A mathematical model quantifying GnRH-induced LH secretion
from gonadotropes
J. Joseph
Blum1,
Michael C.
Reed2,
Jo Ann
Janovick3, and
P. Michael
Conn3
1 Department of Cell Biology,
Duke University Medical Center, Durham 27710;
2 Department of Mathematics,
Duke University Medical Center, Durham, North Carolina 27708; and
3 Oregon Health Sciences
University, Beaverton, Oregon 97006
 |
ABSTRACT |
A mathematical model is developed to investigate the rate
of release of luteinizing hormone (LH) from pituitary gonadotropes in
response to short pulses of gonadotropin-releasing hormone (GnRH). The
model includes binding of the hormone to its receptor, dimerization,
interaction with a G protein, production of inositol 1,4,5-trisphosphate, release of
Ca2+ from the endoplasmic
reticulum, entrance of Ca2+ into
the cytosol via voltage-gated membrane channels, pumping of
Ca2+ out of the cytosol via
membrane and endoplasmic reticulum pumps, and release of LH. Cytosolic
Ca2+ dynamics are simplified
(i.e., oscillations are not included in the model), and it is assumed
that there is only one pool of releasable LH. Despite these and other
simplifications, the model explains the qualitative features of LH
release in response to GnRH pulses of various durations and different
concentrations in the presence and absence of external
Ca2+.
calcium; G protein; endoplasmic reticulum; inositol
1,4,5-trisphosphate; luteinizing hormone
 |
INTRODUCTION |
THE SECRETION of luteinizing hormone (LH) and
follicle-stimulating hormone (FSH) by gonadotropes located in the
anterior pituitary is stimulated by gonadotropin-releasing hormone
(GnRH), a decapeptide that is released by the hypothalamus. It was
suggested by Conn et al. (5, 7) that dimerization of the GnRH receptors
on the surface of the gonadotropes was sufficient to initiate the release of LH, and it has been established that this event occurs in
response to agonist (but not antagonist) occupancy of the receptor (17). Mathematical models have been developed to explore some of the
consequences of dimerization on the response of the gonadotropes to
various concentrations of GnRH and some related peptides (1, 22). Those
models focused primarily on the kinetics of receptor binding and
dimerization and did not include the interaction of the dimerized
receptors with G proteins or the subsequent complex intracellular
signaling systems such as the release of inositol 1,4,5-trisphosphate
(IP3) and the vesicular and cell
membrane Ca2+ dynamics.
Since those early models, our understanding of the signaling systems
between GnRH binding and LH release has greatly increased (18, 39). The
development of perifusion systems and optical methods to study the
changes in cytosolic Ca2+ content
of individual gonadotropes has allowed much new data to be obtained on
the changes in cytosolic Ca2+
concentration (CAC) and the rates of release of LH in response to short
pulses of GnRH. We limit ourselves to modeling the response to short
pulses of GnRH, thus allowing us to ignore the effects of changes in
gene expression that are known to occur in response to long exposure to
GnRH (12, 19). Furthermore, it has become possible to generate
pituitary cell lines expressing different concentrations of GnRH
receptors (27) and, thus, to obtain data on the effect of receptor
number on rates of release of LH and of the change in cytosolic
concentrations of IP3, the
signaling compound that activates release of
Ca2+ from the endoplasmic
reticulum (ER) (26). In view of these advances in our understanding of
the cell physiology of gonadotropes, we believed that it was
appropriate to expand the earlier mathematical models to include the
interaction of the dimerized receptors with a G protein in the cell
membrane, the consequent release of
IP3 and, therefore, of
Ca2+ from the
Ca2+ stores in the ER (CAER),
the opening of Ca2+ channels in
the plasma membrane, and the subsequent active transport of
Ca2+ back into the ER and into the
extracellular fluid.
In this study we have omitted many features of the complex pathway
between the binding of GnRH to its receptors and LH release. Furthermore, we focus only on data concerning LH release in response to
relatively short pulses of GnRH and the effects of varying GnRH
receptor number (GnRHR). In future studies we expect to expand the
model to include receptor internalization (14), the biphasic regulation
of GnRHR that occurs during long exposure to GnRH (6), and the effects
of CAC on the rates of uptake and release of
Ca2+ by the ER and the plasma
membrane (39).
