A mathematical model for LH release in response to continuous and pulsatile exposure of gonadotrophs to GnRH
© Washington et al; licensee BioMed Central Ltd. 2004
Received: 14 June 2004
Accepted: 24 September 2004
Published: 24 September 2004
In a previous study, a model was developed to investigate the release of luteinizing hormone (LH) from pituitary cells in response to a short pulse of gonadotropin-releasing hormone (GnRH). The model included: binding of GnRH to its receptor (R), dimerization and internalization of the hormone receptor complex, interaction with a G protein, production of inositol 1,4,5-trisphosphate (IP3), release of calcium from the endoplasmic reticulum (ER), entrance of calcium into the cytosol via voltage gated membrane channels, pumping of calcium out of the cytosol via membrane and ER pumps, and release of LH. The extended model, presented in this paper, also includes the following physiologically important phenomena: desensitization of calcium channels; internalization of the dimerized receptors and recycling of some of the internalized receptors; an increase in G q concentration near the plasma membrane in response to receptor dimerization; and basal rates of synthesis and degradation of the receptors. With suitable choices of the parameters, good agreement with a variety of experimental data of the LH release pattern in response to pulses of various durations, repetition rates, and concentrations of GnRH were obtained. The mathematical model allows us to assess the effects of internalization and desensitization on the shapes and time courses of LH response curves.
Gonadotropin-releasing hormone (GnRH) is released by the hypothalamus in a pulsatile fashion and stimulates luteinizing hormone (LH) and follicle stimulating hormone (FSH) release by pituitary cells by a complex series of signaling processes. Although there is substantial information about various individual steps in the signaling system, there is less understanding of how these components interact to give rise to the overall behavior of the system. The frequency of pulses varies throughout the menstrual cycle increasing markedly just prior to ovulation. And, it has been observed in in vitro experiments on perifused pituitary cells that pulse frequency and concentration have marked (nonlinear) influences on the release of LH and FSH. The purpose of our work is to use mathematics and machine computation to understand the dynamics of this important and interesting physiological system.
In a prior study, , a mathematical model was developed to investigate the rate of release of luteinizing hormone from pituitary gonadotrophs in response to short pulses of gonadotropin-releasing hormone. The model included binding of the hormone to its receptor, dimerization, interaction with a G-protein, production of inositoltrisphosphate (IP3), release of calcium from the endoplasmic reticulum (ER), entrance of calcium into the cytosol via voltage gated membrane channels, pumping of calcium out of the cytosol via membrane and ER pumps, and the release of luteinizing hormone (LH). Cytosolic calcium dynamics were simplified and it was assumed that there is only one pool of releasable LH. Despite these and other simplifications, the model results matched experimental curves and enabled us to understand the reasons for the qualitative features of the LH release curves in response to GnRH pulses of short durations and different concentrations both in the presence and absence of external calcium. We note that Heinze et al, , created a mathematical model for LH release that reproduces some data for pulsatile administration of GnRH. Their model, however, does not include most of the important intracellular mechanisms known to play important roles; thus, they match data but do not study mechanisms. We also note that mathematical models for other aspects of the reproductive hormone system have been created: Keenan et al, , developed a stochastic systems model for the interactions between GnRH, LH, and testosterone; Gordan et al,  modelled the pulsatile release of GnRH by hypothalamic neurons.
There are four important medium-term effects that were not included in the previous study. Desensitization of the response to GnRH occurs because after GnRH binds to its receptors, some of the bound complexes are internalized and partially degraded . Secondly, prolonged exposure to GnRH desensitizes the outer membrane calcium ion channels, as described in detail by Stojilkovic et al . Thirdly, there exist basal rates of receptor synthesis and degradation. Finally, in response to GnRH, there also occurs an increase in the number of Gq/11proteins closely associated with the plasma membrane . Incorporation of these four phenomena into the previous model allows us to analyze the contrasting effects of desensitization and signal amplification during medium-term continuous and pulsatile exposures to GnRH. We then show that the LH response curves of the enlarged model capture most of the essential features of a large number of experimental studies.
