1. 九大数理集中講義 Comparison, Analysis, and Control of Biological Networks (4) Analysis and Control of Boolean Networks Tatsuya Akutsu Bioinformatics Center Institute for Chemical Research Kyoto University

2. Contents Boolean Network Attractor Detection/Enumeration Algorithms for Singleton Attractor Detection /Enumeration Control of Boolean Networks Integer Linear Programming-based Approach

3. Boolean Network

4. 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]

5. 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 ⇒ 。。。

6. 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

7. 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

8. Attractor Detection

9. 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

10. 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

11. 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

12. Distribution of Attractors in N-K Model Classical conjecture  The number of attractors is Recent 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

13. 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, ）

14. Attractor Detection: Previous Works Around time is enough since there are 2 n global states  But, it cannot be applied to large n  Several heuristics are known, 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 also developed time algorithms for AND-OR BNs [Tamura & Akutsu, FCT07, Trans. IEICE 2009] [Tamura & Akutsu, AB08, Math. in CS 2009] [Melkman, Tamura & Akutsu, 2010]

15. Algorithms for Singleton Attractor Detection/Enumeration

16. Singleton Attractor （ =Attractor with Period 1) attractor

17. Indegree Indegree = the number of input edges = the number of genes directly affecting node v We use K to denote the maximum indgree v indegree ＝２ indegree ＝３ v

18. Simple Recursive Enumeration Algorithm (1) Examine 0-1 assignment one-by-one, and backtrack as soon as some contradiction occurs [Zhang et al., EURASIP JBSB 2007]

19. Illustration of Recursive Algorithm 00 0 0 0 0 0 0 1 0 1 1 Output

20. Simple Recursive Enumeration Algorithm (2) Examine 0-1 assignment one-by-one, and backtrack as soon as some contradiction occurs.  0  00  Ｘ  backtrack  01  010  Ｘ  backtrack  011  Ｘ  backtrack  10 Several variants depending on ordering of nodes Much better than trivial O(n2 n ) time K23456 Basic1.35 n 1.43 n 1.49 n 1.53 n 1.57 n Outdegree- based 1.19 n 1.27 n 1.34 n 1.41 n 1.45 n

21. Analysis of Average Case Time Complexity Probability that v i (0)≠v i (1) is detected when 0-1 assignment for first m bits is examined: Probability that a random assignment for m bits is consistent (with def. of singleton attractor): Expected number of consistent 0-1 assignments for m bits: By taking the maximum of the above for m in [1…n], we can estimate the complexity v1v1 v m-1 vmvm v m+1 t=0t=1 K

22. Computational Experiment Exponential increases, but bases are less than 2 K23456 Basic1.39 n 1.46 n 1.53 n 1.57 n 1.60 n Outdegree- based 1.23 n 1.30 n 1.37 n 1.42 n 1.47 n Empirical Time Complexity

23. Issues on Worst Case Time Complexity Detection of a Singleton Attractor for BNs with indegree K  ( K+1 )-SAT  O(1.322 n ) time for K=2 (randomized)  We developed O((1.322-δ) n ) time algorithm for K=2 Detection problem remains NP-hard even for K=2 O(1.587 n ) time algorithm for BNs with AND/OR nodes (no constraint on K ) [Melkman, Tamura & Akutsu, 2010]

24. Reduction from BN-ATTRACTOR to SAT Detection of Singleton Attractor with Max. Indegree K  (K+1)- SAT (Boolean SATisfiability problem) vivi vjvj vkvk

25. Basic Idea in O(1.587 n ) Time Algorithm Consider recursive assignment of 0-1 values to nodes (A) v=0 ⇒ u=0, v=1 ⇒ w=1 (B) v=0 ⇒ u=0 and w=1 Let f(k) be #(assignments) for BN with k variables By solving the above (like Fibonacci number), f(k) is O(1.4656 n ) However, above procedure cannot be applied to all cases (e.g., not to bipartite networks)  combination with SAT is required  O(1.587 n ) time u v w u v w All nodes are OR NOT input (A) (B)

26. Attractor Detection: Previous Works (2) K=2K=3 AND/OR of literals (any K) Nested canalyzing (any K) Nested canalyzing (any K) in partial k- tree with period p 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) [IEICE, 2009] O(1.587 n ) [IPL, 2010] O(1.871 n ) [JCB, 2013] O(n 2p(w+1) poly(n) ) [TCBB, 2012] 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 )

27. Control of Boolean Network

28. BN-Control: Previous 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] [Machine Learning, 52:169-191, 2003]

29. 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]

30. 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

31. 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

32. Integer Linear Programming- Based Approach

33. 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

34. ILP for Attractor Detection (1) x i : state of v i

35. ILP for Attractor Detection (2) 0

36. ILP for Attractor Detection (3) dummy for using ILP

37. ILP formalization for BN-Control major changes from Attractor Detection

38. Summary

39. Boolean network  A discrete model of a genetic network  Similar to digital circuits Attractor Detection/Enumeration  NP-hard  Much better than a naïve O(2 n ) bound for bounded indegree cases  Identification of cyclic attractors is more difficult Control of Boolean networks  NP-hard  Can be solved by DP algorithm (but, in exponential time) Integer Linear Programming-based Approach  Simple  Flexible for modifications/extensions  Fast if indegree ≦ 2