contribution of a mathematical modelling approach to the...

Model-driven acquisition / Acquisition conduite par le modèle

Contribution of a mathematical modellingapproach to the understanding of the ovarianfunction

Frédérique Clémenta*, Danielle Monniauxb, Jean-Christophe Thalabardc, Daniel Clauded,e,f

a Institut national de recherche en informatique et automatique, unité de recherche de Rocquencourt, domaine deVoluceau, Rocquencourt, BP 105, 78153 Le Chesnay cedex, Franceb Unité de physiologie de la reproduction et des comportements, UMR 6073 INRA–CNRS–université François-Rabelais de Tours, Institut national de la recherche agronomique, 37380 Nouzilly, Francec UFR Necker, université Paris-5, Biostatistiques–informatique médicale, endocrinologie–médecine de lareproduction, groupe hospitalier Necker–Enfants-Malades, 149, rue de Sèvres, 75743 Paris cedex 15, Franced UFR d’Orsay, Université Paris-11, 91405 Orsay cedex, Francee Laboratoire des signaux et systèmes, CNRS–SUPELEC, 91192 Gif-sur-Yvette, Francef INRIA Roquencourt, 78153 Le Chesnay, France

Received 18 July 2001; accepted 20 September 2001

Presented by Michel Thellier

Abstract – The biological meaning of folliculogenesis is to free fertilisable oocytes at the time of ovulation. Weapproached the study of the control of follicular development at the level of follicular granulosa cells, on theexperimental as well as mathematical modelling grounds. We built a mathematical model allowing for the processesof proliferation, differentiation and apoptosis. State variables correspond to the numbers of cells undergoing thesedifferent processes, while control variables correspond to the cellular transition rates. The model results raised thenotion of proliferative resources, which leads to consider the optimal management of these resources and hasmotivated the settling of an experiment investigating the changes in the growth fraction within the granulosathroughout terminal development. We are now investigating the way gonadotrophins, and especially FSH, operate ongranulosa cells, in order to account for the hormonal control of the divergent commitment of granulosa cells towardseither proliferation, differentiation or apoptosis. We are thus focusing on the dynamics of cAMP production, whichappears to be a keypoint in FSH signal transduction.To cite this article: F. Clément et al., C. R. Biologies 325 (2002)473–485. © 2002 Académie des sciences / Éditions scientifiques et médicales Elsevier SAS

cell kinetics / signal transduction / granulosa cell / ovary / follicle / cAMP / control theory

Résumé – Apport de la modélisation mathématique à la compréhension de la fonction ovarienne. La finalitébiologique de la folliculogenèse est la production d’ovocytes aptes à la fécondation et au développement. Nous avonsabordé l’étude des régulations du développement folliculaire terminal au niveau des cellules folliculaires de lagranulosa, tant sur le plan de la modélisation mathématique que sur le plan expérimental. Nous avons construit unmodèle rendant compte des processus de prolifération, différenciation et apoptose, dont les variables d’état sont leseffectifs des différentes catégories de cellules et les variables de contrôle sont les taux de transition entre catégories.L’élaboration et les résultats de ce modèle ont fait émerger la notion de capital prolifératif folliculaire, qui permetd’envisager une gestion optimisée de ce capital, à l’aide de thérapeutiques adéquates, et a motivé la mise en placed’une expérience de mesure de la fraction de croissance au sein de la granulosa. Notre approche se poursuit par l’étudedu mécanisme d’action des gonadotropines (en particulier FSH) sur les cellules folliculaires, afin de rendre comptedu contrôle hormonal exercé sur la composition cellulaire de la granulosa. La dynamique de production d’une

*Correspondence and reprints.E-mail address: [email protected] (F. Clément).

473

C. R. Biologies 325 (2002) 473–485© 2002 Académie des sciences / Éditions scientifiques et médicales Elsevier SAS. Tous droits réservésS1631069102014579/FLA

molécule clé pour la transduction du signal hypophysaire, l’AMPc, fait ainsi l’objet des études en cours. Pour citercet article : F. Clément et al., C. R. Biologies 325 (2002) 473–485. © 2002 Académie des sciences / Éditionsscientifiques et médicales Elsevier SAS

cinétique cellulaire / transduction du signal / cellule de granulosa / ovaire / follicule / AMPc / théorie ducontrôle

Version abrégée

Tout au long de la vie reproductive des femelles demammifères, l’ovaire est le siège d’un phénomène trèsparticulier, appelé folliculogenèse, qui se définit commel’ensemble des processus de croissance et de matura-tion fonctionnelle subis par les follicules ovariensdepuis leur sortie de la réserve de follicules primordi-aux jusqu’à l’ovulation. Sa finalité biologique est laproduction, lors de chaque cycle ovarien, d’un (pour lesespèces mono-ovulantes) ou de plusieurs (pour lesespèces polyo-ovulantes) ovocytes aptes à la féconda-tion et au développement. En fait, la plupart desfollicules n’atteignent jamais le stade ovulatoire, maissubissent un processus de dégénérescence, l’atrésie. Ledéveloppement folliculaire terminal est sous le contrôledes gonadotropines hypophysaires, FSH (follicle-stimulating hormone) et LH (luteinising hormone), quipeuvent être considérées comme des entrées de com-mande agissant sur la fonction ovarienne. L’action desgonadotropines se faisant à l’échelle cellulaire, il estnécessaire de caractériser et de comprendre les change-ments qui s’opèrent au niveau des cellules cibles, ausein du tissu folliculaire appelé la granulosa.

