MODELING IN PHYSIOLOGY
Frequency selectivity, multistability, and oscillations emerge from
models of genetic regulatory systems
Paul
Smolen,
Douglas A.
Baxter, and
John H.
Byrne
Department of Neurobiology and Anatomy, University of Texas
Medical School, Houston, Texas 77030
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ABSTRACT |
To examine the
capability of genetic regulatory systems for complex dynamic activity,
we developed simple kinetic models that incorporate known features of
these systems. These include autoregulation and stimulus-dependent
phosphorylation of transcription factors (TFs), dimerization of TFs,
crosstalk, and feedback. The simplest model manifested multiple stable
steady states, and brief perturbations could switch the model between
these states. Such transitions might explain, for example, how a brief
pulse of hormone or neurotransmitter could elicit a long-lasting
cellular response. In slightly more complex models, oscillatory regimes
were identified. The addition of competition between activating and
repressing TFs provided a plausible explanation for optimal stimulus
frequencies that give maximal transcription. Such optimal frequencies
are suggested by recent experiments comparing training paradigms for
long-term memory formation and examining changes in mRNA levels in
repetitively stimulated cultured cells. In general, the computational
approach illustrated here, combined with appropriate experiments,
provides a conceptual framework for investigating the function of
genetic regulatory systems.
genetic models; transcriptional regulation; optimal stimulus
frequencies; control theory
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INTRODUCTION |
REGULATION OF GENE expression by signals from outside
and within the cell plays important roles in many biological processes, including development (38), the cell cycle (33), hormone action (6),
and neural plasticity (8). As the basic principles of genetic
regulation have been characterized, it has become increasingly evident
that nonlinear interactions, positive and negative feedback within
signaling pathways, time delays, protein oligomerization, and crosstalk
between different pathways need to be considered to fully understand
genetic regulation. A conceptual problem arises, however, of how to
predict the functional properties of these complex, nonlinear
biochemical systems.
Genetic regulatory systems have often been modeled simply as networks
of Boolean logical elements (43, 44). However, the range of nonlinear
behaviors exhibited by such complex biochemical systems can be more
thoroughly understood from an explicitly mathematical, dynamic systems
approach (10, 13, 48). Such an approach allows the array of tools and
concepts developed for analysis of systems of ordinary differential
equations to be brought to bear. For example, bifurcation analysis (13)
is used to determine the steady-state solutions, oscillatory solutions,
and other attracting solutions of a system of ordinary differential
equations as a function of parameters and thereby determine parameter
values at which qualitative transitions in the dynamic behavior of
these systems occur. This is done numerically by means of specialized software such as the package AUTO (5) or in simple cases algebraically. Also, by tracing the solution structure of a system as a function of
parameter values, either through repetitive numerical integration or
more rigorously via bifurcation analysis, one can quantify the
sensitivity of these systems to parameter changes.
Consequences of applying the dynamic systems approach to modeling
of genetic regulatory systems might include identification of multiple
steady states of gene product concentrations. If brief perturbations
can induce transitions between these states, a system with this
structure could act as a switch. Transient exposure to a stimulus, like
a growth factor, might elicit a long-lasting response, such as a switch
from cellular quiescence to growth. In addition, the dynamic systems
approach can identify key physiological control parameters to which the
behavior of specific genetic regulatory systems is particularly
sensitive. Such parameters might provide targets for pharmacological
intervention. Oscillatory regimes and regimes in which transcriptional
behavior is particularly sensitive to initial conditions (i.e.,
apparently chaotic) can also be identified, and the stability of such
regimes can be examined. Mechanistic hypotheses can be tested by first
using simulations to predict behavioral characteristics of a particular
mechanism and then testing for these experimentally.
Modeling biochemical systems, with nonlinear elements and feedback
pathways, as dynamic systems has an extensive history. For example,
circadian rhythms (9) and glycolysis (10) have been modeled as
biochemical oscillators. However, aside from work of Keller (19)
modeling simple generic genetic systems, relatively little application
of dynamic systems techniques has been made to modeling genetic
regulatory systems specifically. Studies modeling a specific
genetically regulated process (e.g., Ref. 36) focus on reproduction of
experimental data pertaining to that process and not on the elucidation
of dynamic properties generic to genetic regulatory systems.
To elucidate dynamic properties of genetic regulation, we have modeled
genetic regulatory systems with different levels of complexity. First,
in a confirmation and extension of Keller's earlier research (19), we
consider a relatively simple model of transcription factors (TFs)
subject to positive and negative autoregulation of their own
transcription. Principles of operation that give rise to multistable
and oscillatory dynamic behavior are discussed. Second, this model is
extended to allow time-dependent phosphorylation of interacting
transcriptional activator and repressor proteins. Simulations with this
extended model identify possible mechanisms for generating maximal
transcription at an optimal stimulus frequency. A similar system with
interacting transcriptional activators and repressors is believed to
mediate early stages of long-term memory (LTM) formation, and the
formation of LTM can be greatest for an optimal frequency of stimulus
presentation. Finally, we point out that the principles identified in
these models are likely to apply to a variety of genetic regulatory systems.
Our models do not include stochastic thermal fluctuations in
molecule numbers. Such fluctuations could, however, be important when
small numbers of important molecular species, e.g., TFs bound to
a limited number of DNA sites, are present. For example, McAdams and
Arkin (26) have suggested that different phenotypes of prokaryotic systems could be selected by stochastic switching between alternative dynamic states. One can say intuitively that fluctuations could destabilize or switch among multiple steady states and would tend to
randomize the amplitude and period of oscillatory solutions. In
specific systems in which the numbers of individual molecular species
could be estimated, further simulations would be necessary to quantify
the importance of such effects.
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PRINCIPLES OF DYNAMIC GENE REGULATION ILLUSTRATED BY SPECIFIC MODELS |
Responsive elements affecting regulatory genes can provide
crosstalk between genetic regulatory systems and feedback within
systems.
