During the terminal part of their development,
ovarian follicles become totally dependent on gonadotropin supply to
pursue their growth and maturation. Both gonadotropins,
follicle-stimulating hormone (FSH) and luteining hormone (LH), operate
mainly through stimulatory G protein-coupled receptors, their signal
being transduced by the activation of the enzyme adenylyl cyclase and
the production of second-messenger cAMP. In this paper, we develop a
mathematical model of the dynamics of the coupling between FSH receptor
stimulation and cAMP synthesis. This model takes the form of a set of
nonlinear, ordinary differential equations that describe the changes in
the different states of FSH receptors (free, bound, phosphorylated, and
internalized), coupling efficiency (activated adenylyl cyclase), and
cAMP response. Classical analysis shows that, in the case of constant
FSH signal input, the system converges to a unique, stable equilibrium
state, whose properties are here investigated. The system also appears
to be robust to nonconstant input. Particular attention is given to the
influence of biologically relevant parameters on cAMP dynamics.
signal transduction; granulosa cells; follicle-stimulating hormone; cyclic adenosine monophosphate
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INTRODUCTION |
FOLLICULOGENESIS IS THE
PROCESS of growth and functional maturation undergone by ovarian
follicles, from the time they leave the pool of primordial (quiescent)
follicles until ovulation, at which point they release a fertilizable
oocyte. Most of the developing follicles never reach the ovulatory
stage but degenerate by a process known as atresia (12).
The gonadotropic hormones follicle-stimulating hormone (FSH) and
luteinizing hormone (LH) play a major role in the regulation of
terminal follicular development through the control of proliferation
and differentiation of the granulosa cells surrounding the oocyte
(29). Gonadotropin secretion is, in turn, modulated by
granulosa cell products such as estradiol and inhibin. During the
follicular phase of the ovarian cycle, negative feedback is responsible
for reducing FSH secretion, leading to the degeneration of all but
those follicles selected for ovulation. Finally, positive feedback is
responsible for triggering the LH ovulatory surge leading to ovulation
(10). Both FSH and LH operate mainly through G
protein-coupled transmembrane receptors, transducing their signal by
activation of the enzyme adenylyl cyclase and production of
second-messenger cyclic adenosine monophosphate (cAMP)
(30).
In previous studies (6, 7), we investigated the divergent
commitment of granulosa cells toward proliferation, differentiation, or
apoptosis in response to their hormonal environment. Under cumulative exposure to gonadotropins, granulosa cells progressively lose their ability to proliferate and acquire a fully differentiated state. The accumulation of intracellular cAMP beyond a threshold seems
to be a key point in cell cycle arrest (26), because it is
believed to lead to the activation of cyclin kinase inhibitors (11). We thus believe that a better understanding of
gonadotropin-induced cAMP production will help gain insight into
changes in the rate of differentiation among granulosa cells during
terminal development (8).
In CONSTRUCTION OF A SIGNAL TRANSDUCTION MODEL, we describe
the mathematical model after stating the biological assumptions on
which it is based; in STABILITY ANALYSIS FOR CONSTANT FSH
INPUT, we focus on the analysis of the model in the case of a
constant FSH level; CONTROL OF FSH-INDUCED CAMP LEVELS is
devoted to the numerical application of the model; the physiological
implications of these results are discussed in DISCUSSION;
and mathematical details are given in APPENDIX.
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CONSTRUCTION OF A SIGNAL TRANSDUCTION MODEL |
Physiological Background: Transduction of the Gonadotropic Signal
in Granulosa Cells
Terminal follicular development is strictly dependent on FSH
supply. Before the selection of the follicle for ovulation, granulosa cells are responsive only to FSH. As follicular maturation progresses, the coupling between FSH receptor stimulation and the activation of
adenylyl cyclase becomes more and more efficient, leading to a steady
increase in cAMP production (13). The accumulation of
FSH-induced cAMP coincides with the appearance of and subsequent dramatic increase in LH receptors, allowing LH to act as a surrogate for FSH in granulosa cells (36). Conversely, when
gonadotropin, and especially FSH, plasma levels are too low to meet the
follicle's trophic requirements, uncoupling of receptor stimulation
with cAMP production is one of the earliest events occurring during granulosa cell death and follicular atresia (15).
