its, this
method can be applied in the context of both logical and differential formalisms. This
approach already led to several interesting results about the relation between the network
structure and the corresponding dynamical properties. In particular, it could be shown that at
least one positive regulatory circuit is necessary to generate multistationarity (i.e., alternative
states of gene expression), whereas at least one negative circuit is necessary to generate a
stable oscillatory behavior. Applications to the analysis of complex gene networks, as well as
to the synthesis of regulatory models to account for global expression data are discussed.
1. Introduction
This decade will probably be remembered as the genome decade. Indeed,
almost a dozen of microorganism sequences have already been completed, including
mainly bacteria but also S. cerevisiae. In addition, many other genomic projects are
well on their way, including those dealing with Man, Mouse, A. thaliana, C.
elegans,
and D. melanogaster. However, there is a long way to go from a complete
genomic sequence to a functional understanding of the corresponding organism.
Even in the case of E. coli, the best characterized free-living organism, the recent
completion of the DNA sequence let us with a lot of open questions regarding gene
function, regulatory mechanisms, or global integration.
Besides genome sequencing, a series of large scale analyses have been initiated,
aiming at uncovering the functional organization of cells. In order to disentangle
gene regulatory networks at the level of the whole organism, several groups started
systematic global studies of gene expression and DNA-protein interactions in
different conditions (1, 39, 40). Clearly, such time or space scale snapshots of gene
expression in various conditions will be of great help in the delineation of the main
regulatory pathways. As a complement to these experimental approaches, there is an
increasing need for efficient theoretical tools and formal frameworks to derive
regulatory structures from partial expression data (3, 4, 15, 23).
About three decades ago, several groups independently started to develop
qualitative tools for the dynamical analysis of gene regulatory networks (5, 8-10). In
this paper, we review the work performed at the Université Libre de Bruxelles,
leading to the development of a set of theoretical concepts and formal tools which - 2 -
could be of some help to the global analysis of gene regulatory networks (25-28,
31-38). In the next section, we introduce the key concept of feedback circuit, as well
as a description of the classification and the properties of these circuits. The third
section is devoted to a brief description of the general logical formalism developed at
Brussels. The fourth section briefly discusses the use of the logical formalism vs.
the use of the more classical differential formalism. Finally, the fifth section
introduces two different uses of our qualitative method: an analytic or deductive
approach which proceeds from the model to its implications, and a synthetic or
inductive approach which proceeds from the experimental data to possible models.
2. Biological and dynamical and roles of feedback circuits
A feedback circuit (or feedback loop) is just a circular chain of interactions.
Most often in biology, these interactions have defined positive or negative signs.
For any circuit, one can easily check that each element exerts an indirect effect on
itself which has the same sign for all elements of the circuit, leading to the
definition of the circuit sign. In fact, this sign only depends on the parity of the
number of negative interactions involved in the circuit: if this number is even, then
the circuit is positive; if this number is odd, then the circuit is negative.
Biologists have been aware of the interesting properties of specific genetic or
biochemical (or mixed) feedback circuits for a long time (e.g., 13, 19). However, the
clear delineation of the two classes of feedback circuits and their strikingly different
properties is more recent (see Table 1). It has been conjectured (34) and more
recently demonstrated (6, 16, 22, 28, 37) that a positive circuit is a necessary
condition for multistationarity, and a negative circuit (with two or more elements)
for stable periodicity. Biologically, this means that positive circuits are required for
differentiative decisions and negative circuits for homeostasis.
Characteristics
Positive circuits
Negative circuits
Number of negative interactions
Even
Odd
Typical dynamical property
Multistationarity
Periodicity
Typical biological property
Differentiation
Homeostasis
Table 1.
Main characteristics of positive and negative feedback circuits.
