn Introduction to Modeling
Biological SystemsBiological SystemsEberhard O.VoitEberhard O.Voit
Department of Biomedical EngineeringDepartment of Biomedical EngineeringGeorgia Institute of Technology and Emory UniversityGeorgia Institute of Technology and Emory University
Atlanta, GeorgiaAtlanta, Georgia
1111thth International Conference on International Conference on Molecular Systems Biology Molecular Systems Biology
June 21-25, 2009 June 21-25, 2009 Shanghai, ChinaShanghai, China
Points to Ponder (1)
What is a model? What is modeling?
Conceptual
Physical
Maps and blueprints
Mathematical models
Does modeling change over time?
Euclid and computers
Points to Ponder (2.1)
Why modeling?
Prediction
Manipulation, optimization
Explanation (counterintuitive behavior;
chains of causes)
Bookkeeping
Organize thoughts
Organize data
Identify outliers
Points to Ponder (2.2)
Why modeling?
(a) (b)
X2
X1
X2
X12.0
212
2.01
15.02
4.0111
XXX
XXXX
1 = 0.9 or 1 = 1.02
X1
X2
0 30 600
0.75
1.5 (c)
time 0 120 2400
1.5
3
X1
X2
(d)
time
Points to Ponder (3.2)
What is a good/bad model?
“The best” model?
Example: Heart
Purpose
Correctness
Simplicity vs. complexity
Degree of detail
Range of applicability
Qualitative vs. quantitative results
Points to Ponder (4)
Limitations of models
Assumptions
Simplifications
Extrapolation
Complexity masking problems, errors
Points to Ponder (5)
Theory of biology
Specific predictions; population vs. individual
General predictions; qualitative vs. quantitative
Design and operating principles
Steps of a Typical Analysis
Model conception and formulation
Parameter estimation
Concept of a steady state
Stability
Sensitivities, gains, robustness
Dynamics
Bolus experiments
Persistent changes in system components
<Optimization>
RealitySimplification and Abstraction
Ignore detailsOmit components, factorsHypothesize
ApproximationRepresent complex processes with
simple(r) functionsLinear, nonlinear, piecewise
Model Conception
Reality Abstraction Graph Equations Analysis Reality
Criteria of a Good Model
Capture the essence of the system under realistic conditions
Be qualitatively and quantitatively consistent with key observations
In principle, allow analyses of arbitrarily large systems
Be generally applicable
Be characterized by measurable quantities
Allow simple translation of results back to subject area
Have a mathematical form that is amenable to analysis
Formulation of a Model for Complex Systems
Tenets of systems analysis :
Each component of the system may potentially dependon all other components and outside factors.
To “understand” the system, we need to know how every component changes over time.
Dynamic changes in a system component are driven by inputs and outputs.
Convenient Math
Two emerging dogmata:
1. Not all mathematical approaches are equally useful.
2. All laws in nature are approximations.
Two pieces of conventional wisdom:
1. In math it’s either right or wrong.
2. Laws in nature are true and absolute.
Components of a Systems Model
VariablesDependentIndependentTime
Change
ProcessesFlow of materialSignals
Parameters and constants
Variables
Dependent: Variable is affected by the action of the systemtypically changes over timemay or may not affect other variables
Names: X, Y4, Zi
Independent: Variable is not affected by the action of the systemtypically constant over timesometimes external and under experim. controlmay or may not affect other variablesexamples: inputs, enzymesmay change from one “experiment” to the next
Change
Mathematics: time is independent variableSystems modeling: time is often implicit
What about time and change?
Typical equation:
outFlux -in Flux 3 dt
dXChange in X3 over time =
Change (cont’d)
Fluxes are functions of variables, thus:
outFlux -in Flux 33 X
dt
dX
new notation
s' ofFunction - s' ofFunction 3 XX X
Don’t see t anymore, but variables do change over time.
Example from enzyme kinetics:
33
33max
22
22max3 XK
XV
XK
XVX
MM
Example: Radioactive Decay
“The change in X is directly proportional to the present amount of X, the proportionality is quantified by k,and the change is in the negative direction (decrease).”
