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Dealing with Discontinuities in the Qualitative Simulation of Genetic Regulatory
Networks
Hidde de Jong, Jean-Luc Gouzé,
Céline Hernandez, Michel Page,
Tewfik Sari, Johannes Geiselmann
INRIA Rhône-Alpes,
655 avenue de l’Europe, Montbonnot, 38334 Saint Ismier CEDEX, France
INRIA Sophia-Antipolis
ESA, Université Pierre Mendès France, Grenoble
Laboratoire
de
Mathématiques,
Université de Mulhouse,
Laboratoire Plasticité et Expression des Génomes Microbiens, Université Joseph Fourier, Grenoble
Hidde.de-Jong, Celine.Hernandez, Michel.Page @inrialpes.fr, [email protected], [email protected],
[email protected]
Abstract
Methods developed for the qualitative simulation of dynamical systems have turned out to be powerful tools for studying genetic regulatory networks. We present a generalization
of a simulation method based on piecewise-linear differential
equation models that is able to deal with discontinuities. The
method is sound and has been implemented in a computer
tool called GNA.
Introduction
Methods developed for the qualitative simulation of dynamical systems have turned out to be powerful tools for studying
the networks of regulatory interactions between genes, proteins, and small molecules which underlie the functioning
of living organisms. The interest in qualitative methods for
analyzing these genetic regulatory networks derives from a
general absence of quantitative information on kinetic parameters and molecular concentrations. As a consequence,
traditional methods for numerical simulation are difficult to
apply (see (de Jong 2002) for a review).
The qualitative simulation method described in (de Jong
et al. 2001b) is able to handle large and complex networks
of regulatory interactions. The method is based on a class
of piecewise-linear (PL) differential equations that has been
well-studied in mathematical biology (Glass & Kauffman
1973; Mestl, Plahte, & Omholt 1995; Snoussi 1989). The PL
models provide a coarse-grained description of genetic regulatory networks, well-adapted to state-of-the-art measurement techniques in genomics. Moreover, they have mathematical properties that favour qualitative analysis of the
steady-state and transient behavior of regulatory systems.
The qualitative simulation method has been implemented in
a publicly-available computer tool, called Genetic Network
Analyzer (GNA) (de Jong et al. 2002). The program has
been used to analyze several genetic regulatory networks of
biological interest, including the network controlling the initiation of sporulation in B. subtilis (de Jong et al. 2001b).
A short version of this paper appears in F. van Harmelen (ed.),
Proceedings of the 15th European Conference on Artificial Intelligence, ECAI 2002, IOS Press, Amsterdam, 2002.
The PL models contain step functions describing the regulatory interactions in a network. This introduces discontinuities in the right-hand side of the differential equations,
which may give rise to nontrivial mathematical problems
(Plahte, Mestl, & Omholt 1994). In the above-mentioned
simulation method, these problems were treated by redefining the discontinuous, piecewise-linear models as a limit
case of continuous, nonlinear models (de Jong et al. 2001b).
Unfortunately, the resulting simulation method is not sound,
in the sense that it is not guaranteed to predict all possible
qualitative behaviors of the system (Kuipers 1994). Soundness is critical for many applications, such as model validation and model discrimination.
Recent work on the generalization of the PL differential
equations to differential inclusions allows the discontinuities
to be dealt with in a mathematically proper and practically
useful manner (Gouzé & Sari 2001). The generalization is
based on an approach developed by Filippov (1988), which
has been widely used in control theory. The major contribution of this paper is to show how the method for the
qualitative simulation of genetic regulatory networks can
be generalized in the line of Filippov’s work. This generalization results in a simulation algorithm with the desired soundness property. Beyond their application in the
context of genetic regulatory networks, the ideas underlying the generalization may be a useful complement to other
methods for the qualitative analysis of dynamical systems
based on the approximation of complex nonlinear functions
by piecewise-linear functions (Nishida & Doshita 1995;
Sacks 1990).
After a brief review of PL models of regulatory networks,
we will discuss the Filippov generalization. The qualitative
simulation method based on this approach is described, followed by its application to a real example, the regulatory
network underlying the initiation of sporulation in B. subtilis. A discussion in the context of related work concludes
the paper.
PL models of genetic regulatory networks
The dynamics of genetic regulatory networks can be modeled by a class of piecewise-linear differential equations
of the following general form (Glass & Kauffman 1973;
Mestl, Plahte, & Omholt 1995; Snoussi 1989):
"!#
consists of a sum of step function expressions, each weighted by a rate parameter, which expresses
I
the logic of gene regulation (Mestl, Plahte,
1995;
+ HJI K & Omholt
FON
Snoussi 1989). The function 7 :MDF
is defined
analogously.
Figure 1 gives an example of a simple genetic regulatory
network. Genes a and b, transcribed from separate promoters, encode proteins A and B, each of which controls the
expression of both genes. 1 More specifically, proteins A and
B repress gene a as well as gene b at different concentrations. Repression of the genes is achieved by binding of the
proteins to regulatory sites overlapping with the promoters.
The network in figure 1 can be described by means of the
following pair of state equations:
]YQ^%,Q
Z \
`] Z % Z '
Z \
(2)
(3)
R#Q$acb
Gene a is expressed at a rate
, if the concentration of protein A is below its threshold X Q and
the con
centration of protein
B below its threshold X Z , that is, if
T U &%[QC*X Q YT U %,Z)*X Z d
T %e_Xf
< . Recall that U
is a step
%hgiX
, and to 0, if %hajX . Protein
function evaluating to 1, if
A is spontaneously degraded at a rate proportional to its own
concentration (] Q a"b is a rate constant). The state equation
of gene b is interpreted analogously.
B
A
a
-
-
b
Figure 1: Example of a genetic regulatory network of two
genes (a and b), each coding for a regulatory protein (A and
B).
The dynamical properties of PL models of the form (1)
H,phase
Ion
space box k can be analyzed in the B -dimensional
'('(' l
:^ %,:Em F
b
%J:
:
+
A
l
k
k , where k
=
=prq)s ,
:
<t=u?v=wB , and pLq(s is a parameter denoting a maximum
concentration for the protein.
In general, a protein encoded by a gene is involved in
different interactions at different threshold
concentrations,
which after ordering are denoted by X : (')'('(_Xyx(: z . The B < dimensional hyperplanes %\:{|Xf: } z , <t=$~ : =S : , divide k
1
n
: % : b
% :
=
n
:e %,: %J:"X

