A simple mathematical model of the bovine estrous cycle: follicle development and endocrine interactions

Transcription

1 A simple mathematical moel of the bovine estrous cycle: follicle evelopment an enocrine interactions H.M.T.Boer a,b,, C.Stötzel c, S.Röblitz c, P.Deuflhar c, R.F.Veerkamp a, H.Woelers a a Animal Breeing an Genomics Center, Wageningen UR Livestock Research, 8200 AB Lelysta, The Netherlans b Aaptation Physiology Group, Department of Animal Sciences, Wageningen University, 6700 AH Wageningen, The Netherlans c Computational Systems Biology Group, Zuse Institute Berlin (ZIB), Takustraße 7, Berlin, Germany Abstract Bovine fertility is the subject of extensive research in animal sciences, especially because fertility of airy cows has ecline uring the last ecaes. The regulation of estrus is controlle by the complex interplay of various organs an hormones. Mathematical moeling of the bovine estrous cycle coul help in unerstaning the ynamics of this complex biological system. In this paper we present a mechanistic mathematical moel of the bovine estrous cycle that inclues the processes of follicle an corpus luteum evelopment an the key hormones that interact to control these processes. The moel generates successive estrous cycles of 21 ays, with three waves of follicle growth per cycle. The moel contains 12 ifferential equations an 54 parameters. Focus in this paper is on evelopment of the moel, but also some simulation results are presente, showing that a set of equations an Tel.: , Fax.: aress: marike.boer@wur.nl (H.M.T.Boer) Preprint submitte to Theoretical Biology December 22, 2010

2 parameters is obtaine that escribes the system consistent with empirical knowlege. Even though the majority of the mechanisms that are inclue in the moel are base on relations that in literature have only been escribe qualitatively (i.e. stimulation an inhibition), the output of the moel is surprisingly well in line with empirical ata. This moel of the bovine estrous cycle coul be use as a basis for more elaborate moels with the ability to stuy effects of external manipulations an genetic ifferences. Keywors: cow, reprouction, hormone patterns, ifferential equations, systems biology 2000 MSC: 92C42, 92C30, 90C31, 65L Introuction Systems biology is a relatively new research area in the fiel of animal sciences. It aims at unerstaning how the various components of a biological system function together, rather than investigating only iniviual parts. One approach is the translation of a conceptual biological moel into a set of mathematical equations that represent the ynamic relations between system components. The purpose of builing such mathematical moels is to interpret an preict the ynamics of complex biological systems, an to ientify new research questions. One example of a ynamic biological system is the bovine estrous cycle, the hormonally controlle recurrent perios when the cow is preparing for reprouction by proucing a fertilizable oocyte. Concurrent with selection for increase milk yiel, a ecrease in airy cow fertility has been observe uring the last ecaes (for reviews see [1, 2]). This ecline in fertility is shown 2

3 by e.g. alterations in hormone patterns uring the estrous cycle, reuce expression of estrous behavior an lower conception rates [3]. However, it is har to unerstan which unerlying mechanisms cause this ecline in fertility. The regulation of estrus is controlle by the interplay of various organs an hormones. Mathematical moeling of the involve mechanisms is expecte to improve insight in the biological processes unerlying the bovine estrous cycle, an coul thereby help to fin causes of ecline fertility in airy cows [4]. Although the enocrine an physiologic regulation of the bovine estrous cycle is stuie extensively, mathematical moels of cycle regulation are scarce an of limite scope [5, 6]. A number of moels have been evelope for other ruminant species, especially ewes [7, 8], but these moels o not contain all the key players that are require to simulate follicle evelopment an the accompanying hormone levels throughout consecutive cycles. A moel that integrates the major tissues an hormones involve, an that is able to simulate the ynamics of follicular evelopment, has been evelope for the human menstrual cycle by Reinecke [9]. This moel, which is base on previous work by Selgrae an colleagues [10, 11, 12], escribes the ynamics of hormones, enzymes, receptors, an follicular phases throughout the cycle in a set of ifferential equations. The objective of the work escribe in this paper was to evelop a mathematical moel of the ynamics of the bovine estrous cycle on iniviual cow level, that is able to simulate follicle evelopment an the accompanying fluctuations in hormone concentrations. Physiologic an enocrine mechanisms that regulate the cycle are very similar between human an cows. There- 3

4 fore, some mechanisms of the human moel in [9] coul be use (although sometimes with simplifications), an extene with other mechanisms like follicular wave emergence an corpus luteum regression. Focus in this paper is on the moel evelopment. Section 2 escribes the biological mechanisms of the bovine estrous cycle an how these mechanisms are incorporate in the moel. In Section 3, the mathematical escription an all moel equations an parameters are given. Simulation results are presente in Section 4, showing that a set of equations an parameters is obtaine that escribes the system consistent with biological ata for cows. In Section 5, it is iscusse how the current moel coul be applie an extene Biological backgroun 2.1. Follicles Two ifferent patterns of follicle evelopment are ientifie in mammals. In humans (an rats an pigs), the evelopment of follicles to ovulatory size occurs only uring the follicular phase, while in cattle (an sheep an horses), evelopment of follicles to ovulatory or near-ovulatory size occurs throughout the cycle [13]. A normal cycle inclues two or three wave-like patterns of follicle evelopment, in which a cohort of follicles start to grow. The average uration of the bovine estrous cycle is 20 ays for 2-wave an 22 ays for 3-wave cycles (reviewe in [14]). Each follicular wave is initiate by an increase of follicle stimulating hormone (FSH) release from the anterior pituitary [15]. The growing follicles prouce estraiol (E2) an inhibin (Inh), which are release into peripheral bloo. In the first one or two 4

