

Characterising metabolic networks

Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)

To study the network characteristics of the metabolism a graph theoretic description needs to

be established.

(a) Here, the graph theoretic description for a simple pathway (catalysed by Mg2+-dependant

enzymes) is illustrated.

(b) In the most abstract approach all interacting metabolites are considered equally. The

links between nodes represent reactions that interconvert one substrate into another. For

many biological applications it is useful to ignore co-factors, such as the high-energy-

phosphate donor ATP, which results

(c) in a second type of mapping that connects only the main source metabolites to the main

products.



Characterising metabolic networks

(d) The degree distribution, P(k) of the metabolic network illustrates its scale-free topology.

(e) The scaling of the clustering coefficient C(k) with the degree k illustrates the hierarchical

architecture of metabolism (The data shown in d and e represent an average over 43

organisms).

(f) The flux distribution in the central metabolism of Escherichia coli follows a power law,

which indicates that most reactions have small metabolic flux, whereas a few reactions, with

high fluxes, carry most of the metabolic activity. It should be noted that on all three plots the

axis is logarithmic and a straight line on such log–log plots indicates a power-law scaling.

CTP, cytidine triphosphate; GLC, aldo-hexose glucose; UDP, uridine diphosphate; UMP,

uridine monophosphate; UTP, uridine triphosphate.Barabasi & Oltvai, Nature Reviews Genetics 5, 101 (2004)



Yeast protein interaction network

A map of protein–protein interactions in

Saccharomyces cerevisiae, which is

based on early yeast two-hybrid

measurements, illustrates that a few

highly connected nodes (which are also

known as hubs) hold the network

together.

The largest cluster, which contains

78% of all proteins, is shown. The colour

of a node indicates the phenotypic effect

of removing the corresponding protein

(red = lethal, green = non-lethal, orange

= slow growth, yellow = unknown).




Degree


The most elementary characteristic of a node is its

degree (or connectivity), k, which tells us how many links

the node has to other nodes. For example, in the

undirected network shown in part a of the figure, node A

has degree k = 5. In networks in which each link has a

selected direction (see figure, part b) there is an

incoming degree, kin, which denotes the number of links

that point to a node, and an outgoing degree, kout, which

denotes the number of links that start from it. For

example, node A in part b of the figure has kin = 4 and

kout = 1. An undirected network with N nodes and L links

is characterized by an average degree <k> = 2L/N

(where <> denotes the average).



Degree distribution


The degree distribution, P(k), gives the probability that a selected node has exactly k links. P(k) is obtained by counting the number o f nodes N(k) with k = 1,2... links and dividing by the total number of nodes N. The degree distribution allows us to distinguish between different classes of networks. For example, a peaked degree distribution, as seen in a random network, indicates that the system has a characteristic degree and that there are no highly connected nodes (which are also known as hubs). By contrast, a power-law degree distribution indicates that a few hubs hold together numerous small nodes.



Network measures


Scale-free networks and the degree exponent

Most biological networks are scale-free, which means that their

degree distribution approximates a power law, P(k) k- , where

is the degree exponent and ~ indicates 'proportional to'. The

value of determines many properties of the system. The

smaller the value of , the more important the role of the hubs

is in the network. Whereas for >3 the hubs are not relevant, for

2> >3 there is a hierarchy of hubs, with the most connected

hub being in contact with a small fraction of all nodes, and for

= 2 a hub-and-spoke network emerges, with the largest hub

being in contact with a large fraction of all nodes. In general, the

unusual properties of scale-free networks are valid only for <

3, when the dispersion of the P(k) distribution, which is defined

as 2 = <k2> - <k>2, increases with the number of nodes (that

is, diverges), resulting in a series of unexpected features,

such as a high degree of robustness against accidental node

failures. For >3, however, most unusual features are absent,

and in many respects the scale-free network behaves like a

random one.



Shortest path and mean path length


Distance in networks is measured with the path length, which

tells us how many links we need to pass through to travel

between two nodes. As there are many alternative paths

between two nodes, the shortest path — the path with the

smallest number of links between the selected nodes — has a

special role. In directed networks, the distance ℓAB from node A

to node B is often different from the distance ℓBA from B to A. For

example, in part b of the figure, ℓBA = 1, whereas ℓAB = 3. Often

there is no direct path between two nodes. As shown in part b of

the figure, although there is a path from C to A, there is no path

from A to C. The mean path length, <ℓ>, represents the average

over the shortest paths between all pairs of nodes and offers a

measure of a network's overall navigability.