Nous avons construit un modèle mathématique ren-dant compte des processus cellulaires de prolifération,différenciation et apoptose (mort cellulaire). Les vari-ables en jeu sont, d’une part, les effectifs des différentescatégories de cellules (variables d’état) et, d’autre part,les taux de transition entre les différents états cellulaires(taux de sortie du cycle cellulaire et taux d’entrée enapoptose). Ces taux sont des variables de contrôle quitraduisent la réponse des cellules folliculaires auxsignaux qu’elles reçoivent de leur environnement hor-monal, et en particulier de FSH et LH. L’élaboration etles résultats de ce modèle ont fait émerger une notionnouvelle, celle de capital prolifératif folliculaire. Cettenotion permet d’envisager une gestion optimisée de cecapital, à l’aide de thérapeutiques adéquates, à des finscliniques (traitement de l’ infertilité anovulatoire) ouzootechniques (superovulation). Elle permet aussi deproposer une hypothèse explicative pour les différencesobservées dans l’espèce ovine entre races mono- et

poly-ovulantes quant à l’effectif cellulaire atteint aumoment de l’ovulation. L’effectif plus faible en cas depoly-ovulation serait dû à un arrêt plus précoce de laprolifération cellulaire, lié à une sortie plus rapide ducycle cellulaire. Cette hypothèse a motivé la mise enplace d’une expérience de mesure comparative de lafraction de croissance (proportion de cellules prolif-érantes parmi l’ensemble des cellules folliculaires) àdifférents stades du développement folliculaire chezdeux races de brebis. Cette expérience a confirmé lacinétique différentielle de sortie du cycle en fonction dutaux d’ovulation. L’ importance de ce résultat doit êtreconsidérée sous l’angle de la physiopathologie com-parée, si l’on sait que dans le syndrome des ovairespoly-kystiques, très fréquemment rencontré lorsd’explorations pour anovulation chez la femme, onconstate un blocage des follicules à un stade intermé-diaire, qui pourrait lui aussi résulter de perturbations dela prolifération cellulaire.

La réflexion engagée se poursuit et concerne actu-ellement la compréhension du mécanisme d’action desgonadotropines sur les cellules folliculaires. En parti-culier, le contrôle de la production d’une molécule clépour la transduction du signal hypophysaire, l’AMPc,fait l’objet des études en cours, tant sur le planexpérimental que sur celui de la modélisation. Nousdisposons d’un modèle décrivant la dynamique de laconcentration intracellulaire en AMPc résultantd’ interactions entre des dynamiques lentes et rapidescontrôlées par FSH. Les points de rencontre de cemodèle avec la démarche expérimentale concernent,d’une part, l’ influence des paramètres du modèle sur lesniveaux d’AMPc intracellulaires atteints à l’équilibreen cas d’exposition à un stimulus FSH constant et,d’autre part, la prédiction de la dynamique de produc-tion d’AMPc observée en réponse à un profil donné del’entrée FSH. Une réponse inadaptée des cellules de lagranulosa vis-à-vis de FSH pouvant avoir des répercus-sions dramatiques sur le développement folliculaire, unmodèle réaliste, caractérisant à la fois la transductionphysiologique et pathologique du signal FSH, devraitse révéler très utile pour simuler de nouvelles approchesthérapeutiques.

474

F. Clément et al. / C. R. Biologies 325 (2002) 473–485

1. Introduction

1.1. Follicular development

In mammal females, the pool of oocytes available forfertilisation is constituted early in life. Ovarian folliclesare spheroidal structures sheltering the maturatingoocyte (Fig. 1). Folliculogenesis is the process of growthand functional maturation undergone by ovarian fol-licles, from the time they leave the pool of primordial(quiescent) follicles until ovulation, at which point theyrelease a fertilisable oocyte. Actually, most of thedeveloping follicles never reach the ovulatory stage butdegenerate by a process known as atresia [1]. Thegonadotrophic hormones, Follicle Stimulating Hor-mone (FSH) and Luteinising Hormone (LH), play amajor role in the regulation of terminal folliculardevelopment through the control of proliferation anddifferentiation of the granulosa cells surrounding theoocyte [2]. Gonadotrophin secretion is, in turn, modu-lated by granulosa cell products such as œstradiol and

inhibin. During the follicular phase of the ovariancycle, negative feedback is responsible for reducingFSH secretion, leading to the degeneration of all butthose follicles selected for ovulation. Finally, positivefeedback is responsible for triggering the LH ovulatorysurge leading to ovulation.

1.2. Biological and medical challenges

The development of ovarian follicles is a crucial,limiting step for the success of reproduction in mam-mals. Yet, the process of follicle selection, the regula-tion of the species-specific ovulation rate and themeaning of the tremendous wasting of follicles throughatresia are still incompletely understood. Resolvingthese fundamental scientific questions correspond toboth clinical and zootechnical challenges. A betterunderstanding of follicular development is required toimprove the control of anovulatory infertility in women,(such as it is encountered in the severe forms of thePoly Cystic Ovary Syndrome) and to control ovulationrate and ovarian cycle chronology in domestic species.