It is well established that GnRH induces oscillations in CAC in
gonadotropes via voltage-gated
Ca2+ channels (VGCCs) located in
the plasma membrane (39). The initial rise in CAC in response to GnRH
is, however, independent of extracellular Ca2+ (CAE) and is due to the rapid
release of Ca2+ from the ER. A
number of studies have shown that, at low and medium concentrations of
GnRH, there is an increase in the initial spike and the subsequent
plateau level of CAC and in the frequency of the subsequent
oscillations with increasing concentrations of GnRH (38). At high GnRH
concentrations, however, there is no further increase in the initial
spike level of CAC, but the frequency does increase. Although there
are well-developed models for Ca2+
oscillatory behavior in a number of cell types (10, 11), none
specifically apply to the CAC oscillations of gonadotropes. Furthermore, most of the published measurements of LH release have been performed over minutes or hours and, thus, represent the
LH released over many oscillations. In view of these data and the close
correlation between the rise in CAC and the amount of LH released (38),
we have chosen to model the CAC (and the CAER) as smooth functions of
time that mimic the initial spike and the subsequent average
Ca2+ levels. Although the rate of
CAC oscillations increases with increasing
IP3 concentration, the
IP3 level, on which the
GnRH-induced Ca2+ responses
depend, does not oscillate (37). Also, considerable evidence indicates
that low CAC facilitates and high CAC inhibits the release of
Ca2+ from the ER (39).
The activation of phospholipase C by the G protein(s) after GnRH binds
to its receptor stimulates the production of
IP3 and diacylglycerol. Because an
inhibitor of diacylglycerol lipase causes dose-dependent inhibition of
LH release, without affecting the ability of arachidonic acid to
facilitate LH release (2, 34), it appears that arachidonic acid is also
involved in the signaling system by which GnRH causes LH release.
Present evidence indicates that this is via a
Ca2+-independent mechanism, and we
have not included the diacylglycerol-arachidonic acid pathway in the
present model.
Recent studies show that multiple proteins are involved in GnRH
signaling (13, 30). Janovick and Conn (16) showed that the G protein
involved in the initiation of the GnRH-activated signaling pathway is
inhibited by cholera toxin, but not by pertussis toxin, and is,
therefore, a Gs guanyl
nucleotide-binding protein. They also obtained data that indicated two
pools of LH in gonadotropes and redistribution of LH by GnRH from a
nonreleasable pool to the releasable pool. Because the kinetics
of transfer of LH from the nonreleasable pool to the releasable pool
have not been studied, we have chosen to treat LH release as occurring
from a single pool.
Glossary
Variables
H |
GnRH concentration (nM)
|
R |
Free GnRHR concentration (nM)
|
HR |
Hormone-receptor complex concentration (nM)
|
HRRH |
Hormone-receptor dimer concentration (nM)
|
E |
Effector concentration (nM)
|
IP3 |
Inositol 1,4,5-trisphosphate concentration (nM)
|
CAC |
Cytosolic Ca2+ concentration (µM)
|
CAER |
ER Ca2+ concentration (µM)
|
CHO |
Fraction of open ER Ca2+ channels
|
LH |
LH concentration (ng)
|
Constants
R0 |
Total receptor concentration (nM)
|
GQ0 |
Total G protein concentration (nM)
|
ERUL = 40 |
Resting Ca2+ concentration in ER
(µM)
|
CAE = 1,000 |
External Ca2+ concentration (µM)
|
= 2 |
Constant in Eq. 10 for
fraction of open ER channels
(nM 1)
|
= 4 |
Constant in Eq. 10 for fraction of open ER
channels (min 1)
|
Rate Constants
k1 = 2.5 |
nM 1 · min 1
|
k 1 = 5 |
min 1
|
k2 = 2,500 |
nM 1 · min 1
|
k 2 = 5 |
min 1
|
k3 = 4,000 |
nM 1 · min 1
|
k 3 = 200 |
min 1
|
k5 = 2 × 107 |
min 1
|
k 5 = 10 |
min 1
|
k6 = 1 |
µM 1 · min 1
|
k66 = 10 |
µM 1 · min 1
|
k666 = 0 |
|
k 6 = 5.0 |
min 1
|
k7 = 2.2 |
µM/min
|
k8 = 0.4 |
nM 1 · min 1
|
k88 = 0 |
|
k888 = 0 |
|
k9 = 0.0002 |
min 1
|
k10 = 5 |
ng/min
|
 |
MODEL DEVELOPMENT |
Our model for the stimulation by GnRH of LH release by pituitary cells
is described in three stages. In the first stage, the hormone binds to
the receptor, dimers are formed, and the production of effectors and
IP3 occurs. In the second stage,
IP3-regulated channels on the ER
allow Ca2+ to be released and
subsequently pumped back into the ER. In the third stage, the
voltage-sensitive cell membrane
Ca2+ channel, leakage
Ca2+ channels, the cell membrane
Ca2+ pump, and the release of LH
are described.
