A few Thoughts on Graphs in Chemistry and Biology Peter Schuster 19 - - PowerPoint PPT Presentation

A few Thoughts on Graphs in Chemistry and Biology

Peter Schuster 19th LL-Seminar on Graph Theory ÖAW, 25.04.2002

Graphs are seen as valuable tools to order and classify information in various scientific disciplines at an intermediate stage of knowledge or level of

• approximation. Such stages are, for example,
• collection or harvesting of data,
• ordering of data according to new categories and

development of models for qualitative analysis

• development of model for quatitative analysis and

accurate predictions.

Graphs are considered here as tools to

• distiguish chemical isomers,
• describe the flux in chemical reaction networks,
• define biological species by their phylogenetic descent, and
• model genotype-phenotype maps in case of neutrality.

Chemists use graphs to distinguish isomers since the second half of the ninteenth century. Atoms are nodes and chemical bonds are edges. In case of hydrocarbons containing exclusively carbon and hydrogen atoms the position of the atom is sufficient to predict its nature: H atoms form one bond and are attached to one edge, whereas C atoms form always four bonds and are connected to four edges.

D.J.Cram and G.S.Hammond, Organic Chemistry, McGraw-Hill, New York 1959, p.18

propane isobutane isopentane n-pentane neopentane n-butane ethane methane

CnH2n+2, n = 1,2,3,4,5

Formulas of the eight simplest alkanes as graphs, which allow for the distinction of isomers, e.g. n- and isobutane, n-, iso- and neo-pentane

C6H6

hexa-2,4-diyne (dimethyl-diacetylene) hexa-1,2,4,5-tetraene (diallene) benzene

Graphs allow for a distinction of single-, double- and triple bonds

C H

2 6O

dimethylether ethanol

Carbon, hydrogen and oxygen atoms are distinguished by the degree of the corresponding nodes: d(H) = 1, d(O) = 2, and d(C) = 4.

C6H6

benzene

The benzene molecule cannot be described by a single graph.

CH3X

methane: X = H methyl chloride: X = Cl methyl bromide: X = Br methyl iodide: X = I methyl fluoride: X = F

Different atoms forming one bond: H, F, Cl, Br, and I

C H6

2

C H4

2

Cl2

ethane 1,1-dichloro ethane 1,2-dichloro ethane

Two isomers that cannot be distinguished by means of their graphs.

Paul Karrer, Lehrbuch der organischen Chemie, Georg Thieme Verlag, Stuttgart 1959, p.737

Paul Karrer, Lehrbuch der organischen Chemie, Georg Thieme Verlag, Stuttgart 1959, p.949

H H H

N C O

1.35 1.22 1.09 1.00 1.00 112.7

• 120
• 122.5
• 124.7
• 121.6
• 118.5
• 1 Å = 10

m

• 10

Molecular structure of the formamide molecule

Molecular structure of an association complex between a protein an a nucleic acid

Chemists use directed graphs to model reaction mechanisms in chemical kinetics.

Paul Karrer, Lehrbuch der organischen Chemie, Georg Thieme Verlag, Stuttgart 1959, p.479

A B C D + + + AB C D + + ABD C + ACD B + ACE B + AD B C + + EC E C + Reaction graph of a kinetic mechanism

A B C D + + + AB C D + + ABD C + ACD B + ACE B + AD B C + + EC E C + k-1 k-2 k-3 k-4 k1 k2 k3 k4 k5 k6 k7 k7 k8 Reaction graph of a kinetic mechanism with rate constants

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).

Biologists use directed graphs in the form of trees to distinguish biological species by their descent. The concept of evolution allows for ordering the wealth of species by means of phylogenetic relation. Direction of development and time ordering is introduced by the fossil record.

time

Charles Darwin, The Origin of Species, 6th edition. Everyman‘s Library, Vol.811, Dent London, pp.121-122.

Phylogenetic tree of animal kingdom

Lynn Margulis & Karlene V. Schwarz, Five Kingdoms. An illustrated guide to the Phyla of Life on Earth. W.H. Freeman & Co., San Francisco, 1982, p. 160.

t3 t2 t1 time

Phylogenetic tree of animal kingdom

Lynn Margulis & Karlene V. Schwarz, Five Kingdoms. An illustrated guide to the Phyla of Life on Earth. W.H. Freeman & Co., San Francisco, 1982, p. 160.

The genotypes or genomes of individuals and species, being reproductively related ensembles of individuals, are DNA

• sequences. They are changing from generation to generation

through mutation and recombination. Genotypes unfold into phenotypes or organisms, which are the targets of the evolutionary selection process. Point mutations are single nucleotide exchanges. The Hamming distance of two sequences is the minimal number of single nucleotide exchanges that mutually converts the two sequence into each other.

