A Complex Mathematical Model of the Human Menstrual Cycle

Transcription

1 Konra-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D Berlin-Dahlem Germany ISABEL REINECKE, PETER DEUFLHARD A Complex Mathematical Moel of the Human Menstrual Cycle ZIB-Report (June 2006)

2 A Complex Mathematical Moel of the Human Menstrual Cycle Isabel Reinecke an Peter Deuflhar June 7, 2006 Abstract This paper aims at presenting the complex couple network of the human menstrual cycle to the intereste community. Beyon the presently popular smaller moels, where important network components arise only as extremely simplifie source terms, we a: the GnRH pulse generator in the hypothalamus, receptor bining, an the biosynthesis in the ovaries. Simulation an parameter ientification are left to a forthcoming paper. Zuse Institute Berlin (ZIB), Takustraße 7, D Berlin, Germany 1

3 Contents 1 Introuction 3 2 Moel Funamentals Basic mechanisms Basic compartment moel GnRH Pulse Generator in the Hypothalamus Pulse frequency an mass GnRH concentration in the pituitary portal system Mechanisms in the Pituitary Receptor bining of GnRH Dynamics of LH an FSH Dynamics in the Ovaries Follicular evelopment Receptor bining of LH an FSH Enzyme concentrations Biosynthesis of the sterois Summary an Outlook 31 Acknowlegement 31 References 31 2

4 Figure 1: Control system of the human menstrual cycle, aapte from [28]. Primarily, the actions on the three levels hypothalamus, pituitary, an ovaries have to be stuie. The GnRH pulse generator in the hypothalamus regulate by ovarian proucts secretes GnRH in a pulsatile manner which influences, together with sterois such as estraiol an progesterone, LH an FSH ynamics. These two gonaotropins, however, regulate the follicular evelopment resulting in the biosynthesis of the sterois. 1 Introuction The term systems biology escribes the fusion of biology, computer science, mathematics, an engineering sciences with the objective of attaining a holistic comprehension of life processes [24]. Being a highly active fiel of research an yieling great hopes an expectations, we aspire to make our moest contribution by stuying an moelling the processes in the female boy associate with the control system of reprouction. The objective of the menstrual cycle is the hormonal regulation of follicular growth an maturation in the ovaries leaing to ovulation at mi-cycle an therefore enabling reprouction. An overview is given in Figure 1. This control system can be isturbe, so that ovulation oes not occur. With ai of hormonal therapies, one can try to restore the normal ovulatory cycle. However, it coul also be the case that the failure of a normal cycle is esire. Then, ovulation can be inhibite by taking hormonal contraceptiva. One possibility for testing meical or pharmaceutical therapies is to conuct clinical stuies. Another possibility coul be the use of a etaile mathematical moel. Such a moel will be evelope in this report. There are alreay some moels escribing the ynamics of the human menstrual cycle. In most cases, focus is on coarse interactions like feeback mechanisms among the most important hormones [16, 42] an follicular masses [3, 4, 17] or, explicitly, on follicular ynamics [6, 7, 8, 10, 25]. Others are concerne with moelling biochemical processes like receptor bining 3

5 [2, 9, 19, 44]. We will present some of these approaches an their integration in this complex moel in the following. Among those moels escribing the entire control system, we focus on the moel presente in [17]. This moel is able to simulate concentrations reproucing quite well the given experimental ata estimate from [30]. Even simple applications are possible. However, the ynamics of important network components like progesterone an estraiol are simplifie an the ynamics in the hypothalamus are not inclue. This moel incorporates the LH an FSH synthesis an release in the pituitary, the follicular ynamics in the ovaries as well as the concentrations of progesterone, estraiol, an inhibin in the bloo as linear combinations of follicular masses. Since the moel in [17] offers a suitable approach for the main components of the system, it is chosen as basis in the subsequent moelling process. The control system of the human menstrual cycle is stuie from a physiological or even biochemical point of view. The ifferent possibilities for expaning the basic compartment moel are iscusse. The paper is organize as follows: In Section 2, some basic remarks concerning the mathematical backgroun an the biochemical basis are mae an we present the moel evelope in the thesis [17]. In the following, the ifferent parts an features which so far have not been consiere as part of moelling the human menstrual cycle but are now incorporate in the moel are presente. In Section 3, the pulse generator for the GnRH release in the hypothalamus an the moel equations for the GnRH concentration in the pituitary portal system being important for the ynamics in the pituitary are erive. The receptor bining of GnRH in the pituitary as well as the ynamics of LH an FSH synthesis an release are presente in Section 4. In Section 5, moelling approaches for the follicular evelopment, receptor an enzyme concentrations, an the biosynthesis of sterois are iscusse. Finally, the results are summarize in Section 6 an an outlook is given. 2 Moel Funamentals At first, we introuce the basic mechanisms use in the moelling process. Then, we present the moel aopte from [17] which serves as a basis for further moelling. 2.1 Basic mechanisms The mathematical basis as well as the mechanisms like Michaelis-Menten kinetics an feeback effects which are use in the moel are provie in the following. Moelling elays. The moel is base upon a eterminstic moelling approach by using a system of ifferential equations. Since the processes take place in ifferent parts of the boy an influence each other with a certain elay (see Figure 2), passing over to elay ifferential equations is a reasonable step: t y(t) = f(t, y(t), y(t τ 1),..., y(t τ m )), y : R R n, f : R R n (m+1) R n, where τ i R +, i = 1,..., m, an n, m N. To escribe the biological variety in an aequate manner, the application of integro-ifferential equations instea of iscrete elays is more accurate, but, concomitantly, more expensive. Biochemical mechanisms. Since many reactions in the human boy are catalyze by enzymes, it is not aequate to assume first-orer reaction kinetics in these cases. The mechanism calle Michaelis-Menten is the simplest an most common approach for enzyme catalyze reactions. 4

6 HYPOTHALAMUS PORTAL SYSTEM PITUITARY BLOOD OVARIES Figure 2: Moel scheme of the human menstrual cycle. The three main compartments hypothalamus, pituitary, an ovaries are connecte through the bloo circulation. The arrows inicate where the elays arise. If the biochemical etails are not establishe an only a stimulatory or inhibitory effect of one hormone on another is known, moelling can be one by use of feeback functions originating from enzyme kinetics. Michaelis-Menten kinetics. Moelling the irreversible enzyme catalyze overall reaction S P, where S stans for the substrate whereas P enotes the prouct, the Michaelis-Menten mechanisms has the form [37] k 1 k E + S (E S) 2 P + E, k 1 where k 1, k 1, k 2 R + enote the reaction constants, E the enzyme, an (ES) the complex. Assuming constant total enzyme concentration an the quasi-stationary state for the enzyme complex concentration, the reaction rate for the prouct is t P = k S 2 E total K M + S, where P enotes the prouct concentration, S the substrate concentration, an E total the total enzyme concentration, with the Michaelis-Menten constant For the reaction rate of substrate S, we obtain K M := k 1 + k 2 k 1. t S = k S 2 E total K M + S. 5