5 waves, a ominant follicle eviates from the cohort of growing follicles that oes not ovulate, but unergoes regression uner influence of progesterone (P4) prouce by the corpus luteum (CL). When the CL is regresse uner influence of PGF2α, the concentration of P4 ecreases [16]. The ominant follicle present at that moment evelops an matures, an ovulation can then take place because the inhibiting effect of P4 on the surge of luteinizing hormone (LH) is remove [17]. Elevate E2 levels increase the secretion of gonaotropin releasing hormone (GnRH), which triggers the LH surge an thereby inuces ovulation. Once an oocyte is successfully ovulate, the remains of the follicle form a new P4-proucing CL. If conception has faile, the CL regresses, P4 levels ecrease, an the cycle restarts (reviewe in [4]). The ovaries contain a pool of small follicles with immature oocytes. Uner influence of FSH, a cohort of 8-41 growing follicles emerge [14]. Approximately two ays after cohort recruitment, one follicle is selecte to become the ominant follicle, an continues to grow [18]. This eviation of the ominant follicle is associate with increase FSH an LH receptor bining, activating the enzymes that catalyze steroiogenesis, resulting in increase E2 prouction an higher E2 serum levels [18]. The ominant follicle expresses more FSH receptors, an it can therefore continue to grow even when FSH serum levels are low [19]. In the moel, the emergence of a follicular wave is inuce when FSH excees a threshol which becomes lower when follicles become larger, representing that larger follicles are more sensitive to FSH. Dominant follicles also secrete increasing amounts of inhibin (Inh). Inh suppresses FSH an, hence, suppresses the growth of suborinate follicles. Ovulation or regression of the ominant follicle eliminates this suppression, 5

6 allowing the onset of the next follicular wave [20, 21]. Small follicles of an emerging cohort may release very small amounts of E2 an Inh per follicle, but taken together, this amount is not negligible. Furthermore, there is always a meium-size or large follicle present [22, 23, 24], which results in a basal hormone prouction throughout the cycle. Different follicles are recruite, growing, an regressing in each cycle an in each wave. However, total E2 an Inh prouction capacity is moele as a continuous function throughout subsequent waves an cycles, representing the total amount of hormone prouction of the follicles present at any moment. Follicle regression is promote by high P4 levels an by the LH surge (Equation 7). The capacity of follicles to prouce E2 an Inh is enote as follicular function in the rest of this paper Corpus luteum The CL evelops within 2-3 ays after ovulation, starting the synthesis an release of P4, which maintains the reainess of the enometrium for receiving the embryo. In absence of a conceptus, the CL will regress at ay of the cycle [25, 26]. In each cycle a new CL evelops, but CL evelopment is moele as a continuous function of P4 proucing tissue, enote as CL function in the rest of this paper. In the moel, CL evelopment is inuce by the LH surge. A threshol an elay are incorporate in the effect of LH on the CL, to account for the time require for the process of transition from follicle to CL [16] an the shift from E2 to P4 prouction [27, 28]. If the CL reaches a certain size, it continues to grow without further stimulation by LH [29]. CL regression is inuce by PGF2α secretion from the uterus (escribe in Section 2.4). Growth an regression of CL function are escribe by 6

7 114 Equation Estraiol an inhibin E2 affects LH synthesis an release [30] an FSH release [19, 31]. E2 serum levels are higher in ovulatory than in non-ovulatory waves [20, 32] an reach peak levels aroun estrus [20, 32, 33, 34, 35, 23]. This suggests that the preovulatory follicle has the largest capacity to prouce an release E2, although its maximum size is not significantly ifferent from the maximum size of non-ovulatory ominant follicles. Consiering the results in [36, 37], where a better vascularity of the ovulatory follicle is reporte, it is reasonable that the ovulatory follicle can secrete more E2 than non-ovulatory follicles an, consequently, E2 serum levels are highest at estrus. In the moel, the rate of E2 prouction an release to the bloo is taken as proportional to follicular function (Equation 11). Inh inhibits FSH synthesis an thus reuces FSH release [21]. Compare to basal Inh serum levels, peak levels are almost ouble in non-ovulatory waves an increase further in ovulatory waves [38]. There are ifferent forms of inhibin, but only inhibin A is consiere in the moel, as it is the preominant form in bovine follicular flui [19]. In the moel, Inh prouction rate is taken as proportional to follicular function (Equation 12) Progesterone an prostaglanin F2α The CL is the main source of P4. Serum P4 concentration is near to zero aroun estrus an high uring the luteal phase [39, 40, 32, 41, 42]. A high correlation between CL iameter an P4 output was reporte in [43, 44, 24]. 7