We consider signal-transduction pathways in which stimuli lead to
second messenger generation and phosphorylation of TFs, which in turn
bind to DNA sequences known as responsive elements and thereby regulate
the transcription of specific genes (18). The regulatory activity of
TFs is often modulated by phosphorylation and by intermolecular
interactions. For example, TFs often bind to DNA as homodimers or as
heterodimers of different TF family members.
A well-known family of TFs is the
Ca2+/adenosine
3',5'-cyclic monophosphate (cAMP)-responsive element
binding protein/activating TF (CREB/ATF) family of homo- and
heterodimers, which bind to Ca2+/cAMP-responsive elements
(CREs). A second well-known group of TFs is the activating protein-1
(AP-1) family of heterodimers of Fos and Jun proteins, which bind to
phorbol ester-responsive elements. Responsive elements that bind TFs
have been found to affect the transcription of genes for diverse TFs
such as Jun, Fos, and CREB (18, 29, 39). As a result, some TFs, such as
Jun, autoregulate their own transcription (28). Moreover, responsive
elements can mediate crosstalk between pathways. For example, members
of the CREB/ATF family of TFs activate transcription of
fos (30), and at least one ATF-Fos
heterodimer binds to CREs, potentially activating CREB/ATF
transcription in turn (14). Thus responsive elements affecting TF gene
transcription can provide crosstalk and positive feedback. Responsive
elements have also been found that regulate genes for potent
transcriptional inhibitors. An example is the inducible
Ca2+/cAMP-responsive early
repressor (ICER) protein, whose transcription is increased on binding
of phosphorylated dimers of CREB to a nearby CRE (31). Such responsive
elements provide negative feedback loops. Given the above interactions,
there is the possibility for rich dynamic activity.
To explore this possibility, we constructed a model (Fig.
1A)
that captures the salient features of TF dimerization, binding, and
phosphorylation-dependent regulation of transcription. Simplifications were made to obtain a model that can be appreciated intuitively. A
single transcriptional activator, which we term TF-A, is considered as
part of a pathway mediating a cellular response to a stimulus. The TF
forms a homodimer that can bind to responsive elements (TF-REs). The
tf-a gene incorporates one of these
responsive elements, and when homodimers bind to this element TF-A
transcription is increased. Binding to the TF-REs is independent of
dimer phosphorylation. Only phosphorylated dimers, however, can
activate transcription. The fraction of dimers phosphorylated is
dependent on the activity of kinases and phosphatases whose activity
can be regulated by external signals. Thus this model incorporates both
signal-activated transcription and positive feedback on the rate of TF
synthesis.

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Fig. 1.
Models of genetic regulation with positive and negative autoregulatory
feedback loops. Genes are indicated by italic type, proteins by
uppercase lettering. A: the
transcription factor (TF) TF-A activates transcription when
phosphorylated (P) and bound as a dimer to specific responsive-element
DNA sequences (TF-REs). Degradation of TFs is also indicated. TF-A can
stimulate transcription of later genes. For simplicity, phosphorylation
and dimerization are not shown as separate kinetic steps.
B: schematic resulting from addition
of a 2nd TF, TF-R, which represses transcription by competing with TF-A
dimer for binding to TF-REs;
k1,d and
k2,d, rate
constants for degradation of TF-A and TF-R, respectively.
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It is further assumed that the transcription rate of
tf-a saturates hyperbolically with
TF-A dimer concentration, to a maximal rate
k1,f,
which is itself proportional to the degree of TF-A phosphorylation
(Eq.
1). This proportionality implies
that a singly phosphorylated TF-A dimer is one-half as effective at
activating transcription as is a doubly phosphorylated dimer. A basal
rate of synthesis of activator
(r1,bas) is present at
negligible dimer concentration. TF-A is degraded with first-order
kinetics, with a rate constant
k1,d. Binding
processes are considered comparatively rapid and close to equilibrium,
so the concentration of homodimer is proportional to the square of TF-A
monomer concentration
([TF-A]2). Here and
subsequently, translation of mRNA into protein is not explicitly
modeled. These simplifications give a model with a single ordinary
differential equation for the concentration of TF-A
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(1)
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Here K1,d is
the dissociation constant of TF-A dimer from TF-REs. Concentrations and
concentration-based parameters such as
K1,d are
dimensionless because effective concentrations of TFs and of DNA
elements in the nucleus are generally not known. Numerical integration
of this and other models was carried out with Gear's adaptive step
size method (7).
Transient phosphorylation of TFs can regulate the dynamics of
genetic systems by switching among multiple stable states.
One intriguing aspect of the simple gene regulatory system described by
Eq. 1
is the emergence of bistability, i.e., two steady-state solutions for
[TF-A] (Fig.
2A,
left). At these solutions,
d[TF-A]/dt = 0. These solutions are termed stable because if [TF-A] is
slightly perturbed from these solutions, it will relax back. For a
relatively wide range of parameters, there is one such solution with
[TF-A] low and its synthesis rate close to
r1,bas and another with
[TF-A] high and its synthesis rate close to
k1,f. Moreover,
appropriate, larger perturbations can switch the model between these
states. It is assumed that only properly phosphorylated dimers can
activate transcription, as is the case for the TFs AP-1 and CREB (12, 42). Signal-dependent activation of protein kinases or phosphatases induces changes in the fraction of dimers phosphorylated. These changes
are modeled as transient increases or decreases in
k1,f. These
perturbations induce upward or downward transitions, respectively, between the steady states. Such transitions could correspond
physiologically to brief stimuli, such as exposure to a hormone,
leading to long-lasting increases or decreases in the levels of
particular proteins.

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Fig. 2.
Complex dynamics of a genetic regulatory system,
Eq.
