Cellular and Molecular Biology 51, 583-594
ISSN 1165-158X
2005 Cell. Mol. Biol.
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DOI 10.1170/T667
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BISTABILITY AND HYSTERESIS IN EPIGENETIC REGULATION
OF THE LACTOSE OPERON.
SINCE DELBRÜCK, A LONG SERIES OF IGNORED MODELS
M. LAURENT1✍, G. CHARVIN1 and J. GUESPIN-MICHEL2
1✍
Dynamique Cellulaire et Modélisation, UMR 8080, Bât. 440, Université Paris-Sud, 91405 Orsay Cedex, France
E-mail: Michel.Laurent@ibaic.u-psud.fr
2 Laboratoire de Microbiologie du froid, EA 2123 Faculté des Sciences et Techniques de Rouen, Mt St Aignan, France
Received March 27, 2005; Accepted April 1, 2005; Published December 14, 2005
Abstract - Bistability is the capacity of a system to switch in an "all-or-none" manner between alternative steady states. This powerful
concept originates from the analysis of non-linear equations driving open systems. It is one of the various patterns of regulation
associated with a particular class of dynamic structures that Glansdorff and Prigogine baptised "dissipative structures". The idea of
discontinuous transitions between alternative states was first formulated much earlier, by Delbrück, in 1949. Cohn and Horibata and
Novick and Weiner confirmed that such transitions occur in experiments on the lactose operon carried out ten years later. Modelling
with non-linear differential equations made it possible to simulate the dynamic behaviour of the lac operon, and modelling by
asynchronous logical analysis elucidated the determinant role played by positive feedback circuits in the emergence of multistationarity.
Nevertheless, these studies were largely ignored until the recent demonstration of the hysteretic nature of the bistable transition between
alternative states of the lac operon. As originally suggested by Delbrück, the pattern of lactose consumption adopted by the bacterium
is controlled epigenetically rather than genetically: the true key determinant is the direction of change of an environmental variable with
respect to the structural components of the operon.
Key words: lac operon, bistability, logical circuits, Boolean logic, dissipative structures, hysteresis
INTRODUCTION
The lac system in E. coli was extensively studied in the
1950’s and 1960’s, primarily as a model of enzyme
induction (11,24,29). The genes responsible for the
synthesis of β-galactosidase are active only in the presence
of a small sugar molecule, the "inducer". As the main
molecular components of the lac operon were identifed
and interactions between them elucidated, this system
became a paradigm for gene regulation, via the concepts of
feedback and autocatalysis (23). However, two distinct
philosophical views of living systems existed at that time
and gave rise to opposing interpretations of this mechanism
of gene regulation. Monod considered the lac operon to be
the archetype of genetic determinism upon enzyme
induction. When genes are switched on, β-galactosides are
metabolised; when they are shut off, lactose-degrading
Abbreviations: gfp: green fluorescent protein; IPTG:
isopropyl-β-D-galactoside; Lac: lactose; OD: optical density;
ODU/s: optical density units per second; ONPG: onitrophenyl-β-D-galactopyranoside; TMG: thiomethyl-β-Dgalactoside; u.bgal: unit of β-galactosidase activity
enzymes are not synthesized and these sugars cannot be
used as a carbon source. The presence or absence of an
external inducer is the deterministic mechanism
responsible for switching the genes on and off. In this
interpretation, the status of the gene controls the fate of the
cell. Cohn and Horibata (9) proposed a more subtle
interpretation of their experiments. They suggested that the
lac system provides an "experimental example of the
Delbrück model" formulated in 1949 (12). According to
Delbrück, biological systems with identical genotypes may
display different behaviour under particular external
conditions. This difference is due to "epigenetic"
differences, which can be transmitted in the cell lineage in
the absence of genetic modification. This hypothesis
corresponds to a very early formulation of the general
principle of phenotypic inheritance. The theoretical and
mechanistic bases of this principle was described more
than 20 years later, by Glansdorff and Prigogine (14): these
epigenetic differences are typical cases of a more general
process, called multistationarity. The lac operon in E. coli
may be considered as a prototype system displaying such
behaviour.
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FEEDBACK LOOP IN THE MECHANISM
OF REGULATION OF THE LAC OPERON
The lactose operon comprises three adjacent genes,
lacZ, lacY and lacA, under the control of a single promoter.
These genes are co-transcribed as a polycistronic
messenger RNA. The proteins encoded by the two first
genes are involved in the uptake and degradation of lactose
and related sugars (Fig. 1). Immediately upstream from the
lac operon is a regulatory domain in which the lacI gene
encodes a constitutively expressed transcriptional
regulator, the lactose repressor lac. The repressor interacts
closely with several operator sites on the DNA, preventing
the binding of the RNA polymerase and subsequent
transcription of the lac operon.
LacZ encodes β-galactosidase, an enzyme responsible
for lactose degradation. Another mechanism converts
lactose into its isomer, allolactose. Allolactose is the natural
inducer of the system: it binds to the lactose repressor and
changes its conformation, causing this protein to loosen its
hold on the DNA, thereby facilitating transcription of the
operon. The product of lacY is a specific permease that
promotes the passage through the membrane of lactose and
some related β-galactosides. Regulation of the lac operon
appears to be autocatalytic: the permease promotes the
passage of lactose through the membrane. This results in
an increase in the intracellular concentration of inducer
and, subsequently, the production of more permease.
