speaker: chuang chieh lin advisor: professor r. c. t. lee national chi-nan university

CSIE in National Chi-Nan University CSIE in National Chi-Nan University 11

How to Reconstruct a Large Genetic How to Reconstruct a Large Genetic Network from Network from nn Gene Perturbations in Gene Perturbations in

fewer than fewer than nn22 Easy Steps Easy Steps

Speaker: Chuang Chieh LinSpeaker: Chuang Chieh Lin

Advisor: Professor R. C. T. LeeAdvisor: Professor R. C. T. Lee

National Chi-Nan UniversityNational Chi-Nan University

Andreas Wagner, Bioinformatics, vol. 17, No. 12, 2001, pp. 1183-1187.

22CSIE in National Chi-Nan University CSIE in National Chi-Nan University

OutlineOutline

Introduction and basic definitionsIntroduction and basic definitions

Graph theoretical frameworkGraph theoretical framework

Parsimonious networkParsimonious network

Algorithm and complexityAlgorithm and complexity

Cycles in genetic networksCycles in genetic networks

ConclusionsConclusions

ReferencesReferences









OutlineOutline



Gene activityGene activity includes whether a gene is expressed or includes whether a gene is expressed or not, as mRNA, as protein etc..not, as mRNA, as protein etc..

Gene networkGene network: In this paper, we define a genetic : In this paper, we define a genetic network as a group of genes in which individual gene network as a group of genes in which individual gene can influence the activity of other genes.can influence the activity of other genes.

The core task of reconstructing genetic networks is to The core task of reconstructing genetic networks is to identify the causal structure of a gene network.identify the causal structure of a gene network.


To reconstruct a genetic networkTo reconstruct a genetic network is to identify, for is to identify, for each network gene, which other genes and their each network gene, which other genes and their activity the gene influences directly.activity the gene influences directly.

Now, let’s see an illustration of genetic network.Now, let’s see an illustration of genetic network.


P

P

DNA Gene 1 Gene 2 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.

transcription factor

protein kinase

protein phosphatase


protein

active

inactive inactive

active


Genetic perturbationGenetic perturbation: an experimental manipulation : an experimental manipulation of gene activity by manipulating either a gene itself of gene activity by manipulating either a gene itself or its product. It includes point mutations, gene or its product. It includes point mutations, gene deletions, or other interference with the activity of the deletions, or other interference with the activity of the product.product.


P

P

DNA Gene 1 Gene 2 Gene 3 Gene 4 Gene 5


protein kinase

protein phosphatase


protein

active

inactive inactive

active

Genetic perturbation: gene deletion Genetic perturbation: gene deletionAspect of gene activity: mRNA expression Aspect of gene activity: phosphorlation state

G1: G2, G5 G1: G3, G4G2: G5 G2: G3, G4G3: G5 G3: G4G4: G5 G4:G5: G5:









OutlineOutline



As the previous instance indicated, we are As the previous instance indicated, we are concerned with qualitative information on gene concerned with qualitative information on gene interaction.interaction.

We consider a “We consider a “digraphdigraph”, a graph representation of ”, a graph representation of genetic networks, to this qualitative information.genetic networks, to this qualitative information.

A digraph is a directed graph consisting of nodes A digraph is a directed graph consisting of nodes and directed edges.and directed edges.

Let’s see an example.Let’s see an example.


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.

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Adjacency listAdjacency list: for each gene : for each gene ii, it simply shows which , it simply shows which genes’ activity state the gene genes’ activity state the gene ii influences directly. influences directly.

We denote We denote AdjAdj ((GG) to be the adjacency list of graph ) to be the adjacency list of graph GG and and AdjAdj ((ii) to be the set of nodes (genes) adjacent to ) to be the set of nodes (genes) adjacent to (directly influenced by) node (directly influenced by) node ii..


