1. Mathematical Modeling of Metabolism, Signaling Pathways,
and Gene Expression
Edda Klipp
Max Planck Institute for Molecular Genetics
Ihnestr. 73, Berlin, Germany

2. Content
Techniques
Introduction to Modeling - Kinetics of individual reactions
Construction of networks - Systems equations,
system simplifications,
conservation relations - Analysis of networks and
cellular reaction systems
- Time constants, time hierarchy - Metabolic control analysis
- Computer based modeling tools 
Datenbases
Biological objects - Enzyme reactions, Ligand interactions
Metabolism
Signal transduction pathways
Gene expression - Cell cycle

3. Systems Biology
Biological research in the 1900s followed a reductionistic approach:
detect unusual phenotype  isolate/purify 1 protein/gene,
 determine its function
However, it is increasingly clear that discrete biological function can only rarely be attributed to an individual molecule.
 The new task is to understanding the structure and dynamics of the complex intercellular web of interactions that contribute to the structure and function of a living cell.

4. Systems Biology
Development of high-throughput data-collection techniques,
e.g. microarrays, protein chips, yeast two-hybrid screens
allow to simultaneously interrogate all cell components at any given time.
 there exists various types of interaction webs/networks
- protein-protein interaction network
- metabolic network
- signaling network
- transcription/regulatory network ...
These networks are not independent
but form „network of networks“.
Barabasi & Oltvai,
Nature Rev Gen (2004)

5. What is a model? Abstraction
Yeast, mouse – as models for human
Verbal explanation
A sequence of letters ATTCGAGGTATA for DNA sequence
Wiring scheme (e.g. Metabolic network)
Mathematical description: Boolean Network
Differential Equations
Stochastic Equations
Abstraction
Simplified representation allowing for understanding

6. Why modeling? Experimental observations:
many simple and complex processes
isolated enzymatic reaction :
temporal prozesses in metabolic networks
pattern of gene expression and regulation
Even the behavior of simple systems can usually not be predicted
intuitively and from experience.
The behavior of complex dynamical processes can not predicted with
sufficient precision just from experience.
For prediction and explanation of processes one needs a model.

7. Why modeling? Advantages Time scales may be stretched or compressed.
Solution algorithms / computer programs can often be used
independently of the actually modeled system.
Costs of modeling are lower than for experiments.
Representation of quantities that are experimentally hidden.
No risk for real systems, no interactions investigation/system.

8. Why modeling? Burning questions (there are more…)
How is cellular response to environmental changes and stress regulated?
How should a cell be treated to yield a high output of a desired product (Biotechnology)
Where should a drug operate to cure a disease (Health care)?
Is our knowledge about a network/pathway complete?

9. Example 1: African Sleeping Disease
Epidemias in East and Central Africa (3-4 Mill. sick)
Parasit (Flagellat) in blood
Carrier: Tsetse fly
NO THERAPY
Glykosom
Parasit
Red
Blood
cell
Mitochondrium
Blood
Of Host

10. Example 2: Metabolic Oscillations
Experimental
Observation:
Addition of KCN
to yeast cells causes
metabolic oscillations
(measured as NADH
fluorescence)
What happens after
mixing of populations
that oscillate out of
phase?
NADH Fluorescence
+ KCN
+ KCN
Time
Time
MIX
Time
NADH Fluoresczence

11. Example 2: Metabolic Oscillations
Explanatory
Model
Oscillations:
What is the Oscillator?
How occurs synchronization?
Ethanol
Acetaldehyde
Glucose
Ethanol
Acetaldehyde
Glucose
Acetaldehyd
Acetaldehyd

12. Example 3: Gene Expression Regulation
Whole-Genome MicroArrays
De Risi et al., 1997 Science 278
During diauxic shift at glucose starvation
Yeast cells switch their metabolism from
fermentation to respiration.
In this time, they also change the expression
pattern of about 1700 of the 6000 genes in an
ordered manner.
- red: induced expression
- green: reduced expression
(measured as fluorescently marked cDNA)

13. Example 3: Gene Expression Regulation
Bioinformatic:
- Clustering of genes according to expression profiles
- Reconstruction of networks
Modeling:
Establishing a dynamic model for changes of
concentrations or activities
- Prediction of changes of metabolism
- Optimization approach: Prediction of optimal
gene expression under certain conditions

14. Example 4: Biosynthesis Optimization
Glucose
Amino acid
Synthesis
Glucose-6-P
Fructose-6-P
6-Phosphogluconat
Histidin
PRPP
Erythrose-4-P
Glycerinaldehyd-3-P Dihydroxyacetonphosphat
Glycin
Serin
Chorismat
Phenylalanin
Cystin
Cystein
Phosphoenolpyruvat
Tyrosin
Tryptophan
Isoleucin
Pyruvat
Alanin
Valin
How to increase the
synthesis of specific
amino acids?
Threonin
Lysin
Leucin
Homoserin
Aspartat
Oxalacetat
Methionin
Asparagin
a-Ketoglutarat
Metabolic models
Flux balance analysis
Metabolic control analysis
Lysin
Hydroxyprolin
Prolin
Glutamat
Arginin
Glutamin