However, if the external concentration of the inducer is
high enough (which is the case in most experimental
conditions), diffusion through the cell envelope is
sufficient to promote its entry.
Lactose is thus both the inducer of the system and a
source of carbon and energy for cells. These two functions
may be experimentally uncoupled, using gratuitous
inducers such as TMG and IPTG. These molecules enter
the cell via the lactose permease. They bind to the lac
repressor but are not metabolised. The uptake of IPTG
therefore induces the synthesis of LacY, which in turn
promotes further IPTG uptake (if the external
concentration is low). Hence, gratuitous inducers can be
used to study the specific effect of the feedback loop on
regulation of the lac operon.
THE KEY EXPERIMENTS OF NOVICK AND
WEINER AND COHN AND HORIBATA
AND DISCOVERY OF THE
"MAINTENANCE" EFFECT
Novick and Weiner (27) and Cohn and Horibata (8-10)
discovered what they described as a ‘maintenance effect’.
As a consequence of the mechanism of enzyme induction,
a single cell may have two alternative states: induced, in
which it can metabolise lactose, or uninduced, in which the
Fig. 1
Schematic representation of the rationale of the
operon model. a) Transcription of the operon is repressed by
the binding of the Lac repressor (encoded by lacI) to the lac
operator, a pseudo-palindromic sequence centred 11 bp
downstream from the transcription start site (25). This
interaction prevents the binding of RNA polymerase, thereby
blocking transcription of the operon. This operon also has an
activator site, which binds to the cAMP-CRP complex,
resulting in the activation of transcription. This mechanism is
involved in catabolic repression. b)Asmall sugar molecule (the
artificial inducer IPTG or the natural inducer allolactose) binds
to the Lac repressor protein and decreases the strength of the
repressor-operator DNA interaction. One of the genes of the
operon, lacY, encodes a specific permease that promotes the
passage through the membrane of both β-galactoside and
inducer. c) At the cellular level, this results in an increase in
intracellular inducer concentration and the production of more
permease, in an autocatalytic manner.
corresponding genes are switched off and lactose
metabolism does not occur. However, the key experiments
of these pioneering authors involved switching on an entire
cell population, rather than a single cell. The protocol was
as follows (Fig. 2) a large amount of inducer was added to
the extracellular medium of a culture of uninduced E. coli
cells. The culture was split into two parts: U (uninduced)
and I (induced). Part U was immediately diluted, resulting
in an immediate large decrease in extracellular inducer
concentration. Part I was diluted after being maintained in
the medium with high inducer concentration for several
minutes. The cells in subculture U remained uninduced
Bistability in lac operon regulation
whereas those in subculture I were induced. Growing
subcultures were then serially diluted with medium
containing a low concentration of inducer: U remained
uninduced and I, induced. The range of external inducer
concentrations at which both induced and uninduced cells
may be present was called the "maintenance"
concentration by Novick and Weiner. The "maintenance
effect" was interpreted as the consequence of a high
permease concentration in induced cells, which would also
have high inducer pumping efficiency. This would enable
these cells to maintain the induced state and to transmit it
to their progeny, even if placed in a medium with a low
concentration of inducer. This interpretation accounts for
the existence of two distinct phenotypes and provides an
explanation as to why induced cells placed in media with
low inducer concentrations remain indefinitely induced,
whereas cells that have never been induced stayed
uninduced at this low concentration of inducer. However,
it does not explain what makes the cells switch between
alternative states. Even Delbrück, with his renowned
intuition, could provide no explanation for this switching,
which remained a mystery until the thermodynamic and
mechanistic bases of dynamic systems were described in
1971, by Glansdorff and Prigogine (14). The next
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challenge was to determine how a graded signal (the
extracellular concentration of inducer) could be converted
into discontinuous changes in gene expression. An
understanding of this conversion is required to explain how
a phenotypic change (an all-or-none transition) in the
behaviour of the lac operon can be inherited following a
transient signal – exposure to a high concentration of
allolactose. This situation corresponds to the modern
concept of bistability – the capacity of a system to achieve
alternative steady states in response to a single set of inputs
and its ability to switch between these states in response to
a signal.
BISTABILITY DUE TO THE EXISTENCE OF
A NON-LINEAR FEEDBACK LOOP
The modelling of relevant processes was required to
unravel the dynamics of lac operon regulation. Babloyantz
and Sanglier (4) and Nicolis and Prigogine (26) proposed
a dynamic model for interpretation of the ‘maintenance
effect’ as the biological facet of the physical process of
multistability. This model, involving a non-linear feedback
loop, accounted for the main behavioural features of the
bistable transition of the lactose operon. However, even
though the mathematical description of the model required
five differential equations (plus one conservation
equation), the model did not take into account the detailed
information available concerning molecular interactions
between components. Fortunately, not all the details are
needed. This non-linear feedback loop has consequences
for the dynamics of lac operon regulation.