Clustering coefficient


In many networks, if node A is connected to B, and B is connected to C,

then it is highly probable that A also has a direct link to C. This

phenomenon can be quantified using the clustering coefficient33 CI =

2nI/k(k-1), where nI is the number of links connecting the kI neighbours of

node I to each other. In other words, CI gives the number of 'triangles'

that go through node I, whereas kI (kI -1)/2 is the total number of triangles

that could pass through node I, should all of node I's neighbours be

connected to each other. For example, only one pair of node A's five

neighbours in part a of the figure are linked together (B and C), which

gives nA = 1 and CA = 2/20. By contrast, none of node F's neighbours link

to each other, giving CF = 0. The average clustering coefficient, <C >,

characterizes the overall tendency of nodes to form clusters or groups.

An important measure of the network's structure is the function C(k),

which is defined as the average clustering coefficient of all nodes with k

links. For many real networks C(k) k-1, which is an indication of a

network's hierarchical character.

The average degree <k>, average path length <ℓ> and average

clustering coefficient <C> depend on the number of nodes and links (N

and L) in the network. By contrast, the P(k) and C(k ) functions are

independent of the network's size and they therefore capture a network's

generic features, which allows them to be used to classify various

networks.

http://www.nature.com/cgi-taf/DynaPage.taf?file=/nrg/journal/v5/n2/full/nrg1272_r.html&filetype=&dynoptions=#B33



Origin of scale-free topology and hubs in biological networks

The origin of the scale-free topology in complex networks can be reduced to two basic mechanisms: growth and preferential attachment. Growth means that the network emerges through the subsequent addition of new nodes, such as the new red node that is added to the network that is shown in part a . Preferential attachment means that new nodes prefer to link to more connected nodes. For example, the probability that the red node will connect to node 1 is twice as large as connecting to node 2, as the degree of node 1 (k1=4) is twice the degree of node 2 (k2 =2). Growth

and preferential attachment generate hubs through a 'rich-gets-richer' mechanism: the more connected a node is, the more likely it is that new nodes will link to it, which allows the highly connected nodes to acquire new links faster than their less connected peers. In protein interaction networks, scale-free topology seems to have its origin in gene duplication. Part b shows a small protein interaction network (blue) and the genes that encode the proteins (green). When cells divide, occasionally one or several genes are copied twice into the offspring's genome (illustrated by the green and red circles). This induces growth in the protein interaction network because now we have an extra gene that encodes a new protein (red circle). The new protein has the same structure as the old one, so they both interact with the same proteins. Ultimately, the proteins that interacted with the original duplicated protein will each gain a new interaction to the new protein. Therefore proteins with a large number of interactions tend to gain links more often, as it is more likely that they interact with the protein that has been duplicated. This is a mechanism that generates preferential attachment in cellular networks. Indeed, in the example that is shown in part b it does not matter which gene is duplicated, the most connected central protein (hub) gains one interaction. In contrast, the square, which has only one link, gains a new link only if the hub is duplicated.





Aa

The Erdös–Rényi (ER) model of a random network starts with N nodes

and connects each pair of nodes with probability p, which creates a

graph with approximately pN (N-1)/2 randomly placed links.

Ab

The node degrees follow a Poisson distribution, which indicates that

most nodes have approximately the same number of links (close to the

average degree <k>). The tail (high k region) of the degree distribution

P(k ) decreases exponentially, which indicates that nodes that

significantly deviate from the average are extremely rare.

Ac

The clustering coefficient is independent of a node's degree, so C(k)

appears as a horizontal line if plotted as a function of k. The mean path

length is proportional to the logarithm of the network size, l log N, which

indicates that it is characterized by the small-world property.

