Computational and Mathematical Challenges from Molecular Life - - PowerPoint PPT Presentation

Computational and Mathematical Challenges from Molecular Life Sciences

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austria and The Santa Fe Institute, Santa Fe, New Mexico, USA

Vortragsreihe Computational Life Sciences Universität Wien, 11.11.2005

Web-Page for further information: http://www.tbi.univie.ac.at/~pks

Fully sequenced genomes Fully sequenced genomes

• Organisms 751

751 projects 153 153 complete (16 A, 118 B, 19 E)

(Eukarya examples: mosquito (pest, malaria), sea squirt, mouse, yeast, homo sapiens, arabidopsis, fly, worm, …)

598 598 ongoing (23 A, 332 B, 243 E)

(Eukarya examples: chimpanzee, turkey, chicken, ape, corn, potato, rice, banana, tomato, cotton, coffee, soybean, pig, rat, cat, sheep, horse, kangaroo, dog, cow, bee, salmon, fugu, frog, …)

• Other structures with genetic information

68 68 phages 1328 1328 viruses 35 35 viroids 472 472 organelles (423 mitochondria, 32 plastids,

14 plasmids, 3 nucleomorphs)

Source: NCBI Source: Integrated Genomics, Inc. August 12th, 2003

M.L. Riley, T. Schmidt, C. Wagner, H.-W. Mewes, D. Frishman. Nucleic Acids Res. 33:D308-D310 (2005)

M.L. Riley, T. Schmidt, C. Wagner, H.-W. Mewes, D. Frishman. Nucleic Acids Res. 33:D308-D310 (2005)

Mathematical models

Discrete methods Continuous methods Enumeration, combinatorics Differentiation, Integration Graph theory, network theory Optimization String matching (sequence comparison)

Mathematical models

Discrete methods Continuous methods Enumeration, combinatorics Differentiation, Integration Graph theory, network theory Optimization String matching (sequence comparison)

Mathematical models

Discrete methods Continuous methods Enumeration, combinatorics Differentiation, Integration Graph theory, network theory Optimization String matching (sequence comparison) Dynamical systems

Stochastic difference equs. Difference equs. Stochastic differential equs. Differential equs. n , t

• dn , t
• n , dt

dn , dt … ODE n , x , t dn , x , t n , dx , dt dn , dx , dt … PDE

Mathematical models

Discrete methods Continuous methods Enumeration, combinatorics Differentiation, Integration Graph theory, network theory Optimization String matching (sequence comparison) Dynamical systems

Stochastic difference equs. Difference equs. Stochastic differential equs. Differential equs. n , t

• dn , t
• n , dt

dn , dt … ODE n , x , t dn , x , t n , dx , dt dn , dx , dt … PDE

Simulation methods

Cellular automata Genetic algorithms Neural networks Simulated annealing Differential equs.

Mathematical models

Discrete methods Continuous methods Enumeration, combinatorics Differentiation, Integration Graph theory, network theory Optimization String matching (sequence comparison) Dynamical systems

Stochastic difference equs. Difference equs. Stochastic differential equs. Differential equs. n , t

• dn , t
• n , dt

dn , dt … ODE n , x , t dn , x , t n , dx , dt dn , dx , dt … PDE

Simulation methods

Cellular automata Genetic algorithms Neural networks Simulated annealing Differential equs.

Structure prediction, optimization and docking

Discrete states Continuous states Dynamic programming Simplex methods Gradient techniques RNA secondary structures, lattice proteins 3D-Structures of (bio)molecules

O CH2 OH O O P O O O

N1

O CH2 OH O P O O O

N2

O CH2 OH O P O O O

N3

O CH2 OH O P O O O

N4

N A U G C

k =

, , ,

3' - end 5' - end Na Na Na Na

5'-end 3’-end

GCGGAU AUUCGC UUA AGUUGGGA G CUGAAGA AGGUC UUCGAUC A ACCA GCUC GAGC CCAGA UCUGG CUGUG CACAG

Definition of RNA structure

5'-End

5'-End 5'-End 3'-End 3'-End

3'-End

70 60 50 40 30 20 10 GCGGAUUUAGCUCAGDDGGGAGAGCMCCAGACUGAAYAUCUGGAGMUCCUGUGTPCGAUCCACAGAAUUCGCACCA

Sequence Secondary structure Symbolic notation

• A symbolic notation of RNA secondary structure that is equivalent to the conventional graphs

