Structure, Function and Evolution of Metabolic Networks (I)

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Structure, Function andEvolution of
MetabolicNetworks (I)

• Jing Zhao
• College of Pharmacy, Second Military Medical
  University
• Shanghai Center for Bioinformation and Technology
• 2009.5.25

Spring school on multiscale methods and modeling
in biophysics and system biology, Shanghai, China

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• Outline
• Reconstruction of metabolic networks
• Network metrics and topological features
• Modularity and network decomposition
• Topological diversity of networks with a given
  degree sequence

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I. Reconstruction of metabolic networks
Zhao J, Yu H, Luo J, Cao Z, Li Y Complex
networks theory for analyzing metabolic networks.
Chinese Science Bulletin 2006, 51(13)1529-1537.
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What is network?
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Examples Internet
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Examples Scientific collaborations
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Examples protein-protein interaction network
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Metabolism
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Examples metabolic network
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How to get genome-specific metabolic reactions?

• Identifying ORFs from the genomic sequence
• (ii) Predicting all the enzyme genes of this
  organism by sequence similarity alignment
• (iii) Comparing the predicted enzymes within this
  organism against the collection of known
• reference pathways to determine all the reactions
  of this organism.

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• Two refined metabolism database for human being
  manually reconstructed
• BiGG database
• Duarte, N. C. Becker, S. A. Jamshidi, N.
  Thiele, I. Mo, M. L. Vo, T. D. Srivas, R.
  Palsson, B. O., Global reconstruction of the
  human metabolic network based on genomic and
  bibliomic data. PNAS 2007, 104, (6), 1777-1782.
• The Edinburgh human metabolic network
• Ma, H. Sorokin, A. Mazein, A. Selkov, A.
  Selkov, E. Demin, O. Goryanin, I., The
  Edinburgh human metabolic network reconstruction
  and its functional analysis. Molecular Systems
  Biology 2007, 3, 135.

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Statistics for BiGG database
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Process for reconstructing the Edinburgh human
metabolic network
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Different graph representations of a simple
metabolic network
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Currency metabolites
Ma H, Zeng A-P Reconstruction of metabolic
networks from genome data and analysis of their
global structure for various organisms.
Bioinformatics 2003, 19(2)270-277.
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Currency metabolites
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Currency metabolites

• Definition
• currency metabolites have high degree
• they make not meaningful shortcuts
• i.e. tie together distant parts of the
  network
• i.e. tie different modules together

Algorithm Remove vertices in order of
(currently) highest degree. The set of removed
vertices that gives the network the highest
modularity is the set of currency metabolites.
Huss M, Holme P Currency and commodity
metabolites Their identification and relation to
the modularity of metabolic networks. IET Systems
Biology 2007, 1280-285.
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Human currency metabolites
Huss M, Holme P Currency and commodity
metabolites Their identification and relation to
the modularity of metabolic networks. IET Systems
Biology 2007, 1280-285.
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• Steps for reconstructing a metabolic network
• Get reaction list
• Generate substrate - product pair list
• Delete currency metabolites
• Generate metabolic network
• Useful tool
• Text2pajek.exe

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II. Network metrics and topological features
Zhao J, Yu H, Luo J, Cao Z, Li Y Complex
networks theory for analyzing metabolic networks.
Chinese Science Bulletin 2006, 51(13)1529-1537.
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network science Measures of network structure.
How does a network that is too large to draw
.look. like? Real-world networks have both
randomness and structure. How can we quantify
network structure? Models of evolving networks.
How do networks get their structure? What
.microscopic. properties are responsible for the
macro-structure of the network. Models of
network changing events. Malicious attacks
overload breakdowns. Classication and functional
prediction. How can we classify vertices and
predict their function in the network? How does
the network structure affect dynamic systems of
the network? Running dynamic simulations on top
of the network and see how dynamic properties
correlates with the network structure.
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• As for 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?

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Degree distribution vs. scale-free networks
Degree distribution p(k) the occurrence
frequency of nodes with degree k, (k1,2,).
Random network
Scale-free network
hub
Barabasi, A.L., Albert, R., Emergence of scaling
in random networks, Science, 1999, 286509-512
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• BA model for network evolution
• (1) Growth the continuous addition of new nodes.
  
