Effects of macromolecular transport and stochastic
fluctuations on dynamics of genetic regulatory systems
Paul
Smolen,
Douglas A.
Baxter, and
John H.
Byrne
Department of Neurobiology and Anatomy, W. M. Keck Center for the
Neurobiology of Learning and Memory, The University of Texas-Houston
Medical School, Houston, Texas 77225
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ABSTRACT |
To predict the dynamics of genetic regulation, it may be
necessary to consider macromolecular transport and stochastic
fluctuations in macromolecule numbers. Transport can be diffusive or
active, and in some cases a time delay might suffice to model active
transport. We characterize major differences in the dynamics of model
genetic systems when diffusive transport of mRNA and protein was
compared with transport modeled as a time delay. Delays allow for
history-dependent, non-Markovian responses to stimuli (i.e.,
"molecular memory"). Diffusion suppresses oscillations, whereas
delays tend to create oscillations. When simulating essential elements
of circadian oscillators, we found the delay between transcription and
translation necessary for oscillations. Stochastic fluctuations tend to
destabilize and thereby mask steady states with few molecules. This
computational approach, combined with experiments, should provide a
fruitful conceptual framework for investigating the function and
dynamic properties of genetic regulatory systems.
transcriptional regulation; transport delays; circadian
oscillations; multistability; genetic modeling
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INTRODUCTION |
COMPUTATIONAL MODELING of biochemical processes often
assumes a homogenous cytoplasmic medium. However, to predict actual dynamics, it may be necessary to consider that transport of
macromolecules between the nucleus and the cytoplasm can take a time on
the order of hours, and the mechanism of transport will therefore help
determine dynamics on this time scale. There are two basic types of
macromolecular transport: passive diffusion and active transport along
cytoskeletal elements mediated by motor proteins such as kinesins or
dyneins (17, 26). A first approximation for modeling active transport might be obtained by assuming a discrete time delay for movement of
macromolecules from their place of synthesis to the location where they
exert an effect. Consideration of qualitative differences in the
behavior of models incorporating diffusion vs. a time delay can be
expected to yield insights into the dynamics of cellular processes that
incorporate diffusional vs. active transport of macromolecules. If a
discrete delay is too drastic a simplification, another approach is to
assume a distributed delay, with the derivative of a concentration
dependent on an integral over a specified range of previous time (27).
Genetic regulation is a process in which the mechanism of transport can
be expected to have a profound influence. Regulation of gene expression
by signals from outside and within the cell plays important roles in
many biological processes, including development (38), hormone action
(9), and neural plasticity (3, 16, 37). Recently, the formation and
movement of single
-actin mRNA transcripts have been visualized by
fluorescent in situ hybridization (10). In about one-half of the cases
examined, the movement of transcripts away from the transcription site
appeared to follow specific tracks. This finding suggests that an
active transport mechanism might be operating to direct mRNAs along
cytoskeletal elements. In the remaining cases, however, the transcripts
appeared to simply diffuse away from their site of formation. These
results motivated us to compare the dynamics of model genetic
regulatory systems incorporating diffusional vs. active modes of
transport for mRNA and protein.
Previous work has examined the dynamics of models of genetic regulatory
systems that use time delays to model the translocation of molecules
within the cell (2, 29-31). These studies focus largely on
determining conditions for steady states to lose stability and
oscillatory solutions to be concurrently formed. Relatively little work
has compared the effect of time delays with the effect of diffusion.
Exceptions are the studies of Busenberg and Mahaffy (5) and Mahaffy and
Pao (31), which considered the stability of steady states in models of
genetic control by repression that incorporate diffusion and delays. As
discrete delays were increased, steady states could lose stability to
oscillatory solutions. Slowing diffusion, by contrast, damped oscillations.
The present study focuses on comparing the dynamic characteristics of
transitions between coexisting steady states in models in which
macromolecular transport is dominated either by diffusion or by active
transport. Active transport was modeled as a discrete or narrowly
distributed time delay. We identified several clear qualitative
differences that could in principle be detected experimentally. We
found that active transport can produce "staircase" transitions of a series of steps in transcription rate or in nuclear concentrations of macromolecules. Moreover, a system dominated by active transport can
possess a type of short-term memory, such that the amplitude of a
response to a brief perturbation depends on the length of time since a
prior change in transcription rate. Also, in a system dominated by
active transport, a single increase in transcription rate due to a
brief stimulus can give a series of distinct subsequent "echoing"
increases in macromolecular concentrations. We also compared
the effects of diffusion with those of active transport on the
capability for oscillations in systems with positive and negative
feedback. A delay tended to create an oscillatory attractor with a
period similar to the delay, whereas diffusion tended to damp oscillations.
To illustrate specific applications of such models, we considered two
genetic systems, the dynamics of which are strongly dependent on
macromolecular transport and time delays: circadian oscillators and
genes responsible for long-term synaptic facilitation (LTF). For
circadian oscillators, we found that time delays are of value for
encapsulating complex biochemical processes but need to be supplemented
by more detailed biochemical models to address specific issues. For
LTF, it was recently suggested that transport of a messenger protein
from active synapses to the nucleus may provide a signal for induction
of essential genes (8). We found that including active
synapse-to-nucleus transport of a messenger protein greatly sharpened
this signal. However, this conclusion was sensitive to parameter
variations within the physiological range.
Finally, we considered the effects of stochastic fluctuations in
molecule numbers. We extended the previous results of McAdams and Arkin
(34) by finding that stochastic fluctuations can preferentially destabilize and thereby mask the existence of steady states
characterized by low concentrations.
Glossary
CRE |
Ca2+/cAMP response element
|
CREB |
CRE binding protein
|
D |
Diffusion coefficient
|
LTF |
Long-term facilitation
|
LTM |
Long-term memory
|
mRNA |
messenger RNA
|
TF |
Transcription factor
|
TF-A |
Transcriptional activator
|
TF-R |
Transcriptional repressor
|
TF-RE |
TF responsive element
|
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RESULTS |
Active Transport (i.e., Models With Delays) Allows Qualitatively Novel
Responses to Stimuli
Genetic regulatory systems often are activated by signal transduction
pathways, in which stimuli (e.g., hormones or neurotransmitters) lead
to phosphorylation of transcription factors (TFs), which in turn bind
to DNA sequences known as responsive elements and thereby regulate the
transcription of specific nearby genes (21). Some TFs, such as Jun and
possibly Ca2+/cAMP response
element (CRE) binding protein (CREB), autoregulate their own
transcription (35, 44). Ubiquity of genetic autoregulation in even
relatively simple organisms is suggested by an inventory of
Escherichia coli
70 promoter regulation,
identifying 21 regulatory proteins that repress their own synthesis and
4 that activate their own synthesis (7).
Stimulus-response properties of models with transport represented by
a time delay.
We added a discrete or distributed delay for macromolecular transport
to a model utilizing a single TF that activates its own transcription
(i.e., positive feedback; Fig.
1A).