7 We observe that assuming Michaelis-Menten kinetics instea of first-orer reaction kinetics leas to a prouct formation rate which is not linearly but sigmoially epenent on the substrate an boune by the maximal rate V max := k 2 E total. The situation is more complicate in the case of reversible reactions like S P, where the Michaelis-Menten mechanism is escribe by [37] E + S k 1 k 1 (E S) k 2 k 2 P + E, k 2 R +. Assuming again constant total enzyme concentration an the quasi-stationary state for the enzyme complex concentration, the reaction rate for the prouct oes not longer epen only on the substrate concentration, but also on the prouct concentration: For the substrate S, the reaction rate is k 1 k 2 t P = E total (k 2 S k 1 P ) K M + S + k. 2 P t S = E total (k 2 S k 1 P ) K M + S + k. 2 P k 1 k 1 k 2 Definition 1. Depenence by Michaelis-Menten kinetics in the case of an irreversible reaction is expresse by the irreversible Michaelis-Menten function f irrev (S, E, p) := p 1 E k 1 S p 2 + S, p R2 + an in the case of a reversible reaction, the reversible Michaelis-Menten function is efine by f rev (P, S, E, p) := p 1 E S p 2 E P p 3 + S + p 4 P, p R 4 +, where P enotes the prouct concentration, S the substrate concentration, an E the total enzyme concentration. Feeback mechanisms. Feeback functions can be erive by enzyme mechanisms [37]. Positive feeback can be escribe by applying Michaelis-Menten kinetics, negative feeback by mechanisms for inhibitory enzymes [37]. Definition 2. The Hill function for positive feeback is efine by h + (H, T, n) := an the Hill function for negative feeback by ( H ) n T 1 + ( ) H n T h 1 (H, T, n) := 1 + ( ) H n, T where H R + {0} enotes the hormone concentration which affects with a positive an negative feeback, respectively, T R + enotes the threshol value, an n 1 the Hill coefficient. If there is more than one hormone responsible for inhibitory an/or stimulatory effects, the question arise how these effects can be combine. Coarsely sai, the feeback functions can be summe if the effects are inepenent an they can be multiplie in the case of epenency [37]. 6

8 PITUITARY LH, FSH OVARIES P 4, E 2, Ih Figure 3: Control system consiere in [17]. 2.2 Basic compartment moel The moel presente in [17] is compose of 13 elay ifferential equations where 4 escribe the ynamics in the pituitary an the resulting FSH an LH concentrations in the bloo an another 9 escribe the ynamics of the follicular masses. Comparison with experimental ata shows that this moel is capable of escribing the ynamics of the human menstrual cycle in a soun way. However, the ynamics of progesterone, estraiol, an inhibin concentrations in the bloo, essential components of the regulatory system, are simply etermine as linear combinations of follicular masses. The activities in the hypothalamus, especially the GnRH pulse generator an the resulting pulsatile secretion of GnRH into the bloo circulation, are not inclue. An overview of the control system consiere in [17] is given in Figure 3. Dynamics in the pituitary. In the basic compartment moel [17], the following assumptions regaring the influence of progesterone, estraiol, an inhibin on the LH an FSH ynamics are mae. High bloo levels of estraiol promote rapi LH synthesis, whereas progesterone inhibits LH synthesis in the luteal phase, an inhibin inhibits the FSH synthesis. Progesterone stimulates the release of FSH an LH when estraiol bloo levels are in a normal range uring the late follicular phase. Estraiol inhibits the release of FSH an LH into the circulation an it has a greater inhibitory effect on FSH release than on LH release. The release rate of LH an FSH is assume to be proportional to the amount of LH an FSH on reserve in the pituitary, respectively. Finally, the clearance rate of LH an FSH is assume to be proportional to the bloo levels of LH an FSH, respectively. Base upon the assumptions mentione in [17], the moel schemes which are shown in Figures 4 an 5 are evelope. The store gonaotropin mass is etermine by synthesis (syn) an release (rel). Once release into the bloo circulation, it is istribute in the bloo volume V B where it is eliminate proportionally to the present concentration (clear). In the following, E 2 enotes the estraiol concentration, P 4 the progesterone concentration, an Ih the inhibin concentration in the bloo. For the ynamics of LH synthesis an release an, finally, the LH concentration in the bloo, the following equations are chosen [17]: t P LH(t) = syn LH (E 2 (t τ E2 ), P 4 (t τ P4 )) rel LH (E 2 (t), P 4 (t), P LH (t)) (1) t LH(t) = 1 rel LH (E 2 (t), P 4 (t), P LH (t)) clear LH (LH(t)), V B (2) 7

9 - PROGESTERONE + + ESTRADIOL - synthesis (LH) release (LH) Figure 4: Moel scheme for LH synthesis an release. - INHIBIN synthesis (FSH) PROGESTERONE ESTRADIOL + - release (FSH) Figure 5: Moel scheme for FSH synthesis an release. where a 1 syn LH (E 2, P 4 ) = a 2 E a 3 2 a a 3 4 +Ea P (3) 4 a 5 rel LH (E 2, P 4, P LH ) = a 6 (1 + a 7 P 4 ) 1 + a 8 E 2 P LH (4) clear LH (LH) = a 9 LH (5) an P LH enotes the mass of store LH in the pituitary, LH the LH concentration in the bloo an τ P4 an τ E2 the elays for the progesterone an estraiol concentration, respectively. The moel equations for FSH are of the form [17]: where t P F SH(t) = syn F SH (Ih(t τ Ih )) rel F SH (E 2 (t), P 4 (t), P F SH (t)) (6) t F SH(t) = 1 rel F SH (E 2 (t), P 4 (t), P F SH (t)) clear F SH (F SH(t)), V B (7) syn F SH (Ih) = a Ih a 11 (8) rel F SH (E 2, P 4, P F SH ) = a 12 (1 + a 13 P 4 ) 1 + a 14 E2 2 P F SH (9) clear F SH (F SH) = a 15 F SH (10) an P F SH enotes the mass of store FSH in the pituitary, F SH the FSH concentration in the bloo an τ Ih the elay for the inhibin concentration. The coefficients a i R +, i = 1,..., 15, are the parameters of the pituitary compartment. Dynamics in the ovaries. On the one han, the moel incorporates the follicular evelopment by efining nine stages an moelling the mass ynamics of follicular an luteal tissue at these stages. On the other han, the plasma concentrations of progesterone, estraiol, an inhibin arise as linear combinations of follicular masses, skipping their prouction an secrection in the follicles. 8

