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Attractor Detection and 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
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Attractor DetectionandControl 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 • 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]
A time t time t+1 A ’ B’ C’ A B C 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 B C 0 1 0 0 0 0 0 1 0 0 A ’ = B 1 0 1 0 1 1 1 0 1 1 0 B ’ = A and C 1 1 1 INPUT OUTPUT C ’ = not A Example of Boolean Network Boolean Network State Transition Table 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 beutilized
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
time t time t+1 A ’ B’ C’ A B C 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 INPUT OUTPUT Attractor(1) State Transition Table • Steady state • Different attractors ⇔ Different cell types • Example • 011 ⇒ 101 ⇒ 010 ⇒ 101 ⇒ 010 ⇒… • 111 ⇒ 110 ⇒ 100 ⇒ 000 ⇒ 001 ⇒ 001 ⇒001 ⇒ …
111 010 000 100 110 011 001 101 time t time t+1 A ’ B’ C’ A B C 0 0 0 0 0 1 0 0 1 0 0 1 0 1 0 1 0 1 0 1 1 1 0 1 0 1 0 0 0 0 0 1 0 0 1 0 1 0 1 1 1 0 1 1 0 1 1 1 INPUT OUTPUT Attractor (2)
indegree=2 indegree=3 v v 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
Distribution of Attractors inN-KModel • 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
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) (or, )
Previous Works and Our Works • Around time is enough since there are2n 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]
Reduction from BN-ATTRACTOR to SAT • Detection of Singleton Attractor with Max. Indegree K (K+1)-SAT (Boolean SATisfiability problem) vj vk vi
Basic Idea of Our Algorithms • Assigning x=0 eliminates three nodes • Assigning x=1 eliminates two nodes ⇒ ⇒ ⇒ need additional work using SAT ⇒ 0 0 1 y OR OR z 0 1 OR OR OR x OR w
Summary of Attractor Detection Algorithms Singleton Attractors Cyclic Attractors (Recursive, Average Case)
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 (iPScells) Control Cancer Cell Normal Cell
Definition of BN-Control • Input • Internal nodes: v1 ,…, vn External nodes:u1 ,…, um • Initial state:v0Desired state: vMBN • 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
Illustration of DP Algorithm D[1,1,1, 2] =1 D[0,0,0, 2] = 0 u1=1, u2=1 DP Computation D[0,1,1, 3] = 1 But, the size of DP table is exponential
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
ILP Representation of Boolean Functions • Variables: either0 or 1 (i.e., integer between 0 and 1) • AND • OR • NOT We applied thismethodology 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
Probabilistic Boolean Network (PBN) [Shmulevich et al., 2002] • 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 A(t+1) = B(t) AND C(t) with Prob.=0.6 A B C A(t+1) = B(t) OR (NOT C(t)) with Prob.=0.4
Example of PBN State Transition Diagram (only for half of nodes) PBN 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) 110 011 001 101 101 101 001 0.1 0.4 0.3 0.2 BN1 BN2 BN3 BN4 BN PBN
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 • Acan be controlled by m boolean variables • Cost functions • Ct(v, u): cost for applying control u for global state vat time t • C(v): cost for final global state v [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 [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 [Akutsu et al., JTB 07] [Akutsu et al., IEEE CDC 09] [Chen et al., BIBM 2010] PSPACE ? Control of PBN Control of BN ILP SAT NP
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
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 !