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Analysing Kauffman Boolean Networks Analysing Kauffman Boolean Networks PAVEL EMELYANOV Institute of Informatics Systems and Novosibirsk State University May 31, 2016 Analysing Kauffman Boolean Networks Plan 1 Introduction 2 Problem

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Analysing Kauffman Boolean Networks

Analysing Kauffman Boolean Networks

PAVEL EMELYANOV

†Institute of Informatics Systems and Novosibirsk State University

May 31, 2016

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Analysing Kauffman Boolean Networks Plan

1 Introduction 2 Problem Definition 3 Method Development 4 Method Evaluation

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Analysing Kauffman Boolean Networks Introduction

Kauffman’s Boolean Networks

Boolean networks (BNs) originated in

  • S. A. Kauffman. Homeostasis and differentiation in random

genetic control networks. Nature, 224(215):177–178, 1969.

  • S. A. Kauffman. Metabolic stability and epigenesis in

randomly constructed genetic nets. J. Theor. Biol., 22:437–467, 1969. are well-known in the scope of modeling complex systems of different kinds: regulatory networks, cell differentiation, evolution, immune response, neural networks, social networks, interactions

  • ver WWW.
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Analysing Kauffman Boolean Networks Introduction

Boolean Network for Flowers of Arabidopsis Thaliana

Y.-E. Sanchez-Corrales, E.R. Alvarez-Buylla, L. Mendoza. The Arabidopsis Thaliana flower organ specification gene regulatory network determines a robust differentiation

  • process. J. Theor. Biol., 264:971-983, 2010.
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Analysing Kauffman Boolean Networks Introduction

Examples of Differentiation for Arabidopsis Thaliana

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Analysing Kauffman Boolean Networks Introduction

Examples of Differentiation for Stem Cells

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Analysing Kauffman Boolean Networks Introduction

Fixpoints of Differentiation for Stem Cells

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Analysing Kauffman Boolean Networks Introduction

One Examples of Differentiation for Stem Cells

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Analysing Kauffman Boolean Networks Problem Definition

Problems of Interest

Given a boolean map F : {0, 1}n → {0, 1}n. We are interested in: finding fixpoints of the map (also called singleton attractors),

  • i. e. finding points x ∈ {0, 1}n such that x = F(x);
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Analysing Kauffman Boolean Networks Problem Definition

Problems of Interest

Given a boolean map F : {0, 1}n → {0, 1}n. We are interested in: finding fixpoints of the map (also called singleton attractors),

  • i. e. finding points x ∈ {0, 1}n such that x = F(x);

finding k-cycles of the map (also called cyclic attractors), i. e. finding points x ∈ {0, 1}n such that x = F k(x) and k > 1

  • beying this identity is minimal, where

F k+1 = F ∘ F k, F 1 = F;

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Analysing Kauffman Boolean Networks Problem Definition

Problems of Interest

Given a boolean map F : {0, 1}n → {0, 1}n. We are interested in: finding fixpoints of the map (also called singleton attractors),

  • i. e. finding points x ∈ {0, 1}n such that x = F(x);

finding k-cycles of the map (also called cyclic attractors), i. e. finding points x ∈ {0, 1}n such that x = F k(x) and k > 1

  • beying this identity is minimal, where

F k+1 = F ∘ F k, F 1 = F; finding basins of attractors, both singleton and cyclic, i. e. finding sets of points x ∈ {0, 1}n such that after some number

  • f iterations of the map F it falls into the corresponding

attractor.

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Analysing Kauffman Boolean Networks Problem Definition

Boolean Map as Functional Network

       x1 = x3 ∨ x4 x2 = x2 x3 = x3 ⊕ (x1 · x2) x4 = x3

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Analysing Kauffman Boolean Networks Problem Definition

Problem Complexity

The NP-hardness of BNs fixpoint problem was independently established in

  • T. Akutsu, S. Kuhara, O. Maruyama, and S. Miyano. A

system for identifying genetic networks from gene expression patterns produced by gene disruptions and overxpressions. Genome Informatics, 9:151–160, 1998.

  • M. Milano and A. Roli. Solving the satisfiability problem

through boolean networks. In Proceedings of the 6th Congress

  • f the Italian Association for Artificial Intelligence on

Advances in Artificial Intelligence, volume 1792 of Lecture Notes in Artificial Intelligence, pages 72–83. Springer-Verlag, 1999.