8 In the moel, the rate of P4 release into the bloo is taken as proportional to CL function (Equation 10). Pulsatile PGF2α release from the uterus inuces CL regression. The rise of P4 early in the cycle initiates a series of events or mechanisms that eventually lea to the rise of PGF2α, followe by a ecline of PGF2α a few ays later. It was shown that aministration of P4 prior to its natural rise resulte in an equally earlier onset of CL regression [45]. Exposure to effective amounts of P4 must last for ays to inuce PGF2α pulses [45, 46, 47, 48]. Peak PGF2α levels are 3-4 times higher than basal levels [49, 50, 51, 52]. PGF2α is regulate by oxytocin (OT), P4 an E2 [53]. P4 first prevents a too early release of PGF2α pulses, but simultaneously stimulates synthesis of enzymes require for PGF2α prouction. In the later luteal phase, change expression of P4 an OT receptors results in a graual ecrease in the suppression of PGF2α [49], leaing to an OT inuce pulsatile release of PGF2α [52, 46]. How these mechanisms are regulating each other is quite complex an not unerstoo in full etail. What is clear is that the rise in P4 levels an the continue presence of P4 above an effective level sets in motion a series of events that lea to CL regression. Hence, we incorporate these series of events as a black box using time elays to obtain the right timing of PGF2α signaling. In the moel, PGF2α increases a specific number of ays (elay τ P 4,1 ) after P4 levels reach a threshol. Similarly, PGF2α eclines another (larger) number of ays (elay τ P 4,2 ) after P4 levels reache a threshol (Equation 8). 8

9 Gonaotropin releasing hormone, luteinizing hormone an follicle stimulating hormone Pulsatile signaling of GnRH regulates LH an FSH secretion [54]. Because GnRH inuces the LH surge, it inirectly inuces ovulation [55]. The GnRH pulse generator is locate in the hypothalamus an is moulate by P4 an E2 [56]. During the luteal phase, both P4 an E2 suppress the activity of the GnRH pulse generator. During pro-estrus however, elevate E2 levels change estrogen receptor signaling, which inuces a GnRH surge [30, 56]. GnRH is release into the portal circulation of the pituitary an bins to GnRH receptors of the anterior pituitary [57]. In the moel, GnRH stimulates LH release, resulting in an LH surge concurrently with the GnRH surge. GnRH synthesis is taken constant as long as the amount of GnRH in the hypothalamus is below a threshol (Equation 1). GnRH release is inhibite when P4 levels are above a threshol an when both P4 an E2 levels are above a threshol. GnRH release is stimulate when P4 levels are low an E2 reaches a threshol (Equation 1b), resulting in a surge of GnRH. GnRH concentration in the pituitary epens on GnRH amount release from the hypothalamus, an is further increase by high E2 levels, representing that E2 up-regulates expression of GnRH receptors [56, 57] (Equation 2). The LH surge at the ay before ovulation inuces ovulation of the ovulatory follicle an formation of the CL. The LH surge will shut own E2 an Inh prouction capacity of the ovulatory follicle [58, 24]. High P4 levels suppress the release of LH via the inhibition of the GnRH pulse generator [59]. Aitionally, high P4 levels ecrease pituitary sensitivity to E2, thereby increasing the amount of E2 require to inuce an LH surge above physio- 9

10 logical levels [56]. Peak LH levels are about five times as high as basal levels or higher [20, 32, 60, 27]. In the moel, LH synthesis is stimulate by E2 an inhibite by P4 (Equation 5a). Besies a small basal LH release, there is a surge of LH when GnRH in the pituitary reaches a threshol (Equation 5b). FSH synthesis is inhibite by Inh [19]. P4 an E2 moulate FSH release via effects on the anterior pituitary an on the GnRH pulse generator in the hypothalamus. Peak FSH serum levels are about three times higher than basal levels [20, 33]. In the moel, FSH synthesis in the pituitary is increase when Inh levels are below a threshol (Equation 3a). FSH release from the pituitary to the bloo is stimulate by P4 an GnRH, an inhibite by E2 (Equation 3b) Mathematical formulation The mathematical approach use for the bovine moel is comparable to the approach use for the moel of the human menstrual cycle, which originally has been evelope at North Carolina State University by Selgrae an colleagues [10, 11, 61, 12], an has been extene at the Zuse Institute [9, 62]. The system is consiere in four compartments: hypothalamus, anterior pituitary, ovaries an uterus, connecte through peripheral an portal bloo (Figure 1). The moel inclues the processes of follicle an CL evelopment an the key hormones that interact to control these processes as escribe in Section 2. The gonaotropin equations are base on synthesis-releaseclearance relations. This structure was first introuce in [11]. The complete 10

11 Hypothalamus GnRH synthesis an release f(peripheral E2 an P4) P4 E2 GnRH Portal bloo system Peripheral bloo Clearance of P4, E2, Inh P4 E2 Inh Pituitary Synthesis of LH an FSH f(p4,e2,inh) Release of LH an FSH f(p4,e2,gnrh) LH FSH Peripheral bloo LH FSH Clearance of LH, FSH P4 E2 Inh Ovaries Follicle growth (E2 synthesis) f(lh,fsh, follicular size) Inhibin synthesis f(follicular size) CL growth f(lh,cl size,pgf2 α) P4 PGF2 α Uterus PGF2 α synthesis f(cl size) Figure 1: Schematic representation of the compartments in the moel of the bovine estrous cycle. 11

12 Figure 2: Complete mechanisms of the bovine moel. Boxes represent the 12 key components of the system. Differential equations are erive for these 12 components. Arrows enote functional epenencies. Stimulating an inhibiting effects are inicate by + an - respectively mechanisms are shown in Figure 2. Base on these mechanisms, 12 orinary ifferential equations (ODEs) with 54 parameters are formulate. If necessary, time elays are incorporate to moel the time between events an their effects, representing the uration of intermeiate steps in biological processes. In this case, the ODE is turne into a elay ifferential equation (DDE). To solve the system of ifferential equations, we use the solver RADAR5 [63], which has been esigne for the solution of stiff elay ifferential equations. 12