• (2) Preferential attachment the rich get
  richer principle.
• The high-degree nodes should appear in the
  earlier stage of network formation.
• Thirteen hub metabolites in E.coli
  metabolic network

Wagner, A., Fell, D.A., The small world inside
large metabolic networks, Proc R Soc Lond B,
2001, 2681803-1810.
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• Performance of scale-free networks
• error tolerance high resistance to random
  perturbations
• attack vulnerability the removal of a few hub
  nodes will destroy the whole network.

Albert, R., Jeong, H., Barabasi, A.-L., Error and
attack tolerance of complex networks, Nature,
2000, 406378-382.
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Jeong, H., Mason, S.P., Barabasi, A.L., Oltvai,
Z.N., Lethality and centrality in protein
networks, Nature, 2001, 41141-42.
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• Notice Computation of the exponent
• cumulative distribution

?
Log-log plot of the degree distribution (A) and
cumulative degree distribution (B) for a network
of 20000 nodes constructed by Barabasi-Albert
preferential attachment model.
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• Clustering coefficient vs. Hierarchical modular
  networks

How many triangles are there in the network?
N(v) the number of links between neighbours of
node v d(v) the degree of node v
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Ravasz E, Somera A L, Mongru D A, Oltvai Z N,
Barabasi A L, Hierarchical organization of
modularity in metabolic networks,
Science,2002,297 1551-1556
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Complex systems usually have a hierarchical
structure, the entities of one level being
compounded into new entities at the next higher
lever, as cells into tissues, tissues into
organs, and organs into functional systems. The
whole is greater than the sum of its parts! At
each new level of complexity in biology new and
unexpected qualities appear, qualities which
apparently cannot be reduced to the properties of
the component parts.
Lifes complex Pyramid from the particular to
the universal
Oltvai, Z.N., Barabási, A.-L., Lifes Complexity
Pyramid, SCIENCE, 2002, 298763-764.
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• Mean path length vs. small-world networks

Small-world network small mean path length
high clustering coefficient
Small-world cell networksgtthe cell may react
quickly to changes of the surroundings
Watts, D.J., Strogatz, S.H., Collective dynamics
of small-world' networks, Nature, 1998,
393440-442.
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• Assortativity coefficient vs. degree-degree
  correlation
• 
  
• Are high-degree vertices connected to other
  high-degree vertices? Or are these vertices
  primarily connected to low-degree vertices?
• ji , ki the degrees of the nodes at the ends of
  the ith edge
• M number of edges in the network
• rgt0 assortative network
• rlt0 disassortative network

Newman , M.E.J., Assortative mixing in networks,
Phys Rev Lett, 2002, 89208701.
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Newman , M.E.J., Assortative mixing in networks,
Phys Rev Lett, 2002, 89208701.
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The average connectivity ltknngt of the nearest
neighbors of a node depending on its connectivity
k for the 1998 snapshot of the Internet, the
generalized BA model and the fitness model.
Romualdo Pastor-Satorras, Alexei Vázquez, and
Alessandro Vespignani, Dynamical and Correlation
Properties of the Internet, PHYSI CAL REV IEW
LETTERS, VOLUME 87, NUMBER 25(2002)
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Correlation profiles of protein interaction
network in yeast. Z-scores for connectivity
correlations Z(K0,K1) (P(K0,K1) -
Pr(K0,K1))/r(K0,K1) where r(K0,K1) is the
standard deviation of Pr(K0,K1) in 1000
realizations of a randomized network.
Maslov, S., Sneppen, K., Specificity and
Stability in Topology of Protein Networks,
Science, 2002, 296910-913.
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Rich-club coefficient and rich-club
phenomenon rich-club coefficient
Notice Rich-club
Assortative mixing
Colizza V, Flammini A, Serrano MA, Vespignani A
Detecting rich-club ordering in complex networks.
Nat Phys 2006, 2(2)110-115.
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Centrality Which nodes are important for
communication on the network? Assumption
Information transmission or material
transportation on the network are along shortest
paths.
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Betweenness centrality

