Logical modelling of cell cycle control in eukaryotes: a comparative
study.
Adrien Fauré1 , Denis Thiery1,2
April 8, 2009
1
Aix-Marseille University & INSERM U928 - TAGC,
immune response [10]...
Unrestricted proliferation is a
Marseille, France.
hallmark of cancer, and cell division plays a role in the
2
development of various diseases. Indeed, therapeutic ap-
CONTRAINTES Project, INRIA-Rocquencourt, Le
Chesnay, France.
proaches [11] that target the cell cycle [12] are considered promising, and related anti-cancer [13, 6, 5], antiviral [14] and anti-fungal/bacterial drugs (antibiotics) [15]
Contact: {faure,thiery}tagc.univ-mrs.fr
have been, or are being developed.
During the cell cycle, a eukaryotic cell goes through
Abstract
a series of well dened phases, in the course of which it
grows and replicates all its components, before eventu-
Dynamical modelling is at the core of the systems biol-
ally dividing into two usually roughly identical daughter
ogy paradigm. However, the daunting complexity of reg-
cells. While cell growth and replication of most cellular
ulatory networks controlling crucial biological processes
components is a continuous process, DNA replication
such a cell division, the paucity of currently available
occurs during the
quantitative data, as well as the limited reproducibility
of large-scale experiments complicate the development
are duplicated in two sister chromatids held together by
qualitative modelling approaches oer a useful alterna-
cohesin. Specialised structures known as kinetochores or
tive or complementary framework to build and analyse
cell pole bodies are also replicated early in the cycle.
This
Separation of the replicated material occurs during
point is illustrated here by analysing recent logical mod-
M phase
els of the molecular networks controlling mitosis in dif-
the course of
introduction to the cell cycle and logical modelling, we
structure that stems from the kinetochores located at
erties of these dierent models. Next, leaning on their
the opposite poles of the cell, and align at the spindle
transposition into a common logical framework, we com-
equatorial plane. The transition to the next phase occurs
pare their functional structure in terms of feedback cir-
only when all chromosomes are properly attached to each
Finally, we conclude this article by discussing
anaphase,
telophase. M phase ends up
with proper cell division, or cytokinesis.
pole and aligned.
assets and prospects of qualitative approaches for the
S and M phases are usually separated by two gap
Introduction
phases,
G1
(between M and S) and
M). A fth phase called
1.1 Overview of the cell cycle
G0
G2
(between S and
can be reached from G1,
that corresponds to a quiescence state of the cell. Gap
phases enable the cell to monitor its environment and
The organisation of cell cycle in eukaryotes
internal state before committing into S or M phase.
The cell cycle is highly regulated.
The cell cycle - the process by which a cell reproduces
Indeed, external
and internal signals may halt the cycle at particular
An important checkpoint called Start in
restriction point in mammalian cells, conG1/S transition. This checkpoint integrates
itself, dividing into two daughter cells - occupies a cen-
checkpoints.
tral place in the life in both unicellular and multicellular
yeast, or the
organisms
trols the
[1, 2, 3, 4] .
Chromosomes separate in
and are decondensed during
modelling of the cell cycle.
1.1.1
prophase. In
metaphase, they are attached at the level of
their centromeres to the mitotic spindle, a microtubular
compare the modelling strategies and dynamical prop-
1
(M for mitosis), itself subdivided into several
subphases. Chromosomes are condensed in
ferent organisms, from yeasts to mammals. After a short
cuits.
(S for synthesis), along with
the new DNA. At the end of this process, chromosomes
of comprehensive quantitative models. In this context,
simplied, but still rigorous dynamical models.
S phase
synthesis of the histones necessary for the packaging of
In animals, cell cycle is con-
nected with dierentiation [5], apoptosis [6], organ size
signals depending on the presence of nutrients, cell size,
[7], tissue maintenance and regeneration [8], ageing [9],
or contact with other cells, thereby coordinating cell pro-
1
liferation with cell growth and the needs of the organism.
thesis, as well as via post-transcriptional modications
In the course of the metaphase to anaphase transition,
controlled by the homologs of the Wee1 kinase and the
the spindle checkpoints monitors chromosome attach-
Cdc25 phosphatase. On the other hand, MPF inhibits
ment to the microtubules, and their alignment on the
its own inhibitors, sometimes called the G1 stabilisers:
metaphase plate. Additional checkpoints monitor DNA
the Cdk inhibitors (CKI) and Cdh1, an activator of the
damage at dierent points of the cycle.
APC (Anaphase Promoting Complex).
CKI sequester
The picture just drawn describes the canonical cell
Cyclin-Cdk complexes, thereby inactivating them. Cdh1
cycle. Specialised variants exist, that present signicant
activates the degradation of the cyclin subunit through
dierences with the classical G1-S-G2-M scheme.
the APC, a ubiquitinating enzymatic complex.
Thus,
In the early stages of development of frog embryos, for
in the course of cell proliferation, states with low MPF
example, the rst divisions involve fast and synchronous
and high CKI and Cdh1 alternate with states with high
successions of S and M phases, with no gap phases be-
MPF and low CKI and Cdh1 activity.
tween them [16]. During Drosophila development, early
How does the cell switch from a state of low cyclin
vice versa ?
divisions are also fast and synchronous, but further lim-
activity to a state of high cyclin activity, and
ited to nuclei within a large syncytium, until gap phases
The cyclin protein identied by Evans in 1983 has later
appear along with true cellularisation around cell cycle
been related to a larger family of cyclins, whose members
13 [17, 18].
peak at dierent time points in the cycle:
G1 cyclins
Various specialised cell types in animals and plants
are active in late G1 and play a key role in the Start
undergo partial or complete endoreduplication cycles en-
transition. These cyclins are represented by members of
the cyclin E family. Homologous members of the Cyclin
A family are activated at the G1 transition and trigger
abling various rounds of replication of (portions of ) chromosomes without intervening nuclear division
[19, 20].
Finally, meiosis can also be considered a specialised
DNA synthesis; their expressions last until mitotic entry.
B-type cyclins
variant of the cell cycle that produces haploid germ cells,
in two rounds of division, from diploid precursors
[21].
are mitotic cyclins, that promote entry
into mitosis and the formation of the mitotic spindle, and
All these events involve a complex machinery of en-
whose degradation triggers mitotic exit and cytokinesis.
zymatic complexes, molecular motors and cytoskeleton.
The cyclin responsible for MPF activity belongs to this
Here, we focus on the delineation of the regulatory net-
family.
work controlling cell division.
1.1.2
G1 and S cyclins play a major role in the transition
from low to high MPF activity. Indeed, G1 cyclins are
The cell cycle molecular engine
not inhibited by the G1 stabilisers [28].
The cell cycle is controlled by a complex network of interacting proteins known as the
cell cycle engine
ing the activation of the S cyclins Clb5 and Clb6 [29, 30].
[22].
Together with the G1 cyclins, they inhibit Cdh1, allow-
Regulatory components contributing to this molecular
ing the accumulation of Clb1 and Clb2, the mitotic cy-
machinery control each other as well as a range of downstream processes necessary for cell duplication.
clins of budding yeast. Clb1 and Clb2 are sucient to
These
maintain their own activity by triggering their own syn-
processes feed back on the engine, forming checkpoints
thesis and inhibiting the G1 stabilisers.
able to halt the progression of the cycle and to ensure
enough time to complete each crucial step.
MPF
In budding
yeast, G1 cyclins Cln3 and Cln2 rst inhibit CKIs, allow-
They further
inhibit the G1 and S cyclins.
(Mat-
The transition from high to low MPF state, which cor-
Discovered in
responds to mitotic exit, is regulated by another negative
1971 for its role in meiotic maturation of frog oocytes
feedback circuit enabling mitotic cyclins to trigger their
[23], MPF was later found to display oscillating activity,
own degradation.
with a period coincident with that of the cell cycle [24].
inactivation of MPF activity, a factor triggering Cyclin
In the course of the 1980s, MPF has been resolved as
degradation, under the control of the mitotic spindle, has
a heterodimer of
(for cyclin-dependent
been suspected early on [31]. It was not until the late
kinase) [25, 26, 27]. Oscillations of the regulatory cyclin
1990s that this factor has been identied as Cdc20, and
subunit, driven by an alternation of synthesis and degra-
its regulator as the checkpoint protein Mad2 [32, 33].
dation phases, control the activity of the enzymatic cdk
The activation of Cdc20 by Cyclin B [34, 35] completes
subunit.
the negative feedback circuit, by which mitotic cyclins
At the core of the cell cycle engine lies the
uration or Mitosis Promoting Factor).
cyclin
and
cdk
Given the role of proteolysis in the
trigger their own degradation. This circuit had already
A combination of positive and negative feedback cirEarly work
been postulated and integrated in mathematical models
had already shown that the cell cycle can be blocked
of the cell cycle [36] prior to the discovery of its molec-
in stable states of high or low MPF activity [24]. The
ular components.
cuits is responsible for these oscillations.
underlying multistable behaviour is ensured by various
The progression of cell cycle is further constrained
positive feedback mechanisms. On the one hand, MPF
by checkpoint mechanisms that condition the activation
self-activates through a positive eect on cyclin syn-
and inactivation of key regulatory components to the
2
Cell cycle phase
molecular denition
cillatory behaviour stems from molecular interactions.