LOGICAL ANALYSIS OF THE
EPIGENETIC REGULATION OF THE
LACTOSE OPERON
Fig. 2
The key experiment of Novick and Weiner (27)
and Cohn and Horibata (9) demonstrating the existence of a
"maintenance" range of concentrations of the inducer. A high
amount of inducer was added into the extracellular medium of
a culture of uninduced E. Coli bacterial cells. A part of the
culture (part U, uninduced) was immediately diluted, so that the
extracellular concentration of inducer dropped to a small value
for the corresponding cells. On the contrary, part I (induced) of
the culture was maintained during several minutes into the
medium containing highly concentrated inducer. Only cells
maintained for long periods in the presence of a high
concentration of inducer are induced. If induced cells are
transferred to a medium with the "maintenance concentration"
of inducer, they – and their progeny – remain induced.
Similarly, if uninduced cells are transferred to a medium with
the "maintenance" concentration, they – and their progeny –
remain uninduced. See the main text for a precise description of
this experiment demonstrating the maintenance of two stable
steady states of β-galactosidase synthesis are maintained by a
genetically homogeneous population growing in a fixed
medium (8).
The models described above have frequently been
ignored by scientists working on the lac operon. A recent
paper (28), for example, starts afresh building on the work
of Novick and Wiener (27) and Cohn and Horribata (9). It
is therefore clear that the theoretical studies carried out
over the last forty years or so, since the work of Glansdorff
and Prigogine (14) have made little impact on the
biological community. Indeed, even the papers by Novick
and Wiener (27) and Cohn and Horribata (9) have been
largely ignored. They are absent from most, if not all,
manuals and treatises of molecular biology and bacterial
genetics, in which the lactose operon remains the
paradigmatic example of gene regulation. The role and
presence of the permease are often omitted from short
descriptions of the lactose operon. It may be that biological
training does not include enough mathematics to enable
biologists to manipulate with ease the non-linear
differential equations required to model and therefore to
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explain experiments showing epigenetic modifications?
However, modelling requiring much less mathematical
knowledge (logical analysis) received no more attention
from biologists until quite recently.
Fig. 3
Classical and asynchronous Boolean analysis of a
simple circuit. a) Logical graph of a two-variable positive
circuit. b) State table corresponding to classical (resp.
asynchronous) Boolean analysis of the above circuit. c)
Classical Boolean state graph. d) Asynchronous Boolean state
graph (see text).
In his early modelling work, René Thomas focused on
the epigenetic modifications following infection of
Escherichia coli with phage lambda. However, as early as
1973 (35), he suggested that Boolean logic could be used
to model the dynamic behaviour of the lactose operon. He
later gave a detailed explanation of the subject (37). The
use of Boolean algebra for modelling the dynamic
behaviour of biological interactions is based on the
typically sigmoidal shape of these interactions, making it
possible to simplify them as all-or-none phenomena: for
example, a protein concentration may be below (0) or
above (1) its threshold of activity, and a gene may be
expressed strongly enough to produce an amount of
protein above (1) or below (0) its threshold of activity. This
simplification amounts to transforming a sigmoid into a
step function (39).
A logical graph represents the interactions (Fig. 3). In
the classical, synchronous Boolean equation, the positive
regulation of a gene y by a protein x is denoted y(t+1) =
x(t), with the vertex on the graph marked by a ‘plus’ sign,
whereas negative regulation is denoted y(t+1)= *x(t), with
the vertex on the graph marked with a ‘minus’ sign The
very simple graph in Fig. 3a shows that x positively
regulates y, which in turn positively regulates x. The
classical Boolean equation is:
x(t+1) = y(t)
y(t+1) = x(t)
Values for these variables can be determined with a
state table (Fig. 3b), in which the left column contains all
the possible combinations of values for the variables at
time t and the right column, the corresponding values of
the variables at time t+1. The lines in which the figures are
equal in both columns correspond to states in which the
system is not changing (stable states). Fig. 3b shows two
stable states: 11 and 00. This state table can be converted
into a graph of states displaying the dynamics of the
system starting from any initial condition (Fig. 3c). There
are two stable states (00 and 11), and one oscillation
01 ➝ 10 ➝ 01.
Biologically, this means that if the initial condition is 01
(protein x below its threshold, protein y above its
threshold), the next step will be 10. However, there is no
reason why the synthesis of protein x, once the gene is
activated, should take the same time as the disappearance
of protein y. Indeed, the turnover of a protein is generally
slower than its synthesis. This led Thomas to propose
"asynchronous logic" (38), in which the variable xi
corresponds to the proteins, and the function Xi
corresponds to the expression of the corresponding gene:
X=y
Y=x
As in the classical formalism, if the values in both
columns of the state table are identical, a stable state has
been reached. In Fig. 3b, states 00 and 11 are the stationary
Bistability in lac operon regulation
states. However, in this case, state 01 is followed either by
the expression of gene X (corresponding to the synthesis of
protein x above it’s threshold, state 11) or the repression of
gene Y followed by a decrease in the amount of protein y
to below it’s threshold (state 00) (Fig. 3d). There is no more
oscillation.
The use of asynchronous logic enabled Thomas to
obtain qualitative results similar to those obtained with
differential equations, despite the gross simplifications it
involves (39). The main advantage of this method, apart
from its great simplicity, is that it does not require values
for the parameters (which are lacking in most biological
situations), to determine the qualitative dynamic behaviour
of the system.