It should be noted that in the present model we ignore the long-term effects that result in changes in DNA, messenger RNA, and protein concentrations (e.g., receptor number) that are known to occur several hours after exposure to GnRH [8–11]. Thus, in the present study, we limit the time of exposure to three hours. We also ignore the long term effects of diacylglycerol which is known to cause an increase in the synthesis of LH α , the α subunit of the LH dimer .
Let H(t) represent the GnRH concentration (nM) in the surrounding medium t minutes after the initiation of the experiment. Initially, the hormone is bound by the receptor, R.
The bound complex HR reacts with itself to form dimers , denoted by HRRH.
A Gq/11protein, denoted GQ, reacts with the dimer to produce an effector, E (e.g., phospholipase C, ).
Glossary of Variables
GnRH concentration (nM)
Free GnRH receptor concentration (nM)
Hormone-receptor complex concentration (nM)
Hormone-receptor dimer concentration (nM)
Gq/11protein concentration (nM)
Effector concentration (nM)
Inositol 1,4,5-trisphosphate concentration (nM)
Cytosolic Ca2+ concentration (μ M)
ER Ca2+ concentration (μ M)
Fraction of open ER Ca2+ channels
LH concentration (ng)
Total receptor concentration (nM)
Total Gq/11protein concentration (nM)
Resting Ca2+ concentration in ER (normally 40 μ M)
External Ca2+ concentration (normally 1000 μ M)
= 2 nM-1, see equation (17)
= 4 min-1, see equation (17)
= 0.02 min-1, see equation (12)
= 0.002 min-1, see equation (12)
= 0.6, fraction internalized receptors returned
= 8.3 × 10-6 nM·min-1, basal rate of receptor synthesis
= 8.3 × 10-4 min-1, basal rate of receptor degradation
= 2.5 nM-1·min-1
= 5 min-1
= 2500 nM-1·min-1
= 5 min-1
= 4000 nM-1·min-1
= 200 min-1
= 2 × 107 min-1
= 10 min-1
= 1 nM-1·min-1
= 10 nM-1·min-1
= 5 min-1
= 2.2 μ M·min-1
= 0.4 nM-1·min-1
= 0.0002 min-1
= 5 ng·min-1
= 0.0008 nM-1·min-1
= 2.7 min-1
The monomers, HR, can also interact with each other to form larger aggregates . Macroaggregation and internalization occur at least 20 minutes after exposure to GnRH . All of the internalized hormone and some of the receptors are then degraded, and the receptors that are not degraded are returned to the membrane [15, 16]. We assume that a fraction of receptors, r0, can be returned intact to the membrane after a time delay of 20 minutes. Consistent with the data of , we assume that r0 = 0.6. Since we are not concerned with the details of the internalization or return processes, we adopt simple first order reactions for these processes. We assume that n monomers, HR, are internalized at a rate k11 and that r0n monomers that have been internalized are available to be returned to the membrane at rate k11.
There is evidence that the macroaggregates consist of an average of n = 100 monomers . In our model, we will choose k11 = 0.08/n = 0.0008 nM-1·min-1. With this choice, 7% of the receptors are internalized after a 5 minute pulse of 1 nM GnRH, and 60 minutes after the initial exposure, approximately half of the internalized receptors have returned. It should be noted that it is only the combination k11n that occurs in the equations.
We make the following simple assumption about the recyling of receptors (consistent with the data of Maya-Nunez et al.  and Table 2 of Conn et al. ). i.e. that the formation of macroaggregates begins 20 minutes after exposure to GnRH and that the internalization and recycling process takes 20 minutes after the formation of the macroaggregates. Let χ(t) be the function that equals 1 for t ≥ 0 and equals 0 for t < 0. Then, at time t, the rate of internalization of receptors is k11n[HR](t) and the rate of return of receptors to the membrane is k11n[HR](t - 40)χ(t - 40). To simplify notation, we write [HR]40 = [HR](t - 40)χ(t - 40).
Since only 60% percent of the internalized receptors are returned to the membrane after exposure to GnRH, there would not be a full recovery of receptors in the membrane. In the model we therefore include a low basal rate of receptor synthesis, P0 = 8.3 × 10-6 nM·min-1, and degradation, γ = 8.3 × 10-4 min-1. The ratio is chosen so that the resting (in the absence of hormone) receptor concentration is R0 = 10-2 nM, and the magnitude of P0 is chosen so that approximately of the resting amount of receptor is produced per hour, thus ensuring a slow recovery to the steady state receptor concentration in the absence of GnRH.
