Comparison Analysis and Control of Biological Networks 5

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

Probabilistic Boolean Network

Probabilistic Boolean Network (PBN) [Shmulevich et al. , 2002] n n Multiple control rules (boolean functions) for each node Control rule is selected randomly at each t according to a given probability distribution q q q B C 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 A(t+1) = B(t) AND C(t) with Prob. =0. 6 A(t+1) = B(t) OR (NOT C(t)) with Prob. =0. 4

Example of PBN State Transition Diagram PBN (only for half of nodes) One of 4(=2× 1× 2) BNs is randomly selected at each time step

BN vs. PBN n n BN: 1 outgoing edge PBN: multiple outgoing edges (with probabilities) 101 0. 3 0. 2 0. 4 BN 1 BN 2 BN 3 001 BN 001 011 101 PBN BN 4 110

BN vs. PBN in State Transition Diagram BN: 1 outgoing edge n ⇔ PBN: multiple outgoing edges (with probabilities) 111 101 0. 2 101 0. 8 110 1. 0 0. 4 1. 0 0. 6 010 011 100 010 0. 7 011 0. 5 100 0. 3 000 0. 5 0. 2 1. 0 001 BN 001 0. 8 PBN

Matrix Representation of a BN State transition table A 0 0 1 1 B 0 0 1 1 C 0 1 0 1 A’ 0 0 1 1 B’ 0 0 0 1 C’ 1 1 0 0 State transition can be written as where

Matrix Representation of a PBN The above means:

PBN-CONTROL: Model n 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 n Cost functions n n n q q Ct(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 n Algorithm Problem: q q n 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

Finding an Optimal Path in PBN Problem: Finding a control sequence for a PBN with the maximum probability Method: Introduction of variable yr, t n yr, t=1 iff r-th BN is selected at time step t Modification of ILP pr: 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 u 1 Control u 2 0. 6 0. 4 t=1 Pr(BN 1)=0. 4 Pr(BN 2)=0. 6 v(0) 0. 4 v 1(1) v 2(1) u 1 u 2 Control 0. 4 t=2 v u : global state 0. 6 0. 4 v 1(2) 0. 6 v 3(1) u 1 0. 4 0. 6 Optimal Path Control (u 2, u 1)

Minimizing the Maximum Cost in PBN

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

Minimizing the Maximum Cost like minimax game v(0) (2) u 1 u 2 0. 6 0. 4 v 1(1) u 1 0. 4 CM(v) 0. 4 0. 6 v 2(1) u 2 Pr(BN 1)=0. 4 Pr(BN 2)=0. 6 u 1 v 3(1) u 2 u 1 0. 6 u 2 0. 4 0. 6 v 1(2) 500 100 200 300

Minimizing the Average Cost v(0) Original Control Problem for PBN u 1 0. 6 0. 4 v 1(1) u 1 0. 4 + v 2(1) u 2 sum + weighted of costs u 2 u 1 0. 6 u 2 0. 6 u 1 + + max v 3(1) u 2 0. 4 0. 6 v 1(2) 500 100 200 300 +

Hardness Results

Limitation of Use of ILP n ILP can be used for Control of BN q Finding an optimal path in PBN q n But, q Control of PBN n q cannot be used for Minimizing the average cost Minimizing the maximum cost n 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 n Control of PBN is harder than Control of BN (NP-complete [Akutsu et al, 2007]) because is believed to be harder than NP n Such technique as ILP, SAT cannot be utilized because these are in NP [Chen et al. , IEEE BIBM, 2010]

Hardness Results (2) n n n Control of BN is NP-complete (for fixed time [Akutsu et al. , JTB 07] steps) Integer linear programming (ILP)-based [Akutsu et al. , IEEE CDC 09] method for control of BN Control of PBN is harder than NP ( -hard) n Such technique as ILP, SAT cannot be utilized

Summary

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