modeling in molecular biologypks/presentation/litschau-05.pdf · modeling in molecular biology...

Modeling in Molecular Biology

Peter Schuster

Institut für Theoretische Chemie, Universität Wien, Austria, andThe 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 modelsDiscrete methods Continuous methods

Enumeration, combinatorics Differentiation, IntegrationGraph theory, network theory OptimizationString matching (sequence comparison)

Dynamical systemsStochastic 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 methodsCellular automata Genetic algorithms Neural networks Simulated annealing Differential equs.

Structure prediction and optimizationDiscrete states Continuous states

Dynamic programming Simplex methods Gradient techniques

RNA secondary structures, lattice proteins 3D-Structures of (bio)molecules

OCH2

OHO

O

PO

O

O

N1

OCH2

OHO

PO

O

O

N2

OCH2

OHO

PO

O

O

N3

OCH2

OHO

PO

O

O

N4

N A U G Ck = , , ,

3' - end

5' - end

Na

Na

Na

Na

5'-end 3’-endGCGGAU AUUCGCUUA AGUUGGGA G CUGAAGA AGGUC UUCGAUC A ACCAGCUC GAGC CCAGA UCUGG CUGUG CACAG

Definition of RNA structure

5'-End

5'-End

5'-End

3'-End

3'-End

3'-End

70

60

50

4030

20

10

GCGGAUUUAGCUCAGDDGGGAGAGCMCCAGACUGAAYAUCUGGAGMUCCUGUGTPCGAUCCACAGAAUUCGCACCASequence

Secondary structure

Symbolic notation

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

RNA sequence

RNA structureof minimal free

energy

RNA folding:

Structural biology,spectroscopy of biomolecules, understanding

molecular functionEmpirical parameters

Biophysical chemistry: thermodynamics and

kinetics

Sequence, structure, and design

G

GG

G

GG

G GG

G

GG

G

GG

G

U

U

U

U

UU

U

U

U

UU

A

A

AA

AA

AA

A

A

A

A

UC

C

CC

C

C

C

C

C

CC

C

5’-end 3’-end

S1(h)

S9(h)

Free

ene

rgy

G

0

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 structureof minimal free

energy

RNA folding:

Structural biology,spectroscopy of biomolecules, understanding

molecular function

Inverse FoldingAlgorithm

Iterative determinationof a sequence for the

given secondarystructure

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 andstructure alignment, kinetic folding, cofolding, …

Minimum free energycriterion

Inverse folding of RNA secondary structures

1st2nd3rd trial4th5th

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

Sk I. = ( )ψfk f Sk = ( )

Sequence space Structure space Real numbers

Mapping from sequence space into structure space and into function

Sk I. = ( )ψfk 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

GUCAAUCAUGUCAAUCAA

GUCAAUCCG

GUCAAUCG

G

GU

CA

AU

CU

G

GU

CA

AU

GA

G

GUC

AAUU

AG

GUCAAUAAGGUCAACCAG

GUCAAGCAG

GUCAAACAG

GUCACUCAG

GUCAGUCAG

GUCAUUCAGGUCCAUCAG GUCGAUCAG

GUCU

AUCA

G

GU

GA

AUC

AG

GU

UA

AU

CA

G

GU

AA

AU

CA

G

GCC

AAUC

AGGGCAAUCAG

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



2. Few common versus many rare structures




n = 100, stem-loop structures

n = 30

RNA secondary structures and Zipf’s law




3. Shape space covering of common structures




3. Shape space covering of common structures

4. Neutral networks of common structures are connected

GkNeutral Network

Structure S k

Gk Ck

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 itssuboptimal structures.

Structure S 0

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 + 9 1013 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 NetworksPerseus Publ., Cambridge, MA, 2002

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

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• 14

10

22

2

2

22

22

2

2

2

2

2 2

333

3

3

33

12

5

5

links # nodes

2 143 65 2

10 112 114 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 or Krebs cycle (enlarged from previous slide).

General conditionsInitial conditions

: T , p , pH , I , ...:

... ... S , u

Boundary conditionsboundary normal unit vector

Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =

Timet

Con

cent

ratio

n

( )x tSolution curves: xi(t)

Kinetic differential equations

);(2 kxfxDtx

+∇=∂∂

),,(;),,(;);( 11 mn kkkxxxkxftdxd

KK ===

Reaction diffusion equations

),(ˆ trgxuux S =∇⋅=

∂∂

Parameter setm,,2,1j;),I,Hp,p,T(j KK =k

The forward problem of chemical reaction kinetics (Level I)


: T , p , pH , I , ...:

... ... S , u


Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =

Timet

Con

cent

ratio

n

( )x tSolution curves: xi(t)


);(2 kxfxDtx

+∇=∂∂


KK ===



∂∂

Parameter setmjIHppTkj ,,2,1;),,,,;I( G KK =

Gen

ome:

Seq

uenc

e I G

The forward problem of cellular reaction kinetics (Level I)

The inverse problem of cellularreaction kinetics (Level I) Time

tCo

ncen

tratio

n

Data from measurements

(t ); = 1, 2, ... , x j Nj

xi (t )j


);(2 kxfxDtx

+∇=∂∂


KK ===



: T , p , pH , I , ...:

... ... S , u


Dirichlet

Neumann

:

:

:

)0(x

),( trgx S =


∂∂

Parameter setmjIHppTk j ,,2,1;),,,,;I( G KK =

Gen

ome:

Seq

uenc

e I G

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

The inverse problem of bifurcationanalysis 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