RNA sequence RNA structure

• f minimal free

energy

RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Empirical parameters Biophysical chemistry: thermodynamics and kinetics

Sequence, structure, and design

G G G G G G G G G G G G G G G G U U U U U U U U U U U A A A A A A A A A A A A U C C C C C C C C C C C C 5’-end 3’-end

S1

(h)

S9

(h)

F r e e e n e r g y G

• Minimum of free energy

Suboptimal conformations

S0

(h) S2

(h)

S3

(h)

S4

(h)

S7

(h)

S6

(h)

S5

(h)

S8

(h)

The minimum free energy structures on a discrete space of conformations

The Vienna RNA-Package: A library of routines for folding, inverse folding, sequence and structure alignment, kinetic folding, cofolding, …

RNA sequence RNA structure

• f minimal free

energy

RNA folding: Structural biology, spectroscopy of biomolecules, understanding molecular function Inverse Folding Algorithm Iterative determination

• f a sequence for the

given secondary structure

Sequence, structure, and design

Inverse folding of RNA: Biotechnology, design of biomolecules with predefined structures and functions

Minimum free energy criterion

Inverse folding of RNA secondary structures

1st 2nd 3rd trial 4th 5th

The inverse folding algorithm searches for sequences that form a given RNA secondary structure under the minimum free energy criterion.

Mapping from sequence space into structure space and into function

Properties of RNA sequence to secondary structure mapping

• 1. More sequences than structures

Properties of RNA sequence to secondary structure mapping

• 1. More sequences than structures

Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures

Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures

n = 100, stem-loop structures n = 30

RNA secondary structures and Zipf’s law

Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures 3. Shape space covering of common structures

Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures 3. Shape space covering of common structures

Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures 3. Shape space covering of common structures 4. Neutral networks of common structures are connected

Properties of RNA sequence to secondary structure mapping 1. More sequences than structures 2. Few common versus many rare structures 3. Shape space covering of common structures 4. Neutral networks of common structures are connected

Reference for postulation and in silico verification of neutral networks

Gk Neutral Network

Structure S

k

Gk C

• k

Compatible Set Ck

The compatible set Ck of a structure Sk consists of all sequences which form Sk as its minimum free energy structure (the neutral network Gk) or one of its suboptimal structures.

Structure S Structure S

1

The intersection of two compatible sets is always non empty: C0 C1

Reference for the definition of the intersection and the proof of the intersection theorem

A ribozyme switch

E.A.Schultes, D.B.Bartel, Science 289 (2000), 448-452

Two ribozymes of chain lengths n = 88 nucleotides: An artificial ligase (A) and a natural cleavage ribozyme of hepatitis--virus (B)

The sequence at the intersection: An RNA molecules which is 88 nucleotides long and can form both structures

Two neutral walks through sequence space with conservation of structure and catalytic activity

Systems biology, cell biology

Gene regulation, cell cycle, metabolic networks, reaction kinetics, homeostasis, ...

The bacterial cell as an example for the simplest form of autonomous life The human body: 1014 cells = 1013 eukaryotic cells + 91013 bacterial (prokaryotic) cells, and 200 eukaryotic cell types

Linear chain Network

Processing of information in cascades and networks

Albert-László Barabási, Linked – The New Science of Networks Perseus Publ., Cambridge, MA, 2002

• Formation of a scale-free network through evolutionary point by point expansion: Step 000

• Formation of a scale-free network through evolutionary point by point expansion: Step 001

• Formation of a scale-free network through evolutionary point by point expansion: Step 002

• Formation of a scale-free network through evolutionary point by point expansion: Step 003

• Formation of a scale-free network through evolutionary point by point expansion: Step 004

• Formation of a scale-free network through evolutionary point by point expansion: Step 005

• Formation of a scale-free network through evolutionary point by point expansion: Step 006

