Some Mathematical Challenges from Life Sciences Part I Peter - - PowerPoint PPT Presentation

Some Mathematical Challenges from Life Sciences

Part I

Peter Schuster, Universität Wien Peter F.Stadler, Universität Leipzig Günter Wagner, Yale University, New Haven, CT Angela Stevens, Max-Planck-Institut für Mathematik in den Naturwissenschaften, Leipzig and Ivo L. Hofacker, Universität Wien Oberwolfach, GE, 16.-21.11.2003

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

1. Mathematics and the life sciences in the 21st century 2. Selection dynamics 3. RNA evolution in silico and optimization of structure and properties

1. Mathematics and the life sciences in the 21st century 2. Selection dynamics 3. RNA evolution in silico and optimization of structure and properties

At the same time people are crying for a new

• biology. They say, they want to make “Integrative

Biology” or “Systems Biology”. Hardly anyone calls it by its proper name: Theoretical Biology. Because it has a bad reputation. I think, however, I can remit the sins of the past and declare: We need a theory, which comprises all that (Molecular, Structural, Cellular, Developmental, ...… , and Evolutionary Biology). Imagine, eventually, we not only need to discuss all this stuff with our expert colleagues, but we have to teach it at universities, at schools, and to the public. How could we manage without a comprehensive theory? This is the challenge we have to meet. Sydney Brenner im Gespräch: „Eine einsame Stimme aus der Prägenomik Ära“. Laborjournal 2002, Heft 4:28 – 33.

BioMedNet

Mathematics in 21st Century's Life Sciences

Genomics and proteomics Large scale data processing, sequence comparison ...

Developmental biology

Gene regulation networks, signal propagation, pattern formation, robustness ...

Cell biology

Regulation of cell cycle, metabolic networks, reaction kinetics, homeostasis, ...

Neurobiology

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

Evolutionary biology

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

+ +

Replication: DNA 2 DNA → T r a n s c r i p t i

• n

: D N A R N A → Metabolism

Food Waste

Nucleotides Amino Acids Lipids Carbohydrates Small Molecules

Translation: RNA Protein →

Protein mRNA

Ribosom

A sketch of cellular DNA metabolism

Five kingdoms.

• L. Margulis, K.V. Schwartz, W.H.Freeman & Co., 1982

Five kingdoms.

• L. Margulis, K.V. Schwartz,

W.H.Freeman & Co., 1982

Genomics and proteomics Large scale data processing, sequence comparison ...

• E. coli:

Length of the Genome 4×106 Nucleotides Number of Cell Types 1 Number of Genes 4 000 Man: Length of the Genome 3×109 Nucleotides Number of Cell Types 200 Number of Genes 30 000 - 100 000

Gerhard Braunitzer, 1929 - 1989

Sequence and structure of

• helices in hemoglobin

Molecular evolution through comparison

• f sequences from different organisms

Hemoglobin sequences in different vertebrates

Evolution at the molecular level.

R.K. Selander, A.G. Clark, T.S. Whittam, eds. Sinauer Associates, 1991.

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

The same section of the microarray is shown in three independent hybridizations. Marked spots refer to: (1) protein disulfide isomerase related protein P5, (2) IL-8 precursor, (3) EST AA057170, and (4) vascular endothelial growth factor Gene expression DNA microarray representing 8613 human genes used to study transcription in the response of human fibroblasts to serum V.R.Iyer et al., Science 283: 83-87, 1999

Wolfgang Wieser. Die Erfindung der Individualität oder die zwei Gesichter der Evolution. Spektrum Akademischer Verlag, Heidelberg 1998. A.C.Wilson. The Molecular Basis of Evolution. Scientific American, Oct.1985, 164-173.

Max Perutz 1994 at the opening of the Max Perutz-Library, Vienna BioCenter

Developmental biology

Gene regulation networks, signal propagation, pattern formation, robustness ...