Hormone binding to IP3 formation.
We denote by H(t) the concentration
of GnRH in the surrounding medium as a function of time, where
time t is measured in minutes. The
receptor concentration per unit volume will be denoted by R. We assume
that the GnRH binds to the receptors via a simple reversible reaction
|
(1)
|
The
bound complex HR reacts reversibly with itself to form dimers, the
concentration of which is denoted by HRRH
|
(2)
|
A
G protein, the concentration of which is denoted by GQ, reacts with the
dimer via a reversible reaction to produce an effector, E. In this
initial study, as discussed in the introduction, it is assumed that E
represents only phospholipase C
|
(3)
|
Although
all these reactions take place on the cell surface, for simplicity all
concentrations are given in nanomoles per unit volume of the
surrounding medium. Because of these assumptions, the concentrations R,
HR, HRRH, GQ, and E satisfy the following set of differential equations
|
(4)
|
|
(5)
|
|
(6)
|
|
(7)
|
|
(8)
|
We
assume that the production of IP3
is proportional to the concentration of E. The subsequent metabolism of
IP3 in gonadotropes is complex
(26), and the kinetics are unknown. Therefore, we assume simply that
IP3 is converted to inactive
metabolites at a rate proportional to its concentration, thus yielding
the differential equation
|
(9)
|
A reasonable estimate for the number of GnRHRs per gonadotrope is
104 (4, 23-25, 27). If we
assume that there are
105-106
cells/ml in a typical experiment on isolated gonadotropes, the GnRHR
concentration would be 1.5-15 pM. Thus we will choose
R0 = 0.01 nM as our
"standard" total receptor concentration. We will study the effect
of varying R0 later. The number of
G proteins per gonadotrope is not known but, in other cells, is
~10-fold or more greater than the number of receptors (28); we have
chosen the total concentration of G protein to be
GQ0 = 0.1 nM. All the variables
HR, HRRH, E, and IP3 have the
value 0, and R = R0 at t = 0.
The standard values of the rate constants in
Eqs. 1-9 are listed
in the Glossary. Because degradation
of GnRH itself has made it difficult to measure the affinity constant
for the binding of the naturally occurring ligand to its receptors, a
precise value is unavailable. A value of ~0.7 × 109
M
1 has been suggested.
Studies using a metabolically stable GnRH superagonist that was 40-fold
more potent than GnRH gave a value of 2 × 1010
M
1, which when divided by
40 gives 0.5 × 109
M
1 (24). The magnitudes of
k1 and
k
1 were
chosen so that most of the binding occurs within 30 s, and the
affinity constant is 0.5 × 109
M
1. The rate constants for
dimerization are unknown. We have chosen k2 and
k
2 such
that there is a high tendency to dimerize and it occurs quite rapidly.
Similarly, we chose the rate constants k3 and
k
3 so that
the binding of the dimer to the G protein is rapid and has high
affinity. The rate constants
k5 and
k
5 were
chosen so that IP3 approaches its
steady-state level in response to H = 0.1 nM in ~2.5 min and to H = 10 nM in ~0.5 min, consistent with the short-pulse data of Morgan et
al. (26).
In Figure 1 we show the responses of the
variables R, HR, HRRH, E, and IP3
to a 5-min pulse of GnRH for H = 0.1, 1, and 10 nM. At H = 0.1 nM, ~95% of the receptors are unoccupied, and thus the amounts
of HRRH, E, and IP3 are also very
small. With a 10-fold increase of H to 1 nM, about two-thirds of the
receptors are occupied, and much more HRRH, E, and
IP3 is produced. With a further
10-fold increase in H, ~96% of the receptors are occupied, and
further significant increases in HRRH, E, and
IP3 occur. At this high hormone
concentration, the formation of HR is so rapid that HR first peaks (at
~0.1 min) and then declines as HRRH is formed. In all three cases,
the concentrations approach their steady-state values in ~1 min and
then relax back to their initial values in 2-5 min after the
hormone is removed at t = 5 min.