Fig. 1. Follicular development. LH: Luteinising hormone; FSH: follicle stimulating hormone; E: œstradiol; GF: growth factors; I: inhibin.The long curved arrow summarises the principal events of folliculogenesis: initiation of growth from the pool of quiescent primordialfollicles, formation of the antrum, acquisition of strict gonadotrophin dependence and apparition of LH receptors on granulosa cells. Thelong left vertical arrow shematically represents the changes in follicular needs thoughout development. The thickness of the right, curvedarrows indicates the contribution of the corresponding follicular maturation stage to the total œstradiol secretion.

475

Pour citer cet article : F. Clément et al., C. R. Biologies 325 (2002) 473–485

Beyond the frame of reproduction physiology, follicu-lar development is a unique instance of rapid andcontrolled development in adult organisms, and folli-cular (particularly granulosa) cells constitute an inter-esting model in cell kinetics studies.

2. Cellular description of folliculardevelopment

We approached the mathematical modelling of folli-cular development from a cellular viewpoint [3], basingon the parallel existing between follicular fate (ie eitherovulation or atresia), and granulosa cell commitmenttowards either proliferation, differentiation or apopto-sis.

2.1. Cell flows in the granulosa

Three cell states are encountered in the granulosaduring terminal follicular development (Fig. 2). Prolif-erating cells are engaged in the division cycle endingup by mitosis completion. Differentiating cells areterminally-maturating cells characterised by their highlevel of sensitivity to gonadotrophins and steroidogenicactivity. Apoptotic cells are expressing a genetic pro-gram leading to cell death and subsequent disappear-ance from the follicle. At any time, proliferative cellsleave the division cycle in an irreversible way, accord-ing to the antagonism between proliferation and differ-entiation observed during terminal follicular develop-ment. The apoptosis risk applies to non-cycling cells,and entry into apoptosis is considered to be irreversible.The dynamics of the granulosa cell population isexpressed as a function of follicular age by the follow-ing system of differential equations:

dNp� t �

dt = �µ − d� t � � Np� t � (1)

dNd� t �dt = d� t � Np� t � − �� t � Nd� t � (2)

dNa� t �dt = �� t � Nd� t �

− �0

s

w� s � �� t − s � Nd� t − s � ds (3)

where Np� t � is the number of proliferating cells, Nd� t �the number of differentiating cells and Na� t � thenumber of apoptotic cells. The rate of cell division µ isinversely proportional to the duration of the cell cycle.The cell cycle exit rate d� t � is an increasing, boundedsigmoid function of age. The apoptosis entry rate �� t �both depends on follicular age and fate.

2.2. Meaning of the cellular transition rates

The two cellular transition rates are the controlvariables mediating the interactions of the follicle withits hormonal environment. They reflect the response offollicular cells to signalling for differentiation or apo-ptosis. The cell cycle exit rate d� t � is modulated by theproportion of granulosa cells that are in the G1 exitwindow, and by the biochemical history of these cellsin terms of exposure and response to mitogenic oranti-mitogenic signalling since last mitosis. The rate ofnon-cycling cells entering apoptosis, �� t �, is similarlymodulated by the balance between pro-apoptotic andsurvival signals integrated at the cellular level.

2.3. Cellular dynamics in ovulatory follicles

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

dNp� t �

dt = �µ − d� t � � Np� t �

dNd� t �dt = d� t � Np� t �

One can then deduce the dynamics of the totalnumber of viable granulosa cells, NT� t �, as well as ofthe growth fraction, GF� t �, representing the relative

Fig. 2. Flow chart of the cellular model. Rectangles represent compartments, straight-lined arrows represent cell flows betweencompartments, the circle arrow marks cell division. Letters correspond to the instantaneous flow rates. µ: constant rate of cell division,d� t �: rate of cell cycle exit, �� t �: rate of cells entering apoptotic death, w� t, s �: rate of phagocytose for a cell that entered apoptosis shours ago.

476


number of proliferating cells amongst the viable cellpopulation:

NT� t � = Np� t � + Nd� t �

G F� t � =Np� t �

NT� t �

dNT� t �dt = µ Np� 0 � exp �

0

t

� µ − d� s � � ds (4)

The values of the five parameters of the sigmoidfunction d� t � can be estimated by fitting the regressionequation (4) on real cell number data.

2.4. Changes in cell numbers (ewe)

The results of the fitting procedure are illustrated inFig. 3. Cell number data were harvested from ewesafter dissection of follicles and counting of their granu-losa cells [4]. The data points are plotted together withthe adjusted curve for the total cell number and itsdecomposition into proliferative and differentiated cells.The number of proliferative cells shows a biphasicpattern: after an initial increase phase, it decreases forthe benefit of the non-cycling population, which endsup constituting the whole population. From an initialmainly proliferative state, the granulosa of the preovu-latory follicle progressively switches to a highly differ-entiated state, so that the growth fraction continuouslydecreases (see insert in Fig. 3). The follicle awaitsovulation with an almost constant number of cells. Inthis sense, it has reached a cellular steady state.However, this is a temporary state, because the onset ofovulation will completely remodel the follicle into acorpus luteum. The number of cells that is reached atthe steady differentiated state is directly related to thecapacity of œstradiol secretion, through which thefollicle takes part in the endocrine interactions betweenthe ovaries and the hypothalamic-pituitary axis. Theobservation that this number is fully determined by thespeed of exhaustion of the proliferative compartmentleads to the notion of follicular proliferative resources.