The number of membrane associated GQ proteins increases in response to a GnRH agonist as described by Cornea et al . For simplicity we assume that the increase of GQ proteins near the membrane depends on the concentration of HRRH in the membrane. The kinetic coefficient k33 is the parameter that determines the rate of increased concentration of GQ at the membrane in response to the formation of HRRH. We are assuming a finite pool of GQ that can be transported from the cytoplasm to the immediate vicinity of the plasma membrane. This pool is assumed to be regulated by the amount of HRRH for only the first 20 minutes, and after this time the rate of increase is negligible . To fit the experimental data, we choose k33 = 2.7 min-1 and multiply the kinetic coefficient k33 by e-t/20. With these parameters, 60 minutes after a constant exposure to 1 nM GnRH, there is a 40% increase of GQ concentration near the membrane and 120 minutes after exposure to the hormone, there is only a 43% increase. The following differential equations reflect the physiological assumptions that we have so far discussed.
We further assume that the production of IP3 is proportional to the concentration of E and that it is converted to inactive metabolites at a rate proportional to its concentration.
As in , the fraction of open channels in the ER, denoted by CHO, depends on IP3 concentration. CHO reaches its maximum 0.25 min after exposure to GnRH and the maximum value of CHO is 0.6. To incorporate multiple pulses, we modify the function CHO from the previous model so that it reaches its maximum 0.25 min after the start of each pulse. Thus we have
where t p is the time after the start of each individual pulse and, as in ,, α = 2 nM-1 and β = 4 min-1.
In response to GnRH, calcium is released from the ER into the cytoplasm with a rate constant ERR and is pumped back into the ER. As discussed in the previous model, the rate constant ERR increases proportionally to cytosolic calcium concentration, CAC, with a rate constant k66 and is inhibited at high CAC at a rate that is proportional to the square of CAC, with a rate constant k666. Just as in, , k6 = 1, k66 = 10, and k666 = 0, i.e., we ignore the inhibitory effects of calcium on reuptake of calcium into the ER.
ERR = k6 + k66[CAC] - k666[CAC]2 (8)
The change in cytosolic calcium concentration, CAER, is then determined by the rate constant ERR, which is the rate of extrusion, multiplied by the fraction of open channels, CHO, and the difference in concentration between the calcium concentration in the cytoplasm and the endoplasmic reticulum. As in Blum et al. , calcium is actively transported back into the ER by pumps with the rate constant k-6 = 5 min-1.
As in the previous model, the volume of the ER is assumed to be 1/20 of the volume of the cytosol. CAC is determined by the rate of calcium efflux through ion channels in the ER membrane minus the rate at which calcium is being pumped back into the ER, plus the rate of calcium entry from the plasma membrane. The function VSR denotes the rate of calcium influx from extracellular calcium into the cytosol and depends on E with rate constant k8  and on CAC with rate constants k88 for the influx at low CAC and k888 for the inhibitory effects at high CAC. There is considerable evidence that desensitization occurs, i.e., the fraction of open calcium channels in the cell membrane decreases soon after exposure to GnRH . Since the precise mechanism of desensitization in unknown, we assume that VSR depends on E and CAC, and that channels slowly become inactive in response to exposure to GnRH, consistent with the experimental data . We further assume that the fraction of open calcium channels in the outer membrane, denoted by VSRO(t), decreases at a linear rate of v1 = 0.02 min-1 when the hormone is applied and has a minimum value of 0. In the absence of hormone, the fraction of open channels increases at a linear rate of v2 = 0.002 min-1 and has a maximum value of 1. Thus, immediately a five minute pulse of 5 nM GnRH, 10% of the channels are in the refractory state and 50 minutes after the removal of the GnRH, all of the channels have recovered, consistent with experimental data; see  for more details. Incorporating calcium influx, pumps and leakage into the cytoplasm from the medium (the term k9 [CAE], we have
VSR(t) = (k8E(t) + k88[CAC](t) - k888([CAC])(t))2) × VSRO(t) (11)
and VSRO satisfies the following.