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Fig. 1.
Response to a 5-min pulse of gonadotropin-releasing hormone (GnRH).
Time variations of R, HR, HRRH, E, and
IP3 are shown during and after a
5-min pulse of 3 concentrations of GnRH. A,
D, and G: responses to
0.1 nM GnRH; B, E, and
H: responses to 1.0 nM GnRH;
C, F, and
I: responses to 10 nM GnRH. See
Glossary for definition of
abbreviations for this and subsequent figures.
|
|
ER.
Ca2+ is stored in the ER and
released when IP3 binds to
receptors on the ER membrane. The detailed dynamics of these receptors are a subject of current research (20) and are thought to play a major
role in the cytosolic Ca2+
oscillations. We do not attempt to model these detailed dynamics but,
instead, assume that the fraction of open channels (CHO) has a given
time course
|
(10)
|
which
depends on IP3 concentration in a
Michaelis-Menten-type saturating fashion. Inside the cells, we measure
concentrations in micromolar (instead of nanomolar), which is the
reason for the factor 10
3
preceding
IP3(t).
The fraction in the first factor on the right approaches a maximum of 1 for high IP3 concentrations. The
factor
te(1
t)
reaches a maximum of 1 when t = 1/
. We choose the parameters
and
of Eq. 10 as
so
this factor reaches its maximum at 0.25 min. Therefore, the maximum of
CHO (the probability of opening) is 0.6, consistent with
the data of Ramos-Franco et al. (29). The time course of CHO(t) is shown in Fig.
2 for the steady-state concentrations of
IP3 = 395, 3,840, and 5,300 nM,
which correspond to GnRH concentrations of 0.1, 1, and 10 nM (Fig. 1).

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Fig. 2.
Probability of IP3-regulated
Ca2+ channel opening. Time course
is shown in response to a 5-min pulse of GnRH at indicated
concentrations.
|
|
We assume that the release of Ca2+
from the ER is jointly proportional to CHO, to the difference (CAER
CAC), and to an intrinsic rate constant ERR, where
The
complicated form of ERR reflects the fact that the release is
facilitated by cytosolic Ca2+ at
low CAC (via parameter
k66) and
inhibited at high CAC (via parameter
k666), as
indicated earlier. We assume that
Ca2+ is pumped back into the ER
at a rate jointly proportional to CAC (via sigmoidal kinetics) and
to (ERUL
CAER), where ERUL is the resting concentration of
Ca2+ in the ER.
Thus
|
(11)
|
Although there are no experimental measurements of the ratio of ER
volume to cell volume in gonadotropes (or most other cells), a
reasonable value is 1/20. Thus the dynamics of CAC (in the absence of the other mechanisms introduced below) would be given by the following equation
|
(12)
|
In most cells, including gonadotropes, the resting level of
cytosolic Ca2+ is 0.05-0.2
µM (11, 38); we have chosen the rate constants in the system so that
the equilibrium value (in the absence of H) is 0.1 µM. The resting
Ca2+ concentration in the ER is
unknown but large, and a reasonable value is ERUL = 40 µM (15,
40).
The rate constant ERR determines the rate of release of
Ca2+ from the ER given by the
first term on the right of Eq. 11.
Initially, CAER = 20 µM and CAC = 0.1 µM, so by choosing
k6 = 5 µM
1 · min
1,
k66 = 50 min
1 · µM
1,
and k666 = 0, we
would obtain an initial rate of decrease of CAER of
if
CHO = 1 (i.e., all the channels were open). The initial rate of
increase of CAC would be 1/20th of that, i.e., 4 µM/min, by our
volume hypothesis discussed earlier. These are overestimates, because,
in fact, for the first few seconds only a small fraction of the ER
channels are open, and the maximum probability of opening is 0.6. With
these choices for
k6 and
k66, the initial
spike in CAC occurs in <1 min and reaches a maximum value at
0.2-1 µM (Fig. 3), consistent with
experimental data. For this initial study, we ignore the inhibitory
effect of high CAC on ERR and take
k666 = 0.

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Fig. 3.
Response to a 5-min pulse of GnRH. Time variation of CAER, CAC, and
rate of luteinizing hormone (LH) release is shown in response to
indicated concentrations of GnRH.
|
|
The second term on the right of Eq. 11
represents pumping of Ca2+ back
into the ER. By choosing
k
6 = 5 min
1, the pumping of
Ca2+ back into the ER is
sufficiently weak (compared with the release of
Ca2+ from the ER), allowing the
initial CAC spike to occur.