Timet

Con

cent

ratio

n

xi (t)

Sequences

Vienna RNA Package

Structures and kinetic parameters

Stoichiometric equations

SBML – systems biology markup language


ODE Integration by means of CVODE

Solution curves

A + B X2 X Y

Y + X D

yxkd

yxkxky

yxkxkbakx

bakba

3

32

2

32

21

1

tdd

tdd

tdd

tdd

tdd

=

−=

−−=

−==

The elements of the simulation tool MiniCellSim

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

ATGCCTTATACGGCAGTCAGGTGCACCATT...GGCTACGGAATATGCCGTCAGTCCACGTGGTAA...CCGDNA string genotype

environment

mRNA

Protein

RNA

Metabolism

RNA and proteinstructures

enzymes and smallmolecules

Recycling of moleculescell membrane

nutrition waste

genotype-p e h p mappinge ynot

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, collectiveproperties, nonlinear

dynamics, signalling, ...

A single neuron signaling to a muscle fiber

The human brain

1011 neurons connected by 1013 to 1014 synapses

BA

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, collectiveproperties, nonlinear

dynamics, signalling, ...

)()()(1 43llKKNaNa

M

VVgVVngVVhmgICtd

Vd−−−−−−=

mmdtdm

mm βα −−= )1(

hhdtdh

hh βα −−= )1(

nndtdn

nn βα −−= )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 43llKKNaNa

M

VVgVVngVVhmgICtd

Vd−−−−−−=

mmdtdm

mm βα −−= )1(

hhdtdh

hh βα −−= )1(

nndtdn

nn βα −−= )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)

nntn

hhth

mmtm

LrVVgVVngVVhmgtVC

xV

R

nn

hh

mm

llKKNaNa

β)1(α

β)1(α

β)1(α

2])()()([1 432

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 + tnnn

hhh

mmm

LrVVgVVngVVhmgVCVR

nn

hh

mm

llKKNaNaM

β)1(αθ

β)1(αθ

β)1(αθ

2])()()([θ1 432

2

−−=∂∂

−−=∂∂

−−=∂∂

−+−+−+∂∂

=∂∂

ξ

ξ

ξ

πξξ

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

50

0

-50

100

1 2 3 4 5 6 [cm]

V [

mV

]

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

0

10

20

30

40

V

[mV

]

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

nnn

mmm

LrVVgVVngVVnnhmgVCVR

nn

mm

llKKNaNaM

β)1(αθ

β)1(αθ

2])()()()([θ1 400

32

2

−−=∂∂

−−=∂∂

−+−+−−++∂∂

=∂∂

ξ

ξ

πξξ

)1(125.0)(exp125.0;αE

α 80800VV

nNa

nV

−≈−=+= 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.......GMutation

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) = lp p

p .......... Error rate per digit

d(i,j) .... Hamming distance between Ii and Ij

........... Chain length of the polynucleotidel

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

QijΣi = 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 of an eigenvalue problem

fxfxnixxQfdtdx n

j jjn

i iijn

j jiji ====−= ∑∑∑ === 111

;1;,,2,1, φφ L

( ) ( ) ( )( ) ( )

)0()0(;,,2,1;exp0

exp01

1

1

0

1

0 ∑∑ ∑

∑=

=

−

=

−

= ==⋅⋅

⋅⋅=

n

i ikikn

j kkn

k jk

kkn

k iki xhcni

tc

tctx L

l

l

λ

λ

{ } { } { }njihHLnjiLnjiQfW ijijiji ,,2,1,;;,,2,1,;;,,2,1,; 1 LLlL ======÷ −

{ }1,,1,0;1 −==Λ=⋅⋅− nkLWL k Lλ

Formation of a quasispeciesin sequence space

Uniform distribution in sequence space

Error rate p = 1-q0.00 0.05 0.10

Quasispecies Uniform distribution

Quasispecies as a function of the replication accuracy q

Chain length and error threshold

npn

pnp

pnpQ n

σ

σσσσ

ln:constant

ln:constant

ln)1(ln1)1(

max

max

≈

≈

−≥−⋅⇒≥⋅−=⋅

K

K

sequencemasterofysuperiorit

lengthchainrateerror

accuracynreplicatio)1(

K

K

K

K

∑ ≠

=

−=

mj j

m

n

ffσ

nppQ

Stock Solution Reaction Mixture

Replication rate constant:

fk = / [ + dS(k)]

dS(k) = dH(Sk,S )

Selection constraint:

# RNA molecules is controlled by the flow

NNtN ±≈)(

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

f5f7

Replication rate constant:

fk = / [ + dS(k)]

dS(k) = dH(Sk,S )

Evaluation of RNA secondary structures yields replication rate constants

spaceSequence

Con

cent

ratio

n

Master sequence

Mutant cloud

“Off-the-cloud” mutations

The molecular quasispeciesin sequence space

S{ = ( )I{

f S{ {ƒ= ( )

S{

f{

I{M

utat

ion

Genotype-Phenotype Mapping

Evaluation of the

Phenotype

Q{jI1

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 on 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

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

AA

AA C

C C CC

C

CC

G G G

G

G

G

G

G U U

U

U

U U5’-

3’-

AAAAA UUUUUU CCCCCCCCG GGGGGGG5’- -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 NationalbankProject No. Nat-7813

European Commission: Contracts No. 98-0189, 12835 (NEST)

Austrian Genome Research Program – GEN-AU: BioinformaticsNetwork (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

modeling in molecular biologypks/presentation/litschau-05.pdf · modeling in molecular biology...

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