Indeed, the very simple logical equation in Fig. 3 may
be sufficient to describe the bistability property of the
lactose operon. Let y be the permease, Y the expression of
the lacY gene (corresponding to expression of the entire
operon), x the internal concentration of inducer (X
corresponds to the entry of the inducer). As the LacI
protein is constitutively synthesised, it is not required as a
variable of the system. If the gratuitous inducer is at
maintenance concentration, the permease allows the entry
of inducer (X = y), which inhibits LacI, leading to the
expression of lacY (and thus the synthesis of permease)
(Y= x). In these conditions, there are two stable states: 00
(the uninduced phenotype, neither permease, nor internal
inducer), and 11 (induced phenotype).
It is very easy to account for the effect of high
concentrations of external inducer. Let us represent the
external inducer concentration as parameter E (E = 1 if the
external inducer concentration is high enough to allow
diffusion through the envelopes, E = 0 if the inducer
concentration is lower). The logical state equation
becomes:
Y = x +E
X=y
When E=0, the table is identical to that shown in Fig.
3b, and there are two possible stable states. When E=1,
protein LacI is always inactive and gene lacY is always
expressed (Y=1), so there is only one stable state, the
induced state (Fig. 4) Thomas (37) also showed that this
method could be used to simulate the more complicated
situation in which lactose is both the inducer and the
carbon source, resulting in its disappearance from the
medium.
This very simple graph clearly illustrates the properties
of the system, as described by Novick and Weiner (27) and
Cohn and Horibata (9). For instance, if the inducer
concentration is increased from maintenance concentration
to high concentration (Fig. 4), we shift from column 2 to
column 3 (E = 1). If the stable state was the uninduced
state, then we observe a shift to the induced state, 11. A
subsequent decrease in inducer concentration (back to
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column 2) does not change the new stationary state. Thus,
in an uninduced population, a transient increase in the
concentration of the inducer shifts the system to the
permanently induced state.
This method made it possible for Thomas to
demonstrate the existence of positive feedback loops in
this system. As explained above, bistability occurs because
the system is non-linear and contains a positive feedback
circuit. An extensive use of this method led Thomas to
realise that what is common to all the cases of bistability
was the presence of a positive feedback circuits (e.g. circuit
with an even number of negative interactions). He
postulated that the existence of a positive feedback was not
only the simplest way to generate multistationnarity, but in
fact a necessary condition of multistationnarity (and that
negative feedback circuits are required for homeostasis
with or without oscillations) (36). This conjecture was later
studied by mathematicians, who showed it to be true for
non-linear systems with sigmoid interactions (7,16,30,32),
and for non-linear conditions in general (34). It was also
recently shown (2) that frequently satisfied mathematical
conditions guaranteed multistability in positive-feedback
systems of arbitrary order. In addition, Thomas showed
that a single positive circuit, regardless of it’s length, can
only generate two stable and one unstable steady state. To
achieve multistationarity with a larger number of
stationary states, a system must have more independent
positive circuits; the number of steady states cannot exceed
3n (with 2n stable states), if n is the number of circuits.
This theorem has two major consequences for studies
of regulatory networks. Firstly, a regulatory network must
Fig. 4
Modelling the dynamic behaviour of the lactose
operon with asynchronous logical methods The state table
corresponds to the asynchroneous Boolean equation:
X=y+E
Y=x
where x = the permease, X = the expression of gene lacY, y =
the internal inducer, Y = the entry of the external inducer, and E
stands for an external inducer concentration high enough (E=1)
or too low (E=0) to allow passive diffusion through the
membrane (see text for explanations).
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contain a positive feedback circuit if it is to display
multistationarity (just as the presence of a negative
feedback circuit is a prerequisite for oscillations). Thus,
although all of the interactions in a network are important,
those involved in circuits are the key interactions for
understanding the dynamics of the system, and the
presence of such circuits in a system may lead to the
formulation of new hypotheses concerning the behaviour
of the system (17,19). Secondly, as these circuits are
necessary but not sufficient for multistationarity, an
experimental demonstration of the phenomenon is needed.
Such a demonstration can be facilitated by computer
analysis of the logical model (6).
Since their early research, Thomas and his co-workers
(32) have developed a more comprehensive logical method
and determined more precisely the conditions required for
multistationarity (40). Both the naïve logical method and
general asynchronous analysis can be used to model the
dynamic behaviour of biological systems with positive
feedback loops (13,31,41).
Comparative differential model of regulation of the
lactose operon
The rate of change over time in the intracellular
concentration of inducer [ai] corresponds to the algebraic
difference between the rate of the process producing
intracellular inducer (vin) and that of the processes
removing this species (vout).
net rate = d[ai]/dt = vin - vout
Eq. 1
Free intracellular inducer is depleted in several
processes (binding of the inducer to the repressor,
catabolism and dilution following cell division). As none of
these removal processes is regulated by ai, the
corresponding rates may be assumed to be simply
proportional to [ai]:
vout = kout [ai]
Eq. 2
where kout is a first-order rate constant.