Ca2+ levels
and LH release.
It is known that the effector E produced from the dimers also activates
voltage-sensitive Ca2+ channels in
the cell membrane (35). We let CAE denote the (constant) Ca2+ concentration in the external
medium and assume that the rate of
Ca2+ influx through the VGCCs is
proportional to (CAE
CAC). The rate constant for the
Ca2+ influx has the form
because
there is evidence (35) that the rate is facilitated by low
concentrations of CAC and inhibited by high concentrations. We assume
that the Ca2+ pumps in the cell
membrane obey second-order Michaelis-Menten kinetics (rate constant
k7) and that
Ca2+ leakage from outside to
inside (rate constant
k9) is a simple first-order process. By addition of these
Ca2+ fluxes to
Eq. 12, the following differential
equation for CAC is obtained
|
(13)
|
Finally, we assume that the rate of release of LH depends on CAC
through second-order Michaelis-Menten kinetics
|
(14)
|
We chose second-order kinetics so that, in the absence of GnRH, the low
CAC fluctuations about 0.1 µM would cause a very low baseline LH
release (15a). CAE = 1 mM is in the normal range of plasma
free Ca2+ concentration.
The balance between
k7 and
k9 was chosen so
that the rate of pumping out of the cell equals the rate of leakage
into the cell when CAC = 0.1 µM, its resting level. The magnitudes
were chosen so that CAC returns to its resting state within a few
minutes after the removal of GnRH. The magnitude of
k8 was chosen to
obtain the typically observed elevated CAC values after the initial
spike while GnRH is still present. In this initial study we ignore the facilitation and inhibition of the VGCCs by CAC, i.e.,
k88 = 0 = k888. We chose
k10 = 5 ng/min to
be consistent with the observed rates of LH release.
Figure 3A shows the time course of the
CAER concentration for 5-min pulses at low, medium, and high
concentrations of GnRH. When H = 0.1 nM, less than one-half of the
Ca2+ in the ER is released into
the cytoplasm, and the time course is slow. A 10-fold increase in the
GnRH concentration results in a marked increase in the rate of dumping
by the ER, then the CAER concentration remains constant for the
duration of the pulse. When H = 10 nM, the dumping is so fast that the
CAER concentration overshoots the equilibrium level. At the end of the
5-min pulse, the dynamics of CAC and CAER are very
complicated, because the IP3-sensitive channels are closing
(Fig. 2) and the membrane calcium channel is shutting down
as E decreases. When the GnRH of the pulse is low, the CAER begins to
recover immediately after the pulse is terminated; at medium and high
GnRH pulse levels, the recovery is preceded by a small further
extrusion of Ca2+ from the ER. The
detailed dynamic behavior is affected by the parameter choices and rate kinetics.
Figure 3B shows the CAC as a function
of time. When H = 0.1 nM, there is an initial rise from 0.1 to ~0.19
µM followed by a gradual decrease. At H = 1 nM, the initial spike
occurs more rapidly and goes to a much higher level, then attains a
plateau while GnRH is still present. At H = 10 nM, the initial rate,
the peak level, and the plateau level are somewhat higher, but the qualitative behavior remains the same. On termination of the pulse at 5 min, CAC returns to its resting level within ~5 min. In all three
cases, the LH production rate follows the CAC. As will be discussed in
detail below, this qualitative behavior is similar to that seen in
numerous experiments.
The mathematical model consists of Eqs. 4-11,
13, and 14. As we have
indicated, many simplifications are inherent in this model and several
known processes are ignored. Furthermore, except for Eq. 10, we have chosen first-order,
rather than Michaelis-Menten, kinetics. This has the advantage of
making the system as simple as possible and reducing the number of
parameters. Some of the parameters in the
Glossary were chosen by reference to,
e.g., measurements of cell volume and receptor number; the others were chosen to obtain model behavior qualitatively similar to experimental results. For convenience, we refer to the set of parameters in the
Glossary as standard parameters.
 |
COMPARISON WITH EXPERIMENTS |
Chang et al. (3) exposed gonadotropes to 10 nM GnRH for 2 min in the
presence and absence of extracellular
Ca2+. In Fig.
