Coupled somatic cell kinetics and germ cell growth: multiscale model-based insight on ovarian follicular development

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1 Coupled somaic cell kineics and germ cell growh: muliscale model-based insigh on ovarian follicular developmen Philippe Michel, Thomas Siehl, Danielle Monniaux, Frédérique Clemen To cie his version: Philippe Michel, Thomas Siehl, Danielle Monniaux, Frédérique Clemen. Coupled somaic cell kineics and germ cell growh: muliscale model-based insigh on ovarian follicular developmen. Muliscale Modeling and Simulaion: A SIAM Inerdisciplinary Journal, Sociey for Indusrial and Applied Mahemaics, 2013, 11 (3), pp <hal v2> HAL Id: hal hps://hal.archives-ouveres.fr/hal v2 Submied on 9 May 2016 HAL is a muli-disciplinary open access archive for he deposi and disseminaion of scienific research documens, wheher hey are published or no. The documens may come from eaching and research insiuions in France or abroad, or from public or privae research ceners. L archive ouvere pluridisciplinaire HAL, es desinée au dépô e à la diffusion de documens scienifiques de niveau recherche, publiés ou non, émanan des éablissemens d enseignemen e de recherche français ou érangers, des laboraoires publics ou privés.

2 COUPLED SOMATIC CELL KINETICS AND GERM CELL GROWTH: MULTISCALE MODEL-BASED INSIGHT ON OVARIAN FOLLICULAR DEVELOPMENT PHILIPPE MICHEL, THOMAS STIEHL, DANIELLE MONNIAUX, AND FRÉDÉRIQUE CLÉMENT Absrac. The developmen of ovarian follicles is a unique insance of a morphogenesis process sill occurring during adul life and resuling from he ineracions beween somaic and germ cells. In mammals, he iniiaion of follicular developmen from he pool of resing follicles is characerized by an increase in he oocye size concomian wih he surrounding somaic cells proliferaing o build an avascular issue called granulosa. We presen a sochasic individual-based model describing he firs sages of follicular developmen, where he cell populaion is srucured wih respec o age (progression wihin he cell cycle) and space (radial disance from he oocye). The model accouns for he molecular dialogue exising beween he oocye and granulosa cells. Three dynamically ineracing scales are considered in he model: (i) a microscopic, local scale corresponding o an individual cell embedded in is immediae environmen, (ii) a mesoscopic, semi-local scale corresponding o anaomical or funcional areas of follicles and (iii) a macroscopic, global scale corresponding o he morphology of he follicle. Numerical simulaions are performed o reproduce he 3D morphogenesis of follicles and follow simulaneously he deailed spaial disribuion of individual granulosa cells, heir organizaion as concenric layers or funcional cell clones and he increase in he follicle size. Deailed quaniaive simulaion resuls are provided in he ovine species, in which well characerized geneic muaions lead o a variey of phenoypic follicle morphogenesis. The model can help o explain pahological siuaions of imbalance beween oocye growh and follicular cell proliferaion Key words. Oocye, granulosa, ovarian follicle, cell proliferaion, sochasic individual based models AMS subjec classificaions. 60G55, 92C17, 92C37, Inroducion. A highly efficien reproducive capaciy is a major advanage for species preservaion, faced wih he naural selecion process, and for individuals wihin species. In mammalian females, ovarian funcion is he subjec of inensive invesigaions wih he aim o improve he reproducive capaciy of domesic and wild animal species and o rea ovarian failures leading o inferiliy in humans. The issues are crucial for boh clinical and zooechnical applicaions. In humans, he prevalence of he polycysic ovarian syndrome, which is a main cause of inferiliy, has been esimaed a up o 10% among reproducive-age women [19]. Improvemen of reproducive bioechnologies, including in viro ferilizaion, inra-cyoplasmic sperm injecion, frozen embryo replacemens and egg donaion, is a key issue for a beer managemen of reproducion. Improving he knowledge upon ovarian funcion and is conrol will help o improve he success of assised reproducive echnologies, hence o preven ovarian failure or hypersimulaion syndrome in women and o manage This work was funded by he INRIA Large Scale Iniiaive REGATE (Regulaion of he GonAdoTropE axis). hps:// Universié de Lyon ; CNRS ; Ecole Cenrale de Lyon, Insiu Camille Jordan, 36 avenue Guy de Collongue Ecully Cedex - France. Inerdisciplinary Cener for Scienic Compuing (IWR). Im Neuenheimer Feld Heidelberg - Germany. INRA, UMR85 Physiologie de la Reproducion e des Comporemens, F Nouzilly, France. CNRS, UMR7247, F Nouzilly, France. Universié François Rabelais de Tours, F Tours, France. IFCE, F Nouzilly, France. INRIA Paris-Rocquencour Research Cenre, Domaine de Voluceau Rocquencour - B.P. 105, Le Chesnay, France. 1