G0/G1
low CycA and CycB activity
As far as cell cycle is concerned, mathematical mod-
S/G2
high CycA, low CycB
elling with systems of Ordinary Dierential Equations
M
high CycB
(ODE) is the most common approach [51, 52, 53]. Each
equation gives the rate of change of the concentration or
Table 1: Cell cycle phases. Components' names based
activity of a given component as a function of the con-
on mammalian nomenclature.
centrations of its regulators (see [54] for a recent review
on ODE modelling).
completion of specic events.
Several articles co-authored by Béla Novák and John
Activation of Cdc20 by
Tyson during the last decades arguably embody the
Cyclin B is controlled by the spindle checkpoint to ensure
state of the art of dierential modelling of cell cycle reg-
that sisters chromatids are not separated before chromo-
ulation
somes are properly attached to the spindle and aligned
on the metaphase plate. Cdc20 promotes mitosis by trig-
eect of over one hundred reported single or multiple
tion of sister chromatids. Additional checkpoint mecha-
mutations, for dierent growth conditions
nisms condition the completion of mitotic exit to the sep-
cycle in four dierent eukaryotes (xenopus, budding and
tors both DNA damage and unreplicated DNA, thereby
ssion yeasts, and mammalian cells [58]).
ensuring that replication is complete before entering M
Numerical
phase. In budding yeast, the morphogenesis checkpoint
integration
and
analysis
of
dierential
models have already led to specic predictions, some al-
conditions the activation of Clb2 to the formation of a
ready experimentally validated (see for example [29] and
bud.
[59], or [36] and [60]). In most cases, however, the results
Consistent with the crucial importance of cell division,
obtained should be considered as qualitative, despite the
cell cycle engine components are highly conserved among
quantitative potential of the modelling method. This is
eukaryotes [37]. Table 2 presents the homology relation-
due to two main issues.
ships existing between key regulatory components of the
On the one hand, the precise
nature of the mathematical relationships and the corre-
cell cycle control network in budding yeast, ssion yeast,
sponding parameters remain dicult to estimate on the
However, sub-
basis of available experimental data. On the other hand,
stantial dierences exist between organisms in terms of
the use of non linear functions complicates the analysis
precise wiring of the network as well as of timing of ex-
and forces the recourse to intrinsically partial numerical
pression and activity pattern of regulatory components
et al.
More
of a generic model enabling consistent simulations of cell
of Cdh1 and CKI by Cdc14. A G2/M checkpoint moni-
In this respect, Jensen
[30].
recently, these authors have supervised the development
aration of sister chromatids by regulating the activation
[38].
Of particular inter-
cell cycle, which enabled consistent simulations of the
gering the degradation of cyclins as well as the separa-
arabidopsis, drosophila and mammals.
[36, 55, 56, 30, 57, 58].
est is their comprehensive model of the budding yeast
approaches (simulations, one or two-dimensional parameter bifurcation, etc.)
[58, 61]. In addition, it is dicult
recently showed that tim-
to further extend large dierential models, as numerical
ing of expression of key players, in particular, cyclins and
instabilities arise when the number of variables and the
Cdc20, is relatively consistent between dierent organ-
complexity of control terms increases.
isms, but that the timing of expression of many other cell
For the same reasons, stochastic modelling is dicult
cycle-regulated proteins is poorly conserved [38]. More-
to apply to large regulatory networks. However, a sig-
over, components that are cyclically expressed or post-
nicant step in this direction is made in
transcriptionally modied in one organism do not appear to be regulated in others. However, Jensen
[62], which
presents a stochastic Petri net model of budding yeast
et al.
cell cycle engine.
showed that such components often take part in molecular complexes involving other cycle regulated subunit(s)
1.2.2
(a principle called just-in-time assembly, [39]). In brief,
Qualitative modeling
although the molecular details may dier, the general
To cope with complex networks and match qualitative
organisation of the regulatory network may still be con-
experimental data, one can rely on qualitative represen-
served.
tations of molecular interactions in terms of graphs [63],
Boolean models and their multilevel extensions [64, 65],
1.2 Dynamical modeling of the cell cycle
regulatory network
standard Petri nets ([66] and references therein), or yet
1.2.1
cell cycle [69, 70, 71, 72, 73, 74].
piecewise linear equations [67, 68].
Among these ap-
proaches, logical modelling is increasingly used to model
Quantitative modeling
Graph-based representations are certainly the most
Rapid progress in the elucidation of the molecular nature
intuitive for biologists, as they arguably formalise their
of the cell cycle engine has triggered the development of
practice of drawing regulatory diagrams.
mathematical models to explain how the observed os-
standardised regulatory graphs, logical models rely on
3
Leaning on
4
Rb
DmE2F-1
Fizzy-related
Fizzy
Cdc25String
Wee1
Rux, Dacapo
cyclin B and B3
Cyclin A
DmCycE
Cyclin D
Drosophila
Arabidopsis
Rb
E2Fa, b, c
Ccs52A1, Ccs52A2
Cdc20, Ccd52B
-
WEE1
KRP1 to KRP7
Cyclin B1, B2 and B3
Cyclin A1, A2 and A3
Nicta;CYCA3;2
Cyclin D1 to D7
Rb
E2F-1, -2, -3
Cdh1
Cdc20
Cdc25B
WEE1/Myt1
p21, p27Kip1
Cyclin B1/2/3
CyclinA1 and A2
Cyclin E
Cyclin D
Mammals
Function
[42, 49, 50]
binds and inactivates E2F
[42, 48, 17, 40, 46]
Transcription factor, controls the G1/S transition
[32, 17, 47]
Activator of the APC, active in late mitosis and G1
[32, 47, 46]
Activator of the APC, active in mitosis
[17, 40]
Phosphatase, activates Cyclin B
[40, 46]
Kinase, inhibits Cyclin B
[17, 40, 45]
Cdk inhibitors
[44, 40, 41]
G2/M transition and intra-M control
[42, 44, 40, 41]
S phase progression, G2/M transition
[42, 43, 41]
G1/S transition
[17, 40, 41]
G1 progression
mammals display plethora of paralogs (cf [41] regarding cyclins in plants for example).
MBF on the other. Not shown in the table, whereas Yeast usually have only one member of each family of component, plants and, to a lesser extent,
Nicta;CYCA3;2 in Plants (related to the Cyclin A family but functional homolog of Cyclin E), or Rb and E2F on the one hand, and Whi5 and SBF and
in some cases, even when sequence homologs have not been found, unrelated components may fulll homologous functions. This is the case for example for
Table 2: Homology relationships between cell cycle regulatory components. Most of the components presented in this table are sequence homologs. However
-
Cdc25
Mih1
Whi5
Mik1
Swe1
MBF
Rum1
Sic1
SBF, MBF
Cdc13
Clb1/2
Ste9
cig2
Clb5/6
Cdh1 (HCT1)
-
Cln1/2
Slp1
Puc1
Cln3
Cdc20
Fission Yeast
Budding Yeast
Boolean rules to dene the eects of dierent combina-
ity levels of a single component
tions of interactions on their targets, thereby enabling
one may use multilevel logical variables [64].
[72, 73]; alternatively,
qualitative simulations. In many cases, regulatory com-
Similarly, arcs can be dened for a particular inter-
ponents can be simply considered as present or absent
val ranging from i (i>0) to the maximum level of the
(or yet as active or inactive), thereby matching qualita-
source node. They can represent dierent types of inu-
tive reasoning pervasive among biologists.
ences, from transcriptional regulation (activation or re-
Logical modelling has been successfully applied to var-
pression of the transcription of a particular gene), to bio-
ious biological regulatory networks, from the bacterio-
chemical reactions (de/phosphorylation, degradation...),
phage lambda lysis/lysogeny switch ([75] and references
to subtler situations (modulation of the activity of an-
therein), to the polarisation of the immune response
other component within a complex, regulation through
[76, 77, 78, 79], the specication of arabidopsis ower
implicit pathways...).