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

d =1

H

d =1

H

d =2

H

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 (S ,S ) =

H 1 2

4 d (S ,S ) = 0

H 1 1

d (S ,S ) = d (S ,S )

H H 1 2 2 1

d (S ,S ) d (S ,S ) + d (S ,S )

H H H 1 3 1 2 2 3

• (i)

(ii) (iii)

The Hamming distance induces a metric in sequence space

4 2 1 8 16 10 19 9 14 6 13 5 11 3 7 12 21 17 22 18 25 20 26 24 28 27 23 15 29 30 31

Binary sequences are encoded by their decimal equivalents: = 0 and = 1, for example, "0" 00000 = "14" 01110 = , "29" 11101 = , etc. ≡ ≡ ≡ , C CCCCC C C C G GGG GGG G

Mutant class

1 2

3 4

5

Sequence space of binary sequences of chain lenght n=5

The RNA model considers RNA sequences as genotypes and simplified RNA structures, called secondary structures, as phenotypes. The mapping from genotypes into phenotypes is many-to-one. Hence, it is redundant and not invertible. Genotypes, i.e. RNA sequences, which are mapped onto the same phenotype, i.e. the same RNA secondary structure, form neutral networks. Neutral networks are represented by graphs in sequence space.

Three-dimensional structure of phenylalanyl-transfer-RNA

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

70 60 50 40 30 20 10 GCGGAU AUUCGC UUA AGDDGGGA M CUGAAYA AGMUC TPCGAUC A ACCA GCUC GAGC CCAGA UCUGG CUGUG CACAG

Sequence Secondary structure Symbolic notation

Definition and formation of the secondary structure of phenylalanyl-tRNA

UUUAGCCAGCGCGAGUCGUGCGGACGGGGUUAUCUCUGUCGGGCUAGGGCGC GUGAGCGCGGGGCACAGUUUCUCAAGGAUGUAAGUUUUUGCCGUUUAUCUGG UUAGCGAGAGAGGAGGCUUCUAGACCCAGCUCUCUGGGUCGUUGCUGAUGCG CAUUGGUGCUAAUGAUAUUAGGGCUGUAUUCCUGUAUAGCGAUCAGUGUCCG GUAGGCCCUCUUGACAUAAGAUUUUUCCAAUGGUGGGAGAUGGCCAUUGCAG

Criterion of Minimum Free Energy

Sequence Space Shape Space

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Phenotype space Non-negative numbers Mapping from sequence space into phenotype space and into fitness values

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Phenotype space Non-negative numbers

Sk I. = ( ) ψ

fk f Sk = ( )

Sequence space Phenotype space Non-negative numbers

Neutral networks of small RNA molecules can be computed by exhaustive folding of complete sequence spaces, i.e. all RNA sequences of a given chain length. This number, N=4n , becomes very large with increasing length, and is prohibitive for numerical computations. Neutral networks can be modelled by random graphs in sequence

• space. In this approach, nodes are inserted randomly into sequence

space until the size of the pre-image, i.e. the number of neutral sequences, matches the neutral network to be studied.

Random graph approach to neutral networks Sketch of sequence space Step 00

Random graph approach to neutral networks Sketch of sequence space Step 01

Random graph approach to neutral networks Sketch of sequence space Step 02

Random graph approach to neutral networks Sketch of sequence space Step 03

Random graph approach to neutral networks Sketch of sequence space Step 04

Random graph approach to neutral networks Sketch of sequence space Step 05

Random graph approach to neutral networks Sketch of sequence space Step 10

Random graph approach to neutral networks Sketch of sequence space Step 15

Random graph approach to neutral networks Sketch of sequence space Step 25

Random graph approach to neutral networks Sketch of sequence space Step 50

Random graph approach to neutral networks Sketch of sequence space Step 75

Random graph approach to neutral networks Sketch of sequence space Step 100

λj = 27 ,

/

12 λk = (k)

j

| | Gk

λ κ

cr = 1 - -1 (

1)

/ κ- λ λ

k cr . . . .

> λ λ

k cr . . . .

< network is connected Gk network is connected not Gk Connectivity threshold: Alphabet size : = 4

• AUGC

G S S

k k k

= ( ) | ( ) =

• 1
• I

I

j j

• cr

2 0.5 3 0.4226 4 0.3700 Mean degree of neutrality and connectivity of neutral networks

Giant Component

A multi-component neutral network

A connected neutral network

Reference for postulation and in silico verification of neutral networks

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

Compatible Incompatible

5’-end 5’-end 3’-end 3’-end

Sequences are compatible or incompatible with structures

G C

k k

Gk

Neutral Network Compatible Set Ck

Neutral networks Gk are embedded in sets of compatible sequences Ck.

:

• C1

C2 :

• C1

C2

G1 G2

Two neutral networks, G1 and G2, are embedded in compatible sets, C1 and C2, respectively. The compatible sets form an intersection consisting of sequences that can form both structures.

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

3’-end

Minimum free energy conformation S0 Suboptimal conformation S1

C G

A sequence at the intersection

• f two structures

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

A ribozyme switch

E.A.Schultes, D.B.Bartel, One sequence, two ribozymes: Implication for the emergence of new ribozyme folds. Science 289 (2000), 448-452

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

Coworkers

Walter Fontana, Santa Fe Institute, NM Christian Reidys, Christian Forst, Los Alamos National Laboratory, NM Peter Stadler, Universität Wien, AT Ivo L.Hofacker Christoph Flamm Bärbel Stadler, Andreas Wernitznig, Universität Wien, AT Michael Kospach, Ulrike Mückstein, Stefanie Widder, Stefan Wuchty Jan Cupal, Kurt Grünberger, Andreas Svrček-Seiler Ulrike Göbel, Institut für Molekulare Biotechnologie, Jena, GE Walter Grüner, Stefan Kopp, Jaqueline Weber