Title: A Maximum Principle for Single-Input Boolean Control Networks
1A Maximum Principle for Single-Input Boolean
Control Networks
- Michael Margaliot
- School of Electrical Engineering
- Tel Aviv University, Israel
- Joint work with Dima Laschov
2Layout
- 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
3Boolean Networks (BNs)
- A BN is a discrete-time logical dynamical
system
- ? A finite number of possible states.
3
4A Brief Review of a Long History
- BNs date back to the early days of switching
theory, artificial neural networks, and cellular
automata.
4
5BNs in Systems Biology
- S. A. Kauffman (1969) suggested using BNs for
modeling gene regulation networks.
- gene
state-variable - expressed/not expressed True/False
- network interactions Boolean
functions -
- stable genetic state
attractor - robustness
basin of attraction -
-
5
6BNs 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
7BNs 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
8SingleInput Boolean Control Networks
where
is a Boolean function
Useful for modeling biological networks with a
controlled input.
8
9Algebraic Representation of BCNs
- State evolution of BCNs
- Daizhan Chen developed an algebraic
- representation for BNs using the semitensor
- product of matrices.
9
10SemiTensor Product of Matrices
- Definition Kronecker product of
-
Let
denote the least common multiplier of
For example,
Definition semi-tensor product of
and
where
10
11SemiTensor Product of Matrices
A generalization of the standard matrix product
to matrices with arbitrary dimensions.
Properties
11
12SemiTensor Product of Matrices
Example Suppose that
Then
All the minterms of the two Boolean variables.
12
13Algebraic 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
14Algebraic Representation of Single-Input BCNs
Theorem Any BCN
may be represented as
where is the transition matrix
of the BCN.
15BCNs as Boolean Switched Systems
16Optimal 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
17Main 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
18Comments 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
19The Singular Case
then there exists an
optimal control satisfying
and there exists an optimal control
satisfying
19
20Proof of the MP Transition Matrix
Recall
so
is called the transition matrix
from time to time corresponding to the
control
20
21Proof of the MP Needle Variation
Suppose that
is an optimal control. Fix a time
and
21
22Proof of the MP Needle Variation
Then
so
22
23Proof of the MP Needle Variation
- Recall the definition of the adjoint
so
This provides an expression for the effect of
the needle variation.
23
24Proof 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
25An Example
Consider the optimal control problem with
and
This amounts to finding a control
steering the system to
25
26An Example
- The algebraic state space form
with
26
27An Example
This means that
so
Now
27
28An Example
This means that
so
Proceeding in this way yields
28
29Conclusions
- 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