10 Follicular evelopment. The follicular ynamics are escribe by moelling active follicular masses at ifferent stages where active mass is efine as mass which is growing an secreting hormones [17]. The following partition into follicular stages is one [17]: Menstrual Follicular Stage (M sf ), representing the mass of several immature follicles Seconary Follicular Stage (SeF ), representing the mass of seconary follicles Preovulatory Follicular Stage (P rf ), representing the mass of the ominant follicle Ovulatory Scar 1 (Sc 1 ), representing the mass uring ovulation Ovulatory Scar 1 (Sc 2 ), representing the mass uring luteinization Luteal Stage i (Lut i ) representing the mass of the corpus luteum at stage i, i = 1,..., 4. The following equations are erive in [17]. For the follicular phase, we obtain ( ( ) ) LH(t) b4 t MsF (t) = b 1 F SH(t) + b 2 F SH(t) b 3 MsF (t) (11) LH 0 ( ) ( LH(t) b4 ( ) ) LH(t) b6 t SeF (t) = b 3 MsF (t) + b 5 b 7 LH(t) SeF (t) (12) LH 0 LH 0 LH 0 t P rf (t) = b 7 LH(t) ( ) LH(t) b9 SeF (t) b 8 P rf (t), (13) LH 0 LH 0 where LH 0 represents the unit of measurement of LH. During ovulation an luteinization, the equations are an, finally, in the luteal phase we get ( ) LH(t) b9 t Sc 1(t) = b 8 P rf (t) b 10 Sc 1 (t) LH 0 (14) t Sc 2(t) = b 10 Sc 1 (t) b 11 Sc 2 (t), (15) t Lut 1(t) = b 11 Sc 2 (t) b 12 Lut 1 (t) (16) t Lut 2(t) = b 12 Lut 1 (t) b 13 Lut 2 (t) (17) t Lut 3(t) = b 13 Lut 2 (t) b 14 Lut 3 (t) (18) t Lut 4(t) = b 14 Lut 3 (t) b 15 Lut 4 (t), (19) where b i R +, i = 1,..., 15, enote the parameters of the ovarian compartment. Estraiol, progesterone, an inhibin concentrations in the bloo. The ovarian hormones estraiol, progesterone, an inhibin are release from the follicles into the bloo where they are eliminate at a constant rate. The following ifferential equations escribe the changes of the hormone s bloo concentration C: t C = 1 V B n c i F i δ C, i=1 9

11 where V B represents the bloo volume, c i the hormone s secretion rate at the follicular stage F i, where F i enotes the corresponing active mass, i = 1,..., 9, an δ the hormone s clearance rate [17]. It is assume that the clearance from the bloo occurs on a faster time scale than the follicular an luteal evelopment, hence, δ is large. Diviing through by δ an efining ε := 1 δ, we obtain: ε t C = 1 V B n i=1 c i δ F i C. Using the pseuo-steay state hypothesis, ε tc 0, we get: n C(t) C i F i, c i δ V B. i=1 where C i := Estraiol is secrete mainly by the seconary follicles an the preovulatory follicle an partially by the corpus luteum. Therefore, the concentration E 2 ( ) may be written as a linear combination of SeF, P rf, an Lut 4 [17]: E 2 (t) = c 1 + c 2 SeF (t) + c 3 P rf (t) + c 4 Lut 4 (t). (20) Progesterone an inhibin are secrete mainly in the luteal phase by the corpus luteum, however, inhibin is also secrete by the preovulatory follicle. Thus, the concentrations of P 4 ( ) an Ih( ) can be written as [17]: P 4 (t) = c 5 Lut 3 (t) + c 6 Lut 4 (t) (21) Ih(t) = c 7 + c 8 P rf (t) + c 9 Lut 3 (t) + c 10 Lut 4 (t), (22) where c i R +, i = 1,..., 10, enote the parameters for these auxiliary equations. 3 GnRH Pulse Generator in the Hypothalamus The gonaotropin-releasing hormone (GnRH) is a hypothalamic hormone regulate by ovarian sterois which stimulates the pituitary system. The GnRH pulse generator in the hypothalamus is an essential component of the reprouctive control system. If it fails, it may have serious consequences for the entire cycle. Continuous GnRH aministration, for example, leas to esensitization of the GnRH receptors in the pituitary. That in turn, results in a ecrease of LH an FSH release into the bloo circulation, wherefore a normal follicular evelopment is not longer possible. Hence, the question of eveloping therapy methos with ai of a mathematical moel in the case of isfunction of the pulse generator is of great interest. There are about 1000 [34] up to 1500 [26] GnRH neurons proucing GnRH in the hypothalamus. They represent the final output neurones of an integrate neural network [20]. Regulation of the pulse frequency an pulse mass is mainly affecte by estrogens an progesterone. Estrogens act on many components of the GnRH network incluing the brainstem A2 neurones [20]. It bins to the estrogen receptors leaing to an increase norarenaline secretion in the case of A2 neurons an, thereby, facilitating the activity within the GnRH neural network [20]. Thus, the expression of GnRH-mRNA is enhance resulting eventually in an increase GnRH secretion [20]. In contrast, progesterone inhibits the GnRH pulses via progesterone receptors [39]. There have been some approaches for moelling the neural network explicitly [14]. Since the processes in the hypothalamus are not known in etail [20, 39] an, moreover, since the simulation of the GnRH neural network is very expensive, another approach for the incorporation of the pulse generator into the moel is chosen. 10