13 Hill functions Because the exact mechanisms are often not known or more specific than necessary, Hill functions are use to moel stimulatory an inhibitory effects of the hormones. They are use whenever there is a nonlinear relation between two substances. A Hill function is a sigmoial function between zero an one, which switches at a specifie threshol from one level to the other with a specifie steepness. Positive Hill functions are use for stimulating effects an are efine as h + (S(t); T, n) := S(t) n T n + S(t) n S(t) represents the effector, T the threshol for change of behavior, an n controls the steepness of the curve. Negative Hill functions are use for inhibitory effects an are efine as h (S(t); T, n) := T n T n + S(t) n Here, the value of the function has its maximum at the lowest value of the initiating substrate S(t), an switches to zero if this substrate passes the threshol T. Whenever a Hill function is use, it is provie with another parameter m that controls the height of the switch. This parameter serves as maximum stimulatory respectively inhibitory effect. For abbreviation of notation, we use H + (S) instea of m h + (S(t); T, n). We usually choose the steepness coefficient n = 2, but, when appropriate, we set n = 1, 5, or 10 to capture smoother or steeper effects. The complete set of Hill functions is specifie in Appenix A, an parameter values can be foun in Appenix B. 13

14 H S t,t,n m n 1 n 2 n 5 n 10 Figure 3: Scale positive Hill functions with ifferent steepness Moel equations The amount of GnRH in the hypothalamus is a result of synthesis in the hypothalamus an release into the pituitary, t GnRH Hypo(t) = Syn GnRH (t) Rel GnRH (t). (1) GnRH synthesis epens on its current level in the hypothalamus. If this level approaches a specifie threshol, synthesis ecreases to zero. This effect is moele as Syn GnRH (t) = c GnRH,1 ( 1 GnRH ) Hypo(t) GnRH max. (1a) Hypo As long as GnRH is far below its maximum, the factor 1 GnRH Hypo(t) has GnRH max Hypo only a small impact. The release of GnRH from the hypothalamus to the pituitary is epenent on its current level in the hypothalamus. E2 inhibits GnRH release uring the luteal phase, i.e. if P4 an E2 are high at the same time, escribe by H 1 (P4&E2). H 1 (P4&E2) enotes the sum of two Hill functions minus their prouct, an inhibits GnRH release only if both 14

15 substrates are above their threshol. Aitionally, the release of GnRH is inhibite by P4 only, Rel GnRH (t) = (H 1 (P4&E2) + H 2 (P4)) GnRH Hypo (t). (1b) Changes in GnRH amount in the pituitary are epenent on the release amount from the hypothalamus, but also on the presence of E2. E2 increases the number of GnRH receptors in the pituitary. This effect is inclue in the equation as a positive Hill function. GnRH clearance from pituitary portal bloo is proportional to the GnRH level in the pituitary, i.e. GnRH clearance is represente by c GnRH,2 GnRH P it (t), in which c GnRH,2 is a constant, t GnRH Pit(t) = Rel GnRH (t) H + 3 (E2) c GnRH,2 GnRH Pit (t). (2) 259 FSH is synthesize in the pituitary an release into the bloo, t FSH Pit(t) = Syn FSH (t) Rel FSH (t). (3) FSH synthesis rate in the pituitary is only epenent on elaye Inh, as in [61]. FSH is synthesize when the Inh level is low, i.e. high Inh levels inhibit FSH synthesis, which is inclue as a negative Hill function, Syn FSH = H 4 (Inh τ ). (3a) The inex τ stans for a elaye effect of Inh, i.e. Inh is consiere at time t τ. FSH release from the pituitary to the bloo is stimulate by P4 an GnRH, an inhibite by E2, Rel FSH = (H + 5 (P4) + H 6 (E2) + H + 7 (GnRH Pit )) FSH Pit (t). (3b) 15

16 Concluing, FSH serum level is a result of the ifference between the release amount from the pituitary an clearance in the bloo, t FSH Bloo(t) = Rel FSH (t) c FSH FSH Bloo (t), (4) where c FSH is the FSH clearance rate constant. Like FSH, the LH serum level epens on synthesis in the pituitary, re- lease into the bloo an clearance thereof, t LH Pit(t) = Syn LH (t) Rel LH (t). (5) 271 LH synthesis in the pituitary is stimulate by E2 an inhibite by P4, Syn LH (t) = H + 8 (E2) + H 9 (P4). (5a) We assume a low constant basal LH release b LH from the pituitary into the bloo. On top of that, LH release is stimulate by GnRH, Rel LH (t) = (b LH + H + 10(GnRH Pit )) LH Pit (t). (5b) 274 Summarizing, LH in the bloo is obtaine as t LH Bloo(t) = Rel LH (t) c LH LH Bloo (t), (6) where c LH is the LH clearance rate constant. Follicular function is stimulate by FSH, whereas its ecrease is promote by P4 an the LH surge, t Foll(t) = H+ 11(FSH ) (H + 12(P4) + H + 13(LH Bloo )) Foll(t). (7) The sensitivity of the follicles to respon to FSH grows with their size. In the moel, the threshol of FSH to stimulate the follicular function ecreases 16