The rate of production of intracellular inducer, vin, is
proportional to the concentration of inducer in the
extracellular medium ([ae]). However, as stated above,
intracellular inducer facilitates the transcription of
permease, and permease promotes the transport of inducer
from the external medium into the bacterial cell. Therefore,
the intracellular inducer indirectly activates the entry of
external inducer into the cell. For a given concentration [ae]
of external inducer, the rate, vin, displays a sigmoidal
dependence on [ai], indicating both the existence of a
saturating phenomenon and the non-linearity of the
response. This is well approximated by an empirical Hilllike equation (21,22):

[a i ]n
vin = [a e ] k 0 +
K + [ai ]n





Eq. 3
The parameter k0 in Eq. 3 corresponds to passive
diffusion of the inducer throughout the cell membrane. In
the absence of this phenomenon (or in absence of a basal
level of protein synthesis in the absence of inducer), there
would be no permease at all and the first molecule of
inducer could never enter the cell, preventing permease
synthesis. Indeed, basal transcription was found
experimentally to correspond to the presence in the cell of
1 to 10 molecules of β-galactosidase in the absence of
inducer (15).
In Eq. 3, the non-linearity of the dependence of the rate
vin on intracellular inducer concentration [ai] is crucial (see
Fig. 5 below). According to Muller et al. (25) and Ozbudak
et al. (28), this non-linearity is due to binding of the inducer
to the LacI tetramer (co-operative binding with a Hill
number n close to 2). More generally, Vilar et al. (42)
suggested that the rate of production of permease is a nonlinear function of intracellular inducer concentration.
Whatever the molecular basis of the co-operative
phenomenon in regulation of the lac operon, the
consequence is that the intracellular inducer exerts nonlinear, positive feedback, controlling its own entry.
Fig. 5 shows the dependence of vin and vout on
intracellular inducer concentration [ai]. There are three ai
concentrations for which vin = vout. These particular values
of [ai] define three steady states (SS1, SS2 and SS3) for the
system because, in each, [ai] does not change with time
(d[ai]/dt = vin - vout = 0). Small fluctuations disappear
around the SS1 and SS3 steady states corresponding to
extreme values of [ai], whereas they are amplified around
the intermediate SS2 steady state (Fig. 5). Thus, SS1 and
SS3 act as attractors (they are stable steady states) whereas
SS2 behaves like a repellent (it is unstable). The lac operon
therefore behaves like a bistable system, with the two
stable steady states separated by one unstable steady state.
The two stable steady states of the system correspond to
two quite different regimes of lactose consumption. In the
SS1 steady state, [ai] is low: the repressor protein is bound
to operator sites on the lac promoter and the operon is
repressed. In contrast, the high stationary ai concentration
in the SS3 state facilitates the binding of the inducer
molecule to the repressor protein, thereby removing it from
the DNA and allowing transcription to proceed: the
corresponding cells are therefore induced.
Hysteresis as the ultimate explanation of the
"maintenance" effect
Steady-state concentrations of ai are characteristic of
the system for a given set of parameter values. To
understand fully the behaviour of the lac operon and to
Bistability in lac operon regulation
interpret the original experiment of Novick and Weiner
(27), we must describe what these steady states become
when the concentration of inducer in the extracellular
medium [ae] is changed. This involves calculating, for any
[ae], the value(s) of [ai] as follows:
d[ai]/dt = vin - vout = 0
Eq. 4
The corresponding steady states values of [ai] are thus the
roots of the equation:
n


[a e ] k 0 + [a i ] n  − k out [ai ]= 0
K + [ai ] 

Eq. 5
This equation may be solved numerically for [ai], for a
given set of parameter values (k0, kout, n and K) and for
each of ae concentration. For low values of [ae] (medium
grey region in Fig. 6), the system has only one steady state,
which corresponds to a low [ai]. In these conditions, the
operon is repressed. For a very high [ae] (light grey region
in Fig. 6), the system still has only one steady state, but this
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steady state corresponds to a high concentration of ai (i.e.
the operon is derepressed). At intermediate [ae] values, the
repressed and derepressed stable steady states coexist
(regions of positive slope in the dark grey region in Fig. 6)
together with an intermediate unstable steady state
(corresponding to the branch with negative slope in Fig. 6).
This intermediate situation corresponds to that previously
described in Fig. 5.
Let us suppose that the external concentration of
inducer [ae] is intially very low (the lac operon is repressed)
and that a continuous increase in [ae] occurs (black arrows
in Fig. 6). Until the stationary state lies on the branch of
positive slope, the stationary [ai] is slightly readjusted (it
increases slightly) but remains low. However, if [ae]
exceeds the threshold value (corresponding to the change
in sign of the slope), the system jumps suddenly towards
the branch of stable steady states corresponding to high [ai]
values: the lac operon becomes derepressed. Thus, a major
discontinuity appears for the steady state concentration of
ai, following a continuous increase in [ae].