The behavior of this model without delay was considered previously
(42). The factor, TF-A, forms a homodimer that can bind to responsive elements (TF-REs). The tf-a gene
incorporates a TF-RE, and when homodimers bind to this element,
tf-a transcription is increased. Responses to stimuli are modeled by varying the degree of TF-A phosphorylation. The transcription rate saturates with TF-A dimer concentration to a maximal rate
kf, which is
proportional to TF-A phosphorylation. At negligible dimer
concentration, the synthesis rate is
Rbas. TF-A is eliminated with a
rate constant kd.
Binding processes are considered comparatively rapid, so the
concentration of dimer is proportional to
[TF-A]2. The variable
[TF-A] is the nuclear concentration of TF-A. The rate
constants kf and
kd, which set the
time scale for [TF-A] equilibration, are fairly rapid
(e.g., kd = 0.1 min
1) in simulations
below (Fig. 2). This assumption of fairly
rapid equilibration of [TF-A] would probably not be
reasonable for overall cellular [TF-A], because the
equilibration time would be on the order of the degradation time for
TF-A protein. However, a short time scale for equilibration is more
likely for nuclear [TF-A]. This is because the rate
constants kf and
kd include
implicitly entrance and exit of TF-A protein from the relatively small
nuclear volume and are thus larger than those governing the dynamics of overall cellular [TF-A]. The model incorporated a time
delay
=
1 +
2, with
1 the time taken for the
transcription of tf-a mRNA and its
movement to translation and
2
the time required for movement of TF-A protein to the nucleus. This
delay appears between any change in the level of nuclear TF-A and the
appearance in the nucleus of TF-A synthesized in response to that
change.

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Fig. 1.
A: phosphorylated dimers of TF-A
activate tf-a transcription when bound
to specific DNA sequences (TF-REs). Degradation is also indicated
(kd).
B: bistability in model in
A. For 0.6 min 1 < kf < 2.5 min 1, 2 stable steady-state
solutions of [TF-A] exist
(top and
bottom of
d[TF-A]/dt = 0 curve) with
an unstable solution between them
(middle, dashed portion of curve).
Outside this region there is a single steady-state solution. Other
parameters are as follows: Rbas = 0.01 min 1,
kd = 0.1 min 1, and
Kd = 10 nM2. See
Glossary and
RESULTS for definition of
abbreviations.
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Fig. 2.
Responses of model with discrete delays.
A: model of Fig.
1A exhibits a "staircase"
transition between states. Starting from a lower steady state of
Eq. 2 with
Rbas = 0.02 min 1,
kd = 0.2 min 1,
Kd = 10 nM2, = 120 min, and = 20 min; kf is
increased at t = 200 min from 2 to 20 min 1. After approximately
the delay , a small step in [TF-A] occurs. This is
followed by successive steps to a new steady state.
B: state-dependent responses to
perturbations in model of Fig. 1A. Equation 2 is initially in upper steady state of Fig.
1B, with
kf = 1 min 1, = 120 min, and
= 20 min. At t = (210 )
min (arrow a),
kf is increased
to 10 min 1 for 2 min,
generating a large increase in [TF-A] at
t = 210 min. At
t = (400 ) min,
kf is decreased
to a new baseline of 0.1 min 1 (bar). Thus, at
t = 400 min, [TF-A]
decreases most of the way toward a new, low steady state. At
t = (500 ) min
(arrow b),
kf is increased
to 10 min 1 for 2 min. A
large increase in [TF-A] results at
t = 500 min. At
t = (800 ) min
(arrow c),
kf is again
increased to 10 min 1 for 2 min. An imperceptible increase in [TF-A] occurs at
t = 800 min. * Small increases
in [TF-A] that are "echoes" of larger excursions (see
results).
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These considerations yield a model consisting of a single delay
differential equation for the concentration of TF-A monomer in the
nucleus. For discrete delay
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(1)
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with Kd the
dissociation constant of dimer from TF-REs. The first term (in braces)
on the right-hand side is evaluated at a time
previous to the time
when d[TF-A]/dt is computed.
The simplest reasonable distributed delay assumes that
[TF-A] is averaged over a time interval, and this average
is used to calculate
d[TF-A]/dt at a later
time. This corresponds to assuming that the times required for
individual tf-a mRNAs to be
transcribed and translated are equally likely to lie anywhere within a
specific interval and never lie outside it. The same assumption is made for TF-A protein movement. This results in
|
(2)
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where
the angle brackets average [TF-A] over a window with a
width of
min centered a time
before the time at which
d[TF-A]/dt is computed.
For integration of Eqs. 1 and 2, the program XPP (B. Ermentrout,
University of Pittsburgh) was used, with the Gear integration method
selected. For other simulations, the simple forward-Euler method was
implemented in FORTRAN, with storage of delayed variables for use in
subsequent calculations.
With zero delay, we have shown (42) that the genetic regulatory system
described by Eq. 1 is bistable. There
are two steady-state solutions, or fixed points, for
[TF-A], which are asymptotically stable, in that if
[TF-A] is slightly perturbed from these solutions, it will
relax back. At these solutions,
d[TF-A]/dt = 0. Therefore, a fixed point in the absence of delay will remain a fixed point when
finite delay is assumed, and no new fixed points can be created by
adding a delay. As Fig. 1B
illustrates, for
kf between 0.6 and 2.5 min
1, there are
three values of [TF-A] that are steady states of the system. For the middle (unstable) steady state, a small perturbation of
[TF-A] will grow until [TF-A] moves to either
of the other two steady states. Dimerization of TF-A to activate its
own transcription is essential for bistability in this system. In the
variant of Eq. 1 where only first
powers of [TF-A] are present, corresponding to activation
of transcription by TF-A monomers, there is only a single nonzero fixed point.
Large perturbations of [TF-A] can switch this model between
stable steady states. Such perturbations can be induced by brief changes in the strength
kf by which
[TF-A] activates its own transcription. A change in
kf could
correspond to a change in the fraction of phosphorylated TF-A. The
resulting state transitions could correspond physiologically to
short-lived stimuli, such as exposure to a neurotransmitter or hormone,
leading to long-lasting changes in the levels of proteins. A brief
change in the basal transcription rate,
Rbas, of the
tf-a promoter could be similarly
interpreted and also can give a state transition. The model of Fig.
1A without delay would switch
between states on a brief (~10-min) perturbation (42). However, with
a discrete delay
, or with a narrowly distributed delay with a mean
of
, the perturbation must be applied for considerably longer than
. This difference comes about because, with no delay, positive
feedback of increased [TF-A] on TF-A synthesis can occur immediately and accentuate the effect of even a brief perturbation. With a delay, however, positive feedback cannot begin to act until the
delay has passed. Thus inclusion of a delay does not change the
property of multiple coexisting steady states, but the nature of the
perturbations required to induce transitions and the nature of the
transitions themselves are strongly affected.
Figure 2A illustrates the nature of
the transition between steady states of [TF-A] on an
increase in kf.
There is a series of five steps, each separated by the delay, caused by
successive increases in positive feedback due to previous jumps in
[TF-A]. The appearance of staircase steps does
require that the time scale of [TF-A] equilibration be
shorter than the transport delay
, as reflected in the parameter
values chosen for Fig. 2A
(equilibration time constant = 1/kd = 10 min).