3 2 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT ovulaion rae in farm species. The ovarian follicles consiue he funcional unis wihin he ovary. Follicular developmen, namely folliculogenesis, is a mulisage process involving growh and funcional mauraion. In mammals, i commences wih he recruimen of primordial follicles from a resing pool formed early in life and ends wih eiher ovulaion or follicular deah by aresia [30]. Follicular developmen is ighly and finely regulaed by hormones and cell ineracions, and can be schemaically divided ino wo phases, corresponding o basal and erminal developmen [32]. Terminal follicular developmen has been exensively sudied and he role of he piuiary gonadoropins in is conrol is now well esablished [14, 20, 47]. Recen daa from biologiss and clinicians have also poined a he imporance of basal follicular developmen in seing up he ovarian reserve of gonadorophin responsive follicles. In humans as in domesic species of agronomical ineres, his ovarian reserve is he direc arge of assised reproducive bioechnologies and reamens for feriliy [46, 5, 37]. However, despie an increasing ineres and recen idenificaion of imporan regulaing facors, mechanisms regulaing basal follicular developmen are no fully undersood ye [21, 15]. On he microscopic scale, each primordial follicle is composed of a germ cell named oocye, surrounded by a single layer of flaened resing somaic cells named granulosa cells. The iniiaion of growh is characerized by hree main evens, namely a change in he shape of granulosa cells from flaened o cuboidal, an increase in he number of granulosa cells which begin o proliferae, and enlargemen of he oocye. Thereafer, during basal follicular developmen muliple layers of proliferaing granulosa cells develop around he oocye while he oocye furher enlarges (see Figure 1.1). In sheep, he granulosa cell populaions double some imes before anrum formaion and he ime aken o complee his process varies beween 50 and 150 days wih very lile follicular aresia [31]. On he macroscopic scale, he so-called preanral follicle grows in he ovarian corex as a compac cellular aggregae wih a spheroidal shape. Imporanly, his developmen is coordinaed by complex regulaions involving direc physical conacs ha occur a he granulosa cell-oocye inerface and he molecular dialogue exising beween he oocye and is surrounding granulosa cells [1, 29, 23]. The oocye synhesizes and secrees specific facors, such as growh and differeniaion facor 9 (GDF9) and bone morphogeneic proein 15 (BMP15), which ac on granulosa cells o modify heir proliferaion, funcion and differeniaion [38, 25]. Conversely, granulosa cell-derived facors, mainly he Ki Ligand (KITLG), play a deerminan role in simulaing oocye growh [22, 26, 45]. The objecive of his sudy is o propose a model explaining how he ineracions beween he oocye and is surrounding granulosa cells conribue o build a preanral follicle, i.e. a cellular srucure wih a specific morphological and funcional organizaion. The proposed model gives a sochasic and discree descripion of basal follicular developmen before anrum formaion. I suppors he previously proposed hypohesis ha follicles are consruced by he non-random, radial proliferaion of granulosa cell clones, which form long, hin, unbranched columns across he follicle wall [43, 3]. The model has been developed o reproduce qualiaively and quaniaively basal follicular developmen in sheep and o explain some pahological cases of disconnecion beween oocye growh and proliferaion of is surrounding granulosa cells, which have been repored in ewes carrying muaions in he fecundiy genes BMP15 or is

4 Muliscale model-based insigh on ovarian follicular developmen 3 Fig Developmen of preanral follicles. Afer iniiaion of growh, each primary follicle is composed of a germ cell named oocye, surrounded by a single layer of cuboidal cells. During follicular developmen, he oocye enlarges, and he granulosa cells proliferae and form gradually muliple concenric layers around he oocye. These secondary and primary follicles consiue a pool of so-called preanral follicles. Their developmen, called basal follicular developmen, is coordinaed by igh ineracions exising beween he oocyes and heir surrounding granulosa cells. recepor BMPRIB [4, 48, 2]. Sheep is a species of agronomical ineres which presens also imporan similariies wih he human model for various characerisics of female reproducion [32]. For hese reasons and he availabiliy of quaniaive daa on basal follicular developmen of boh normal wild-ype ewes [28] and ewes carrying muaions [4, 48], sheep was a choice species for a model. 2. Model design. In his secion, we inroduce an Individual Based Model (Markovian IBM coninuous in ime and discree in space: see [6]) ha describes he evoluion of he granulosa cell populaion using a probabilisic formalism previously used in [13, 16, 6]. Each granulosa cell is represened by is age and locaion in space (secion 2.2). The changes in he follicle morphology involve emporal changes in granulosa cell proliferaion and displacemen (secion 2.3) coupled wih oocye growh (secion 2.4). The cell dynamics are governed by a sochasic differenial equaion (SDE) defined on a probabiliy space (Ω, F, P) derived from he IBM model [13, 16, 7, 6] (secion 2.5). We recall ha Ω is he space of all possible evens relaed o follicle growh (cell division, cell displacemen... ), F is he se of admissible evens, i.e. hose whose probabiliy can be assessed hrough he probabiliy funcion P. The noaions used for variables and funcions in his secion are summarized in Table Muliscale approach. The model is designed o sudying he muliscale basis of he morphogenesis of a preanral follicle. The follicle morphology a a given sage of basal developmen can be characerized by he follicle diameer, he diameer of he oocye a is cener, and he number and locaion of he granulosa cells surrounding he oocye. To undersand how he ineracions beween he oocye and surrounding follicular cells can give rise o a ighly uned morphogeneic process, where cell proliferaion and oocye growh are conrolled boh in exen and ime according o a precise and species-specific program, we chose o ake up a muliscale approach ha considers 3 differen levels: 1. The microscopic, local scale corresponds o an individual cell embedded in is immediae environmen. I is srucured in space in a discree, deerminisic

5 4 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT manner hrough a collecion of ses (defined below as L (j,i) (ω)), as well as in age in a coninuous, sochasic manner (disribuion of he cell age a division); 2. The mesoscopic, semi-local scale corresponds o a union of variable size of such ses. On he morphological ground, hese ses are pooled in concenric layers cenered on he oocye (defined below as L i (ω)) ha organize he follicle growh (Cf Figure 2.1). On a funcional ground, hey can be pooled in small adjacen subses (Figure 2.4) ha organize he neighboring of a given cell; 3. The macroscopic, global scale corresponds o variables ha describe he morphology of he follicle as a whole, eiher direcly (oocye diameer d O (), follicle diameer d F (), number of layers), or indirecly (cell number, Z ) Spaial domain. The model is based on he biologically-based hypoheses ha granulosa cell proliferaion is conrolled by facors of he BMP family (BMP15, GDF9) secreed by he oocye, and ha oocye growh is in urn under he conrol of granulosa cell-derived facors (KITLG) (see [27, 18]). These assumpions amoun o making boh he average cell cycle duraion of each granulosa cell and he effec of each granulosa cell on oocye growh depend on he cell disance from he surface of he oocye. Geomery of he Oocye and Granulosa cells. For each ime 0 and for almos every (a.e.) ω Ω, he oocye is assumed o be a compac (incompressible) ball of diameer d O (, ω) cenered on he origin (in he spaial domain) ha we denoe by B N (0, d O (, ω)) R N (N = 1, 2 or 3 is he dimension of he physical space). B N (0, d O (, ω)) := { (x 1,, x N ) saisfying N i=1 } x 2 i d O(, ω)/2. Granulosa cells are locaed in differen cell layers around he oocye and are represened as balls of equal size whose volume (and herefore diameer d G ) is nearly consan in ime (see Figure 1.1). Physical space. We le E (ω) be he space around he oocye, i.e. he whole domain excep he space occupied by he oocye, E (ω) := R N /B N (0, d O (, ω)), N = 1, 2 or 3 fixed, E (ω) = { (x 1,, x N ) saisfying N i=1 } x 2 i > d O(, ω)/2. The mos basic physical space srucure ha is organized radially from he oocye is a layer surrounding he oocye. More precisely, each granulosa cell has o be in a layer defined as follows. Definiion 2.1. (Layer : basic space pariion) The i h layer L i (ω) surrounding he oocye a ime (and ω Ω) is given by (see Figure 2.1) L i (ω) : = B N (0, d O (, ω) + 2i d G )/B N (0, d O (, ω) + 2(i 1)d G ) { = (x 1,, x N ) s d O(, ω) + (i 1)d G N x 2 i < 1 2 d O(, ω) + i d G }. i=1

