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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