Why does this describe radioactive decay over time?“Solution” to the differential equation is X(t) =X0 exp(–kt),because
kXktkXXdt
dX )exp(0
kXX Equation:
Change (cont’d)
Processes
Very important to distinguish
Flow of material (mass is moving): Solid, heavy arrows
andFlow of information (signals, modulation) :
Dashed, thin arrows
Essentially any interaction between variables orbetween system and environment
Confusion may lead to wrong model structure;often difficult to diagnose.
Formulation of a Model for Complex Systems
Translation into math :
X1
V1+ V1
–
Xi
Vi+ Vi
–
),...,,,...,,( 121 mnnnii XXXXXVV
inside outside
very complex
111 VVdt
dXX 1
iii VVdt
dXX i
Formulation of a Model for Complex Systems
),...,,,...,,( 121 mnnnii XXXXXVV
mn
j
gji
gmn
ggii
ijmniii XXXXV1
21,21 ...
Savageau: Approximate it per Taylor but in log-space
Result:
What can we do with this “very complex” function
?
iii VVdt
dXX i
S-systems
mniiimniii hmn
hhi
gmn
ggii XXXXXXX
,21,21 ... ... 2121
The change in each system component is described asa difference between two terms, one describing allcontributions to growth or increase in the variable,the other one describing all contributions to loss or decrease in the variable.
Each term is represented as a product of power-functions.
Each term contains and only those variables that have a direct effect; others have exponents of 0 and drop out.
’s and ’s are rate constants, g’s and h’s kinetic orders.
Alternative Power-Law Formulations
mniiimniii hmn
hhi
gmn
ggii XXXXXXX
,21,21 ... ... 2121
S-system Form:
Xi
Vi1+ Vi1
–
Vi,p+ Vi,q
–
ijiji
i VVdt
dXX
Generalized Mass Action Form:
ijkfjiki XX
Meaning of Parameters
Kinetic orders g i j , h i j :
Effect of variable Xj on production or degradationof variable Xi.
Rate constants i and i:
Magnitude of production and degradation fluxes of Xi.
Meaning of Parameters
Kinetic orders g i j , h i j :
Effect of variable Xj on production or degradationof variable Xi.
Rate constants i and i:
Magnitude of production and degradation fluxes of Xi.
Parameter Values
Experience and educated guesses.
Data needs, advantages and limitations of the various approaches.
Estimation of parameters from traditional rate laws.
Estimation of kinetic orders from steady-state data. Estimation of parameters from dynamic data.
Talk during Conference.
Comment on Parameter Estimation
Parameter estimation is arguably the hardest part of
modeling
Very different options:
flux-versus-concentration data
rate laws
dynamic data
Dynamic data contain the most information, but are
the most difficult to evaluate
Steps of a “Typical Analysis”
Model conception and formulation
Parameter estimation
Computations at a steady state
Stability
Sensitivities, gains, robustness
Dynamics
Bolus experiments
Persistent changes in system components
<Optimization>
Can we compute the steadystate(s) of the system?
Does the system have a steadystate, where no variable changes in value?
How is the steady state ofthe system affected by inputs?(“Gains”)
Steady-State Analysis
Justitia of Biel
Can the system tolerate a slightly changed structure?(“Sensitivities”)
Can the system tolerate alarge perturbation?(Change in environment)
Can the system tolerate asmall perturbation?(Normal fluctuations inmilieu)
Can the system tolerate a slightly changed structure?(Mutation, Disease)
Stability
Castellers of Nens del Vendrell
Method: Eigenvalue analysis.
Rules of Thumb Stable system + “small” temporary perturbation system returns to old steady state Stable system + “large” temporary perturbation ? ? ?
almost anything may happen: system may approach new steady state, die, blow up, start to oscillate, …
Unstable system + any temporary perturbation ? ? ?
almost anything may happen … Stable system + “small” persistent perturbation system moves to new steady state Stable system + “large” persistent perturbation ? ? ?
structure may be changed significantly … Unstable system + any persistent perturbation ? ? ?
almost anything may happen …
How does the system respondto changed input?