1. A,
left: perturbations switch system
between steady states. TF-A concentration ([TF-A]) is
initially in a low steady state. Parameter values are maximal rate of
TF-A synthesis
(k1,f) = 10 min 1, basal rate of TF-A
synthesis (r1,bas) = 0.1 min 1,
k1,d = 1 min 1, and dissociation
constant of TF-A dimer from TF-REs
(K1,d) = 10. Brief changes in
k1,f (50 min 1 for
t between 30 and 31 min, 1 min 1 for
t between 60 and 63 min) give small
excursions of [TF-A] from low steady state (dotted circles
mark imperceptible transients in [TF-A]). At
t = 90 min,
k1,f is increased
to 100 min 1 for 1 min,
switching [TF-A] to an upper steady state. At
t = 120 min,
k1,f is increased
to 50 min 1 for 1 min. At
t = 150 min,
k1,f is decreased
to 1 min 1 for 3 min.
[TF-A] then returns to lower steady state.
A,
right:
k1,f is kept at 5 min 1 except for brief
perturbations (here taken as identical to those at
left). Only transient excursions of
[TF-A] occur. Note change in scale for [TF-A]
between left and
right.
B: bifurcation analysis of
multistability in this model as a function of
k1,f; other
parameters are as in A,
left. For 6.1 min 1 < k1,f < 25.1 min 1, there exist 2 stable
steady-state solutions of [TF-A] (solid portions of
d[TF-A]/dt curve) with an
unstable solution between them (dashed curve). Outside this region
there is only a single steady-state solution.
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The transient responses of this system are also state dependent (Fig.
2A,
left). With the system in the upper
steady state, a perturbation of
k1,f which gave
only a minute excursion of [TF-A] above the lower steady
state (dotted circle at left) now
gives a very large excursion above the upper steady state. This
difference in response magnitude following a state transition is a
model for "priming" of a system to respond more vigorously to
subsequent stimuli.
The range of parameters permitting bistability can be more precisely
delineated by bifurcation analysis. Figure
2B, derived algebraically,
demonstrates this technique for varying
k1,f in Eq.
1. For each value of
k1,f between 6.1 min
1 and 25.1 min
1, there are three values of
[TF-A] that are steady states of
Eq. 1. The upper and lower states are
those illustrated in Fig. 2A. For the
middle steady state, a small perturbation of [TF-A] will grow until [TF-A] moves to either of the other two steady
states. Thus the middle state is unstable and would not be expected to be observed experimentally. For
k1,f < 6.1 min
1 or > 25.1 min
1,
Eq. 1
has only one stable steady-state solution for [TF-A].
Figure 2A,
right, demonstrates that, for
parameters outside the permissible range supporting multiple stable
steady states, brief perturbations in
k1,f only yield
transient excursions in [TF-A], which return to the single
steady state.
Combined positive and negative feedback, or time delays, can
generate oscillations and complex transients in genetic systems.
The model of Fig. 1A is similar to a
model analyzed by Keller (case B of Ref. 19), which also exhibits
bistability. Indeed, in that work, five other models based on
autoactivation or autorepression of transcription by binding of gene
product to DNA were also analyzed. All these models exhibit bistability
over significant parameter ranges. Thus biochemical architectures
capable of supporting bistability may be common in genetic regulatory
systems. However, models similar to that of Fig.
1A have only one type of feedback:
either positive or negative. Without both types of feedback, such a
model cannot be expected to support oscillations. For example, positive
feedback can act to drive [TF-A] to high levels, but then
there is no process to bring [TF-A] back down. To
investigate the effect of negative feedback, we introduced a protein,
TF-R, that represses transcription by binding to TF-REs. Its rate of
synthesis is increased by binding of the TF-A dimer to a TF-RE (Fig.
1B). TF-R competitively inhibits binding of TF-A dimers to TF-REs; thus it inhibits the transcription of
the genes tf-a and
tf-r. The equations for this model
are
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(2)
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(3)
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Parameters in Eq.
2 have the same meaning as in the
model of Eq.
1.
KR,d is the
dissociation constant of TF-R monomers from TF-REs. Parameters in
Eq. 3
are analogous to those in Eq.
2
(k2,f, k2,d, and
K2,d are maximal
synthesis rate, degradation rate constant, and dissociation constant
from TF-REs of TF-A dimers). Robust oscillations are readily generated
by this model (Fig.
3A).
Both TFs oscillate approximately in phase, with a period on the order of 1 h. Moreover, physiologically plausible variations in parameters can give disproportionate changes in oscillation amplitude and mean,
thus providing for a signal-dependent modulation of the oscillations.
In Fig. 3A, at
t = 160 min the maximal rate of TF-A synthesis k1,f is
reduced from 10.5 min
1 to
10 min
1. This modest
parameter change gives a large change in the pattern of oscillation.

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Fig. 3.
A: addition of repressor protein
dynamics (Fig. 1B) to simple model
of Fig. 1A allows oscillations to
occur. Parameter values are
k1,f = 10.5 min 1,
r1,bas = 0.4 min 1,
k1,d = 1 min 1,
K1,d = 10, maximal rate of TF-R synthesis
(k2,f) = 0.3 min 1,
k2,d = 0.2 min 1,
K2,d = 10, and KR,d = 0.2. At t = 160 min,
k1,f is decreased
to 10 min 1. This change
induces a large damping of oscillations.
B: bifurcation analysis showing
oscillatory solution in [TF-A] as a function of
k1,f; other
parameters are as in A. For
k1,f < 9.9 min 1 or
k1,f > 10.8 min 1 there is only the
stable steady state shown; for
k1,f between
these values there is a stable oscillatory state. Solid curves
illustrate maxima and minima of oscillations, which surround an
unstable steady state (dashed curve).
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Bifurcation analysis can precisely determine the parameter range over
which oscillations exist. We used the numerical software package AUTO
(5) to trace the amplitude of oscillations as a function of parameters.