Three-dimensional structure of the complex between the regulatory protein cro-repressor and the binding site on

• phage B-DNA

) , ( t r xi

n i k k k x x x r F x D t x

m n i i i i

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

2 1 2 1 2

K K K r = + ∇ = ∂ ∂

Autocatalytic chemical reactions A + 2 X 3 X

• Direct,

, or hidden in the reaction mechanism (Belousow-Zhabotinskii reaction). Multiple steady states Oscillations in homogeneous solution Deterministic chaos Turing patterns Spatiotemporal patterns (spirals) Deterministic chaos in space and time

Pattern formation in reaction-diffusion systems

Development of the fruit fly drosophila melanogaster: Genetics, experiment, and imago

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

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).

Parameter set

m j x x x I H p p T k

n j

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

2 1

K K K =

Time t Concentration ( ); = 1, 2, ... , x t i n

i

Solution curves: xi Kinetic differential equations

n i k k k x x x f x D t x

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

K K K = + ∇ = ∂ ∂ n i k k k x x x f t d x d

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

K K K = =

Reaction diffusion equations

General conditions: , , pH , , ... Initial conditions: Boundary conditions: boundary ... normal unit vector ... Dirichlet , Neumann , T p I s u n i xi , , 2 , 1 ; ) ( K = n i t r f xs

i

, , 2 , 1 ; ) , ( K = =

• n

i t r f x u u x

s i i

, , 2 , 1 ; ) , ( ˆ K r

r

= = ∇ ⋅ = ∂ ∂

• The forward-problem of chemical reaction kinetics

The inverse-problem of chemical reaction kinetics

Parameter set

m j x x x I H p p T k

n j

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

2 1

K K K =

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

i k

xi Kinetic differential equations

n i k k k x x x f x D t x

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

K K K = + ∇ = ∂ ∂ n i k k k x x x f t d x d

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

K K K = =

Reaction diffusion equations

General conditions: , , pH , , ... Initial conditions: Boundary conditions: boundary ... normal unit vector ... Dirichlet , Neumann , T p I s u n i xi , , 2 , 1 ; ) ( K = n i t r f x s

, , 2 , 1 ; ) , ( K

= =

• n

i t r f x u u x

, , 2 , 1 ; ) , ( ˆ K r

= = ∇ ⋅ = ∂ ∂

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

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

l l K K Na Na

π 2 ) ( ) ( ) ( 1

4 3 2 2

− + − + − + ∂ ∂ = ∂ ∂ m m t m

m m

β α − − = ∂ ∂ ) 1 ( h h t h

h h

β α − − = ∂ ∂ ) 1 ( n n t n

n n

β α − − = ∂ ∂ ) 1 (

Hodgkin-Huxley PDEquations Travelling pulse solution: V(x,t) = W( ) with

• = x
• t

Hodgkin-Huxley equations describing pulse propagation along nerve fibers

The human brain 1011 neurons connected by 1013 to 1014 synapses

Evolutionary biology

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

Generation time 10 000 generations 106 generations 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 Higher multicelluar

• rganisms

10 d 20 a 274 a 20 000 a 27 380 a 2 × 107 a 273 800 a 2 × 108 a

Time scales of evolutionary change

Bacterial Evolution

• S. F. Elena, V. S. Cooper, R. E. Lenski. Punctuated evolution caused by selection of

rare beneficial mutants. Science 272 (1996), 1802-1804

• D. Papadopoulos, D. Schneider, J. Meier-Eiss, W. Arber, R. E. Lenski, M. Blot.

Genomic evolution during a 10,000-generation experiment with bacteria. Proc.Natl.Acad.Sci.USA 96 (1999), 3807-3812

24 h 24 h

Serial transfer of Escherichia coli cultures in Petri dishes

1 day 6.67 generations 1 month 200 generations

• 1 year 2400 generations
• lawn of E.coli

nutrient agar

1 year

Epochal evolution of bacteria in serial transfer experiments under constant conditions

• S. F. Elena, V. S. Cooper, R. E. Lenski. Punctuated evolution caused by selection of rare beneficial mutants.