The regulatory graph is a useful tool to get a rst
organs [80, 81, 82], the delineation of the segmentation
pattern in drosophila embryo
the development of sense organs in drosophila
dierentiation of keratinocytes
glance of a logical model, but a logical model fundamen-
[83, 84, 85, 86, 87, 88],
[90]...
tally consists in a set of
[89], the
logical rules, each directing the
evolution of one component.
All these appli-
In our context, one can
cations deal with decision-making systems enabling the
distinguish two main approaches to dene these logical
selection of specic cell fates, and can thus be analysed
rules. On the one hand, logical rules can be dened as an
in terms of alternative stable states.
arithmetic sum of (weighted) positive and negative inu-
In contrast, in the case of the cell cycle, what mat-
ences received by each node. Depending on whether the
ters is the actual sequence of transitions, the succession
sum lies above, below or at a particular threshold, the
of changes in the levels of activity of components con-
value of the node considered will tend toward 1, 0, or re-
trolling the progression of the cell through the dierent
main unchanged [70, 90, 72, 73]. Alternatively, Boolean
phases of the cycle. This does not impede many regula-
formulae combining Boolean variables, NOT, AND and
tory components to adopt a switch-like behaviour [91] ,
OR operators can be used to dene the target value of
which can be approximated by logical variables.
a component depending on the values of its regulators
(see, e.g., [71] or
Proper logical modelling of cell cycle engine is still in
its infancy.
[74]).
However, a series of Boolean models have
been recently proposed, from [69] presenting a very sim-
1.3.2
plied logical model of the mammalian cell cycle engine,
State updating, stable states, and mutant
simulations
to increasingly sophisticated models for budding yeast
To study the dynamical behaviour of a logical model, we
[70, 74], ssion yeast [72, 73], and mammals [71].
have to further select an updating schedule. Indeed, at
Before turning to the description and comparison of
a given
specic models in more details, the following section en-
logical state
(dened as a specic combinations
of values for the network components), the logical rules
capsulates the main aspects and variants of the Boolean
may imply
modelling framework.
updating calls
for several components (there
is an updating call on a component whenever the target
1.3 Logical modeling
value is dierent from the current one).
1.3.1
updating calls are updated simultaneously at each step
The simplest assumption considers that all current
Regulatory graph and Boolean rules
synchronous schedule ).
(
The resulting behaviour is de-
The skeleton of a logical model is often represented in
terministic since each state can have at most one succes-
terms of an oriented graph, where the nodes denote regu-
sor state (although several states may lead to the same
latory components, and the arcs denote cross-regulatory
state).
Alternatively, one may consider that one component
interactions. A logical variable is associated with each
ements, from molecular compounds (genes, mRNA, pro-
asynchronous
schedul e), either considering all possible single changes,
teins, protein complexes...)
or using a deterministic transition order, or yet using
at most can be updated at each time step (
node, which can represent dierent types of biological elto phenomenological vari-
probabilistic selection rules.
ables (e.g., cellular mass, DNA synthesis, or cell division). These variables take integer values within an in-
For a given schedule, the dynamical behaviour of a
terval ranging from 0 (absence of a component or unde-
logical model can be represented in the form of a state
tectable activity) to a maximum activity level. In most
transition graph, where the nodes denote dierent states
cases, this maximum value is equal to one (
of the system and the arcs denote (allowed) transitions
binary
vari-
ables). If all variables are binary (i.e., if they take their
from one state to another
values within the [0;1] interval), we face a
tively, a particular trajectory corresponding to a linear
Boolean model.
[70, 71, 72, 73].
Alterna-
However, it is often necessary to consider intermediary
sequence of transitions may be represented in a table,
activity levels for some nodes. One possibility is to use
where the columns correspond to the dierent compo-
multiple binary variables to represent the dierent activ-
nents and successive rows to successive dynamical steps
5
[74].
reverse-engineer Boolean networks from dynamical data
[100].
Within a logical framework, it is relatively straightfor-
CellNetAnalyser
implements several methods to
ward to dene and simulate mutants and other kinds of
analyse uxes in metabolic or signal transduction net-
perturbations, provided that they correspond to clear-
works represented in terms of bipartite graphs [92]. Fi-
cut qualitative outcome [71, 74].
nally,
GINsim
supports the denition, simulation and
analysis of Boolean and multilevel logical models for
Various solutions have been proposed to rene logical simulations while avoiding the complexity of fully
regulatory networks [101].
asynchronous schemes, for example by distinguishing be-
here have been encoded or adapted using
tween fast and slow subnetworks, simulated sequentially
are made available in the companion model repository
[92, 93, 86], or by grouping transitions into ranked syn-
(www.gin.univ-mrs.fr).
chronous or asynchronous classes
GINsim
and
[71], or yet by intro-
2
ducing delaying nodes [74].
Whatever the updating schedule, a logical model conserves the same
All the models reviewed
stable states (states for which each com-
ponent has a target value identical to its current one,
Logical models of the cell cycle
engine in eukaryotes
2.1 Budding and ssion yeast cell cycle
models
and thus from which no transition is possible). In this respect, powerful algorithms have been recently proposed
to compute stable states in large regulatory networks
This is particularly useful to exclude the
Yeast, and most particularly budding yeast, has for long
occurrence of stable state for a viable cell cycle model,
been a reference model system for the study of the cell
or yet to identify specic mutant arrest states
cycle control mechanism.
[90, 94, 95].
[71, 74].
Saccharomyces cerevisiae
is a
As we shall see, in all the models considered here,
simple, unicellular organism, used from time immemo-
the underlying network wiring appears tight and robust
rial for brewing and bakery - hence its common name
enough to enable the generation of fully synchronous
of
state transitions remarkably consistent with available bi-
mutation screens have been carried on, facilitated by the
ological data.
possibility to grow haploid cells, thereby avoiding the
brewer's
or
baker's yeast.
During the 1970s, extensive
pitfall of recessive mutations.
1.3.3
A
Regulatory circuits and their dynamical
roles
regulatory circuit
Many key regulators of
cell division have been discovered in this context
[102].
Consequently, yeast cell cycle also constitutes a system
of choice to develop or assess computational methods
dedicated to the dynamical modeling of biological regu-
is formally dened as a simple cir-
latory networks.
cular path in the regulatory graph dened above. The
sign of a circuit is given by the product of the signs of its
The model of the budding yeast cell cycle presented
constitutive interactions and reect the (indirect) eect
in [70] gathers all the usual suspects of cell cycle reg-
of each component of the circuit on itself
[64, 96, 97].
ulation: four groups of cyclins with dierent functions
It is now well established that positive regulatory cir-
(Cln3 activates Cln1 and Cln2, which inhibits Cdh1 and
cuits are necessary for multiple stable states or attrac-
Sic1, allowing on the one hand Clb5 and Clb6 to activate
tors, whereas negative circuits are needed to generate
S phase, and on the other hand, Clb1 and Clb2 to acti-
sustained oscillations
vate M phase); the Cdk inhibitor Sic1; activators of the
[98].
As mentioned earlier, negative and positive regula-
APC Cdc20 and Cdh1; plus several transcription fac-
tory circuits have been regularly involved at the core of
tors (SBF and MBF for Cln1,2 and Clb5,6, Mcm1/SFF
the cell cycle engine, but rigorous delineation of circuit
for Clb1,2, and Swi5 for Sic1).
roles is complicated by the increasing number of cross-
ers are abstracted (Pds1) or lumped together with other
regulations in recent models, implying an exponential
components (Cdc14 with Cdc20). This scheme matches
increase of the number of regulatory circuits.
current textbook descriptions of the core engine of the
In this
A few other key play-
budding yeast cell cycle (see, e.g., [103] Figure 3-34).
respect, within the logical framework, it is possible to
In their model, Li
identify all feedback circuits embedded into a complex
et al
represent all components by
regulatory networks and determine the role of each of
Boolean variables and use generic rules (arithmetic sum
these circuits depending on the levels of external regu-
of positive or negative inuences, with weight 1 or -1,
lators
respectively, and a threshold xed at 0) to compute the
[94].
response of by each component to multiple regulatory
1.3.4
inputs.
