A Maximum Principle for Single-Input Boolean

Control Networks

Michael MargaliotSchool of Electrical EngineeringTel Aviv University, Israel

Joint work with Dima Laschov

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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

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Boolean Networks (BNs)

A BN is a discrete-time logical dynamical system:

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1 1 1

1

( 1) ( ( ), ( )),

(1)

( 1) ( ( ), ( )),

n

n n n

x k f x k x k

x k f x k x k

1x

or

2x

and

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.

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BNs in Systems Biology

S. A. Kauffman (1969) suggested using BNs for modeling gene regulation networks.

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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).

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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).

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Single—Input Boolean Control Networks

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1 1 1

1

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

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

n

n n n

x k f x k x k u k

x k f x k x k u k

where:

is a Boolean function

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

if

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.

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

Semi—Tensor Product of MatricesDefinition Kronecker product of

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and

p qB R

11 1( ) ( )

1

.n

mp nq

n nn

a B a B

A B R

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

â

Semi—Tensor Product of Matrices

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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

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/ /( )( ).α n α pA B A I B I â

Example Suppose that0 1

, { , }.1 0

a b

Then

2 1

0

0( )( )

0

0

v vw

v w v w v w vwa b I I

v w v w v w vw

v vw

â â

All the minterms of the two Boolean variables.

Algebraic Representation of Boolean Functions

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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

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( 1) ( ) ( ) x k L u k x k â â

1 1 1

1

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

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

n

n n n

x k f x k x k u k

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.

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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

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0.N

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

2 .n

r R

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

Uu .J

),( uNx

( , ).Tr x N u

(0)x

N

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

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*u

2: 1,n

N R

( ) ( ( )) ( 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

1

2

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

, if ( ) 0.

e m su s

e m 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.

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1

2

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

, if ( ) 0.

e m su s

e m s

The Singular Case

Theorem If

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1

2

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

, if ( ) 0.

e m su s

e m 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

*w

Proof of the MP: Transition Matrix

More generally,

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( 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

kj .u

( , ; )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

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*u

,( )

*( ), otherwise.

v j pu j

u j

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

j

( )u j

p

v

1N 0

Proof of the MP: Needle Variation

This yields

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*( ) ( , 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

â â â

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

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

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

T

T

J u J u r x N x N

r C N p u L u p v x p

â â â

?

Proof of the MP: Needle Variation

Recall the definition of the adjoint

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so

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

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

T

T

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) ( *( ) ) *( ).

T

T

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

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If

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

so

*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

0,

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

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1 2

2 1 2

( 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

3N

This amounts to finding a control

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

An ExampleThe algebraic state space form:

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with

( 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

L

An ExampleAnalysis using the MP:

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*

*

*

1(2) (3) (2)

1

1(2)

1

[0 1 0 1] (2).

T

T

m L x

r L x

x

â â â

â â â

â

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

[0 1 0 1] .

T

T

L u

â

An ExampleWe can now calculate

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*

*

1(1) (2) (1)

1

[ 1 0 0 0] (1).

Tm L x

x

â 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/

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