Otherwise, equilibration cannot occur rapidly enough to allow a sharp
relaxation to a plateau of concentration, which accounts for the shape
of the step. Indeed, in Fig. 2A, one
could as well plot the actual rate of
tf-a transcription, which would have
to itself be undergoing well-defined staircase steps, as newly
synthesized TF-A protein is actively transported to the vicinity of the
tf-a gene, where it can rapidly
activate transcription. Such steps in transcription rate might be
experimentally visualized by fluorescent in situ hybridization (10).
For Eq. 2 with parameter values as in
Fig. 2A, if the width
of the
distributed delay is less than ~30 min, steps in [TF-A] are evident during the transition from the lower to the upper state,
whereas if
is greater than ~45 min, only a smooth transition is
seen (result not shown). Thus, if the distribution of times required
for individual mRNA molecules to be transported and translated and for
the corresponding TF-A protein molecules to move to the nucleus was not
too broad, distinct steps in tf-a
transcription rate might be expected.
The transient responses of the model of Eqs.
1 and 2 are also state
dependent. In Fig. 2B a perturbation
of kf at
arrow a gives a large excursion of
[TF-A] above its original value (the upper steady state of
Fig. 1B) at
t = 210 min. This same perturbation would give only a minute excursion of [TF-A] above the
lower steady state of Fig. 1B (too
small to see on the scale of Fig.
2B). This difference in response
magnitude after a state transition is a model for "priming" a
system to respond more vigorously to subsequent stimuli (42). In
addition, the presence of a delay gives the system a type of
"memory." The memory can be described as follows: after a
sustained change in the degree of TF-A phosphorylation (i.e., a change
in kf), the
response to a subsequent change in kf depends
strongly on whether the interval between the two changes is less than
or greater than the delay
. Figure
2B also illustrates this point. An
abrupt decrease in
kf from 1 to 0.1 min
1 at
t = 280 min gave a rapid transition of
[TF-A] to a lower steady state at
t = 400 min. If a brief increase in
kf was applied
within one delay time
after the decrease, at arrow
b, then a large excursion in [TF-A]
resulted at t = 500 min, even though
just before this excursion [TF-A] had decreased most of the
way to the lower steady state. The same brief increase in
kf applied after
a time greater than
had elapsed, at arrow
c, gave an excursion in [TF-A] too small to
see at t = 800 min. The memory is due
to the delay for a change in mRNA synthesis to propagate to a
corresponding change in nuclear [TF-A]. The large
[TF-A] excursion at t = 500 min was due to a brief increase in
kf at
t = 380 min, at which time
[TF-A] was still high. The high [TF-A] combined
with the increased
kf to strongly
activate tf-a transcription, with the resulting large peak in [TF-A] not occurring until
t = 500 min.
The response to a single perturbation can "echo" many times when
a discrete delay characterizes transport. A brief increase in
kf, insufficient
to give a state transition, will elicit a brief perturbation in
[TF-A]. However, this response, in turn, briefly increases
the synthesis of tf-a mRNA, thus
yielding a second, although diminished, increase in [TF-A].
Within a significant range of parameters, many successive echoes can be
obtained. A larger brief initial perturbation can cause successive
echoes that grow until the system undergoes a permanent state
transition. Distinct echoes require that the time scale for
[TF-A] equilibration be shorter than the delay
, so that
each perturbation relaxes close to baseline before the following
perturbation occurs. As discussed above, this is more likely for
nuclear than for overall cellular [TF-A], and also for the
actual rate of tf-a transcription, which is here thought of as undergoing repeated distinct increases, as
newly synthesized TF-A protein is actively transported to the vicinity
of the tf-a gene. With
Eq. 2 the echo perturbations are fairly robust to the width
of the distributed delay. Typically, significant echoes remain evident, even with
on the order of 45 min.
To examine whether the qualitative dynamic properties found above apply
to a somewhat more detailed and realistic model and to allow for
comparison with subsequent simulations that assume diffusive transport
of mRNA and protein (see below), the model of
Eq. 1 was extended to
explicitly include tf-a mRNA.
Translation was modeled as a simple first-order process, and a delay
was included between mRNA transcription and changes in nuclear
[TF-A]. The differential equations are
|
(3)
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(4)
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The dynamics of this model are qualitatively similar to
those of Eq. 1. In particular, if the
time scale characterizing the variation in
tf-a mRNA concentration
([tf-a mRNA]) is rapid,
the model reduces to that of Eq. 1.
For a relatively wide range of parameters, there is bistability.
Perturbations in
k1,f have to be
of significant length to cause state transitions. Repeated echoes in mRNA and protein levels can be elicited by a single perturbation.
Stimulus-response properties of models with diffusion.
To compare the above dynamics with those produced by an analogous model
with diffusive transport, we considered a cell divided into
N spherical shells, with the innermost
region, shell 1, a small sphere at the
center (Fig.
3A). To
obtain insights into the dynamics, it sufficed to take
N = 10, because larger values of
N did not alter the qualitative
conclusions discussed below. Equations for diffusion with spherical
symmetry (4) were used. In each shell, there are first-order
degradation terms for mRNA and protein:
kRNA
mRNAN and
kp
PN, respectively. Transcription
of mRNA is assumed to occur only in shell
1 and translation of protein only in the outermost
shell N, leading to the equations
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(5)
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(6)
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Integration
of models with diffusion was done by the forward-Euler method.

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Fig. 3.
Dynamics of model based on transport modeled as diffusion.
A: schematic of model. Transcription
of mRNA takes place in innermost spherical shell, and translation takes
place in outermost shell (shell boundaries indicated by dashed
circles). B: at
t = 100 min, increasing
Rbas from 0.08 to 3.75 nM/min for
140 min (bar) gives a smooth state transition to a new stable state.
Other parameters are as follows:
ktranscription = 3.75 nM/min, kRNA = 0.004 min 1,
kp = 0.08 min 1,
Kd = 10 nM2,
ktranslation = 2.0 min 1,
N = 10, and
DRNA = DP = 5 µm2/s.
C: oscillations of TF-A protein and
mRNA in model incorporating diffusion as well as positive and negative
feedback via Eqs. 11 and 12. Concentrations in nuclear region
are shown. Parameter values are as follows:
N = 10, ksynA = 10.5 nM/min, Rbas = 0.4 nM/min,
kARdeg = 0.01 min 1,
ksynR = 3.0 nM/min, kRRdeg = 0.002 min 1,
ksynP = 2.22 min 1,
kdegP = 0.06 min 1,
Kd = 10 nM2,
KR,d = 2.0 nM,
and DRNA = DP = 5 µm2/s.