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Analysing Kauffman Boolean Networks Problem Definition

Solving BNs Fixpoint Problems

To solve the fixpoint problem different techniques were developed (non-exhaustive reference list): boolean formulae satisfiability with SAT-solvers [Dubrova, Teslenko, 2011]; abstract interpretation of dynamical systems [Paulev´ e, Magnin, Roux, 2012]; Petri nets modeling [Steggles, Banks, Shaw, Wipat, 2007]; matrix algebras computations [Cheng, Qi, Zhao, 2012]; graph-theoretical decompositions [Zhang, Hayashida, Akutsu, Ching, Ng, 2007; Soranzo, Iacono, Ramezani, Altafini, 2012].

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Analysing Kauffman Boolean Networks Problem Definition

What We Do

Our approach consists of: decomposition of an original network into smaller networks (this talk); solving Fixpoint Problem for each small network; reconstruction solution for the entire network from “small” sub-solutions.

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Analysing Kauffman Boolean Networks Method Development

Idea: Acyclic Case

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Analysing Kauffman Boolean Networks Method Development

Idea: Acyclic Case

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Analysing Kauffman Boolean Networks Method Development

Idea: Acyclic Case

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Analysing Kauffman Boolean Networks Method Development

Idea: Acyclic Case

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Analysing Kauffman Boolean Networks Method Development

Idea: Acyclic Case

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Analysing Kauffman Boolean Networks Method Development

Idea: Add One Feedback Arc

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Analysing Kauffman Boolean Networks Method Development

Idea: Add One Feedback Arc

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Analysing Kauffman Boolean Networks Method Development

Idea: Add One Feedback Arc

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Analysing Kauffman Boolean Networks Method Development

Feedback (Arc) Region

For each feedback arc ST let us consider next vertices belonging to the graph G without all feedback arcs: upper cone of the arc end Con+(T) is a set of all vertices being reachable from the end of the feedback arc; lower cone of the arc start Con−(S) is a set of all vertices reaching the start of the feedback arc. Reg(ST) = Con+(T) ∩ Con−(S).

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Analysing Kauffman Boolean Networks Method Development

Feedback Region

T S Con+(T) Con-(S) Reg(ST)

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Analysing Kauffman Boolean Networks Method Development

Big Region

Con+(T) S T Con-(S) Q R

Big Region includes all vertices of feedback upper and lower cones which are disjoint.

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Analysing Kauffman Boolean Networks Method Development

Region Interaction: Simple Influence

T R Q S

One region is contained within a zone of influence (upper cone) of another region: R ∈ Con+(T) ∨ T ∈ Con+(R)

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Analysing Kauffman Boolean Networks Method Development

Region Interaction: Tangled

T R Q S

One region is partially influenced by another (only part of a region lies within upper cone

  • f another):

R ̸∈ Con+(T) ∧ T ̸∈ Con+(R) ∧ (Q ∈ Con+(T) ∨ S ∈ Con+(R))

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Analysing Kauffman Boolean Networks Method Development

Region Interaction: Disjoint

S T Q R

Regions are disjoint and do not interact: Con−(S) ∩ Con+(R) = ∅ ∧ Con+(T) ∩ Con−(Q) = ∅

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Analysing Kauffman Boolean Networks Method Development

Feedback Arc Set Problem

Given a directed graph G = (V , A).

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Analysing Kauffman Boolean Networks Method Development

Feedback Arc Set Problem

Given a directed graph G = (V , A). We need to find its maximum acyclic spanning subgraph G1 = (V , A0).

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Analysing Kauffman Boolean Networks Method Development

Feedback Arc Set Problem

Given a directed graph G = (V , A). We need to find its maximum acyclic spanning subgraph G1 = (V , A0). Arcs excluded from this subgraph (A1 = A ∖ A0) are called feedback arcs. This gives a name for the complementary problem: finding a minimum feedback arc set (MinFAS). It is not unique.

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Analysing Kauffman Boolean Networks Method Development

Feedback Arc Set Problem

Given a directed graph G = (V , A). We need to find its maximum acyclic spanning subgraph G1 = (V , A0). Arcs excluded from this subgraph (A1 = A ∖ A0) are called feedback arcs. This gives a name for the complementary problem: finding a minimum feedback arc set (MinFAS). It is not unique. In general this problem is hard to solve. Therefore we need an efficient algorithm finding a correct (upper) approximation of FAS.

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Analysing Kauffman Boolean Networks Method Development

FAS Algorithmics

  • R. M. Karp. Reducibility among combinatorial problems. In
  • R. Miller and J. Thatcher, editors, Complexity of Computer

Computations, pages 85–103. Plenum Press, New York, 1972.

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Analysing Kauffman Boolean Networks Method Development

FAS Algorithmics

  • R. M. Karp. Reducibility among combinatorial problems. In
  • R. Miller and J. Thatcher, editors, Complexity of Computer

Computations, pages 85–103. Plenum Press, New York, 1972.

