Modeling in Molecular Biology Peter Schuster Institut fr - - PowerPoint PPT Presentation

Modeling in Molecular Biology

Peter Schuster

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

Third GEN-AU Summer School: Ultra-Sensitive Proteomics and Genomics Litschau, 29.– 31.08.2005

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

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 and optimization

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

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

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

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.

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Structure space Real numbers Mapping from sequence space into structure space and into function

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Structure space Real numbers

Sk I. = ( ) ψ

Sequence space Structure space

Sk I. = ( ) ψ

Sequence space Structure space

The pre-image of the structure Sk in sequence space is the neutral network Gk

AUCAAUCAG GUCAAUCAC GUCAAUCAU GUCAAUCAA G U C A A U C C G G U C A A U C G G GUCAAUCUG G U C A A U G A G G U C A A U U A G GUCAAUAAG GUCAACCAG G U C A A G C A G GUCAAACAG GUCACUCAG G U C A G U C A G GUCAUUCAG GUCCAUCAG GUCGAUCAG GUCUAUCAG GUGAAUCAG GUUAAUCAG GUAAAUCAG GCCAAUCAG GGCAAUCAG GACAAUCAG UUCAAUCAG CUCAAUCAG

GUCAAUCAG

One-error neighborhood

The surrounding of GUCAAUCAG in sequence space

Degree of neutrality of neutral networks and the connectivity threshold

A multi-component neutral network formed by a rare structure: < cr

A connected neutral network formed by a common structure: > cr

Reference for postulation and in silico verification of neutral networks

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

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

Cell biology

Regulation of 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).

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

The forward problem of bifurcation analysis in cellular dynamics (Level II)

The inverse problem of bifurcation analysis in cellular dynamics (Level II)

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

Time t Concentration xi (t)

Sequences

Vienna RNA Package

Structures and kinetic parameters Stoichiometric equations

SBML – systems biology markup language

Kinetic differential equations

ODE Integration by means of CVODE

Solution curves

A + B X 2 X Y Y + X D

y x k d y x k x k y y x k x k b a k x b a k b a

3 3 2 2 3 2 2 1 1

t d d t d d t d d t d d t d d = − = − − = − = =

The elements of the simulation tool MiniCellSim

SBML: Bioinformatics 19:524-531, 2003; CVODE: Computers in Physics 10:138-143, 1996

ATGCCTTATACGGCAGTCAGGTGCACCATT...GGC TACGGAATATGCCGTCAGTCCACGTGGTAA...CCG DNA string genotype environment mRNA Protein RNA

Metabolism

RNA and protein structures enzymes and small molecules Recycling of molecules cell membrane nutrition waste genotype-p e h p mapping e y not genetic regulation network metabolic reaction network transport system

The regulatory logic of MiniCellSym

The model regulatory gene in MiniCellSim

The model structural gene in MiniCellSim

Neurobiology

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

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.

Neurobiology

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

) ( ) ( ) ( 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 n n h h h m m m L r V V g V V n g V V h m g V C V R

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

β ) 1 ( α θ β ) 1 ( α θ β ) 1 ( α θ 2 ] ) ( ) ( ) ( [ θ 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 n n m m m L r V V g V V n g V V n n h m g V C V R

n n m m l l K K Na Na M

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

Mutation-selection equation: [Ii] = xi 0, fi > 0, Qij 0 Solutions are obtained after integrating factor transformation by means

• f an eigenvalue problem

f x f x n i x x Q f dt dx

n j j j n i i i j n j ji j i

= = = = − =

∑ ∑ ∑

= = = 1 1 1

; 1 ; , , 2 , 1 , φ φ L

( ) ( ) ( ) ( ) ( )

) ( ) ( ; , , 2 , 1 ; exp exp

1 1 1 1

∑ ∑ ∑ ∑

= = − = − =

= = ⋅ ⋅ ⋅ ⋅ =

n i i ki k n j k k n k jk k k n k ik i

x h c n i t c t c t x L l l λ λ

{ } { } { }

n j i h H L n j i L n j i Q f W

ij ij ij i

, , 2 , 1 , ; ; , , 2 , 1 , ; ; , , 2 , 1 , ;

1

L L l L = = = = = = ÷

−

{ }

1 , , 1 , ;

1

− = = Λ = ⋅ ⋅

−

n k L W L

k

L λ

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Formation of a quasispecies in sequence space

Uniform distribution in sequence space

Error rate p = 1-q

0.00 0.05 0.10

Quasispecies Uniform distribution Quasispecies as a function of the replication accuracy q

Chain length and error threshold

n p n p n p p n p Q

n

σ σ σ σ σ ln : constant ln : constant ln ) 1 ( ln 1 ) 1 (

max max

≈ ≈ − ≥ − ⋅ ⇒ ≥ ⋅ − = ⋅ K K

sequence master

• f

y superiorit length chain rate error accuracy n replicatio ) 1 ( K K K K

∑ ≠

= − =

m j j m n

f f σ n p p Q

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

Phenylalanyl-tRNA as target structure Randomly chosen initial structure

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

space Sequence Concentration

Master sequence Mutant cloud “Off-the-cloud” mutations

The molecular quasispecies in sequence space

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

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, Stefan Wuchty, Universität Wien, AT Jan Cupal, Stefan Bernhart, Lukas Endler, Ulrike Langhammer, Rainer Machne, Ulrike Mückstein, Hakim Tafer, Thomas Taylor, Universität Wien, AT

Universität Wien

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