2.5. Follicular sensitivity towards the cell cycle exitrate: proliferating resources

The corresponding curve for the exit rate is shown inFig. 4. The minimal value of d� t � roughly equals halfthe value of µ, while its maximal value roughly reachestwice the value of µ. The time when the value of d� t �exceeds that of µ (tE in Fig. 4) is of critical importance,as it corresponds to the number of proliferating cellsstarting decreasing. Starting from similar initial condi-

tions, the earlier this time occurs, the earlier and loweris the peak in the proliferating cell number Np� t �. As aresult, the number of fully-differentiated cells availableat the end of follicular development is lower, and thedecrease in the growth fraction is faster (see Fig. 5). Inthe ovine species, follicles of poly-ovulating breedsovulate at a smaller size, associated with a lowernumber of granulosa cells compared to mono-ovulatingbreeds. According to the model, the smaller number ofgranulosa cells in the preovulatory follicles of poly-ovulating breeds could be accounted for by a fasterincrease in the cell cycle exit rate, corresponding todifferential exploitation of the follicular proliferativeresources.

2.6. Experimental measurement of the growthfraction

This experiment [5] aimed at (i) determining thechanges in the growth fraction of granulosa cells in the‘Î le-de-France’ mono-ovulating sheep breed comparedto the ‘Romanov’ poly-ovulating breed and (ii) study-ing the effect of overstimulating granulosa cells withFSH on these changes. The former objective corre-sponds to the testing of the hypothesis derived from themodel results on differential increases in the cell cycleexit rate according to the breed, while the latter is a firststep in understanding the way FSH acts on the cellcycle exit rate. The experimental steps included ova-riectomy of control and FSH-treated adult cyclic ewes,dissection of follicles and pooling of granulosa cellsaccording to follicular diameter, granulosa cell isola-

Fig. 3. Changes in cell numbers. Main plot: changes in thenumber of viable cells NT� t � (solid line), proliferating cellsNp� t � (dotted line) and differentiating cells Nd� t � (dashed line),as a function of the follicular age t. Insert: changes in the growthfraction GF� t � of the granulosa cell population as a function ofthe follicular age t.

477

To cite this article: F. Clément et al., C. R. Biologies 325 (2002) 473–485

tion, and continuous [3H] thymidine labelling andcolcemid administration to cultured cells. The determi-nation of the growth fraction was based on the proce-dure of Maekawa and Tsuchiya [6], which was initiallyproposed to determine the durations of the different cellcycle phases. This method is also appropriate to deter-mine the growth fraction in cell populations. Briefly, itconsists in the continuous and combined administrationof [3H] thymidine and colcemid, respectively to labelall proliferating cells lying in the S phase of the cycleand to block mitosis. After a delay corresponding to theduration of the G2 phase, the first labelled mitosisappear so that four states can be detected: labelled orunlabelled dividing cells, labelled or unlabelled non-dividing cells. The proportion of cells in the unlabelled,non-dividing state decreases before reaching a plateauafter a delay corresponding to the duration of the G1phase. The height of this plateau (modulo unity) finallygives an estimation of the growth fraction.

Fig. 4. Changes in the cell cycle exit rate d� t � as a function ofthe follicular age t compared to the constant value of the celldivision rate µ (horizontal line). E: intersection point whered� t � = µ, I: inflexion point.

Fig. 5. Follicular sensitivity towards the cell cycle exit rate. Changes in the cell cycle exit rate (top left panel, expressed in h−1 ),proliferating cells (top right panel, expressed in 106 cells), total viable cell number (bottom left panel, expressed in 106 cells) and growthfraction (bottom right panel) as a function of follicular age. The dotted line corresponds to the intersection time E as defined in Fig. 4,occuring earlier compared to the solid line.

478


2.7. Experimental results

2.7.1. Comparison ‘Romanov’/‘Île-de-France’ inbasal conditions

Several lessons can be drawn from the results obtainedin basal (non stimulated) conditions (left part of Fig. 6).We can first observe that, whatever the folliculardiameter, the growth fraction is strictly lower thanunity, which means that the granulosa populationappears always heterogeneous during terminal follicu-lar development. Further, the growth fraction continu-ously decreases. The drop is particularly markedbetween 3 and 4 mm diameter, which corresponds tothe period when granulosa cells acquire LH receptors.Finally, the speed of decrease (as measured by the slopeof the regression line between the growth fraction anddiameter after logit transformation) is higher in follicles

from ‘Romanov’ compared to ‘Î le-de-France’ ewes. Inagreement with the model prediction, this result sug-gests that the increase in the cell cycle exit rate duringterminal follicular development is faster in ‘Romanov’ ,compared to ‘Î le-de-France’ follicles. The impact ofthis result is emphasized by comparative physiology.Indeed, in women suffering from severe forms of thePolyCystic Ovary Syndrome, follicles stop their devel-opment at an intermediary stage. Their failing ovulationmight be related to alterations in the control of granu-losa cells proliferation such as those observed inRomanov ewes.