12 P 4 - GnRH + > T freq E 2 E 2 Figure 6: Moel scheme for the regulation of the pulse frequency. T freq E 2 : threshol value for the estraiol concentration in the case of frequency regulation. 3.1 Pulse frequency an mass The GnRH pulse frequency is not constant uring the menstrual cycle. In the follicular phase, GnRH is release at approximately constant intervals. The length of interval between two pulses is about 80 minutes [45] or 90 minutes [21]. At mi-cycle, the gaps between the pulses shorten [21, 45]. As a result of the higher pulse frequency at mi-cycle, it is assume that estraiol, at least at high concentrations, stimulates the GnRH pulse frequency. In the luteal phase, the interval between the pulses is prolonge to several hours [21]. Progesterone is the main factor of this reuction of pulse frequency, inhibiting the GnRH pulse generator [15, 21]. In the early luteal phase, the pulse frequency iminishes to one pulse every 2 to 4 hours, in the mi-luteal phase to every 4 to 6 hours an in the late luteal phase to every 8 to 12 hours [5]. The moel scheme is presente in Figure 6. To moel the pulsatility in an aequate manner, a stochastic approach is chosen. A similar approach for a pulse generator has been presente in [22, 23], in the context of moelling the male reprouctive hormone system. Let {S j } j N be a sequence of inepenent, non-negative ranom variables with the same istribution function F, where F (0) < 1. The pulse time points {T j } j N0 are then moelle by a renewal process: T 0 := 0, T j+1 := T j + S j+1, j N 0. The associate renewal count process is escribe by an inhomogenous Poisson process. A homogenous Poisson process woul not be aequate in this case since the pulse frequency is regulate by non-constant feeback. For the survival time between two pulses, the Weibull ensity is chosen: f(s) = P [s T j 1, λ( )] ( ) γ 1 ( ( s ) γ ) s = γ λ(s) λ(r)r exp λ(r)r. T j 1 T j 1 Suppose that the survival time is S j, starting at the pulse point time T j 1, j N. Then we obtain: F (S j ) = Tj 1 +S j T j 1 = 1 exp ( f(s)s = P [T j 1 < s T j T j 1, λ( )] ( Tj T j 1 λ(r)r ) γ ). 11

13 E 2 > T mass E 2 < T mass E GnRH Figure 7: Moel scheme for the regulation of the GnRH pulse mass. TE mass 2 : threshol value for estraiol concentration in the case of mass regulation. To calculate T j, first the ranom variable U j U[0, 1] has to be generate. Then, it has to be equate with F (S j ). It follows Tj T j 1 λ(r)r = ( ln(1 U j )) 1/γ. (23) The function λ( ) escribes the pulse intensity, affecte by progesterone (negative feeback) an estraiol (positive feeback at high concentration): ( ) ( ( )) λ(t) = h P 4 (t τ P4 ), T freq P 4, n freq P h + E 2 (t τ E2 ), T freq E 2, n freq E 2 λ max, (24) where τ P4, τ E2 R + enote the elays, T freq P 4, T freq E 2 R + the threshol values, n freq P 4, n freq E 2 R + the hill coefficients, an λ max R +. The GnRH mass which is store between two pulses is affecte by estraiol as well. Estraiol inhibits the GnRH amplitue [33], whereas progesterone has no significant effect [39]. The GnRH surge, in turn, is triggere by rising estraiol upon the hypothalamus [5]. Hence, it is reasonable to assume that estraiol is inhibitory at low concentrations an stimulatory at high concentrations [5, 43, 45]. The moel scheme is presente in Figure 7. The pulse mass is store between two pulses an is completely release at the next pulse time point: M GnRH,j = Tj T j 1 ( h (E 2 (t τ E2 ), T E2,1, n E2,1) + h + (E 2 (t τ E2 ), T E2,2, n E2,2) ) M max t (25) where T E2,1, n E2,1, T E2,2, n E2,2, M max R + an T mass E 2 := s (see Figure 7) where ( h (E 2 (s), T E2,1, n E2,1) + h + (E 2 (s), T E2,2, n E2,2) ) = 0. t 3.2 GnRH concentration in the pituitary portal system Except for the pituitary portal system, the GnRH concentration oes not play a major role in the bloo system. The GnRH concentration in the bloo circulation is not etectable [31] an the half life for GnRH in bloo plasma is eclare as 4 minutes [32]. It can be assume that there is a basal GnRH secretion at the constant rate b GnRH inepenent of feeback mechanisms since even at high GnRH pulse frequency the GnRH concentration eclines not to zero values but to a baseline niveau [31]. Moreover, the pulse mass is not release at once, but release into the pituitary portal system in a istribute manner. The clearance of GnRH is assume to be proportional to the present GnRH concentration in the pituitary portal system enote by α GnRH. Bining of GnRH to its receptor locate in the pituitary is proportional to the present GnRH concentration in the pituitary portal system an to the GnRH receptor concentration. 12

14 It follows for the GnRH concentration in the pituitary portal system: t GnRH(t) = α GnRH GnRH(t) β GnRH GnRH(t) R GnRH (t) (26) + b GnRH V P P S + M GnRH,j V P P S ψ(t T j ), (27) where V P P S R + enotes the bloo volume of the pituitary portal system. We choose the Gamma ensity for the mass istribution: ψ(t) = am Γ(m) tm 1 e a t, a, m R +. To be accurate, the pulse masses of the foregoing pulses have to be consiere as well, see [22, 23]: t GnRH(t) = α GnRH GnRH(t) β GnRH GnRH(t) R GnRH (t) (28) + b j GnRH M GnRH,i + ψ(t T i ). V P P S V P P S (29) i=1 Since the remaining masses are small in comparison with the new pulse mass, we assume that they can be neglecte. 4 Mechanisms in the Pituitary GnRH is transporte over the pituitary portal system into the pituitary where it bins to its receptor. Activation of the receptor, finally, leas to the release of the gonaotropins luteinizing hormone (LH) an follicle stimulating hormone (FSH). Both gonaotropins are store in the pituitary an the secretion from this pool is regulate by GnRH an by the sterois prouce in the ovaries. 4.1 Receptor bining of GnRH GnRH stimulates the release of LH un FSH, but it has no effect on other pituitary hormones [15]. Within a few secons after bining of the peptie to its receptor, a massive calcium entrance into the cell occurs [32]. This leas after a couple of minutes to the release of store FSH an LH [32]. The amplitue of the GnRH pulses (amplitue moulation) an, as fine regulation, the frequence of GnRH pulses (frequency moulation) regulate the LH an FSH concentrations over the cycle [32]. Reactions on increase GnRH concentrations occur in the case of LH faster an more istinctive than in the case of FSH [15]. The number of GnRH receptors is not constant. In the early follicular phase, the number is higher than in the late follicular phase [12]. The pulsatility of GnRH secretion is necessary, since continuous aministration of GnRH leas to ecreasing LH an FSH concentrations [29]. This phenomenon can be explaine by the negative feeback of GnRH on its receptor [15]. Declining response of the GnRH receptors on GnRH is calle esensitization [12]. In [19], a mathematical moel for the GnRH receptor bining, incluing the receptor recycling, which leas to the release of LH is evelope. Unfortunately, the moel is not esigne for the human case, but for ovines, which is why the results have to be hanle with care. The moel scheme for the receptor bining in the pituitary is aopte from [19], shown in Figure 8. After bining to its receptor, the complex stimulates the release of LH from the LH pool. 13

