The development of ovarian follicles is a crucial, limiting step for the success of reproduction in mammals. Yet, the process of follicle selection, the regulation of the species-specific ovulation rate and the meaning of the tremendous wasting of follicles through atresia are still incompletely understood. Resolving these fundamental scientific questions correspond to both clinical and zootechnical challenges. A better understanding of follicular development is required to improve the control of anovulatory infertility in women, (such as it is encountered in the severe forms of the Poly Cystic Ovary Syndrome) and to control ovulation rate and ovarian cycle chronology in domestic species. Beyond the frame of reproduction physiology, follicular development is a unique instance of rapid and controlled development in adult organisms, and follicular (particularly granulosa) cells constitute an interesting model in cell kinetics studies.

Three cell states are encountered in the granulosa during terminal follicular development (Fig. 2). Proliferating cells are engaged in the division cycle ending up by mitosis completion. Differentiating cells are terminally-maturating cells characterised by their high level of sensitivity to gonadotrophins and steroidogenic activity. Apoptotic cells are expressing a genetic program leading to cell death and subsequent disappearance from the follicle. At any time, proliferative cells leave the division cycle in an irreversible way, according to the antagonism between proliferation and differentiation observed during terminal follicular development. The apoptosis risk applies to non-cycling cells, and entry into apoptosis is considered to be irreversible. The dynamics of the granulosa cell population is expressed as a function of follicular age by the following system of differential equations:

The two cellular transition rates are the control variables mediating the interactions of the follicle with its hormonal environment. They reflect the response of follicular cells to signalling for differentiation or apoptosis. The cell cycle exit rate $\delta \left(t\right)$ is modulated by the proportion of granulosa cells that are in the G1 exit window, and by the biochemical history of these cells in terms of exposure and response to mitogenic or anti-mitogenic signalling since last mitosis. The rate of non-cycling cells entering apoptosis, $\alpha \left(t\right),$ is similarly modulated by the balance between pro-apoptotic and survival signals integrated at the cellular level.

From now on, we focus on the course of an ovulatory follicle, for which the apoptosis entry rate can be neglected. The first two equations of system (1) – (3) become:

One can then deduce the dynamics of the total number of viable granulosa cells, $N_{\mathrm{T}}\left(t\right),$ as well as of the growth fraction, $\mathrm{GF}\left(t\right),$ representing the relative number of proliferating cells amongst the viable cell population:

The values of the five parameters of the sigmoid function $\delta \left(t\right)$ can be estimated by fitting the regression equation (4) on real cell number data.

The results of the fitting procedure are illustrated in Fig. 3. Cell number data were harvested from ewes after dissection of follicles and counting of their granulosa cells 〚4〛. The data points are plotted together with the adjusted curve for the total cell number and its decomposition into proliferative and differentiated cells. The number of proliferative cells shows a biphasic pattern: after an initial increase phase, it decreases for the benefit of the non-cycling population, which ends up constituting the whole population. From an initial mainly proliferative state, the granulosa of the preovulatory follicle progressively switches to a highly differentiated state, so that the growth fraction continuously decreases (see insert in Fig. 3). The follicle awaits ovulation with an almost constant number of cells. In this sense, it has reached a cellular steady state. However, this is a temporary state, because the onset of ovulation will completely remodel the follicle into a corpus luteum. The number of cells that is reached at the steady differentiated state is directly related to the capacity of œstradiol secretion, through which the follicle takes part in the endocrine interactions between the ovaries and the hypothalamic-pituitary axis. The observation that this number is fully determined by the speed of exhaustion of the proliferative compartment leads to the notion of follicular proliferative resources.

