Download How to reconstruct a large genetic network from n gene

How to Reconstruct a Large Genetic
Network from n Gene Perturbations in
fewer than n2 Easy Steps
Andreas Wagner, Bioinformatics, vol. 17, No. 12, 2001, pp. 1183-1187.
Speaker: Chuang Chieh Lin
Advisor: Professor R. C. T. Lee
National Chi-Nan University
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Outline
Introduction and basic definitions
Graph theoretical framework
Parsimonious network
Algorithm and complexity
Cycles in genetic networks
Conclusions
References
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Outline
Introduction and basic definitions
Graph theoretical framework
Parsimonious network
Algorithm and complexity
Cycles in genetic networks
Conclusions
References
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Introduction and basic definitions
Gene activity includes whether a gene is expressed or
not, as mRNA, as protein etc..
Gene network: In this paper, we define a genetic
network as a group of genes in which individual gene
can influence the activity of other genes.
The core task of reconstructing genetic networks is to
identify the causal structure of a gene network.
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To reconstruct a genetic network is to identify, for
each network gene, which other genes and their
activity the gene influences directly.
Now, let’s see an illustration of genetic network.
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transcription
factor
protein
kinase
protein
transcription
phosphatase
factor
inactive
inactive
P
protein
P
DNA
Gene 1
Gene 2
active
active
Gene 3
Gene 4
Gene 5
This is a hypothetical biochemical pathway involving two
transcription factors, a protein kinase and a protein phosphatase,
as well as the genes encoding them.
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Genetic perturbation: an experimental manipulation
of gene activity by manipulating either a gene itself or
its product. It includes point mutations, gene
deletions, or other interference with the activity of the
product.
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transcription
factor
protein
kinase
protein
transcription
phosphatase
factor
inactive
inactive
P
protein
P
DNA
Gene 1
Gene 2
Genetic perturbation: gene deletion
Aspect of gene activity: mRNA expression
G1:
G2:
G3:
G4:
G5:
active
active
Gene 3
Gene 4
Gene 5
Genetic perturbation: gene deletion
Aspect of gene activity: phosphorlation state
G2, G5
G5
G5
G5
G1:
G2:
G3:
G4:
G5:
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G3, G4
G3, G4
G4
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Outline
Introduction and basic definitions
Graph theoretical framework
Parsimonious network
Algorithm and complexity
Cycles in genetic networks
Conclusions
References
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Graph theoretical framework
As the previous instance indicated, we are
concerned with qualitative information on gene
interaction.
We consider a “digraph”, a graph representation of
genetic networks, to this qualitative information.
A digraph is a directed graph consisting of nodes
and directed edges.
Let’s see an example.
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We use a → b to mean that gene a influence the activity of gene
b directly. For brevity, genes will be labeled by numbers from
now on.
1
13
17
18
4
8
7
19
9
6
3
2
11
20
10
15
5
12
16
0
14
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Adjacency list: for each gene i, it simply shows which
genes’ activity state the gene i influences directly.
We denote Adj(G) to be the adjacency list of graph G
and Adj(i) to be the set of nodes (genes) adjacent to
(directly influenced by) node i.
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Adjacency list of G:
1
13
17
18
4
8
7
19
9
6
3
2
11
20
10
15
5
12
16
0
G
14
0:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
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2 5 8
12
5 12
2 17
10 15
1 20
20
14
8 17
0
0
2
8
8
6 18
13
Accessibility list: the list of perturbation effects or the
list of regulatory effects. It shows all nodes (genes)
that can be accessed (influenced in their activity state)
from a given gene by paths of direct interactions.
We denote Acc(G) to be the accessibility list of the
graph G and Acc(i) to be the set of nodes that can be
reached (influenced) from node (gene) i.