n
: % : X : gj%

(1)
where $&% ('(')'*%,+-/. is a vector of cellular protein concentrations, and 0132 ('()' '(425+ . , 6 diag 87 (')'(')/79+ .
The rate of change of each concentration %;: , <>=@?A=@B , is
defined as the difference of the rate of synthesis 2,:*3 and
7 : 3C% :
HJI
the rate+ HJof
of the protein. The function
ILK degradation
F
2 :EDGF
% P QMSR,QTVUW&%,Q*X Q Y
,
T9UO&%[Z)*X
% P Z SR Z T U % Q _X Q -T U &% Z _X
into regions that are called domains. More precisely, a domain k is defined by l '(')' l + , where every
:
, <M=i?;=iB , is defined by one of the equations below:
As a notational convention, names of genes are printed in italic
and names of proteins start with a capital.

g
X :
: : giX : ')'('
(4)
n
:e ,
% : y
X x: z gi%,:
: '

=pLq(s If for a domain , there are n some ? , <i=?=B , < =

:
:W
%J:
%J:uXy:
=" , such that
, then is called a
%O:
switching domain. The corresponding variables are called
switching variables. The order of a switching domain is a
number between 1 and B , equal to the number of switching
variables. A domain that is not a switching domain is called
a regulatory domain. denotes the set of domains in k .
In figure 2(a) the two-dimensional phase space box k
for the example network is shown. As proteins A and B
each have two thresholds, the phase space box is partitioned
into
16n switching domains.
For example,
9 regulatory
&% Q *% Z Land
m F b
% Q gX Q Wb
% Z g|X Z
=
=
%,QC_%,ZMm F n b is
=
a regulatory
domain, whereas
%,QgiX Q \%,Z;SX Z
is a (first-order) switching domain.
When evaluating the step function expressions in (1) in
a regulatory domain, 2 : and 7 : reduce to sums of rate conn
stants. More precisely, in
domain , 2,: reduces
a regulatory
to some ^: m@: 2V:/ n ! = =6 , and 7:
79:*3
!
=
=
. Inside
to some Y: m:
, the state equations thus simplify to linear and decoupled
differential equations
S `
)'(')' + *.