0 ≤ VSRO(t) ≤ 1 (13)
Finally, the rate of release of LH depends on cytosolic calcium concentration (see Blum et al.  for details). Although there is evidence that there are three pools of LH in gonadotrophs, one pool, comprising of only 2% of the total LH, is released within one minute after exposure to GnRH, and the third pool is not released during continuous exposure to GnRH (Naor et al.,). Therefore, as in the previous model , we treat LH as being released from a single pool.
The mathematical model consists of equations (1) – (14). These non-linear equations cannot be solved analytically but solutions can be obtained by machine computation. To do this, we used the solver ODE45 in Matlab.
The values of the rate constants are given in Table 2. The values for many of them are discussed in detail, with references, in our original paper, . The values of the rate constants for the signalling mechanisms introduced in this paper were discussed (above) as the mechanisms were introduced. In some cases the rate constants were taken from experimental data (references given) and in other cases, where direct experimental data does not yet exist, we explained the rationale for our choices. Since the resulting model captures and explains many experimental studies (see below), these choices provide useful predictions for future experimental studies.
Figures 3A and 3C show the concentrations of free receptors and receptors bound to the hormone. It can be seen that, initially in both the present and previous models, there is a very rapid decline in free receptors, R, and a very rapid increase of receptors to which GnRH has bound (HR) but have not yet dimerized. This is immediately followed, as shown in Figure 3D, by the formation of the dimers (HRRH). After this initial reaction, the concentrations of HR and HRRH remain constant in the previous model, but decline in the present model due to internalization and degradation. The recycling of receptors was assumed to start at 40 minutes (see equation (1)), which is why the rates of decrease of HR and HRRH decline at that time. Because of degradation, only a fraction (r0 = 0.6) of the internalized receptors are returned to the membrane. Thus, in the presence of continuous exposure to GnRH, the total number of receptors in the membrane continues to decline as shown in Figure 3B. The rate of change of IP3 (Fig. 3E) is closely related to the rate of change of HRRH as shown in Fig. 3D. Finally, Fig. 3F shows that there is a slow increase of approximately 43% of the concentration GQ associated with the membrane during the exposure.
The previous model (Blum et al, ) was intended to explain the short term response of gonadotrophs to GnRH. The success of the previous model in the first few minutes is not visible in Figures 1, 2, 3, and 6 because the long time scale compresses the first five minutes. The present model, which includes the four important medium-term processes discussed in the Introduction, now enables us to study the effects of these intracellular processes on medium-term responses, including the responses to pulses of GnRH. From now on, when we refer to the "model", we mean the present expanded model.
As shown in Figure 7B, the LH release rate decreases appreciably after the first pulse, and then continues to decrease slowly with each subsequent pulse. This arises (see equation (14)) because of the decline in the size of the cytosolic calcium pulse after each GnRH pulse as shown in Figure 7C. The ER is able to refill its calcium store to almost the same level as the preceding pulse, although the amount remaining in the ER after each pulse decreases appreciably (Figure 7D). Notice that the fraction of open channels in the outer membrane (Figure 7F) declines dramatically, while the fraction of open ER channels declines only slightly with each pulse (Figure 7E). This suggests that the primary cause of decline in the amount LH release with each GnRH pulse is the desensitization of the outer membrane. We examine this hypothesis further below.
To understand why the number of open ER channels does not decrease markedly from pulse to pulse, we refer to Figure 8. Note that the total number of receptors (Figure 8B) declines steadily by approximately 1/3 in the course of the experiment as does the number of free receptors (Figure 8A). The decline in the HRRH peaks is much greater (approximtely 40%, Figure 8D) because the formation of these dimers depends on the square of [HR]. However, the decline in the effector, E, which leads to the formation of IP3 (see equation (6)) is only 25% (data not shown) because of the substantial, rapid rise in GQ (Figure 8F) in response to the first pulse of GnRH. Thus, the IP3 peaks decline only about 25% (Figure 8E). Because of the Michaelis-Menten kinetics of the interaction between IP3 and the ER channels, there is an even smaller change in the fraction of open ER channels (CHO) in response to each GnRH pulse. This explains why the internalization and degradation of receptors does not have a more profound effect.