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A total cell radius of 10 µm was assumed for most of the simulations;
thus a shell thickness was 1 µm. To determine plausible values for
diffusion coefficients (D), we
considered recent experimental observations of diffusion of dextran
(39) and green fluorescent protein (50). Unless otherwise noted,
D = 5 µm2/s was used. Transcription
depends on TF-A dimer, as in Eq. 1; translation is simply proportional to the concentration of mRNA. Thus
in Eqs. 5 and 6 we take
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(7)
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(8)
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where ktranscription represents a
maximal rate of transcription,
ktranslation is a
proportionality constant, Rbas
represents a basal transcription rate in the absence of TF-A, and
Kd represents the
dissociation constant for TF-A dimer binding to the
tf-a promoter.
Bistability could be obtained (Fig.
3B) as for the version without
diffusion (i.e., Eqs. 3 and 4). For parameters as in Fig. 3B and for small
Rbas (0.08 min
1), the model is
bistable for 3.5 < ktranscription < 10. If concentrations are in the nanomolar range, these parameter
values correspond to a maximal gene transcription rate of ~10
mRNAs/min, which is reasonable for a strongly expressed gene (20, 47).
Perturbations of appreciable length in
ktranscription or
Rbas (Fig.
3B) were needed to bring about state
transitions, because newly synthesized mRNA must be translated and the
protein must reenter the nucleus before it can bind to DNA and initiate
positive feedback. When state transitions did occur, the time course of
protein and mRNA concentrations always remained smooth (Fig.
3B). There was no staircase pattern
of steps to increasing levels. As with delay, bistability was observed
only when dimerization of TF-A is assumed necessary for its effect on transcription.
We also examined responses to brief perturbations in
Rbas and
ktranscription.
With physiologically reasonable parameter values, a brief increase in
either parameter gave a relatively abrupt increase in mRNA and protein
levels followed by a slow decline (not shown). The time scale of the
decline is set by the slow mRNA degradation rate. There was a
significant lag of ~30 min between peak mRNA and peak protein levels,
which is due to the time scale of protein equilibration set by the slow
protein degradation rate constant.
With diffusive transport, as opposed to a simple delay, the spreading
out of any excess of mRNA or protein in one region disrupts the
distinct peaks that are necessary for propagation of echo perturbations
(i.e., the occurrence of more than one significant perturbation of
protein or mRNA levels after a brief perturbation in a parameter). No
echoes were observed with physiologically reasonable parameter values.
Time Delays and Diffusion Suppress Oscillations Present When
Macromolecular Transport Is Neglected; Time Delays, but Not Diffusion,
Can Create New Oscillatory Modes
We compared the effect of diffusion with the effect of delay on the
capability for stable oscillations in models of genetic regulation.
These models incorporate positive and negative feedback via TF-As and
TF-Rs.
Oscillations with only positive or only negative feedback.
In our earlier study (42), we noted that models analogous to
Eq. 1 with only positive or only
negative feedback do not have oscillatory solutions when the time taken
for macromolecular transport is neglected. However, with a discrete
delay, it has been proven that models with only negative feedback often
possess oscillatory solutions (1). What if only positive feedback is
present with a delay: can there be stable oscillations?
We are not aware of a general theorem for this case. However, we
considered whether stable oscillations can be supported by Eq. 1 or by the simpler analog in
which transcription is activated by monomeric TF-A. If oscillatory
solutions came into existence as
was increased from zero, they
would be expected to appear via Hopf bifurcations from fixed points
(46). Such a bifurcation requires a concomitant change in stability of
the fixed point: stable to unstable or vice versa. As demonstrated in
the APPENDIX, however, the fixed
points of Eq. 1 and the analog have
the same stability properties with and without delay. In addition, our arguments appear to generalize (see
APPENDIX) to rule out changes in
stability of fixed points as
is varied and creation of stable oscillatory solutions via Hopf bifurcations for all models in a class
defined by three conditions. First, the models are represented by a
single first-order differential equation comprised of a degradation term and a synthesis term for a single variable. Second, a discrete delay affects only the synthesis term. Third, the fixed points satisfy
two not very restrictive constraints (see
APPENDIX). Furthermore, if addition
of a discrete delay cannot destabilize a fixed point, neither can it be
destabilized by a biologically reasonable distributed delay (27). Thus
Eq. 2 and any model in the above class
are not expected to exhibit stable oscillations for distributed delay.
Oscillations with positive and negative feedback.
To investigate the effect of concurrent negative and positive feedback,
we introduced into the model of Fig.
1A an additional gene,
tf-r, the transcription rate of which
is increased by binding of the TF-A dimer to a TF-RE. TF-R monomer
represses transcription by competitively inhibiting binding of TF-A
dimers to TF-REs. The equations for this model are
|
(9)
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(10)
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Parameters in Eqs. 9 and 10 are analogous to those in
Eq. 1.
KR,d is the
dissociation constant of TF-R monomers from TF-REs. Robust oscillations
are readily generated by this model when
= 0 (42). Discrete delays
of an order reasonable for macromolecular transport (
~ 120 min)
suppress this oscillatory pattern. However, a new oscillatory attractor
with a period on the order of the delay is created. For example, if
= 120 min, a lengthy transient of complex oscillations is seen (not
shown). After ~150 h, the transient evolves to a stable limit cycle.
For distributed delay, if
60 min, these oscillations were abolished.
To contrast the above results with the dynamics obtained by assuming
diffusional transport of TF-A and TF-R protein and mRNA, we modeled
diffusion of these species within a spherically symmetrical cell.
Transcription rates for tf-a and
tf-r mRNA were given by
|
(11)
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(12)
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Degradation of all macromolecular species is assumed
to take place in every shell. Degradation of tf-a and
tf-r mRNA is given by
kARdeg[tf-a
mRNA]N and
kRRdeg[tf-r
mRNA]N, respectively. The rate
of protein synthesis was directly proportional to the mRNA
concentration in the outermost shell, and the rate of degradation of
protein was directly proportional to protein concentration. The
proportionality constants are
ksynP and
kdegP. With
D of ~5
µm2/s, oscillatory behavior was
readily obtained (Fig. 3C). The
period of the oscillations was ~30 h. Further simulations found that slowing diffusion by lowering D always
acts to suppress oscillations. For suppression, diffusion had to be
slowed considerably, because the time scale for movement of
macromolecules between nucleus and cytoplasm had to become comparable
to that for other chemical processes (i.e., macromolecular degradation
and synthesis). For the simulation of Fig.
3C, oscillations were not suppressed
until D for all species was lowered
from 5 to ~0.2 µm2/s.
We repeated the simulation of Fig. 3C,
reducing D between the second and
third spherical shells to simulate the effect of a nuclear membrane.
The modified coefficient was determined by using the equation that
describes diffusion in the radial direction, x2 = 4Dt/
, where
x is the mean distance traveled in
time t given D (6). We assumed that a molecule
moves one shell width (1 µm), crossing the membrane, in a time on the
order of 1 min. Therefore, we reduced
D between the second and third
spherical shells to 0.04 µm2/s
for protein and mRNA. This reduction sufficed to abolish oscillations.
Dynamics of Two Specific Genetic Systems Are Strongly Influenced by
Time Delays
Molecular processes underlying circadian oscillations.