Figure 3B gives an example. When
k1,f is varied,
with all other parameter values as in Fig.
3A, oscillations exist only in the
interval 9.9 min
1 < k1,f < 10.8 min
1.
Biological oscillations with periods on the order of hours might be
explained by such genetic models. Examples could be oscillatory secretion of hormones (3), such as human growth hormone (35) or
leutinizing hormone-releasing hormone (21). In secretory cells, the
genetic regulatory system utilizing CREB and related TFs has been
postulated to yield oscillations of transcription rate (47). A negative
feedback loop on gene expression, dependent on TF phosphorylation, has
also been postulated within a molecular model for circadian rhythm
generation (9).
In various models of biological phenomena, e.g., population dynamics
(32), time delays serve as another way to generate oscillations or
complex transients. In genetic regulatory systems, there are ubiquitous
time delays associated with, for example, the transport of mRNA and
proteins between the cytoplasm and the nucleus. To briefly explore the
effect of introducing time delays into a simple model, a delay of
several minutes was introduced in the model of Fig.
1A. The delay was between changes in
TF-A concentration and the resultant changes in the rate of formation of new TF-A due to tf-a transcription.
This converted Eq.
1 into a delay differential equation,
which was integrated by the fourth-order Runge-Kutta method. As a
result of incorporating the delay, a brief increase in
k1,f leads to 10 or more large oscillations in the TF concentrations, with a period
comparable to the delay, before the system stabilizes in the upper
state (not shown).
Protein oligomerization can underlie complexity, such as
multistability and oscillations, in the dynamics of genetic regulatory
systems.
Dimerization of TFs is essential for bistability and oscillations in
these models. Figure 4 illustrates a graph
of d[TF-A]/dt vs.
[TF-A] for the simple model of
Eq.
1. The synthesis rate of TF-A is a
nonlinear function of [TF-A]. This allows the graph to have
a complex shape, with three steady states of [TF-A] with d[TF-A]/dt = 0. The
nonlinearity is due to the square of [TF-A] in
Eq.
1, which represents dimer
concentration. The middle state with
d[TF-A]/dt = 0 is unstable
to small perturbations in [TF-A], whereas the lower and
upper states are stable and represent the two steady states illustrated
in Fig. 2A,
left.

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Fig. 4.
TF dimerization is important for bistability. Solid curve,
d[TF-A]/dt vs.
[TF-A] for model of Eq.
1, with parameters as in Fig.
2A. The 3 crossings of 0 on ordinate
correspond to lower, middle, and upper steady states (SS1-SS3,
respectively). Dashed curve,
d[TF-A]/dt vs.
[TF-A] for model obtained by replacing 2nd powers of
[TF-A] in Eq.
1 with 1st powers; parameter values
are as in Fig. 2A. In this model, SS1
and SS3 have been eliminated; single steady state remaining is SS2.
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Qualitatively different results are obtained with a variant of our
model in which just monomers of TF-A bind to the TF-REs. When
Eq. 1
is changed so that the activation of
tf-a transcription is proportional
simply to [TF-A], the graph of
d[TF-A]/dt vs. [TF-A] is qualitatively altered (Fig. 4) such that only one
stable state at which
d[TF-A]/dt = 0 remains.
We also find that in the model with repressor TF-R,
Eqs.
2 and 3, dimerization of TF-A is essential
for oscillations in TF concentration. If transcription of TFs depends
only on the concentration of monomeric TF-A, the system always settles
to a steady state. Thus it can be inferred that protein oligomerization
provides an important mechanism for achieving complex dynamics in
transcription.
Stimulus frequency can be decoded in complex ways into variations in
response strength.
The above models do not consider further dynamic behaviors of genetic
regulatory systems that could result from concurrent modification of
transcriptional activator and repressor efficacy by stimulus-dependent
phosphorylation. We therefore added an additional level of complexity
to the model of Eqs.
2 and 3. The fractions of repressor and
activator proteins that are phosphorylated are made dynamic variables.
Only when phosphorylated can these proteins affect transcription.
Different kinetics of phosphorylation and dephosphorylation for the two
TFs can allow for different sensitivities to stimuli, such that stimuli
sufficient to saturate the phosphorylation of one TF may not do so for
the other.
Such a model might be able to simulate experiments that have
demonstrated optimal stimulus frequencies for activation, or repression, of transcription. For example, transcription of the cell
adhesion molecule L1 in cultured neurons is strongly repressed by
imposed, continual 0.1-Hz electrical stimulation but not repressed significantly by 0.3-Hz stimulation (17). Also,
c-fos transcription in cultured
neurons is enhanced almost 200% by bursts of 6 electrical stimuli at
10 Hz with an interburst interval of 1 min but not significantly
enhanced by bursts of 12 stimuli with an interburst interval of 2 min
(41). Continuous 0.1-Hz stimuli gave a 70% enhancement. A model in
which an intermediate intensity or frequency of stimulation
phosphorylated and activated one TF only, whereas a higher intensity of
stimulation activated also a second TF that counteracted the effect of
the first, might explain these phenomena. Moreover, a similar model
might apply to initial steps in LTM formation. It has recently been
proposed that changes in the relative phosphorylation of
transcriptional activators and repressors may be important for
induction of transcription required for the formation of LTM (49). The
relationship between stimulus frequency and amount of LTM formation,
and by inference amount of transcription, appears sometimes to be
nonmonotonic. In Drosophila, spaced
olfactory stimulus presentations, with a relatively long interstimulus
interval (ISI), yield much more LTM than do massed presentations (a
short ISI) even given the same total training time (thus more massed presentations) (46). In this system, there is some optimal stimulus frequency for formation of LTM, and by inference for activation of
transcription. Optimal stimulus frequencies appear also to exist
for types of task learning by humans (20, 24).