Random networks




Scale-free networks Scale-free networks are characterized by a power-law degree

distribution; the probability that a node has k links follows P(k) ~ k- ,

where is the degree exponent. The probability that a node is highly

connected is statistically more significant than in a random graph, the

network's properties often being determined by a relatively small number

of highly connected nodes that are known as hubs (see figure, part Ba;

blue nodes). In the Barabási–Albert model of a scale-free network, at

each time point a node with M links is added to the network, which

connects to an already existing node I with probability I = kI/JkJ,

where kI is the degree of node I and J is the index denoting the sum over

network nodes. The network that is generated by this growth process has

a power-law degree distribution that is characterized by the degree

exponent = 3.

Bb Such distributions are seen as a straight line on a log–log plot. The

network that is created by the Barabási–Albert model does not have an

inherent modularity, so C(k) is independent of k (Bc). Scale-free

networks with degree exponents 2< <3, a range that is observed in

most biological and non-biological networks, are ultra-small, with the

average path length following ℓ ~ log log N, which is significantly shorter

than log N that characterizes random small-world networks.




Hierarchical networks To account for the coexistence of modularity, local clustering and scale-

free topology in many real systems it has to be assumed that clusters

combine in an iterative manner, generating a hierarchical network.

The starting point of this construction is a small cluster of four densely

linked nodes (see the four central nodes in Ca). Next, three replicas of

this module are generated and the three external nodes of the replicated

clusters connected to the central node of the old cluster, which produces

a large 16-node module. Three replicas of this 16-node module are then

generated and the 16 peripheral nodes connected to the central node of

the old module, which produces a new module of 64 nodes. The

hierarchical network model seamlessly integrates a scale-free topology

with an inherent modular structure by generating a network that has a

power-law degree distribution with degree exponent = 1 + ln4/ln3 =

2.26 (see Cb) and a large, system-size independent average clustering

coefficient <C> ~ 0.6.

The most important signature of hierarchical modularity is the scaling of

the clustering coefficient, which follows C(k) ~ k-1 a straight line of slope -

1 on a log–log plot (see Cc). A hierarchical architecture implies that

sparsely connected nodes are part of highly clustered areas, with

communication between the different highly clustered neighbourhoods

being maintained by a few hubs (see Ca).



Reminder A few remarks on the past lectures ...



V14: Prediction of P-P interaction from correlated mutationsResults obtained by i2h in a set of 14 two domain proteins of known structure = proteins with two interacting domains. Treat the 2 domains as different proteins.

A: Interaction index for the 133 pairs with 11 or more sequences in common. The true positive hits are highlighted with filled squares.

B: Representation of i2h results, reminiscent of those obtained in the experimental yeast two-hybrid system. The diameter of the black circles is proportional to the interaction index; true pairs are highlighted with gray squares. Empty spaces correspond to those cases in which the i2h system could not be applied, because they contained <11 sequences from different species in common for the two domains.

In most cases, i2h scored the correct pair of protein domains above all other possible interactions.

Pazos, Valencia, Proteins 47, 219 (2002)



V14: Co-localization of interaction partners

Use localization data to assess the

quality of prediction because two

predicted interacting partners

sharing the same subcellular

location are more likely to form a

true interaction.

Comparison of colocalization index

(defined as the ratio of the number

of protein pairs in which both

partners have the same subcellular

localization to the number of

protein pairs where both partners

have any sub-cellular localization

annotation).

Lu, ..., Skolnick, Genome Res 13, 1146 (2003)

Multithreading predictions (MTA) are

less reliable than high-confidence inter-

actions, but score quite well amongst

predictions + HTS screens.



V14:Do partners have the same function?

Proteins from different groups of

biological functions may interact with

each other.

However, the degree to which interacting

proteins are annotated to the same

functional category is a measure of

quality for predicted interactions.

Here, the predictions cluster fairly well

along the diagonal.

Lu, ..., Skolnick, Genome Res 13, 1146 (2003)



V15: Statistical significance of complexes and modules

Number of complete cliques (Q = 1) as

a function of clique size enumerated in

the network of protein interactions

(red) and in randomly rewired graphs

(blue, averaged >1,000 graphs where

number of interactions for each protein

is preserved).

Inset shows the same plot in log-

normal scale. Note the dramatic

enrichment in the number of cliques in

the protein-interaction graph

compared with the random graphs.

Most of these cliques are parts of

bigger complexes and modules.