6 Muliscale model-based insigh on ovarian follicular developmen 5 This layer is conained in he ball of diameer d O (, ω) + 2i d G bu i lies ouside he ball of diameer d O (, ω) + 2(i 1)d G, where d G is he diameer of granulosa cells. Fig Layer definiion and 3D pariion. Panel A: hisological slice of a growing follicle showing he oocye and is surrounding granulosa cells. Panel B: superimposiion of ideal layers (delimied by blue circles) whose common deph is in he order of he diameer of a granulosa cell. Boom C and D panels: 3D pariion. To evaluae he number of cells in he firs (second...) layer surrounding he oocye, we consruc an ideal layer pariion of he space. In spherical coordinaes, a pariion of he layer i is L (.,i) (ω) := {(r, θ, φ) [d O (, ω)/2 + (i 1)d G, d O (, ω)/2 + id G ] [(.)δ θ, (.+1)δ θ ] [(.)δ φ, (.+1)δ φ ], where δ θ and δ φ are in he same order as he diameer of a granulosa cell (d G ). Since he follicle diameer increases wih ime, he volume of L (.,i) (ω) also increases, in he same order as d O (, ω)d 2 G, and he number of granulosa cells locaed in his pariion evolves as d O (, ω)/d G. Definiion ( 2.2. )(Layer pariion) A pariion of he i h layer L i (ω) is a collecion of ses (ω) (N i N ) which saisfies : Almos surely (a.s.) ω Ω, L (j,i) j [0,N i] for all 0, for all j [0, N i ] i) L (j,i) (ω) is a conneced (i canno be wrien as he union of wo nonempy separaed ses) and measurable se (i is possible o evaluae is volume), which is included in L i (ω), ii) L (j,i) (ω) pariions L i (ω), a.s. ω Ω, 0, j j : L i (ω) = L (j,i) (ω), 0, L (j,i) (ω) L (j,i) (ω) =. j [0,N i] iii) L (j,i) (ω) is ime coninuous : a.s. ω Ω, T > 0, i N, j [0, N i ] B

7 6 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT B(R N ) he funcion B T L (j,i) (ω) 1dx (he volume of B L (j,i) (ω)) belongs o C 0 ([0, T ]). Each se L (j,i) is a pariion of layer L i and can be consruced from spherical coordinaes (see Figure 2.1). Since, in case of overcrowding, a granulosa cell can be hrown ou of he curren pariion i belongs o (L (j,i) (ω)), one need boh o locae he neighboring pariions ino which he cell can be displaced when i has o moved, and o assess he volume of his pariion. Definiion 2.3. (Neighborhood) We define he neighborhood of he se L (j,i) V (j,i) (ω) = (ω) as { (j, i ) : for all ɛ > 0 here exiss x R N so ha (ω) B N (x, ɛ) and L (j,i ) (ω) } B N (x, ɛ), L (j,i) which means ha here exiss a sequence u k L (j,i) (ω) and v k L (j,i ) (ω) such ha he disance from u k o v k converges o 0 as k goes o infiniy. Definiion 2.4. (Volume) Since L (j,i) (ω) is a measurable se we can compue is volume by V ol (j,i) (ω) = 1dx. L (j,i) (ω) Variable d O () d F () X k () A k () L (j,i) V ol (j,i) Z Definiion Table 2.1 Variables and funcions. Oocye diameer Follicular diameer Locaion of cell k Age of cell k Pariion of space Volume of space pariion L (j,i) Granulosa cell populaion N Granulosa cell number V (j,i) Neighbourhood of space pariion L (j,i) x j i Densiy of cells in pariion,i ) L(j Funcion Definiion (see secion 2.3 for more deails) p(l (j,i), Z, ) Displacemen probabiliy for cell locaed in he pariion L (j,i) m((j, i), (j, i ), Z, ) Probabiliy of displacemen from se L (j,i) o se L (j,i ) b(k, Z, ) Division probabiliy for cell k 2.3. Granulosa cells dynamics. Definiion 2.5. (Granulosa Cell) Each granulosa cell can be characerized by is locaion in he physical space and age. X k (), X k N 2, denoes he locaion of granulosa cell k in he physical space: X =(j, i) means ha he granulosa cell is locaed in L (j,i) (ω) 1 1 For sake of simpliciy, we will drop he ω in he sequel.

