A Maximum Principle for Single-Input Boolean Control Networks

transcript

A Maximum Principle for Single-Input Boolean

Control Networks

Michael MargaliotSchool of Electrical EngineeringTel Aviv University, Israel

Joint work with Dima Laschov

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

Boolean Networks (BNs)

A BN is a discrete-time logical dynamical system:

( 1) ( ( ), ( )),

x k f x k x k

where and is a Boolean function. ( ) {True,False}ix if

→ A finite number of possible states.

A Brief Review of a Long History

BNs date back to the early days of switching theory, artificial neural networks, and cellular automata.

BNs 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

Modeling

Analysis

stable genetic state attractor

robustness basin of attraction

BNs in Systems BiologyBNs 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).

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).

Single—Input Boolean Control Networks

( 1) ( ( ), ( ), ( )),

x k f x k x k u k

where:

is a Boolean function

( ), ( ) {True,False}ix u

Useful for modeling biological networks with a

controlled input.

Algebraic Representation of BCNsState evolution of BCNs:

Daizhan Chen developed an algebraic

representation for BNs using the semi—tensor

product of matrices.

( ) ... ( ( (0), (0)), (1))...).x k f f f x u u

Semi—Tensor Product of MatricesDefinition Kronecker product of

p qB R

11 1( ) ( )

a B a B

m nA R

m nA R Definition semi-tensor product of and p qB R

/ /( )( )α n α pA B A I B I â

where ( , ).α lcm n p

Let lcm( , )a b

lcm(6,8) 24.denote the least common multiplier of , .a b

For example,

( / ) ( / ) ( / ) ( / )

( ) ( )α n α p

mα n nα n pα p qα p

A B A I B I

Semi—Tensor Product of Matrices

A generalization of the standard matrix product to

matrices with arbitrary dimensions.

/ /( )( ).α n α pA B A I B I â

Properties:

( ) ( )A B C A B Câ â â â

( ) ( ) ( )A B C A C B C â â â

Semi—Tensor Product of Matrices

/ /( )( ).α n α pA B A I B I â

Example Suppose that0 1

, { , }.1 0

0( )( )

v w v w v w vwa b I I

v w v w v w vw

All the minterms of the two Boolean variables.

Algebraic Representation of Boolean Functions

Represent Boolean values as:

1 21 0True , False .

0 1e e

1 1 2( ,..., ) ...n f nf x x M x x x â â â â

Theorem (Cheng & Qi, 2010). Any Boolean function 1 2 1 2:{ , } { , }nf e e e e

may be represented as

where is the structure matrix of .f2 2nxfM R

Proof This is the sum of products representation of .f

Algebraic Representation of Single-Input BCNs

( 1) ( ) ( ) x k L u k x k â â

( 1) ( ( ), ( ), ( )),

x k f x k x k u k

Theorem Any BCN

may be represented as

12 2n n

LRwhere is the transition matrix of the BCN.

BCNs as Boolean Switched Systems

( 1) ( ) ( )x k L u k x k â â

1( )u k e 2( )u k e

1( 1) ( ) x k L x k â 2( 1) ( ) x k L x k â

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

1 2(0), ( 1) , ( ) , .U u u N u i e e

( ) ( , ). (4)TJ u r x N u

),( uNx

( , ).Tr x N u

Main Result: A Maximum Principle Theorem Let be an optimal control.

Define the adjoint by:

and the switching function by:

The MP provides a necessary condition for optimality in terms of the switching function

2: 1,n

( ) ( ( )) ( 1), ( ) , (5)Ts L u s s N r â â

: 0,1 1m N R

*1( ) ( 1) ( ).

1Tm s s L x s

â â â

, if ( ) 0,*( ) (6)

, if ( ) 0.

e m su s

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.

, if ( ) 0,*( ) (6)

, if ( ) 0.

e m su s

The Singular Case

Theorem If

, if ( ) 0,*( ) (6)

, if ( ) 0.

e m su s

( ) 0,m s then there exists an

optimal control satisfying

and there exists an optimal control

satisfying

*u 1*( ) ,u s e

2*( ) .w s e

Proof of the MP: Transition Matrix

More generally,

( 1) ( ) ( )x k L u k x k â â

is called the transition matrix from

time to time corresponding to the control

Recall

( ; ) ( , ; ) ( ; )x k u C k j u x j u â

( , ; ) ( 1) ( 2) ... ( ).C k j u L u k L u k L u j â â â â â â

( , ; )C k j u

so ( 2) ( 1) ( 1) ( 1) ( ) ( ).x k L u k x k L u k L u k x k â â â â â â

Proof of the MP: Needle Variation

Define

*( ), otherwise.

v j pu j

Suppose that is an optimal control. Fix a time{0,1,..., 1}p N and 1 2{ , }.v e e

( )u j

This yields

*( ) ( , 1; *) *( ) *( )x N C N p u L u p x p â â âThen

( ) ( , 1; ) ( ) ( )

( , 1; *) *( )

x N C N p u L u p x p

C N p u L v x p

â â â

*( ) ( ) ( , 1; *) ( *( ) ) *( )x N x N C N p u L u p v x p â â âso

( *) ( ) ( *( ) ( ))

( , 1; *) ( *( ) ) *( )

J u J u r x N x N

r C N p u L u p v x p

â â â

Recall the definition of the adjoint

( *) ( ) ( *( ) ( ))

( , 1; *) ( *( ) ) *( ).

J u J u r x N x N

r C N p u L u p v x p

â â â

( ) ( ( )) ( 1), ( ) , Ts L u s s N r â â

( *) ( ) ( *( ) ( ))

( 1) ( *( ) ) *( ).

J u J u r x N x N

p L u p v x p

â â â

This provides an expression for the effect of the

needle variation.

Proof of the MP

Suppose that

( *) ( ) ( 1) ( *( ) ) *( ).TJ u J u p L u p v x p â â â

*1( ) ( 1) ( ) 0.

1Tm p p L x p

â â â

1*( ) ,u p e take

2.v e Then 1 0

( *) ( ) ( 1) ( ) *( )0 1

TJ u J u p L x p

â â â

u is also optimal. This proves the result in

the singular case. The proof of the MP is similar.

An ExampleConsider the BCN

( 1) ( ) ( )

( 1) ( ) ( ) ( )

x k u k x k

x k u k x k x k

1 2(0) (0) False.x x

Consider the optimal control problem with

and [1 0 0 0] .Tr

This amounts to finding a control

steering the system to 1 2(3) (3) True.x x

An ExampleThe algebraic state space form:

( 1) ( ) ( )x k L u k x k â â

(0) [0 0 0 1]Tx

0 1 0 1 0 0 0 0

0 0 0 0 0 0 0 0.

1 0 1 0 0 1 1 1

0 0 0 0 1 0 0 0

An ExampleAnalysis using the MP:

1(2) (3) (2)

[0 1 0 1] (2).

â â â

This means that (2) 0,m so 1*(2) .u e Now*(2) ( (2)) (3)

[0 1 0 1] .

An ExampleWe can now calculate

1(1) (2) (1)

[ 1 0 0 0] (1).

Tm L x

â â â

This means that (1) 0,m so 2*(1) .u e

1 2 1*(2) , *(1) , *(0) .u e u e u e

Proceeding in this way yields

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/

A Maximum Principle for Single-Input Boolean Control Networks

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