• Node betweenness measures the degree to which
  a vertex is participating in the communication
  between pairs of other vertices

the number of shortest paths from s
to t the number of shortest paths
from s to t with v as an inner vertex
Holme P, Kim BJ, Yoon CN, Han SK Attack
vulnerability of complex networks. Phys Rev E
2002, 65056109.
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Edge betweenness measures the degree to which
an edge is participating in the communication
between pairs of other vertices
the number of shortest paths from s
to t the number of shortest paths
from s to t with v as an inner vertex
Holme P, Kim BJ, Yoon CN, Han SK Attack
vulnerability of complex networks. Phys Rev E
2002, 65056109.
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• Nodes and edges of high betweenness centrality
  could be bottlenecks of the network, thus could
  be important enzymes or metabolites.
• Edges of high betweenness centrality could be
  bridges of modules.

Rahman, S.A., Schomburg, D., Observing local and
global properties of metabolic pathways 'load
points' and 'choke points' in the metabolic
networks, Bioinformatics, 2006,
221767-1774. Girvan M, Newman MEJ Community
structure in social and biological networks. Proc
Natl Acad Sci 2002, 99(12)7821-7826.
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Closeness centrality
Closeness centrality measures the degree to which
a vertex is close to other vertices on average.
Service facility locating problem Find the
location of a shopping mall that the average
driving distance to the mall is minimal.
Solution the nodes which have the biggest
closeness centrality
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Center Emergency facility locating problem
find the optimal location of a firehouse such
that the worst-case response distance of a fire
engine is minimal.
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• k-core

1, 2 and 3-core. Two basic properties of cores
first, cores may be disconnected subgraphs
second, cores are nested for igtj, an i-core is a
subgraph of a j-core of the same graph.
gt The probability of nodes both being essential
and evolutionary conserved successively increases
toward the innermost cores.
Wuchty, S., Almaas, E., Peeling the yeast protein
network, Proteomics, 2005, 5444-449.
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Reciprocity metric
aij 1 if there is an arc from nodes i to
j, aij 0 otherwise L the number of
total arcs in the network N the number of total
nodes in the network
? -1 for purely unidirectional networks ?
1 for purely bidirectional networks
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Network null models

• Network structures are always relative
• Network structures how the network differs from
  a random network, or a null model
• One has to be clear about what to compare with a
  null model
• Null model 1 random graphs (Poisson random
  graphs,
• Erdos-Renyi graphs)
• Null model 2 random graphs constrained to
  the set of
• degrees of the original graph

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Null Models random rewiring
Maslov, S., Sneppen, K., Specificity and
Stability in Topology of Protein Networks,
Science, 2002, 296910-913.
Maslov S, Sneppen K, Zaliznyak A Detection of
topological patterns in complex networks
correlation profile of the internet. Physica A
Statistical and Theoretical Physics 2004,
333529-540.
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Z-score
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Graph analysis and visualization software
Pajek http//vlado.fmf.uni-lj.si/pub/
networks/pajek/ txt2pajek.exe
pajek.exe UCINET http//www.analytictec
h.com/downloaduc6.htm NetMiner
http//www.netminer.com/NetMiner/home_01.jsp
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• III. Modularity and network decomposition