17 with increasing follicular function. For this effect of a rising FSH sensitivity, a negative Hill function is inclue to control the threshol of FSH, T Foll FSH (t) := T Foll FSH h (Foll(t); T FSH Foll, 1), 278 an the Hill function for the effect of FSH on follicular function becomes H 11(FSH + ) := m Foll FSH h + Foll (FSH Bloo (t); T FSH (t), 2). (7a) PGF2α initiates the functional regression of the CL, an thereby the ecrease in P4 levels. After a large time elay, PGF2α synthesis is stimulate by elevate P4 levels above a specifie threshol value. The PGF2α level eclines a couple of ays after its rise, which is inclue as a elaye positive effect of P4 on the ecay of PGF2α, t PGF2 α(t) = H+ 14(P4 τ1 ) H + 15(P4 τ2 ) PGF2 α(t). (8) The LH peak initiates growth of the CL with a specifie elay. After reaching a certain size, the CL continues to grow on its own as long as PGF2α is low. The CL starts to regress when PGF2α levels rise above a threshol, t CL(t) = H+ 16(LH τ ) + H + 17(CL) H + 18(PGF2 α) CL(t). (9) The prouction of P4 in the ovary is assume to be proportional to CL function, an the prouction of E2 an Inh is assume to be proportional to follicular function. Therefore, the equations for P4, E2, an Inh o not 17

18 contain any Hill functions, P4(t) t = cp4 CL CL(t) c P4 P4(t), (10) E2(t) t = ce2 Foll Foll(t) c E2 E2(t), (11) Inh(t) t = cinh Foll Foll(t) c Inh Inh(t). (12) The parameters c P4, c E2 an c Inh enote the respective clearance rate con- stants. Figure 2 gives an overview of all mechanisms escribe by the moel equa- tions. Detaile notations for the Hill functions, parameters, an equations are given in Appenix A, Appenix B, an Appenix C respectively Parameter ientification an sensitivity analysis The main ifficulty is not to simulate the system, i.e. to solve the if- ferential equations, but to ientify the unknown parameters. Unfortunately, many of the parameters are not measurable. Sometimes the range of values is known, but some parameters are completely unknown. The techniques for parameter estimation that are use in this moel are implemente in the software packages PARKIN [64, 65] an NLSCON [66], which have been evelope at the Zuse Institute for many years. These programs take into ac- count parameter sensitivities an linear epenencies, an inclue a number of optimization methos such as, for example, affine covariant Gauss-Newton methos [67]. A renewe version of this software, especially aapte to pa- rameter ientification in orinary ifferential equation moels, has been use throughout the paper. The mathematical backgroun is escribe in [67]. To obtain a goo initial guess for the parameter optimization proceure, we use a moel ecomposition approach an successively enlarge the set of 18

19 estimate parameters. The first step is to efine input curves representing the evelopment of Inh, P4, an E2 levels in the bloo over time. This use of explicit functions, which simplifies parameter ientification, was alreay suggeste by Schlosser [11]. Composition of these input curves is base on publishe ata for enocrine profiles of cows with a normal estrous cycle, see for example [68]. 1 1 GnRH 0.5 FSH ays ays (a) GnRH (b) FSH 1 1 LH 0.5 P ays ays (c) LH () P4 1 1 E2 0.5 Inh ays ays (e) E2 (f) Inh Figure 4: Simulate curves of the close moel together with the ata points use for parameter estimation. Panels 4(a), 4(b) an 4(c) show ata points base on qualitative behavior of hormones as escribe in literature ([68]). Panels 4(), 4(e) an 4(f) show ata points obtaine from the input curves. Day zero correspons to the ay of LH peak Following the approach in [61], we use the input curves to successively fit the profiles of the other components. The etaile proceure can be foun 19

20 in [69]. In the last step, the input curves for P4, E2, an Inh are replace by their original ODE/DDE escription to obtain a close network. The final parameter values are liste in Table B.1, an the corresponing simulation results are illustrate in Figure 4. A sensitivity analysis has been performe with the techniques esribe in [67]. A more etaile escription incluing column norms of the sensitivity matrix an subconitions, which provie information about the sensitivities an the epenencies of the parameters, can be foun in [69]. It turns out that among the most sensitive an best preictable parameters are p 36 = τ P 4,1, p 11 = τ Inh, p 20 = c F SH, an p 39 = τ P 4, Simulation results The figures in this section show the compute ynamics of follicle an CL evelopment an accompanying fluctuations in hormone levels over consecutive cycles. The simulation results show that the current set of moel parameters generates curves consistent with empirical knowlege for cows with a normal estrous cycle with three follicular waves. Notice that the moel generates consecutive cycles that are not entirely ientical (quasi-perioic behavior), but that vary slightly in patterns an peak heights between cycles. Small ifferences in moel output at the en of a cycle result in a ifferent starting point of the next cycle, which leas to variation between the curves. This variation in hormone levels between cycles coul well resemble variation within a cow over consecutive cycles. However, a ifferent parameterization can be use to prouce a stable limit cycle. Each estrous cycle contains three waves of follicular growth (Figure 5). 20

21 1 0.8 Foll CL ays (a) Foll CL ays (b) Figure 5: Output curves of follicular function (Foll) an CL function (CL) over time for one cycle (a) an in consecutive cycles (b) The CL starts to grow a few ays after ovulation an is large uring the first two follicular waves, which suppresses follicle growth. As the larger follicles become more sensitive to P4, at a certain size the effect of P4 becomes so large that it inuces follicle regression. After regression of the CL, the ominant follicle of the thir follicular wave can continue to grow, leaing to ovulation, which causes a sharp ecline in follicular function. The pattern of serum E2 levels is a result of follicular function (Figures 5 an 6). The thir wave of follicular growth takes place when P4 levels 21