15. Model and Modeling The analysis of model systems generates a language,
to describe complex processes.
Investigation starts with:
- Ordering, classification
- Choice of prototypes
- Systematics
e.g. Enzyme reactions
They may have 1, 2, 3,... substrates; they catalyze transfer of different groups,...
Phosphofructokinase is an example for kinases
Enzyme nomenclature (EC )

16. Model and Modeling Distinction of complex processes:
reversible – irreversible
periodic – non-periodic
deterministic – stochastic
If the system is known at one time
then it is known forever
Probability distribution for
system states
Other system-relevant categories:
stable – unstable
robust – fragile
active - inactive

17. Adequateness of Models
Theoretical models are frequently mathematical models.
but mathematical formalism is not first aim of a model,
instead start with consideration of problem.
Translation of the biological problem into a model.
Models are not “right” or “wrong”.
Their usefulness is a compromise between
Adequatness (reflection of reality) and handling/Simplicity.
„If we don‘t make the models,
we will not know, why they are wrong.“

18. Purpose of a Model
A model reflects only specific aspects of the reality.
Aim of a model: give answers to specific questions!!!
Model development is subjective and selective.
Prediction of system behavior: precise results for input/output
but system itself might maybe be treated as black box.
Function of an object: model be very realistic,
parts/structure/relations of the object/process are important (glass box).
Generality: model application to many different objects/processes
Simplicity: e.g. for mathematical treatment – contradiction to realistic

19. Model Development in 5 Steps
Defining the scope –
- Formulating the PROBLEM
Distribution of molecules on
both sides of a membrane
Ai Ao
dAi/dt = f(Ai, Ao, C, p)
Establishing a simple model
as „Cartoon“
in mathematical terms
- Model verification: can it in principle explain the observation?
- Solving the related (mathematical) problems
- Model validation: determine parameters from experimental data
- Sensitivity analysis: depends the result on parameter choice?
Prediction / Comparison of results with real systems (EXPERIMENT)
- Difference ?!?
- Iterative improvement of model (model structure, parameters, …)

20. Structure of the system
fast
Sext S1 S2 S3 S4 S5
Smito S6
Boundary of the system
slow
slow
Variables, parameters, constants
State variables - set of variables describing the system completely
Dimension of the systems = number of independent state variables
How many variables are used in my model?
Units of variables and parameters etc. fit together?

21. Variables, Parameters, Constants
V, d – Variable, p – Constant
– Parameter, v, S – Variables
Variables – variable quantities, for which the model establishes a relation
Constant – quantity with a fixed value
Parameter – quantity to which a value can be assigned
This value is changeable and depends on measurements.
If a quantity is a variable, a parameter or a constant depends on the model.

22. Analysis of Dimensions
State variables are a set of variables that describe a system completely.
They are independent of each other; each of them is necessary to describe
the system states completey.
Dimension of a systems = number of independent state variables
How many variables has my model?
too few – system is underdetermined
too many – system is underdetermined and probably contratictory
Question/Task
Can we reduce the number of state variables, e.g. by changing system limits?
Which units do the variables have?

23. What determines system behavior?
Two fundamental causes for changes of the system behavior:
Influences of the environment (input)
Processes within the system
– the structure of the system determines how exogenous or endogenous
signals will be processed
General rule: assign system limits such that the output does NOT feedback
To the input of the system.
General rule: different system structures can produce similar system behavior
Structure determines behavior, not the other way around.

24. Direction of discovery
known to be predicted
Structure Function
Protein interactions Biochemical action
Metabolic pathways Concentration changes
Enzyme sets Influence of perturbations
Possible behavior, bifurcations
: :
Function Structure
Transmission of a signal Sequence of signaling compounds
Time course of concentrations Possible protein interactions

25. Concept of state
The state of a system is a snapshot of the system at a given time that
contains enough information to predict the behaviour of the system for
all future times. The state of the system is described by the set of variables that must kept track of in a model.
Different models of gene regulation have different representations of the state:
Boolean model: a state is a list containing for each gene involved, of whether
it is expressed („1“) or not expressed („0“)
Differential equation model: a list of concentrations of each chemical entity
Probabilistic model: a current probability distribution and/or a list of actual
numbers of molecules of a type
Each model defines what it means by the state of a system.
Given the current state the model predicts what state/s can occur next.