We developed a model with two antagonistic TFs for the purpose of
testing qualitatively to what extent a single model could provide a
unified explanation of these results concerning optimal stimulus
frequencies and groupings. In the model, the dimeric character of TF-A
and TF-R allows different efficacies of activation, or repression, by
singly vs. doubly phosphorylated dimers. As before, monomer-dimer
equilibria and the existence of heterodimers are neglected. Only
homodimers of TF-A and TF-R, which compete for binding to responsive
elements (TF-REs), are considered. Phosphorylation of dimers of TF-R is
considered necessary for binding to TF-REs. Phosphorylation does not
affect binding of TF-A to TF-REs, in accordance with data for the
specific transcriptional activator CREB implicated in the formation of
LTM (37). The amount of gene transcription during a given time interval
is assumed to be proportional to the concentration of phosphorylated,
TF-RE-bound TF-A integrated over that interval. The overall flow of our
model from stimulus to the transcription of genes that mediate a
cellular response is diagrammed in Fig.
5A, and
the binding and phosphorylation processes are schematized in more
detail in Fig. 5B. The corresponding equations as given in the APPENDIX are
more complex than Eqs.
2 and 3 because phosphorylation of TF-R
affects binding to DNA so that separate differential equations are
needed for each TF-R species.

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Fig. 5.
Schematic of TF-A and TF-R regulatory model.
A: overall flow of model from stimulus
to gene transcription. A stimulus induces phosphorylation of both TF-A
and TF-R, which compete for binding to TF-RE. In some simulations
(dashed box), product of gene regulated by TF-A and TF-R itself
represses transcription of a further gene product (e.g., L1).
B: binding and phosphorylation. Dimers
of TF-A and of TF-R are phosphorylated in response to a stimulus. TF-A
binds to TF-REs irrespective of phosphorylation but can only activate
transcription when phosphorylated. TF-R binds competitively to TF-REs
but only when phosphorylated.
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TF phosphorylation and dephosphorylation were simulated with
Michaelis-Menten kinetics. For example, TF-A phosphorylation has a
time-dependent maximal velocity
kA,f (t) and a Michaelis constant KA,ph; TF-R phosphorylation has a
maximal velocity kR,f (t).
Relatively low values were chosen for the Michaelis constants of
phosphorylation. This choice introduces zero-order ultrasensitivity, defined generally as an amplification of the response of a reversible covalent modification system to perturbations in enzymatic activity when the opposing enzymes operate in a zero-order kinetic regime (11).
Specifically, a modest increase in the maximal velocity of
phosphorylation of either TF, from somewhat less than to somewhat greater than the maximal velocity of dephosphorylation of that for TF,
results in a large increase in the fraction of the TF phosphorylated.
Further details of equations and parameters are given in the
APPENDIX. In general, the maximal
velocities of TF phosphorylation would be dynamic variables whose
values depend on the activation of kinases, located in the nucleus, by
stimulus applications. These stimuli are transduced in a possibly
complex, time-dependent manner from the cell membrane (where stimuli
are sensed) to these kinases. However, for simulation of ISIs on the order of seconds, as are utilized in Ref. 17, we assume that temporal
averaging occurs during this transduction so that different stimulus frequencies correspond to different constant values of the
maximal velocities of TF phosphorylation.
As illustrated in Fig.
6A,
when either kA,f (t) or
kR,f (t) is varied alone, the
transcription rate varies monotonically. However, let us now specialize
to the case in which the first-order dephosphorylation rate constant
for TF-A is less than that for TF-R and increase together both
phosphorylation maximal velocities (keeping them equal). One then sees
a sharp rise in transcription rate to a peak as the maximal velocity
for TF-A phosphorylation passes that for TF-A dephosphorylation and
most of the TF-A becomes phosphorylated, followed by a sharp decline in
transcription rate as the maximal velocity of TF-R phosphorylation
exceeds that for dephosphorylation and most of the TF-R becomes
phosphorylated. Figure 6B illustrates
that the sharpness of this tuning curve depends strongly on the
Michaelis constants for phosphorylation and dephosphorylation being
small enough to allow rapid shifts in TF phosphorylation states. Larger
Michaelis constants give a flatter curve. Also, we investigated
whether TF dimerization was important for the qualitative dynamics of
this model as had been found for the models of Fig. 1. Figure
6B illustrates that, indeed, if
TF-R is assumed to be monomeric, the sharpness of the tuning curve is
greatly reduced.

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Fig. 6.
Dependence of transcription rate (in units of
min 1) in model of Fig. 5
on rate constants for TF phosphorylation. The 1st set of parameter
values in APPENDIX is used, except as
noted. A: increasing only forward
phosphorylation rate constant for TF-A
(kA,f) gives a
monotonically increasing transcription rate [ ; forward rate
constant for TF-R phosphorylation
(kR,f) fixed to
equal backward rate constant for TF-R phosphorylation
(kR,b)],
and increasing only
kR,f for TF-R
gives a monotonically decreasing transcription rate ( ;
kA,f fixed at 0.1 min 1), but increasing
both rate constants while keeping their ratio at 1 yields an optimum
( ). B: peak of transcription rate
( , same as corresponding curve in
A) is greatly reduced by increasing
Michaelis constants for TF-A and TF-R phosphorylation and
dephosphorylation to 10 ( ). Also, sharpness is greatly reduced by
assuming monomeric TF-R ( ), with all other parameters unchanged,
i.e., phosphorylated TF-R monomers bind to DNA with on and off rate
constants k2,f
and k2,b, and
Rtot is a TF-R monomer
concentration. C: with transcription
of cell adhesion molecule L1 added, L1
transcription and [L1] as a function of stimulus frequency
have pronounced minima.
kR,f and
kA,f exhibit
small oscillations about mean values. These values, in units of
min 1, are both assumed to
be equal to value of stimulus frequency in Hz.
|
|
We have qualitatively simulated the data of Itoh et al. (17) that
demonstrate suppression of transcription of the cell adhesion molecule
L1 by continuous 0.1-Hz but not 0.3-Hz stimuli. As indicated in Fig.