The binding of FSH to its transmembrane receptors triggers an
intracellular signal via the heterotrimeric G proteins. The FSH-bound
receptor activates the G
-stimulatory (Gs)
subunit, which interacts with adenylyl cyclase to generate
an increase in cyclic AMP. Once cAMP is synthesized, it either binds
and activates specific protein kinases such as protein kinase A or is
degraded by cyclic nucleotide phosphodiesterase (PDE)
(30).
The control of cAMP levels in granulosa cells involves both fast
biochemical processes, such as binding and desensitization, occurring
on a time scale of a few minutes, and slower physiological processes
lasting hours or even a few days, which result mainly in changing the
efficiency of the enhancement of cAMP synthesis by stimulated FSH
receptors via adenylyl cyclase activation. The increase in this
coupling efficiency is a progressive, hormonally regulated process
(29), so that the degree of maturation of a follicle can
be characterized by the average cAMP level in its granulosa cells. The
design of our model follows from the interactions between these
contrasting biochemical and physiological dynamics. From here onward,
we will focus on the dynamics of intracellular cAMP in an average
granulosa cell from the time the follicle becomes able to respond to
FSH in term of cAMP production.
Biological Assumptions
The model is based on the following assumptions, which are
supported by the available biological knowledge on FSH signal
transduction in granulosa cells during the first part of terminal
follicular development.
Binding of FSH to its receptor (RFSH) results in
the formation of an active complex (XFSH)
Bound receptors activate adenylyl cyclase (E) through a
conformational change in the associated G protein
Activated adenylyl cyclase (EFSH)
synthesizes cAMP from the substrate Mg2+ ATP
cAMP is hydrolyzed into AMP by PDE
Bound receptors are subjected to a desensitization process through
cAMP-mediated phosphorylation
Phosphorylated inactive complexes (XpFSH)
undergo internalization into the cell, where receptors are dissociated
from FSH
Internalized receptors (Ri) are recycled
back to the cell membrane, whereas FSH is hydrolyzed
Consideration of only those reactions relevant to follicular
development allows some simplifications to be made. Reactions generating short-lived intermediary species are neglected. In particular, the cycle of G protein activation/deactivation is not
modeled explicitly. The process of receptor synthesis is assumed to
compensate both for intracellular receptor degradation and for the
depletion of the receptor pool during cell division, so that the total
number of FSH receptors in different states (free, active,
phosphorylated, and internalized) remains constant (4), leading to the following cellular cycle for FSH receptors under different states
Finally, cAMP-independent desensitization is not taken into
consideration, because its behavior during the maturation of granulosa
cells is not yet known. In addition, the amount of FSH is assumed to be
sufficiently large that its concentration is unaffected by binding to receptors.
Model Equations
Let RFSH, XFSH,
XpFSH, and Ri be,
respectively, the concentrations of free, bound active, bound inactive
(phosphorylated), and internalized FSH receptors (italics indicate
concentrations). Let EFSH be the
concentration of activated adenylyl cyclase, and let cAMP be
the concentration of intracellular cAMP. Let k+, k
, ki, and
kr be the rate constants for FSH binding, FSH unbinding, bound complex internalization, and receptor recycling to the
cell membrane, respectively. The function 
describes the (cAMP-dependent) rate of receptor desensitization. The rates of change
of the concentrations are given by the following ordinary differential
equations
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(1)
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(2)
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![<FR><NU>d<IT>E</IT><SUB>FSH</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=&bgr;</IT>[<IT>&sfgr;X</IT><SUB>FSH</SUB><IT>−E</IT><SUB>FSH</SUB>]<IT>E</IT><SUB>FSH</SUB>](/content/vol281/issue1/fulltext/E35/img011.gif)
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(3)
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(4)
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(5)
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(6)
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Equations 1, 4, and 6 result from applying
the principle of mass action. In Eq. 4, ATP is treated as a
nonlimiting substrate at a constant concentration, and its effect is
included in the kinetic constant 
. In the same way, the
concentration of the enzyme PDE is included in
kPDE.