15 H R H k 3 b LH P LH k 1 (H-R H ) k 2 R H - a 2 a 1 LH H Figure 8: Moel scheme for the receptor bining in the pituitary [19]. H: GnRH, R H : GnRH receptor, (H-R H ): complex, R H -: esensitize receptor. The activate receptor oes not return irectly to the unboun state, but becomes esensitize. There is a basal LH synthesis rate, b LH, an parts of LH are release at a constant rate inepenent from GnRH, a 1. The resulting moel equations for the receptor recycling are, slightly moifie from [19]: t R GnRH = k 1 GnRH R GnRH + k 3 R GnRH - (30) t (GnRH-R GnRH) = k 1 GnRH R GnRH k 2 (GnRH-R GnRH ) (31) t R GnRH- = k 2 (GnRH-R GnRH ) k 3 R GnRH -, (32) where k 1, k 2, k 3, R +. For the LH ynamics, we obtain t P LH = b LH a 1 P LH a 2 P LH (GnRH-R GnRH ) (33) t LH = 1 (a 1 P LH + a 2 P LH (GnRH-R GnRH )) c LH, V B (34) where b LH, a 1, a 2, c R +. To incorporate the negative feeback of GnRH on its receptor, the ifferential equation for the free receptor concentration coul be written as t R GnRH = k 1 GnRH T GnRH 1 + GnRH + k 3 R GnRH - k 1 GnRH R GnRH, (35) T GnRH where k, T GnRH R +. We obtain for the total receptor concentration: R GnRH,total = R GnRH + R GnRH - + (GnRH-R GnRH ). (36) The moel presente in [2] an [44] inclues aitionally the calcium ynamics. The moel is escribe as moel schemes in Figures 9 an 10 whereby the moel equations are taken from [44]. Again, it is assumen that, initially, GnRH is boun by the receptor. The boun complex reacts with itself to form imers. Formation of larger aggregates is possible. Macroaggregation an internalization occur with a elay after exposure to GnRH [44]. A G protein reacts with 14

16 H R H H R H (H-R H ) (H-R H ) GQ (H-R H ) 2 E CA E IP 3 MM CHO CA C CA ER MM LH Figure 9: Moel scheme for receptor bining in the pituitary incluing calcium ynamics in the cell [2, 44]. H: GnRH, R GnRH : free GnRH receptor, (H-R H ): hormone-receptor complex, (H-R H ) 2 : hormone-receptor imer, GQ: G protein, E: effector, IP 3 : Inositol 1, 4, 5-triphosphate, CA E : external Ca 2+, CA C : cytosolic Ca 2+, CA ER : enoplasmic reticulum (ER) Ca 2+, CHO: fraction of open ER Ca 2+ channels, MM: Michaelis-Menten kinetics. the imer to prouce an effector. It is assume that a fraction of receptors, r 0, can be returne intact to the membrane after a time elay. There is a low basal rate of receptor synthesis, P 0, an egraation, γ. From these assumptions, we obtain the following ifferential equations [44]: t R GnRH = k 1 GnRH R GnRH + k 1 (GnRH-R GnRH ) (37) + r 0 k 11 n (GnRH-R GnRH ) 40 + P 0 γ R GnRH (38) t (GnRH-R GnRH) = k 1 GnRH R GnRH k 1 (GnRH-R GnRH ) + 2 k 2 (GnRH-R GnRH ) 2 2 k 2 (GnRH-R GnRH ) 2 k 11 n (GnRH-R GnRH ) (40) t (GnRH-R GnRH) 2 = k 2 (GnRH-R GnRH ) 2 + k 2 (GnRH-R GnRH ) 2 (41) k 3 GQ (GnRH-R GnRH ) 2 + k 3 E (42) t GQ = k 3 GQ (GnRH-R GnRH ) 2 + k 3 E + k 33 e t/20 (GnRH-R GnRH ) 2 t E = k 3 GQ (GnRH-R GnRH ) 2 k 3 E, (44) where (39) (43) (GnRH-R GnRH ) 40 := (GnRH-R GnRH )(t τ) χ(t τ) (45) { 1 if t 0, χ(t) := (46) 0 if t < 0 an k 1, k 1, k 11, k 2, k 2, k 3, k 3, k 33, P 0, γ, τ R +, r 0 (0, 1), n N. It is further assume that the prouction of inositol 1,4,5-triphosphate, IP 3, is proportional to the concentration of E an that it is eliminate at a rate proportional to its concentration 15

17 H R H (H-R H ) r r 0 R H - R H H Figure 10: Moel scheme for the receptor recycling in the pituitary [44], slightly moifie by ignoring macroaggregation an incluing the state of the esensitize receptor R H -. H: GnRH, R H : GnRH receptor, (H-R H ): complex, H : egrae hormone, R H : egrae receptor. [44]. t IP 3 = k 5 E k 5 IP 3, k 5, k 5 R +. (47) Ca 2+ is store in the enoplasmic reticulum, ER, an it is release when IP 3 bins to its receptor on the ER membrane. It is assume that the fraction of open channels, CHO, epens on IP 3 through Michaelis-Menten kinetics [44]: CHO(t) = α 10 3 IP 3 (t) 1 + α 10 3 IP 3 (t) ( β t p e 1 β tp ), α, β R +, (48) where t p R + enotes the time after the start of each iniviual pulse. The Ca 2+ concentration ynamics in the enoplasmic reticulum, CA ER, an the concentration of the cytosolic Ca 2+ ynamics, CA C, are escribe by the ifferential equations where t CA ER = CHO ERR (CA ER CA C ) (49) 2 CA 2 C + k CA 2 (ERUL CA ER ) C (50) t CA C = 0.05 CHO ERR (CA ER CA C ) (51) 2 CA 2 C 0.05 k CA 2 (ERUL CA ER ) C (52) + V SR (CA E CA C ) k 7 CA 2 C CA 2 C + k 9 CA E (53) ERR = k 6 + k 66 CA C k 666 CA 2 C (54) V SR = k 8 E + k 88 CA C k 888 CA 2 C V SRO (55) { t V SRO = v 1 if GnRH > 0, (56) v 2 if GnRH = 0, an k 6, k 6, k 66, k 666, k, k 8, k 88, k 888, k 9, ERR, ERUL, v 1, v 2 R +, whereas the external Ca 2+ concentration, CA E, remains constant [44]. 16