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0: 161:2:3: 2 5 84:5: 126: 5 127: 2 178:9: 10 1510: 1 2011: 2012: 1413: 8 1714: 015: 016: 217: 818:19: 820: 6 18

Adjacency list of G:

G


Accessibility listAccessibility list: the list of perturbation effects or the : the list of perturbation effects or the list of regulatory effects. It shows all nodes (genes) list of regulatory effects. It shows all nodes (genes) that can be accessed (influenced in their activity state) that can be accessed (influenced in their activity state) from a given gene by paths of direct interactions.from a given gene by paths of direct interactions.

We denote We denote AccAcc ((GG) to be the accessibility list of the ) to be the accessibility list of the graph graph GG and and AccAcc ((ii) to be the set of nodes that can be ) to be the set of nodes that can be reached (influenced) from node (gene) reached (influenced) from node (gene) ii..


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0: 2 161:2:3: 0 2 5 8 12 14 164:5: 0 2 12 14 166: 0 2 5 12 14 167: 2 8 178:9: 0 1 2 5 6 10 12 14 15 16 18 2010: 0 1 2 5 6 12 14 16 18 2011: 0 2 5 6 12 14 16 18 2012: 0 2 14 1613: 8 1714: 0 2 1615: 0 2 1616: 217: 818:19: 820: 0 2 5 6 12 14 16 18

Accessibility list of G:

G









OutlineOutline


Before proceeding with the algorithm, we have to Before proceeding with the algorithm, we have to give some concepts and theorems first.give some concepts and theorems first.


The most parsimonious networkThe most parsimonious network

An acyclic digraph defines its accessibility list, but an An acyclic digraph defines its accessibility list, but an accessibility list may have more than one accessibility list may have more than one corresponding acyclic digraph.corresponding acyclic digraph.

Let’s see an example first.Let’s see an example first.


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(a) (b)

(c) (d)

(d) is the most parsimonious network of Acc, i.e., (a).


An accessibility list An accessibility list AccAcc and a digraph and a digraph GG are are compatiblecompatible if if GG has has AccAcc as its accessibility list. as its accessibility list. AccAcc is the accessibility list is the accessibility list inducedinduced by by GG..

GGparspars is called is called the most parsimonious networkthe most parsimonious network compatible with compatible with AccAcc..


We prefer simplest or most parsimonious one of gene We prefer simplest or most parsimonious one of gene network.network.

For any accessibility list For any accessibility list AccAcc of a digraph of a digraph GG, there , there exists a most parsimonious network exists a most parsimonious network GGparspars. (From a . (From a result of a theorem.) Therefore result of a theorem.) Therefore GGparspars is the core of all is the core of all the corresponding digraphs.the corresponding digraphs.

More complicated digraphs make people confused.More complicated digraphs make people confused.

Why we prefer the most parsimonious Why we prefer the most parsimonious network?network?


Theorem 1Theorem 1

Let Let AccAcc be the accessibility list of an acyclic digraph. be the accessibility list of an acyclic digraph. Then there exists exactly one graph Then there exists exactly one graph GGparspars that has that has AccAcc as its accessibility list and that has fewer edges than as its accessibility list and that has fewer edges than any other graph any other graph GG with Acc as its accessibility list. with Acc as its accessibility list.

Before starting the proof, we need to introduce some Before starting the proof, we need to introduce some terminology.terminology.


Range and shortcutRange and shortcut

Consider two nodes Consider two nodes ii and and jj of a digraph that are of a digraph that are connected by an edge connected by an edge ee. The . The rangerange rr of the edge of the edge ee is is the length of the shortest path between the length of the shortest path between ii and and jj in the in the absence of absence of ee. If there is no other path connecting . If there is no other path connecting ii and and jj, then , then rr : = . : = .

An edge An edge ee with range with range rr ≥ 2 but is called a ≥ 2 but is called a shortcutshortcut..

Let’s see an example.Let’s see an example.


ij

zk

zk-1

zk-2

z2

z1

e

r (e) = k + 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.