Science 272 (1996), 1802-1804

2000 4000 6000 8000 Time 5 10 15 20 25 Hamming distance to ancestor Generations

Variation of genotypes in a bacterial serial transfer experiment

• D. Papadopoulos, D. Schneider, J. Meier-Eiss, W. Arber, R. E. Lenski, M. Blot. Genomic evolution during a

10,000-generation experiment with bacteria. Proc.Natl.Acad.Sci.USA 96 (1999), 3807-3812

In evolution variation occurs on genotypes but selection operates on the phenotype. Mappings from genotypes into phenotypes are highly complex objects. The only computationally accessible case is in the evolution of RNA molecules. The mapping from RNA sequences into secondary structures and function, sequence structure function, is used as a model for the complex relations between genotypes and phenotypes. Fertile progeny measured in terms of fitness in population biology is determined quantitatively by replication rate constants of RNA molecules.

Population biology Molecular genetics Evolution of RNA molecules Genotype Genome RNA sequence Phenotype Organism RNA structure and function Fitness Reproductive success Replication rate constant

The RNA model

.... GC UC .... CA .... GC UC .... GU .... GC UC .... GA .... GC UC .... CU

d =1

H

d =1

H

d =2

H

City-block distance in sequence space 2D Sketch of sequence space

Single point mutations as moves in sequence space

CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... CGTCGTTACAATTTA GTTATGTGCGAATTC CAAATT AAAA ACAAGAG..... G A G T A C A C

Hamming distance d (I ,I ) =

H 1 2

4 d (I ,I ) = 0

H 1 1

d (I ,I ) = d (I ,I )

H H 1 2 2 1

d (I ,I ) d (I ,I ) + d (I ,I )

H H H 1 3 1 2 2 3

• (i)

(ii) (iii)

The Hamming distance between sequences induces a metric in sequence space

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

fk f Sk = ( )

Sequence space Structure space Real numbers

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

RNA

RNA as scaffold for supramolecular complexes

ribosome ? ? ? ? ?

RNA as adapter molecule

GAC ... CUG ...

leu genetic code

RNA as transmitter of genetic information

DNA

messenger-RNA protein transcription translation RNA as

• f genetic information

working copy

RNA as carrier of genetic information RNA RNA viruses and retroviruses as information carrier in evolution and evolutionary biotechnology in vitro

RNA as catalyst ribozyme

The RNA DNA protein world as a precursor of the current + biology

RNA as regulator of gene expression

gene silencing by small interfering RNAs

RNA is modified by epigenetic control RNA RNA editing Alternative splicing of messenger RNA is the catalytic subunit in

supramolecular complexes

Functions of RNA molecules

N1

N2

N3

N4

N A U G C

k =

, , ,

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

nd 3’-end

GCGGAU AUUCGC UUA AGUUGGGA G CUGAAGA AGGUC UUCGAUC A ACCA GCUC GAGC CCAGA UCUGG CUGUG CACAG 3'-end 5’-end

70 60 50 40 30 20 10

Definition of RNA structure

5'-e

Stacking of free nucleobases or other planar heterocyclic compounds (N6,N9-dimethyl-adenine)

The stacking interaction as driving force of structure formation in nucleic acids

Stacking of nucleic acid single strands (poly-A)

James D. Watson and Francis H.C. Crick Nobel prize 1962 1953 – 2003 fifty years double helix Stacking of base pairs in nucleic acid double helices (B-DNA)

U = A C G

• U D
• Three Watson-Crick type base pairs

G G G G C C C G C C G C C G C C G C C G C C C C G G G G G C G C

Plus Strand Plus Strand Minus Strand Plus Strand Plus Strand Minus Strand

3' 3' 3' 3' 3' 5' 5' 5' 3' 3' 5' 5' 5' +

Complex Dissociation Synthesis Synthesis

Complementary replication as the simplest copying mechanism of RNA Complementarity is determined by Watson-Crick base pairs: G C and A=U