Models incorporating negative-feedback loops have been used to simulate
circadian rhythms (14, 25). Circadian "clocks" involve negative
feedback of one or a few core genes on their own expression. In
Drosophila and mammals, accumulation
of PER and TIM proteins represses per and tim
transcription. Later, the repression is relieved by PER and TIM
degradation subsequent to multiple slow phosphorylation steps, so that
the next cycle can begin. In the fungus
Neurospora the FRQ protein exhibits
analogous dynamics (36). Recent data should allow for the development of more comprehensive models for these circadian oscillators. In
particular, prior models have assumed that phosphorylation of the core
gene product must precede the onset of transcriptional repression (14,
25). However, autorepression by FRQ protein of its own transcription
occurs rapidly after frq induction and may therefore be independent of slow FRQ phosphorylation and
degradation (36). Autorepression by PER may also be independent of slow PER phosphorylation (11). We have carried out preliminary simulations with a generic model of a circadian rhythm generator to examine whether
independence of transcriptional repression from phosphorylation implies
that any specific biochemical features are necessary for the production
of robust circadian oscillations.
Multiple protein phosphorylations could proceed sequentially (e.g.,
phosphorylation 1 must precede
phosphorylation 2) or independently. We simulated both possibilities. In the model schematized in Fig. 4A, a
protein undergoes an obligatory series of sequential phosphorylations before degradation. A key aspect of the model is that all species of
protein are equally capable of repressing their own transcription. Repression occurs by binding and sequestration of a putative
transcriptional activator, the total concentration of which is a fixed
parameter Atot. The rate of
transcription is a sigmoidal function of free activator with a Hill
coefficient of 3, maximal velocity
vRNA, and
half-maximal transcription occurring at a free activator concentration Kact. The rate of
translation is proportional to mRNA concentration with proportionality
constant ktrans.
No delay is assumed between synthesis of protein and its repression of
transcription. However, a delay
of 90 min is assumed between
transcription and synthesis of protein. The states of increasing
protein phosphorylation are denoted
P0,...,Pm.
The concentration of each phosphorylation state requires a separate
differential equation; however, the kinetics of each phosphorylation
are assumed identical (Fig. 4B). The
model simulated circadian oscillations as illustrated in Fig. 4B, which displays time courses of
mRNA and of total protein concentration. Thus circadian rhythms can be
generated by a model in which all species of protein are equally
capable of repressing their own transcription. Circadian oscillations
could only be simulated if many phosphorylation states (>8) were
included. For the simulation of Fig.
4B, 10 phosphorylations were assumed.

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Fig. 4.
Circadian oscillations generated by a model incorporating sequential
protein phosphorylations. A: schematic
of model. Transcription of a circadian gene is inhibited by its protein
product P. Parameter values not given in text are as follows.
All phosphorylations are irreversible and first order with identical
rate constant
kphos = 3.0 h 1, and degradation of
fully phosphorylated protein is irreversible and Michaelis-Menten with
maximal reaction velocity
(Vmax) = 1.7 µM/h and Michaelis-Menten constant
(Km) = 0.02 µM. Degradation of mRNA is Michaelis-Menten with
Vmax = 12 µM/h
and Km = 0.8 µM. Binding of P to TF-A is bimolecular with forward and reverse rate
constants 20 µM 1 · h 1
and 14 h 1, respectively.
Other parameters are as follows:
vRNA = 40 µM/h,
ktrans = 0.3 µM/h, Atot = 0.21 µM, and
Kact = 0.1 µM.
B: circadian oscillations with period
close to 24 h are simulated with parameter values in
A.
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Not only were many phosphorylations essential for obtaining circadian
oscillations, but also the delay
was required. Only relaxation to a
steady state was seen if
was removed. This result is not
surprising, because the slow phosphorylations are not within the
negative-feedback loop of protein synthesis followed by transcriptional repression. Oscillations within a pure negative-feedback loop require a
delay within the loop or a very high Hill coefficient of feedback along
with at least three variables in the loop (1).
The above simulations assuming sequential phosphorylations were
compared with analogous simulations in which the individual phosphorylations were allowed to proceed independently of each other
(not shown). Oscillations were also obtained in the latter case.
However, in both cases, oscillations were always of a very short period
unless many (~10) slow phosphorylations were assumed necessary for
degradation. Thus not only the delay
but also the slow
phosphorylations are required for simulation of long-period (~24-h)
circadian oscillations. Therefore, these simulations yield the specific
prediction that many obligatory phosphorylations must precede protein
degradation in the circadian clocks of
Neurospora and
Drosophila.
Molecular processes underlying LTF.
CREB protein can bind to CREs to induce the transcription of
immediate-early genes crucial for LTF and the formation of long-term memory (LTM) (32, 48). Active transport of CREB protein appears to
occur in neurites. Fluorescently tagged CREB protein perfused into
dendrites of hippocampal neurons is rapidly and unidirectionally transported to the nucleus, whereas CREB protein perfused into the
nucleus remains there (8). Transport of newly phosphorylated CREB
protein from active neuronal synapses to the nucleus is hypothesized to
provide a signal for the transcription of genes necessary for LTF (8).
Active transport of CREB protein might be preferred over passive
diffusion for signal transmission, because it could preserve
concentration peaks. Suppose synaptic activity causes a brief episode
of local CREB protein translation and phosphorylation. If CREB protein
spread by passive diffusion throughout the neuronal volume, only a very
small and slow change in its concentration might occur at the nucleus.
However, we have seen that active transport, which might be modeled as
a time delay, could preserve concentration peaks, so that a sharp peak
of phosphorylated CREB protein might arrive at the nucleus in response
to a brief episode of synaptic CREB protein translation. This peak of
CREB protein could then initiate the transcription of genes important
for synaptic modification.
To place these considerations on a more quantitative footing, transport
of phosphorylated CREB was simulated within a simple neuronal geometry.
A cubic "soma" with a volume of 1,000 µm3 was considered, with four
attached "dendrites," each 1 µm thick and 350 µm long.
Hippocampal pyramidal cells can have dendrites of similar length and
mean thickness (19). However, this simple morphology only suffices for
a preliminary estimation of whether adding active transport to
diffusive transport, both with rates similar to experimental data, is
likely to significantly sharpen a pulse of messenger protein arriving
in the soma after a synaptic event. More detailed modeling would be
required for any specific cell type.
All dendrite lengths were measured along the
x coordinate, dendrites were modeled
as long boxes with y and
z dimensions of 1 µm, and diffusive
movement was allowed along x,
y, and
z coordinates. D in the range of 2-8
µm2/s was assumed (39, 50); 2 µm2/s is rather slow diffusion;
however, diffusion in fine dendrites might be expected to be slowed by
organelles and other obstructions. For each coordinate, spatial step
lengths were chosen from a Gaussian distribution with a mean determined
by D and the simulation time step
dt; i.e., the mean step length
xavg =
(6). Reflection
of particles at boundaries was implemented. At each boundary,
reflection only requires retrograde movement along the coordinate
perpendicular to the boundary; the steps in the other two coordinates
are unaltered. The retrograde movement was chosen, such that the sum of
absolute values of the anterograde and retrograde movements gave the
total step length that would have occurred in the absence of
reflection. In some simulations, active transport was superimposed on
diffusion. Active transport occurred along the
x coordinate as a constant movement
toward the soma of 300 µm/h, similar to some reported rates of fast
mRNA transport within dendrites (45). CREB was assumed to be
dephosphorylated with a time constant on the order of 10 h. CREB
molecules were placed at the end of one dendrite, corresponding to the
abrupt phosphorylation of CREB that might be engendered by a synaptic event. The arrival of phosphorylated CREB in the soma and its accumulation and dephosphorylation were simulated.