Let us suppose that, [ae] is initially very high (the lac
operon is derepressed) and that a continuous decrease in
Fig. 5
Dynamic behaviour of the lac operon. Individual rates of production (vin) and removal (vout) of the intracellular inducer ai. The
corresponding kinetic laws are described in the main text. The curves were obtained using the following set of parameter values: kout=0.6, n=2,
K=5, k0=0.02, [ae]=3 (arbitrary units). The net rate of ai-production is the algebraic difference between production (vin) and removal (vout)
processes for the ai species. Steady-state conditions correspond to ai concentrations for which the net rate of ai production is zero. Insets show
analysis of the local stability properties of the three steady states SS1, SS2 and SS3. The local stability properties of the system are examined
by slightly modifying the concentration of ai (arrows) around each of the three steady states. For instance, let us consider the perturbation -d[ai]
from the steady state SS1 or SS3. In these conditions, the system reaches a state in which vin is greater than vout (i.e. the net rate of ai production
is positive). Thus, more ai is produced than removed: the fluctuation -d[ai] disappears. Small fluctuations disappear around SS1 or SS3, whereas
they are amplified around SS2. Thus, SS1 and SS3 are stable steady states whereas SS2 is unstable.
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Fig. 6
Jump-like transitions, threshold effects and hysteretic behaviour on the trajectory of steady states of the lac operon upon variations
in the extracellular concentration of allolactose, [ae]. The trajectory of steady states (S-shaped curve in the [ae]-[ai] plane) was determined by
numerical methods, as explained in the main text. Stable steady states correspond to the two parts of the trajectory that have a positive slope,
whereas the middle part of the trajectory with a negative slope corresponds to an unstable steady state. Three distinct areas may be defined from
this trajectory, corresponding to different ranges in the extracellular concentration of inducer: the lower part (mid grey), in which steady states
are stable and are associated with low [ai] values, and therefore with repressed states; the upper part (light grey), in which steady states are stable
but associated with high [ai] values, corresponding to derepressed states. Finally, in the middle part of the plane (dark grey), three steady states
(two stable and one unstable) are associated with each of the [ae] values. For each of these three distinct areas, an example of the kinetic
behaviour of the system is given, in terms of individual rate of production and degradation of the intracellular inducer. This figure illustrates the
behaviour of the system when [ae] is varied: when [ae] increases from low to high values (black arrows), the system undergoes a transition
(horizontal black arrows) from the low branch of stable steady states to the branch of stable steady states corresponding to high [ai] values; a
similar but reverse transition is observed if [ae] values are decreased (light arrows). However, the forward and backward transitions occur for
different threshold [ae] values (Y-axis values corresponding to horizontal arrows), according to the direction (black or grey arrows) of movement
along the trajectory: the system remembers the characteristics of its previous state (hysteretic phenomenon). Equations and parameter values
are as in Fig. 5.
[ae] occurs (Fig. 6, grey arrows). Once again, a switch is
observed between the derepressed and repressed states, but
the jump-like transition occurs for a lower threshold value
of [ae] than that triggering the forward transition. The
threshold values of [ae] are therefore different for the
forward and backward switches.
Consider now intermediate values of [ae]
(corresponding to the dark grey region in Fig. 6). What
governs the choice between the possible regimes (the
repressed state corresponding to low stationary [ai] or the
derepressed state corresponding to high [ai]) in this range
of [ae] values? This choice depends entirely on which
branch of stability the system was in before the control
parameter [ae] reached its current value (black or grey
arrows): the system behaves as if it remembers the
characteristics of its previous state (a hysteretic
phenomenon). Hence, the range of [ae] values in which two
stable steady states coexist (the hysteretic region) also
corresponds to the "maintenance" concentration range of
Novick and Weiner (27).
Studies of dynamic processes in the regulation of the
lac operon provide a clear theoretical basis to Delbrück’s
intuitive explanations and make it possible to interpret the
work of Novick and Weiner. A positive feedback circuit is
required for multistationarity, accounting for the binary
response of transcription of the lac operon. In terms of
"dissipative structures", we can also explain why the
quantity of β-galactosidase enzyme in the bacteria jumps
discontinuously between the uninduced state and the
induced state. This behaviour results purely from the
dynamics of regulation for this operon, producing a true
discontinuity in the stationary concentration of internal
Bistability in lac operon regulation
inducer: the difference between the induced and uninduced
regimes of the lac operon corresponds to a difference in the
cellular environment of the operon ([ai]) and not to the
molecular properties of the lac operon itself. For a given set
of environmental conditions, the lac operon system may
adopt either of the regimes of lactose consumption in the
absence of changes in the molecular characteristics of the
genes. The history of the system is the true deterministic
key: it is not genetically coded and is instead based on the
direction of variation of the environmental variable with
respect to the operon itself.
According to Delbrück (1949), cells can exist in
functionally different steady states "with no change in
genes, plasmagenes, enzymes or other structural units". He
suggested that changes between states could result from
changes in environmental conditions. Similarly, Cohn and
Horibata (8) observed that two stable steady states of βgalactosidase synthesis could be maintained by a
genetically homogeneous population growing in a fixed
medium.