18 Finally, it is assume that the release of LH epens on CA C through Michaelis-Menten kinetics [44]: t LH = k 10 CA 2 C 2 + CA 2, k 10 R +. (57) C This moel focus on the GnRH-inuce secretion of LH, since GnRH has effect only on gonaotropin release an not on synthesis. The release stimulation is escribe in more etail which is why it offers a suitable alternative. In Figure 10, the receptor recycling process, as it is erive in [44], slightly moifie, is presente. Aitionally to [2] which is constraine to the receptor bining leaing to calcium ynamics an, finally, to the LH ynamics, the recycling process is inclue in [44]. Thus, assuming receptor ynamics like in [19], even if erive for the ovine case, is reasonable even for women. The research focus on the case of LH, since it is a key element at mi-cycle. Therefore, we can only assume that the release of FSH meiate by GnRH is comparable with LH release [45]. 4.2 Dynamics of LH an FSH In aition to the assumptions presente in [17] an in Section 2, we can fin the following eviations. Probably, the inhibitory effect of estraiol oes not remain uring the entire cycle. Actually, it has the ability to exert both negative an positive feeback on the secretion of the gonaotropins [1, 33, 43, 45]. Since the initiation of the LH an FSH surge is the consequence of the stimulation by estraiol [45], it is assume that estraiol is inhibitory at low concentrations an stimulatory at high concentrations. This biphasic feeback is also part of the regulation of the GnRH pulse generator presente in Section 3. The stimulatory effect of estraiol is amplifie by progesterone [15, 43]. In contrast to the assume stimulatory effect of progesterone, it is possible that estraiol can inhibit FSH synthesis [1] an progesterone can, in combination with estraiol, exert a negative feeback on the FSH secretion [45]. Inhibin inhibits the FSH synthesis [15, 30] an, moreover, the FSH release [21, 27], whereas activin enhances FSH secretion [21]. These feeback mechanisms are meiate by receptor bining which is not completely clarifie. If we consier these aitional assumptions (see Figures 11 an 12), the moel equations for the LH ynamics have the following moifie form, in comparison to Equations 1 to 5, assuming epenence of effects: where t P LH = syn LH (E 2,τE2, P 4,τP4 ) rel LH (E 2,τE2, P 4,τP4, (GnRH-R GnRH ), P LH ) (58) t LH = 1 rel LH (E 2,τE2 +τ V LH, P 4,τP4 +τ LH, (GnRH-R GnRH ) τlh, P LH,τLH ) B (59) clear LH (LH), (60) syn LH = b LH + h 1 (P 4) h + 1 (E 2) syn LH,max (61) rel LH = ( h 2 (E 2) + h + 2 (E 2) ) h + 3 (P 4) rel LH,max (GnRH-R GnRH ) P LH (62) clear LH = α LH LH. (63) The parameter rel LH,max is comparable to a 2 an c is analog to α LH in Equations 33 an

19 - PROGESTERONE + < T LH ESTRADIOL > T LH + synthesis (LH) release (LH) Figure 11: Regulation of LH synthesis an release by progesterone an estraiol. - INHIBIN synthesis (FSH) - PROGESTERONE ESTRADIOL + < T - F SH + > T F SH release (FSH) Figure 12: Regulation of FSH synthesis an release by progesterone, estraiol, an inhibin. In the case of FSH, we have: t P F SH = syn F SH (Ih τih ) rel F SH (E 2,τE2, P 4,τP4, Ih τih, (GnRH-R GnRH ), P F SH ) (64) t F SH = 1 rel F SH (E 2,τE2 +τ V F SH, P 4,τP4 +τ F SH, Ih τih +τ F SH, (GnRH-R GnRH ) τf SH, P F SH,τF SH ) B where clear F SH (F SH) (66) syn F SH = b F SH + h 3 (Ih) syn F SH,max (67) rel F SH = ( h 4 (E 2) + h + 4 (E 2) ) h + 5 (P 4) h 5 (Ih) rel F SH,max (GnRH-R GnRH ) P F SH (68) clear F SH = α F SH F SH. In both cases, h + i (H) := h+ i (H, T i, n i ) an h i (H) := h i (H, T i, n i ), i = 1,..., 5, enote the hill functions as well as H τ := H(t τ) for H = E 2, P 4, Ih, (GnRH-R GnRH ), P LH, P F SH an τ = τ E2, τ P4, τ E2 + τ LH, τ Ih, τ E2 + τ F SH, τ P4 + τ F SH, τ Ih + τ F SH. 5 Dynamics in the Ovaries The follicular evelopment is primarily controlle by LH an FSH where FSH plays an important role especially at the beginning while LH promotes follicular growth up to ovulation an maintains the activities of the corpus luteum in the secon half of the cycle. The process of evelopment can be ivie into several stages characterize by the LH an FSH receptor concentrations, the enzyme prouction, an the biosynthesis of sterois, hormones eriving from cholesterol. 18 (65) (69)

20 Figure 13: Follicular evelopment, aapte from [28]. For simplification, it can be assume that the total receptor concentration is constant. The enzyme concentrations in the follicles epen on the LH an FSH bloo concentrations an on the concentration of activate LH an FSH receptors. By assuming simple Michaelis-Menten kinetics, we can evelop moel equations for the steroiogenesis, the steroi biosynthesis in the ovaries, from which the steroi concentrations in the bloo arise. In the following subsections, there are three main questions to be clarifie: Which parts of the steroiogenesis occurs in which cells an at which stage. Shortly expresse, we want to know what happens where an when. 5.1 Follicular evelopment The purpose of regulation carrie out by reprouctive hormones is eventually the follicular evelopment in the ovaries which is shown in Figure 13. Out of the pool of immature follicles in the ovaries, a couple of them enter the menstrual cycle an run through the single follicular stages, unless they unergo atresia. The prouction of sterois epens largely upon the follicles growth an maturation. There are mainly two cell types which are responsible for the ifferent processes leaing to the two cell theory. Two cell theory. There are mainly two cell types which are important in the pathway of follicular evelopment, the granulosa an the theca cells. They iffer in, for instance, receptor expression an enzyme prouction. The basic features of the two cell theory are shown in Figure 14. The anrogen synthesis in the theca cells is mainly meiate by the enzymes cholesterol sie chain cleavage (P450scc), 17α-hyroxylase (P OH ), an 3β-hyroxysteroi ehyrogenase (3β-HSD) [5]. In granulosa cells, the 17β-hyroxysteroi ehyrogenase enzyme (17β-HSD) transforms anrostenione into testosterone, an aromatase (P450arom) catalyzes the reaction from testosterone to estraiol [5]. The enzymes P450scc an 3β-HSD are prouce in the granulosa cells as well. Follicular growth an maturation. The following approaches are conceivable an have been evelope to some extent in the past: 19