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Accessibility list of G:
1
13
17
18
4
8
7
19
9
6
3
2
11
20
10
15
5
12
16
0
14
0:
1:
2:
3:
4:
5:
6:
7:
8:
9:
10:
11:
12:
13:
14:
15:
16:
17:
18:
19:
20:
2 16
0 2 5 8 12 14 16
0 2 12 14 16
0 2 5 12 14 16
2 8 17
0
0
0
0
8
0
0
2
8
1 2 5 6 10 12 14 15 16 18 20
1 2 5 6 12 14 16 18 20
2 5 6 12 14 16 18 20
2 14 16
17
2 16
2 16
8
0 2 5 6 12 14 16 18
G
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Outline
Introduction and basic definitions
Graph theoretical framework
Parsimonious network
Algorithm and complexity
Cycles in genetic networks
Conclusions
References
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Before proceeding with the algorithm, we have to
give some concepts and theorems first.
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The most parsimonious network
An acyclic digraph defines its accessibility list, but an
accessibility list may have more than one
corresponding acyclic digraph.
Let’s see an example first.
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(d) is the most parsimonious network of Acc, i.e., (a).
0:
1:
2:
3:
4:
5:
0
1 2 3 4 5
2 3 4 5
3 4 5
1
2
5
4
(a)
5
(b)
0
0
1
1
2
2
4
3
3
5
4
3
5
(c)
(d)
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An accessibility list Acc and a digraph G are
compatible if G has Acc as its accessibility list. Acc
is the accessibility list induced by G.
Gpars is called the most parsimonious network
compatible with Acc.
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Why we prefer the most parsimonious
network?
We prefer simplest or most parsimonious one of gene
network.
For any accessibility list Acc of a digraph G, there
exists a most parsimonious network Gpars. (From a
result of a theorem.) Therefore Gpars is the core of all
the corresponding digraphs.
More complicated digraphs make people confused.
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Theorem 1
Let Acc be the accessibility list of an acyclic digraph.
Then there exists exactly one graph Gpars that has Acc
as its accessibility list and that has fewer edges than
any other graph G with Acc as its accessibility list.
Before starting the proof, we need to introduce some
terminology.
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Range and shortcut
Consider two nodes i and j of a digraph that are
connected by an edge e. The range r of the edge e is
the length of the shortest path between i and j in the
absence of e. If there is no other path connecting i
and j, then r : = .
An edge e with range r ≥ 2 but
shortcut.
is called a
Let’s see an example.
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e
i
j
r (e) = k + 1
zk
z1
zk-1
e is a shortcut. When eliminating e, i
and j are still connected by a path of
length k + 1, so r (e) = k + 1.
z2
zk-2
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Lemma 1
For any accessibility list Acc of a digraph, there exists
a compatible graph Gpars that is free of shortcuts.
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Proof of Lemma 1
Assume that there is no such graph Gpars.
ei
yi
xi
yi
xi
deleting ei
Pi
Pi
Length of Pi is greater than 1.
If there exists a shortcut ei between xi and yi , delete ei . Then
by the definition of shortcut, we’ll derive that xi and yi are still
connected via Pi , whose length is greater than 1.
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Suppose that we have n possible (xi , yi), i.e., (x1 ,
y1), …, (x1, xn). After repeating all possible (xi , yi), i
= 1, …, n, we’ll derive a shortcut-free graph
compatible with the accessibility list. This is a
contradiction to the assumption made in the
beginning of this proof.
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Lemma 2
Assume that Acc is the accessibility list of a digraph
G. For each node x, the adjacency list Adj(x) of a
shortcut-free graph Gpar compatible with Acc is a
subset of the adjacency list Adj(x) of any graph
compatible with Acc.
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Proof of Lemma 2
Assume that Lemma 2 is false.
W. L. O. G., suppose that a shortcut-free graph Gpars
and some other graph G induce Acc.
By assumption, Gpars contains at least one node x so
that Adj(x) of Gpars contains at least one node y that
isn’t in Adj(x) of G.
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Because G and Gpars have the same accessibility list
Acc, there must exist some path x → z1 → z2 → … →
zk → y from x to y in G. For the same reason, z1 is
accessible from x in Gpars, z2 from z1 in Gpars, … and
zk from zk-1 in Gpars.
Therefore we can find two paths (x →…→y) in Gpars:
(1) the edge e between x and y
(2) the path x → z1 →z2 →… →zk →y
This is in contradiction to the assumption that Gpars is
shortcut-free because e is a shortcut.
Let’s see an example!