2.7.2. Comparison ‘Romanov’/‘Île-de-France’ afterstimulation by FSH

Under FSH overstimulating condition, the pattern ofthe growth fraction remains almost unchanged in fol-

Fig. 6. Changes in the growth fraction throughout follicular development. Changes in the growth fraction in follicles from untreated‘Romanov’ (top left panel), untreated ‘Î le-de-France’ (bottom left panel), stimulated ‘Romanov’ (top right panel) and stimulated‘Î le-de-France’ (bottom right panel) ewes, as a function of follicular diameter (from C. Pisselet et al., Reprod. Nutr. Dev. (2000), withpermission).

479


licles from ‘Î le-de-France’ ewes, while the decrease inthe growth fraction is slowed down in the follicles ofRomanov ewes. Hence, in such condition, there is nosignificant difference between the mono and poly-ovulating breeds. Romanov granulosa cells appearmore sensitive to exogenous FSH in terms of prolifera-tion. Alternatively, they may be less sensitive to endog-enous FSH. In any cases, the underlying mechanismsneed to be investigated. They could imply either theslowing-down of the increase in the cell cycle exit rate,or a reentry of cells from the G0 phase into the cellcycle.

3. Optimal strategy for exploitingfollicular resources

The notion of follicular proliferative resources leadsto question whether there exists an optimal strategy ofexploiting these resources to reach ovulation. Thisstrategy would consist in controlling the cellular tran-sition rates in an optimal way. Hence, both the cellcycle exit and apoptosis entry rates are no longer givenby parametric functions of the follicular age, but are apriori unknown functions to be determined, respec-tively designated as d*� t � and �*� t �. They are onlyassumed to take positive and bounded values(dmin ≤ d*� t � ≤ dmax and �min ≤ �*� t � ≤ �max).

Optimality is defined in regards of the physiologicalconditions for ovulation, or rather, for the onset of theovulatory LH surge. The exact mechanism leading tothe LH surge is unknown. The only indubitable pointlies in the importance of rising levels of plasmaticœstradiol [7], which signals the increase in the maturitylevel of ovarian follicles. The LH surge thus resultsfrom a tight synchronisation between follicular cells, onthe one hand, and hypothalamo-pituitary cells, on theother hand. This constraint of synchronisation has beenintroduced in terms of a constraint on the differentiationlevel of the follicle at the surge time � tf � ; reaching athreshold � Nc � on follicular differentiated cells isneeded for triggering the surge. We studied threedifferent modes for the œstradiol action upon LHrelease, corresponding to three expressions of thethreshold. The threshold concerns either the instanta-neous number of differentiated cells (relation (5)), orthe cumulative number of differentiated cells (relation(6)), or the weighted cumulative number with greatestweighting for the contributions that are close to thesurge instant (relation (7)).

Nd� tf � = Nc (5)

�t0

tfNd� t � dt = Nc (6)

�t0

tfexp

− c � tf − t �Nd� t � dt = Nc (7)

We finally sought for the optimal cellular transitionrates allowing to reach the target value Nc in minimaltime.

3.1. Optimal cellular dynamics

Whatever the target may be, the optimal solutionconsists in applying the minimal apoptosis rate �min andswitching the cell cycle exit rate from its minimalbound dmin to its maximal bound dmax at a given time ts(see Fig. 7). The optimal cell cycle exit rate can thus beconsidered as a limiting case of a sigmoid function witha very steep increase. The chronology of folliculardevelopment is fully determined by both the switchingand surge times, whose values are explicitly given asfunctions of the initial values and other parameters (seereference [8] for details). The switching time ts occursall the more lately than the conditions for reaching thetarget are demanding (high value for Nc), while thedelay between the switching and surge times dependson the target (lowest delay with instantaneous target,highest with cumulated one).

The number of viable cells (NT (t) = Np (t) +Nd (t)) can be expressed as a function of the parametersof the control problem:

dNT� t �

dt

= � µ Np� 0 � exp � � µ − dmin � t � if t < ts

µ Np� ts � exp � � µ − dmax �� t − ts � � if t > ts

Fig. 7. Optimal ‘bang-bang’ cell cycle exit rate. Thin line: valueof the constant division rate µ , thick segments: optimal cellcycle exit rate value as a function of follicular age (ts switchingtime, tf surge time).

480


This allows in turn to fit the values of the optimalstrategy parameters � dmin, dmax, ts � from the experimen-tal cell counts described in section 2.4 (see Fig. 8) andto get information back on the surge time tf andthreshold Nc.

