Transcript An Introduction of Complex networks

Structure, Function and
Evolution of Metabolic
Networks (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
Outline
I. Reconstruction of metabolic networks
II. Network metrics and topological features
III.Modularity and network decomposition
IV.Topological diversity of networks with a
given degree sequence
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.
What is network?
Examples: Internet
Examples: Scientific
collaborations
Examples: protein-protein
interaction network
Metabolism
Examples: metabolic network
How to get genome-specific metabolic reactions?
(i) 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.
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.
Statistics for BiGG database
Process for reconstructing the Edinburgh human metabolic network
Different graph representations of a simple metabolic network
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.
Currency metabolites
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,
1:280-285.
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,
1:280-285.
Steps for reconstructing a metabolic network
1. Get reaction list
2. Generate substrate - product pair list
3. Delete currency metabolites
4. Generate metabolic network
Useful tool:
Text2pajek.exe
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.
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.
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?
Degree distribution vs. scale-free networks
Degree distribution p(k) :
the occurrence frequency of nodes with degree k,
(k=1,2,…).
Random
network
Scale-free
network
hub
Barabasi, A.L., Albert, R., Emergence of scaling in random networks, Science, 1999,
286:509-512
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, 268:1803-1810.
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, 406:378-382.
Jeong, H., Mason, S.P., Barabasi, A.L., Oltvai, Z.N., Lethality and centrality in protein
networks, Nature, 2001, 411:41-42.
Notice: Computation of the exponent
cumulative distribution : P( x  k )   p(i)

i k
p ( k ) ~ k 

P( x  k ) ~ k ( 1)
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.
Clustering coefficient vs. Hierarchical modular
networks
How many triangles are there in the network?
2 N (v )
CC (v) 
d (v)(d (v)  1)
N(v): the number of links between neighbours of node v
d(v) :the degree of node v
C (k ) ~ k 1
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
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!
Life’s complex Pyramid: from the particular to
the universal
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.
Oltvai, Z.N., Barabási, A.-L., Life’s Complexity Pyramid, SCIENCE, 2002, 298:763764.
Mean path length vs. small-world networks
Small-world network: small mean path length; high clustering
coefficient
Small-world cell networks=>the cell may react quickly to changes of the
surroundings
Watts, D.J., Strogatz, S.H., Collective dynamics of `small-world' networks, Nature, 1998,
393:440-442.
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
r>0: assortative network
r<0: disassortative network
Newman , M.E.J., Assortative mixing in networks, Phys Rev Lett, 2002, 89:208701.
Newman , M.E.J., Assortative mixing in networks, Phys Rev Lett, 2002, 89:208701.
The average connectivity <knn> 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)
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, 296:910-913.
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.
Centrality:
Which nodes are important for communication
on the network?
Assumption: Information transmission or material
transportation on the network are along shortest paths.
Betweenness centrality
Node betweenness measures the degree to
which a vertex is participating in the
communication between pairs of other vertices
 v ( s, t )
C B (v )  
s  t  ( s, t )
 ( s, t ) :the number of shortest paths from s to t
 v (s, 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, 65:056109.
Edge betweenness measures the degree to which an
edge is participating in the communication between pairs
of other vertices
 e ( s, t )
C B ( e)  
s  t  ( s, t )
 ( s, t ) :the number of shortest paths from s to t
 e (s, 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, 65:056109.
•
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, 22:1767-1774.
Girvan M, Newman MEJ: Community structure in social and biological networks. Proc
Natl Acad Sci 2002, 99(12):7821-7826.
Closeness centrality
Closeness centrality measures the degree to which a
vertex is close to other vertices on average.
CC (v) 
1
 d (v, t )
t v
“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
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.
C1(G )  {x  V max d ( x, y )  min (max d ( x, y ))}
yV
xV
yV
k-core
1, 2 and 3-core. Two basic properties of cores: first, cores may be disconnected
subgraphs; second, cores are nested: for i>j, an i-core is a subgraph of a j-core of
the same graph.
=> 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, 5:444-449.
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
ρ = -1 for purely unidirectional networks
ρ = 1 for purely bidirectional networks
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
Null Models : random rewiring
Maslov, S., Sneppen, K., Specificity and Stability in Topology of Protein Networks, Science,
2002, 296:910-913.
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, 333:529-540.
Z-score
P  Pr
Z 
Pr
Graph analysis and visualization software:
Pajek:
http://vlado.fmf.uni-lj.si/pub/networks/pajek/
txt2pajek.exe; pajek.exe
UCINET:
http://www.analytictech.com/downloaduc6.htm
NetMiner:
http://www.netminer.com/NetMiner/home_01.jsp
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.
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, 402:C47-C52.
Papin JA, Reed JL, Palsson BO: Hierarchical thinking in network biology: the unbiased
modularization of biochemical networks ,Trends in Biochemical Sciences 2004, 29:641-647.
For a given decomposition of a network, the modularity metric is defined as:
r
M  [eii  ( eij ) 2 ]
i 1
j
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, 38:321-330.
Guimera R, Sales-Pardo M, Amaral LAN: Modularity from fluctuations in random graphs and
complex networks. Physical Review E 2004, 70:025101.
2.2 Simulated annealing method:
r
max M  max [eii  ( eij ) 2 ]
i 1
j
Guimera R, Nunes Amaral LA: Functional cartography of complex metabolic networks. Nature 2005,
433(7028):895-900.
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.
Topological overlap algorithm: Substrate graph
OT (i, j ) 
J n (i, j )
min(k i , k j )
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
Shortest path algorithm: enzyme graph
dissimilarity(i, j )  min(d (i, j ), d ( j, i))
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.
Betweenness method: substrate-reaction bipartite graph
C B (r ) 
 r ( s, t )
1

k in (r ) s t  ( s, t )
 r (s, t ) is the number of shortest paths between s and t that passes
through r,  ( s, t ) is the total number of shortest paths between s and
t, kin (r) 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.
Corrected Euclidean-like dissimilarity algorithm: substrate graph
D (i, j )  (d ij  d ji ) 
2
N
2
2
[(
d

d
)

(
d

d
)
]
 ki kj
ik
jk
k 1
k i , j
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 2006:7:386.
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.
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
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.
Constructing network ensemble from the extreme networks
Assortative coefficient (r) as function of the randomization fraction (p).
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.
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.
Relationship between modularity(M) and assortative
coefficient (r). The data shown in the figures are averaged
over 10 random realizations of the rewiring process.
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.
Holme P, Zhao J: Exploring the assortativity-clustering space of a network's degree
sequence. Phys Rev E 2007, 75 046111.
Thanks!