Figure 5B
compares simulations incorporating slow diffusion
(D = 2 µm2/s, bottom
trace) with simulations incorporating active
transport superimposed on slow diffusion (top
trace). For generation of each of these traces, 50 CREB molecules were placed at the end of a dendrite at
t = 20 h. It is evident that inclusion
of active transport increased the peak level of somatic CREB by a
factor of 2-5. The time required to reach the peak was decreased
considerably, from ~4 h to 2 h. However, for faster diffusion
(D = 8 µm2/s), inclusion of active
transport produced only a modest improvement in the peak CREB level (a
factor of ~2; Fig. 5C). The time
to peak was decreased only slightly. These simulations indicate that the presence of active synapse-to-nucleus transport of a putative messenger protein, such as phosphorylated CREB, could considerably sharpen the nuclear signal for synaptic activity compared with slow
diffusive transport. We note that this conclusion is sensitive to
parameter variations within a physiologically reasonable range. Sometimes, if diffusion is rapid, the presence of active transport would give little advantage. However, the simple geometry used in these
simulations underestimates the dilution of messenger protein that would
be expected to occur by diffusion in neurons with complex morphologies
in the absence of active transport. Also, if the dephosphorylation of
CREB proceeded more rapidly, e.g., with a time constant of 3 h, the advantage of including active transport became greater (not
shown). For slower diffusion (D = 2 µm2/s), virtually no
phosphorylated CREB arrived at the nucleus if diffusive transport alone
was assumed.

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Fig. 5.
A: schematic of model neuron used to
simulate transport of protein from an active synaptic zone, assumed to
be located at end of a dendrite, to nucleus.
B: time courses for number of protein
molecules present in nucleus after placement of 50 molecules at end of
a dendrite. At t = 20 h, molecules
were placed and allowed to move. Movement was by diffusion combined
with active transport (top curve) or
by diffusion alone (bottom curve).
Relatively slow diffusion (D = 2 µm2/s) was assumed. Population
of molecules decayed with a time constant of 10 h. Simulation time step
was fixed at 1 s. At each time step and for each molecule, a random
number was chosen between 0 and 1, and if that number was less than
deterministic probability of molecule decaying, as determined from time
constant and time step, that molecule was eliminated from simulation.
C: same as
B, except
D is raised to 8 µm2/s.
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Stochastic Fluctuations in Molecule Numbers Can Mask the Existence
of Stable Steady States
Stochastic fluctuations are generally significant in reacting systems
when relatively small numbers of molecules are present (22, 34). A
thorough analysis is difficult, because transport should be treated
stochastically. Here, as a preliminary step, stochasticity was added to
biochemical reactions in the model of Eqs.
3 and 4 for synthesis
and degradation of a TF that activates its own transcription, but
transport was still modeled as a distributed delay. Given
physiologically reasonable parameters that allow multiple stable steady
states, do fluctuations due to stochasticity cause state transitions?
The variables [tf-a mRNA]
and [TF-A] were reinterpreted as molecule numbers rather
than as concentrations by rescaling parameter values. Deterministic
rates for synthesis and degradation processes are given by
|
(13)
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(14)
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(15)
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(16)
|
The expression for
Rtranslation incorporates a
distributed delay for macromolecular transport. To include fluctuations
due to association and dissociation of TF-A monomers, we dropped the assumption that TF-A dimer concentration is proportional to
[TF-A]2. Deterministic
rates for association and dissociation were assigned Rmonomer-dimer = kf([TF-A]monomer)2
and Rdimer-monomer = kb[TF-A]dimer,
respectively. The reciprocals of the deterministic rates are the
average time intervals between reactions. Denote such a time interval
by Tavg. If a
particular biochemical reaction occurs at
t = 0, the probability
P(t)
that the next reaction of that type will occur within a specific short time interval
t centered at a later
time t is (13)
|
(17)
|
At each time step
t, a separate
random number was chosen for each elementary process of
synthesis or degradation of mRNA or protein and TF-A association
and dissociation. Each random number was drawn from a uniform
distribution on {0,1}. For any elementary
process, if the random number was less than the product of
t and the average rate
(Eqs. 13-16), we assumed
that the process occurred once. If the products of
t and the average rates are kept
small (<0.1), then this scheme approximates Eq. 17.
Equations 13-16 were also used to formulate
ordinary differential equations for
[tf-a mRNA],
[TF-A]monomer, and
[TF-A]dimer.
Simulations with this "deterministic" model were compared with
simulations that included fluctuations. The deterministic model
exhibits bistability for a significant range of parameters. One stable
solution has [tf-a mRNA]
low and its synthesis rate close to
Rbas, and the other has
[tf-a mRNA] high and its
synthesis rate close to
k1,f. Figure
6A
illustrates an example in which the deterministic model is stable in
the lower steady state until a temporary increase in
Rbas switches it to a new stable
state. The stochastic variant of the model was then initialized near
the lower steady state. Within ~40 h, random fluctuations accumulated
and enabled a transition to the upper state (Fig.
6B). This behavior occurred with
physiologically reasonable parameter values, similar to those used by
McAdams and Arkin (34). For example, in the simulation of Fig.
6B, the maximal mean transcription
rate k1,f of 3.8 mRNA molecules per minute is reasonable for a strongly activated
promoter. A reverse transition, from the upper to the lower state, was
not seen even after 400 h of simulated time. Upward transitions within
10-100 h of simulated time, with no downward transitions after 400 h, were seen for five different random number sequences. Thus the lower
stable state can be "masked" by fluctuations, in that the system
would not ordinarily be observed near it. These results were
essentially unchanged when
was varied over the range 5-100 min. However, assuming a discrete delay (
= 0) resulted in the appearance of much larger and spurious fluctuations due to
amplification by fluctuations exactly one delay time earlier (not
shown).

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Fig. 6.
Stochastic fluctuations in molecule numbers can destabilize steady
states of genetic regulatory systems.
A: without stochastic fluctuations,
deterministic model (Eqs.
13-16) is bistable. Initial levels of RNA and
protein are low (<1 nM) and steady. Parameters are as follows:
k1,f = 0.2 nM/min, k1,d = 0.02 min 1,
k2,f = 0.2 nM/min, k2,d = 0.03 min 1,
Rbas = 0.0015 nM/min,
kf = 2 nM/min,
kb = 20 min 1, = 90 min, and
Kd = 10 nM. At
t = 10 h (arrow),
Rbas is increased to 0.1 nM/min
for 500 min. Increase causes a transition to a stable state of high RNA
and protein levels. B: initial state
of A is spontaneously destabilized
when stochastic fluctuations are incorporated. Parameters as in
A, except
1) = 5 min and
2) to convert nanomolar
concentrations to molecule numbers within a nuclear region assumed to
have a 2-µm radius,
k1,f,
Rbas, and
Kd were
multiplied by a numerical factor of 19.2 nM 1 and
kf was divided by
this factor.