• Formation of a scale-free network through evolutionary point by point expansion: Step 007

• Formation of a scale-free network through evolutionary point by point expansion: Step 008

• Formation of a scale-free network through evolutionary point by point expansion: Step 009

• Formation of a scale-free network through evolutionary point by point expansion: Step 010

• Formation of a scale-free network through evolutionary point by point expansion: Step 011

• Formation of a scale-free network through evolutionary point by point expansion: Step 012

• Formation of a scale-free network through evolutionary point by point expansion: Step 024

• 14

10 2 2 2 2 2 2 2 2 2 2 2 2 2 2 3 3 3 3 3 3 12 5 5 links # nodes 2 14 3 6 5 2 10 1 12 1 14 1

Analysis of nodes and links in a step by step evolved network

A B C D E F G H I J K L 1

Biochemical Pathways

2 3 4 5 6 7 8 9 10

The reaction network of cellular metabolism published by Boehringer-Ingelheim.

The citric acid

• r Krebs cycle

(enlarged from previous slide).

Stefan Bornholdt. Less is more in modeling large genetic networks. Science 310, 449-450 (2005)

General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) ( x

) , ( t r g x S =

• Time

t Concentration ( ) x t Solution curves: xi(t) Kinetic differential equations ) ; (

2

k x f x D t x + ∇ = ∂ ∂

) , , ( ; ) , , ( ; ) ; (

1 1 m n

k k k x x x k x f t d x d

K K

= = = Reaction diffusion equations

) , ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m , , 2 , 1 j ; ) , I , H p , p , T (

j

K K = k

The forward problem of chemical reaction kinetics (Level I)

General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) ( x

) , ( t r g x S =

• Time

t Concentration ( ) x t Solution curves: xi(t) Kinetic differential equations ) ; (

2

k x f x D t x + ∇ = ∂ ∂ ) , , ( ; ) , , ( ; ) ; (

1 1 m n

k k k x x x k x f t d x d K K = = = Reaction diffusion equations

) , ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m j I H p p T kj , , 2 , 1 ; ) , , , , ; I ( G K K =

Genome: Sequence IG

The forward problem of cellular reaction kinetics (Level I)

The inverse problem of cellular reaction kinetics (Level I)

Time t Concentration Data from measurements (t ); = 1, 2, ... , x j N

j

xi (t )

j

Kinetic differential equations

) ; (

2

k x f x D t x + ∇ = ∂ ∂ ) , , ( ; ) , , ( ; ) ; (

1 1 m n

k k k x x x k x f t d x d

K K

= = = Reaction diffusion equations General conditions Initial conditions : T , p , pH , I , ... :

...

... S ,

u

Boundary conditions

boundary normal unit vector Dirichlet Neumann :

:

:

) ( x

) , ( t r g x S =

• )

, ( ˆ t r g x u u x

S =

∇ ⋅ = ∂ ∂

Parameter set

m j I H p p T k j , , 2 , 1 ; ) , , , , ; I ( G K K

=

Genome: Sequence IG

1 2 3 4 5 6 7 8 9 10 11 12 Regulatory protein or RNA Enzyme Metabolite Regulatory gene Structural gene

A model genome with 12 genes

Sketch of a genetic and metabolic network

The model regulatory gene in MiniCellSim

The model structural gene in MiniCellSim

Evolutionary time: 0000 Number of genes 12 : + 06 structural 06 regulatory Number of interactions 15 : + + 04 inhibitory + 10 activating 1 self-activating

A genabolic network formed from a genotype of n = 200 nucleotides

100 1000 10000 1e+05 5 10 15 20 TF00 TF01 TF02 TF03 SP04 TF05 SP06 SP07 SP08 SP09 TF10 SP11

Evolutionary time 0000 , initial network : Intracellular time Stationary state Intracellular time scale Evolutionary time scale [generations]: 0000 initial network

Evolution of a genabolic network:

Initial genome: random sequence of length n = 200, AUGC alphabet Gene length: n = 25 Simulation with mutation rate: p = 0.01 Evolutionary time unit >> intracellular time unit

Number of genes: total / structural genes regulatory genes

Evolution of a genabolic network:

Initial genome: random sequence of length n = 200, AUGC alphabet Gene length: n = 25 Simulation with mutation rate: p = 0.01 Evolutionary time unit >> intracellular time unit Recorded events: (i) Loss of a gene through corruption of the start signal “TA” (analogue of the “TATA Box”), (ii) creation of a gene, (iii) change in the edges through mutation-induced changes in the affinities of translation products to the binding sites, and (iv) change in the class of genes (tf sp).