5A, an extra kinetic step is
necessary, with TF-A and TF-R regulating the transcription of a gene
whose protein product represses the transcription of the
L1 gene. This converts the maximum of
transcription of the gene regulated by TF-A and TF-R into a minimum of
L1 transcription. We assume that, in
response to stimuli applied at the membrane, maximal velocities of TF
phosphorylation vary on a time scale considerably longer than the
intervals between stimuli in the protocols of Itoh et al. (17).
Therefore, after an initial transient, these velocities exhibit a
steady-state behavior of small oscillations about mean values. Figure
6C illustrates that, with these mean values taken as
directly proportional to stimulus frequency, the model predicts that
0.1-Hz stimulation produces an ~90% reduction in
L1 transcription from the basal rate.
In contrast, 0.3-Hz stimulation gives only a 7% reduction. Stimulation
at 0.1 Hz suffices to phosphorylate TF-A only, which activates the
transcription of the repressor of L1
transcription. Stimulation at 0.3 Hz phosphorylates TF-R as well, so
the repressor of L1 transcription has
its own transcription repressed. Small values for the Michaelis
constants of TF-R phosphorylation and dephosphorylation are necessary.
Only then can the model give a virtually complete cancellation of
strong transcriptional repression in response to a threefold increase
in stimulus frequency (from 0.1 to 0.3 Hz).
As previously mentioned, we had also hoped to simulate experiments
concerning c-fos transcription (41).
Given a fixed average stimulus frequency, bursts of 6 stimuli at 10 Hz
repeated every minute were reported to yield much more transcription
than evenly spaced 0.1-Hz stimuli or bursts of 12 stimuli at 10 Hz
repeated every 2 min. However, our current model cannot simulate these results. If it is assumed, as above, that velocities of TF
phosphorylation are approximately proportional to stimulus frequency,
the stimulus paradigms would be expected to yield, on average, the same
velocities of phosphorylation of the TFs, and the same transcription
rates, because the number of stimuli averaged over time is the same in all paradigms. A more detailed model of stimulus coupling to nuclear events, considering nonlinear kinetics of particular second messenger systems, may be required.
To explain why massed stimulus presentations are less effective than
spaced presentations in producing LTM in
Drosophila, Yin et al. (49) proposed
the same generic mechanism that is considered here. An intermediate
intensity or frequency of stimulation phosphorylates and activates a TF
that activates transcription of genes essential for LTM formation,
while a higher frequency of stimulation activates also a second TF that
counteracts the effect of the first. However, rather than assuming
fixed average values for phosphorylation velocities, Yin et al. (49)
assumed that the net dephosphorylation rate for the repressor TF-R is faster than that of the activator TF-A during ISIs. Then, during spaced
stimuli, the net difference (phosphorylated activator
phosphorylated repressor) becomes large during the long ISIs, but
during massed stimuli TF-A phosphorylation is always approximately canceled out by TF-R phosphorylation. The kinetic scheme of Fig. 5 is
again used to test this hypothesis. Because the ISIs are now on the
order of minutes rather than seconds, we assume that each stimulus
abruptly resets the phosphorylation rate constants to maximal values
that decay exponentially.
As Fig. 7 demonstrates, our model predicts
an optimal stimulus frequency for transcription, and by inference for
LTM formation, when maximal velocities for TF dephosphorylation are
chosen in accordance with the hypothesis of Yin et al. (49). We also
found (not shown) that qualitatively similar results are obtained if both TFs are dephosphorylated at identical rates and it is assumed instead that the phosphorylation rate of TF-R is slower than that of
TF-A during exposure to a stimulus. Then TF-R is again only able to
become highly phosphorylated during massed stimuli. In addition,
alternative kinetic schemes utilizing only one TF were also found to
give an optimal stimulus frequency for transcription. One such model
variant postulates both activating and inhibiting phosphorylation sites
on TF-A, with the inhibiting site only becoming significantly
phosphorylated by massed stimuli. Another model variant relies on
competing kinase and phosphatase activities. In principle, these model
variants could also explain the aspects of
L1 transcriptional regulation
simulated above (Fig. 6C). It may be
inferred that the existence of two competing processes, such as
activator and repressor phosphorylation, that have different sensitivity to stimuli and opposing effects on transcription of a
specific gene could provide a general mechanism for tuning the response
of a genetic system to an optimum stimulus frequency.

View larger version (16K):
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|
Fig. 7.
Nonmonotonic dependence of transcription rate on stimulus frequency in
model of Fig. 5 with kinetic parameters consistent with hypothesis of
Yin et al. (49) for explaining greater efficacy of spaced stimuli in
formation of LTM (APPENDIX, 2nd set of
parameter values). A: time course of
transcription rate (in units of
min 1) during spaced
stimuli. B: massed stimuli are used,
and in all other respects simulation and graph are as in
A, including time scale. Comparing
A and
B demonstrates that, over 100 min, 100 massed stimuli [interstimulus interval (ISI) = 1 min]
produce considerably less transcription, and by inference less LTM
formation, than 8 spaced stimuli (ISI = 15 min).
C: dependence of transcription rate on
ISI for 2 cases. Top curve, dephosphorylation rate constant
for TF-A (kA,b;
1.0 min 1) < kR,b (7.0 min 1), as in hypothesis
of Yin et al. (49). Bottom curve, dephosphorylation rate
constants are equal (1.0 min 1).
|
|
 |
DISCUSSION |
Biochemical nonlinearities such as dimerization, feedback loops, and
time delays are common in genetic regulatory systems (10). Our results
indicate that incorporating these features into models of relatively
simple genetic regulatory systems can give rise to complex dynamic
activity and nonmonotonic dependence of response strength on stimulus.
