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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