Statistics of one thousand generations Total number of genes: 11.67 2.69 Regulatory genes: 5.97 2.22 Structural genes: 5.70 2.17

Neurobiology

Neural networks, collective properties, signalling, ... dynamics, nonlinear

A single neuron signaling to a muscle fiber

The human brain 1011 neurons connected by 1013 to 1014 synapses

B A

Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

Christof Koch, Biophysics of Computation. Information Processing in single neurons. Oxford University Press, New York 1999.

) ( ) ( ) ( 1

4 3 l l K K Na Na M

V V g V V n g V V h m g I C t d V d − − − − − − =

m m dt dm

m m

β α − − = ) 1 ( h h dt dh

h h

β α − − = ) 1 ( n n dt dn

n n

β α − − = ) 1 (

Hogdkin-Huxley OD equations

A single neuron signaling to a muscle fiber

Gating functions of the Hodgkin-Huxley equations

Temperature dependence of the Hodgkin-Huxley equations

) ( ) ( ) ( 1

4 3 l l K K Na Na M

V V g V V n g V V h m g I C t d V d − − − − − − =

m m dt dm

m m

β α − − = ) 1 ( h h dt dh

h h

β α − − = ) 1 ( n n dt dn

n n

β α − − = ) 1 (

Hogdkin-Huxley OD equations

Hhsim.lnk

Simulation of space independent Hodgkin-Huxley equations: Voltage clamp and constant current

Hodgkin-Huxley partial differential equations (PDE)

n n t n h h t h m m t m L r V V g V V n g V V h m g t V C x V R

n n h h m m l l K K Na Na

β ) 1 ( α β ) 1 ( α β ) 1 ( α 2 ] ) ( ) ( ) ( [ 1

4 3 2 2

− − = ∂ ∂ − − = ∂ ∂ − − = ∂ ∂ − + − + − + ∂ ∂ = ∂ ∂ π

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

Hodgkin-Huxley ordinary differential equations (ODE) Travelling pulse solution: V(x,t) = V() with = x + t

n β n α d n d θ h β h α d h d θ m β m α d m d θ L r V V g V V n g V V h m g d V d θ C d V d R

n n h h m m l l K K Na Na M

− − = − − = − − = − + − + − + = ) 1 ( ) 1 ( ) 1 ( 2 ] ) ( ) ( ) ( [ 1

4 3 2 2

ξ ξ ξ π ξ ξ

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

50

• 50

100 1 2 3 4 5 6 [cm] V [ m V ]

T = 18.5 C; θ = 1873.33 cm / sec

T = 18.5 C; θ = 1873.3324514717698 cm / sec

T = 18.5 C; θ = 1873.3324514717697 cm / sec

• 10

10 20 30 40 V [ m V ] 6 8 10 12 14 16 18 [cm]

T = 18.5 C; θ = 544.070 cm / sec

T = 18.5 C; θ = 554.070286919319 cm/sec

T = 18.5 C; θ = 554.070286919320 cm/sec

Propagating wave solutions of the Hodgkin-Huxley equations

Hodgkin-Huxley ordinary differential equations (ODE) Travelling pulse solution: V(x,t) = V() with = x + t

n β n α d n d θ m β m α d m d θ L r V V g V V n g V V n n h m g d V d θ C d V d R

n n m m l l K K Na Na M

− − = − − = − + − + − − + + = ) 1 ( ) 1 ( 2 ] ) ( ) ( ) ( ) ( [ 1

4 3 2 2

ξ ξ π ξ ξ

) 1 ( 125 . ) ( exp 125 . ; α E α

80 80 V V n Na n

V − ≈ − = + = β

An approximation to the Hodgkin-Huxley equations

Propagating wave solutions of approximations to the Hodgkin-Huxley equations

Evolutionary biology

Optimization through variation and selection, relation between genotype, phenotype, and function, ...