  • B. Berger and P. W. Shor. Approximation algorithms for the

maximum acyclic subgraph problem. In Proceedings of First ACM–SIAM Symposium on Discrete Algorithms, pages 236–243. ACM Press, 1990.

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Analysing Kauffman Boolean Networks Method Development

FAS Algorithmics

  • R. M. Karp. Reducibility among combinatorial problems. In
  • R. Miller and J. Thatcher, editors, Complexity of Computer

Computations, pages 85–103. Plenum Press, New York, 1972.

  • B. Berger and P. W. Shor. Approximation algorithms for the

maximum acyclic subgraph problem. In Proceedings of First ACM–SIAM Symposium on Discrete Algorithms, pages 236–243. ACM Press, 1990.

  • P. Eades, X. Lin, and W. F. Smyth. A fast and effective

heuristic for the feedback arc set problem. Information Processing Letters, 47(6):319–323, 1993.

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Analysing Kauffman Boolean Networks Method Development

FAS Algorithmics

  • R. M. Karp. Reducibility among combinatorial problems. In
  • R. Miller and J. Thatcher, editors, Complexity of Computer

Computations, pages 85–103. Plenum Press, New York, 1972.

  • B. Berger and P. W. Shor. Approximation algorithms for the

maximum acyclic subgraph problem. In Proceedings of First ACM–SIAM Symposium on Discrete Algorithms, pages 236–243. ACM Press, 1990.

  • P. Eades, X. Lin, and W. F. Smyth. A fast and effective

heuristic for the feedback arc set problem. Information Processing Letters, 47(6):319–323, 1993.

  • G. Even, J. Noar, B. Schieber, and M. Sudan. Approximating

minimum feedback sets and multicuts in directed graphs. Algorithmica, 20:151–174, 1998.

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Analysing Kauffman Boolean Networks Method Development

Algorithm Characteristics

It can solve weighted and unweighted problems. At the moment, its unweighted version is used. The weighted version is investigated in the scope how to take into account information about network properties. It consists of relatively simple matrix manipulations being highly parallelizable. If n is number of graph vertices, then overall complexity of this method is O(𝜗−2n2M(n)log2 n), where 𝜗 is a parameter relating to approximation quality, M(n) – complexity of an n × n matrices multiplication. It is widely known to be one of the best approximations what is important for successful use of our method.

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Analysing Kauffman Boolean Networks Method Development

Network Example: 19 vertices

19 39 11 29 26 6 27 24 28 17 25 20 24 9 17 18 14 13 12 12 15 8 2 47 26 11 4 1 43 3 5 4 28 5 6 10 23 19 22 21 16 20 7 44 31 33 15 30 37 35 36 48 40 41 18 38

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Analysing Kauffman Boolean Networks Method Development

Network Example: 19 vertices, 𝜗 = 1.0

19 39 11 29 26 6 27 24 28 17 25 20 24 9 17 18 14 13 12 12 15 8 2 47 26 11 4 1 43 3 5 4 28 5 6 10 23 19 22 21 16 20 7 44 31 33 15 30 37 35 36 48 40 41 18 38

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Analysing Kauffman Boolean Networks Method Development

Network Example: 19 vertices, 𝜗 = 0.01

19 39 11 29 26 6 27 24 28 17 25 20 24 9 17 18 14 13 12 12 15 8 2 47 26 11 4 1 43 3 5 4 28 5 6 10 23 19 22 21 16 20 7 44 31 33 15 30 37 35 36 48 40 41 18 38

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Analysing Kauffman Boolean Networks Method Development

Network Example: 100 vertices

57 115 86 114 17 1 15 2 40 3 37 4 23 5 27 81 38 6 13 36 7 93 11 7 26 56 34 42 42 15 3 27 14 53 16 98 20 20 6 21 36 62 22 23 19 33 5 26 8 28 21 32 12 43 28 35 41 38 33 88 39 83 45 29 46 18 48 2 96 9 52 24 80 35 58 32 77 10 64 25 67 31 69 4 71 30 97 1 8 51 9 49 76 52 82 55 53 74 50 70 56 54 43 66 125 126 44 89 45 55 46 141 47 91 153 48 18 67 31 57 60 11 118 60 87 12 147 22 61 113 68 13 128 47 62 99 63 122 64 80 81 85 16 145 65 30 84 17 110 66 39 96 99 19 159 68 69 41 71 79 70 158 146 112 94 154 72 136 59 73 120 121 34 89 73 74 51 108 75 25 142 78 84 77 61 119 88 79 10 58 95 59 82 37 29 92 93 94 83 85 86 78 138 149 87 100 50 107 90 95 40 148 75 97 91 151 98 143 102 44 101 103 104 105 156 24 76 72 132 109 90 150 140 116 117 134 49 106 14 63 137 123 127 155 129 130 131 133 144 135 139 54 111 92 152 65 124 157
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Analysing Kauffman Boolean Networks Method Development