The desensitization rate in Eqs. 2 and 5 is a
Hill function of intracellular cAMP
with saturation value (), half-saturating cAMP concentration
(
), and slope of the increase in (
) as real parameters.
This sigmoidal dependence accounts in a compact way for the
phosphorylation cascade occurring downstream of cAMP, including transmembrane receptors as phosphorylation targets. Thus
phosphorylation in the model is assumed to be cAMP mediated in a
dose-dependent, increasing, and saturated manner. On qualitative
grounds, this choice was substantiated by the critical importance of
cAMP-dependent postreceptor events for desensitization
(21). On quantitative grounds, the Hill function allows
either for a progressive effect of cAMP level or for a rather
all-or-nothing effect, depending on the value of the slope parameter
. Besides, the phosphorylation rate is assumed to be bounded by the
saturation value , which reflects the limits in the phosphorylation
capacity resulting from the balance between phosphorylation through
kinases and dephosphorylation through phosphatases.
Equation 3 governs the change in the coupling variable
EFSH and is designed to be understood from a
physiological rather than a biochemical viewpoint. 
acts as a time
scale parameter. Whenever it takes a low value ( 
1), the changes
in the coupling variable EFSH are slower than
those of the other variables of the model. The amplification parameter
measures the degree of signal amplification and represents the
average number of adenylyl cyclase molecules activated by one bound
receptor at steady state.
The choice for the right-hand term of Eq. 3 is subject to
the following physiological constraints, which make it specific to
granulosa cells: 1) there is a basal concentration of
activated adenylyl cyclase (E0 > 0), due
to minor constitutive activity of G proteins; 2) under
cumulative exposure to FSH, the capacity for cAMP production in
response to FSH stimulation increases during terminal follicular
development (EFSH is an increasing function as
long as XFSH > EFSH); 3) the increase in the
efficiency of coupling is correlated with an increase in the
follicle's vulnerability toward FSH supply (as soon as
XFSH < EFSH,
EFSH starts decreasing); 4) coupling
and uncoupling are autoamplified processes, due to paracrine and
autocrine mechanisms enhancing the follicular sensitivity to FSH (right
EFSH term of amplification).
Model Reduction
The total number of receptors remains constant; hence, Eqs.
1, 2, 5, and 6 are subject to the conservation law
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(7)
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where RT is the constant size of the global receptor
pool. We can thus replace Ri in Eq. 1
by RT 
(RFSH + XFSH + XpFSH), reducing the system to five equations
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(8)
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(9)
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![<FR><NU>d<IT>E</IT><SUB>FSH</SUB></NU><DE>d<IT>t</IT></DE></FR><IT>=&bgr;</IT>[<IT>&sfgr;X</IT><SUB>FSH</SUB><IT>−E</IT><SUB>FSH</SUB>]<IT>E</IT><SUB>FSH</SUB>](/content/vol281/issue1/fulltext/E35/img020.gif)
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(10)
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(11)
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(12)
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Initial values of the variables will be denoted, respectively, as
R0, X0,
E0, cAMP0, and
Xp0.
Boundedness
A basic requirement for a physiological model to be plausible is
that solutions should remain bounded for all time and that concentrations should remain nonnegative. It is easy to verify (for
details see APPENDIX, Upper Bounds of Solutions)
that, as long as kPDE > 0, this is the
case in the above model for constant FSH input, so that, for any
> 0, there exists a T 
0, such that for all
t T
If kPDE = 0, there is no mechanism for
removing cAMP from the system; hence, cAMP concentrations can grow
without bound. This is obviously not a physiologically realistic case.