and #
#
(5)
where
diag Y
)'('(') + .
Since the step functions are not defined at the thresholds, the
state equations are not defined in the switching domains.
m
For every regulatory domain
, we define the
function : #:¡ : . Analysis of (5) shows that
all solution trajectories in monotonically tend towards
a£ target equilibrium,
a stable equilibrium given by ¢
£
¤ (')'(') +Y/.
(Edwards et al. 2001;
, with

Snoussi 1989).
Q bYR Q
Q ] Q
In the example, we have
for
b-R Z
] Z,
Z Z protein A, and
,
for protein B.
In regulatory domain
in figure
the trajectories tend
£ 2(a),
towards the target equilibrium ¥¦R Q ] Q 4R Z ] Z .
Different regulatory domains generally have different
target
equilibria. For instance, in regulatory domain , the target
equilibrium is given by b-R^Z ]YZ_ .
The global solution of (1) could be obtained by piecing together the local solutions in regulatory domains, in
such a way as to guarantee continuity of the global solution across the threshold hyperplanes (Edwards et al. 2001;
Snoussi 1989). This works fine as long as trajectories arriving at a threshold hyperplane can be continued in another
regulatory domain,
e.g., trajectories arriving at
the switching
domain
from the regulatory domain
(figure 2(a)).
However, when the trajectories on both sides of a threshold hyperplane evolve towards this plane, as in the case of
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,
²³ ´Jµ·¶
® ¯
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°
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(a)
(b)
(c)
£ £ £ Figure 2: (a) Phase space box for the
example
network
in
figure
1.
,
,
denote
the
target
equilibria
of

the regulatory domains , , , and are assumed to lie in the upperright, upperleft, and lowerleft
regulatory
domains,
respectively. In addition, the figure
shows the
discontinuities at the switching domains
and . (b)-(c) Determination of
the target equilibrium sets Ñ
and Ñ
.
trajectories arriving from
and
at , mathematical
perplexities arise.
There
is
no
indication
on how the local
solutions in
and
can be continued.
Analysis of discontinuities in PL models
Û$
The troubles at the threshold hyperplanes are caused by discontinuities in the right-hand side of (1), due to the use of
step functions. In order to deal with these discontinuities, we
will use a method originally proposed by Filippov (1988).
This method, recently applied by Gouzé and Sari (2001) to
PL systems of the form (1), consists of extending a system
of differential equations with discontinuous right-hand sides
into a system of differential inclusions.
Let be a switching domain of order ~ . Let Ò be the
hyperplane of dimension B ~ containing . The boundary
of in Ò is the set Ó of all points m Ò , such that each
ball ÓGÔ /Õ9 in Ò of center and radius Õha0b intersects
both and ÒwÖG (Kelley 1969). In the case that is a
regulatory domain, Ò equals k .
Now, for every m we define the sets
× Ø × ;

n
. m

. m

n
.

iÓ
.

Notice that, since the set Ûu3 contains a single element,
the extension of the PL system agrees with the original system in the regulatory domains. If is a switching domain,
Ûu3
is defined by
and
.
regulatory domain, iÓ '
Ø contains the domains in the boundary of , whereas

contains the regulatory domains that have in their
boundary.
In the× case
of the regulatory
domain × in figure 2(a),
L

tÚ5 .
we find Ø > >Ù >Ø Ú5 , while
O and

.
Furthermore,
The basic idea of the Filippov approach is to extend the
differential equations (1) into differential inclusions
mÜÛ|
i
(6)
KÞÝ
D
k
k
where Û
is a set-valued function. 2
jm
Ûu3
, and a regulatory domain, we define
For
simply as
2
Û$;