Network Example: 100 vertices, 𝜗 = 1.0

57 115 86 114 17 1 15 2 40 3 37 4 23 5 27 81 38 6 13 36 7 93 11 7 26 56 34 42 42 15 3 27 14 53 16 98 20 20 6 21 36 62 22 23 19 33 5 26 8 28 21 32 12 43 28 35 41 38 33 88 39 83 45 29 46 18 48 2 96 9 52 24 80 35 58 32 77 10 64 25 67 31 69 4 71 30 97 1 8 51 9 49 76 52 82 55 53 74 50 70 56 54 43 66 125 126 44 89 45 55 46 141 47 91 153 48 18 67 31 57 60 11 118 60 87 12 147 22 61 113 68 13 128 47 62 99 63 122 64 80 81 85 16 145 65 30 84 17 110 66 39 96 99 19 159 68 69 41 71 79 70 158 146 112 94 154 72 136 59 73 120 121 34 89 73 74 51 108 75 25 142 78 84 77 61 119 88 79 10 58 95 59 82 37 29 92 93 94 83 85 86 78 138 149 87 100 50 107 90 95 40 148 75 97 91 151 98 143 102 44 101 103 104 105 156 24 76 72 132 109 90 150 140 116 117 134 49 106 14 63 137 123 127 155 129 130 131 133 144 135 139 54 111 92 152 65 124 157
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Analysing Kauffman Boolean Networks Method Development

Network Example: 100 vertices, 𝜗 = 0.01

57 115 86 114 17 1 15 2 40 3 37 4 23 5 27 81 38 6 13 36 7 93 11 7 26 56 34 42 42 15 3 27 14 53 16 98 20 20 6 21 36 62 22 23 19 33 5 26 8 28 21 32 12 43 28 35 41 38 33 88 39 83 45 29 46 18 48 2 96 9 52 24 80 35 58 32 77 10 64 25 67 31 69 4 71 30 97 1 8 51 9 49 76 52 82 55 53 74 50 70 56 54 43 66 125 126 44 89 45 55 46 141 47 91 153 48 18 67 31 57 60 11 118 60 87 12 147 22 61 113 68 13 128 47 62 99 63 122 64 80 81 85 16 145 65 30 84 17 110 66 39 96 99 19 159 68 69 41 71 79 70 158 146 112 94 154 72 136 59 73 120 121 34 89 73 74 51 108 75 25 142 78 84 77 61 119 88 79 10 58 95 59 82 37 29 92 93 94 83 85 86 78 138 149 87 100 50 107 90 95 40 148 75 97 91 151 98 143 102 44 101 103 104 105 156 24 76 72 132 109 90 150 140 116 117 134 49 106 14 63 137 123 127 155 129 130 131 133 144 135 139 54 111 92 152 65 124 157
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Analysing Kauffman Boolean Networks Method Evaluation

Network Statistics Examples

𝜗 #Vert’s #Arcs #Reg’s Big Reg. #Vert’s per Reg. 0.01 30 45 2 no 4 / 5.50 / 7 0.5 30 45 4 yes 4 / 11.5 / 20 0.01 30 54 4 no 2 / 3.00 / 5 0.5 30 54 6 yes 2 / 7.50 / 25 0.01 100 160 4 no 3 / 5.00 / 7 0.5 100 160 6 yes 2 / 8.67 / 20

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Analysing Kauffman Boolean Networks Method Evaluation

Network Statistics Examples: 100 Vertices

𝜗 #Arcs #Reg’s Big Reg. #Vert’s per Reg. 0.01 144 5 no 3 / 8.40 / 20 0.5 144 7 yes 3 / 16.71 / 56 1 144 7 yes 3 / 16.71 / 56 0.01 153 7 no 2 / 3.86 / 9 0.5 153 10 yes 2 / 12.8 / 41 1 153 13 yes 2 / 10.54 / 28 0.01 179 9 no 3 / 8.78 / 33 0.5 179 12 yes 2 / 10.17 / 51 1 179 13 yes 3 / 10.85 / 32 0.01 185 8 no 2 / 10.50 / 21 0.5 185 10 yes 2 / 10.50 / 23 1 185 11 yes 2 / 12.36 / 23