8 Muliscale model-based insigh on ovarian follicular developmen 7 A k (), A k R +, denoes he age of granulosa cell k a ime, which represens he ime elapsed since is laes miosis. The populaion a ime, denoed by Z, is he se of all individuals alive a ime : (2.1) N Z = k=1 δ (Xk (),A k ()), where N is he (discree) number of granulosa cells a ime. The populaion a iniial ime ( = 0) is given by: N 0 Z 0 = k=1 δ (Xk (0),A k (0)). The disance of a given granulosa cell (X k () = (j, i); A k ()) (locaed he i h layer, i 1) from he oocye is simply given by (i 1)d G, where d G denoes he (consan) cell diameer (while V ol G denoes is volume). Granulosa cells are subjec o boh miosis and displacemen evens (here is neiher cell deah nor quiescence): 1. b(k, Z, ) denoes he probabiliy for he k h cell of populaion Z o undergo miosis a ime. 2. p(l (j,i), Z, ) denoes similarly he probabiliy for he k h cell, locaed in he pariion L (j,i) (X k () = (j, i)), o move wihin he physical space a ime. 3. m((j, i), (j, i ), Z, ) denoes he probabiliy ha he displacemen be operaed from se L (j,i) o se L (j,i ) V (j,i). Definiion 2.6. (Average cell cycle duraion : {λ i } i ) The cell cycle duraion is a sochasic variable following a Poisson disribuion, so ha: (2.2) b(k, Z, ) = 1 e A k()/λ i, X k () = (j, i), where λ i is he average cell cycle duraion (i.e. he ime elapsed beween wo consecuive miosis or from he cell birh o he firs miosis) for any cell locaed in se L (j,i). λ i only depends on he layer index ; i increases wih respec o i, which means ha he proliferaion rae is lowered as he cell disance from he oocye surface increases. Definiion 2.7. (Tolerance o overcrowding : (µ, σ)) The k h cell is displaced as soon as he cell number in se L X k()=(j,i) becomes oo large compared o is volume (see Figure 2.2). The displacemen ime follows a logisic disribuion, so ha he probabiliy of displacemen reads: (2.3) p(l (j,i), Z, ) = e x ji µ σ, where x ji = Z, L (j,i) V ol G /V ol (j,i), wih Z, L (j,i) is he cell number in L (j,i), µ accouns for he local level of overcrowding, and variance σ can be inerpreed as he degree of olerance o overcrowding.

9 8 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Fre de rique CLE MENT 0 0 Fig Local displacemen of granulosa cells from L(j,i) (yellow area) o L(j,i ) V (j,i) (red areas). When he cell number becomes oo large compared o he capaciy of an area (he yellow one here), a cell can be displaced o anoher area in is neighborhood (in red) wih a probabiliy proporional o he free space in he arge area. Lef panel: 3D view showing all possible adjacen arge areas from he curren area. Middle panel: 3D view showing wo over six of he adjacen possible arge areas in he longiude direcion. Righ panel: 2D view of a follicle (ransversal) secion showing four over six of he adjacen possible arge areas, in eiher he radius or he circumference direcion. The green circle corresponds o he oocye a he cener of he follicle. Definiion 2.8. (Follicle diameer : df ) The follicle diameer df can be esimaed from he number of granulosa cells N, he diameer of a granulosa cell dg and he oocye diameer do (): 1/3 N 3 (2.4) dg + do ()3, df () = µ where µ, he average filling coefficien, is he average number of cell per volume uni (see secion for deails). The displacemen is preferenially operaed o he less overcrowded neighboring ses. X () A he ime when he k h cell is displaced, i moves from se L k o anoher se 0 0 (j,i ) (j,i) L V according o he following discree law derived from eq. (2.3): (j 0,i0 ) (2.5) 0 1 p(l, Z, ). X (j 00,i00 ) (1 p(l, Z, )) 0 m((j, i), (j, i ), Z, ) = (j 00,i00 ) L (j,i) V 2.4. Oocye cell dynamics. Oocye growh rae is known o be simulaed by he KITLG facor which is expressed as boh diffusible (KITLG1) and membrane bound (KITLG2) isoforms by granulosa cells (see [35, 44]). Through heir direc conacs wih he oocye membrane, he granulosa cells of he firs layer have a higher conribuion o oocye growh han more disan cells, especially since KITLG2 signals more efficienly han KITLG1. Accordingly, in he model, oocye growh rae depends on granulosa cell-derived signals in a layer specific and oocye diameer-dependen manner. Definiion 2.9. (Granulosa cell-derived growh facor : ( λκii )i ) Each granulosa cell conribues o he growh of he oocye. Parameer κi and more precisely he normalized parameer λκii represens he inensiy of KITLG secreion from any granulosa cell locaed in he ih layer surrounding he oocye. κi is decreasing wih respec o i, i.e., o he disance from he oocye (see secion 5.1).

10 Muliscale model-based insigh on ovarian follicular developmen Markovian IBM - ime evoluion model. The dynamics of he oocye diameer, d O (), are ruled by a sochasic differenial equaion (SDE): (2.6) d O () = d O (0) + 0 ( do (s ) ) α( 1 do (s ) ) β i 1 κ i log 2 (e)λ i Z s, L i s ds, where d O (0) is he oocye diameer a iniial ime, Z s, L i s is he cell number in layer L i s, and (α, β) are real parameers. The precise choice of α and β values deermines he shape of he growh law. Boh miosis and displacemen are assumed o be very fas evens, so ha heir duraion can be negleced. Hence, hey occur insananeously compared o he main ime scale of he model (ha of cell cycle duraion and oocye growh rae, respecively parameerized by A k and κ k ). Following [13, 16, 7], we chose a sochasic Markov poin process which capures he probabilisic dynamics of each even (birh or displacemen). In he markovian framework, he birh and displacemen evens are boh independen and asynchronous. Using he noaions from above, and inroducing ε := N + R + E, we can describe he evoluion of he cell populaion (Z ) 0 by a SDE driven by a Poisson poin process. Le Q(ds, n(dk), dθ, J(dj)) be a Poisson poin measure (P.P.M.) on R + N R + N 2 whose inensiy q(ds, n(dk), dθ, J(dj)) := ds n(dk) dθ J(dj) is independen of Z 0 (see [7, 6] for more deails). Then, we have N 0 Z = δ (Xk (0),A k (0)+) + (2.7) where k=1 0 ε [ 1 k<ns Q(ds, n(dk), dθ, J(dj)) (2δ (Xk (s ), s) δ (Xk (s ),A k (s )+ s))1 0 Θ<m1(s,Z s,k) +(δ (J,Ak (s )+ s) δ (Xk (s ),A k (s )+ s))1 m1(s,z s,k) Θ<m 2(s,Z s,k,j) ] (2.8) m 1 (s, Z s, k) = b(k, Z s, s ), (2.9) m 2 (s, Z s, k, (j, i )) = m 1 (s, Z s, k)+ p(l (j,i) s, Z s, s )m(x k (s ), (j, i ), Z s, s ). We can inerpree he erms of (2.7) as follows. To describe he populaion a a given ime : 1. We firs consider he aging of he cell populaion from is iniial sae, where each cell is aged by A k (0) and he whole populaion is represened by he erm N 0 k=1 δ (X k (0),A k (0)). As ime increases, he cell age increases, so ha he iniial cell populaion has become N 0 k=1 δ (X k (0),A k (0)+) a ime. 2. We hen consider he cell division evens, and coun he cells ha were born beween he iniial and curren ime. If a birh occured a ime s, 0 s <, and since he age is rese o 0 a birh, we add wo new cells of age ( s) a ime. In compensaion, since he ne yield of cell division is one addiional cell, we have o remove from he populaion he moher cell, whose age had