Tools
The logical rules can thus be directly derived
from the topology of the network. However, in the case
Dierent tools are available to simulate and analyse
of components that are not negatively regulated (namely
Boolean networks.
computation
of
the
Boolean networks
For example,
attractors
[99].
of
DDLab
enables the
Cln3, Cln1,2, Mcm1/SFF, Swi5 and Cdc20&Cdc14), the
large
synchronous
authors have introduced self-degradation loops to inac-
oers means to
tivate them when all their activators are OFF. Simula-
REVEAL
6
tions are carried out synchronously. The authors simu-
documented mutants.
late the behaviour of their model from all possible initial
Cln3 yields a cyclic attractor whose sequence corre-
states, showing that most trajectories converge towards
sponds to the cell cycle (even under the synchronous
a main attractor, a stable state corresponding to the G1
updating mode), consistent with biological data. As al-
phase. Other stable states attract minor portions of the
ready hinted by [74], this was not the case in the original
state space. Cell cycle simulations are performed start-
model, where overexpression of Cln3 yielded a M-phase
ing from the main stable state, and switching Cln3 on:
arrest. This cyclic behaviour is consistent with the result
the resulting sequence of synchronous transitions match-
of a feedback circuit analysis, leading to the identica-
ing cell cycle evolution up to returning to G1 resting
tion of functional positive and negative circuits in the
state.
adapted model (see Table 3).
The authors further analysed the eect of per-
In particular, overexpression of
turbation by introducing slight changes in the network
In [72], Davidich and Bornholdt proposed a Boolean
topology (adding, suppressing or changing the sign of
model of the ssion yeast cell cycle, dened in terms
one arrow). However they did not systematically anal-
of equations similar to those found in [70] . Consistent
yse the eect of reported mutations.
with biological data showing that a single B-type cyclin,
We have transcribed the logical rules used by the au-
Cdc13, is sucient for both S and M phase progression
thors into our formalism. The resulting model is shown
[107], and with the preponderance of the G2 phase in the
on Figure 1, top left. A colour code highlights protein
ssion yeast cell cycle [108], this model puts the empha-
homology relationships with other organisms. Although
sis on the control of Cdc13 activity. The starter kinase
the global topology of the network is preserved, our
SK is the only component representing G1 or S cyclins
transcription highlights the presence of positive regula-
in this model. Consistent with the results of [107] and
tory circuits on several components, namely SBF, MBF,
with the quantitative model published by [57], Cdc13 is
Clb5/6 and Clb1/2, Cdh1 and Sic1 (i.e., all variables
assigned two activity levels, represented by two distinct
that have both positive and negative regulators).
Boolean variables.
In
contrast, the negative loops introduced by the authors
We have transcribed this model using
GINsim (Figure
to match available kinetic data do not appear in our
1, bottom left). Similar in its construction, the model
regulatory graph, as they do not correspond to true self-
yields results comparable to those reported in [70]:
degradation or inhibitory mechanisms; in our transpo-
main attractor corresponding to G1 gathers most trajec-
sition, the corresponding components are assigned zero
tories; switching on the start signal from this attrac-
basal value to implement spontaneous decays.
tor triggers a sequence of state transitions qualitatively
Analy-
a
sis of the functionality of the feedback circuits of this
matching cell cycle progression.
model conrms that the positive self-activating feedback
also suers from similar drawbacks, as it generates sev-
loops play an important role in its dynamics, in particu-
eral spurious stable states, due to the introduction of
However, this model
lar regarding the maintenance of alternative, artefactual
ad hoc
stable states.
budding yeast model presented by [70], however, circuit
ad hoc
In contrast with the
analysis of this ssion yeast model identies functional
This observation led us to simplify Li's model by eliminating all
positive feedback loops.
positive as well as negative feedback circuits.
positive loops (Figure 1, top right).
Accord-
For proper logical rules, the resulting model converges
ingly, overexpressing the Starter kinase SK, which cor-
towards a single stable state corresponding to G1. How-
responds to both Cln3 and Cln1/2 in the budding yeast
ever, in the absence of SBF and MBF autocatalytic
model, does yield a cyclic attractor.
feedback loops, the start signal is not maintained long
tions do not aect the G1 stabilisers Ste9 and Rum1.
However, oscilla-
enough to ensure a proper sequence of state transitions
To get rid of the spurious stable states, we have
towards the G1 stable state under the fully synchronous
adapted Davidich & Bornholdt's model. The resulting
updating mode. This problem can be compensated by
network is shown Figure1, bottom right. All
the introduction of delays for SBF/ MBF switching OFF
tive feedback loops have been removed and compensated
(using the priority system introduced in [71]). Another
by the introduction of priorities to account for the main-
possibility would be to include additional biological data
tenance of the start signal and its eect on Rum1 and
showing that Cln1/2 and Clb5/6 may regulate their own
Ste9.
transcription factors, and that there is some redundancy
Cdc13 variables by a single, ternary one.
ad hoc posi-
Furthermore, we have replaced the two Boolean
between SBF and MBF [104, 105, 106]. Such feedbacks
The resulting model reproduces the cell cycle se-
would ensure that Cln2 and Clb5 signals are maintained
quence, provided priorities are used to maintain the
long enough for Clb2 to take up the repression of Sic1
Start signal active long enough. It has three functional
and Cdh1. However, integration of these data into the
two-elements circuits:
original modelling framework would largely perturb the
(Table 3). Overexpression of SK yields a cyclic attractor,
behaviour of the system, due to the rigidity of the dy-
where all downstream components oscillate, and whose
namical rules.
sequence corresponds to the cell cycle.
Using this revised model, we could simulate various
one negative and two positive
Recently, a more detailed model for the budding yeast
7
Figure 1:
Top:
Budding yeast cell cycle. Left: model transcribed from [70]; dynamical rules are generated from
the network structure as the sum of the positive and negative inuences exerted on each nodes by its regulators.
Right: an adaptation of the same model using adjusted logical rules. See text for details.
Bottom:
Fission yeast
cell cycle. Left: model transcribed from [72]. Dynamical rules are generated as in [70]. Note that two variables
are used to represent dierent levels of activity for Cdc13 in a strictly Boolean formalism. Right: an adaptation of
the same model using adjusted logical rules. Cdc13 is now represented by a single, ternary variable.
conventions:
Graphical
ovals: Boolean variables; rectangles: multilevel variables; green arrows: activations; red blunt arrows:
inhibitions; node colours emphasise homology relationships.
8
cell cycle control network has been published by Irons
ing aside redundancies and specicities, several groups
[74]. This model encompasses several checkpoint mod-
are working on the delineation of generic models of the
ules in addition to the core cycling engine, albeit in a
mammalian cell cycle, leaning in part on current knowl-
very schematic form.
edge about the yeast cell cycle engine (see, e.g., [69], [56]
The FEAR and MEN pathways
are present, but reduced to single variables, and the mor-
and [71]).
phogenesis checkpoint, that conditions entry into mitosis
In 2006, we have proposed a generic Boolean model
to the formation of a BUD, as well as the spindle check-
for the mammalian cell cycle [71], based on the quan-
point, that blocks exit from mitosis until chromosomes
titative model presented in
are properly aligned on the metaphase plate, are rep-
model encompasses the key players of the original model:
resented by direct eect from the variables representing
four cyclins (D, E, A and B), the CKI p27Kip1, the tran-
the bud and M-phase onto the targets of these check-
scription factor E2F and its inhibitor Rb, as well as the
points.
activators of the APC Cdc20 and Cdh1.
Several of these modules have been previously
modelled in the dierential framework
[56] (Figure 3, left).
This
The eects
[109, 30, 110],
of growth factors are represented by the activation of
but, to date, these modules have not yet been integrated
Cyclin D. More importantly, we introduced the recently
into a single comprehensive model.
Iron's model thus
discovered UbcH10 [122] to account for the fact that Cy-
provides a simple, but relatively extended view of the
clin A is a target of Cdh1 in late mitosis or early G1,
budding yeast cell cycle.
but not in late G1, thereby allowing the rise of CycA
Irons' model is dened in terms of a series of logical
and the concomitant inactivation of Cdh1.
rules, completed with a system of temporisation based
Compared to the models described above, an original
on the introduction of additional nodes to delay the ac-
aspect lies in the implicit representation of molecular
tivation or degradation of particular components. Con-
complexes. For example, the sequestration of cyclin-cdk
trary to previous models ([70] and [72]), this one yields
a single, cyclic attractor.
complexes by the CKI (p27 in the case of the mam-
Both fully synchronous and
malian cell cycle, shown in yellow in the Figures of this
temporised dynamics have been considered, yielding re-
paper) is represented by direct inhibitions in
sults qualitatively consistent with available kinetic data.
and
Finally, Irons further simulated the behaviour of several
[74].
