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
Layout 2 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
Boolean Networks (BNs) A BN is a discrete-time logical dynamical system: and is a Boolean function. where → A finite number of possible states. 3
A Brief Review of a Long History BNs date back to the early days of switching theory, artificial neural networks, and cellular automata. 4
BNs in Systems Biology S. A. Kauffman (1969) suggested using BNs for modeling gene regulation networks. Modeling gene expressed/not expressed network interactions Analysis stable genetic state robustness 5 state-variable True/False Boolean functions attractor basin of attraction
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
BNs in Systems Biology 5. 6. 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).
Single—Input Boolean Control Networks where: is a Boolean function Useful for modeling biological networks with a controlled input. 8
Algebraic Representation of BCNs State evolution of BCNs: Daizhan Chen developed an algebraic representation for BNs using the semi—tensor product of matrices. 9
Semi—Tensor Product of Matrices Definition Kronecker product of and Let denote the least common multiplier of For example, Definition semi-tensor product of where 10 and
Semi—Tensor Product of Matrices A generalization of the standard matrix product to matrices with arbitrary dimensions. Properties: 11
Semi—Tensor Product of Matrices Example Suppose that Then All the minterms of the two Boolean variables. 12
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
Algebraic Representation of Single-Input BCNs Theorem Any BCN may be represented as where 14 is the transition matrix of the BCN.
BCNs as Boolean Switched Systems 15
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
Main Result: A Maximum Principle Theorem Let be an optimal control. Define the adjoint by: and the switching function by: Then 17
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.
The Singular Case Theorem If optimal control then there exists an satisfying and there exists an optimal control satisfying 19
Proof of the MP: Transition Matrix Recall so More generally, time 20 is called the transition matrix from to time corresponding to the control
Proof of the MP: Needle Variation Suppose that Define 21 is an optimal control. Fix a time and
Proof of the MP: Needle Variation Then This yields so ? 22
Proof of the MP: Needle Variation ? Recall the definition of the adjoint so This provides an expression for the effect of the needle variation. 23
Proof of the MP Suppose that If take Then so is also optimal. This proves the result in the singular case. The proof of the MP is similar. 24
An Example Consider the BCN Consider the optimal control problem with and This amounts to finding a control steering the system to 25
An Example The algebraic state space form: with 26
An Example Analysis using the MP: This means that 27 so Now
An Example We can now calculate This means that so Proceeding in this way yields 28
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
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