4 we show the rate of LH release predicted
by the model and CAC as a function of time in the presence (1,000 µM)
and absence (5 µM) of external
Ca2+. In the presence of
Ca2+, an initial rapid rise in LH
release rate is followed by a rapid decline, a brief pause, and then
decay to zero in ~5 min, closely resembling the experimental results
shown in Fig. 5 of Chang et al. (3). In the absence of
Ca2+, the peak level of LH release
is considerably lower and there is no pause, also as observed by Chang
et al. In both cases, the rate of LH release is closely correlated to
cytosolic Ca2+, as expected.

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Fig. 4.
Effect of external Ca2+. Time
variation of CAC and of rate of LH release is shown in response to a
2-min pulse of GnRH (10 nM) in presence
(A, CAE = 1,000 µM) and absence
(B, CAE = 5 µM) of external
Ca2+.
|
|
Experiments using 7-min pulses were performed by Stoljilkovic et al.
(38) at many different GnRH levels. Figure
5 shows the rate of LH release in the model
for a wide range of GnRH concentrations. For each GnRH concentration,
the rate increases rapidly, reaches a peak, declines rapidly to a
plateau level during the pulse, and then declines to resting level
within a few minutes. The rate of LH release tracks the CAC (see Fig.
8). The rate of initial rise, the peak height, and the plateau level
increase with increasing GnRH concentration. These results are very
similar to the graphs shown in Fig. 7 of Stojilkovic et al. (38).
However, the LH release by the model does not vary over as wide a GnRH
range, as in these experiments. In particular, the model graph at 500 nM GnRH (not shown) differs only slightly from the graph at 50 nM GnRH,
whereas Stojilkovic et al. found a small but noticeable difference.
Also, Stojilkovic et al. observed a very small release at 0.005 nM
GnRH, whereas the model has essentially no release. Thus the present
model does not reproduce their results at extremely high or extremely
low GnRH concentrations.

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Fig. 5.
LH release rate. Rate of LH release is shown in response to 7-min
pulses of GnRH at indicated concentrations.
|
|
Figure 6 shows the LH release peak and
plateau levels as a function of GnRH concentration on a logarithmic
scale. As in the experiments of Stojilkovic et al. (38), one obtains a
sigmoidal saturating curve, and the ratio of plateau level to peak
level at high GnRH concentrations is ~0.3, comparable to the
experimental observation. Figure 7 shows
the relationship between
LH and
CAC at the peak and at the
plateau level at the end of the pulse. As in the experimental data, the
close relationship between LH release and CAC is clear.

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Fig. 6.
Percent maximal LH release rate. Rates of LH release at peak and
plateau phases are shown as a function of GnRH concentration.
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Fig. 7.
Relation between change in LH release rate and change in cytosolic
Ca2+ level. LH release rate is
plotted as a function of CAC in peak and plateau phase for a 7-min
pulse of GnRH.
|
|
Numerous pulse experiments have compared LH release in the presence and
absence of external Ca2+. Figure
8 shows the model LH release rate and model
CAC as a function of time for normal external
Ca2+ (1,000 µM) and very low
external Ca2+ (5 µM). The only
difference between Fig. 8 and Fig. 4 is that the duration of the pulse
is 2 min in Fig. 4 and 7 min in Fig. 8; this allows the plateau to
develop. At low external Ca2+, the
LHR peak and the CAC peak are appreciably lower, and both concentrations decline slowly toward zero during the pulse, rather than
reach a plateau. These results are comparable to those shown in Fig. 5
of Stojilkovic et al. (38) and Fig. 5 (100 nM GnRH) of Iida et al.
(15a).

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Fig. 8.
Effect of external Ca2+. Time
variation of CAC and of rate of LH release is shown in response to a
2-min pulse of GnRH (10 nM) in presence
(A, CAE = 1,000 µM) and absence
(B, CAE = 5 µM) of external
Ca2+.
|
|
Recently, it has become possible to vary the number of GnRH receptors
on gonadotropes by using an antagonist,
NAcD2Na1-D4ClPhe-D3Pal-Ser-NMetyr-DLys-Leu-(isp)Lys-Pro-ALANH2 (PAL), that binds to but does not activate the receptor
and by creating pituitary cell lines that express different receptor concentrations. Pinter et al. (27) obtained data for LH release in
response to the continuous presence of 10 nM GnRH for up to 3 h at
various concentrations of PAL (i.e., reduced number of available
receptors). Because we model only short-time effects in this initial
study, we examined the effect of reduced receptor number on LH release
only for the first 15 min. Figure 9 shows that after ~5 min the total LH increases linearly with time, as expected. Decreasing the receptor number in the model decreases the
amount of LH produced, as seen experimentally. The percent reduction in
LH release is greater than the percent reduction in receptor number,
because the rate of dimerization (a 2nd-order reaction) is very
sensitive to receptor concentration.