[70],
[72]
In contrast, in [71], we implicitly consider
that such complexes are formed whenever all their com-
mutants in terms of viability or arrest in a particular
ponents are activated.
phase of the cycle, as well as checkpoint-mediated ar-
For example, sequestering and
inactivation of cyclin-cdk complexes by CKI are mod-
rest.
elled by arcs from CKI onto cyclin targets, along with
We have transposed the logical rules using GIN-
logical rules restricting cyclins activities on their targets
sim and simulated Iron's model under the synchronous
to the absence of p27 (see
mode. The corresponding regulatory graph is shown in
representation has two main advantages: (i) a lowering
Figure 2 and enables the recapitulation wild-type and
of the number of logical components to consider; (ii)
mutant simulation results for the rules dened by Irons.
immediate component activation following their release
Using our feedback circuit analysis tool, we could further
from a sequestering complex.
identify the underlying functional circuits (3) .
[71] for more details). This
Simulation of this model yields a cyclic attractor qualitatively matching reported cell cycle oscillations.
2.2 Logical cell cycle models for multicellular eukaryotes
In
addition, several mutant phenotypes could be properly
recapitulated (cf.
[71]).
However, this very simplied
Boolean model has several drawbacks. In particular, Rb
Proper dynamical modelling and simulation of cell cycle
remains OFF in the wild-type cycle, while some mutant
in multicellular eukaryotes still constitutes a daunting
simulations disagree with reported phenotype (for ex-
challenge, in particular regarding mammalian systems.
ample, p27- and CycEop cells should cycle in absence
Beyond the involvement of additional components such
of CycD, whereas in our model they remain arrested in
as Rb and E2F proteins, complexity stems from the high
G1). The consideration of additional components or of
level of redundancy within each family of components.
multiple levels for some regulatory factors (such as Rb)
Indeed, databases [111, 112, 113] and publications point
could in principle resolve these discrepancies.
to innumerable cyclins [114, 115], cyclin-dependent ki-
Mammals display a large variety of cell types, some
nases [116], cdk inhibitors [117], E2F transcription fac-
of which do not follow the canonical G1-S-G2-M pat-
tors [118], pocket proteins [119], Cdc25 phosphatases
tern of cell division:
[120], etc. Most of these components are in fact essen-
meiosis... Such variations happen in various organisms.
tial in particular cell types or growth conditions, but
However they have been particularly well studied in the
dispensable in others. The overwhelming complexity of
context of drosophila development
the mammalian cell cycle network (see, e.g., [63, 121])
currently developing a model of the cell cycle control
discourage the integration of all details into a compre-
network in drosophila, with the aim of accounting for
hensive dynamical model. However, provisionally leav-
developmental variants of mitosis:
9
asymmetric division, endocycles,
[17, 123].
We are
namely, endocycles
Figure 2: Budding yeast cell cycle regulatory graph build on the basis of [74]. This model reproduce the simulations
presented in Figure 3B of [74] for the same logical rules, under the synchronous updating mode (without delaying
nodes). Graphical convention as in Figure 1.
10
Figure 3:
Left:
A generic Boolean model for the mammalian cell cycle (for details, see [71]).
Right:
Drosophila
cell cycle model. Graphical conventions: thinner arrows indicate indirect regulation through complex formation,
other graphical conventions as in Figure 1.
3
and syncytial cycles.
3.1 Similarities and variations and in
cell cycle models
Endocycles (or endore(du)plication cycle) happen in
particular drosophila cell lineages, including larval salivary glands and chorion (eggshell) cells
[124].
Discussion
They
Beyond
homology
relationships
between
regulators,
consist in nested S phases without intervening mitosis,
mathematical models of the cell cycle highlight the con-
which results in polyploid or polytene cells, characterised
servation of the regulatory circuits governing cell di-
by gene amplications enabling very high production of
vision.
specic proteins.
Novák et al.
In 1998, using ordinary dierential equations,
[128] proposed a prospective cell cycle
model, emphasising the universality of the corresponding
regulatory mechanisms.
More recently, [58] published
Syncytial cycles occur during the early developmen-
a generic, modular model of the eukaryotic cell cycle,
tal stages of the drosophila egg, where S and M phases
enabling the recapitulation of the results obtained by
succeed at a very fast pace, in absence of growth, gap
Novák's and Tyson's groups for dierent organisms, by
phases or cytokinesis. The nucleus divide synchronously
considering specic subsets of modules and tuning ap-
until the thirteenth round of division, after which cellu-
propriate parameters.
larisation occurs.
Here, we have focused on recent Boolean models of
the eukaryotic cell cycle. These models have been published by dierent groups, and designed to reproduce the
We have derived a model using published data on
behaviour of the core cycle oscillator in dierent organ-
drosophila cell cycle, completed when necessary with in-
isms. Still, their comparison reveals that, in spite of the
formation transferred from other organisms, principally
specicity of the wiring of each model and some dier-
mammals, using orthology relationships between regula-
ences in modelling assumptions (e.g. regarding the use
tory components. Shown in Figure 3, right, this model
of Boolean versus multilevel components, or the repre-
can be used to simulate the canonical cell cycle, syn-
sentation of protein complexes), model behaviour rely
cytial cycles, as well as endocycles.
on similar sets of conserved functional circuits.
Transition from
canonical cycles to endocycles is regulated by Notch and
Indeed, once transposed into a common logical frame-
Archipelago [125, 126], while the transition from syncy-
work, these dierent models can be compared in terms
tial cycles to canonical cycles originates from the titra-
of functional circuits. The results of this comparison are
tion of maternally expressed cyclins
summarised in Table 3.
de novo
[127], along with
zygotic expression of canonical cell cycle com-
A general observation is that
only a relatively low number of positive and negative
circuits are found functional, even in the most complex
ponents.
11
models analysed. Strikingly, two circuits are conserved
or the spindle assembly checkpoint and metaphase to
in all ve models: one negative circuit involving the ho-
anaphase transition [141, 142].
mologs of Cyclin B and Cdc20, and one positive circuit
involving the homologs of Cyclin B and Cdh1.
However, the complexity of the mammalian engine re-
Addi-
mains daunting: beyond the introduction of components
tional circuits are conserved in subsets of organisms.
specic of higher eukaryotes, such as Rb and E2F factors, complexity stems from the high level of redundancy
Following cyclin discovery in the 1980s[25], and the
within each component family. Databases [111, 112, 113]
demonstration [26] that it corresponds to Masui and
and publications point to innumerable cyclins [114, 115],
Markert's MPF [23], oscillations of cyclins, and most
cyclin-dependent kinases [116], cdk inhibitors [117], E2F
particularly of B-type cyclins, have been placed at the
transcription factors [118], pocket proteins [119], Cdc25
heart of the molecular control of the cell cycle.
phosphatases [120]...
Cy-
Most of these redundant family
clin activity oscillate between a stable low state, cor-
members are in fact essential components in particular
responding to interphase, to a stable high state, corre-
cell types, particular pathways, but dispensable in oth-
sponding to mitosis (see [31] for an early review). The
ers. Due to the lack of knowledge on the specicity of
positive feedback between Cyclin B and Cdh1 (an ac-
these family members, and to the overwhelming com-
tivator of APC) has been considered at the basis of B
plexity of a comprehensive map of the cell cycle network
cyclin bistable behaviour since long [129].
Other pos-
[63, 121], the integration of all these details into a com-
itive feedbacks have been identied, most particularly
prehensive dynamical model is presently out of reach.
feedbacks involving the Cdk modiers Cdc25 (activator)
In this respect, modular approaches may help to design
and Wee1 (inhibitor) [36, 130, 131]. However, these two
comprehensive models or to simplify regulatory graph
factors have not been considered in most of the models
by selecting representative members of each family (cf.
reviewed here, with the exception of the ssion yeast and
[121] for a rst step in this direction).
drosophila models.
To cope with such complexity, the logical framework
The negative feedback circuit between Cdc20 and Cy-
has several assets.
Logical models are relatively easy
clin B singled our by our analysis is also generally con-
to dene, and do not necessitate detailed quantitative
sidered as crucial. This circuit had already been singled
data. As mentioned above, various tools are available to
out [31], and integrated in mathematical models of the
study the dynamics of logical models, in particular for
cell cycle [36] well before the deciphering of the corre-
the identication of stable states, as well as for the anal-
sponding molecular mechanisms [32, 34, 33, 35]. Inter-
ysis of the dynamical roles of regulatory circuits. Logical
estingly, Cdc20 has been outlined by Jensen
[38] as
models can also be easily used to predict known or novel
one of the components for which periodicity is the most
behaviour in response to various types of perturbations.
conserved among eukaryotes.
et al.
Our comparative study
Furthermore, logical modelling has interesting compo-
conrms that this negative feedback circuit is conserved
sitional properties, which could be exploited to develop
and functional in all four organisms considered.
incremental modelling strategies.
