九大数理集中講義 Comparison, Analysis, and Control of Biological Networks (5) Control of Probabilistic Boolean Networks Tatsuya Akutsu Bioinformatics Center Institute for Chemical Research Kyoto University

Contents Boolean Network Probabilistic Boolean Network (PBN)  Formulation, Control of PBN Finding an Optimal Path in PBN  ILP-based method Minimizing the Maximum Cost in PBN  Optimize the worst case control cost Hardness of Control of PBN  Control of PBN is ∑ 2 p -hard Summary

Probabilistic Boolean Network

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]

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 step

BN vs. PBN BN: 1 outgoing edge PBN: multiple outgoing edges (with probabilities) BNPBN BN 1 BN 2 BN 3 BN

BN vs. PBN in State Transition Diagram BN: 1 outgoing edge ⇔ PBN: multiple outgoing edges (with probabilities) BN PBN

Matrix Representation of a BN A ’ B ’ C ’ A B C State transition can be written as where State transition table

Matrix Representation of a PBN The above means:

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]

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]

PBN-CONTROL: Dynamic Programming But, it takes too long CPU time because A is 2 n ×2 n matrix

Finding an Optimal Path in PBN

Problem: Finding a control sequence for a PBN with the maximum probability Method: Introduction of variable y r,t y r,t =1 iff r -th BN is selected at time step t Modification of ILP p r : probability of selecting r -th BN We also developed DP (Dynamic Programming) algorithm (with exponential time and space) [Chen et al., IEEE BIBM, 2010]

Optimal Path Problem t=0 t=1 t=2 Control v(0) v 1 (1)v 2 (1)v 3 (1) u2u2 v 1 (2) u2u2 u1u Optimal Path Control (u2,u1)(u2,u1) u1u u1u1 0.4 Pr(BN 1 )=0.4 Pr(BN 2 )=0.6 v i, u j : global state

Minimizing the Maximum Cost in PBN

Minimizing the Maximum Cost (1) 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 Minimizing the worst case cost is important Method: dynamic programming (with exponential time/space) F(v t,u t ) : set of states that can be reached from v t with control u t [Chen et al., IEEE BIBM, 2010]

Minimizing the Maximum Cost (2) Pr(BN 1 )=0.4 Pr(BN 2 )=0.6 v(0) v 1 (1)v 2 (1)v 3 (1) u2u2 0.4 u1u1 u1u1 u2u2 u1u1 u2u2 0.6 u1u1 u2u2 v 1 (2) CM(v)CM(v) like minimax game

Minimizing the Average Cost v(0) v 1 (1)v 2 (1)v 3 (1) u2u2 0.4 u1u1 u1u1 u2u2 u1u1 u2u2 0.6 u1u1 u2u2 v 1 (2) Original Control Problem for PBN weighted sum of costs max +

Hardness Results

Limitation of Use of ILP ILP can be used for  Control of BN  Finding an optimal path in PBN But, cannot be used for  Control of PBN Minimizing the average cost  Minimizing the maximum cost Why ?

Hardness Results (1) Thm. 1: Minimizing the maximum cost in control of PBN is -hard. Thm. 2: Minimizing the average cost in control of PBN (i.e., original control problem for PBN) is -hard. Proof: Reduction from QBF for 3-DNF Implications Control of PBN is harder than Control of BN (NP-complete [Akutsu et al, 2007] ) because is believed to be harder than NP Such technique as ILP, SAT cannot be utilized because these are in NP [Chen et al., IEEE BIBM, 2010]

Hardness Results (2) Control of BN is NP-complete (for fixed time steps) 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]

Summary

Probabilistic Boolean Network  Probabilistic extension of BN  DP algorithm for control of PBN Finding an Optimal Path in PBN  ILP-based method Minimizing the Maximum Cost  DP algorithm Hardness of Control of PBN  Harder than NP and Control of BN