'
For a set ß , àá&ßAâ represents the power set of ß .
(7)
. m

Ø
ãä3æ
'
(8)
æ
The smallest closed convex set
of a set is the intersection of all closed convex sets containing æ (Filippov
1988). In the case of switching domains, Û| will not
generally be single-valued.
Let ç è*'(')'(*X: } z )'('('º/. , é è*'(')'(RJ:Íê/(')'('º/. , and ë 8] (')'(')/]+Y/.
be
I numerical parameter values. Furthermore,
m
I
An
k represent the initial conditions.
let bf`¢
absolutely continuous function &ì*íe&ìbY ç é ë is
b
a solution I of (6) in the sense of Filippov on î b-*ï î , ï¥a6
, if
W3bf@
and for almost all ìAm î bY*ï î it holds that Wì*vm
Ûu3&ì**
(Filippov 1988). The qualification ‘for almost all
ì>m
bY_ï
î
î ’ means that the set of time-points for which the
I
condition
does not hold is of measure 0. For all initial values
m
bY*ï
k there exists a solution of (6) on î
î . However, this
solution is not guaranteed to be unique.
For every domain , a so-called target equilibrium set
Ñ
can be defined. If is a regulatory domain, then

n
ãä ;å ` å Ñ

£
'
(9)
If is a switching domain, the definition is a little bit more
complicated. Let be a switching domain of order ~ , contained in the B ~ -dimensional hyperplane Ò . Then
Ñ

Ò"ð
ã(äW £
.
n

. m
Ø
'
(10)
That is, Ñ
is the smallest closed convex set of the target equilibria of regulatory domains . having in their
boundary, intersected with the hyperplane containing .
A solution may instantaneously cross a switching domain
or remains in it for some time ïñaòb , sliding along the
threshold hyperplane containing the domain. Gouzé and
Sari (2001) have shown that the latter sliding mode solutions
. The sliding
exist in a switching domain , iff Ñ ó
mode solutions monotonically tends towards Ñ (Gouzé
& Sari 2001). Because Ñ does not generally include a
single point, the behavior of the system is not uniquely determined
by the differential inclusion (6).
ô
Consider the examples
in figure 2(b)-(c). The target equi
librium set Ñ of the switching domain
£ is£ defined,
following (10), by the intersection of ãä X Z
and the threshold hyperplane %Z
. The smallest
closed convex set consists of the linear segment connecting
the points 3R Q ] Q R Z ] Z and b-R Z ] Z . Ñ and the
threshold
plane %eZ1X Z do not intersect in the figure,
so
Ñ
and all solutions instantaneously

cross
.
This is different in the
case of . Here, the tar
get equilibrium
set Ñ is given by the intersection of
ãä £ £

, the linear segment connecting the
points bY4R Z ] Z and b-_b , and
the threshold
hyperplane
%,ZG0X Z
. Consequently, Ñ equals b-*X Z , and there
exists a (unique)
sliding mode solution in , tending to
wards 3bY_X Z . Because the target equilibrium lies inside ,
it is also a steady state of the
system. Closer analysis reveals
that the equilibrium b-*X Z is stable. Notice the intuitive
validity of the Filippov approach: solutions arriving at
from
or
slide along the threshold plane towards the
equilibrium.
Method for qualitative simulation
Qualitative constraints on parameters
Most of the time, precise numerical values for the threshold
and rate parameters in (1) will not be available. Instead, we
will specify qualitative constraints on the parameter values,
as explained in (de Jong et al. 2001b). These constraints,
having the form of algebraic inequalities, can usually be inferred from biological data.
The first constraint is obtained by ordering the : threshold concentrations of gene ? , yielding the threshold inequalities. In the case
of protein
A, there are two threshold con
centrations: X Q and X Q . Assuming the first to be lower than
the
second, we obtain the threshold inequalities bg|X Q g
X Q g
Q
prq)s . The
of the thresholds of protein B
ordering
give rise to brgjX Z gjX Z g pLq(s Z .
Second, the possible target equilibrium levels W: :
of % : in different regulatory domains m can be ordered with respect to the threshold concentrations. The resulting
equilibrium inequalities for %\Q in the example are
Q
X Q gR Q ] Q g
pLq(s . In the absence of protein B, while
protein A has not yet reached its highest level, gene a is expressed at a rate R Q . The corresponding target equilibrium
value R[Q ]Q of %[Q must be above the second threshold X Q ,
otherwise the concentration of the protein would not be able
to reach or maintain a level at which the observed negative
autoregulation
of gene a occurs. In a similar way, we set
X Z g"RJZ ]YZg
Z
%õZ
prq)s for .
A quantitative PL model of a genetic regulatory network
consists of state equations (1) and numerical parameter values ç é ë . In a qualitative PL model, on the other hand,
the state equations are supplemented by threshold and equilibrium inequalities. Every quantitative PL model can be
abstracted into a unique qualitative PL model, while a qualitative PL model corresponds to a set of quantitative PL models.
Qualitative states and behaviors
An intuitive qualitative description of the state of a regulatory system consists of the domain in which the system
resides, supplemented by the position with respect to this
domain of the target equilibrium set to which the state of the
system tends. A qualitative behavior is then given by the
sequence of qualitative states traversed by
K the
+
system.
_bY
We first define a function ö D l k
<
<V that
maps a domain and a point ÷ to a sign vector ø describing
the relative position of and ÷ . If % : is a non-switching
variable, then
ú
ù : ûü
b
if ý : "þ_ÿ
if J : g ý
if ý : =J
:
<
<
,
: gjþ_ÿ
:

:
,
.
On the other hand, if % : is a switching variable, then : Xy:
, and
ú
ù :
ûü
b
: a Xy:
if ý ,
: "Xy:
if ý
,
if ý : g Xy: .
<
<
D
Generalizing
Ý
KÞÝ the definition,+ we obtain the set function _bY
l
that maps a domain n and a set

k
<
<V
æ
to a set of sign vectors:
_æ ö ÷ ÷ mæ .
I
Let Wì*;íõì_b-_ ç é ë be the solution of a quantitative PL model describing a regulatory network on the timeinterval î b-*ï î . Now suppose that for some ì , b = ì{g"ï , we
have &ì*>m , m . The point &ì* corresponds to a
qualitative state of the system defined by
*ì*

*'
I
&ì*
The solution
on î b-*ï î passes through a sequence of
(')'('(
domains
. The corresponding sequence of qualitative states is called the qualitative behavior of the system
on the time-interval. More specifically, a qualitative behavior of the system is defined by
áâ "
á!"$#%&Má"$#'(Wá"$#ââ)+*+*,*+-!"$./&Má"$./(Wá"$.{ââ)â
Consider the solution trajectory in figure 3(a), obtained
for given exact
parameter values, which moves from
an ini
tial state in
towards a stable equilibrium in . Following the above definitions, the
can
into
be abstracted
10 solution
¤ ,
the qualitative
behavior
2
2
where
,
,

<
<

3bY < , and 3 4 3bY_b . For regulatory
3R Q ] Q R Z ] Z domain , we have Ñ v
. For the
R Q ] Q aX Q aX Q parameter values
in
figure
3,
we
find
and
RJZ ]YZÜaX Z añX Z _t

Ñ
. As a consequence,
5 < < , and hence

< < . In the case of
the switching domain , the
closed convex set
smallest
of the target equilibria in
and
consists of the linear segment connecting R^Q ]YQfR,Z ]Z_ and bY4R[Z ]Z . For
the parameter values in figure 3, this segment
does not
in
% Z X Z
_ tersect
with
,
so
that
and

3 6
. The other qualitative states are derived
analogously.
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´JÁ
´-¸
´JÂ
78
²³ ´JµÏ¶
9:A@
9:>?
7
9:>=
´,¹
9:<;
® ¯
°
§ °
®±
°
(a)
ª;«/¬ °
9,:D
9:BC9,: ; ;
9: ; B
(b)
Figure 3: (a) Solution trajectory of the PL model (2)-(3), obtained for given numerical values for the parameters and initial conditions. (b) Transition graph resulting from a simulation of the example system starting in the domain . Qualitative states associated with regulatory domains and switching domains are indicated by unfilled and filled dots, respectively. Qualitative states associated with domains containing
an equilibrium point are circled (de Jong et al. 2001a).
The example shows how a numerical solution of the system can be abstracted into a qualitative behavior consisting
of a sequence of qualitative states. More generally, every
solution of a quantitative PL model on a time-interval
bY*ï
î 10 î can be abstracted into a unique qualitative behavior
3_bY_ï-
.
Simulation algorithm
Given a qualitative PL model and qualitativeI initial conditions consisting of a regulatory domain , one can ask
what are the possible qualitative behaviors of the system.
Determining these qualitative behaviors is the aim of qualitative
simulation. Phrased in a different way, denoting by
E
the set of solutions ì* on some time-interval î b-*ï î of
I
all quantitative PL models corresponding
to the qualitative
I
3bfr
m
, the aim of qualitative
model, such that