3.2. Ovulation/anovulation

Deviation from the optimal response can result froma shift in the time schedule for the switching instant, asit is illustrated in Fig. 9. The way of shifting theswitching time has great repercussions on the issue ofthe problem. Indeed, when the actual switching timeoccurs earlier than the optimal one, the proliferativeresources of the follicular granulosa may be exhaustedtoo quickly. In the instantaneous case, the target mayremain beyond reach. In the cumulated case, the delayin reaching the target may hamper the normal progressof the ovarian cycle. The corresponding follicularresponse is thus completely non-optimal. On the con-trary, when the switching time occurs later than theoptimal one, the target can be reached with limiteddelay, but at the expense of producing follicular cells inexcess. The corresponding follicular response is thusonly suboptimal.

4. Mechanistic approach of the cellulartransition rates

The previous sections have highlighted the crucialrole of the dynamics of the cellular transition rates inthe outcome of follicular development. In the long run,we aim at expressing these rates, no more as functionsof the follicular age, but as functions of the historicalbackground of granulosa cells in terms of exposure to

gonadotrophins. Hence, to further understand the con-trol of the granulosa dynamics, we need to investigatethe action of gonadotrophins on the cellular transitionrates.

Both FSH and LH mainly operate through G protein-coupled transmembrane receptors, transducing theirsignal by the activation of the enzyme adenylyl cyclaseand the production of second messenger cyclic Adenos-ine Mono-Phosphate (cAMP) [9]. The binding of FSHto its transmembrane receptors triggers an intracellularsignal via the heterotrimeric G proteins. The FSH-bound receptor activates the � G stimulatory � � Gs �protein, which interacts with adenylyl cyclase to gen-erate an increase in cyclic AMP. Once cAMP issynthesised, it either binds and activates specific pro-tein kinases such as protein kinase A or is degraded bycyclic nucleotide phosphodiesterase [9]. The accumu-lation of FSH-induced cAMP coincides with the appear-ance and subsequent dramatic increase in LH receptors,allowing LH to act synergistically or as a surrogate forFSH in granulosa cells [10]. As it also appears to be akeypoint in cell cycle arrest [11], a better understandingof gonadotrophin-induced cAMP production wouldhelp to gain insight into the dynamics of the cell cycleexit rate amongst granulosa cells during terminal devel-opment. We now focus on the FSH branch of the cAMPcascade, as it is the first to operate in granulosa cells[12].

Fig. 8. Optimal changes in cell numbers.

Fig. 9. Consequences of a backward or forward shift in theswitching time upon the changes in the total number of granulosacells. Solid line: changes in the total cell number when applyingoptimal switching time � ts = 110 h �. Dotted lines: changes inthe total cell number when applying shifted switching time;dotted lines upper than the solid line correspond to a forward-shifted switching time (ts > 110 h, by 10 h increment fromdownwards to upwards), while dotted lines lower than the solidline correspond to a backward-shifted switching time (ts > 110 h,by −10 h increment from upwards to downwards). The dashedline joins the values of the time � tf � elapsed before reaching asame cumulated target condition � Nc �.

481


4.1. FSH signal processing by granulosa cells

The biochemical model includes the steps of FSHsignal detection, relay and amplification, and extinc-tion:– binding of FSH to its receptor � RFSH � results in theformation of an active complex � XFSH �

FSH + RFSH\k−

k+XFSH

– bound receptors activate adenylyl cyclase (E) througha conformational change in the associated G protein

E EFSH

XFSH

– activated adenylyl cyclase � EFSH � synthesises cAMPfrom the substrate Mg+ + ATP (adenosine triphosphate)

Mg+ + ATP + EFSH →x cAMP

– cAMP is hydrolysed into AMP by phosphodiesterase

cAMP →kpde

AMP

– bound receptors are subject to a desensitisationprocess through cAMP-mediated phosphorylation

XFSH ––––→q� cAMP �

XpFSH

– phosphorylated inactive complexes � XpFSH � undergointernalisation into the cell, where receptors are disso-ciated from FSH

XpFSH →ki

Ri

– internalised receptors Ri are recycled back to the cellmembrane, while FSH is hydrolysed

Ri →kr

RFSH

Considering only reactions relevant to folliculardevelopment allows some simplifications to be made.Reactions generating short-life intermediary species areneglected. In particular, the cycle of G-proteinactivation/deactivation is not modelled explicitly. Theprocess of receptor synthesis is assumed to compensateboth for intracellular receptor degradation and for thedepletion of the receptor pool during cell division, sothat the total number of FSH receptors in differentstates (free, active, phosphorylated and internalised)remains constant [13], leading to the following cellularcycle for FSH receptors under different states:

FSH + RFSH \ XFSH

↑ ↓

Ri ← XpFSH

Finally, cAMP-independent desensitisation is nottaken into consideration, as its behaviour during thematuration of granulosa cells is not yet known. Inaddition, the amount of FSH is assumed to be suffi-ciently large so that its concentration is unaffected bybinding to receptors.

