Attractor Detectionand

Control of Boolean Networks

Tatsuya Akutsu

Bioinformatics Center

Institute for Chemical Research

Kyoto University

Contents Boolean Network

Attractor Detection Definition and Algorithms

Control of Boolean Network Definition and DP algorithm

Integer Programming-based Approach PBN and its Control Conclusion

Acknowledgment Tamura Takeyuki, Morihiro Hayashida

[Kyoto U.] Masaki Yamamoto [Kwansei Gakuin U.] Wai-Ki Ching, Shuqin Zhang, Xi Chen [U.

Hong Kong] Michael Ng [Hong Kong Baptist U.] Avraham A. Melkman [Ben-Gurion

University of the Negev]

Boolean Network

Boolean Network Mathematical model of genetic networks node⇔gene

State of node ：　 1 (active) / 0 (inactive) Regulation rules

Boolean function (AND, OR, NOT …) Edge from y to x ⇔ y directly controls x

Synchronized update Almost the same as digital circuits (with

clocks)[Kauffman, The Origin of Order, 1993]

Example of Boolean Network

A

B C A’ = B

B’ = A and C

C’ = not A

State Transition TableBoolean Network

A’ B’ C’

time ｔ time ｔ＋１

A B C

0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0

INPUT OUTPUT

Example of state transition ：１１１　⇒　１１０　⇒　１００　⇒　０００　⇒　００１　⇒　００１　⇒　００１　⇒　。。。

Why Boolean Networks ? Criticism that BN is too simplified

Unless simplified, difficult for theoretical analysis, inference, and control though complex models can be used for simulation

Maybe useful for qualitative analyses

One of most simple non-linear models Negative results on BN suggest negative results on

more general (non-linear) models

Almost the same as digital circuits Theories and techniques in computer science can

be utilized

Our Focus: Time Complexity Many problems for BN are NP-hard

NP-hard means that there is no polynomial time algorithm (unless P=NP)

It will take O(2n) time or more if we use naïve methods

But, we want to solve much better Because we can solve the cases of

n=300 for O(1.1n) n=600 for O(1.05n)

Important for coping with large-scale networks

Attractor Detection

Attractor (1)

Steady state Different attractors ⇔

Different cell types Example

011 ⇒ 101 010 ⇒ ⇒101 010 …⇒ ⇒

111 110 100 ⇒ ⇒ ⇒000 ⇒ 001 001 ⇒ ⇒ 001 …⇒

A’ B’ C’

time ｔ time ｔ＋１

A B C

0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0

INPUT OUTPUT

State Transition Table

Attractor (2)

A’ B’ C’

time ｔ time ｔ＋１

A B C

0 0 0 0 0 1 0 1 0 0 1 1 1 0 0 1 0 1 1 1 0 1 1 1

0 0 1 0 0 1 1 0 1 1 0 1 0 0 0 0 1 0 1 0 0 1 1 0

INPUT OUTPUT

000

010

001

101100

110

011111

N-K Model (Kauffman Network) N: Number of nodes (We use n instead of N) K: Indegree

Indegree = the number of input edges = the number of genes directly affecting node v

Each node has (maximum or average) indegree K Boolean function assigned to each node is randomly

selected

v

indegree ＝２ indegree ＝３

v

Distribution of Attractors in N-K Model Classical conjecture

The number of attractors is Some results suggest that this conjecture may

not be true Superpolynomial growth ( > nγ for any γ) of the number of

attractors (Samuelsson & Troein, PRL, 2003) Superpolynomial growth of the average size of attractors (Drossel et al., PRL, 2005)

No conclusive result is known

)(O n

Singleton Attractor (or Point Attractor) Biological interpretation of attractors

Different attractors 　⇔　 Different cell types Point attractor

Attractor with period 1 Corresponding to a steady state Definition: satisfying

Attractor Detection Input: Boolean Network Output: Point Attractor (if any)

))(,),(()( 1 tvtvt nv

)()1( tt vv （ or, 　 　　　　　　　　　）)0()1( vv

Previous Works and Our Works Around time is enough

since there are 2n global states Several heuristics, but no theoretical guarantee 　 [Irons, Pysica D, 2006], [Devloo et al., Bull. Math. Biol. 2003], …

