1 A Pontryagin Maximum Principle for Multi–Input Boolean Control Networks ⋆ Dmitriy Laschov and Michael Margaliot ⋆⋆ School of Electrical Engineering–Systems, Tel Aviv University, Israel 69978. Summary. A Boolean network consists of a set of Boolean variables whose state is deter- mined by other variables in the network. Boolean networks have been studied extensively as models for simple artificial neural networks. Recently, Boolean networks gained con- siderable interest as models for biological systems composed of elements that can be in one of two possible states. Examples include genetic regulation networks, where the ON (OFF) state corresponds to the transcribed (quiescent) state of a gene, and cellular net- works where the two possible logic states may represent the open/closed state of an ion channel, basal/high activity of an enzyme, two possible conformational states of a pro- tein, etc. Daizhan Cheng developed an algebraic state-space representation for Boolean control networks using the semi–tensor product of matrices. This representation proved quite useful for studying Boolean control networks in a control-theoretic framework. Using this representation, we consider a Mayer-type optimal control problem for Boolean control networks. Our main result is a necessary condition for optimality. This provides a parallel of Pontryagin’s maximum principle to Boolean control networks. 1 Introduction A Boolean network consists of a set of Boolean variables whose state is determined by other variables in the network. Cellular automata, with two possible states per ⋆ Research supported in part by the Israel Science Foundation (ISF). ⋆⋆ Corresponding author: Prof. Michael Margaliot, School of Electrical Engineering– Systems, Tel Aviv University, Israel 69978. Homepage: www.eng.tau.ac.il/ ~ michaelm Email: [email protected]
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1
A Pontryagin MaximumPrinciple for Multi–InputBoolean Control Networks⋆
Dmitriy Laschov and Michael Margaliot⋆⋆
School of Electrical Engineering–Systems, Tel Aviv University, Israel 69978.
Summary. A Boolean network consists of a set of Boolean variables whose state is deter-
mined by other variables in the network. Boolean networks have been studied extensively
as models for simple artificial neural networks. Recently, Boolean networks gained con-
siderable interest as models for biological systems composed of elements that can be in
one of two possible states. Examples include genetic regulation networks, where the ON
(OFF) state corresponds to the transcribed (quiescent) state of a gene, and cellular net-
works where the two possible logic states may represent the open/closed state of an ion
channel, basal/high activity of an enzyme, two possible conformational states of a pro-
tein, etc. Daizhan Cheng developed an algebraic state-space representation for Boolean
control networks using the semi–tensor product of matrices. This representation proved
quite useful for studying Boolean control networks in a control-theoretic framework. Using
this representation, we consider a Mayer-type optimal control problem for Boolean control
networks. Our main result is a necessary condition for optimality. This provides a parallel
of Pontryagin’s maximum principle to Boolean control networks.
1 Introduction
A Boolean network consists of a set of Boolean variables whose state is determined
by other variables in the network. Cellular automata, with two possible states per
⋆ Research supported in part by the Israel Science Foundation (ISF).⋆⋆ Corresponding author: Prof. Michael Margaliot, School of Electrical Engineering–
Systems, Tel Aviv University, Israel 69978. Homepage: www.eng.tau.ac.il/~michaelmEmail: [email protected]
2 Dmitriy Laschov and Michael Margaliot
cell, are a particular case of Boolean networks. Here the state of each variable at
time k+1 is determined by the state of its spatial neighbors at time k [1]. A Boolean
network with n variables has 2n possible states and therefore the dynamics for any
initial condition must fall into an attractor.
Boolean networks have been studied extensively as models for simple artificial
neural networks (see, e.g. [2]). Here each neuron realizes a threshold function that
attains the values zero or one. More recently, Boolean networks gained renewed
interest as models for biological systems composed of elements that can be in one
of two possible states (i.e., ON or OFF). S. A. Kauffman [3] modeled a gene as a
binary device, and studied the behavior of large, randomly constructed nets of these
binary genes. Kauffman’s simulations indicate that if each network node has two
or three inputs, then the dynamical behavior of the network demonstrates order
and stability. Kauffman also related the behavior of the random nets to various
cellular control processes including cell differentiation. The key idea being to view
each stable attractor as representing one possible cell type.
Kauffman’s pioneering ideas stimulated research in several directions. One di-
rection is the theoretical analysis of the dynamics of Boolean networks, espe-
cially using tools from the theory of complex systems and statistical physics (see,
e.g. [4, 5, 6, 7, 8, 9, 10]).