21 T H E C A G R A N U L O S A chol P450scc preg 3β-HSD prog P OH P OH 17α-preg 17α-prog P OH P OH DHEA 3β-HSD anr 17β-HSD chol anr test P450scc P450arom P450arom 17β-HSD preg estro estra 3β-HSD prog LH FSH LH Figure 14: Two cell theory [11, 15, 40, 41]. chol: Cholesterol, preg: Pregnenolone, prog: Progesterone, 17α-preg: 17α-Hyroxypregnenolone, 17α-prog: 17α-Hyroxyprogesterone, DHEA: Dehyroepianrosterone, anr: Anrostenione (Anrost-4-ene-3,17-ione), test: Testosterone, estro: Estrone, estra: Estraiol (Estraiol-17β). Moelling the ynamics of single follicles. The time points where follicles enter the maturing process are generate by a stochastic process. The case where the ynamics in the follicles are represente by the estraiol concentrations is treate in [6, 25]. The ynamics are escribe by a system of ifferential equations. This moel is able to characterize ovulatory as well as anovulatory cycles [25]. In [6], the moel is moifie in orer to moel the PCO synrome. Moelling on a cellular level. There are mainly two ifferent cell types in the follicles: theca an granulosa cells. By moelling the cell growth an maturation process as well as the receptor complex concentration, the steroi synthesis an release ynamics can be etermine. The case of growth kinetics of the granulosa cell population has been treate in [7, 8]. Moelling follicular masses in iscrete stages. The number of follicles in every single stage is not important, only its mass. Every stage has its characteristic enzyme an steroi prouction. This approach is use in [17, 18, 36, 38]. The iea of moelling follicular masses coul be combine with the two-cell theory. That means that we o not consier total follicular masses, but a granulosa cell mass an a theca cell mass since granulosa an theca cells are the locations where steroiogenesis takes place. The approach mentione at thir position is chosen in [17], our basic compartment moel. The iea mentione last coul be incorporate into the basic moel. By incluing aitional information, the moel for the follicular evelopment will be moifie. In the following, we stuy the influence of the gonaotropins FSH an LH. The ovaries contain a pool of immature primorial follicles consisting of approximately 400,000 follicles at puberty [15, 21]. They possess an oocyte, an immature egg-cell, which is surroune by a single layer of flattene, poorly ifferentiate cells calle granulosa cells [15, 21, 41]. Out of this pool, 1 to 15 [5] or even 30 [21] enter the follicular growth an maturation process of the menstrual cycle. It is assume that these first steps of the so-calle initial recruitment [41] is inepenent of the gonaotropins FSH an LH [5, 21, 35, 41, 46], but regulate by intra-ovarian factors [41]. 20

22 When the primorial follicle has left the pool of resting follicles, the size of the oocyte begins to increase, the zona pellucia, a membrane that will eventually surroun the oocyte, begins to form, an the granulosa cells assume a cuboial shape [15, 41]. The follicle at this stage is calle primary follicle [15, 41]. It is assume that the follicular growth is still inepenent of FSH an LH [46]. As the so-calle seconary follicle or preantral follicle [15, 41] is being forme, the granulosa cells express FSH receptors [21, 41]. But to start the gonaotropin-epenent follicular evelopment an to ensure the further evelopment, a certain, iniviually varying threshol value for FSH has to be exceee [21, 41]. Henceforth, the follicles are gonaotropin-sensitive [15, 21]. The transition from the seconary follicle to the next stage is promote by FSH [41], now calle the tertiary follicle or antral follicle [15]. The granulosa cells rapily proliferate an the follicle continues to enlarge [15, 41]. The prouction of IGF-2 starts which is iffuning to the surrouning ovarian stroma inucing the ifferentiation of the theca cells [21]. These express LH receptors an become LH-sensitive [21, 41]. Since the thecal layer is forme, LH can also have a stimulating effect on the follicular evelopment. Within the granulosa layer, flui is accumulating leaing to the formation of the so-calle antrum. The follicles are not longer only FSH responsive but FSH epenent [35, 41]. Among the recruite follicles, selection of normally one ominant follicle takes place an the suborinate follicles unergo atresia [21, 41]. The mechanism leaing to this selection is still unknown [21]. Upon the completion of this growth phase, the follicle, now referre to as Graafian follicle or preovulatory follicle, is prepare for ovulation [15, 41]. FSH inuces LH receptors in the granulosa cells an, consequently, LH as well as FSH promote follicular maturation [21, 41]. The growth an maturation process is complete an the follicle is reay for the ovulation when speaking of the ovulatory follicle. LH stimulates this process where the follicle ruptures an releases the ovum. Then, the cells of the rupture follicle change morphologically which is calle luteinization [41]. The follicle at this stage is referre to as luteinizing follicle [41]. The correct term from now on is corpus luteum when speaking of the follicle [21] which can be ivie into the stages early corpus luteum, mature corpus luteum, an late corpus luteum. At the beginning of this phase, it is referre to as early corpus luteum. LH has no influence on these changes, the corpus luteum is establishing autonomously [21]. In the mature corpus luteum as well as in the late corpus luteum, LH becomes more an more inispensable in the maintainance of corpus luteum activities [41]. The regression of the corpus luteum ens, unless pregnancy occurs, as an avascular scar, the corpus albicans [41]. There are other factors influencing follicular evelopment which have been neglecte at this state of moelling since FSH an LH are the main hormones regulating the follicular growth an maturation. Two essential processes which have to be istinguishe can be ientifie: the maturation of the follicles or, unerstoo in a iscrete manner, the transistion to the next follicular stage, an the growth, the proliferation of cells within one stage. The physiological classification can be use as basis for the etermination of the mathematical partioning. To improve the quality, each stage coul be ivie into several steps as it was one in [17] for the luteal phase. The exact assignment of the influence of FSH an LH on the transition between stages an on the growth at the stages is not possible, since the specifications in the literature can vary. Simulation of the ifferent possibilities coul help to ecie which approach is to be chosen. The assume influence of the gonaotropins at the ifferent stages is shown in Figure 15 schematically. 21