Detection of a singleton attractor is NP-hard　　　　　　　　　　　　　　　　　　　　 　

[Akutsu et al., GIW 1998]

We developed algorithms with average case theoretical bounds [Zhang et al., EURASIP JBSB 2007]

We developed algorithms for singleton attractor detection time algorithm for AND-OR BNs [Melkman, Tamura & Akutsu, 2010]

time algorithm for nested canalyzing BNs

[Akutsu, Melkman, Tamura & Yamamoto, 2011]

)2( nO

)587.1( nO

)799.1( nO

Reduction from BN-ATTRACTOR to SAT

Detection of Singleton Attractor with Max. Indegree K (K+1)-SAT (Boolean SATisfiability problem)

vi

vj vk

Basic Idea of Our Algorithms

y z

x

w

OR

OR

OR

OR

0

0 0OR

OR

1

1

Assigning x=0 eliminates three nodes Assigning x=1 eliminates two nodes

⇒ ⇒⇒ need additional work using SAT ⇒

)3()2()( nfnfnf nnf 325.1)(

)587.1( nO

Summary of Attractor Detection Algorithms

K=2 K=3 AND/OR of literals(any K)

Canalyzing(any K)

AND/OR of literals(Planar, any K)

Recursive(Ave. Time)

O(1.19n) O(1.27n)

SAT based (detection)

O(1.323n) O(1.474n) N/A N/A N/A

Our algorithms(detection)

O((1.323-δ)n)(δ=0.00004)

O(1.587n) O(1.799n) O((1+ε)n)

Singleton Attractors

Cyclic Attractors (Recursive, Average Case)

K=2 K=3 K=4 K=5

period=2 O(1.57n) O(1.70n) O(1.78n) O(1.83n)

period=3 O(1.72n) O(1.86n) O(1.92n) O(1.95n)

Control of Boolean Network

Control Theory for Biological Systems One of the main targets of Systems Biology Though control theory is well established for linear

systems, biological systems have non-linear components

May lead to new drugs and treatment methods Introduction of 4 genes turns normal cells into

induced pluripotent stem cells (iPS cells)

制御

がん細胞 正常細胞 Cancer Cell Normal Cell

Control

Definition of BN-Control Input

Internal nodes: v1 ,…, vn 　　 External nodes ： u1 ,…, um Initial state: v0 　　 Desired state: vM 　　　　　 BN

Output Sequence of states of external nodes ： u(0), u(1), …, u(M)

v(0)=v0, v(M)=vM 　　　（ leading to the desired state at time M ）

[Akutsu et al., J. Theo. Biol. 2007]

BN-Control: Related Works Datta et al. defined a problem of control of PBN

(Probabilistic Extension of BN) and proposed a dynamic programming based method They also proposed various extensions But, their method must handle 2n×2n matrices

BN-Control (also PBN-Control) is NP-hard BN-Control can be solved in polynomial time if the

network has a tree structure [Akutsu et al., JTB 2007]

Practical approach based on Model Checking/SAT

[Langmund & Jha, APBC 2008, JBCB 2009]

Theoretical studies using Semi-Tensor Product [Cheng, 2009, 2010, …]

[Machine Learning, 52:169-191, 2003]

Dynamic Programming for Control of BN BN version of the algorithm by Datta et al.

DP table: takes 1 if there is a control seq. leading to the

target state can be computed by

],,,,[ 21 tbbbD n

otherwise ,0

],,[ if ,1],,,,[ 1

21

Mn

n

bbMbbbD

v

otherwise ,0

),( and 1][

such that ),( is thereif ,1]1,,,,[

121 xbfc

xc

,t,c,cDtbbbDnn

Illustration of DP Algorithm

D[0,1,1, 3] = 1

D[1,1,1, 2] =1

u1=1, u2=1

D[0,0,0, 2] = 0

DPComputation

otherwise ,0),( and 1][

such that ),( is thereif ,1

]1,,,,[

1

21

xbfc

xc

,t,c,cD

tbbbD

n

n

But, the size of DP table isexponential

Integer Linear Programming-Based Approach

Integer Programming Linear Programming (LP)

Maximize (or minimize) an objective linear function under constraints of linear inequalities

Integer Linear Programming (ILP) LP + constraints that specified variables must take integer value Several efficient solvers: CPLEX, Gurobi Used for solving various NP-hard problems

integers:,,0,0

10020100

50050100

subject to

32maxize

　yxyx

yx

yx

yx

ILP Representation of Boolean Functions Variables ：　 either ０ or １　

(i.e., integer between 0 and 1) ＡＮＤ

OR

NOT

1,, yxzyzxzyxz AND

yxzyzxz ,,yxz OR

xz 1xz NOT

We applied this methodology to BN-control.