Where is the system goingfrom here?
How does the system respondto a slightly changed structure?
Dynamics
How can we optimize the performance of the system ?
How can we intervene in thefunction of the system?
Georgio de Chirico (1914)
Dynamical Analyses
Types:BolusPersistent change in inputExogenous supply of (dependent) metaboliteChanges in structure
Methods:Algebraic analysesNumerical analysesSimulations
Almost all done per computer!
This is the fun part!
Dynamical Analyses, Based on (Numerical) Solutions Solve (“run”) system
Plot or tabulate results Study transients and approach of steady state Study oscillations and other responses
Change initial values and run again
Corresponds to system that is temporarily perturbed; eat sugar, … Question: Are effects of the change significant? Instability?
Change independent variables and run again
Corresponds to change in environmental conditions Exogenous supply (input), increase temperature, …
Change parameters and run again
Corresponds to permanent change in system structure Mutation, remove kidney, stronger feedback, build a dam, …
Compare various aspects of systems and their solutions
Example
X 1X 2X 5
X 3
X 4
Question: What affects production of X1?Answer: X5 and X3
Thus: 1513
5311gg XXV
11111hXV Degradation analogously:
Example (numerical)
X 1X 2X 5
X 3
X 4
5.0
75.0)0( 58
9.0)0( 25.12
5.0)0( 105
1.1)0(510
5
45.0
45.0
24
35.0
35.0
23
25.0
25.0
12
15.0
151.0
31
X
XXXX
XXXX
XXXX
XXXXX
X1 X2X5
X3
X4
5.0
75.0)0( 58
9.0)0( 25.12
5.0)0( 105
1.1)0(510
5
45.0
45.0
24
35.0
35.0
23
25.0
25.0
12
15.0
153113
X
XXXX
XXXX
XXXX
XXXXX g
Example: Focus on Inhibition
X1 X2X5
X3
X4
5.0
75.0)0( 58
9.0)0( 25.12
5.0)0( 105
1.1)0(510
5
45.0
45.0
24
35.0
35.0
23
25.0
25.0
12
15.0
153113
X
XXXX
XXXX
XXXX
XXXXX g
g13 is the parameter that characterizes the strength of the inhibition. g13 is negative or zero. If g13=0, then there is no inhibition.
Example: Focus on Inhibition
F i r s t E x p l o r a t i o n : g 1 3 = 0 ( n o i n h i b i t i o n ; b a s e l i n e )
0 2 4
0
. 7 5
1 . 5
t i m e
X 1
X 2
X 3
X 4Con
cent
rati
on
J u s t f o r c o m p l e t e n e s s : C h e c k e i g e n v a l u e s : a l l n e g a t i v e r e a l s t a b l e s y s t e m
T r y g 1 3 = – 1 : W h a t i s y o u r p r e d i c t i o n ?
g 1 3 = – 1 :
0 2 4
0
. 7 5
1 . 5
t i m e
X 1
X 2
X 3
X 4
Con
cent
rati
on
C o m p a r e w i t h p r e v i o u s g r a p h ! F o r i n s t a n c e , X 1 g o e s u p !
A g a i n , c h e c k e i g e n v a l u e s : a l l n e g a t i v e r e a l s t a b l e s y s t e mi m a g i n a r y p a r t s o s c i l l a t i o n s p o s s i b l e
W i t h t h i s n e w k n o w l e d g e , w h a t i s y o u r p r e d i c t i o n f o r s t r o n g e r i n h i b i t i o n ?
g 1 3 = – 4 :
0 2 4
0
1
2
t i m e
X 1
X 2
X 3 X 4
Con
cent
rati
on
A g a i n , c h e c k e i g e n v a l u e s : a l l n e g a t i v e r e a l s t a b l e s y s t e m i m a g i n a r y p a r t s o s c i l l a t i o n s
W h a t i s y o u r p r e d i c t i o n f o r s t r o n g e r i n h i b i t i o n ?