EXPERIMENTAL CHARACTERISATION OF
HYSTERESIS IN SINGLE CELLS AND
IN CELL POPULATIONS
Discontinuous transitions between different states were
interpreted by Cohn and Horibata (9) in their experiments
with the lactose operon. Subsequent theoretical studies
have shown that hysteresis is an inherent property of
bistable systems. We need to know the precise conditions
in which hysteresis occurs if we are to characterise this
phenomenon experimentally. These conditions do not
resemble the usual experimental protocols of molecular
biologists and the corresponding properties have not been
recognised or sought in many cases. This is because, unlike
other aspects of the regulation of dynamic systems such as
oscillatory and chaotic behaviour, the experimental
characterisation of hysteresis requires both observation of
the system and well-defined modification of that system
(i.e. by varying one control parameter, increasing and
decreasing it).
Despite the "maintenance" effect of Novick and
Weiner, can the induction observed in E. coli cells be
reversed? This question is difficult to answer
experimentally because it is not possible to control
intracellular inducer concentration directly. Moreover, the
gratuitous inducer IPTG is not metabolised by cells.
Imagine that growing, induced cells are serially diluted
over numerous generations in a medium containing no
inducer at all. When induced cells are removed from a
medium containing a high concentration of IPTG and
transferred to a medium containing no IPTG, they divide,
resulting in an increase in total cell volume. Thus, the
intracellular concentration of IPTG decreases in each
591
generation. Furthermore, the passive diffusion of IPTG
through the cell membrane may accelerate the decrease in
intracellular IPTG concentration with generation time. The
intracellular concentration of inducer gradually decreases
as a result of both cell division and metabolic degradation
mechanisms.
Fig. 7 (upper part) shows the original results obtained in
this way for the wild-type E. coli K12 strain. As expected,
for increasing concentrations of IPTG (induction curve), an
abrupt increase in enzymatic activity was observed beyond
a threshold value (around 1.2 x 10–6 M IPTG). At higher
IPTG concentrations, enzymatic activity was stable. When
grown in a medium containing IPTG at concentrations of
3 x 10–7 M to 1.4 x 10–6 M (range "R"), fully induced (10–4
M IPTG) cells remained induced. In contrast, cells that had
never been induced remained uninduced at these
concentrations of IPTG. At concentrations in the range R,
extracellular IPTG cannot induce uninduced cells can
maintain induction in cells that are already induced. This
confirms the existence of a "maintenance" concentration.
Observations of ‘de-induction’ in the range R of IPTG
concentrations were not an artefact of a system in a
transitory regime: pre-induced cells placed in a medium at
10–7 M became totally uninduced (negative control). Cells
grown in medium containing 10–4 M IPTG for ‘deinduction’ retained their initial level of induction (positive
control). Finally, cells grown at IPTG concentrations
between 2 x 10–6 M and 10–4 M IPTG behaved like cells
that had not previously been induced cells. Pre-induced
cells can be maintained in an induced state for at least 24
generations in medium containing 10–6 M IPTG, with no
loss of β-galactosidase activity. So, in the range R, there are
two alternative states for the lac operon, resulting in high
(induced) and low (uninduced state) levels of βgalactosidase activity. The state adopted by a given cell
depends exclusively on the history of that cell (i.e. its
previous state), a feature typical of hysteretic behaviour.
Reversion is observed not only if the extracellular
concentration of inducer is zero, but also if this
concentration falls below a non-zero threshold. In addition,
the threshold value required for the reverse transition,
induced ➝ uninduced, differs from that for the forward
transition, uninduced ➝ induced. As previously discussed,
this property is also characteristic of hysteretic systems. By
contrast, a lactose-negative, presumably lacY– (but
possibly simply lacking active permease) cryptic strain,
CIP 54157, always returned to induction values when deinduced, with no hysteresis (lower part of Fig. 6). As
expected, the bistability of the lac operon is lost in the E.
coli cryptic strain because this behaviour is due specifically
to the action of the permease.
The transition between alternative steady states
observed experimentally does not appear to be as sharp as
would be expected for an all-or-none phenomenon.
592
M. Laurent et al.
Fig. 7
Induction (black symbols) and de-induction (white symbols) curve for K12 cells (upper figure) and for strain 54157 (lower figure).
Experiments were performed five times for each strain (except for the deinduction curve for 54157, formerly known as ML3 in the Pasteur
Institute classification (19), for which the whole curve was traced once only, but reproducibility was assessed at several concentrations in the
neighbourhood of the transition). For the other experiments, means and standard deviations are shown. The induction of enzyme activity was
observed beyond a threshold value around 1.2 x 10–6 M IPTG for the wild-type K12 strain. The de-induction curve displays marked hysteresis.
For the cryptic 54157 strain, only the 30 highest concentrations of IPTG gave significant induction and no hysteresis was observed.
