Download - What? Why? How? An Introduction to Modeling Biological Systems Eberhard O.Voit Department of Biomedical Engineering Georgia Institute of Technology and

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


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


What is a good/bad model?


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


Limitations of models

Assumptions

Simplifications

Extrapolation

Complexity masking problems, errors


Theory of biology

Specific predictions; population vs. individual

General predictions; qualitative vs. quantitative

Design and operating principles


Type of model

Components

Methods

Use

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.

Needs

Language / Notation:

“Convenient” math

Theory

Methods of analysis

Must be mathematics

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.


Translation into a diagram :

X1

V1+ V1

–

Xi

Vi+ Vi

–


Translation into math :

X1

V1+ V1

–

Xi

Vi+ Vi

–

),...,,,...,,( 121 mnnnii XXXXXVV

inside outside

very complex

111 VVdt

dXX 1

iii VVdt

dXX i


),...,,,...,,( 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.

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

Modeling Process

Ideas

Reality Check Draft (?) Model

Refinement

Data

Analyze

Data

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.

Summary

Given fully parameterized model equations, study:

Steady-stateSensitivities, gainsStabilityDynamics:

bolus, mutation, scenarios, simulations

Some analyses could be done by hand, but computer analysis is much more convenient

Download - What? Why? How? An Introduction to Modeling Biological Systems Eberhard O.Voit Department of Biomedical Engineering Georgia Institute of Technology and

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