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Fig. 9.
Effect of receptor number. Total LH release is shown as a function of
time during exposure to 10 nM GnRH at indicated percentage of normal
receptor number.
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|
 |
DISCUSSION |
Although we have ignored much of the complexity of the signaling system
between the binding of GnRH to its receptors and release of LH (and
FSH) from gonadotropes, the present model nevertheless captures many
of the significant short-term features of this system. The shape and
time course of LH release in response to GnRH pulses in the model are
very similar to those seen experimentally, in the presence and absence
of external Ca2+. Furthermore, the
behavior of the LH release varies as significant parameters (such as
GnRH concentration and receptor number) are changed in ways that are
quite similar to experimental observations. The present model
allows one to predict, for the first time, the effects of varying
concentrations, e.g., external
Ca2+ concentration (Fig. 4) or
receptor number (Fig. 9), and to compare the predictions with
experimental data. When more information becomes available about the
kinetic parameters and mechanisms of the internal signaling system,
the model can be used to understand the quantitative consequences of
its structural organization.
Among the important short-term mechanisms that we have ignored are
1) facilitatory and inhibitory
mechanisms for Ca2+ entry, as a
result of setting
k666,
k88, and
k888 equal to
zero, 2) the roles of diacylglycerol
and arachidonic acid in LH release (3),
3) the roles of multiple G proteins
(32), 4) the effect of protein
kinase C on Ca2+ channels and on
the unmasking of cryptic receptors (31),
5) the mechanisms of degradation of
IP3 and/or its conversion to other
inositol phosphates that may play a role in the system, 6) receptor endocytosis and
recycling, and 7) two pools of LH. Thus the present model demonstrates that the major features of the
short-term responses can be understood using only the mechanisms that
are included in the model, i.e., binding to the receptor, dimerization,
interaction of the dimerized receptor with a G protein, production of
an effector that opens the VSCC in the cell membrane and catalyzes the
formation of IP3 (which opens the
Ca2+ channels in the ER), and the
Ca2+-dependent release of LH.
Because we have used such a simple model for the
IP3-ER interactions, our model
does not exhibit the rapid Ca2+
oscillations seen experimentally in certain GnRH concentration ranges.
An important implication is that the
Ca2+ oscillations are not
necessary to obtain the LH release profiles over periods of >2 min.
Although the Ca2+ oscillations are
presumably of great importance to avoid damage to the cell, it is their
time average, captured in this model, that governs LH release.
Korngreen et al. (21) proposed a somewhat similar but less detailed
model for biphasic Ca2+ response
to signaling.
We plan to use this mathematical and computational model to perform
many investigations of short-term behavior. For example, it is known
that the time course of CAC in response to endothelin is similar to the
response to GnRH with some differences (36, 38). Aside from the
obvious changes in
k1 and
k
1, it would be interesting to know whether small changes in subsequent rate constants [e.g., the dimerization or the rate of reaction of the dimerized receptor with the G protein(s)] would explain the observed differences. The model can be easily expanded to include
two pools of releasable hormones so that the differences between LH
release and FSH release can be compared.
We also plan to expand the model to include mechanisms that affect
long-term behavior, such as 1)
endocytosis of the dimers and recycling of the receptors (8),
2) synthesis of new receptors and
more LH (and FSH) by the nucleus (9, 19), and
3) conversion of a nonreleasable
pool of LH to a releasable pool (16). This will enable us investigate
adaptation in multipulse experiments and in response to continuous
long-term exposure to GnRH and endothelin.
 |
ACKNOWLEDGEMENTS |
We are grateful to John Davies for software support and to Kayne
Arthurs for preparation of the figures.
 |
FOOTNOTES |
This research has been supported by National Science Foundation Grant
DMS-9709608 and National Institute of Child Health and Human
Development Grant HD-19899.
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. J. Blum,
Dept. of Cell Biology, Duke University Medical Center, Durham, NC 27710 (E-mail: j.blum{at}cellbio.duke.edu).
Received 24 May 1999; accepted in final form 27 September 1999.
 |
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