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Acc:
x: z1 z2 y
z1 : z2 y
z2 : y
Adj(Gpars):
x
z1
x: z1 y
z1: z2
z2 : y
Adj(G):
x: z1 z2
z1: z2
z2 : y
x
z1
A shortcut!
z2
y
z2
y
Gpars
G
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Corollary 1
The shortcut-free graph Gpars compatible with Acc is a
unique graph with the fewest edges among all graphs
G compatible with Acc.
This corollary follows immediately from Lemma 2.
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Now, we can proceed to the algorithm.
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Outline
Introduction and basic definitions
Graph theoretical framework
Parsimonious network
Algorithm and complexity
Cycles in genetic networks
Conclusions
References
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1:
2:
for all nodes i of G
Adj(i) = Acc(i)
3:
4:
5:
6:
for all nodes i of G
if node i hasn’t been visited
call PRUNE_ACC(i)
end if
7:
8:
9:
10:
11:
12:
13:
PRUNE_ACC(i)
for all nodes j Acc(i)
if Acc(j) =
declare j as visited.
else
call PRUNE_ACC(j)
end if
14:
15:
16:
17:
18:
19:
20:
for all nodes j Acc(i)
for all nodes k Adj(j)
if k Acc(i)
delete k from Adj(i)
end if
declare node i as visited
end PRUNE_ACC(i)
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A recursive pruning algorithm
to reconstruct the most
parsimonious graph from an
accessibility list.
35
This algorithm is based on the following theorem, so
we have to get something from the theorem.
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Theorem 2
Let Acc(G) be the accessibility list of an acyclic
digraph, Gpars its most parsimonious graph, and V(Gpars)
the set of all nodes of Gpars. Then the following identity
holds:
In stead of proving the theorem, we give an example
later.
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0:
1:
2:
3:
4:
5:
1 2 3 4 5
2 3 4 5
3 4 5
0:
1:
2:
3:
4:
5:
5
1
2 3 4 5
3 4 5
0:
1:
2:
3:
4:
5:
5
1
2
3 4 5
5
Original Acc(G)
0
1 via 2, 3, 4, 5
0 via 1, 2, 3, 4, 5
1
1
1
2
4
0
0
3
5
2
2
4
3
5
4
3
5
A possible corresponding G
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0:
1:
2:
3:
4:
5:
0:
1:
2:
3:
4:
5:
1
2
3 4 5
5
2 via 3, 4, 5
1
5
1
2
3 4
5
0
4 via 5
1
1
2
2
5
0:
1:
2:
3:
4:
5:
0
0
4
1
2
3 4
3
4
2
4
3
5
3
5
The most parsimonious network
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Actually, the aforementioned example is an
illustration of our algorithm.
From this theorem, we can derive Corollary 2.
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Corollary 2
Let i, j and k be any three pairwise different nodes of
an acyclic directed shortcut-free graph G. If j is
accessible from i, then no node k accessible from j is
adjacent to i.
i
A shortcut !!
j
k
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Computational complexity
Let k < n − 1 be the average number of entries in a
node’s accessibility list.
Assume that there are n genes, that is, n entries.
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During execution, each node accessible from a node j
induces one recursive call of PRUNE_ACC, after which
the node accessed from j is declared as visited. Thus
each entry of the accessibility list of a node is explored
no more than once.
Line 15 of the algorithm loops over all nodes adjacent to
a node j. Let a denotes the average number of entries in
Adj(j).
The overall computational complexity would be O(nka).
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For practical matters, large scale experimental gene
perturbations in the yeast Saccharomyces cerevisiae
(n ≈ 6300) suggest that k < 50 ([HMJRS2000]), a ≤ 1
([W2001a]) and thus nka << n2.
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Storage complexity
The algorithm stores two copies of the accessibility
list, as well as a list of the nodes that has been visited.
Because the graph is acyclic, the recursion depth can
be no greater than n − 1.
Note that k < n − 1 is the average number of entries in
a node’s accessibility list.
The overall storage requirements are O(nk).
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Outline
Introduction and basic definitions
Graph theoretical framework
Parsimonious network
Algorithm and complexity
Cycles in genetic networks
Conclusions
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Dealing with cycles
All we have mentioned are restricted on acyclic graphs.
