Attractor Detection and Control of Boolean Networks

1. Attractor DetectionandControl of Boolean Networks Tatsuya Akutsu Bioinformatics Center Institute for Chemical Research Kyoto University

2. 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

3. 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]

4. Boolean Network

5. 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]

6. 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　⇒　。。。

7. 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

8. 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

9. Attractor Detection

10. 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 ⇒ …

11. 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)

12. 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

13. 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

14. 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, 　 　　　　　　　　　）

15. 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]

16. Reduction from BN-ATTRACTOR to SAT • Detection of Singleton Attractor with Max. Indegree K (K+1)-SAT (Boolean SATisfiability problem) vj vk vi

17. 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

18. Summary of Attractor Detection Algorithms Singleton Attractors Cyclic Attractors (Recursive, Average Case)

19. Control of Boolean Network

20. 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

21. 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]

22. 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]

23. 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

24. 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

25. Integer Linear Programming-Based Approach

26. 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

27. 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]

28. 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

29. 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

30. PBN and its Control

31. 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

32. 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

33. 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

34. 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]

35. 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]

36. 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

37. Conclusion

38. 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

39. 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 !