g 1 3 = – 8 :
0 2 4
0
1
2
t i m e
X 1
X 2
X 3
X 4
Con
cent
rati
on
A g a i n , c h e c k e i g e n v a l u e s : a l l n e g a t i v e r e a l , b u t s m a l l e r i n m a g n i t u d e s t a b l e s y s t e mi m a g i n a r y p a r t s o s c i l l a t i o n s
I n w o r d s : s t r o n g e r i n h i b i t i o n s t r o n g e r o s c i l l a t i o n s , b u t s t e a d y - s t a t e u n c h a n g e d
g 1 3 = – 1 6 :
0 2 4
0
2 . 5
5
t i m e
X 1
X 2
X 3
X 4
Con
cent
rati
on
A g a i n , c h e c k e i g e n v a l u e s : t w o p o s i t i v e r e a l u n s t a b l e s y s t e m !E x p l a i n !
S t r o n g e r i n h i b i t i o n ? T r y i t o u t i n t h e w o r k s e s s i o n !
T a k e h o m e m e s s a g e : S y s t e m l o s e s s t a b i l i t y w h e n i n h i b i t i o n b e c o m e s t o o s t r o n g .
Case Study: Purine Metabolism
Start: Lots of data (kinetic, physiological, clinical, …) Which data are (most) relevant?
Decent idea about pathway structure
Questions:Do pieces fit together?Can we make reliable predictions?How do diseases relate to metabolism?What would be good drug targets?
First Model PRPP
IMP
Xa
UA
GMP
GDP
GTP
S-AMP
AMP
ADP
ATP
HX
XMP
5--P
v16
v15
v14v13
v12
v11
v10
v8
v7
v9
v3
v2
v1
v6
v5
v4
EquationsParametersAnalysis
Stoichiometry faulty
Verdict: Revise!
Second Model PRPP
IMP
Xa
UA
Ade
dGMP
dGDP
dGTP
GMP
GDP
GTP
Gua
Guo
dGuo
dAdo
dAMP
dADP
dATP
AdoAMP
ADP
ATP
HX
Ino
dIno
vprpps
vaprt
vgprt
vhprt
vadrnr
vua
vhprt
vimpd
vden
vgmprvampd
vpolyam
vasuc
vgnuc vgdrnr
vdgnuc
vinuc
vgprt
vgua vhxd
vxd
vada
vdada
S-AMP
SAM
XMP
-700
-600
-500
-400
-300
-200
-100
0
100
200
300
X1 X3 X5 X7 X9 X11 X13
51
79
1113
3
Dependent Variables Rate Constants
Sen
siti
viti
es
RefinementsMore DataAnalysis
Verdict: Revise!
“Final” Model
Numerous iterations of refinements and comparisons
Verdict: Cautious optimism
PRPP
IMP S-AMPXMP
RNA
DNA
Xa
UA
Ade
SAM
dGMP
dGDP
dGTP
GMP
GDP
GTP
Gua
Guo
dGuo
dAdo
dAMP
dADP
dATP
Ado
AMP
ADP
ATP
HX
Ino
dIno
R5P
vprppsvpyr
vaprt
vade
vgprt
vhprt
vadrnr
vx
vua
vhprt
vimpdvgmps
vden
vgrna
vrnag
vgmpr
varna
vrnaa
vtrans
vaslivampd
vpolyam
vasuc
vmat
vgnuc
vgdrnr
vdgnuc
vdnag
vgdna
vdnaa
vadna
vinuc
vhx
vgprt
vgua vhxd
vxd
vada
vdada
Pi
Pi
Pi
Pi Result:Model consistent with literature informationNew classification of purine-related mental diseases
What can we do with such a model?