11 10 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT jus overcome ( s) a he miosis ime. The balance beween adding wo daugher cells and removing one moher cell is accouned for by he wo firs erms in brackes on he second line of (2.7) (see also Figure 2.3). Fig Cell division and aging in he populaion. A ime =0, we have represened a single granulosa cell ha has jus divided (red line). A ime = he red cell divides and gives birh o a second granulosa cell (green line). A ime = 2 1.5, he red cell divides again and gives birh o a hird granulosa cell (blue line). Then, a ime = 3 4.3, he blue cell also divides and gives birh o a fourh granulosa cell (black line). Before he firs miosis, 0 1, he age of he red cell is equal o. Beween he firs and second miosis, 1 2, he age of he red cell is equal o 1. Beween he second and hird miosis, 2 3, he age of boh he red and blue cells is equal o 2. Afer he hird miosis, 3, he age of he red cell remains unchanged, while he age of he blue cell reses o 3, which is also he age of he new black cell. Since he green cell has no overcome any division during he simulaion ime, is age remains equal o 1 for all ime We finally consider he displacemen evens and coun he cells ha moved once or more from heir iniial locaion beween he iniial and curren ime. When a displacemen occurs a ime s, 0 s <, we have o add a cell a he arge locaion and o remove one from he original locaion A k (s ), leing he cell age unchanged, as i is accouned for by he firs erm in parenheses on he hird line of (2.7). 3. Simulaion resuls. In his secion, we describe he simulaion resuls obained from he biological specificaions provided in secion 3.1 and he parameer consrains explained in secion 5.1. The simulaion algorihm is deailed sep by sep in secion Numerical specificaions based on available biological knowledge. A generic issue in parameer fiing for physiologically-oriened muliscale mahemaical models is ha alhough quie precise funcional knowledge is available on he lower scale, quaniaive experimenal daa are raher available on he higher scale. In our case, he quesion is how o infer he parameers enering he microscopic funcions (µ, σ, α, β, λ i, κ i, cf Table 3.1) from macroscopic morphological variables (oocye and follicle diameers, cell number) only. Moreover, we have o face an addiional difficuly, due o he fac ha hese variables are available as relaionships beween

12 Muliscale model-based insigh on ovarian follicular developmen 11 one variable and anoher, bu no direcly as funcions of ime; in oher words, we do no dispose of rue kineic daa. In sheep, numerical daa are available from morphological sudies based on hisological analysis of ovarian secions conaining preanral follicles of differen sizes. The inerdependence beween morphological properies is easily accessible o biological sudies, since i can be deduced from hisological analysis of muliple follicles a a single poin in ime. These daa describe he concomian changes in oocye diameer, number of granulosa cells per follicle, and follicle diameer, in ovaries of Romney [28, 4] and Merinos [48] wild-ype ewes, and in ovaries of ewes carrying muaions in he fecundiy genes BMP15 (Inverdale Romney ewes, [4]) and is recepor BMPR1B (Booroola Merinos ewes, [48]). Table 3.1 Parameers and nominal values. Parameer Definiion Value (Uni) d G Diameer of a granulosa cell a V ol G Volume of a granulosa cell b d O (0) Oocye diameer (iniial value) a µ Tolerance o overcrowding (average) 0.52 σ Tolerance o overcrowding (sandard-deviaion) 0.03 (α, β) Oocye growh parameers ( , 0) λ i Average cell cycle duraion in layer i 500( (i 1)) κ i Oocye growh facor in layer i /(1 + 2(i 1)) a 10 2 µm b 10 8 µm 3 Follicular growh in wild-ype ewes. In wild-ype ewes, he growh of preanral follicles from he primordial sage o anrum formaion (from 40 o 250 µm diameer) occurs hrough harmonious growh of he oocye (from 34 o 100 µm diameer) and proliferaion of granulosa cells (from 16 o abou 4400 cells per follicle) [28]. A furher analysis of he hree ses of available daa [28, 4, 48] showed similar relaionships beween he parameers describing follicular developmen in Romney and Merinos wild-ype ewes (Figures 3.1 A and B). Effecs of muaions in fecundiy Fec genes on follicular growh. The presence of naural muaions in he fecundiy Fec genes leads o a parial or a complee disconnecion beween oocye growh and granulosa cell proliferaion, suggesing ha he molecular dialogue exising beween oocye and granulosa cells has been srongly affeced or disruped, respecively. Inverdale ewes are carriers of he F ecx I muaion in he gene encoding he growh facor BMP15 [17]. Homozygous F ecx I carrier ewes are serile wih small underdeveloped ovaries conaining follicles wih no more han one layer of granulosa cells. Abnormal follicles beyond he primordial sage of developmen up o 120 µm diameer conain very large healhy-looking oocyes wih prominen zona pellucida and a single layer of abnormal granulosa cells which never exceed 30 cells per follicular hisological secion [4], corresponding o no more han 140 cells per follicle from our esimaes using he available daa [28, 4]. The effecs of he F ecx I muaion on he relaionships exising beween follicle growh, oocye growh and granulosa cell proliferaion are illusraed on Figures 3.1 C and D (daa se II).