Spirin, Mirny, PNAS 100, 12123 (2003)



V15: Architecture of protein network

Fragment of the protein network. Nodes

and interactions in discovered clusters

are shown in bold. Nodes are colored by

functional categories in MIPS:

red, transcription regulation;

blue, cell-cycle/cell-fate control;

green, RNA processing; and

yellow, protein transport.

Complexes shown are the SAGA/TFIID

complex (red), the anaphase-promoting

complex (blue), and the TRAPP complex

(yellow).

Spirin, Mirny, PNAS 100, 12123 (2003)



V15: Evolution of the yeast protein interaction network

Isotemporal categories are designed

through a binary (b) coding scheme.

The b code represents the

distribution of each yeast protein's

orthologs in the universal tree of life.

Bit value 1 indicates the presence of

at least one orthologous hit for a

yeast protein in a corresponding

group of genomes, and bit value 0

indicates the absence of any

orthologous hit. The presented

example is 110011 in the b format

and 51 in the d format. Orthologous

identifications are based on COGs at

NCBI and in von Mering et al. (2002).

Qin et al. PNAS 100, 12820 (2003)

Previously, phylogenetic profileswere used to detect proteininteraction partners.Here, use phylogenetic profiles to detect modules.



V15: Evolution of the yeast protein interaction network

Interaction patterns.

Z scores for all possible interactions

of the isotemporal categories in the

protein interaction network.

For categories i and j,

Zi,j = (Fi,jobs – Fi,j

mean)/i,j

where Fi,jobs is the observed number

of interactions, and Fi,jmean and i,j are

the average number of interactions

and the SD, respectively, in 10,000

MS02 null models.

Qin et al. PNAS 100, 12820 (2003)

The diagonal distribution of large positive Z scores indicates that yeast proteins

tend to interact with proteins from the same or closely related isotemperal

categories.



V19: Flux balancingmass conservation.

Therefore one may analyze metabolic systems by requiring mass conservation.

Only required: knowledge about stoichiometry of metabolic pathways and

metabolic demands

For each metabolite:

Under steady-state conditions, the mass balance constraints in a metabolic

network can be represented mathematically by the matrix equation:

S · v = 0

where the matrix S is the m n stoichiometric matrix,

m = the number of metabolites and n = the number of reactions in the network.

The vector v represents all fluxes in the metabolic network, including the internal

fluxes, transport fluxes and the growth flux.

)( dtransporteuseddegradeddsynthesizei

i VVVVdt

dXv

Any chemical reaction requires



V19: Flux balance analysis

Since the number of metabolites is generally smaller than the number of reactions

(m < n) the flux-balance equation is typically underdetermined.

Therefore there are generally multiple feasible flux distributions that satisfy the mass

balance constraints.

The set of solutions are confined to the nullspace of matrix S.

To find the „true“ biological flux in cells ( e.g. Heinzle, Huber, UdS) one needs

additional (experimental) information,

or one may impose constraints

on the magnitude of each individual metabolic flux.

The intersection of the nullspace and the region defined by those linear inequalities

defines a region in flux space = the feasible set of fluxes.

iii v



V19: Rerouting of metabolic fluxes

(Black) Flux distribution for the wild-type.

(Red) zwf- mutant. Biomass yield is 99% of

wild-type result.

(Blue) zwf- pnt- mutant. Biomass yield is 92% of

wildtype result. The solid lines represent

enzymes that are being used, with the

corresponding flux value noted.

Note how E.coli in silico circumvents removal of

one critical reaction (red arrow) by increasing

the flux through the alternative G6P P6P

reaction.

Edwards & Palsson PNAS 97, 5528 (2000)



V20: Extreme Pathwaysintroduced into metabolic analysis by the lab of Bernard Palsson

(Dept. of Bioengineering, UC San Diego). The publications of this lab

are available at http://gcrg.ucsd.edu/publications/index.html

Extreme pathway

technique is based

on the stoichiometric

matrix representation

of metabolic networks.

All external fluxes are

defined as pointing outwards.

Schilling, Letscher, Palsson,

J. theor. Biol. 203, 229 (2000)



V20: Feasible solution set for a metabolic reaction network

(A) The steady-state operation of the metabolic network is restricted to the region

within a cone, defined as the feasible set. The feasible set contains all flux vectors

that satisfy the physicochemical constrains. Thus, the feasible set defines the

capabilities of the metabolic network. All feasible metabolic flux distributions lie

within the feasible set, and

(B) in the limiting case, where all constraints on the metabolic network are known,

such as the enzyme kinetics and gene regulation, the feasible set may be reduced

to a single point. This single point must lie within the feasible set.