26. Kinetics – change of state
A B
Deterministic, continuous time and state: e.g. ODE model
concentration of A decreases and concentration of B increases. Concentration change in per time interval dt is given by
Probabilistic, discrete time and state : transformation of a molecule of type A into a molecule of type Sorte B. The probability of this event in a time interval dt is given by
a – number of molecules of type A
Deterministic, discrete time and state : e.g. Boolean network model
Presence (or activity) of B at time t+1 depends on presence (or activity) of A at time t

27. Biological processes are complex phenomena
Central dogma of molecular biology:
Gene
mRNA
Proteines
Cellular processes

28. Cellular Reaction Systems are Networks
Behavior is dependent on
the individual kinetics of single reactions
the topology of the network
Metabolism
Graph theoretical: Nodes + Edges
 Routes, Measures for Connectivity
Stoichiometric: + number of molecules
Moiety conservation, Flux distribution
Kinetic: + Kinetics + Concentrations
Time-dependent behavior, Steady States,
Control Analysis, Bifurcations
Signaling

29. Basic Elements of Biochemical Networks
Transport
Reaction
Reaction
Glucose1-P v1
Glucose6-P v2
Fructose6-P
Metabolite
Metabolite
Glucose-
Phosphat-
isomerase
Phospho-
glucomutase
Metabolite
Design of structured dynamic models
1. Determination of system limits
G1P
G6P
F6P v1 v2
System
extern
Concentration change =
Production – Degradation + Transport
2. Balancing
Rate as function of
concentrations and parameters
3. Assignment of
Kinetics

30. Basic Elements of Biochemical Networksv1 v2 v3 v4 v5
Systems equations
r – number of reactions
Si – metabolite concentrations
vj – reaction rates
nij – stoichiometric coefficients
Network properties
Individual reaction properties
Matrix representation

31. Data Bases GO (Gene Ontology)
functional description of gene products
KEGG (Kyoto Enzyclopedia of Genes and Genomes)
reference knowledge base offering information about genes and proteins, biochemical compounds and reactions, and pathways
BRENDA (Comprehensive Enzyme Information System)
curated database containing functional data for individual enzymes
NCBI (National Center for Biotechnology)
,provides several databases:
- molecular databases, with information about nucleotide sequences, proteins, genes,
molecular structures, and gene expression
- taxonomy database: names and lineages of more than 130,000 organisms
SPAD (Signaling PAthway Database)
information about signaling pathways (schemes, links)
JWS Online, Model database
published models,implemented in Mathematica®
Models can be simulated
Biomodels, Model database
published models,implemented in SBML

32. Modeling Tools http://sbml.org BALSA BASIS BIOCHAM BioCharon
biocyc2SBML
BioGrid
BioModels
BioNetGen
BioPathway Explorer
Bio Sketch Pad
BioSens
BioSPICE Dashboard
BioSpreadsheet
BioTapestry
BioUML
BSTLab
CADLIVE
CellDesigner
Cellerator
CellML2SBML
Cellware
CL-SBML
COPASI
Cytoscape
DBsolve
Dizzy
E-CELL
ecellJ
ESS
FluxAnalyzer
Fluxor
Gepasi
INSILICO discovery
JACOBIAN
Jarnac
JDesigner
JigCell
JWS Online
Karyote*
KEGG2SBML
Kinsolver*
libSBML
MathSBML
MesoRD
MetaboLogica
MetaFluxNet
MMT2
Modesto
Moleculizer
Monod
Narrator
NetBuilder
Oscill8
PANTHER Pathway
PathArt
PathScout
PathwayLab
Pathway Tools
PathwayBuilder
PaVESy
PNK
Reactome
ProcessDB
PROTON
pysbml
PySCeS
runSBML
SBML ODE Solver
SBMLeditor
SBMLmerge
SBMLR
SBMLSim
SBMLToolbox
SBToolbox
SBW
SCIpath
Sigmoid*
SigPath
SigTran
SIMBA
SimBiology
Simpathica
SimWiz
SmartCell
SRS Pathway Editor
StochSim
STOCKS
TERANODE Suite
Trelis
Virtual Cell
WinSCAMP
XPPAUT

33. Modeling of biochemical reaction networks
Content (metabolism):
Components
Main properties of catalyzed reactions
Structure of networks
Stoichiometric analysis, flux balance analysis
System simplifications
Control theory
Definitions, Theorems
Typical control structures in metabolic networks

34. Literature Mathematical Physiology, Springer, 1998
James Keener, James Sneyd
Computational Cell Biology, Springer, 2002
Editors: Christopher Fall, Eric Marland, John Wagner, John Tyson
Systems Biology in Practice. Concepts, Implementation and Application, VCH Wiley, 2005
Edda Klipp, Ralf Herwig, Axel Kowald, Christoph Wierling, Hans Lehrach
Systems Biology. Properties of Reconstructed Networks, Cambridge Univ Press, 2006
Bernhard Palsson
System Modeling in Cellular Biology. From concepts to nuts and bolts, MIT Press, 2006
Editors: Zoltan Szallai, Jörg Stelling, Vipul Periwal
An Introduction to Systems Biology. CRC Press, 2006
Uri Alon
Heinrich, R. & Schuster, S., 1996, The Regulation of Cellular Systems, Chapman & Hall
Fell, D. 1997, Understanding the Control of Metabolism, Portland Press