simulation is to find the
E set of qualitative behaviors that ab.
stract from some m
The simulation algorithm generates a set of qualitative behaviors by recursively determining transitions between qualI
itative states, starting from the qualitative state associated
(de Jong et al. 2001a). In order
with the initial domain
to achieve this, two issues need to be addressed. First of all,
how can we determine the qualitative state associated with
a domain from the constraints on the parameters? Second,
how can we find the possible transitions from this qualitative
state?
In order to determine the qualitative state associated with
a domain , we need to derive Ñ * from the threshold and equilibrium inequalities. This is achieved by computing -F _ , where F k is a hyperrectangular, closed convex set that is sure to include Ñ , but
that may be an overapproximation of the latter. The details
of the procedure, which consists of a rather straightforward
comparison of the upper
lowerbounds of and£ either
£ and
or
the target equilibria .¾ ,
the target
equilibrium

.dm Ø
, depending on whether is a regulatory or
switching domain, are given in (de Jong et al. 2001a).
The possible transitions are defined by two transition
rules. The relative position of the domains and . is
given by .Ê . As can be easily verified, .8
always
consists of a single sign vector, that is, .¾
'G
.
with qualitative
. The
3 domains
3 . and are associated
and
, calculated to be +F * and
states
.3 .+F .¾* , respectively.
×
3
.{m
Rule
is a transition from
to
3 . 1 Let -F _ . There
ó % :
, if (1)
is a switch
, and (2) if
ing variable in . , but not in , then there is some H m
-F _ , such that ö :JI:^ < .
m × .8
Rule
. There is a transition from

2. Let
.-F .¾* ó
%#:
to
, if (1)
is a

, and (2) if
switching variable in , but not in . , then there is some
ó H .õm .-F .¾* , such that ö :. I : 0
< .
Intuitively, the first transition rule says that, in order to
enter a switching domain . in the boundary of , some
trajectories must tend towards . (condition (2)). If is
a switching domain, then there must exist sliding mode trajectories in (condition (1)). The second transition rule
says that, in order to enter a domain . from a switching
domain in the boundary of . , the trajectories in . must
not tend towards (condition (2)). If . is a switching domain, then there must exist sliding mode trajectories in .
I
(condition (1)).
Given an initialI domain , describing the initial protein
I
algorithm computes the
concentrations , the simulation
I initial qualitative state Ý
, and then determines all possito successor qualitative states by
ble transitions from K
means of the rules above (de Jong et al. 2001a). The generation of successor states is repeated in a recursive manner.
This results in a directed graph of qualitative states and transitions, the state transition graph, which contains all qualitative states reachable from the initial qualitative state. The
simulation algorithm has been implemented in a new version
of the program GNA (de Jong et al. 2002) (figure 4).
Figure 3(b) shows the transition graph for a qualitative
simulation of the
example system, starting in the regulathe possible transitions from
tory domain . Consider
associated with regulatory domain
the qualitative state
to × qualitative
with the boundary do
associated
states
mains >

>Ú L NM5 . We have to verify whether the conditions
(1) and (2) of
rule 1 are verified. -F * is calculated to be 3bY < , by means
of the procedure
in (de Jong et al. 2001a), while bY %õZ
equals
a switching variable in , but not
< . With
in , we find that (1) and (2) are
Consequently,
satisfied.
there
exists
a
transition
from
to
.
Transitions
from
3
to the other candidate successor states are ruled out,
because they violate condition (2).
Soundness
I
Given a qualitative PL model and an initial regulatory domain , what can be said about the correctness of the behaviors produced
by qualitative simulation? We demand that
Sm E
,
the transition
a qualitative
for every 1
0
O0 1graph
0 contains
bY_ï-
, such that
(soundness).
behavior
Figure 4: Screen capture of the qualitative simulator GNA. The leftmost window shows the genes and interactions of the
example network in figure 1, the middle window the state transition graph resulting from simulation of the network, and the
rightmost window the temporal sequence of qualitative states in one selected path in the state transition graph.
Theorem 1 The qualitative simulation algorithm is sound.
E
I
I
Proof sketch Let be a solution in . On î bY*ï î , I &ì* tra(')'('(
m
, where