o Analyze normal metabolic state: study responses
o Bolus experiments: study response to inputs
o Changes in enzyme activities: study metabolic diseases
o Change fluxes: Screen for drug treatments
Intended effects
Side effects
o Redistribute fluxes
diseases
metabolic engineering
Optimization(Citric acid again)
GLUCOSE Ext., X19
TRP1, X20
HK, X21ATP, X17
ADP
G6P, X2 G6Pdh, X22
PGI, X23
F6P, X3 F2,6P, X4
FBPase, X24
PFK2, X25
PFK1, X26
CITc, X16
NH4+, X27
2 PEP, X5
PK, X28
2 NAD+
2 NADH, X18
2 PYRc, X6
TRP2, X34
OXAc, X7
PC, X29 MDHc, X32
OXYG, X41RESP, X42
PYRm, X11
CIT, X14
ISC, X15
AcCoA, X13
CoA, X12
TRP5, X44
TRP4, X43
PDH, X36
ACN, X39
ISCDH, X40
CS, X37
MEDIUM
CYTOPLASM
MITOCHONDRION
PEPc, X30
ALAt , X35
GLUCOSE Int., X1
GOT, X31
NADHX18
ATPase, X46
PC, X29
PFK1, X26
AK, X45
PK, X28
HK, X21
PFK2, X25
PEPc, X30
NADHase, X47
MDHc, X32
RESP, X42
PK, X28
MDHm, X38
PDH, X36
ALA
ATP, X17
ATP, X17
ADP
ADP
+++
+++
–––
ADP
4 ADP
4 ATP, X17
ATP, X17 ADP
ATP, X17
NADH, X18 NAD+
MALc, X8
TRP3, X33
NADH, X18 NAD+
MDHm, X38
MALm, X9OXAm, X10
NADH, X18
NAD+
–––
–––
NADH, X18
NAD+
ATPase, X46
AK, X45
NADHase, X47
ATP, X17 + AMP
ATP, X17 ADP
2 ADP
NADH, X18 NAD+
ATPase, X46
AK, X45
NADHase, X47
ATP, X17 + AMP
ATP, X17 ADP
2 ADP
NADH, X18 NAD+
ATPX17
Task:Reroute flux in anoptimal fashion;e.g., maximizecitric acid output
Pathway Optimization with S-systems (Voit, 1992)
Optimization under steady-state (batch) conditions becomes
Linear Program
even though (nonlinear) kinetics is taken into account:
maximize log(flux) [or log(variable)]
subject to:
Steady-state conditions in log(variables)
Constraints on log(variables)
Constraints on log(fluxes)
Pathway Optimization (cont’d)
Hatzimanikatis, Bailey, Floudas, 1996: Use these features foroptimization of pathway structure
Great Advantage:
Methods of Operations Research applicable• very well understood • applicable for over 1,000 simultaneous variables• robust and efficient• incomparably faster than nonlinear methods
Torres, Voit, …: Applications (e.g., citric acid, ethanol, glycerol, L-carnitine)
Pathway Optimization (cont’d)
Recent extensions:
Optimize dynamics over time horizons(with Ernandi-Radhakrishnan)
Optimize Generalized Mass Action systems (alternative power-law systems), using dynamic programming andbranch-and-bound methods(with E. Gatzke, USC)
Method of ControlledMathematical Comparisons
Crucial consequence for many purposes: Structure determined by parameter values Identification of structure becomes parameter estimation Comparison of two alternative systems allows
characterization of the role of some mechanism
In contrast to models such as polynomials, the relationship between S-system parameters and structural features of a pathway is essentially one-to-one.
Recall: Search for Design Principles
X4
X7
X1
X5 X2
X6 X3
X4
X7
X1
X5 X2
X6 X3
Exploration of Design Principles:What is the effect of feedback inhibition,
everything else being equal?
Controlled Mathematical Comparisons
1. Construct a system model and an alternative that differsin one feature of interest (e.g., feedback inhibition).
2. Select all parameter values the same but adjust parameter values associated with this feature such that both systems have same steady state and as many other features as possible.
3. Study sensitivities, dynamics etc. (e.g., response time)4. Differences are caused by the parameter (mechanism) of
interest.5. New variations on this theme in Schwacke and Voit (2004).6. Applications mainly in gene circuitry (Savageau et al.)
and some generic metabolic pathways.7. Results are almost independent of specific parameters and
elucidate general design and operating principles.
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