Lemma 1Lemma 1

For any accessibility list For any accessibility list AccAcc of a digraph, there exists of a digraph, there exists a compatible graph a compatible graph GGparspars that is free of shortcuts. that is free of shortcuts.


Proof of Lemma 1Proof of Lemma 1

Assume that there is no such graph Assume that there is no such graph GGparspars..

xi

yi

ei

Pi

Length of Pi is greater than 1.

xi

yi

Pi

deleting ei

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.


Suppose that we have Suppose that we have nn possible ( possible (xxi i , , yyii), i.e., (), i.e., (xx11 , , yy11), ),

…, (…, (xx11, , xxnn). After repeating all possible (). After repeating all possible (xxi i , , yyii), ), ii = 1, = 1,

…, …, nn, we’ll derive a shortcut-free graph compatible , we’ll derive a shortcut-free graph compatible with the accessibility list. This is a contradiction to with the accessibility list. This is a contradiction to the assumption made in the beginning of this proof.the assumption made in the beginning of this proof.


Lemma 2Lemma 2

Assume that Assume that AccAcc is the accessibility list of a digraph is the accessibility list of a digraph GG. For each node . For each node xx, the adjacency list , the adjacency list AdjAdj ((xx) of a ) of a shortcut-free graph shortcut-free graph GGparpar compatible with compatible with AccAcc is a is a subset of the adjacency list subset of the adjacency list AdjAdj ((xx) of any graph ) of any graph compatible with compatible with AccAcc..


Assume that Lemma 2 is false.Assume that Lemma 2 is false.

W. L. O. G., suppose that a shortcut-free graph W. L. O. G., suppose that a shortcut-free graph GGparspars and some other graph and some other graph GG induce induce AccAcc..

By assumption, By assumption, GGparspars contains at least one node contains at least one node xx so so that that AdjAdj((xx) of ) of GGpars pars contains at least one node contains at least one node yy that that isn’t in isn’t in AdjAdj((xx) of ) of GG..

Proof of Lemma 2Proof of Lemma 2


Because Because GG and and GGparspars have the same accessibility list have the same accessibility list AccAcc, there must exist some path , there must exist some path x x → → zz11 → → zz22 → … → → … → zzkk → → y y from from xx to to yy in in GG. For the same reason, . For the same reason, zz11 is is accessible from accessible from xx in in GGparspars, , zz22 from from zz11 in in GGparspars, … and , … and zzkk from from zzkk-1-1 in in GGparspars..

Therefore we can find two paths (Therefore we can find two paths (xx →…→ →…→yy) in ) in GGparspars::(1) the edge (1) the edge ee between between xx and and yy

(2) the path (2) the path xx → → zz1 1 →→zz2 2 →… →→… →zzkk → →yy

This is in contradiction to the assumption that This is in contradiction to the assumption that GGparspars is is shortcut-free because shortcut-free because ee is a shortcut. is a shortcut.

Let’s see an example!


x: z1 z2 yz1: z2 yz2: y

Acc: Adj(Gpars)

:

x: z1 y z1: z2

z2: yAdj(G):

x: z1 z2

z1: z2

z2: y

x

y

z2

z1

G

x

y

z2

z1

Gpars

A shortcut!


Corollary 1Corollary 1

The shortcut-free graph The shortcut-free graph GGparspars compatible with compatible with AccAcc is a is a unique graph with the fewest edges among all graphs unique graph with the fewest edges among all graphs GG compatible with compatible with AccAcc..

This corollary follows immediately from Lemma 2.This corollary follows immediately from Lemma 2.


Now, we can proceed to the algorithm.Now, we can proceed to the algorithm.