Edwards & Palsson PNAS 97, 5528 (2000)



V21: Reconfigured Network

Klamt & Stelling Trends Biotech 21, 64 (2003)

A C P

B

D

A(ext) B(ext) C(ext)R1 R2 R3

R5

R4 R8

R9

R6

R7bR7f

3 EFMs are not systemically independent:EFM1 = EP4 + EP5EFM2 = EP3 + EP5EFM4 = EP2 + EP3



V21: relation between EP and EFMsWhat is the consequence of when all exchange fluxes (and hence all

reactions in the network) are irreversible?

Klamt & Stelling Trends Biotech 21, 64 (2003)

EFMs and EPs always co-incide!



V22: Correlation between genes of same metabolic pathway

Ihmels, Levy, Barkai, Nat. Biotech 22, 86 (2004)

Genes of the glycolysis pathway

(according KEGG) were clustered

and ordered based on the correlation

in their expression profiles.

Shown here is the matrix of their

pair-wise correlations.

The cluster of highly correlated

genes (orange frame) corresponds

to genes that encode the central

glycolysis enzymes.

The linear arrangement of these

genes along the pathway is shown at

right.

Of the 46 genes assigned to the

glycolysis pathway in the KEGG

database, only 24 show a correlated

expression pattern.

In general, the coregulated genes

belong to the central pieces of

pathways.



The connectivity of a given metabolite

is defined as the number of reactions

connecting it to other metabolites.

Shown are the distributions of

connectivity between metabolites in an

unrestricted network () and in a

network where only correlated

reactions are considered ().

In accordance with previous results

(Jeong et al. 2000) , the connectivity

distribution between metabolites

follows a power law (log-log plot).


V22: Connectivity of metabolites

In contrast, when coexpression is

used as a criterion to distinguish

functional links, the connectivity

distribution becomes exponential

(log-linear plot).



V22: Co-expression of transporters

Transporter genes are

co-expressed with the relevant

metabolic pathways providing

the pathways with its metabolites.

Co-expression is marked in green.




V25: Combination of data sets into probabilistic interactomes

(B) Combination of data sets into

probabilistic interactomes.

The 4 interaction data sets

from HT experiments were

combined into 1 PIE.

The PIE represents a

transformation of the

individual binary-valued

interaction sets into a data

set where every protein pair

is weighed according to the

likelihood that it exists in a

complex. A „naïve” Bayesian network is used to model

the PIP data. These information sets hardly

overlap.


Because the 4 experimental

interaction data sets contain

correlated evidence, a fully

connected Bayesian network

is used.



V25: Static naive Bayesian Networks

The likelihood ratio L defined as

relates prior and posterior odds according to Bayes‘ rule:

negffP

posffPffL

N

NN ...

......

1

11

priorNpost OffLO ...1

In the special case that the N features are conditionally independent

(i.e. they provide uncorrelated evidence) the Bayesian network is a so-called

„naïve” network, and L can be simplified to:

N

i

N

i i

iiN negfP

posfPfLffL

1 11...




V25: PIP vs. the information sources

Ratio of true to false positives (TP/FP) increases

monotonically with Lcut, confirming L as an

appropriate measure of the odds of a real

interaction.

The ratio is computed as:

Protein pairs with Lcut > 600 have a > 50%

chance of being in the same complex.Jansen et al. Science 302, 449 (2003)

cut

cut

LL

LL

cut

cut

Lneg

Lpos

LFP

LTP



V25: PIE vs. the information sources

9897 interactions are predicted from PIP and

163 from PIE.

In contrast, likelihood ratios derived from single

genomic factors (e.g. mRNA coexpression) or

from individual interaction experiments (e.g. the

Ho data set) did no exceed the cutoff when used

alone.

This demonstrates that information sources that,

taken alone, are only weak predictors of

interactions can yield reliable predictions when

combined.


network wrap-up

Documents

degree barabasi oltvai

degree distribution

average degree

nature reviews genetics

incoming degree

outgoing degree

undirected network

network characteristics