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STABILITY ANALYSIS FOR CONSTANT FSH INPUT |
Quasi-Steady-State Model and Steady States
When 1, the changes in RFSH,
XFSH, cAMP and
XpFSH can be considered fast compared with the
change in EFSH. Applying a quasi-steady-state
approximation to Eqs. 9, 11, and 12 in case of constant FSH input leads respectively to the following relations
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(13)
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(14)
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(15)
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Substituting this into Eq. 8, we obtain
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(16)
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where
Hence, the quasi-steady-state assumption defines a
quasi-steady-state model reducing system 8-12 to a
one-variable, nonlinear differential equation
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(17)
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Figure 1 illustrates the
degree of discrepancy, as far as the changes in
EFSH are concerned, between the complete and the reduced models.
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Fig. 1.
Comparison between approximate and exact solution of activated
adenylyl cyclase concentration (EFSH). Solid
lines correspond to the solutions obtained from Eq. 10 in
the complete model; dotted lines represent the solutions obtained from
Eq. 17, assuming quasi-steady state on the other variables.
From left to right, the 3 pairs of curves are
respectively associated with time scale parameter () values of 0.1, 0.01, and 0.005. Time unit is 102 s, and
E0 = 0.1 × 104
molecules/cell. Other parameter values are displayed in Table 2. The
amplitude of the discrepancy between the reduced and complete models
increases as value increases, whereas the length of the transient
period increases with decreasing value.
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Although the transient behavior of EFSH is
under the control of , its steady-state
E
is not. The steady state corresponding
to E0 > 0 is characterized by
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(18)
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with X a solution of
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(19)
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By use of simple geometric reasoning, it is possible to prove that
Eq. 19 always admits a unique, positive real root (see APPENDIX, Existence and Uniqueness of Strictly
Positive Roots of Eq. 19). This root defines the unique
equilibrium state of the system. The level of intracellular cAMP at
this steady state is an increasing function of the following
parameters: FSH input, size of receptor pool
RT and constants k+,
ki kr, and . Conversely it is a decreasing function of k,
kPDE, and (see proof in
APPENDIX, Control of the Steady-State Level
cAMP*). The influence of the slope parameter is not univocal:
cAMP steady-state level is either an increasing function of if
or a decreasing one in the opposite case.
Stability of the Steady State
Linear stability of systems of ordinary differential equations
such as those arising in this paper is determined by the roots of a
polynomial. The stability analysis involves the linearization of
system 8-12 in the form
where q is the vector of the time-dependent
concentrations (RFSH,
XFSH, XpFSH,
EFSH, cAMP), and Mj is
the matrix of the linearized nonlinear terms evaluated at the steady
state, which is defined as the Jacobian matrix and is given by
using for simplicity the notation
Solutions are obtained by setting
where q0 is the constant vector of initial
values, and the eigenvalues 
are the roots of a characteristic
polynomial | Mj I | = 0, with I the identity matrix.
The steady state is stable if all roots have a negative real part.
Because formal calculation did not allow us to carry through the study
of the real part signs, we made use of the Hurwitz criterion
(5), which derives necessary and sufficient conditions for
negativity. In the case where steady-state , *, is saturated and
can be approximated by the constant value , the Hurwitz criterion shows that the eigenvalues of Mj have strictly negative
real parts, so that the steady state is asymptotically stable (see
details in APPENDIX, Hurwitz Criterion for Linear
Stability Analysis). Application of the criterion is not so
straightforward when the dependence of * on cAMP* is
taken into account, so that linear stability analysis in the general
case remains to be studied. However, note that, because the equilibrium
in the case of a constant is hyperbolic, it is locally structurally
stable; hence, it will also be asymptotically stable whenever the
dependence of on cAMP is weak (16).
Controllability Analysis
Roughly speaking, a dynamic system is said to be controllable if,
starting from given initial conditions, one can find an admissible
control variable (here FSH) such that there exists a time for which the
state variables will be steered to prescribed values. Controllability
is an important feature of the model, because if the system were not
controllable, the equilibrium values would be reached independently of
FSH, which would be unsatisfactory for a model that we want to use for
control purposes.