It is essential to clearly realize that appropriate circuits are necessary but not
sufficient conditions. Indeed, in order to actually manifest multistationarity or stable
periodicity, the system must also display appropriate nonlinearities and proper
parameter values. We say that a feedback circuit is functional when it actually
generates the dynamics corresponding to its sign. - 3 -
In the context of the logical description, we associate a characteristic state to
each feedback circuit (or union of circuits), which is defined as the state located at
the threshold values involved in the circuit. In fact, it can be shown that the
parameter conditions required to have a circuit functional are identical to those
required to have the corresponding characteristic state stationary (21).
This concept of characteristic state can be extended to continuous descriptions as
follows. When a positive circuit is functional, it usually generates a separatrix. On
this separatrix is found one of the steady states of the system, e.g., a saddle point for
a two-variable positive circuit. This unstable steady state, always found in
association with the property of multistationarity, is thus called the characteristic
state of the circuit. Similarly, when a negative circuit is functional, one finds a
steady state associated with the periodic motion. This steady state is typically a
focus (although it may be an unstable node) in the case of a two-element negative
circuit (see also 35 and 38).
3. Kinetic logic and its application to gene networks
Biological regulatory interactions are usually nonlinear, thus rendering analytical
approaches problematic. This is why Sugita (24), Kauffman (8-10) and others
looked for a qualitative representation of regulatory networks. Our group was led to
develop an asynchronous logical formalization whose generalized version can be
characterized as follows:
1) Asynchronous updating of the state vector (31).
2) Whenever needed, use of multilevel variables (x
i
) and functions (X
i
) (33).
1
3) Explicit consideration of threshold values for the variable and functions (35).
Thus, x
i
and X
i
{0, s
(1)
, 1, s
(2)
, 2, ...}; states involving only integer values (0, 1,
2, ...) are called regular states (e.g., 00, 01, etc.), whereas states involving one or
more threshold values are called singular states (e.g., 0 s
(1)
, s
(2)
s
(1)
, etc.).
4) Use of logical parameters to quantify single interaction or combinations of
interactions exerted on a same element (e.g., K
i.i
, K
i.ij
, K
i.ijk
, etc.); these
paperameters can take the same values as the corresponding variable x
i
(20, 35).
Formally, a regulatory network can be fully described by a set of three matrices,
which contains the signs of the interactions, the thresholds associated to these
interactions, and the values of the corresponding logical parameters, respectively.
As an illustration, we present below the matrices associated to a simple three-
element network, whose concrete concrete nature will be described in section 5. The
matrix of interactions is:

1
Even though logical variables and functions have the same dimension, it is
convenient here to assimilate the variables to the presence/absence of the gene
products, and the functions to the state on or off of the genes. - 4 -
a
b
c
a
+
+
-
b
+
+
c
-
+
+
in which box 11 (first row, first column) tells us that gene a regulates itself
positively; box 12 (first row, second column) tells us that gene b activates the
expression of gene a, etc. Just by looking at this matrix, we can identify all the
feedback circuits of the system: two positive one-element circuits involving genes a
and c, respectively; three positive two-element circuits involving genes a and b, a
and c, and b and c, respectively; plus two negative three-element circuits: abc and
acb
. Now, let us consider the following threshold matrix:
a
b
c
a
1
1
1
b
2
2
c
1
1
1
in which box 11 tells us that auto-regulation of gene a occurs over the first
functional threshold of its product; box 21 (row 2, column 1) tells us that the
activation of b by a occurs over the second functional threshold of product a.
2
Note
that to both genes a and c are associated three-level logical variables (taking the
integer values 0, 1 and 2), whereas a Boolean variable is associated to gene b. We
use multilevel variables only when it is biologically justified (see section 5).
Finally, we introduce the following matrix of logical parameters:
K
i
K
i.1
K
i.2
K
i.3
K
i.12
K
i.13
K
i.23
K
i.123
a
(i=1)
0
0
0
0
0
1
1
2
b
(i=2)
0
1
0
1
1
c
(i=3)
0
0
0
0
1
1
0
2
in which the first column (K
i
s) gives the logical weight of the basal expression,
i.e., in the ab