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|
 |
DISCUSSION |
We have considered the dynamics of model genetic regulatory systems in
which TFs activate or repress their own transcription. Differences have
been illustrated that depend on whether transport of mRNA and protein
is described by diffusion or by active transport modeled as a time
delay. Active transport was modeled by a narrowly distributed time
delay. For example, a time delay leads to a type of memory. After a
sustained change in transcription rate, the amplitude of the excursion
in protein concentration due to a brief change in transcription rate
depends strongly on whether the interval between the two changes is
less than or greater than the delay (Fig.
2B). This memory is due to the delay
for a change in mRNA synthesis to propagate through translation to a
corresponding change in nuclear protein level. Also, given appropriate
parameters (large kf and kd in
Eq. 1), models with time delays can exhibit staircase-like
transitions from one steady state of macromolecular concentrations to
another in response to a sustained stimulus (Fig.
2A) or repeated perturbations of
concentrations in response to a single brief stimulus. If newly
synthesized TF was actively transported to the vicinity of its own
gene, then staircase transitions in transcription rate or repeated
perturbations of this rate after a single stimulus might be visualized
by fluorescent in situ hybridization (10). Observation of such
phenomena would constitute strong evidence for active transport of mRNA
and protein. It does not seem that a system reliant on diffusional
transport would exhibit either of these phenomena.
The dynamics exemplified by Fig. 2 might help determine the relative
efficacies of different training paradigms in forming LTM (33, 42). If
transcription of CREB or of another TF that activates its own
transcription is essential for plasticity underlying the formation of
LTM, then stimuli separated by short time intervals might be expected
to be relatively ineffective at forming LTM. A longer time interval
after the first stimulus might be required to allow for TF
transcription, translation, and translocation to the nucleus. Thus a
larger amount of nuclear TF would be available at the arrival of the
second stimulus, and, when activated, that TF could strongly promote
the transcription of itself and of other genes essential for LTM
formation. Such considerations could help explain the usual superiority
of temporally spaced, as opposed to massed, training sessions in
forming LTM.
Two issues arise concerning the validity and importance of models
incorporating different transport mechanisms. First, what conditions
might justify modeling macromolecular transport with a time delay?
Second, how useful is this type of modeling, in conjunction with
experiments, for understanding the dynamics of genetic systems?
Can Active or Diffusive Transport Be Plausibly Modeled as a Narrowly
Distributed Time Delay?
Numerous examples of active mRNA and protein transport in the cytoplasm
are now known, including microtubule-dependent movement of
Vg1 mRNA in
Xenopus oocytes (49), the
myosin-dependent segregation of the lineage-determining PAR-1, PAR-2,
and PAR-3 proteins in Caenorhabditis
elegans (15), microtubule-dependent localization of
several maternal mRNAs for proteins that direct embryonic development (such as bicoid, bicuadal-D, and
oskar) in
Drosophila oocytes (24), and
retrograde transport of proteins with a nuclear localization signal in
Aplysia neurons (40).
Recent experimental studies of active transport by motor proteins
suggest that the distribution of times taken by individual macromolecules to be transported a given intracellular distance could
be quite narrow. The kinetics of motion along microtubules have
recently been examined by use of latex beads with single kinesin
molecules attached. The motion is comprised of single steps ~8 nm
long (18, 41). The distribution of distances traveled in a given time
can be characterized by a randomness parameter r, defined as the variance divided by
the product of the mean and the single-step distance (41). At >1 mM
ATP, similar to that in vivo, r
1/2 (41). The variance can be estimated for typical
intracellular distances of transport. With use of the definition of
r, if a population of macromolecules
moves a mean distance of 20 µm and if the movement is driven by
kinesin with a single step length of 8 nm, then with
r
1/2 the variance in the
distance moved by individual molecules will be 0.08 µm2. A variance as small as 0.08 µm2 could be modeled as a
discrete time delay, because for small variances the ratio of the
variance in arrival times to the mean travel time is the same as the
ratio of the variance in position to the mean distance moved. In
contrast, analogous calculations demonstrate that diffusive transport
could not be modeled as a time delay, because the variance in the
distance moved by individual macromolecules is much greater.
However, it is not generally known whether successive mRNAs from a
given gene are transported to the same translation site or closely
grouped sites. If successive mRNAs were directed to a variety of
translation sites at different distances from the gene, there would be
a distribution of times to reach the sites. Also, it can be expected
that some diffusional movement would be required even if active
transport predominated, because each mRNA would probably need to move
along more than one cytoskeletal element to reach its destination, and
"jumps" between elements might be via diffusion. A model
formulation using a distributed delay to describe transport might be an
appropriate approximation for such complications. In our simulations,
oscillations were abolished if the width of the distribution was
20-30% of the average delay. Repeated perturbations and
transitions via concentration steps were not abolished until somewhat
larger distribution widths. Whether distributions this narrow are
reasonable characterizations of intracellular transport is an issue
requiring experimental investigation.
Specific Issues for Further Investigation by Modeling and Experiment
Our modeling of stochastic fluctuations in genetic regulatory systems,
which illustrates that such fluctuations could preferentially destabilize and mask the existence of steady states with low
concentrations of macromolecules, could certainly be further developed.
One issue of interest is whether fluctuations tend to be significantly
smaller or to have a different frequency spectrum when diffusional
transport dominates over active transport. It appears that if
simulations such as those of McAdams and Arkin (34) are to predict the
degrees of variability in the behavior of actual genetic systems,
transport and perhaps its stochastic nature will often have to be
included. However, the inclusion of transport may be less necessary in
prokaryotic systems such as those modeled by McAdams and Arkin. If
transport is diffusional, the small dimensions of many prokaryotic
cells and the lack of a nuclear membrane imply that, on the time scale of minutes, concentrations of species not rapidly degraded can be
regarded as homogenous.
Transport that is partly diffusive and partly active could be described
in a more quantitative, spatially resolved manner by use of a
stochastic ordinary differential (Langevin) equation in combination
with a partial differential (Fokker-Planck) equation (12). A Langevin
equation can be formulated to describe the average and fluctuating
components of motion of an actively transported macromolecule. The
corresponding Fokker-Planck equation describes the evolution of the
spatial distribution of an ensemble of macromolecules. It has a
convective term for the mean flow due to active transport and a
diffusive term for thermal fluctuations. An additional term could be
added to describe ordinary diffusion. Given estimates of kinetic
parameters, numerical simulations using this equation could predict the
variance in distances traveled by members of an ensemble of
macromolecules. Related statistical quantities, such as a correlation
function for overlap of concentration perturbations due to separate
genetic induction events, could also be predicted.