13 12 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT Booroola ewes are carriers of he F ecb B muaion in he gene encoding he BMP recepor BMPR1B [48, 33, 39]. Oocyes in small follicles of homozygous F ecb B carrier ewes grow larger han in equivalen sized follicles of wild-ype ewes, suggesing he exisence of an imbalance beween granulosa cell proliferaion and oocye growh [48]. For insance, from our esimaes using he available daa [48, 28], in follicles wih a diameer of 225 µm, oocye diameer was found o be 30 µm larger and he number of granulosa cells 350 cells lower in F ecb B carrier ewes, compared o wild-ype ewes. The effecs of he F ecb B muaion on he relaionships exising beween follicle growh, oocye growh and granulosa cell proliferaion are illusraed on Figures 3.1 C and D (daa se BB) Physiological balance beween oocye growh and granulosa cell proliferaion. The numerical simulaions of he model are useful o following he dynamics of he growing follicles on differen scales. On he microscopic scale, he main simulaion oupus consis of (i) he deailed spaial disribuion of individual granulosa cells, ha is ruled by he sequence of miosis and displacemen evens driven by eq. (2.7), and (ii) he growh rae of he oocye according o eq. (2.6). Togeher, he concurren dynamics of he granulosa cells and he oocye underlie he morphological changes in he follicle. For insance, as long as he mos ouside layer is no fully filled wih cells, he ouer conour of he follicle remains quie irregular, whereas i ends o a more spheric shape once his layer is pleny of cells. On he macroscopic scale, he simulaion oupus allow one o compue direcly variables characerizing eiher he whole cell disribuion (oal cell number, mean cell disance from he oocye or mean cell age) or he follicle size (follicle diameer on a 2D secion, raio of he oocye o follicle diameer). On an inermediary, mesoscopic scale, we can also ge informaion abou a given subpopulaion of cells. For insance, all he daugher cells issued from one of he granulosa cells presen a iniial ime in he firs layer can be locaed a any ime, and he corresponding clonal erriories can be visualized a final ime. The macroscopic oupus obained wih he nominal values of he parameers (Table 1) are illusraed on panels A, B and C of Figure 3.2. The oocye diameer increases from is iniial value of 33µm o a final value of 100µm. In he same ime, he number of granulosa cells increases from 22 cells, all belonging o he firs granulosa layer, o 3400 cells disribued over 6 cell layers. The corresponding final follicular diameer reaches 220µm. The model provides a realisic morphogeneic descripion of follicle growh. The granulosa builds as a compac issue around he oocye (see cenral panel of Figure 3.3), and he ouer conour of he follicle can be more or less regular depending on he degree of filling of he ouside layer. An insance of clonal erriory emanaing from a given progenior cell is shown on Figure 3.4. We used a color code o help locae he clonal subpopulaion; he inensiy of blue saining is relaed o he proporion of clonal cells wihin he whole populaion (here are no clonal cells in he whie area, while hey are prevailing over oher cells in he blue one). The relaive size of each of he clones issued from he 22 progenior granulosa cells changes in he course of a simulaion (see Figure 3.5). The non-uniform disribuion of he clone sizes seems o ensue from differences beween he ime of firs miosis of he moher cells. Indeed, afer soring he clone disribuion wih respec o he firs miosis imes, he Spearman saisical es on rank correlaion gave a negaive (rs = 0.64, 0.77, 0.46, 0.59 and 0.44) Spearman correlaion coefficien wih significance less han 0.05 (P = , < , , and ) on five

14 Muliscale model-based insigh on ovarian follicular developmen 13 independen simulaion oupus. Hence, he vialiy of a given clone appears o be even sronger han he progenior cell undergoes is firs miosis early Impaired balance induced by geneic muaions. For a fixed value of parameer λ 1, he growh rae of he oocye depends only on parameer κ 1. In he ime needed o reach he arge oocye diameer of 100µm (which corresponds o he end of he preanral growh in ewes, hence o he sopping ime in he model), he increase in he cell number will be higher han in he wild-ype case for relaively low values of κ 1 (since he oocye grows more slowly). In a similar way, he cell increase will be lower for relaively high values of κ 1, since in his case he oocye grows more quickly. Accordingly, we can delimi hree disinc domains in he (λ 1, κ 1 ) parameer space wih respec o he raio of oocye o follicle diameer (see Figure 3.6). The middle domain conains couples of (λ 1, κ 1 ) parameers consisen wih a wild-ype like relaion beween he oocye and follicle diameer (such as ha shown on panel A of Figure 3.1). This domain separaes he large oocye domain (delimied by grey verical sripes) ha lies above and conains parameer couples consisen wih he geneic muaions naurally encounered in ewes (BB and II cases), from he small oocye domain (delimied by grey horizonal sripes), ha lies below and can be relaed o imbalanced follicular growh encounered in oher species as discussed in secion 4. To build Figure 3.6, we firs fixed he value of λ 1 and chose wo differen values of κ 1 defining an inerval [κ 1min, κ 1max ] such ha we clearly go over- or undersized oocyes. Searching wihin his inerval by dichoomy, we go anoher κ 1 value, such ha he follicle neiher belongs o he large oocye domain nor o he small oocye one. More precisely, his means ha he normalized difference beween he observed diameer, d sim O, and he expeced (experimenally observed) diameer, dexp O (on a se of experimenal daa {(d exp (i), dexp(i)), i}) exceeds a hreshold value: O Number of experimenal daa i=1 F (d sim O ((dexp F (i))) dexp O (i) ) 2 (i) 0.1 d exp O We hen compared he simulaion oupus corresponding o his (κ 1, λ 1 ) couple wih he experimenal poins, by visual examinaion. We repeaed his process ieraively while sweeping he range of λ 1 values. Amongs he 350 simulaions performed, abou 10% were compaible wih he wild-ype daa and could be sored as wild-ype like cases. We hen superimposed on he domains five poins corresponding o differen ypes of follicular developmen obained from he simulaion resuls displayed on Figure 3.2, for he wild-ype (++), F ecb B homozygous carrier (BB) and F ecx I homozygous carrier (II) ewes. To deermine reference coordinaes for he wild-ype case, we chose o fix λ 1 o 500 o reach a compromise beween wo numerical consrains. On one side, he larger λ 1 is, he longer is he compuing ime (simulaion duraion from several minues o several days). On he oher side, wih oo small values of λ 1, here is a grea sensiiviy o he value of κ 1 ha hampers he proper idenificaion of parameer couples consisen wih wild-ype ewes case. I is worh noicing ha here are several parameer combinaions leading o large oocye cases (he same is rue for wild-ype and small oocye cases). Saring from he (λ 1, κ 1 ) couple corresponding o he wild-ype reference case (orange cross on Figure 3.6), we can increase he value of κ 1 o ge a BB like phenoype (BB1 case: open pink square a κ 1 = ) or a more severe II like phenoype (open cyan diamond a