22 2 1.5 GnRH LH E ays (a) GnRH LH E ays (b) Figure 6: Output curves of serum concentrations of E2 an LH, an portal concentration of GnRH over time for one cycle (a) an in consecutive cycles (b) are low, resulting in increase E2 levels. These increase E2 levels inuce a steep GnRH an LH surge, which is the trigger for ovulation. Notice that the height of the GnRH surge is etermine by the E2 peak level. During the remaining cycle, GnRH an LH levels are low, representing the lower pulse frequency an amplitue compare to the surge. Increase FSH levels inuce the growth of a follicular wave an thereby the start of Inh increase, but FSH is suppresse when Inh levels are above a certain level (Figure 7). Notice that FSH peak levels in the thir wave of 22

23 FSH Inh ays (a) FSH Inh ays (b) Figure 7: Output curves of serum concentrations of Inh an FSH over time for one cycle (a) an in consecutive cycles (b) the cycle iffer in consecutive cycles because of corresponing ifferences in height of the GnRH surge (Figures 6 an 7). When Inh has ecline ue to follicular regression, FSH increases again an inuces the next follicular wave. Because follicular growth is moele in three waves, also Inh levels rise in three waves in a cycle. P4 serum levels are proportional to CL function. P4 concentration is small uring the first ays of the cycle an rises when the CL starts to grow (Figure 5). Notice that a lower LH peak height results in a less steep 23

24 1 0.5 PGF2α P ays (a) PGF2α P ays (b) Figure 8: Output curves of serum concentrations of P4 an PGF2α over time for one cycle (a) an in consecutive cycles (b) P4 increase an lower levels of P4 in the following cycle (Figures 6 an 8). Increase P4 levels inuce a rise in PGF2α after a couple of ays, which causes CL regression an eclining P Discussion an Outlook The current mathematical moel escribes the interaction between a number of key physiological processes of the bovine estrous cycle. The moel is able to simulate the ynamics of follicle an CL growth an evelopment, 24

25 as well as the associate hormone level changes in consecutive cycles. The current moel comprises 12 equations an 54 parameters. The estrous cycles generate by the moel are not entirely ientical an coul well resemble variations within a cow over consecutive cycles. The above simulations show a quasi-perioic behavior, but a ifferent parameterization (not liste in this paper) coul be use to prouce a stable limit cycle. This shows that the variations between simulate cycles are not an intrinsic characteristic of the moel, but epen on the parameterization. However, the cycles of a real cow are usually quite irregular, an we think this is not ue to changes in external factors for that cow but rather arises from the fact that each cycle presents slightly new an somewhat ifferent starting values for the next cycle, which we think that our moel mimics. Alternatively, one coul a a stochastic component to the regular system (representing small variations in external factors) to inuce variations in consecutive cycles, but this was not in the scope of our work. The sensitivity analysis shows that parameters 36, 20 an 11 are the most sensitive parameters of the moel, which means that a small change in the value of one of these parameters will have a large effect on the moel solution. Parameter 36 (elay of P4 until stimulating PGF2α increase) is possibly a sensitive parameter because CL life span is critical for the uration of the cycle. Parameter 20 (FSH clearance rate constant) an Parameter 11 (elay of Inh in FSH synthesis) are possibly sensitive parameters because FSH an Inh serum levels have an important effect on the progress of follicle evelopment. The moeling metho with ODEs/DDEs as use for the presente moel 25

26 of the bovine estrous cycle was also use for the moel of the human menstrual cycle [10, 11, 61, 9]. As we aime at the evelopment of a moel for the ynamical changes of a biological system, incluing the information about how components influence the rates of change of other components, our approach to moel the system with ifferential equations appears to be the most reasonable. Maybe qualitative results coul have been obtaine with other methos such as, for example, boolean networks, but ifferential equations allow for a simulation of quantitative profiles of the involve components. To our knowlege, no comparable moels of the bovine estrous cycle are available. The current moel escribes the mechanisms of an iealize cow, base on average numbers obtaine from several ata sources. It woul in principle be possible to fit the moel to measurement ata of an iniviual cow that woul show small eviations of the cycle, or even a pathological abnormal cycle ue to certain isorers. This woul represent the next step in the moeling approach. Because empirical ata are usually noisy, parameter optimization woul then also have to take measurement errors into account. Although the current moel coul thus offer possibilities to simulate fertility isorers, its preictive ability may be limite in those parts an for those aspects in which the moel is not entirely mechanistic but rather escriptive. One example thereof is the moeling of PGF2α. Because the etaile biological mechanisms that inuce the rise of PGF2α are very complex an not completely unerstoo, we chose to restrict the number of state variables for this part of the moel, an to inclue time elays. This mimics the situation in cows that the rise of P4 early in the cycle starts a series of 26