OutlineOutline


1:1: for all nodes for all nodes ii of of GG2:2: AdjAdj((ii) = ) = AccAcc((ii))

3:3: for all nodes for all nodes ii of of GG4:4: if node if node ii hasn’t been visited hasn’t been visited5:5: call PRUNE_ACC(call PRUNE_ACC(ii))6:6: end ifend if

7:7: PRUNE_ACC(PRUNE_ACC(ii))8:8: for all nodes for all nodes j Accj Acc((ii))9:9: if if AccAcc((jj) =) =10:10: declare declare jj as visited. as visited.11:11: elseelse12:12: call PRUNE_ACC(call PRUNE_ACC(jj))13:13: end ifend if

14:14: for all nodes for all nodes jj AccAcc((ii))15:15: for all nodes for all nodes k k AdjAdj((jj))16:16: if if k k AccAcc((ii))17:17: delete delete kk from from AdjAdj((ii))18:18: end ifend if19:19: declare node declare node ii as visited as visited20:20: end PRUNE_ACC(end PRUNE_ACC(ii))

A recursive pruning algorithm to reconstruct the most parsimonious graph from an accessibility list.


This algorithm is based on the following theorem, so This algorithm is based on the following theorem, so we have to get something from the theorem.we have to get something from the theorem.


Theorem 2Theorem 2

Let Let AccAcc ((GG) be the accessibility list of an acyclic ) be the accessibility list of an acyclic digraph, digraph, GGparspars its most parsimonious graph, and its most parsimonious graph, and VV ((GGparspars) ) the set of all nodes of the set of all nodes of GGparspars. Then the following identity . Then the following identity holds:holds:

In stead of proving the theorem, we give an example In stead of proving the theorem, we give an example later.later.


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Original Acc(G)

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A possible corresponding G

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0 via 1, 2, 3, 4, 5 1 via 2, 3, 4, 50

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The most parsimonious network


Actually, the aforementioned example is an Actually, the aforementioned example is an illustration of our algorithm.illustration of our algorithm.

From this theorem, we can derive Corollary 2.From this theorem, we can derive Corollary 2.



Let Let ii, , jj and and kk be any three pairwise different nodes of be any three pairwise different nodes of an acyclic directed shortcut-free graph an acyclic directed shortcut-free graph GG. If . If jj is is accessible from accessible from ii, then no node , then no node kk accessible from accessible from j j is is adjacent to adjacent to ii..

i

j

k

A shortcut !!


Computational complexityComputational complexity

Let Let kk < < nn − − 1 be the average number of entries in a 1 be the average number of entries in a node’s accessibility list.node’s accessibility list.

Assume that there are Assume that there are nn genes, that is, genes, that is, nn entries. entries.


During execution, each node accessible from a node During execution, each node accessible from a node jj induces one recursive call of PRUNE_ACC, after which induces one recursive call of PRUNE_ACC, after which the node accessed from the node accessed from jj is declared as visited. Thus is declared as visited. Thus each entry of the accessibility list of a node is explored each entry of the accessibility list of a node is explored no more than once.no more than once.

Line 15 Line 15 of the algorithm loops over all nodes adjacent to of the algorithm loops over all nodes adjacent to a node a node jj. Let . Let aa denotes the average number of entries in denotes the average number of entries in AdjAdj ((jj).).

The overall computational complexity would be The overall computational complexity would be OO ((nkanka).).


For practical matters, large scale experimental gene For practical matters, large scale experimental gene perturbations in the yeast perturbations in the yeast Saccharomyces cerevisiaeSaccharomyces cerevisiae ((nn ≈ ≈ 6300) suggest that 6300) suggest that kk < 50 ([HMJRS2000]), < 50 ([HMJRS2000]), aa ≤ 1 ≤ 1 ([W2001a]) and thus ([W2001a]) and thus nkanka << << nn22..


Storage complexityStorage complexity

The algorithm stores two copies of the accessibility The algorithm stores two copies of the accessibility list, as well as a list of the nodes that has been visited.list, as well as a list of the nodes that has been visited.

Because the graph is acyclic, the recursion depth can Because the graph is acyclic, the recursion depth can be no greater than be no greater than n n − − 1.1.

Note that Note that kk < < nn − − 1 is the average number of entries in 1 is the average number of entries in a node’s accessibility list.a node’s accessibility list.