The decline is much smaller for pulse period of 30 minutes (Panel B). For a pulse period of 1 hour, the same amount of LH is released in response to each pulse for each GnRH concentration (Panel C). In vivo, one would not expect desensitization, so this result is consistent with experimental observations that LH pulses of the same magnitude occur approximately once an hour except just prior to ovulation (Kaiser et al,). Note also that at the medium concentration of 1nM there is less desensitization at both period 15 and period 30 minutes than at the high concentration. These results are consistent with the experimental results seen by Hawes et al,  (our Figure 4) and Baird et al.,  (our Figure 5), and Janovick & Conn, [, see their Figures 1,2,3,4].
In all of our previous simulations, except those in Figure 13 where we compared the two mechanisms for desensitization of LH release, the parameters in the model were never varied. We now discuss two situations where the modification of parameters gives good fits to the data and possibly new insights.
A similar result can be acheived by introducing desensitization of both the outer membrane and ER calcium channels instead of changing the internalization and recycling of the receptors. The parameters for the desensitization of the calcium channels in the outer membrane were increased from 0.02 min-1 to 0.4 min-1. This resulted in approximately 70% decrease in the magnitude of response to the second pulse of ET, but further increase in v1 did not cause any further reduction in magnitude. Since there is evidence suggesting that the calcium channels in the ER desensitize in response to GnRH (Conn et al ), we introduced this desensitization into the model to see if ER desensitization might also be occuring in response to ET. For simplicity, the rates of desensitization and of recovery of the ER calcium channels were chosen to be identical to that of the desensitization of the outer calcium channels. By including desensitization of both the outer membrane and ER calcium channels, the amount of LH released in response to the second pulse of ET was as small as was observed experimentally (data not shown). Thus, our current model, with few parameter changes, appears capable of explaining the responses to endothelin. However, in the absence of more detailed experimental data (for example responses to pulses of different durations and frequencies, etc.) we cannot at present distinguish between the two above proposed mechanisms.
We have extended our previous model to include receptor internalization and partial degradation, outer membrane calcium channel desensitization, basal levels of receptor synthesis and destruction, and an increase in the number of Gq/11proteins closely associated with the plasma membrane. With these additions we are now able to examine the behavior of the model system over medium term (up to three hours) exposures to GnRH and to a variety of pulsatile exposures. We have compared the model behavior to many such different experiments and found that it shows the essential response properties of the gonadotrophs. Furthermore, since the model includes many of the intracellualar physiological processes, we have used the model to investigate and understand the mechanisms that give rise to the various experimental results.
We used parameter variation to investigate whether receptor internalization or outer membrane calcium channel desensitization plays the major role in LH release desensitization and concluded that outer membrane calcium channel desensitization is more important, at least in the experiments of Janovick and Conn . We also used parameter variation to show that changing two parameters (the rate of recovery of the outer membrane channels and the rate of receptor internalization) the model gives good matches to the data of Stoljilkovic et al , on LH responses to pulses of endothelin. This strongly suggests that the same intracellular mechanisms are primarily responsible for the LH responses to GnRH and endothelin.
It is important to note that the model ignores a number of processes that play a role in the long-term response to GnRH. In gonadotrophs, depending on the frequency and duration of exposure to pulses of GnRH, there may be an increase or decrease in the number of receptors in the cell membrane due to changes in gene expression and/or mRNA translation [28, 9, 8, 29, 30]. These long-term effects are not important for the current study but will be included in future work. It is also known that there is activation of protein kinase C in gonadotrophs exposed to GnRH , but while PKC may not be involved in GnRH-mediated LH release , PKC may have other roles in the pituitary, such as to modulate gonadotroph responsiveness to GnRH . Another aspect that our model ignores is the rapid calcium concentration oscillations in the cytosol. As shown by Stojilkovic and Tomic , the frequency of the oscillations affect the LH release. In the present model, as in the previous model , for simplicity we have assumed that the average cytosolic calcium concentration is an adequate approximation to the rapid oscillatory responses. Finally, we note (Stanislaus et al, ) that there is evidence that the GnRH receptor interacts with more than one G protein and Stanislaus et al,, have proposed that this underlies the differential regulation of the release of luteinizing hormone and follicle stimulating hormone. We plan to address these questions in future work.
This research was supported by National Science Foundation grant DMS-0109872 and National Institues of Health grant HD19899. We are grateful to Dr. Paula Budu for helping us prepare some of the figues.
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