The study of the controllability matrix associated with the linearized
system at steady state is more easily tractable (24) than
that of the Jacobian matrix and allows us to conclude that the
nonlinear system 8-12 is locally strongly accessible
everywhere except if E0 = 0 (details in
APPENDIX, Local Control of the System). The
controllability analysis does not require the assumption of a constant
FSH input, so it leads to quite general results regarding FSH input shape.
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CONTROL OF FSH-INDUCED cAMP LEVELS |
Dimension of Model Variables and Parameters
To handle the model equations from a numerical viewpoint, we need
to know the dimensions and ranges of both variables and parameters so
as to confine cAMP output values within physiological limits. As far as
variables are concerned (Table 1),
granulosa-specific information is available. During terminal follicular
development, there are ~103-104 FSH binding
sites per cell (17, 27). Experimental measurements of
cAMP concentration in granulosa cells (1, 13, 19, 20) under different conditions lie in a range from 0.1 to 10 pmol/106 cells, roughly corresponding to 0.2 to 20 × 104 molecules per cell (molecular mass of cAMP is 327 Da).
As far as parameters are concerned (Table
2), FSH binds its receptors with high
affinity; the equilibrium dissociation constant
Kd = k/k+ is on the order of
1010 M (23). The other kinetic constants are
assigned ranges of values consistent with published biochemical models
in other cell types (14, 32). We used physiological FSH
plasma concentrations as inputs, lying in the range from 1 to 10 ng/ml,
with 3 ng/ml corresponding to 1010 M on the basis of an
average FSH molecular mass of 3 × 104 Da
(35). The lowest FSH values correspond to tonic secretion, whereas the highest rather correspond to the surge secretion or the
level used in stimulation protocols or in vitro experiments.
Physiological Meaning of Variations in Parameter Values
Variations in the model parameter values correspond to
physiological or pathological alterations in the different steps of FSH
signal processing by granulosa cells. Binding equilibrium parameters
(k+, k) might vary
among species in relation to species-specific genetic differences or
even intraspecies as a result of functional mutations or receptor
polymorphism affecting the extracellular domain of the FSH receptor
that contains the binding site. The number of FSH receptors available
for binding can be experimentally altered. For instance, the treatment
of granulosa cells with neuramidase, which catalyzes the removal of
cell surface sialic acid, increases specific FSH binding
(25). In the model, such a treatment would result in an
increase in the size of the global receptor pool
(RT). The level of adenylyl cyclase activation
might differ according to genetic differences affecting the FSH
receptor domain(s) responsible for G protein activation, the degree of
Gs-intrinsic GTPase activity, or the use of adenylyl
cyclase activators such as forskolin. Variations in the amplification
parameter () may partly account for such processes, as this
parameter is related to the average number of adenylyl cyclase
molecules activated by one FSH-bound receptor during its lifetime as an
active form. Different values of the cAMP synthesis parameter ()
could correspond to different types of adenylyl cyclase, as
several of them have been identified (33). Signal
extinction in the model is ensured by the hydrolysis of cAMP and the phosphorylation-induced desensitization of bound receptors. Variations in the hydrolysis rate
(kPDE) can be experimentally achieved through
chronic infusion with a PDE inhibitor such as isobutylmethylxanthine
(IBMX) or, conversely, through constrained overexpression of PDE in
cultured cells. Similarly, infusion of kinase inhibitors such as
staurosporine alters the balance between kinase and phosphatase
activities and can be related to variations in the parameters of the
phosphorylation rate, especially its saturation value (). The rate
of renewal of free FSH receptors results from a dynamic equilibrium
between the processes of internalization, degradation, recycling, and
synthesis. In the model, renewal is dependent on both the
internalization (ki,) and the recycling (kr) rates. Finally, the time scale parameter
() measures the speed of amplification of FSH signal in granulosa
cells. It integrates the role of cross talks with different signaling
pathways, notably paracrine and autocrine signaling through growth
factors and steroids.
For a given combination of the model parameters, variations in FSH
input help to determine the range of values where the model is the most
sensitive to changes in FSH levels. Besides, increasing the level of
the constant FSH input illustrates how the cell is protected against an
overflow in intracellular cAMP.