27 events or mechanisms that eventually lea to the rise of PGF2α, followe by a ecline of PGF2α several ays later. The time elays are thus a black box where the intermeiate events that regulate PGF2α levels are not escribe. In this way, we were able to obtain the right time point of CL regression even though we on t know the biological mechanisms exactly. By reucing the elays, the uration of the luteal phase can be reuce. This coul mean that P4 serum levels alreay ecline uring the secon wave of follicle evelopment, which coul then become the ovulatory wave. The shorter elays coul thus result in a shorter cycle with only two follicular waves. However, the consequence of the chosen approach is that the preictive abilities for this part of the moel are limite. Moel improvement an refinement of this sub-moel will play an important role in future work. Apart from fitting of the moel to iniviual cow ata, mentione above, we plan to use this moel to etermine the level of control exerte by various system components on the functioning of the system. Examples of such moel applications are to explore the mechanisms that influence the pattern of follicular waves, or to stuy hormone patterns associate with subfertility. Also, the moel can serve as a basis for more elaborate moels an simulations, with the ability to stuy effects of external manipulations an genetic ifferences. Possible extensions of the moel coul be in the fiel of energy metabolism, stress, isease, an factors affecting the expression of estrous behavior. There are relationships between regulation of the estrous cycle an energy balance, which can cause fertility problems in high proucing airy cows in negative energy balance (for reviews see [70, 71]). Changes in reprouctive performance that are associate with high milk prouction may in 27

28 part be explaine by elevate P4 an E2 clearance rates, as escribe in the physiological moel of [3]. In this physiological moel, clearance rates of hormones by the liver of cows with high milk prouction are increase as a result of elevate fee intake, leaing to an increase liver bloo flow an metabolic activity. With a similar level of hormone prouction, circulating hormone levels woul thus be lower. Lameness, an example of a stress inucing conition, was foun to inhibit the LH surge an ovulation, whereas incience of estrous behavior (although with less intensity) was not reuce. These observations suggest that stress, cause by lameness, reuces P4 exposure before estrus an/or E2 prouction by the ominant follicle [72, 73]. Further, a normal enocrinological cycle is prerequisite for appropriate expression of estrous behavior. The relationships foun between P4, E2 an intensity of estrous behavior show that hormones involve in regulation of the estrous cycle also affect the expression of estrous behavior [74, 75]. These an other finings an hypotheses about regulation of the bovine estrous cycle coul be translate into mathematical equations or moifie parameterization an incorporate in the current moel Acknowlegements The authors woul like to thank Dr B. Beera, Prof Dr B. Kemp an Prof Dr M. Smits for their helpful comments on the manuscript. These results are obtaine through IP/OP: Systems Biology, financially supporte by the Dutch Ministry of Agriculture, Nature an Foo Quality (BAS no ). S. Röblitz an C. Stötzel have been supporte by the DFG Research Center Matheon Mathematics for Key Technologies in Berlin. 28

29 474 Appenix A. List of Hill functions The Hill functions liste below are the full notations of the Hill functions mentione in Section 3.2 an represent the mechanisms shown in Figure 2. ( H1 (P4&E2) := m P4&E2 h GnRH,1 (P4(t); TP4, 2) + h (E2(t), T GnRH,1 E2, 2) ) h GnRH,1 (P4(t); TP4, 2) h GnRH,1 (E2(t), TE2, 2) H2 GnRH,2 (P4) := mp4 h GnRH,2 (P4(t), TP4, 2) H 3 + GnRH,2 (E2) := me2 h + GnRH,2 (E2(t), TE2, 5) H 4 (Inh τ ) := m FSH Inh H + 5 (P4) := m FSH P4 H 6 (E2) := m FSH E2 h (Inh(t τ Inh ), T FSH Inh, 2) h + (P4(t); T FSH P4, 2) h (E2(t); T FSH E2, 2) H + 7 (GnRH Pit ) := m FSH GnRH h (GnRH Pit (t); T FSH GnRH, 1) H + 8 (E2) := m LH E2 h + (E2(t); T LH E2, 2) H 9 (P4) := m LH P4 h (P4(t); T LH P4, 2) H + 10(GnRH Pit ) := m LH GnRH h + (GnRH Pit (t); T LH GnRH, 2) H 11(FSH + ) := m Foll FSH h + Foll (FSH Bloo (t); T FSH (t), 2), H + 12(P4) := m Foll P4 H + 13(LH ) := m Foll LH T Foll FSH (t) := T Foll FSH h (Foll(t); T FSH Foll, 1) h + (P4(t); T Foll P4, 2) h (LH Bloo (t); T Foll LH, 2) H 14(P4 + PGF2 α,1 τ ) := mp4 h + (P4(t τ P4,1 ), TP4 PGF2 α, 2) H 15(P4 + PGF2 α,2 τ ) := mp4 h + (P4(t τ P4,2 ), TP4 PGF2 α, 10) H + 16(LH τ ) := m CL LH h + (LH Bloo (t τ LH ); T CL LH, 2) H + 17(CL) := m CL CL h + (CL(t); T CL CL, 2) H + 18(PGF2 α) := m CL PGF2 α h + (PGF2 α(t); T CL PGF2 α, 1) 29

30 Appenix B. List of parameters In our moel, [ ] stans for the unit of the substance, usually a concentration, an can be specifie from measurements. Typical units are [FSH]=[LH]=IU/l, [P4]=ng/ml, an [E2]=pg/ml. t enotes time ; in our moel [t] stans for ays. Table B.1: List of parameters. No. Symbol Value Quantity Explanation 1 GnRH max Hypo 20 2 c GnRH, m P4&E [GnRH Hypo ] maximum value for GnRH in the [GnRH Hypo ] [t] [GnRH Hypo ] [t] hypothalamus synthesis rate constant of GnRH in the hypothalamus maximum part of GnRH synthesis rate constant inhibite by E2 an GnRH,1 4 TE [E2] threshol of E2 to suppress GnRH P4 release GnRH,1 5 TP [P4] threshol of P4 to allow E2 suppression of GnRH release GnRH,2 6 mp /[t] maximum part of GnRH synthesis rate constant inhibite by P4 GnRH,2 7 TP [P4] threshol of P4 to inhibit GnRH release irectly Continue on next page... 30