Thus the dynamic principles illustrated are likely to be important in
phenomena in which regulation of transcription has an essential role,
such as development and the formation of LTM.
It is of value to compare the method of modeling genetic regulatory
systems as sets of ordinary differential equations, used herein, with
other approaches. Genetic regulatory systems have also been modeled as
networks of Boolean logical elements. In this approach, genes are
considered as either on or off and biochemical connectivities are
reduced to logical update rules for determining the set of genes that
will be on at a given time step as a function of the genes that were on
at the preceding time step. Relatively large time steps are used in
numerical simulations. By consideration of the mathematical properties
of such networks, significant insights concerning the expected dynamics
of genetic systems have been obtained, e.g., by Thomas and colleagues
(44, 45). These authors point out, for example, the importance of
negative feedback loops for maintaining homeostasis in levels of gene
products and of positive feedback loops for allowing multiple stable
states of these levels. More concretely, Somogyi and Sniegoski (43)
have developed a computational method for efficiently modeling large genetic networks in a Boolean fashion, which is designed to determine connectivity relations from simultaneously measured experimental expression time courses of sets of genes.
However, although the dynamic systems approach is more computationally
intensive than the Boolean approach, we prefer the former because it is
a more physically correct representation. Not only are there
quantitative differences, but, more importantly, it has been found that
simple biochemical models, when expressed as a network of Boolean
logical elements instead of as a system of ordinary differential
equations, can exhibit qualitatively different, and spurious,
attractors, i.e., different stable oscillatory solutions (1). Although
it has not been specifically demonstrated that more complex biochemical
and/or genetic Boolean network models also suffer
from this flaw, it seems plausible that a significant number would.
However, it should be noted that in investigating complex systems that
would require more differential equations than could be computationally
simulated in a reasonable amount of time, applying the Boolean network
approach while keeping in mind the above caveat may be the best
practical alternative.
Some of the phenomena illustrated in this work can also be dealt with
by modeling within the framework of electrical circuit analysis, as
McAdams and Shapiro (27) have done with a recent model of the
phage-Escherichia coli genetic
regulatory system. These authors constructed a model of this system in
which many nonlinear biochemical processes were represented as Boolean
logical switches (the product of a reaction such as transcription of a given gene is either present or absent, depending on the value of an
effector variable). However, other biochemical reactions were still
modeled as continuous input-output relations and numerically integrated
as ordinary differential equations. Time delays were also incorporated
in some kinetic steps between effector concentration changes and rate
changes. This method is therefore a hybrid of the continuous approach
based on ordinary differential equations and the approach based on
modeling genes as Boolean logical elements.
One specific application of the techniques discussed in this work could
be to model the genetic regulatory system responsible for initial steps
in LTM formation in greater detail. Signals that raise levels of cAMP,
such as pulses of neurotransmitter, induce phosphorylation of the TF
CREB (39). CREB can then induce the transcription of immediate early
genes crucial for neuronal plasticity and formation of LTM (2, 34, 50).
Proteins related to CREB, such as CREB2, are transcriptional repressors
that bind to the same CRE sequence as CREB (39). They are generally
phosphorylated by the same signals that phosphorylate CREB. Although
the functional relevance of repressor phosphorylation has not yet been
established, the basic architecture of the model of Fig. 5 appears to
be present.
Recent data, obtained in Aplysia,
actually suggest that phosphorylation of CREB2 by mitogen-activated
protein kinase reduces its repressing activity (25). This would
contradict the mechanism for generation of an optimal stimulus
frequency hypothesized by Yin et al. (49) and simulated in Fig. 7,
which relies on phosphorylation enhancing repression. An optimal
stimulus frequency for transcription in this system might still,
however, be explained in terms of two competing processes that have
different sensitivity to stimuli and opposing effects on transcription.
For example, CREB has both activating and inhibiting phosphorylation
sites (Ser-133 and Ser-142, respectively), and, if the inhibiting site
only became significantly phosphorylated in response to massed stimuli,
an optimum could result.
Additional biochemical elements of the CREB genetic system might allow
qualitatively new dynamics. For example, positive feedback exists via
binding of CREB to CREs affecting its own transcription. Negative
feedback exists in the form of a repressor protein, ICER, whose
transcription is induced by CREB. These feedback loops could create
multistability or oscillatory behavior. There is some empirical indication for complex dynamics mediated by these elements.
Oscillations in CREB mRNA have been reported in secretory cells, and
the above feedback loops have been proposed as essential components of
the oscillatory mechanism (47).
More generally, other biochemical architectures that have been observed
in genetic regulatory systems but not captured in any of our models so
far include convergence of different signaling pathways through
distinct TFs onto a particular gene [e.g., TFs of the ETS family
and pituitary TF Pit-1 onto the prolactin gene (15)],
heterodimerization of TFs in two separate pathways with a common third
TF [e.g., thyroid hormone receptor and peroxisome proliferator-activated receptor heterodimerize with the retinoic acid
receptor (16)], and conditional regulation by a single TF
[e.g., enhancement of basal transcription of the dopamine
-hydroxylase gene along with suppression of cAMP-induced
transcription by the TF YY1 (40)]. Such architectures may be
expected to provide alternative mechanisms for generating dynamic
phenomena such as multistability and oscillations.
It is unlikely that all transcriptionally regulated genes will exhibit
the behaviors illustrated by our models. However, the diversity of TFs
and their interactions suggest that behaviors such as these will be
identified. Indeed, our models are simplifications of the actual
kinetic schemes characterizing genetic systems. MacLeod (23) has
recently proposed that epigenetic, heritable changes in gene expression
following exposure to chemicals might play a role in carcinogenesis.
Such changes would correspond dynamically to perturbations of genetic
regulatory systems from one steady state to another.