The roles of the connected positive and negative
Indeed, the core cell
cycle engine can be divided in specic modules control-
circuits highlighted by our study has been previously
ling each transition.
emphasized
by
checkpoints inuence the core engine to regulate cell cy-
further proposed that
cle progression, including response to growth factors or
as
crucial
Pomerening[132].
Cross
for
cell
et al.
cycle
oscillations
this combination could be widespread among eukaryotes
Moreover, various pathways and
DNA damage.
[133].
However, logical model cannot generate truly quantitative predictions. It is thus important to build bridges
3.2 Outlook
between qualitative and quantitative approaches.
For
example,[71] transposes Novák & Tyson's generic model
A limitation or our comparative analysis stems from the
for the mammalian cell cycle in the logical framework,
limits of the models themselves.
As discussed above,
whereas [73] proposes a more formal translation method
published logical models are relatively simple, focusing
and applies it to Novák's ssion yeast cell cycle model
on the core engine of the cell cycle.
[143]. Referencing to other biological applications, sev-
nents and interactions are missing.
Various compo-
eral software suites, including
For example, the
SQUAD
[144] and
Odefy
logical mammalian model [71] inspired by [56] overlooks
[145], even automate the translation of logical models
the possibility that p27 also sequesters Cyclin B-Cdk1
into generic dierential equations.
complexes [134].
models have also been adapted into probabilistic frame-
In parallel, ODE
works, such as stochastic Petri nets[62].
As previously mentioned, dierential modelling has
been the most popular approach as far as cell cycle mod-
Beyond the core cycling network, numerous control
elling is concerned. Recent eorts focus on the develop-
modules could be integrated to the mammalian model,
ment of less comprehensive, but more detailed models,
starting with the heavily studied and modeled the DNA
focusing on the G1 phase [135, 136, 137, 138, 139, 140],
damage checkpoint [146, 147, 148, 149, 150, 151, 152,
12
13
64
Total number of
Cdc2_Cdc13[1]/Cdc25
Clb1_2/Cdh1,
Sic1/Clb5_6
positive circuits
Clb2/Cdh1
CKI/Clb2
Cdc20/Clb2
CycB/Wee1
CycA/ Rb
Rb/CycE
CycB/Cdh1
CycB/Stg
CycA/Cdh1
Rb/CycA
Cdc20/CycB
132
[71]
Mammals
CycA/Fzr
Fzr/CycB
CycE/Rb
CycE/Dap
Fzy/CycB
CycA/E2F
199
Drosophila
Table 3: Functional regulatory circuits identied in logical models for the core regulatory network controlling cell cycle in budding yeast, ssion yeast,
drosophila, and mammals (all auto-regulatory loops are functional but not listed here). A color code emphasises homology relationships between key regulatory
factors (cf. Table 2).
Cdc2_Cdc13/Rum1
Sic1/Clb1_2
Functional
Ste9/Cdc2_Cdc13
CKI/Clb2/MEN/Cdc14
Clb1_2/Cdc20_Cdc14
Clb2/MEN/Cdc14/Cdh1
SFF/Swi5/CKI
Clb1_2/MBF/Clb5_6
Cdc20_Cdc14/Clb5_6/Mcm1_SFF
Cdc20/FEAR/Cdc14/CKI/Clb2
701
Budding yeast [74]
Functional
Cdc2_Cdc13/Slp1
9
from [72])
Fission yeast (adapted
negative circuits
circuits
Budding yeast (adapted from [70]
Organism
153]. Recently, Abou-Jaoudé
et al.
proposed an analysis
discovery, 2008, 7, 32438.
checkpoint in mammals) that takes advantage of three
dierent modelling approaches [153].
39,
mdm2 network can display dierent behaviours depend-
3,
They then turn to the dierential and stochastic for-
24,
as periods of oscillations.
Other groups are tackling other modules, including
105,
clock [154], apoptosis [155, 156, 157], as well as various
other pathways inuencing cell cycle progression [158,
24,
To date, however, predictive models for cell cycle vari-
these variants remain to be developed.
130,
2005,
Reproduction,
76181.
[22] A. W. Murray,
References
Nature, 1992, 359, 599604.
[23] Y. Masui and C. L. Markert,
177,
Cell, 2000, 100, 718.
J. Exp. Zool., 1971,
12945.
[24] W. J. Wasserman and L. D. Smith,
J. Cell. Sci.,
2002,
1978,
115,
24614.
78,
J. Cell Biol.,
R1522.
[25] T. Evans, E. T. Rosenthal, J. Youngblom, D. Dis-
Cell, 2004, 116, 22134.
[3] A. W. Murray,
[4] J. Bloom and F. Cross,
tel and T. Hunt,
Nat. Rev. Mol. Cell Biol.,
Cancer Res. Clin. Oncol., 2008, 134, 72541.
339,
J.
[27] A.
[7] B. Z. Stanger,
W.
Murray,
M.
J.
Solomon
Nature, 1989, 339, 2806.
77,
Drug Resist. Up-
11,
2004,
W.
Cell, 1994,
[29] K. C. Chen, A. Csikasz-Nagy, B. Gyory, J. Val,
Cell Cycle, 2008, 7, 31824.
Cell Cycle,
M.
103750.
B. Novak and J. J. Tyson,
[8] Y. Dor and D. A. Melton,
and
[28] A. Amon, S. Irniger and K. Nasmyth,
jothy, K. Weglarczyk, A. Zuse, M. Eshraghi, K. D.
dat., 2007, 10, 1329.
Nature, 1989,
27580.
Kirschner,
[6] S. Maddika, S. R. Ande, S. Panigrahi, T. ParanManda, E. Wiechec and M. Los,
Cell, 1983, 33, 38996.
[26] A. W. Murray and M. W. Kirschner,
14960.
[5] K.-H. von Wangenheim and H.-P. Peterson,
Mol. Biol. Cell,
2000,
36991.
[30] K. Chen, L. Calzone, A. Csikasz-Nagy, F. Cross,
3,
B. Novak and J. Tyson,
11046.
Mol. Biol. Cell, 2004, 15,
384162.
[9] D. van Heemst, P. M. den Reijer and R. G. J.
Westendorp,
Eur. J. Cancer, 2007, 43, 214452.
[10] A. D. Wells,
Semin. Immunol., 2007, 19, 1739.
[11] G. Brooks and L. T. NB,
[31] A. W. Murray and M. W. Kirschner,
1989,
278,
455464.
10,
Philos.
1999,
354,
158390.
[34] S. Prinz, E. S. Hwang, R. Visintin and A. Amon,
Curr. Biol., 1998, 8, 75060.
Cell Prolif., 2003, 36, 13149.
Biochim. Biophys. Acta,
Science, 1997,
4603.
Trans. R. Soc. Lond., B, Biol. Sci.,
[13] K. Vermeulen, D. R. Van Bockstaele and Z. N.
[14] L. M. Schang,
Science,
61421.
[33] G. Fang, H. Yu and M. W. Kirschner,
Drug Resist. Updat.,
16281.
Berneman,
246,
[32] R. Visintin, S. Prinz and A. Amon,
Drug Discov. Today,
[12] M. Schmidt and H. Bastians,
1697,
Oncogene,
276575.
2005,
[162, 163], and for the regulation of transitions between
2007,
2001,
[21] M. A. Morelli and P. E. Cohen,
ants in mammals, such as meiosis [161] or endocycles
4,
Cell,
297306.
[20] M. A. Lilly and R. J. Duronio,
159, 160] .
[2] M. Dorée and T. Hunt,
Bioessays, 2002,
101222.
[19] B. A. Edgar and T. L. Orr-Weaver,
the relationship between the cell cycle and the circadian
1999,
Nat. Cell Biol., 2001,
E359.
[18] A. Mazumdar and M. Mazumdar,
malisms to analyse more quantitative properties, such
8,
919.
[17] S. J. Vidwans and T. T. Su,
ing on clues that are both stress and cell type specic.
2007,
Mol. Biotechnol., 2008,
[16] A. Philpott and P. R. Yew,
Starting from a
logical model of their system, they show that the p53-
[1] P. Nurse,
Nature reviews. Drug
[15] R. L. Lock and E. J. Harry,
of the p53-Mdm2 network (involved in a DNA damage
2004,
[35] A. D. Rudner and A. W. Murray,
197209.
2000,
14
149,
137790.
J. Cell Biol.,
[36] B. Novak and J. J. Tyson,
( Pt 4),
J. Cell. Sci., 1993, 106
[55] A. Ciliberto and J. J. Tyson,
115368.
2000,
Philos. Trans. R. Soc. Lond., B, Biol.
Sci., 1995, 349, 27181.