Control of cAMP Steady-State and Transient Levels
Study of cAMP steady-state levels.
Given a fixed value of FSH input, every parameter except the time
scale parameter () affects the value of the cAMP steady-state level
cAMP*. This value is an increasing function of FSH input, the size of receptor pool RT, the rate constants
k+, ki, and
kr, and the half-saturating cAMP concentration
. Conversely it is a decreasing function of the unbinding rate
k, the hydrolysis rate
kPDE, and the saturation value . The way
cAMP* is influenced by a parameter is analyzed formally in
APPENDIX (Control of the Steady-State Level
cAMP*). Interestingly, the -parameter, which rules the rate of
increase in the phosphorylation rate (the slope of the Hill function),
has a nonunivocal influence on cAMP*, depending on the value
of RT compared with a threshold value given by
For values of RT lower than
RThresh, cAMP* is an increasing function of ,
whereas it is a decreasing function for values >RThresh.
When RT = RThresh, altering the
value of simply has no effect. This means that, if the receptor
pool is small, the cAMP steady-state level rather benefits from an
almost all-or-nothing effect of cAMP level on the phosphorylation
process than from a progressive, smoother effect.
Beyond this qualitative study, quantitative dose-effect-like curves can
be constructed from Eq. 19, which amounts, in term of
cAMP*, to
These curves are displayed in Figs.
2 and 3.
The range of variations for some parameters has been deliberately
exaggerated beyond physiological values so as to examine a large range
of cAMP steady-state-reachable levels for a given parameter.
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Fig. 2.
Influence of follicle-stimulating hormone (FSH), size of
receptor pool (RT), FSH binding rate
(k+), FSH unbinding rate
(k), amplification parameter (), cAMP
synthesis rate (), cAMP hydrolysis rate by phosphodiesterase
(kPDE), phosphorylated receptor internalization
rate (ki), and internalized receptor recycling
rate (kr) on cAMP steady-state
(cAMP*) level. Panels illustrate the influence of the FSH
input and other parameter values from left to
right and top to bottom on
cAMP*.
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Fig. 3.
Influence of saturation value of phosphorylation rate (), slope parameter (), and half-saturating cAMP concentration
() (depending on RT value) on
cAMP*. Panels illustrate the influence of the parameters of
the phosphorylation rate on cAMP*. Top:
influence of () (left) and (right);
bottom: influence of when either
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(left)
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or
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(right)
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Study of cAMP transient levels.
To retrace the history of FSH-induced cAMP production, starting from a
quiescent initial state, we also need to understand the transient
behavior of the system before its reaching steady state. To do so, we
performed a series of numerical simulations of the model using a
computer program written in C language. The differential equations were
integrated by means of a Runge-Kutta method of order 4 (28), with a step of 0.01 s. cAMP transient levels
depend in a complicated manner on the values of the model parameters
and FSH input. The same steady state can in particular be achieved in
different ways depending on the value of the time scale parameter .
This parameter is the only one in the model that does not affect the
steady state but instead exerts a substantial influence on the
transient behavior, especially of the coupling variable
EFSH.
Initial values.
The start of the simulation was assumed to be the point in time when
FSH receptors become efficiently coupled to G proteins and start
influencing intracellular cAMP production, which corresponds to the
follicle's entering the FSH-responsive stage. Before this point, it is
assumed that there is a minor basal level of activated adenylyl
cyclase, resulting in a low basal level of cAMP production uncoupled
with FSH input. Initial conditions for the differential equations
Eqs. 8-12 are set to
These correspond to the binding equilibrium between
RFSH and XFSH and steady
state for Eq. 11 with activated cyclase level E0 decoupled from receptor stimulation.
Influence of the receptor pool size.
Figure 4 (top) illustrates the
effects of varying the size RT within the
physiological range of 0.5-2 × 104
receptors/cell. As expected, decreasing RT leads
to a lower cAMP steady-state level. With the smallest
RT value (0.5 × 104), this
cAMP level corresponds to a steady-state value of the desensitization
rate being much lower (0.02/s) than the saturation value (0.06/s).