The overall storage requirements are The overall storage requirements are OO ((nknk).).








OutlineOutline


Dealing with cyclesDealing with cycles

All we have mentioned are restricted on acyclic graphs.All we have mentioned are restricted on acyclic graphs.

Now let us go to see the problems brought by cyclic Now let us go to see the problems brought by cyclic graphs.graphs.


Problems that single gene perturbation Problems that single gene perturbation can’t solvecan’t solve

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They have the same accessibility list. Therefore, we can not reconstruct the gene network uniquely.


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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 Instead of solving this problem, we collapse the nodes which form a cycle into which form a cycle into a single groupa single group of nodes with of nodes with indistinguishable order of regulatory interactions.indistinguishable order of regulatory interactions.

Such a single group can be also called a Such a single group can be also called a strongly strongly connected componentconnected component or or strong component strong component of a of a directed graph directed graph GG. Every two nodes in a strong . Every two nodes in a strong component are mutually accessible.component are mutually accessible.

Let us see an example.Let us see an example.


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A single group

A single group

This graph is called a condensation of G.


How do we construct a condensation of a gene How do we construct a condensation of a gene network?network?

There are a theorem and a corollary before our There are a theorem and a corollary before our presenting the algorithm constructing a condensation presenting the algorithm constructing a condensation of a gene network.of a gene network.


Theorem 3Theorem 3

Let Let PP be the accessibility matrix of a digraph be the accessibility matrix of a digraph GG with n with n nodes, nodes, xx11, …, , …, xxnn. The strong component containing . The strong component containing xxii is is

determined by the unit entries of determined by the unit entries of iith row in the matrix th row in the matrix . .

xi



Let Let ii and and jj ( (ii ≠≠ jj) be two nodes of a digraph ) be two nodes of a digraph GG. . ii and and jj are in the same component iff are in the same component iff and and

We use corollary 3 because we will work with accessibility lists, not matrices.

Now we are going to present the algorithm.


1:1: for all nodes for all nodes ii of of GG2:2: if if componentcomponent [[ii] has not been defined] has not been defined3:3: create new node create new node xx of of GG**

4:4: componentcomponent [[ii] = ] = xx5:5: for all nodes for all nodes jj AccAcc ((ii))6:6: if if i i AccAcc ((jj))7:7: componentcomponent [[jj] = ] = xx8:8: end ifend if9:9: end ifend if

10:10: for all nodes for all nodes ii of of GG**

11:11:12:12: for all nodes for all nodes ii of of GG13:13: for all nodes for all nodes jj AccAcc ((ii))14:14: if if componentcomponent [ [ii] ≠ ] ≠ componentcomponent [[jj]]15:15: if if componentcomponent [[jj] ] 16:16: add add componentcomponent [[jj] to ] to 17:17: end ifend if18:18: end ifend if


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Storage and time complexityStorage and time complexity

The graph The graph GG** has at most the same number of nodes has at most the same number of nodes and accessibility list.and accessibility list.

The algorithm generates only one copy of The algorithm generates only one copy of GG** and its and its accessibility list.accessibility list.

Therefore both time and storage complexity are Therefore both time and storage complexity are OO ((kk), ), where where kk is the average number of entries of the is the average number of entries of the accessibility list. (accessibility list. (kk < < nn22))









OutlineOutline



Genetics is concerned with identifying the gene Genetics is concerned with identifying the gene interactions and their biological significance.interactions and their biological significance.

Function genomics takes this concern to the next Function genomics takes this concern to the next level, that is, identifying gene interactions among level, that is, identifying gene interactions among thousands of genes in a genome.thousands of genes in a genome.

There are other ways to simplify gene networks, such There are other ways to simplify gene networks, such as Boolean logic design, reduction in symbolic logic, as Boolean logic design, reduction in symbolic logic, graph theory, and etc..graph theory, and etc..



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speaker: chuang chieh lin advisor: professor r. c. t. lee national chi-nan university

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