[Akutsu et al., IEEE CDC 2009]

Result on Attractor Detection Data: randomly generated BNs

with cases of indegree=2 and indegree=3 n: #nodes

3GHz Xeon CPU + ILOG CPLEX Result ： quite fast if indegree=2

Result on BN-Control Data: randomly generated BNs

with cases of indegree=2 and indegree=3 n: #internal nodes, m: #external nodes, M: #steps

Result ： fast if indegree=2

but, not so fast if indegree=3

PBN and its Control

Probabilistic Boolean Network (PBN)

Multiple control rules (boolean functions) for each node

Control rule is selected randomly at each t according to a given probability distribution Almost equivalent to Dynamic Bayesian Network Pros: Capable of noise. Can be modeled as Markov

process. Cons ： Not scalable since it takes O(2n) or more time

for almost all problems on PBN

AB

C

A(t+1) = B(t) AND C(t)

A(t+1) = B(t) OR (NOT C(t))

with Prob.=0.6

with Prob.=0.4

[Shmulevich et al., 2002]

Example of PBN

PBN

State Transition Diagram(only for half of nodes)

One of 4(=2×1×2) BNs is randomly selected at each time setp

BN vs. PBN BN: 1 outgoing edge PBN: multiple outgoing edges (with probabilities)

BN PBN

101

001

101

001 011 101 110

BN1 BN2 BN3 BN4

0.1 0.20.3 0.4

PBN-CONTROL: Model Probabilistic Boolean network (PBN, an extension of Boolean

network) Global state at time t: Probabilistic regulation rule is given as a 2n×2n matrix A A can be controlled by m boolean variables

Cost functions Ct(v, u): cost for applying control u for global state v at time t C(v): cost for final global state v

nn tvtvt }1,0{))(,),(()( 1 v

))(,),(()( 1 tutut mu

xw,uAxvwv ))(())(|)1(Pr( ttt

[Datta et al., Machine Learning, 2003]

PBN-CONTROL: Problem and Algorithm Problem: Given initial state v(0), control rule A(u(t)), target time M ,

and cost functions, Find a first control action u(0) minimizing

Can be solved by dynamic programming

M

tt ttCMCE

0

))(),(())(( uvv

)(),1())(())(|)1(Pr(

..

ttttt

ts

vvuAvv

111

000

*1,

}1,0{

*

*

)()(),(min)(

)()(

xxv

uxuAuvv

vv

ttt

M

JCJ

CJ

m

[Datta et al., Machine Learning, 2003]

Hardness Results Control of BN is NP-complete Integer linear programming (ILP)-based

method for control of BN Control of PBN is harder than NP ( -hard)

Such technique as ILP, SAT cannot be utilized

PSPACE

NP

p2

p2 Control of BN

ILP SAT

Control of PBN?

[Akutsu et al., JTB 07]

[Akutsu et al., IEEE CDC 09]

[Chen et al., BIBM 2010]

Conclusion

Conclusion Boolean network

A discrete model of a genetic network Similar to digital circuits

Attractor Detection/Enumeration NP-hard Much better than naïve O(2n) bound for several cases

Control of Boolean Networks NP-hard

Integer Linear Programming-based Approach Simple, Flexible for modifications/extensions

Control of Probabilistic Boolean Networks -hard ⇒ SAT or IP cannot be utilized p

2

Future Work Development of Non-trivial Algorithms for

Periodic Attractor Detection In progress

Control of Boolean Network Break O(2n) bound !

Control of PBN How to cope with -hardness

Development of Hybrid Model/Theory Combining Boolean and Linear Models

Thank you !

p2