On a more general level, the diversity of transcription factors and
their interactions suggest that behaviors such as those considered here
(e.g., long-lasting state transitions in response to perturbations or
oscillations dependent on time delays) will be identified. Thus the
dynamic principles illustrated here are likely to be important in
phenomena where regulation of transcription has an essential role, such
as development or the formation of LTM. It has recently been proposed
(28) that epigenetic, heritable changes in gene expression after
exposure to chemicals might play a role in carcinogenesis. Such changes
correspond dynamically to perturbations of genetic regulatory systems
from one steady state to another, such as we have modeled. An
outstanding issue will be to determine whether the parameters of
particular genetic systems in vivo are permissive for specific types of
dynamic behavior. To this end, we suggest that the rate of movement and
the variance in distance moved in a given time should be examined for a
few TFs of particular importance and for their mRNA transcripts. This can help determine whether diffusional or active transport dominates in
particular systems. We believe that as the dynamic behaviors of gene
networks are explored empirically the present work can provide a
conceptual framework for the interpretation of such experiments.
 |
APPENDIX |
Here we demonstrate the mathematical points stated in
RESULTS. First, we show that the fixed
points of Eq. 1 and of its analog where first powers of [TF-A] replace second powers have the
same stability properties with discrete delay as without. In the
analog, nondimensionalization of [TF-A] and time
successively sets
Kd and
kf to 1. Rbas is also assumed to be small
and has little effect on the dynamics about the nonzero fixed point
(simulations have verified this for several combinations of parameter
values) and can be neglected. The analog then reduces to
|
(18)
|
Here we have introduced a notation we use throughout the
APPENDIX. For a variable
X, a discrete delay
, and present
time t, Xdel
X(t
), whereas X
X(t).
Equation 18 has a single
nonzero asymptotically stable fixed point at
X = (1
a)/a.
Biological relevance requires X > 0 at this fixed point, so a < 1. If
0, then in general, if
g(X,Xdel)
denotes the right-hand side of a first-order delay differential
equation, a characteristic equation governing stability of the fixed
point is derived by linearizing the differential equation about the
fixed point and then substituting
X(t) = exp(
t) into the linearized
equation (see Ref. 27 for details)
|
(19)
|
The derivatives in Eq. 19
are to be evaluated at the fixed point. At the nonzero fixed point of
Eq. 18, Eq. 19 gives
a2exp(

)
a
= 0. If the real
part of the eigenvalue
is negative (respectively, positive), the
fixed point is stable (respectively, unstable). A switch in stability
is only possible when the real part of
is 0 for some
> 0.
Substituting a pure imaginary
= ki
into Eq. 19 and evaluating at the
fixed point of Eq. 18 give two
simultaneous equations
Using the Pythagorean identity gives
which cannot be satisfied if
a < 1, as is required for
X > 0 at the fixed point. Thus the
fixed point remains stable for all
.
For Eq. 1, nondimensionalization of
[TF-A] and t and neglect
of Rbas gives
|
(20)
|
X = 0 is a fixed point of
Eq. 20, and in addition, there are two
positive fixed points if 0 < a < 1/2. If
= 0 the upper positive fixed point is stable, the
lower is unstable. If
> 0, we again look for a purely imaginary
that solves Eq. 19. For
= ki, we have
g/
Xdel
must be greater than
(
g/
X)
to admit a real solution (k,
). For
Eq. 20,
(
g/
X) = a, and for
a < 1/2, numerical calculation shows that
g/
Xdel >
(
g/
X)
for only the lower positive fixed point. Thus only this fixed point
could switch stability as
varies. The other fixed points must
remain stable for all
.
We consider further whether the lower nonzero fixed point could become
stable for any
. Suppose at time 0 an upward perturbation in X is made
from this point, small enough to remain within the neighborhood where
g/
Xdel >
(
g/
X)
(a similar argument applies for downward perturbations). The
perturbation is free until time
to relax toward the fixed point.
Denote the value of X at
by
X1. At time
,
X will begin to increase. This occurs
because the positive contribution to
dX/dt,
X2del/(X2del + 1), abruptly becomes greater than the negative contribution aX1. Later, if
X "tries" to decrease below
X1, it will be
unable to, because
X2del/(X2del + 1) will remain greater than
aX1 (since
Xdel > X1). We now
choose a value X = X2, above
X1 and the peak
of the small perturbation but still within the region where
g/
Xdel >
(
g/
X).
Suppose X never increases above
X2 for subsequent
t. If so, one can average dX/dt
from time
to time t and, since
X must always remain in the
interval
(X1,X2),
this average must approach zero for large t.
This average equals the difference of the average of
X2/(X2 + 1) from time 0 to time
(t
) and the average of
aX from time
to time
t. For sufficiently large
t, one can neglect the relatively small intervals from 0 to
and from
(t
) to
t when taking these averages. Thus
the average of
[X2/(X2 + 1)]
aX from time
to time t must approach zero for
large t. However,
X lies within
(X1,X2)
from time
to time t. We recall that 1)
g(X,Xdel)
[X2/(X2 + 1)]
aX = 0 at the
lower fixed point and 2)
g/
Xdel >
(
g/
X) for all X between this fixed point and
X2. These
conditions imply that
[X2/(X2 + 1)]
aX is positive and
monotonically increasing from
X1 to X2, which in turn
implies that the average of
[X2/(X2 + 1)]
aX from time
to time t cannot approach zero for
large t. This contradiction implies
that, rather than staying within (X1,X2)
for all t,
X must eventually increase above
X2 and above its
original upward perturbation from the lower nonzero fixed point. Thus
this fixed point is unstable, regardless of the value of
. We
conclude that none of the fixed points of Eq. 20 can switch stability as
varies.
The arguments of the preceding three paragraphs appear, for a class of
models, to rule out switches of stability of fixed points when the
length of the discrete delay is varied. This class satisfies the
following conditions: 1) there is a
single dependent variable X, and
dX/dt
g(X,Xdel)
is the sum of two terms, a synthesis term dependent only on
Xdel and a
degradation term dependent only on X;
2) all fixed points are either
asymptotically stable or unstable in the absence of delay; and
3)
g/
Xdel >
(
g/
X)
[respectively,
g/
Xdel <
(
g/
X)]
at those fixed points that are unstable (respectively, asymptotically
stable) in the absence of delay.
 |
ACKNOWLEDGEMENTS |
We thank C. Canavier and R. Butera for comments on the manuscript.
 |
FOOTNOTES |
This work was supported by National Institutes of Health Grants T32
NS-07373 and R01 RR-11626 and by Texas Higher Education Coordination
Board Grant 011618-048.
The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement"
in accordance with 18 U.S.C. §1734 solely to indicate this fact.
Address for reprint requests and other correspondence: J. H. Byrne,
Dept. of Neurobiology and Anatomy, W. M. Keck Center for the
Neurobiology of Learning and Memory, The University of Texas-Houston
Medical School, PO Box 20708, Houston, TX 77225 (E-mail:
jbyrne{at}nba19.med.uth.tmc.edu).
Received 4 December 1998; accepted in final form 20 May 1999.
 |
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