Zhao J, Yu H, Luo J, Cao Z, Li Y Complex
networks theory for analyzing metabolic networks.
Chinese Science Bulletin 2006, 51(13)1529-1537.
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2.1 Modularity From functional
view Modularity the system can be decomposed
in parts (modules), such that each part has its
own relatively independent function, while
different parts have some communications with
each other. From topological view Assumption
A densely connected subnetwork ? "part with
complex function." Modularity network could be
divided into groups of vertices that have a high
density of edges within them, with a lower
density of edges between groups.
Hartwell LH, Hopfield JJ, Leibler S, Murray AW
From molecular to modular cell biology. Nature
1999, 402C47-C52. Papin JA, Reed JL, Palsson BO
Hierarchical thinking in network biology the
unbiased modularization of biochemical networks
,Trends in Biochemical Sciences 2004, 29641-647.
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For a given decomposition of a network, the
modularity metric is defined as
the sum is over the a partition into clusters and
eij is the fraction of edges that leads between
vertices of cluster i and j
The modularity metric of a network is
defined as the largest modularity metric of all
possible partitions of the network. The
modularity of networks must always be compared to
the null case of a random graph.
Newman M Detecting community structure in
networks EurPhysJB 2004, 38321-330. Guimera R,
Sales-Pardo M, Amaral LAN Modularity from
fluctuations in random graphs and complex
networks. Physical Review E 2004, 70025101.
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2.2 Simulated annealing method
Guimera R, Nunes Amaral LA Functional
cartography of complex metabolic networks. Nature
2005, 433(7028)895-900.
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2.3 Hierarchical clustering method Similarity
index(or dissimilarity index) to signify the
extent to which two nodes would like in the same
cluster. Agglomerative method to start off with
each node being its own cluster. At each step, it
combines the two most similar clusters to form a
new larger cluster until all nodes have been
combined into one cluster. Divisive method to
begin with one cluster including all the nodes,
and attempts to find the splitting point at which
two clusters are as dissimilar as possible.
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Topological overlap algorithm Substrate graph
Jn(i,j) denotes the number of nodes to which
both i and j are linked ( plus 1 if there is a
direct link between i and j ) ki, kj is the
degree of i and j, respectively. Agglomerative
method.
Ravasz E, Somera AL, Mongru DA, Oltvai ZN,
Barabasi AL Hierarchical Organization of
Modularity in Metabolic Networks. Science 2002,
297(5586)1551-1555
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Shortest path algorithm enzyme graph
d(i, j) is the number of arcs in the shortest
directed path from i to j . Agglomerative
method.
Ma H-W, Zhao X-M, Yuan Y-J, Zeng A-P
Decomposition of metabolic network into
functional modules based on the global
connectivity structure of reaction graph.
Bioinformatics 2004, 20(12)1870-1876.
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Betweenness method substrate-reaction bipartite
graph
is the number of shortest
paths between s and t that passes through r,
is the total number of shortest paths
between s and t, is the in-degree of
node r. Divisive method.
Holme P, Huss M, Jeong H Subnetwork hierarchies
of biochemical pathways. Bioinformatics 2003,
19(4)532-538.
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Corrected Euclidean-like dissimilarity algorithm
substrate graph
d(i, j) is the number of arcs in the shortest
directed path from i to j . Agglomerative
method.
Zhao J, Yu H, Luo J, Cao Z, Li Y Hierarchical
modularity of nested bow-ties in metabolic
networks. BMC Bioinformatics 20067386.
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• IV. Topological diversity of networks with a
  given degree sequence

Zhao J, Tao L, Yu H, Luo J-H, Cao Z-W, Li Y-X
The effects of degree correlations on network
topologies and robustness. Chinese Physics 2007,
16.
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• Seed networks
• Seed network A the hierarchically modular
  network constructed by Ravasz et al. (RB model)
  in the 3rd iteration.
• Seed network B a model network constructed by
  the BA preferential attachment model .
• Seed network C the biggest connected cluster of
  the E.coli metabolic
• Seed network D the biggest connected cluster of
  the protein interaction network CCSB-HI1

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Extreme networks of degree correlation
The Smax graph (A) and Smin graph (B) for a small
seed network. Nodes with different degrees are
shown in different colours.
Graphs with the same degree sequence have
significantly topological diversity.
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Constructing network ensemble from the extreme
networks
Assortative coefficient (r) as function of the
randomization fraction (p).
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Relationship between mean path length (L) and
assortative coefficient (r). The data shown in
the figures are averaged over 10 random
realizations of the rewiring process.
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Relationship between clustering coefficient(C)
and assortative coefficient (r). The data shown
in the figure are averaged over 10 random
realizations of the rewiring process.
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Relationship between modularity(M) and
assortative coefficient (r). The data shown in
the figures are averaged over 10 random
realizations of the rewiring process.
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The effect of degree correlation on network
robustness. Figures in the first and second row
depict the robustness under attacks and failures
as a function of assortativity, respectively. The
data shown in the figures are averaged over 10
random realizations of the rewiring process.
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Holme P, Zhao J Exploring the assortativity-clust
ering space of a network's degree sequence. Phys
Rev E 2007, 75 046111.
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• Thanks!