Now let us go to see the problems brought by cyclic
graphs.
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Problems that single gene perturbation
can’t solve
1
2
4
2
3
1
0
4
3
0:
1:
2:
3:
4:
0
1
0
0
0
0
2
2
1
1
1
3
3
3
2
2
4
4
4
4
3
They have the same accessibility list.
Therefore, we can not reconstruct the
gene network uniquely.
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1
2
4
2
3
1
0
4
3
0:
1:
2:
3:
4:
0
3
4
1
2
0
0:
1:
2:
3:
4:
1
2
3
4
0
Note that the order of direct regulatory interactions in these
two networks is different, as reflected in the adjacency lists.
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Instead of solving this problem, we collapse the
nodes which form a cycle into a single group of
nodes with indistinguishable order of regulatory
interactions.
Such a single group can be also called a strongly
connected component or strong component of a
directed graph G. Every two nodes in a strong
component are mutually accessible.
Let us see an example.
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1
3
5
2
4
15
7
9
14
6
0
8
10
12
13
11
A single group
This graph is called a
condensation of G.
1, 3, 4, 5, 15
2
7
6 , 9, 12
14
0
10
8
A single group
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13
51
How do we construct a condensation of a gene
network?
There are a theorem and a corollary before our
presenting the algorithm constructing a condensation
of a gene network.
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Theorem 3
Let P be the accessibility matrix of a digraph G with n
nodes, x1, …, xn. The strong component containing xi is
determined by the unit entries of ith row in the
matrix
.
xi
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Corollary 3
Let i and j (i ≠ j) be two nodes of a digraph G.
i and j are in the same component iff
and
We use corollary 3 because we will work with
accessibility lists, not matrices.
Now we are going to present the algorithm.
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1:
2:
3:
4:
5:
6:
7:
8:
9:
for all nodes i of G
if component [i] has not been defined
create new node x of G*
component [i] = x
for all nodes j Acc (i)
if i Acc (j)
component [j] = x
end if
end if
10: for all nodes i of G*
11:
12: for all nodes i of G
13:
for all nodes j Acc (i)
14:
if component [i] ≠ component [j]
15:
if component [j]
16:
add component [j] to
17:
end if
18:
end if
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1:
2:
3:
4:
5:
6:
7:
1
2
7
5
4
3
6
2
1
1
5
6
5
5
3
3
2
6
7
7
6
4 5 6 7
4 5 6 7
4 5 6 7
7
1
2
7
5
4
3
6
x1
x3
x2
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1
2
7
5
4
3
6
x1
1:
2:
3:
4:
5:
6:
7:
2
1
1
5
6
5
5
3
3
2
6
7
7
6
4 5 6 7
4 5 6 7
4 5 6 7
7
x3
x2
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Storage and time complexity
The graph G* has at most the same number of nodes
and accessibility list.
The algorithm generates only one copy of G* and its
accessibility list.
Therefore both time and storage complexity are O(k),
where k is the average number of entries of the
accessibility list. (k < n2)
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Outline
Introduction and basic definitions
Graph theoretical framework
Parsimonious network
Algorithm and complexity
Cycles in genetic networks
Conclusions
References
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Conclusions
Genetics is concerned with identifying the gene
interactions and their biological significance.
Function genomics takes this concern to the next
level, that is, identifying gene interactions among
thousands of genes in a genome.
There are other ways to simplify gene networks, such
as Boolean logic design, reduction in symbolic logic,
graph theory, and etc..
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by Systematic RNA Interference, Fraser, A. G., Kamath, R. S., Zipperlen, P.,
MartinezCampos, M. and Sohrmann, M., Nature, Vol. 408, pp. 325-330.
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61
[GEOCJ2000] Functional Genomic Analysis of Cell Division in C.
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Nature Genet., Vol. 22, 1999, pp. 281-285.
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[W2001b] The Yeast Protein Interaction Network Evolves Rapidly and
Contains Few Redundant Duplicate Genes, Wagner, A., Mol. Bio. Evol.,
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[WF2001] The Small World Inside Large Metabolic Networks, Wagner, A.
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