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A Maximum Principle for Single-Input Boolean Control Networks

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Title: 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)
  • A BN is a discrete-time logical dynamical
    system
  • ? A finite number of possible states.

3
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.
  • Modeling
  • gene
    state-variable
  • expressed/not expressed True/False
  • network interactions Boolean
    functions
  • Analysis
  • stable genetic state
    attractor
  • robustness
    basin of attraction

5
6
BNs in Systems Biology
  • 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).

6
7
BNs in Systems Biology
  • 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).

7
8
SingleInput Boolean Control Networks

where

is a Boolean function
Useful for modeling biological networks with a
controlled input.
8
9
Algebraic Representation of BCNs
  • State evolution of BCNs
  • Daizhan Chen developed an algebraic
  • representation for BNs using the semitensor
  • product of matrices.

9
10
SemiTensor Product of Matrices
  • Definition Kronecker product of
  • and

Let
denote the least common multiplier of
For example,
Definition semi-tensor product of
and
where
10
11
SemiTensor Product of Matrices
A generalization of the standard matrix product
to matrices with arbitrary dimensions.
Properties
11
12
SemiTensor 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
where
is the structure matrix of
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
  • 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

then there exists an
optimal control satisfying
and there exists an optimal control
satisfying
19
20
Proof of the MP Transition Matrix
Recall
so
  • More generally,

is called the transition matrix
from time to time corresponding to the
control
20
21
Proof of the MP Needle Variation
Suppose that
is an optimal control. Fix a time
and
  • Define

21
22
Proof of the MP Needle Variation
Then
  • This yields

so
22
23
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
24
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
25
An Example
  • Consider the BCN

Consider the optimal control problem with
and
This amounts to finding a control
steering the system to
25
26
An Example
  • The algebraic state space form

with
26
27
An Example
  • Analysis using the MP

This means that
so
Now
27
28
An Example
  • We can now calculate

This means that
so
Proceeding in this way yields
28
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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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