An outstanding major issue for future investigation will be to
determine whether the parameters of specific genetic systems in vivo
are permissive for specific types of dynamic behavior. Experiments to
help determine this might include monitoring transcription of
transfected reporter gene constructs, with defined promoters subject to
regulation by TFs, in cultured cells during specific patterns of
hormone or neurotransmitter application. A prolactin promoter-luciferase gene construct has been used to provide real-time quantification of promoter activity in cultured secretory cells (4).
Another relevant technique is polymerase chain reaction amplification
and quantitation of specific mRNAs from tissue samples (43); however,
this technique does not resolve dynamics at the single cell level. We
believe that, as the dynamic behaviors of gene networks are explored
empirically, the present work can provide a conceptual framework for
the analysis and interpretation of such experiments.
 |
APPENDIX |
Details of equations and parameters for simulations with the model of
Fig. 5 follow. We make some assumptions consistent with experimental
results concerning the dynamics of a specific system with competing TFs
that is thought to mediate the initial steps in LTM formation. In
particular, we have been guided by analyses of competitive interactions
of the transcriptional activator CREB and related repressors for their
target DNA sequences, i.e., CRE sites. It is assumed that
1) total amounts of TF-A and TF-R
remain constant (2), 2)
phosphorylation does not affect binding of TF-A to CRE sites (37), and
3) singly phosphorylated TF-A dimers have one-half the activity of doubly phosphorylated ones, with unphosphorylated dimers inactive (22).
We denote the concentration of free CRE sites by
Gfree. For brevity, in equations
the activator TF-A is denoted by A and the repressor TF-R by R. [AP] is used to denote the concentration of phosphorylated
TF-A sites, and [AA] is used to denote the concentration of
free TF-A dimers. [AAB] denotes the concentration of TF-A
dimer bound to DNA. Atot denotes
the total concentration of TF-A dimers. The total concentration of
repressor dimers is Rtot.
[RR] is the concentration of free, unphosphorylated TF-R
dimers. RRP and RRPP denote single or double phosphorylation; RRPPB
denotes bound, doubly phosphorylated R dimers. Because phosphorylation
of TF-A is independent of binding to DNA, the rate of change of
phosphorylated TF-A can be described by a single differential equation
for the concentration of phosphorylated TF-A sites
|
(A1)
|
where KA,deph
is the Michaelis constant for TF-A dephosphorylation and
kA,b is the
backward rate constant for TF-A phosphorylation.
A single differential equation describes the rates of change of free
and bound TF-A dimers because of conservation of total dimers. The
association (forward) and dissociation (backward) rate constants are
k1,f and
k1,b,
respectively
|
(A2)
|
where
[AAB] = Atot
[AA].
Separate equations are needed for the rates of change of each species
of TF-R because binding and phosphorylation are not independent. Total
phosphorylation and dephosphorylation rates are first expressed in
terms of site concentrations, and then, in the differential equations,
fractions of these total rates appropriate to each molecular species
are used
|
(A3)
|
|
(A4)
|
|
(A5)
|
|
(A6)
|
|
(A7)
|
|
(A8)
|
where KR,ph
and KR,deph are
Michaelis constants for TF-R phosphorylation and dephosphorylation,
respectively, and
kR,b and k2,b are the
backward rate constants for TF-R phosphorylation and TF-R binding to
DNA, respectively.
The rate of transcription of the target gene for whose promoter region
TF-A and TF-R compete (rRep) is
taken as proportional to the concentration of bound TF-A dimers
multiplied by the fraction of phosphorylated TF-A sites, with a rate
constant
kRep
|
(A9)
|
There is a conservation condition on the total number of DNA binding
sequences
|
(A10)
|
where
Gtot is the total number of CRE
sites.
For modeling the data of Itoh et al. (17) indicating an optimal
frequency for repression of transcription of the cell adhesion molecule
L1, an additional kinetic step is needed. The target gene for TF
regulation is assumed to express a protein Rep that represses
transcription of the L1 gene.
L1 transcription is assumed to proceed
at a basal rate rL1 in the absence of Rep. L1 transcription only occurs if Rep is
not bound to a promoter for the L1
gene. Rep dimers bind to this promoter with dissociation constant
KRep
|
(A11)
|
|
(A12)
|
In simulations of the formation of LTM, we posited that each stimulus
immediately set kA,f (t) and
kR,f (t) to maximal values kA,max and kR,max,
respectively. After a stimulus,
kA,f (t) and kR,f (t) decayed to zero with
time constants
2 and
1, respectively.
All simulations used parameter values from one of the two following
sets. Concentrations are left dimensionless due to lack of sufficient
experimental data. Parameters marked "varies" have values given
in the text or in Fig. 6 or 7.
For the simulations of Fig. 6
For the simulations of Fig. 6C,
kA,f and
kR,f were assumed
to execute small oscillations about the mean values in the figure legend. In the absence of data to construct a kinetic model for these
oscillations, we merely assumed sinusoidal oscillations with a
frequency equal to the stimulus frequency and an amplitude of 5% of
the mean value.
For the simulations of Fig. 7
 |
ACKNOWLEDGEMENTS |
We thank Pramod Dash, Ron Dror, and Shogo Endo for comments on an
earlier draft of the paper and B. Ermentrout for use of his XPP program
for simulations.
 |
FOOTNOTES |
This work was supported by Office of Naval Research Grant
N0014-95-1-0579, by National Institutes of Health Grants K05-MH-00649, T32-NS-07373, and R01-RR-11626, and by Texas Higher Education Coordination Board Grant 011618-048.
Address for reprint requests: J. H. Byrne, Dept. of Neurobiology and
Anatomy, University of Texas Medical School, 6431 Fannin St., Suite
7.046, Houston, TX 77030.
Received 30 April 1997; accepted in final form 27 October 1997.
 |
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AJP Cell Physiol 274(2):C531-C542
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