The corresponding curve for the exit rate is shown in Fig. 4. The minimal value of $\delta \left(t\right)$ roughly equals half the value of μ, while its maximal value roughly reaches twice the value of μ. The time when the value of $\delta \left(t\right)$ exceeds that of μ (t_E in Fig. 4) is of critical importance, as it corresponds to the number of proliferating cells starting decreasing. Starting from similar initial conditions, the earlier this time occurs, the earlier and lower is the peak in the proliferating cell number $N_{\mathrm{p}}\left(t\right).$ As a result, the number of fully-differentiated cells available at the end of follicular development is lower, and the decrease in the growth fraction is faster (see Fig. 5). In the ovine species, follicles of poly-ovulating breeds ovulate at a smaller size, associated with a lower number of granulosa cells compared to mono-ovulating breeds. According to the model, the smaller number of granulosa cells in the preovulatory follicles of poly-ovulating breeds could be accounted for by a faster increase in the cell cycle exit rate, corresponding to differential exploitation of the follicular proliferative resources.

This experiment 〚5〛 aimed at (i) determining the changes in the growth fraction of granulosa cells in the ‘Île-de-France’ mono-ovulating sheep breed compared to the ‘Romanov’ poly-ovulating breed and (ii) studying the effect of overstimulating granulosa cells with FSH on these changes. The former objective corresponds to the testing of the hypothesis derived from the model results on differential increases in the cell cycle exit rate according to the breed, while the latter is a first step in understanding the way FSH acts on the cell cycle exit rate. The experimental steps included ovariectomy of control and FSH-treated adult cyclic ewes, dissection of follicles and pooling of granulosa cells according to follicular diameter, granulosa cell isolation, and continuous 〚³H〛 thymidine labelling and colcemid administration to cultured cells. The determination of the growth fraction was based on the procedure of Maekawa and Tsuchiya 〚6〛, which was initially proposed to determine the durations of the different cell cycle phases. This method is also appropriate to determine the growth fraction in cell populations. Briefly, it consists in the continuous and combined administration of 〚³H〛 thymidine and colcemid, respectively to label all proliferating cells lying in the S phase of the cycle and to block mitosis. After a delay corresponding to the duration of the G2 phase, the first labelled mitosis appear so that four states can be detected: labelled or unlabelled dividing cells, labelled or unlabelled non-dividing cells. The proportion of cells in the unlabelled, non-dividing state decreases before reaching a plateau after a delay corresponding to the duration of the G1 phase. The height of this plateau (modulo unity) finally gives an estimation of the growth fraction.

2.7.1 Comparison ‘Romanov’/‘Île-de-France’ in basal conditions

Several lessons can be drawn from the results obtained in basal (non stimulated) conditions (left part of Fig. 6). We can first observe that, whatever the follicular diameter, the growth fraction is strictly lower than unity, which means that the granulosa population appears always heterogeneous during terminal follicular development. Further, the growth fraction continuously decreases. The drop is particularly marked between 3 and 4 mm diameter, which corresponds to the period when granulosa cells acquire LH receptors. Finally, the speed of decrease (as measured by the slope of the regression line between the growth fraction and diameter after logit transformation) is higher in follicles from ‘Romanov’ compared to ‘Île-de-France’ ewes. In agreement with the model prediction, this result suggests that the increase in the cell cycle exit rate during terminal follicular development is faster in ‘Romanov’, compared to ‘Île-de-France’ follicles. The impact of this result is emphasized by comparative physiology. Indeed, in women suffering from severe forms of the PolyCystic Ovary Syndrome, follicles stop their development at an intermediary stage. Their failing ovulation might be related to alterations in the control of granulosa cells proliferation such as those observed in Romanov ewes.

2.7.2 Comparison ‘Romanov’/‘Île-de-France’ after stimulation by FSH

Whatever the target may be, the optimal solution consists in applying the minimal apoptosis rate α_min and switching the cell cycle exit rate from its minimal bound δ_min to its maximal bound δ_max at a given time t_s (see Fig. 7). The optimal cell cycle exit rate can thus be considered as a limiting case of a sigmoid function with a very steep increase. The chronology of follicular development is fully determined by both the switching and surge times, whose values are explicitly given as functions of the initial values and other parameters (see reference 〚8〛 for details). The switching time t_s occurs all the more lately than the conditions for reaching the target are demanding (high value for N_c), while the delay between the switching and surge times depends on the target (lowest delay with instantaneous target, highest with cumulated one).