15 14 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT κ 1 = ). We can also follow a curve (dashed line on Figure 3.6) along which he raio κ 1 /λ 1 is kep unchanged, so ha he oocye growh rae remains consan (BB2 case: yellow square a λ 1 = 4720 and κ 1 = ). Anoher couple of parameers consisen wih he BB case, ha lies ouside his curve bu needs a far lower increase in λ 1 can be found for insance a λ 1 = 1000 and κ 1 = (BB3 case: open grey square on Figure 3.6). The 3 uppermos panels of Figure 3.3 illusrae differen seps of follicle growh in a large oocye (BB1) case. The iniial condiions were he same han in he wildype case (oocye diameer of 33µm and 22 cells in he firs layer). The final oocye diameer is idenical o he wild-ype case (100µm), bu boh he final number of granulosa cells (1319) and cell layers (3) are far lower. As a resul, he final follicular diameer only reaches 168µm (see doed curves on panels D, E and F of Figure 3.2). The oher BB cases are shown in Figure 3.7. The BB2 case ends up wih 1306 cells disribued over 3 layers and a resuling follicle diameer of 168µm, while he BB3 case ends up wih 1106 cells disribued over 3 layers and a resuling follicle diameer of 161µm. The severe II phenoype ends up wih only 382 cells, all conained in a single layer corresponding o a follicle diameer of 127µm (see solid curves on panels D, E and F of Figure 3.2). The 3 lowermos panels of Figure 3.3 illusrae differen seps of follicle growh in a small oocye case. The simulaion was performed wih κ 1 = (open blue circle on Figure 3.6). The iniial condiions were again he same han in he wild-ype case. While he final oocye diameer reaches only 75µm, boh he number of granulosa cells (6497) and cell layers (11) are far greaer han in he wild-ype case. As a resul, he final follicle diameer reaches 267µm (see panels G, H and I of Figure 3.2). 4. Discussion. Our model gives, for he firs ime, a mahemaical descripion of he coupled kineics of oocye growh and granulosa cell proliferaion during he developmen of an ovarian follicle. I is convenien o describe he morphogenesis of follicles as spheroids, by locaing granulosa cells in elemenary volumes wihin cell layers around he oocye and auhorizing heir displacemen ino adjacen ones, and i gives new spaial informaion on he clonal expansion of granulosa cells wih ime. Moreover, i can reproduce he concomian changes occurring in granulosa cell numbers and oocye diameer during early follicle developmen in he ovine species. The model disinguishes beween hree differen scales. These 3 scales are inricaely merged on he dynamical ground, since he main evens (cell division or displacemen), as well as he oocye growh law involve a leas 2 differen scales. Namely, (i) he probabiliy for cell division depends boh on he local age variable and he semilocal layer index, (ii) he probabiliy for cell displacemen depends on boh he local noion of overcrowding and he semi-local one of cell neighboring and (iii) he oocye growh rae is defined from he global cell number, bu in a semi-locally srucured way (weighed conribuions of cells according o he layer index). In his individual-based populaion model, inspired from previous works in populaion dynamics [13], evens as displacemen and miosis are boh independen and asynchronous. They happen during subsequen incremens of imes whose duraion changes wih he size of he populaion. Since no more han one even can happen during a ime incremen d, he duraion decreases as he cell number increases, o he order of d exp(1/2n ). On he numerical ground, he main advanage of his formalism is ha he algorihmic procedure (see secion 5.2) gives an exac numerical soluion o he underlying sochasic process (2.6)-(2.7). The main drawback is ha he number of evens per physical ime uni increases drasically wih he cell number,

16 Muliscale model-based insigh on ovarian follicular developmen 15 Fig Relaionships beween follicle and oocye growh in sheep follicles and effecs of muaions in he F ecx or F ecb genes. Panels A and C illusrae he relaionships beween follicle and oocye diameer, panels B and D illusrae he relaionships beween granulosa cell number, follicle diameer and oocye diameer. A and B: daa ses from wild-ype sheep (daa se ++ 1: [28]; daa se ++ 2: [4]; daa se ++ 3: [48]). The 3 daa ses from wild-ype sheep agree closely for he sudied relaionships. C and D: daa ses from homozygous carriers of F ecx I (daa se II: [4]) or F ecb B (daa se BB: [48]) genes. For each given follicle diameer, he presence of F ecb B or F ecx I muaions leads o large (BB), or very large (II) oocyes, respecively, compared o wild-ype (++) oocyes, and low (BB), or very low (II) granulosa cell numbers, compared o wild-ype (++) granulosa cell numbers. so ha he compuing ime may become limiaive. On he heoreical ground, he well-posedness of he problem (exisence and uniqueness of soluions) is guaraneed under generic (smoohness) assumpions on he bounds of funcions p, b, and m, ha are saisfied in our model. As a consequence, one can pass o he limi when he cell number a iniial ime is large enough and approximae he discree sochasic process

17 16 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT Oocye diameer Oocye diameer Oocye diameer Follicle diameer 40 Follicle 1 Follicle A Follicle diameer Follicle diameer Diameer Diameer Follicle Oocye B Cell number D E F G Diameer 50 Follicle 1 Follicle 2 0 Oocye 1 Oocye H Cell number 50 Follicle Oocye Cell number Layer number Layer number Layer number Cell number Follicle 1 Follicle C I Cell number Cell number Fig Simulaion resuls corresponding o differen oocye and granulosa cell growh raes in sheep follicles. The panels A, D and G illusrae he relaionships beween follicle and oocye diameer, he panels B, E and H illusrae he relaionships beween granulosa cell number, follicle diameer and oocye diameer, and he panels C, F and I illusrae he relaionships beween he number of cell layers surrounding he oocye and he number of granulosa cells. A, B and C: resuls of one simulaion fiing wih he daa se of wild-ype (++) sheep. The poins in panels A and B correspond o he 3 daa ses ++ 1, ++ 2 and ++ 3, represened in Figure 7. D, E and F: resuls of wo simulaions corresponding o (1) large oocyes and low granulosa cell numbers corresponding o daa from BB sheep (doed curves), (2) very large oocyes and very low granulosa cell numbers corresponding o daa from II sheep (solid curves). G, H and I: resuls of one simulaion corresponding o small oocyes and high granulosa cell numbers. by a coninuous deerminisic equaion (parial differenial equaions of conservaion law ype) wih a consan ime incremen. Ye, in our case, we will have o deal wih a complex limi problem, due o he 2D characer of he model. In he same ime ha he cell number ends o infiniy, boh he cell diameer d G and he volume V ol (j,i) (ω) of pariion se L (j,i) have o end o zero o accoun for he seric consrains. This observaion is a he source of he quie complex definiion of he space domain(using geomery and measure heory) in secion 2.2. Recen resuls [6] sugges ha such a complex limi problem can be racable.

