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

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

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 ｔ＋１ A B C INPUTOUTPUT 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(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

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 ｔ＋１ A B C INPUTOUTPUT State Transition Table

Attractor (2) A ’ B ’ C ’ time ｔ ｔ＋１ A B C INPUTOUTPUT

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 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

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, ）

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]

Algorithms for Singleton Attractor Detection/Enumeration

Singleton Attractor （ =Attractor with Period 1) attractor

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

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]

Illustration of Recursive Algorithm Output

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

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

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

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]

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

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( 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)

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= ) [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 )

Control of Boolean Network

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: , 2003]

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]

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[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

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

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

ILP for Attractor Detection (2) 0

ILP for Attractor Detection (3) dummy for using ILP

ILP formalization for BN-Control major changes from Attractor Detection

Summary

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