Generation time Selection and adaptation 10 000 generations Genetic drift in small populations 106 generations Genetic drift in large populations 107 generations RNA molecules 10 sec 1 min 27.8 h = 1.16 d 6.94 d 115.7 d 1.90 a 3.17 a 19.01 a Bacteria 20 min 10 h 138.9 d 11.40 a 38.03 a 1 140 a 380 a 11 408 a Multicelluar organisms 10 d 20 a 274 a 200 000 a 27 380 a 2 × 107 a 273 800 a 2 × 108 a

Time scales of evolutionary change

Genotype = Genome

GGCUAUCGUACGUUUACCCAAAAAGUCUACGUUGGACCCAGGCAUUGGAC.......G

Mutation Fitness in reproduction: Number of genotypes in the next generation Unfolding of the genotype: RNA structure formation Phenotype Selection

Evolution of phenotypes

Ij In I2 Ii I1 I j I j I j I j I j I j

+ + + + +

(A) + fj Qj1 fj Qj2 fj Qji fj Qjj fj Qjn Q (1- )

ij

• d(i,j)

d(i,j)

=

l

p p

p .......... Error rate per digit d(i,j) .... Hamming distance between Ii and Ij ........... Chain length of the polynucleotide l

dx / dt = x - x x

i j j i j j

Σ

; Σ = 1 ; f f x

j j j i

Φ Φ = Σ Qji Qij

Σi

= 1 [A] = a = constant [Ii] = xi 0 ;

• i =1,2,...,n ;

Chemical kinetics of replication and mutation as parallel reactions

Stock Solution Reaction Mixture

Replication rate constant: fk = / [ + dS

(k)]

dS

(k) = dH(Sk,S)

Selection constraint: # RNA molecules is controlled by the flow N N t N ± ≈ ) ( The flowreactor as a device for studies of evolution in vitro and in silico

f0 f f1 f2 f3 f4 f6 f5 f7

Replication rate constant: fk = / [ + dS

(k)]

dS

(k) = dH(Sk,S)

Evaluation of RNA secondary structures yields replication rate constants

S{ = ( ) I{ f S

{ {

ƒ = ( )

S{ f{ I{

Mutation Genotype-Phenotype Mapping Evaluation of the Phenotype

Q{

j

I1 I2 I3 I4 I5 In

Q

f1 f2 f3 f4 f5 fn

I1 I2 I3 I4 I5 I{ In+1 f1 f2 f3 f4 f5 f{ fn+1

Q

Evolutionary dynamics including molecular phenotypes

Phenylalanyl-tRNA as target structure Randomly chosen initial structure

In silico optimization in the flow reactor: Evolutionary Trajectory

28 neutral point mutations during a long quasi-stationary epoch Transition inducing point mutations change the molecular structure Neutral point mutations leave the molecular structure unchanged

Neutral genotype evolution during phenotypic stasis

Evolutionary trajectory Spreading of the population

• n neutral networks

Drift of the population center in sequence space

Spreading and evolution of a population on a neutral network: t = 150

Spreading and evolution of a population on a neutral network : t = 170

Spreading and evolution of a population on a neutral network : t = 200

Spreading and evolution of a population on a neutral network : t = 350

Spreading and evolution of a population on a neutral network : t = 500

Spreading and evolution of a population on a neutral network : t = 650

Spreading and evolution of a population on a neutral network : t = 820

Spreading and evolution of a population on a neutral network : t = 825

Spreading and evolution of a population on a neutral network : t = 830

Spreading and evolution of a population on a neutral network : t = 835

Spreading and evolution of a population on a neutral network : t = 840

Spreading and evolution of a population on a neutral network : t = 845

Spreading and evolution of a population on a neutral network : t = 850

Spreading and evolution of a population on a neutral network : t = 855

Evolutionary design of RNA molecules

D.B.Bartel, J.W.Szostak, In vitro selection of RNA molecules that bind specific ligands. Nature 346 (1990), 818-822 C.Tuerk, L.Gold, SELEX - Systematic evolution of ligands by exponential enrichment: RNA ligands to bacteriophage T4 DNA polymerase. Science 249 (1990), 505-510 D.P.Bartel, J.W.Szostak, Isolation of new ribozymes from a large pool of random sequences. Science 261 (1993), 1411-1418 R.D.Jenison, S.C.Gill, A.Pardi, B.Poliski, High-resolution molecular discrimination by RNA. Science 263 (1994), 1425-1429