18 Muliscale model-based insigh on ovarian follicular developmen 17 Fig D represenaion of follicle growh in 3 examples of resuls from model simulaion. In his figure, a window has been opened o see he oocye (yellow) surrounded by he granulosa cells (green).the 3 upper panels illusrae differen seps of follicle growh for a follicle conaining a large oocye, as observed in BB sheep. The 3 cenral panels illusrae differen seps of follicle growh for a follicle conaining a medium-size oocye, as observed in wild-ype (++) sheep. The 3 lower panels illusrae differen seps of follicle growh for a follicle conaining a small oocye (an animaed version of he figure is provided as supplemenal maerial). The model is based on our curren knowledge of he molecular dialogue exising beween he oocye and granulosa cells, involving oocye-derived facors of he BMP family which can simulae granulosa cell proliferaion, and granulosa cell-derived facors such as KITLG which can enhance oocye growh via signaling hrough he KIT recepors presen on he oocye cell membrane [26, 18]. In he model, he parameers affecing he dynamics of follicular developmen are he average cell cycle duraion in granulosa cells (λ), and heir inensiy of secreion (κ) of a signal conribuing o oocye growh. The value of λ was assumed o increase wih he disance of he granulosa cell o he oocye, in agreemen wih he lower proliferaive aciviy observed in granulosa cells as hey move away from he oocye and are exposed o

19 18 Philippe MICHEL, Thomas STIEHL, Danielle MONNIAUX and Frédérique CLÉMENT Fig D represenaion of clonal proliferaion of he granulosa cells. In his figure, a window has been opened o see he oocye (yellow) surrounded by he granulosa cells. The 3 panels illusrae differen seps of follicle growh. The granulosa cells originaing from a same progenior cell formed a clone (blue cells). The blue saining inensiy represens he proporion of clonal cells in each pariion of he spaial domain, i.e. he probabiliy for each granulosa cells o belong o he lineage of he progenior cell. Fig Disribuion of he clone sizes, expressed as he proporion of cells emanaing from a same progenior cell wihin he whole populaion a final ime. The lef hisogram represens he unsored disribuion of clones issued from he progenior cells (cells locaed in he firs layer a iniial ime). The righ hisogram represens he disribuion of clone sizes wih respec o he ime of firs miosis of he progenior cell. lower and lower concenraions of oocye-derived miogenic facors [18, 24]. The value of κ was assumed o decrease when he disance of he granulosa cell o he oocye increases, illusraing boh he diluion of he soluble KITLG form due o diffusion, and he high physiological imporance of he membrane-bound KITLG form expressed by he firs granulosa cell layer, in close conac wih he oocye [42, 44]. During he erminal sages of heir developmen, ovarian follicles become sricly dependen on he supply of he piuiary hormone FSH (follicle simulaing hormone), so ha he endocrine conrol of follicular growh becomes prominen over he paracrine conrol. However, he follicular cells in he viciniy of he oocye (locaed in he so-called

20 Muliscale model-based insigh on ovarian follicular developmen 19 Fig Sensiiviy sudy in he (λ 1, κ 1 ) parameer space. The middle (whie) domain conains couples of (λ 1, κ 1 ) parameers consisen wih a wild-ype like relaion beween he oocye and follicle diameer. The large oocye domain (grey verical sripes) and small oocye domain (grey horizonal sripes) conain couples of (λ 1, κ 1 ) parameers leading respecively o a higher or lower oocye o follicle diameer raio compared o he wild-ype case (see body ex for more explanaion on he figure building). The dashed line corresponds o an unchanged κ 1 o λ 1 raio. Orange cross: reference poin for he wild-ype phenoype (++). Open squares: insances of mild phenoype wih large oocyes corresponding o BB sheep. Open cyan diamond: insance of severe phenoype wih large oocyes corresponding o II sheep. Blue open circle: insance of an opposie phenoype wih small oocyes. cumulus) remain in close ineracions wih i up o he ovulaion even [16, 17, 41]. Coupling he sochasic IBM model presened in his paper wih he already exising models of erminal follicular developmen [10, 11, 36] would be a naural way of embedding he oocye dynamics in hose models. In he law describing oocye growh, he oocye radius depends on boh he number of is surrounding granulosa cells (and hus indirecly, on heir cell cycle duraion λ) and heir average inensiy of secreion κ. The values of he parameer couple (λ 1, κ 1 ) deermines he morphogenesis of he follicle, since i conrols he balance beween oocye growh and granulosa cell proliferaion. Our simulaion resuls have shown ha numerous couples are compaible wih a wild-ype like follicular developmen, all belonging o he physiological domain defined in Figure 3.6. Two oher domains wih unbalanced follicular growh, characerized by relaively large or small oocyes, were also defined. The simulaion oupus have reproduced he effecs of differen muaions in he fecundiy Fec genes on follicular developmen in sheep, giving rise o follicles wih abnormally large oocyes. From our model, for a given value of λ 1 fixed o 500, a 10 fold increase in he value of κ 1 (from is iniial value in wild-ype ewes) led o abnormal follicles conaining very large oocyes surrounded by a single layer of low proliferaing granulosa cells, mimicking he ovarian phenoype of he homozygous F ecx I carrier Inverdale ewes [4]. This severe phenoype associaed wih inferiliy is due o he absence of producion of biologically acive BMP15 in he muan females [17], and i