23 F SH F SH > T F SH F po F p F s F SH, LH LH F SH F SH, LH F t F g LH M o L a L l L m L e M l LH LH Figure 15: Gonaotropin-epenence uring the follicular evelopment. The gonaotropins influence the growth an maturation at certain stages, growth stimulation is expresse by an arrow towars the box, maturation (transition) stimulation by an arrow to the transition arrow. F po : mass of primorial follicles, F p : mass of primary follicles, F s : mass of seconary follicles, F t : mass of tertiary follicles, F g : mass of Graafian follicle, M o : mass of follicle at ovulation, M l : mass of follicle at luteinization, L e : mass of early corpus luteum, L m : mass of mature corpus luteum, L l : mass of late corpus luteum, L a : mass of corpus albicans. Accoring to the moel scheme, we obtain for the follicular phase: t F s(t) = k 1 h + (F SH(t τ F SH ), T F SH, n F SH ) + k 2 f 1 (F SH(t τ F SH )) F s (t) (70) k 3 f 1 (F SH(t τ F SH )) F s (t) (71) t F t(t) = k 3 f 1 (F SH(t τ F SH )) F s (t) + k 4 f 2 (F SH(t τ F SH ), LH(t τ LH )) F t (t) (72) k 5 f 2 (F SH(t τ F SH ), LH(t τ LH )) F t (t) (73) t F g(t) = k 5 f 2 (F SH(t τ F SH ), LH(t τ LH )) F t (t) + k 6 f 1 (LH(t τ LH )) F t (t) (74) k 7 f 1 (LH(t τ LH )) F g (t), (75) uring ovulation an luteinization: an for the luteal phase: t M o(t) = k 7 f 1 (LH(t τ LH )) F g (t) k 8 M o (t) (76) t M l(t) = k 8 M o (t) k 9 M l (t), (77) t L e(t) = k 9 M l (t) k 10 f 1 (LH(t τ LH )) L e (t) (78) t L m(t) = k 10 f 1 (LH(t τ LH )) L e (t) k 11 f 1 (LH(t τ LH )) L m (t) (79) t L l(t) = k 11 f 1 (LH(t τ LH )) L m (t) k 12 L l (t) (80) t L a(t) = k 12 L l (t) k 13 L a (t), (81) where τ F SH, τ LH R + enote the elays, T F SH R + the threshol value for FSH, n F SH R + the hill coefficient, an k i R +, i = 1,..., 13. The approach by the functions f 1 (H) := H α, α R + 22

24 an f 2 (H 1, H 2 ) := H α 1 1 H α 2 2, α 1, α 2 R +, where H, H 1, an H 2 enote the imensionless FSH or LH concentration, represents only one of the possibilities accoring to the basic compartment moel. These moel equations coul be written seperately for the cases of granulosa an theca cells if esire. 5.2 Receptor bining of LH an FSH In orer to fin out more about the effects of the gonaotropins on the events in the ovary, it is necessary to ientify the interface between the two compartments pituitary an ovaries. FSH an LH are release into the bloo an reach the ovaries. Stimulation of follicular growth an maturation an, therewith, biosynthesis of the sterois is enhance mainly by gonaotropin coupling to its receptor which activates a cascae of reactions. The gonaotropins FSH an LH bin to their receptors an form a complex. By interacting with the G protein an transforming of GDP to GTP, the enzyme aenylate cyclase (or aenylyl cyclase) gets activate. The activate AC reacts with ATP, forming camp. By influence of camp, protein kinase A (PKA) gets activate leaing to higher enzyme synthesis an therefore to stimulation of the steroiogenesis. FSH-inuce camp prouction. In [9], a moel for the FSH-inuce camp prouction in ovarian follicles is erive. An overview of the reaction scheme is given in Figure 16. FSH bins to its receptor, R F SH, with the constant rate k +. This results in the formation of an active complex, (FSH-R F SH ), which can issociate with the constant rate k. Boun receptors activate aenylate cyclase, AC i. The activation epens on the complex concentration which is multiplie by the amplification parameter σ representing the average number of aenylate cyclase activate by one boun receptor. By reaction of the activate aenylate cyclase, AC a, with ATP, camp is synthesize activating the protein kinase A. In the moel presente in [9], the cycle of G protein activation/eactivation is not moele explicitly, neither the camp-inepenent esensitization. Moreover, it is assume that the total number of FSH receptors (free, active, phosphorylate, an internalize) remains constant, an that the amount of FSH is sufficiently large so that its concentration is unaffecte by bining to receptors. We obtain the following equations [9]: where t R F SH = k (F SH-R F SH ) + k r R i k + F SH R F SH (82) t (F SH-R F SH) = k + F SH R F SH (ρ + k ) (F SH-R F SH ) (83) t AC F SH = β (σ (F SH-R F SH ) AC F SH ) AC F SH (84) t camp = ω AC F SH k P DE camp (85) t (F SH-R F SH-p) = ρ (F SH-R F SH ) k i (F SH-R F SH -p) (86) t R i = k i (F SH-R F SH -p) k r R i, (87) ρ(camp ) = α camp γ δ γ + camp γ. 23

25 AC i FSH k +,k (FSH-R F SH ) R F SH k r R i ki ρ(camp) (FSH-R F SH -p) PDE AC a ω camp k P DE AMP ATP PKA i PKA a Figure 16: Moel scheme for the FSH-inuce camp prouction [9]. R F SH : FSH receptor, (FSH-R F SH ): receptor complex, (FSH-R F SH -p): phosphorylate receptor, R i : internalize receptor, AC i : inactive aenylate cyclase, AC a : active aenylate cyclase, PDE: phosphoiesterase, PKA i : inactive protein kinase A, PKA a : active protein kinase A. The total receptor concentration is R F SH,total = R F SH + (F SH-R F SH ) + (F SH-R F SH -p) + R i. (88) Receptor ynamics. The total receptor concentration epens on the follicular stage. FSH receptors are only expresse in the granulosa cells [5, 12, 15] while LH receptors are expresse in the theca cells an, at a later stage, in the granulosa cells as well. The receptor concentration in the granulosa cells epens also on the FSH concentration, an the receptor concentration in the theca cells on the LH concentration. These informations are not sufficient for escribing the etaile receptor ynamics mathematically. Coarsely, the ynamics can be escribe by: t R LH,total = f(ft theca,..., L theca a ; LH) + f(fg gran,..., L gran a ; F SH) (89) t R F SH,total = f(fs gran,..., L gran a ; F SH), (90) where f is not specifie. Fpo gran,..., L gran a enote the granulosa cell masses, Fpo theca,..., L theca a the theca cell masses. Generally, it can be assume that the bining of LH to its receptor procees in a similar manner as in the case of FSH, see [9]. The concentration of free LH an FSH receptors is regulate by the LH an FSH concentration, respectively, an the receptor recycling process. Since there are several possibilities how the signal transuction can be exerte, not only via the prouction of camp, we focus on the receptor recycling process. As a start, the total receptor concentration is assume to be constant uring the cycle as it is one in [9]. 24