A Maximum Principle for Single-Input Boolean Control Networks

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

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

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

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

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

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

7. BNs in Systems Biology 7 Signaling network controlling the stomatal closure in plants (Li et al, PLos Biology, 2006); 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).

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

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

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

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

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

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

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

15. BCNs as Boolean Switched Systems

16. Optimal Control Problem for BCNs 16 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

17. Main Result: A Maximum Principle 17 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

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

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

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

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

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

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

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

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

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

27. An Example This means that so Now 27 Analysis using the MP:

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

29. Conclusions 29 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/