Network modularity, currency metabolites and graph representations

metabolic networks, intronetworks (physicist style?)

modularity & currency metabolitessubnetwork hierarchies

Network modularity, currency metabolites andgraph representations of metabolism

Petter Holme

KTH, CSC, Computational Biology

December 17, 2007, Mathematics for biological networks

http://www.csc.kth.se/∼pholme/

Petter Holme Network modularity, currency metabolites and graph representations



wallpapers




B. burgdorferi, 2000




human, 2006




complex network studies

. . . of biochemical networks. What questions can we ask?

how can the large-scale organization be characterized?

are there any universal features over different species?

do the differences tell us something about evolution?

can we identify functional modules?

. . the functions of molecules?






















































what systems?

What systems can be analyzed with complex network methods?

items are, naturally, coupled pairwise MAYBE?

the network is relatively sparse (the average degree isconstant) ALMOST TRUE

there is a dynamic system on the network TRUE!

the time scale of this dynamics is faster than thedynamics of network evolution TRUE!




what systems?









what systems?









what systems?









what systems?









what systems?









what systems?









what systems?









what systems?









representations

ATP + NADH↔ ADP + NADPH

NADH kinase




representations


vertices: substrates

edges: between products / substrates

on different sidesPetter Holme Network modularity, currency metabolites and graph representations



representations



edges: between products / substrates

on the same sidePetter Holme Network modularity, currency metabolites and graph representations



representations



edges: between all products / substratesPetter Holme Network modularity, currency metabolites and graph representations



representations


vertices: substrates + enzymes (reactions)

edges: between substance / reaction vertices

NADH kinase




representations

different representations, give different information

most common representation = substance graphs—followingatoms, the number of conversions between two molecules aresmall⇔ the graph distance is small




representations






representations






what is network structure?

how the network differs a random network

to be more precise: how the network differs from a nullmodel
















network null-models

Network structures are relative . . .

. . . so one has to be clear about what to compare with . . . anull model

Null model 1: random graphs (Poisson random graphs,Erdos-Renyi graphs)

Null model 2: random graphs constrained to the set ofdegrees of the original graph




network null-models








network null-models








network null-models








random graphs

with probability p, add an edgefor each pair of vertices,




random graphs





random graphs





random graphs





random graphs





random graphs





random graphs





random graphs





random graphs





random graphs





random graphs





random graphs





random rewiring

start from the original graphchoose edge pairs, and swap them




random rewiring





random rewiring





random rewiring





random rewiring





the dogmas of network science

real networks have both structure and randomness

the network structure relates to the function of the network
















degree distribution




network structure of metabolism

Power-law degree distributions.

Increasing average degree.

Network modularity?







Network modularity?







Network modularity?







Network modularity?




modularity (of a partition)

Definition

Q =∑

i

eii −

∑

j

eij

2

(1)

the sum is over the a partition into clusters and eij is the fraction ofedges that leads between vertices of cluster i and j

1

2

Q ≈ 0.48




modularity (of a graph)

Definition

Q(G) = maxpartitions

Q (2)

(null model: random graphs)

Definition


Q −

⟨

maxpartitions

Q

⟩

(3)

(null model: random graphs with the same degree sequence)




modularity (of a graph)

Definition


Q (2)

(null model: random graphs)

Definition


Q −

⟨

maxpartitions

Q

⟩

(3)

(null model: random graphs with the same degree sequence)




modularity (the biological idea)

(Bhattacharyya et al., 2006, Annu. Rev. Biochem. 75, pp. 655–80)




modularity (the biological idea)

(Bhattacharyya et al., 2006, Annu. Rev. Biochem. 75, pp. 655–80)




modularity and robustness

robustness—sensitivity of a relevant dynamic system on thenetwork to perturbationsperturbations = changes in:

network topologyconcentrationsdelays in periodic input

in spreading of harmful things (like disease), modularityincreases robustness

in other systems where a flow is needed throughout thenetworks, modularity might decrease robustness





























































mo

du

lar

red

un

dan

cy

(Kitano, 2004, Nat. Rev. Genet. 5, pp. 826–837.)





distributed redundancy (A. Wagner):

In distributed robustness, many parts of a systemcontribute to its function, but all of these parts havedifferent roles. When one part fails or is changed throughmutations, other parts can compensate for this failure,but not simply by standing in for the failed part.




currency metabolites

ATP

gluc

ose

gluc

ose

6−P

fruct

ose

6−P

fruct

ose

1,6−

Pgl

ycer

alde

hyde

3−P

1,3−

P gl

ycer

ate

3−P

glyc

erat

e2−

P gl

ycer

ate

phos

phoe

nolp

yruv

ate

pyru

vate




currency metabolites

Wagner & Fell, 2001 Schuster et al., 2002 Ma & Zeng, 2003ATP ATP ATPADP ADP ADP

NADP NADP NADPNADPH NADPH

NAD NADNADH

Pi Pi

H2O H2OH+

PPi

CMPCO2

O2

NH3Petter Holme Network modularity, currency metabolites and graph representations



currency metabolites (a definition)

1 currency metabolites have high degree2 they make not meaningful shortcuts3 i.e. tie together distant parts of the network4 i.e. tie different modules together

. . . let’s turn this around to a definition . . .



