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Fig. 4.
Influence of RT, , and
kPDE. Top: influence of
RT on changes in cAMP levels (left)
and on changes in (right); inset: nos. are in
molecules/cell. Middle: influence of on changes in
activated adenylyl cyclase levels (left) and on changes in
cAMP levels (right); inset: nos. are
dimensionless. Bottom: influence of
kPDE on changes in cAMP levels (left)
and on changes in the phosphorylation rate (right);
inset: nos. are in kPDE (/s). In this
figure and in Figs. 5-9, the various panels give the time
evolution of the various concentrations that constitute the variables
in the model. These are expressed in units of 104
molecules/cell (see Table 1). The phosphorylation rate is shown as
a function of time. Figures 5-8, top left, show the
pattern of applied FSH input (1010 M; this is constant in
Figs. 5-7). Differently styled lines on each plot of the same
figure correspond to different values of the model parameter under
study (displayed in insets), the other parameters being kept
unchanged. The basic set of common parameter values is summarized in
Table 2, and common initial values are RT = 2.0, basal level of adenylyl cyclase (E0) = 0.01, and
cAMP = 0.25 (104 molecules/cell), with constant
FSH input of 3 × 1010 M. Time units are 102
s, so the simulations correspond to periods of between ~8 h and 5 days.
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Influence of the binding dissociation constant.
The increase in the dissociation constant,
Kd = k/k+, leads to an
increase in the free receptor concentration together with a decrease in
the concentration of bound receptors and its derived (phosphorylated
and internalized) forms. This again affects the steady-state values
cAMP* and *, with the highest value of
Kd corresponding to the lowest cAMP level.
Influence of the amplification parameter.
Figure 4 (middle) illustrates the role of the amplification
parameter . The patterns of changes in EFSH
and cAMP are almost superimposed. The scale of the cAMP
value range is dramatically increased as increases.
Influence of the hydrolysis parameter.
A nonzero value of kPDE is necessary for the
cAMP concentration to reach a steady-state value. If
kPDE = 0, as can be seen on the solid line
of Fig. 4 (bottom), signal turn-off is mediated only
by the phosphorylation function , which quickly reaches its
saturation value () and cannot control the exponential increase in
cAMP concentration. Conversely, increasing the value of
kPDE affects cAMP levels so as to stabilize at a value far below saturation.
Influence of the phosphorylation saturation parameter.
Changes in the saturation capacity of the desensitization function
affect not only the steady-state level of cAMP but also the different
forms of FSH receptors, as can be seen in Fig.
5. Increasing the value of reduces
the number of FSH receptors in the bound active state
XFSH in favor of the phosphorylated state
XpFSH. The associated increase in internalized
receptors cannot compensate for this imbalance even if the
internalization process is at the source of free receptor renewal and
hence, indirectly, of active bound receptors.
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Fig. 5.
Influence of the desensitization saturation capacity (/s).
Top left: changes in the levels of active bound FSH
receptors (XFSH); top right: changes
in cAMP levels; bottom left: changes in the
phosphorylation rate; bottom right: changes in the levels of
phosphorylated bound FSH receptors (XpFSH).
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Influence of receptor renewal.
Increasing either the internalization rate ki or
the recycling rate kr allows for a quicker
renewal of free FSH receptors from phosphorylated bound receptors, thus
enhancing the FSH signal.
Influence of the time scale parameter.
Low values ( 1) lead to a marked contrast between the
dynamics of fast (RFSH,
XFSH, cAMP, and
XpFSH) and slow (EFSH) variables, whereas high values (i.e., not much lower than 1) tend to
homogenize the time scales of all the variables. As increases from
a small value toward 1, the time required to reach equilibrium is
significantly decreased, as can be seen in Fig.
6. Thus, for close to 1, the
steady state is reached in a few minutes, whereas, for small values (as
low as 103 in this instance), it can take several days.