[37] K. Nasmyth,
[57] A. Sveiczer, J. J. Tyson and B. Novak,
Nature, 2006, 443, 5947.
functional genomics & proteomics,
[58] A. Csikász-Nagy,
2002,
14,
[59] F. R. Cross,
M. Klovstad,
[41] G. Wang, H. Kong, Y. Sun, X. Zhang, W. Zhang,
N. Altman, C. W. DePamphilis and H. Ma,
Plant
108 ( Pt 9),
J. Cell. Sci.,
[62] I. Mura and A. Csikász-Nagy,
2008,
Genome Biol., 2002, 3, RESEARCH0070.
254,
The origins of order: self organization and selection in evolution, Oxford University Press US, 1993.
[47] K. Fülöp, S. Tarayre, Z. Kelemen, G. Horváth,
[66] C. Chaouiya,
Z. Kevei, K. Nikovics, L. Bakó, S. Brown, A. Kon2005,
4,
[67] L. Glass and S. A. Kauman,
Hateboer,
A.
Wobst,
B.
O.
1973,
Petersen,
Cell. Biol., 1998, 18, 667997.
Mol.
Cell,
2004,
117,
66,
261,
88,
Proc. Natl. Acad. Sci. U.S.A., 2004, 101, 47816.
Mol. Cell. Biol., 2004, 24, 36609.
Science, 1991, 251, 10768.
[71] A. Fauré, A. Naldi, C. Chaouiya and D. Thiery,
Bioinformatics, 2006, 22, e12431.
Proc. Natl. Acad. Sci. U.S.A., 1991,
[72] M. I. Davidich and S. Bornholdt,
910711.
[53] J. J. Tyson,
Proc. Natl. Acad. Sci. U.S.A.,
3,
[73] M. Davidich and S. Bornholdt,
Methods,
2007,
2008,
41,
23847.
255,
[74] D. Irons,
15
PLoS ONE, 2008,
e1672.
1991,
732832.
[54] J. C. Sible and J. J. Tyson,
Exp. Cell Res., 2000,
91103.
[70] F. Li, T. Long, Y. Lu, Q. Ouyang and C. Tang,
[50] S. Ahmed, C. Palermo, S. Wan and N. C. Wal-
[52] A. Goldbeter,
Bull. Math. Biol., 2004,
30140.
[69] S. Huang and D. E. Ingber,
899
913.
[51] R. Norel and Z. Agur,
J. Theor. Biol.,
10329.
T. Sari and J. Geiselmann,
man, O. Schub, K. Breitkreuz, D. Dewar, I. Rupes,
B. Andrews and M. Tyers,
39,
[68] H. De Jong, J.-L. Gouzé, C. Hernandez, M. Page,
[49] M. Costanzo, J. L. Nishikawa, X. Tang, J. S. Mill-
88,
Brief. Bioinformatics, 2007, 8, 210
9.
108492.
worth,
Biological Feedback,
[65] S. A. Kauman,
Annu. Rev. Genet., 2005, 39, 6994.
L. Le Cam, E. Vigo, C. Sardet and K. Helin,
J. Theor. Biol.,
CRC Press, Inc., 1990, p. 316.
Biochem. J., 2005, 387, 63947.
Cell Cycle,
BMC
Mol. Biol. Cell, 1999, 10, 270334.
[64] R. Thomas and R. D'Ari,
P. Coccetti, P. Fantucci, M. Vanoni and L. Al-
[48] G.
Proc. Natl. Acad.
85060.
[63] K. W. Kohn,
[45] M. Barberis, L. De Gioia, M. Ruzzene, S. Sarno,
dorosi and E. Kondorosi,
90,
M. Miller and
Genomics, 2006, 7, 108.
[44] C. A. Nieduszynski, J. Murray and M. Carrington,
[46] J. Bähler,
V. Archambault,
[61] P. J. Ingram, M. P. H. Stumpf and J. Stark,
294554.
Plant Cell, 2003, 15, 276377.
berghina,
2006,
Mol. Biol. Cell, 2002, 13, 5270.
Sci. U.S.A., 2003, 100, 97580.
[43] Y. Yu, A. Steinmetz, D. Meyer, S. Brown and W.H. Shen,
K. C. Chen,
Biophys. J.,
Yi, J. J. Tyson and J. C. Sible,
D. Glover and N. B. La Thangue,
298
[60] W. Sha, J. Moore, K. Chen, A. D. Lassaletta, C.-S.
[42] X. F. Hao, L. Alphey, L. R. Bandara, E. W. Lam,
1995,
2,
436179.
90316.
Physiol., 2004, 135, 108499.
D. Battogtokh,
B. Novák and J. J. Tyson,
[40] K. Vandepoele, J. Raes, L. De Veylder, P. Rouzé,
Plant Cell,
Briengs in
2004,
307.
Science, 2005, 307, 7247.
S. Rombauts and D. Inzé,
J. Theor. Biol., 2004, 230,
56379.
[39] U. de Lichtenberg, L. J. Jensen, S. Brunak and
P. Bork,
Bull. Math. Biol.,
3759.
[56] B. Novák and J. Tyson,
[38] L. J. Jensen, T. S. Jensen, U. de Lichtenberg,
S. Brunak and P. Bork,
62,
26977.
J. Theor. Biol., 2009.
J. Theor. Biol.,
Bull. Math. Biol.,
[75] D. Thiery and R. Thomas,
1995,
57,
[76] M. Kaufman, J. Urbain and R. Thomas,
Biol., 1985, 114, 52761.
Math. Biol., 1995, 57, 24776.
129,
J. Theor.
[97] D. Thiery,
Bioinformatics,
2008,
24,
i2206.
14162.
Pacic Symposium on Biocomputing. Pacic Symposium on Biocomputing, 1998,
[99] A. Wuensche,
[78] E. Muraille, D. Thiery, O. Leo and M. Kaufman,
J. Theor. Biol., 1996, 183, 285305.
[79] L. Mendoza,
Bull.
Brief. Bioinformatics, 2007, 8, 2205.
[98] E. Remy and P. Ruet,
J. Theor. Biol.,
[77] M. Kaufman and R. Thomas,
1987,
[96] R. Thomas, D. Thiery and M. Kaufman,
27797.
89102.
Pacic
Symposium on Biocomputing. Pacic Symposium
on Biocomputing, 1998, 1829.
BioSystems, 2006, 84, 10114.
[80] L. Mendoza and E. R. Alvarez-Buylla,
Biol., 1998, 193, 30719.
[100] S. Liang, S. Fuhrman and R. Somogyi,
J. Theor.
[101] A. Gonzalez, A. Naldi, L. Sánchez, D. Thiery and
[81] L. Mendoza, D. Thiery and E. R. Alvarez-Buylla,
C. Chaouiya,
Bioinformatics, 1999, 15, 593606.
[102] G. Simchen,
[82] C. Espinosa-Soto, P. Padilla-Longoria and E. R.
Alvarez-Buylla,
211,
223,
J. Theor. Biol., 2001,
[104] L. Dirick and K. Nasmyth,
J. Theor. Biol., 2003,
J. A. DeCaprio,
2005,
Systems
171,
[107] D. L. Fisher and P. Nurse,
[108] J. Bähler and S. Svetina,
237,
S7184.
Dev. Biol., 2008, 52, 105975.
Bioessays,
EMBO J.,
1996,
15,
85060.
Journal of the Royal Society, Interface / the Royal Society, 2008, 5 Suppl
[88] L. Sánchez, C. Chaouiya and D. Thiery,
Genetics,
4961.
[87] M. Chaves and R. Albert,
1,
351,
Nat. Cell Biol., 2001, 3, 104350.
[106] J. M. Bean, E. D. Siggia and F. R. Cross,
51737.
biology, 2006, 153, 15467.
1991,
[105] J. Ayté, C. Schweitzer, P. Zarzov, P. Nurse and
J. Theor. Biol., 2003,
[86] M. Chaves, E. D. Sontag and R. Albert,
Nature,
7547.
118.
[85] L. Sánchez and D. Thiery,
224,
[103] D. O. Morgan,
11541.
[84] R. Albert and H. G. Othmer,
Annu. Rev. Genet., 1978, 12, 16191.
The Cell Cycle: Principles of Control, New Science Press, 2007.
Plant Cell, 2004, 16, 292339.
[83] L. Sánchez and D. Thiery,
BioSystems, 2006, 84, 91100.
Int. J.
J. Theor. Biol.,
2005,
2108.
[109] A. Ciliberto, B. Novak and J. Tyson,
2003,
163,
J. Cell Biol.,
124354.