Definition

Remove vertices in order of (currently) highest degree. The set ofremoved vertices that gives the network the highest modularity isthe set of currency metabolites.




human currency metabolites

currency metabolitescommodity metabolites

number of deleted vertices

∆Q

0.08

0.10

0.12

0.14

0.16

10 30 400 20




detected currency metabolites

Wagner & Fell, 2001 Schuster et al., 2002 Ma & Zeng, 2003ATP ATP ATPADP ADP ADP

NADP NADP NADPNADPH NADPH

NAD NADNADH

Pi Pi

H2O H2OH+

PPi

CMPCO2

O2

NH3Petter Holme Network modularity, currency metabolites and graph representations




i. nitrogen metabolism

iv. citric acid cycle / porphyrin synthesis

vii. metabolism of xenobiotics

x.

ii. amino acid metabolism / protein synthesis

v. fatty acid metabolism

viii. tyrosine and tryptamine metabolism

xi. glycerophospholipid and arachidonic acid metabolism

sugar metabolismiii.

vi. steroid metabolism

ix. DNA metabolism / glycolysis

nucleoside / nucleotide metabolism

com

mod

ity m

etab

olite

s

curr

ency

met

abol

ites

i

vii

viii

ix

x

iii

iixi

v

vi

iv




different organisms

organism samples nodes edges c. m. modularityanimals 5 1621 4662 6.2 0.157

plants 1 1561 4302 1 0.144fungi 2 1281 3654 1.5 0.150

bacteria 99 1050 2739 1.7 0.140




conclusions

proposed graph based definition of currency metabolites

metabolic networks are modular, but not so much

the reason they are not more modular might be giverobustness

(Huss & Holme, 2007, IET Syst. Biol. 1, pp. 280–5.)




conclusions








conclusions








conclusions








conclusions








subnetwork hierarchies

P. Holme, M. Huss & H. Jeong, 2003, Bioinformatics 19, pp. 532–8.

Start with a directed bipartite networks, with both reaction andsubstance vertices (keep the currency metabolites).

Iteratively remove the reaction vertices with the highestbetweenness (fraction of shortest paths passing through avertex).

Study the dendrogram of this process.

0

( )S2 h0 ( )h0S1hh

h0

max









0

( )S2 h0 ( )h0S1hh

h0

max









0

( )S2 h0 ( )h0S1hh

h0

max









0

( )S2 h0 ( )h0S1hh

h0

max




shell- vs community-type ordering

(a) (b)




dendrogram

(b)

(a)




modules

CoAD−glucosamine 1−phosphate

−acetyldihydrolipoamideacetyl−CoA

dihydrolipoamidedeoxyguanosine

2−deoxy−D−ribose 1−phosphatedeoxyadenosine

guanineguanosine

−D−ribose 1−phosphate

adenineadenosine

hypoxanthineinosine

orthophosphate

−D−ribose 1−pyrophosphate

−acetyl−D−glucosamine 1−phosphate

αα

SN

iiiii

i




modules

α

N

α

H O2

pyruvate, CO2

H O2CO2

ATP, ADP

pyrophosphateNADPH, NADH

substratereaction nodelinkin−flowout−flow

−D−ribose 1−phosphate

CoA

orthophosphateadenine

deoxyadenosine

inosineadenosine

hypoxanthine

guanine

guanosine

deoxyguanosine

1−phosphate−acetyl−D−glucosamine

acetyl−CoA

−D−ribose 1−pyrophosphate

pyrophosphate

2−deoxy−D−ribose 1−phosphate

iiiiii




modules

primosome complex

5.99.1.3.DNA topoisomerase II2.7.7.7.DNA polymerase III

DNA helicase II2.7.7.7.DNA polymerase I

SSBopen prepriming complex

6.5.1.2.DNA ligaseRep

5.99.1.2.DNA topoisomerase I

−phosphohistidinepN

−phosphohistidine

−phosphohistidineNHPr protein N−pros−phosphohistidine

pN

enzyme IIIGlcenzyme IIIMan

enzyme IIIScrenzyme IIIMan

Glcenzyme IIIpNenzyme IIIFru

Screnzyme IIIenzyme IIIFru

−phosphohistidine

HPr protein histidinep




modules

pyruvate

−phosphohistidine

−phosphohistidineenzyme IIIGlc

pN

enzyme IIIFrupN

−phosphohistidineenzyme IIIScr

pN

enzyme IIIMan

enzyme IIIMan

enzyme IIIGlc

enzyme IIIFru

HPr proteinphosphohistidine

N−pros−

enzyme IIIScr

pN −phosphohistidine

phosphoenolpyruvate

CTPGTP

ortophosphate

ATPADP

SSB

DNA helicase II

6.5.1.2.DNA ligase

2.7.7.7.DNA polymerase I

open prepriming complexRep

prepriming complex

RNA primer−primosome complex

UTP

ortophosphate

5.99.1.2.DNA topoisomerase I2.7.7.7.DNA polymerase III

HPr protein histidine

primosome complex5.99.1.3.DNA topoisomerase II




dentrogram statistics

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

S1~

2~SS1 / S2

10 20 30 0 40 80 120 1600 5 10 15 20 25 30 0

0 20 40 60 800 10 20 30 40 50 60 70 50 100 150 200 2500

h h h

h h h

0

0.2

0.4

0.6

0.8

1

0

0.2

0.4

0.6

0.8

1

,,

S2

S1

/ S2

S1~

~,

,S

2S

1/ S

2S

1~~ ,

,S

2S

1/ S

2S

1~~

,,

S2

S1

/ S2

S1~

~

(b) (c)

(f)(e)(d)

(a)




conclusions

Shell- rather than community-type ordering

A few, seemingly clear, modules




conclusions






conclusions




Network modularity, currency metabolites and graph representations

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