1. Attractor Detection and Control 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. 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. 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 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 INPUTOUTPUT 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 be utilized

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(2 n ) 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(1.1 n ) n=600 for O(1.05 n ) Important for coping with large-scale networks

9. Attractor Detection

10. 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 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 INPUTOUTPUT State Transition Table

11. Attractor (2) A ’ B ’ C ’ time ｔ 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 INPUTOUTPUT 000 010 001 101 100 110 011 111

12. 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

13. 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 ( > n γ 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 are 2 n 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) vivi vjvj vkvk

17. Basic Idea of Our Algorithms y z x w OR 0 00 1 1 Assigning x=0 eliminates three nodes Assigning x=1 eliminates two nodes ⇒ ⇒ need additional work using SAT ⇒

18. Summary of Attractor Detection Algorithms K=2K=3 AND/OR of literals (any K) Canalyzing (any K) AND/OR of literals (Planar, any K) Recursive (Ave. Time) O(1.19 n )O(1.27 n ) SAT based (detection) O(1.323 n )O(1.474 n ) N/A Our algorithms (detection) O((1.323-δ) n ) (δ=0.00004) O(1.587 n )O(1.799 n )O((1+ε) n ) Singleton Attractors Cyclic Attractors ( Recursive, Average Case ) K=2K=3K=4K=5 period=2 O(1.57 n )O(1.70 n )O(1.78 n )O(1.83 n ) period=3 O(1.72 n )O(1.86 n )O(1.92 n )O(1.95 n )

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 (iPS cells) Cancer CellNormal Cell Control

21. Definition of BN-Control Input  Internal nodes: v 1,…, v n External nodes ： u 1,…, u m  Initial state: v 0 Desired state: v M BN Output  Sequence of states of external nodes ： u(0), u(1), …, u(M) v(0)= v 0, v(M)=v M （ 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 2 n ×2 n 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[0,1,1, 3] = 1 D[1,1,1, 2] =1 u 1 =1, u 2 =1 D[0,0,0, 2] = 0 DP Computation 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 ： either ０ or １ (i.e., integer between 0 and 1) ＡＮＤ OR NOT We applied this methodology 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) 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(2 n ) or more time for almost all problems on PBN A B 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]

32. 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

33. BN vs. PBN BN: 1 outgoing edge PBN: multiple outgoing edges (with probabilities) BNPBN 101001101001011101110 BN 1 BN 2 BN 3 BN 4 0.1 0.2 0.3 0.4

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 2 n ×2 n matrix A A can be controlled by m boolean variables Cost functions  C t (v, u) : cost for applying control u for global state v at 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 PSPACE NP Control of BN ILPSAT Control of PBN ? [Akutsu et al., JTB 07] [Akutsu et al., IEEE CDC 09] [Chen et al., BIBM 2010]

37. Conclusion

38. Boolean network  A discrete model of a genetic network  Similar to digital circuits Attractor Detection/Enumeration  NP-hard  Much better than naïve O(2 n ) 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(2 n ) bound ! Control of PBN How to cope with -hardness Development of Hybrid Model/Theory Combining Boolean and Linear Models Thank you !