The number of viable cells ($N_{\mathrm{T}}\left(t\right)={\mathrm{N}}_{\mathrm{p}}\left(t\right)+{\mathrm{N}}_{\mathrm{d}}\left(t\right)$) can be expressed as a function of the parameters of the control problem:

This allows in turn to fit the values of the optimal strategy parameters $\left({\delta }_{\min },{\delta }_{\max },t_{\mathrm{s}}\right)$ from the experimental cell counts described in section 2.4 (see Fig. 8) and to get information back on the surge time t_f and threshold N_c.

Deviation from the optimal response can result from a shift in the time schedule for the switching instant, as it is illustrated in Fig. 9. The way of shifting the switching time has great repercussions on the issue of the problem. Indeed, when the actual switching time occurs earlier than the optimal one, the proliferative resources of the follicular granulosa may be exhausted too quickly. In the instantaneous case, the target may remain beyond reach. In the cumulated case, the delay in reaching the target may hamper the normal progress of the ovarian cycle. The corresponding follicular response is thus completely non-optimal. On the contrary, when the switching time occurs later than the optimal one, the target can be reached with limited delay, but at the expense of producing follicular cells in excess. The corresponding follicular response is thus only suboptimal.

The biochemical model includes the steps of FSH signal detection, relay and amplification, and extinction:

• • binding of FSH to its receptor $\left({\mathrm{R}}_{\mathrm{FSH}}\right)$ results in the formation of an active complex $\left({\mathrm{X}}_{\mathrm{FSH}}\right)$

  $$\mathrm{FSH}+{\mathrm{R}}_{\mathrm{FSH}}{\rightleftharpoons }_{k_{-}}^{k_{+}}{\mathrm{X}}_{\mathrm{FSH}}$$

• • bound receptors activate adenylyl cyclase (E) through a conformational change in the associated G protein

• • activated adenylyl cyclase $\left({\mathrm{E}}_{\mathrm{FSH}}\right)$ synthesises cAMP from the substrate ${\mathrm{Mg}}^{++}\mathrm{ATP}$ (adenosine triphosphate)

  $${\mathrm{Mg}}^{++}\mathrm{ATP}+{\mathrm{E}}_{\mathrm{FSH}}{\to }^{\omega }\mathrm{cAMP}$$

• • cAMP is hydrolysed into AMP by phosphodiesterase

  $$\mathrm{cAMP}{\to }^{{\mathrm{k}}_{\mathrm{pde}}}\mathrm{AMP}$$

• • bound receptors are subject to a desensitisation process through cAMP-mediated phosphorylation

  $${\mathrm{X}}_{\mathrm{FSH}}{\to }^{\rho \left(\mathrm{cAMP}\right)}{\mathrm{X}}_{\mathrm{pFSH}}$$

• • phosphorylated inactive complexes $\left({\mathrm{X}}_{\mathrm{pFSH}}\right)$ undergo internalisation into the cell, where receptors are dissociated from FSH

  $${\mathrm{X}}_{\mathrm{pFSH}}{\to }^{k_{\mathrm{i}}}{\mathrm{R}}_{\mathrm{i}}$$

• • internalised receptors R_i are recycled back to the cell membrane, while FSH is hydrolysed

  $${\mathrm{R}}_{\mathrm{i}}{\to }^{k_{\mathrm{r}}}{\mathrm{R}}_{\mathrm{FSH}}$$

Considering only reactions relevant to follicular development allows some simplifications to be made. Reactions generating short-life intermediary species are neglected. In particular, the cycle of G-protein activation/deactivation is not modelled 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 internalised) remains constant 〚13〛, leading to the following cellular cycle for FSH receptors under different states:

Finally, cAMP-independent desensitisation is not taken into consideration, as its behaviour during the maturation of granulosa cells is not yet known. In addition, the amount of FSH is assumed to be sufficiently large so that its concentration is unaffected by binding to receptors.