4.2. Biochemical dynamics

The control of cAMP levels in granulosa cellsinvolves both fast biochemical processes, occurring ona time scale of a few minutes, such as binding anddesensitisation, and slower physiological processes last-ing hours or even a few days, which mainly result inchanging the efficiency of the enhancement of cAMPsynthesis by stimulated FSH receptors via adenylylcyclase activation. The design of our model followsfrom the interactions between these contrasting bio-chemical and physiological dynamics. The followingsystem intends to render the dynamics of intracellularcAMP in an average granulosa cell in response to FSH:

dRFSH

dt = k− XFSH + kr Ri − k+ FSH RFSH (8)

dXFSH

dt = k+ FSH RFSH − � q + k− � XFSH (9)

dEFSH

dt = b �rXFSH − EFSH � EFSH (10)

dcAMPdt = x EFSH − kpde cAMP (11)

dXpFSH

dt = q XFSH − ki XpFSH (12)

dRi

dt = ki XpFSH − kr Ri (13)

RFSH, XFSH, XpFSH, Ri are respectively the concentra-tions of free, bound-active, bound-inactive (phosphory-lated) and internalised FSH receptors. EFSH is theconcentration of activated adenylyl cyclase and cAMPthe concentration of intracellular cAMP. k+, k−, k i, k rare respectively the rate constants for FSH-binding,FSH-unbinding, bound-complex internalisation andreceptor recycling to the cell membrane.

Equations (8), (11), and (13) result from applying theprinciple of mass-action. The concentrations of ATPand phosphodiesterase enzyme are assumed to beconstant and non-limiting, so that their effects areincluded in the respective kinetic constants x and kpde.Desensitisation through phosphorylation is assumed tobe cAMP-mediated in a dose-dependent, increasing andsaturated manner. It follows that the desensitisation rate

482


q in equations (9) and (12) is a Hill function ofintracellular cAMP:

q � cAMP � = k cAMPc

nc + cAMPc

with k, n and c real, strictly positive parameters.

The dynamics of the coupling variable EFSH, whichwe assimilated to activated adenylyl cyclase, incorpo-rates specific features of FSH signal transduction ingranulosa cells. It accounts for both FSH-dependentand auto-amplified settling and efficiency of the cAMPcascade through follicular development. The b param-eter acts as a time scale parameter. Whenever it takes alow value � b ! 1 �, the changes in the coupling vari-able EFSH are slower than those of the other variables ofthe model. The r parameter measures the degree ofsignal amplification and represents the average number

of adenylyl cyclase molecules activated by one boundreceptor at steady state.

Finally, equations (8) to (12) are subject to theconservation law

RT = RFSH + XFSH + XpFSH + Ri

where RT is the constant size of the global receptorpool. This allows us to reduce the model by oneequation.

4.3. Control of FSH-induced cAMP levels

The cAMP modelling approach allows two meetingpoints with possible experimental investigations. Thefirst concerns the influence of the model parameters onthe cAMP steady-state level reached after exposure toconstant FSH input, while the second concerns thepattern of cAMP dynamics in response to a givenpattern of FSH input.

Fig. 10. Control of cAMP steady-state levels. The different panels illustrate the influence of the FSH input (in M, left top panel), the sizeof the receptor pool RT (in 104 molecules/cell, middle top panel), and other parameter values: k+ � M−1 s−1

�, k− � s−1�, r (dimensionless),

x � s−1�, kpde � s−1

�, ki � s−1� and kr � s−1

�, from left to right and top to bottom on cAMP steady-state level.

483


4.3.1. Control of FSH-induced cAMP steady-statelevels (constant input)

The way cAMP steady state level (cAMP*) is influ-enced by a parameter can be analysed formally (seedetails in [12]). This value is an increasing function ofFSH input, the size of receptor pool RT, the rateconstants k+, ki and kr and the half-saturating cAMPconcentration n. Conversely it is a decreasing functionof the unbinding rate k−, the hydrolysis rate kpde and thesaturation value k. Interestingly, the c parameter, whichrules the rate of increase in the phosphorylation rate(the slope of the Hill function), has a non univocalinfluence on cAMP*, depending on the value of RTcompared to a threshold value RThresh, given by:

RThres =n kpde

x r ��1 +k−

k+ FSH�+ k

2 � 1ki

+ 1kr

+ 1k+ FSH ��

For values of RT lower than RThres, cAMP* is anincreasing function of c, while it is a decreasingfunction for values higher than RThres. This means that,if the receptor pool is small, the cAMP steady-statelevel rather benefits from an almost all-or-nothingeffect of cAMP level on the phosphorylation process,rather than from a progressive, smoother effect.

Beyond this qualitative study, quantitative dose-effect-like curves can be constructed, such as those dis-plaid in Fig. 10. As variations in the model parametervalues correspond to physiological, pathological orpharmacological alterations in the different steps of FSHsignal processing by granulosa cells, these theoreticalcurves should be at least partly experimentally testable.

4.3.2. Control of FSH-induced cAMP transientlevels

The model retraces the long-term behaviour of cAMPin granulosa cells in response to FSH alone. From any

Fig. 11. Response to non-constant FSH stimulation. From left to right and top to bottom: FSH input (M), changes in the levels (in 104

molecules/cell) of free FSH receptors, bound FSH receptors, activated adenylyl cyclase, cAMP, phosphorylated receptors, internalisedreceptors and in phosphorylation rate � s−1

�. Time unit is 102 s.

484


given FSH input, the cAMP output pattern can bepredicted, such as it is illustrated in Fig. 11 with realFSH data [14]. As cAMP is an accessible variable onthe experimental ground, the predicted output could beconfronted with experimental measures. On the con-trary, the intermediary species (as the different states ofFSH receptors) do not seem susceptible of easy mea-sure on the long term, even if they have a directbiochemical meaning.