verses a sequence of domains
.
Like in the proof of the soundness of QSIM (Kuipers 1994),
it
10 can be shown by induction that the qualitative behavior
3_bY_ï-
is generated by the algorithm. The proof in (de
Jong et al. 2001a) rests on two properties of the simulation
algorithm. First of all, the set F used in the previous
section is sure to include Ñ . Second, the transition rules
cover all solutions
a domain
× leaving
× .¾ and entering a do
main . , .õm
or m
.

P
Application
The use of GNA will be illustrated by modeling and simulating the regulatory network underlying the initiation of
sporulation in B. subtilis. While nutrients are plentiful,
B. subtilis divides as fast as possible in order to efficiently compete with its neighbors. However, when conditions become unfavorable, the bacterium protects itself by
forming environmentally-resistant spores (Grossman 1995;
Hoch 1993; Stragier & Losick 1996). A graphical representation of the network, following the conventions proposed
by Kohn (2001), is shown in figure 5.
On the basis of the extensive literature on B. subtilis, and
information contained in the database SubtiList (Moszer,
Glaser, & Danchin 1995), a model of the regulatory network controlling the initiation of sporulation has been constructed. The model of the network comprises 9 state variables and 2 input variables. The state equations are supplemented by 32 threshold inequalities and 34 equilibrium
inequalities.
The network is centered around a phosphorylation pathway, the so-called phosphorelay, which integrates a variety
of environmental, cell-cycle, and metabolic signals. Under conditions appropriate for sporulation, the phosphorelay transfers a phosphate to the Spo0A regulator via a sequence of phosphorylation steps modulated by kinase and
phosphatase proteins. The phosphorelay has been simplified
in this paper by ignoring intermediate steps in the transfer of
phosphate to Spo0A.
When input signals in favor of sporulation arrive, the
concentration of Spo0A Q P reaches a threshold value above
which it activates various genes that commit the bacterium to
sporulation. In order to produce a critical level of Spo0A Q P,
signals arriving at the phosphorelay need to be amplified and
stabilized. This is achieved by a number of positive and negative feedback loops controlling the activity of the phosphorelay through transcriptional regulation of its components.
GNA has been used to simulate the network underlying
the initiation of sporulation from initial conditions reflecting
a perturbation of the vegetative growth conditions. The perturbation consists in an external signal indicating a state of
nutritional deprivation, which causes KinA to autophosphorylate.
Simulation of the network takes a few seconds to complete on an average PC, and gives rise to a transition graph of
465 states. Eliminating the states corresponding to switching domains without sliding mode solutions, which are of
limited interest from a biological point of view, yields a reduced transition graph of 82 states. All qualitative behaviors
end up in a single qualitative equilibrium state, associated
with a regulatory domain containing a stable equilibrium of
the system.
The qualitative equilibrium state corresponds to the spo U
phenotype, because the concentration of RTS , a transcription
factor encoded by the spoIIA operon that is essential for the
development of the forespore (Stragier & Losick 1996), has
not reached the threshold above which it directs the transcription of its target genes. Whereas in some qualitative
behaviors the system directly reaches the qualitative equilibrium state, in others it first passes through a period of
transient sigF expression, which can be interpreted as corresponding to the spo U phenotype. We conclude that the transition graph faithfully represents the two possible responses
VX
KinA
kinA
Signal
Legenda
VW
Spo0E
Spo0A
phosphorelay
spo0E
spo0A
Z
A
VAX V W
Synthesis of protein A from gene a
a
Spo0A P
Activation
Inhibition
SinI
Z
A B
A
AbrB
abrB
VW
VWVW
VW
VAW
Covalent binding of A and B
sinR
VAX
VAY
sigH
sigF
VW
sinI
B
SinR
VAX
Hpr
hpr
Figure 5: Key genes, proteins, and regulatory interactions making up the network involved in B. subtilis sporulation. In order to
improve the legibility of the figure, the control of transcription by the sigma factors R\[ and R ] has been represented implicitly,
by annotating the promoter with the sigma factor in question.
to nutrient depletion that are open to the cell: either it continues vegetative growth or it enters sporulation.
Discussion
We have presented a method for the qualitative simulation of genetic regulatory networks described by a class of
piecewise-linear (PL) differential equations. The method is