Another research direction is modeling various biological processes using Boolean
networks. Analyzing the behavior of the Boolean network may provide considerable
insight on the original biological process. This is a vast area of research and we
review here only a few examples.
1.1 Boolean networks modeling in biology
Boolean networks seem especially suitable for modeling genetic regulation networks
where the ON (OFF) state corresponds to the transcribed (quiescent) state of the
gene. There are several other motivations [11] for using Boolean networks in this con-
text, including the fact that many metabolic and genetic networks demonstrate some
form of bi-stability. An important example are epigenetic switches (see, e.g. [12, 13]).
Specific examples of genetic regulation networks modeled using Boolean networks
include: the cell–cycle regulatory network of the budding yeast [14]; the yeast tran-
scriptional network [15]; the network controlling the segment polarity genes in the
fly Drosophila melanogaster [16, 17]; the ABC network determining floral organ cell
fate in Arabidopsis [18] (see also [19]);
Boolean networks were also used for modeling various cellular processes. In this
context the two possible logic states may represent the open/closed state of an ion
channel, basal/high activity of an enzyme, two possible conformational states of
a protein, etc. Specific examples include: a detailed model for the highly complex
cellular signaling network controlling stomatal closure in plants [20]; and a model of
the molecular pathway between two neurotransmitter systems, the dopamine and
glutamate receptors [21].
1 A Pontryagin Maximum Principle for Multi–Input Boolean Control Networks 3
Szallasi and Liang [22] discuss the use of Boolean networks in modeling carcino-
genesis and for analyzing the effect of therapeutic intervention (see also [23]).
These studies suggest that Boolean networks provide a highly efficient modeling
tool for large–scale biological networks. These models are able to reproduce the
main characteristics of the biological network dynamics: attractors of the Boolean
network correspond to stationary biological states; large attraction basins indicate
robustness of the biological state, and so on.
Modeling using Boolean networks requires only coarse–grained qualitative infor-
mation (e.g., an interaction between two genes is either activating or inhibiting).
This is in sharp contrast to other models, for example, those based on differential
equations, that require knowledge of numerous parameter values (e.g., rate con-
stants). For a general exposition on various approaches for modeling gene regulation
networks, see [24].
Modeling a biological system involves considerable uncertainty. This is due to
the noise and perturbations that affect the biological system, and to the inaccuracies
of the measuring equipment. One approach for tackling this uncertainty is by using
Probabilistic Boolean Networks (PBNs) [25, 26]. These may be viewed as a collection
of (deterministic) Boolean networks combined with a probabilistic switching rule
determining which network is active at each time instant.
It is natural to extend the idea of Boolean networks to include input variables.
For example, an input may represent the dosage that is administered to a patient.
Boolean networks with (binary) inputs variables are referred to as Boolean Control
Networks (BCNs). PBNs with inputs were used to design and analyze therapeutic
intervention strategies. The idea here is to find a control that shifts the network from
an undesirable location (representing a “diseased” state) to a desirable state. Such
problems can be cast as stochastic optimal control problems, and solved numerically
using dynamic programming [27, 28].
Daizhan Cheng and his colleagues developed an algebraic state–space representa-
tion of BCNs using the semi–tensor product of matrices. This representation proved
quite useful for studying BCNs in a control–theoretic framework. Examples include
the analysis of disturbance decoupling [29], controllability and observability [30],
realization theory [31], and more [32, 33, 34].
Here we make use of this state–space representation to analyze a Mayer–type
optimal control problem for BCNs. Our main result is a necessary condition for a
control to be optimal. This provides a parallel of the celebrated Pontryagin max-
imum principle (PMP) (see, e.g., [35, 36, 37]) for BCNs. The proof of our main
result is motivated by the simple proof of a special case of the PMP used in the
variational analysis of switched systems [38] (see also [39, 40, 41]). The first result
in this direction appeared in our recent paper [42] describing a maximum principle
for the special case of single–input BCNs.
The remainder of this chapter is organized as follows. Section 2 reviews BCNs.
Section 3 describes Cheng’s algebraic state–space representation of BCNs using
the semi–tensor product of matrices. Section 4 details our main result which is a
4 Dmitriy Laschov and Michael Margaliot
x2x1
and
u2u1
andor
Fig. 1 Graphical representation of the BCN in Example 1.
new maximum principle (MP) for BCNs. Section 5 includes the proof of our main
result. In Section 6 we consider the so–called singular case where the MP does not
provide any direct information on the optimal control. Several synthetic examples
demonstrate the application of the new MP.
2 Boolean control networks
A Boolean control network is a discrete–time logical dynamic control system in the