25,
[110] E. Queralt, C. Lehane, B. Novak and F. Uhlmann,
[90] M. A. Schaub, T. A. Henzinger and J. Fisher,
[111] R. Aleri, I. Merelli, E. Mosca and L. Milanesi,
[89] A. Ghysen and R. Thomas,
2003,
Cell, 2006, 125, 71932.
8027.
BMC systems biology, 2007, 1, 35.
BMC systems biology, 2007, 1, 4.
[91] J. J. Tyson, K. C. Chen and B. Novak,
Cell Biol., 2003, 15, 22131.
[92] S. Klamt,
J. Saez-Rodriguez,
L. Simeoni and E. D. Gilles,
ics, 2006, 7, 56.
Curr. Opin.
2000,
[113] G.
J. A. Lindquist,
BMC Bioinformat-
[93] M. Chaves, R. Albert and E. D. Sontag,
Biol., 2005, 235, 43149.
[112] M. Kanehisa and S. Goto,
Joshi-Tope,
M.
Gillespie,
I.
Vastrik,
sal, G. R. Gopinath, G. R. Wu, L. Matthews,
S. Lewis, E. Birney and L. Stein,
Res., 2005, 33, D42832.
Nucleic Acids
[114] Y. Geng, Q. Yu, E. Sicinska, M. Das, J. E. Schnei-
Lecture
der, S. Bhattacharya, W. M. Rideout, R. T. Bron-
Notes in Computer Science, 2007, 4695, 233247.
son, H. Gardner and P. Sicinski,
Cell,
43143.
[95] A. Garg, A. Di Cara, I. Xenarios, L. Mendoza and
G. De Micheli,
Nucleic Acids Res.,
2730.
P. D'Eustachio, E. Schmidt, B. de Bono, B. Jas-
J. Theor.
[94] A. Naldi, D. Thiery and C. Chaouiya,
28,
Bioinformatics, 2008, 24, 191725.
[115] F. Traganos,
16
Cell Cycle, 2004, 3, 324.
2003,
114,
[116] D. Santamaría, C. Barrière, A. Cerqueira, S. Hunt,
[137] M. Barberis, E. Klipp, M. Vanoni and L. Al-
C. Tardy, K. Newton, J. F. Cáceres, P. Dubus,
M. Malumbres and M. Barbacid,
448,
Nature,
Cell, 2008, 14, 15969.
[118] J. DeGregori and D. G. Johnson,
6,
2006,
N. Yoshioka, A. Dhiman, R. Miller, R. Gendel-
Dev.
man, S. V. Aksenov, I. G. Khalil and S. F. Dowdy,
Mol. Syst. Biol., 2007, 3, 84.
Curr. Mol. Med.,
63,
Cell. Mol. Life Sci.,
[120] R. Boutros, C. Dozier and B. Ducommun,
Opin. Cell Biol., 2006, 18, 18591.
Curr.
and
Tashima,
trich,
[142] B.
Mol. Syst. Biol., 2008, 4, 173.
[122] M. Rape and M. Kirschner,
Nature,
2004,
432,
Genet., 2005, 21, 14962.
Okamoto
PLoS ONE, 2008, 3, e1555.
Ibrahim,
P.
Dittrich,
S.
Diekmann
and
Biophys. Chem., 2008, 134, 93100.
Theoretical biology &
medical modelling, 2006, 3, 13.
[144] L. Mendoza and I. Xenarios,
Trends
[145] D. M. Wittmann, J. Krumsiek, J. Saez-Rodriguez,
D. A. Lauenburger, S. Klamt and F. J. Theis,
preparation, 2008.
Curr. Biol., 2004,
[146] A. Ciliberto, B. Novak and J. J. Tyson,
6306.
2005,
[126] H. R. Shcherbata, C. Althauser, S. D. Findley and
H. Ruohola-Baker,
M.
Chaos (Woodbury, N.Y.), 2001, 11, 277286.
[125] V. Schaeer, C. Althauser, H. R. Shcherbata, W.M. Deng and H. Ruohola-Baker,
Hamada,
[143] B. Novak, Z. Pataki, A. Ciliberto and J. Tyson,
Dev. Cell, 2004, 6, 3217.
[124] J. M. Claycomb and T. L. Orr-Weaver,
H.
J. Biosci. Bioeng., 2008, 106, 36874.
E. Schmitt,
58895.
14,
[140] Y.
[141] B. Ibrahim, S. Diekmann, E. Schmitt and P. Dit-
[121] L. Calzone, A. Gelay, A. Zinovyev, F. Radvanyi
[123] J. A. Coman,
Cell
T. Hanai,
76780.
and E. Barillot,
[139] B. Pfeuty, T. David-Pfeuty and K. Kaneko,
Cycle, 2008, 7, 324657.
73948.
[119] K. A. Wikenheiser-Brokamp,
PLoS Comput. Biol., 2007, 3, e64.
[138] T. Haberichter, B. Mädge, R. A. Christopher,
8115.
[117] A. Besson, S. F. Dowdy and J. M. Roberts,
2006,
berghina,
2007,
Development, 2004, 131, 3169
[147] A.
4,
Doi,
in
Cell Cycle,
48893.
M.
Nagasaki,
K.
Ueno,
H.
Matsuno
Genome informatics. International
Conference on Genome Informatics, 2006, 17,
and S. Miyano,
81.
11223.
[127] L. Calzone, D. Thiery, J. J. Tyson and B. Novak,
Mol. Syst. Biol., 2007, 3, 131.
[148] M. Lupi, G. Matera, C. Natoli, V. Colombo and
P. Ubezio,
[128] B. Novak, A. Csikasz-Nagy, B. Gyory, K. Nas-
Philos. Trans. R. Soc.
Lond., B, Biol. Sci., 1998, 353, 206376.
myth and J. J. Tyson,
[129] K. Nasmyth,
Trends Genet., 1996, 12, 40512.
[130] C. D. Thron,
Biophys. Chem., 1996, 57, 23951.
[131] C. D. Thron,
Oncogene, 1997, 15, 31725.
Cell Cycle, 2007, 6, 94350.
[149] J.-P. Qi, S.-H. Shao, D.-D. Li and G.-P. Zhou,
Amino Acids, 2007, 33, 7583.
[150] K. Iwamoto, Y. Tashima, H. Hamada, Y. Eguchi
and M. Okamoto,
BioSystems, 2008, 94, 10917.
[151] C. J. Proctor and D. A. Gray,
ogy, 2008, 2, 75.
BMC systems biol-
[132] J. R. Pomerening, S. Y. Kim and J. E. Ferrell,
Cell, 2005, 122, 56578.
[133] F. R. Cross,
[152] J. E. Toettcher, A. Loewer, G. J. Ostheimer, M. B.
Yae, B. Tidor and G. Lahav,
Dev. Cell, 2003, 4, 74152.
[134] E. Aleem, H. Kiyokawa and P. Kaldis,
Biol., 2005, 7, 8316.
Sci. U.S.A., 2009, 106, 78590.
Nat. Cell
[153] W. Abou-Jaoudé, D. Ouattara and M. Kaufman,
J. Theor. Biol., 2009.
Am. J. Physiol., Cell Physiol., 2003, 284, C34964.
[135] Z. Qu, J. Weiss and W. MacLellan,
[136] M. Swat, A. Kel and H. Herzel,
2004,
20,
Proc. Natl. Acad.
[154] B. Kang, Y.-Y. Li, X. Chang, L. Liu and Y.-X. Li,
PLoS Comput. Biol., 2008, 4, e1000019.
Bioinformatics,
[155] T. Alarcón, R. Marches and K. M. Page,
Biol., 2006, 240, 5471.
150611.
17
J. Theor.
[156] S. Legewie, N. Blüthgen and H. Herzel,
Comput. Biol., 2006, 2, e120.
PLoS
[157] K. A. Janes, H. C. Reinhardt and M. B. Yae,
Cell, 2008, 135, 34354.
[158] H. Fuss, W. Dubitzky, S. Downes and M. J. Kurth,
Bioinformatics, 2006, 22, e15865.
[159] H. Fuss, W. Dubitzky, C. S. Downes and M. J.
Kurth,
Biophys. J., 2008, 94, 19952006.
[160] H. Li, C. Y. Ung, X. H. Ma, B. W. Li, B. C. Low,
Z. W. Cao and Y. Z. Chen,
25,
Bioinformatics, 2009,
35864.
Results and problems in cell dierentiation, 2006, 42, 34367.
[161] C. Rajesh and D. L. Pittman,
[162] J. Erenpreisa, M. Kalejs and M. S. Cragg,
Biol. Int., 2005, 29, 10128.
[163] D. M. J. Martindill and P. R. Riley,
2008,
7,
Cell
Cell Cycle,
1723.
18