• Y. Wang, R.R.Rando, Specific binding of aminoglycoside antibiotics to RNA. Chemistry &

Biology 2 (1995), 281-290 Jiang, A. K. Suri, R. Fiala, D. J. Patel, Saccharide-RNA recognition in an aminoglycoside antibiotic-RNA aptamer complex. Chemistry & Biology 4 (1997), 35-50

An example of ‘artificial selection’ with RNA molecules or ‘breeding’ of biomolecules

The SELEX technique for the evolutionary preparation of aptamers

Aptamer binding to aminoglycosid antibiotics: Structure of ligands

• Y. Wang, R.R.Rando, Specific binding of aminoglycoside antibiotics to RNA. Chemistry & Biology 2

(1995), 281-290

tobramycin

A A A A A C C C C C C C C G G G G G G G G U U U U U U

5’- 3’-

A A A A A U U U U U U C C C C C C C C G G G G G G G G

5’-

• 3’

RNA aptamer

Formation of secondary structure of the tobramycin binding RNA aptamer with KD = 9 nM

• L. Jiang, A. K. Suri, R. Fiala, D. J. Patel, Saccharide-RNA recognition in an aminoglycoside antibiotic-

RNA aptamer complex. Chemistry & Biology 4:35-50 (1997)

The three-dimensional structure of the tobramycin aptamer complex

• L. Jiang, A. K. Suri, R. Fiala, D. J. Patel,

Chemistry & Biology 4:35-50 (1997)

Acknowledgement of support

Fonds zur Förderung der wissenschaftlichen Forschung (FWF) Projects No. 09942, 10578, 11065, 13093 13887, and 14898 Wiener Wissenschafts-, Forschungs- und Technologiefonds (WWTF) Project No. Mat05 Jubiläumsfonds der Österreichischen Nationalbank Project No. Nat-7813 European Commission: Contracts No. 98-0189, 12835 (NEST) Austrian Genome Research Program – GEN-AU: Bioinformatics Network (BIN) Österreichische Akademie der Wissenschaften Siemens AG, Austria Universität Wien and the Santa Fe Institute

Universität Wien

Coworkers

Peter Stadler, Bärbel M. Stadler, Universität Leipzig, GE Paul E. Phillipson, University of Colorado at Boulder, CO Heinz Engl, Philipp Kügler, James Lu, Stefan Müller, RICAM Linz, AT Jord Nagel, Kees Pleij, Universiteit Leiden, NL Walter Fontana, Harvard Medical School, MA Christian Reidys, Christian Forst, Los Alamos National Laboratory, NM Ulrike Göbel, Walter Grüner, Stefan Kopp, Jaqueline Weber, Institut für Molekulare Biotechnologie, Jena, GE Ivo L.Hofacker, Christoph Flamm, Andreas Svrček-Seiler, Universität Wien, AT Kurt Grünberger, Michael Kospach , Andreas Wernitznig, Stefanie Widder, Michael Wolfinger, Stefan Wuchty, Universität Wien, AT Jan Cupal, Stefan Bernhart, Lukas Endler, Ulrike Langhammer, Rainer Machne, Ulrike Mückstein, Hakim Tafer, Thomas Taylor, Universität Wien, AT Martin Nowak, Harvard University, MA and Sebastian Bonhoeffer, ETH-Zürich, CH Thomas Wiehe, Universität Köln and Erich Bornberg-Bauer, Universität Münster

Universität Wien

Web-Page for further information: http://www.tbi.univie.ac.at/~pks