The maximal value reached by EFSH and
cAMP can significantly overshoot its steady-state value, and
this effect also becomes more pronounced as increases toward 1. The
transient response is sensitive to even small variations in the value
of , especially for given parameter combinations. This is
illustrated in Fig. 7, where the
hydrolysis rate is about twice what its value is in other figures
(0.09/s). The time derivative of EFSH
changes signs, and thus crosses its steady-state value, one or more
times depending on the precise value of , subsequently leading to a
variety of cAMP-transient patterns.
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Fig. 6.
Influence of the time scale parameter (large
variations), showing from left to right and from
top to bottom the correspondence of each panel to
the next: FSH input, changes in the levels of free FSH receptors, bound
FSH receptors, activated adenylyl cyclase, cAMP, phosphorylated
receptors, internalized receptors, phosphorylation rate. Inset: nos.
are dimensionless.
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Fig. 7.
Influence of (small variations), showing from
left to right and from top to
bottom the correspondence of each panel to the next: FSH
input, changes in the levels of free FSH receptors, bound FSH
receptors, activated adenylyl cyclase, cAMP, phosphorylated receptors,
internalized receptors, phosphorylation rate. Inset: nos.
are dimensionless.
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Level and pattern of FSH input.
In the physiological range from 0.3 to 3.0 × 1010
M, a 10-fold variation in FSH concentration (Fig.
8, top) results in a twofold variation in cAMP level. Beyond a given level of FSH (depending on the
values of the other parameters), increasing the FSH level will have
almost no effect on the cAMP response. We also investigated the pattern
of cAMP response to nonconstant FSH inputs (results not shown). In the
case of a square-shaped FSH input, the system switches from one steady
state to another as FSH switches between its high and low values. The
reaction of the system to the changes in FSH input is again under the
control of . The smaller is, the smoother the changes in the
system variables, to the extent that the effects of the variation in
FSH input may be perceptible only in the behavior of the receptor
species (RFSH, XFSH,
XpFSH, Ri). Other
simulations with exponentially decreasing or sinusoidal FSH input
yielded qualitatively similar conclusions. We show in Fig.
9 an example of the model behavior in
response to real FSH data taken from Adams et al. (2; Fig. 1B,
p. 631). The changes in FSH input are mirrored in those of free
FSH receptor concentration, whereas they are quite tightly tracked by
those in bound FSH receptors. The changes in phosphorylated and
internalized receptors are nearly similar and follow the
phosphorylation rate, which starts rising only when significant cAMP
levels have been reached. The changes in activated adenylyl cyclase and
cAMP are smoother. After increasing in a continuous way, they end up
oscillating in a dampened manner around a steady-state value.
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Fig. 8.
Response to different levels of FSH stimulation, showing
from left to right and from top to
bott |
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TOP
ABSTRACT
INTRODUCTION
![∂&rgr;*=<FENCE><FR><NU>∂&rgr;</NU><DE>∂cAMP</DE></FR></FENCE><SUB>cAMP*</SUB>=<FR><NU>&ggr;&agr;&dgr;<SUP>&ggr;</SUP>cAMP*<SUP>(&ggr;−1)</SUP></NU><DE>[&dgr;<SUP>&ggr;</SUP>+cAMP*<SUP>&ggr;</SUP>]<SUP>2</SUP></DE></FR>=<FR><NU>&ggr;&agr;&dgr;<SUP>&ggr;</SUP>+<FENCE><FR><NU>&ohgr;&sfgr;</NU><DE>k<SUB>PDE</SUB></DE></FR><IT> X<SUP>*</SUP></IT><SUB>FSH</SUB></FENCE><SUP><IT>&ggr;−1</IT></SUP></NU><DE><FENCE><IT>&dgr;<SUP>&ggr;</SUP>+</IT><FENCE><FR><NU><IT>&ohgr;&sfgr;</IT></NU><DE><IT>k</IT><SUB>PDE</SUB></DE></FR><IT> X<SUP>*</SUP></IT><SUB>FSH</SUB></FENCE><SUP><IT>&ggr;</IT></SUP></FENCE><SUP><IT>2</IT></SUP></DE></FR>](/content/vol281/issue1/fulltext/E35/img042.gif)
