1. A Maximum Principle for Single-Input Boolean Control Networks Michael Margaliot School of Electrical Engineering Tel Aviv University, Israel Joint work with Dima Laschov 1

2. Layout  Boolean Networks (BNs)  Applications of BNs in systems biology  Boolean Control Networks (BCNs)  Algebraic representation of BCNs  An optimal control problem  A maximum principle  An example  Conclusions 2

3. Boolean Networks (BNs) A BN is a discrete-time logical dynamical system: 3 where andis a Boolean function. → A finite number of possible states.

4. A Brief Review of a Long History BNs date back to the early days of switching theory, artificial neural networks, and cellular automata. 4

5. BNs in Systems Biology  S. A. Kauffman (1969) suggested using BNs for modeling gene regulation networks. 5 gene state-variable expressed/not expressed True/False network interactions Boolean functions Modeling Analysis stable genetic state attractor robustness basin of attraction

6. BNs in Systems Biology BNs have been used for modeling numerous genetic and cellular networks: 1. Cell-cycle regulatory network of the budding yeast (F. Li et al, PNAS, 2004); 2. Transcriptional network of the yeast (Kauffman et al, PNAS, 2003); 3. Segment polarity genes in Drosophila melanogaster (R. Albert et al, JTB, 2003); 4. ABC network controlling floral organ cell fate in Arabidopsis (C. Espinosa-Soto, Plant Cell, 2004). 6

7. BNs in Systems Biology 5. Signaling network controlling the stomatal closure in plants (Li et al, PLos Biology, 2006); 6. Molecular pathway between dopamine and glutamate receptors (Gupta et al, JTB, 2007);  BNs with control inputs have been used to design and analyze therapeutic intervention strategies (Datta et al., IEEE MAG. SP, 2010, Liu et al., IET Systems Biol., 2010). 7

8. Single—Input Boolean Control Networks 8 where: is a Boolean function Useful for modeling biological networks with a controlled input.

9. Algebraic Representation of BCNs State evolution of BCNs: Daizhan Chen developed an algebraic representation for BNs using the semi—tensor product of matrices. 9

10. Semi—Tensor Product of Matrices Definition Kronecker product of 10 and Definition semi-tensor product ofand where Let denote the least common multiplier of For example,

11. Semi—Tensor Product of Matrices 11 A generalization of the standard matrix product to matrices with arbitrary dimensions. Properties:

12. Semi—Tensor Product of Matrices 12 Example Suppose thatThen All the minterms of the two Boolean variables.

13. Algebraic Representation of Boolean Functions 13 Represent Boolean values as: Theorem (Cheng & Qi, 2010). Any Boolean function may be represented as where is the structure matrix of Proof This is the sum of products representation of

14. Algebraic Representation of Single-Input BCNs 14 Theorem Any BCN may be represented as where is the transition matrix of the BCN.

15. 15 BCNs as Boolean Switched Systems

16. Optimal Control Problem for BCNs  Fix an arbitrary and an arbitrary final time  Denote  Fix a vector  Define a cost-functional:  Problem: find a control that maximizes Since contains all minterms, any Boolean function of the state at time may be represented as 16

17. Main Result: A Maximum Principle  Theorem Let be an optimal control. Define the adjoint by: and the switching function by: Then  The MP provides a necessary condition for optimality in terms of the switching function 17

18. Comments on the Maximum Principle  The MP provides a necessary condition for optimality.  Structurally similar to the Pontryagin MP: adjoint, switching function, two-point boundary value problem. 18

19. The Singular Case Theorem If 19 then there exists an optimal control satisfying and there exists an optimal control satisfying

20. Proof of the MP: Transition Matrix More generally, 20 is called the transition matrix from time to time corresponding to the control Recall so

21. Proof of the MP: Needle Variation Define 21 Suppose thatis an optimal control. Fix a time and

22. Proof of the MP: Needle Variation This yields 22 Then so ?

23. Proof of the MP: Needle Variation Recall the definition of the adjoint 23 so ? This provides an expression for the effect of the needle variation.

24. Proof of the MP Suppose that 24 If so takeThen is also optimal. This proves the result in the singular case. The proof of the MP is similar.

25. An Example Consider the BCN 25 Consider the optimal control problem with andThis amounts to finding a control steering the system to

26. An Example The algebraic state space form: 26 with

27. An Example Analysis using the MP: 27 This means thatsoNow

28. An Example We can now calculate 28 This means thatso Proceeding in this way yields

29. Conclusions  We considered a Mayer –type optimal control problem for single –input BCNs.  We derived a necessary condition for optimality in the form of an MP.  Further research: (1) analysis of optimal controls in BCNs that model real biological systems, (2) developing a geometric theory of optimal control for BCNs.  For more information, see http://www.eng.tau.ac.il/~michaelm/ 29