an extension of (de Jong et al. 2001b), which allows one
to deal with discontinuities in the right-hand side of the differential equations, occasioned by the use of step functions.
The qualitative simulation method is supported by the computer tool GNA, which has been used to analyze the network underlying the initiation of sporulation in B. subtilis,
described in (de Jong et al. 2002).
Several ways to deal with the step function discontinuities
in (1) have been proposed in the literature, such as restricting
the analysis to an easy-to-handle subclass of PL models (Edwards et al. 2001) or relaxing the discontinuous PL models
to continuous nonlinear models (Mestl, Plahte, & Omholt
1995; Nishida & Doshita 1987). The approach presented
here has the advantage of putting no restrictions on the class
of genetic regulatory networks that can be handled, while
explicitly defining the behavior of the system in the threshold planes by means of simple-to-analyze PL models. On a
formal level, the generalized logical method of Thomas and
colleagues (Thomas, Thieffry, & Kaufman 1995) is related
to the method presented here. For a subclass of the PL models (1), the logical method can identify equilibrium points
in the threshold hyperplanes (Snoussi & Thomas 1993), but
a general way to deal with the discontinuities is currently
missing.
PL models of the form (1) can be interpreted as representing a class of hybrid systems (Ghosh & Tomlin 2001;
Alur et al. 2001), consisting of modes in which the system
evolves in a continuous way and discrete transitions between
the modes controlled by a switching logic. Ghosh and Tomlin give a hybrid-system formalization of genetic regulatory
networks described by (1), but do not deal with the discontinuity problems discussed in this paper. In order to handle
discontinuities entailed by mode transitions, hybrid-system
simulation methods based on Filippov solutions have been
developed (Mosterman, Zhao, & Biswas 1999). These methods are suitable for (semi-)quantitative, but not for qualitative PL models.
The qualitative simulation method discussed in this paper
applies to PL models of the form (1), which are applicable in a range of domains. In addition to genetic networks,
they have been used for modeling food webs, neural networks, and biological computers. However, the ideas underlying the generalization of the method seem more widely
applicable in qualitative reasoning, especially when nonlinear functions are approximated by piecewise-linear functions (e.g., (Nishida & Doshita 1995; Sacks 1990)). This
may give rise to discontinuities on the boundaries separating
the regions where the system behaves linearly, as for the PL
models described in this paper. The methods for the analysis of piecewise-linear systems cited above are not able to
deal with the problems occasioned by sliding modes (figure 2(a)). The same holds for general qualitative simulation
methods like QSIM (Kuipers 1994). The extension of the
Filippov approach to other classes of piecewise-linear differential equations, or even to nonlinear differential equations
in general, leads to a host of difficult but interesting research
problems.
A simulation study using randomly-generated PL models
of genetic regulatory networks, carried out with a previous
version of the simulation method, has shown that large and
comp
^ lex networks can be handled (de Jong & Page 2000).
As would be expected, the generalized method presented
in this paper produces transition graphs that are considerably larger than before. In addition to qualitative states associated with regulatory domains, we now also distinguish
qualitative states associated with switching domains. However, qualitative states corresponding to switching domains
without sliding mode solutions can be ignored, and hence
eliminated from the transition graph. Qualitative simulation of the sporulation network leads to a transition graph
with 465 states, only 82 of which are associated with regulatory domains or switching domains with sliding mode solutions. Further work should establish whether the upscalability property is retained more generally.
The soundness of the qualitative simulation method guarantees that no solution of a quantitative PL model consistent
with the qualitative PL model is omitted. In fact, to each
such solution there corresponds a qualitative behavior in the
transition graph. The soundness of the simulation algorithm
is critical for future extensions of the method towards model
validation and model discrimination.
Acknowledgments The authors would like to thank
Grégory Batt and Ivayla Vatcheva for comments on a previous version of this paper.
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