Methods: Induction: Cells were incubated with shaking at 37°C in tubes containing 5 ml medium. In these conditions, the generation time was
60 min. β-galactosidase assays were performed with cells at the beginning of the saturation state. We checked that this protocol did not affect
the induction state of the operon: in the stationary regime of induction, enzyme activity was proportional to cell concentration, for both
exponentially growing cells and saturation-growth cells. We worked with cell populations corresponding to about 1 ODU (‘standard
concentration’), as determined by turbidimetry with a Shimadzu spectrophotometer at 600 nm. This corresponded to a density of 4 x 108
cells/ml. Relaxation to a lower-induced state, ‘de-induction’: We removed 1 µl of induced cells (cell suspension at 1 ODU) from their
medium, which contained 10–4 M IPTG (K12 strain) or 10–3 M IPTG (54157 strain) and placed it in tubes containing 5 ml of medium
containing various lower concentrations of IPTG concentrations. Solutions were incubated overnight (almost 12 generations) and residual
enzyme activities were then measured. Measurement of β-galactosidase activity: We used the classical in vivo kinetics of ONPG (Sigma)
hydrolysis method, modified as described by Beggs and Rogers (5), using toluene treatment to permeabilise cell membranes: o-nitrophenol
formation was assayed continuously at 420 nm. The initial slope of the hydrolysis curve, proportional to enzyme activity, was determined in
ODU/s: for the standard concentration, 1 ODU/s corresponded to 0.35 µmoles of ONPG hydrolysed per second in a cuvette 1 cm long. One
u.βgal corresponds (3) to 10–9 moles ONPG hydrolysed per min. Kinetics were followed for 100 sec for each sample. Under these conditions,
the minimum activity we were able to detect was 10–4 ODU/s = 2.1 u.βgal.
Bistability in lac operon regulation
Nevertheless, it has been shown (27) that E. coli
populations were very heterogeneous in terms of lac
induction around transition. The "concentration width" of
the transition may also correspond to the width of the
distribution of induction state in the population, assuming
that kinetic parameters are significantly different between
bacteria. This interpretation is supported by recent
observations (28) of the induction and de-induction curves
obtained for another gratuitous inducer, TMG. With TMG,
lac operon transcription was followed directly by
measuring fluorescence because the bacteria were
transformed with a plasmid construct containing the
reporter gene encoding green fluorescent protein (gfp)
under control of the lac promoter. The bacterial response to
TMG induction is hysteretic, but smooth transitions are
observed between minimal and maximal levels of
fluorescence (Fig. 8). However, microscopy of cell
populations showed that the distribution of fluorescence
levels was bimodal, with induced cells having more than
100 times the fluorescence levels of uninduced cells.
Hence, individual cells never displayed intermediate
fluorescence
levels,
but
population-averaged
measurements tended to obscure this.
Fig. 8
Hysteretic behaviour of a series of E. coli
populations. The fluorescence detected was that of the gfp
protein, produced from a gene under the control of the lac
promoter. Cells initially uninduced (lower figure) or fully
induced (upper figure) for lac expression were then grown in
media containing various amounts of the gratuitous inducer,
TMG. TMG concentration must exceed a threshold of 30 µM
to switch on expression in cells that were initially uninduced
(lower figure), and must decrease below a threshold of 3 µM to
turn off expression in initially induced cells (upper figure).
White arrows indicate the initial fluorescence level of the cell
populations in each panel. The grey region corresponds to the
"maintenance" concentration of the inducer ? the range of
concentrations for which the system displays hysteretic
behaviour (data adapted from ref. 28).
593
CONCLUSION
Regulation of the lac operon is the prototype
mechanism for the negative control of gene expression.
In the concluding chapter of the 1961 session of the Cold
Spring Harbor symposium on quantitative biology,
Monod and Jacob (23) gave a striking description of the
fields opened up by advances in the understanding of
gene and protein regulation. They highlighted the
strength of Delbrück’s ideas on multistationarity and
explored the possible dynamic significance of several
regulatory networks, corresponding to Novick and
Weiner’s work and to hypothetical networks including
what we now refer to as positive and negative feedback
circuits. However, this paper was subsequently
overlooked. Consequently, although the lac operon has
been thoroughly studied for more than 40 years, very
little effort has been devoted to studies of the dynamic
aspect of this non-linear system and substantial progress
in this area could still be made. Ozbudak et al. (28)
recently described a genetic construct in E. coli making it
possible to assess lacZ expression and the level of
catabolic repression simultaneously. Catabolic repression
is the repression of the lactose operon by external
glucose. Studies of this artificial system confirmed the
bistable nature of the transition controlled by the
repressor.
Other molecular control systems may also
quantitatively contribute to the adaptation of cells to the
availability of lactose. For instance, it has been shown (1)
that SsrA RNA can be used to tag the mRNA encoding
the lac repressor and that this process may play a role in
regulation of the lac operon. Additional mechanisms of
regulation may act at the translational or transcriptional
levels. Half a century after the discovery of the lac
operon and the publication of Delbrück’s ideas, there are
still a number of aspects of the dynamics of the regulation
of this operon that remain to be investigated.
Just as the molecular mechanisms of regulation of this
metabolism led to countless discoveries, giving rise to the
concept of regulatory networks, dynamic studies of this
operon could provide a stimulus for a large field of
studies concerned with the dynamics of these regulatory
networks. Thus, after being ignored for up to half a
century, the pioneering works of Delbrück (12, 1959),
Novick and Weiner (27, 1957), Cohn and Horibata (8-10,
1959), and of Sanglier, Babloyantz, Glansdorff,
Prigogine and Nicolis (1971-1976), and Thomas (19731981), may now open up a very fruitful new field of
research in the regulation of living systems.
Acknowledgments – We would like to thank René Thomas and
Marcelline Kaufman for helpful discussions and critical reading of the
manuscript.
594
M. Laurent et al.
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