The control of cAMP levels in granulosa cells involves both fast biochemical processes, occurring on a time scale of a few minutes, such as binding and desensitisation, and slower physiological processes lasting hours or even a few days, which mainly result in changing the efficiency of the enhancement of cAMP synthesis by stimulated FSH receptors via adenylyl cyclase activation. The design of our model follows from the interactions between these contrasting biochemical and physiological dynamics. The following system intends to render the dynamics of intracellular cAMP in an average granulosa cell in response to FSH:

$R_{\mathrm{FSH}},X_{\mathrm{FSH}},X_{\mathrm{pFSH}},R_{\mathrm{i}}$ are respectively the concentrations of free, bound-active, bound-inactive (phosphorylated) and internalised FSH receptors. E_FSH is the concentration of activated adenylyl cyclase and cAMP the concentration of intracellular cAMP. $k_{+},k_{-},k_{\mathrm{i}},k_{\mathrm{r}}$ are respectively the rate constants for FSH-binding, FSH-unbinding, bound-complex internalisation and receptor recycling to the cell membrane.

Equations (8), (11), and (13) result from applying the principle of mass-action. The concentrations of ATP and phosphodiesterase enzyme are assumed to be constant and non-limiting, so that their effects are included in the respective kinetic constants ω and k_pde. Desensitisation through phosphorylation is assumed to be cAMP-mediated in a dose-dependent, increasing and saturated manner. It follows that the desensitisation rate ρ in equations (9) and (12) is a Hill function of intracellular cAMP:

The dynamics of the coupling variable E_FSH, which we assimilated to activated adenylyl cyclase, incorporates specific features of FSH signal transduction in granulosa cells. It accounts for both FSH-dependent and auto-amplified settling and efficiency of the cAMP cascade through follicular development. The $ϐ$ parameter acts as a time scale parameter. Whenever it takes a low value $\left(ϐ⪡1\right),$ the changes in the coupling variable E_FSH are slower than those of the other variables of the model. The σ parameter measures the degree of signal amplification and represents the average number of adenylyl cyclase molecules activated by one bound receptor at steady state.

Finally, equations (8) to (12) are subject to the conservation law

The cAMP modelling approach allows two meeting points with possible experimental investigations. The first concerns the influence of the model parameters on the cAMP steady-state level reached after exposure to constant FSH input, while the second concerns the pattern of cAMP dynamics in response to a given pattern of FSH input.

4.3.1 Control of FSH-induced cAMP steady-state levels (constant input)

Beyond this qualitative study, quantitative dose-effect-like curves can be constructed, such as those displaid in Fig. 10. As variations in the model parameter values correspond to physiological, pathological or pharmacological alterations in the different steps of FSH signal processing by granulosa cells, these theoretical curves should be at least partly experimentally testable.

4.3.2 Control of FSH-induced cAMP transient levels

The model retraces the long-term behaviour of cAMP in granulosa cells in response to FSH alone. From any given FSH input, the cAMP output pattern can be predicted, such as it is illustrated in Fig. 11 with real FSH data 〚14〛. As cAMP is an accessible variable on the experimental ground, the predicted output could be confronted with experimental measures. On the contrary, the intermediary species (as the different states of FSH receptors) do not seem susceptible of easy measure on the long term, even if they have a direct biochemical meaning.

The most appropriate data would consist in repeated measurements of intracellular cAMP throughout the development of dynamically-monitored follicles. Concomitantly, FSH levels should be measured. To track cAMP production until the ovulatory stage, one need to be placed in controlled situations. As far as physiological situations are concerned, the luteal phase in ruminant species would be the most appropriate window of the ovarian cycle to harvest data. In such species, terminal follicular development during luteal phase is mainly FSH-dependent and thus fulfills the requirements for investigating FSH-induced cAMP production throughout terminal development. As far as pharmacological situations are concerned, appropriate conditions can be artificially reproduced by means of either previous desensitisation with GnRH agonists, or use of GnRH antagonists, and administration of recombinant FSH with known bioactivity. The most limiting point in both situations is the need for dynamical non-invasive measurements of intracellular cAMP. This might be achievable in the future through repeated ultrasound-guided follicular cell pick-up as follicular development progresses. In domestic animals, follicular fluid pick-up are already running well and technical progress of devices could render direct cell pick-up feasible in the medium term.