4.4. Possible experimental investigations

The most appropriate data would consist in repeatedmeasurements of intracellular cAMP throughout thedevelopment of dynamically-monitored follicles. Con-comitantly, FSH levels should be measured. To trackcAMP production until the ovulatory stage, one need tobe placed in controlled situations. As far as physiologi-cal situations are concerned, the luteal phase in rumi-nant species would be the most appropriate window ofthe ovarian cycle to harvest data. In such species,terminal follicular development during luteal phase ismainly FSH-dependent and thus fulfills the require-ments for investigating FSH-induced cAMP productionthroughout terminal development. As far as pharmaco-logical situations are concerned, appropriate conditionscan be artificially reproduced by means of eitherprevious desensitisation with GnRH agonists, or use of

GnRH antagonists, and administration of recombinantFSH with known bioactivity. The most limiting point inboth situations is the need for dynamical non-invasivemeasurements of intracellular cAMP. This might beachievable in the future through repeated ultrasound-guided follicular cell pick-up as follicular developmentprogresses. In domestic animals, follicular fluid pick-upare already running well and technical progress ofdevices could render direct cell pick-up feasible in themedium term.

5. Further modelling and conclusion

Future improvement of the model could include thecoupling of both FSH and LH signalling pathways andthe design of a more biochemically-based formulationto the equation ruling the changes in the couplingefficiency. We intend to use the resulting cAMP dynam-ics as a control variable on the cellular transition rates,ruling the commitment of granulosa cells towardseither proliferation, differentiation or apoptosis. Asinadequate response of granulosa cells to gonadotro-phin signals may have great repercussions on folliculardevelopment and may even lead to infertility, a realisticmodel characterising both physiological and pathologi-cal signal transduction would be very useful in simu-lating new therapeutic strategies.

References

[1] G. Greenwald, S. Roy, Follicular development and its control, in:E. Knobil, J.D. Neill (Eds.), The Physiology of Reproduction, Raven Press,New York, 1994, pp. 629–724.

[2] J.S. Richards, Maturation of ovarian follicles: actions and interactionsof pituitary and ovarian hormones on follicular cell differentiation, Physiol.Rev. 15 (1980) 51–89.

[3] F. Clément, M.A. Gruet, P. Monget, M. Terqui, E. Jolivet, D. Monni-aux, Growth kinetics of the granulosa cell population in ovarian follicles: anapproach by mathematical modelling, Cell Prolif. 30 (1997) 255–270.

[4] C.G. Tsonis, R.S. Carson, J.K. Findlay, Relationships between aro-matase activity, follicular fluid œstradiol 17(�) and testosterone concentra-tions, and diameter and atresia of individual ovarian follicles, J. Reprod.Fertil. 72 (1984) 153–163.

[5] C. Pisselet, F. Clément, D. Monniaux, Fraction of proliferating cells ingranulosa during terminal development in high and low prolific sheep breeds,Reprod. Nutr. Dev. 40 (2000) 295–304.

[6] T. Maekawa, J. Tsuchiya, A method for the direct estimation of thelength of G1, S and G2 phases, Exp. Cell Res. 53 (1968) 55–64.

[7] R. Goodman, Neuroendocrine control of the ovine estrous cycle, in:E. Knobil, J. Neill (Eds.), The Physiology of Reproduction, Raven Press, NewYork, 1994, pp. 659–709.

[8] F. Clément, Optimal control of the cell dynamics in the granulosa ofovulatory follicles, Math. Biosci. 152 (1998) 123–142.

[9] J.S. Richards, L. Hedin, Molecular aspects of hormone action inovarian follicular development, ovulation and luteinization, Annu. Rev.Physiol. 50 (1988) 441–463.

[10] R. Webb, B.G. England, Identification of the ovulatory follicle in theewe: associated changes in follicular size, thecal and granulosa cell luteiniz-ing hormone receptors, antral fluid steroids, and circulating hormones duringthe preovulatory period, Endocrinol. 110 (1982) 873–881.

[11] J.J. Peluso, A. Pappalardo, B.A. White, Control of rat cell mitosis byphorbol ester-, cyclic amp- and œstradiol-17-�-dependent pathways, Biol.Reprod. 49 (1993) 416–422.

[12] F. Clément, D. Monniaux, J. Stark, K. Hardy, J. Thalabard, S. Franks,D. Claude, A mathematical model of FSH-induced cAMP production inovarian follicles, Am. J. Physiol. Endocrinol. Metab. 281 (2001) E35–E53.

[13] R.S. Carson, J.K. Findlay, H.G. Burger, A.O. Trounson, Gonadotropinreceptors of the ovine ovarian follicle during follicular growth and atresia,Biol. Reprod. 21 (1979) 75–87.

[14] G.P. Adams, R.L. Matteri, O.J. Ginther, Effect of progesterone onovarian follicles, emergence of follicular waves and circulating follicle-stimulating hormone in heifers, J. Reprod. Fert. 96 (1992) 627–640.

485


contribution of a mathematical modelling approach to the...

Documents