31 Table B.1 continue from previous page No. Symbol Value Quantity Explanation GnRH,2 8 me2 1.5 [GnRH Pit ] [GnRH Hypo ] maximum scaling of pituitary sensitivity for GnRH GnRH,2 9 TE [E2] threshol of E2 to increase pituitary sensitivity for GnRH 10 c GnRH, /[t] GnRH clearance rate constant in the pituitary 11 τ Inh 1.5 [t] elay of Inh in FSH synthesis 12 m FSH Inh 1 [FSH]/[t] maximum FSH synthesis rate in the pituitary in the absence of Inh 13 T FSH Inh 0.06 [Inh] threshol of Inh for inhibition of FSH synthesis 14 m FSH P4 2 1/[t] maximum part of FSH release rate that is stimulate by P4 15 T F SH P [P4] threshol of P4 to stimulate FSH release 16 m FSH E /[t] maximum part of FSH release rate that is inhibite by E2 17 T FSH E [E2] threshol of E2 to inhibit FSH release 18 m FSH GnRH 3 1/[t] maximum part of FSH release rate that is stimulate by GnRH Continue on next page... 31

32 Table B.1 continue from previous page No. Symbol Value Quantity Explanation 19 T FSH GnRH 0.4 [GnRH] threshol of GnRH to stimulate FSH release 20 c FSH 0.8 1/[t] FSH clearance rate constant 21 m LH E2 1.5 [LH]/[t] maximum part of LH synthesis that is stimulate by E2 22 T LH E2 0.1 [E2] threshol of E2 to stimulate LH synthesis 23 m LH P4 4.5 [LH]/[t] maximum part of LH synthesis that is inhibite by P4 24 T LH P [P4] threshol of P4 to inhibit LH synthesis 25 m LH GnRH 4 1/[t] maximum part of LH release rate that is stimulate by GnRH 26 T LH GnRH 4 [GnRH] threshol of GnRH to stimulate LH release 27 b LH /[t] basal LH release rate constant 28 c LH 11 1/[t] LH clearance rate constant 29 m Foll FSH 0.8 [Foll]/[t] maximum increase of follicular function stimulate by FSH 30 T Foll FSH 0.8 [FSH] threshol of FSH to stimulate follicular function Continue on next page... 32

33 Table B.1 continue from previous page No. Symbol Value Quantity Explanation 31 T FSH Foll 0.3 [Foll] threshol of follicular function to ownscale FSH threshol 32 m Foll P /[t] maximum part of follicular ecay stimulate by P4 33 T Foll P [P4] threshol of P4 to stimulate ecrease of follicular function 34 m Foll LH 2.8 1/[t] maximum part of follicular ecay stimulate by LH 35 T Foll LH [LH] threshol of LH to stimulate ecrease of follicular function 36 τ P4,1 12 [t] elay of P4 until stimulating PGF2α increase PGF2 α,1 37 mp4 0.3 [PGF2α]/[t] maximum growth rate of PGF2α PGF2 α,1 38 TP [P4] threshol of P4 to stimulate PGF2α increase 39 τ P4,2 17 [t] elay of P4 until stimulating PGF2α ecrease PGF2 α,2 40 mp4 11 [PGF2α]/[t] maximum ecay rate of PGF2α PGF2 α,2 41 TP [P4] threshol of P4 to stimulate PGF2α ecrease 42 τ LH 4.5 [t] elay of LH in CL Continue on next page... 33

34 Table B.1 continue from previous page No. Symbol Value Quantity Explanation 43 m CL LH [CL]/[t] maximum increase of CL stimulate by LH 44 T CL LH 1.2 [LH] threshol of LH to stimulate CL increase 45 m CL CL [CL]/[t] maximum increase of CL stimulate by itself 46 T CL CL [CL] threshol of CL to stimulate selfgrowth 47 m CL PGF2 α /[t] maximum ecrease of CL stimulate by PGF2α 49 c P4 CL [P4]/[CL] 1/[t] 48 T CL PGF2 α 2 [PGF2α] threshol of PGF2α to initiate ecrease of CL proportionality factor of CL in P4 increase 50 c P /[t] P4 clearance rate constant 51 c E2 Foll 1.9 [E2]/[Foll] 1/[t] proportionality factor of follicular function in E2 increase 52 c E /[t] E2 clearance rate constant of 53 c Inh Foll 4.8 [Inh]/[Foll] 1/[t] proportionality factor of elaye follicular function in Inh increase 54 c Inh 4 1/[t] Inh clearance rate constant 34

35 Appenix C. List of equations The equations liste below are the full notations of the equations evelope in Section 3.2. Parameters are enote with p an are numbere accoring to Table B.1. Components numbering an initial values can be foun in Table C.2. no component initial value 1 GnRH Pit GnRH Bloo FSH Pit FSH Bloo LH Pit LH Bloo Foll P GF 2α CL P E Inh Table C.2: Initial values 35

36 ( t y 1(t) = p 2 1 y ) ( 1(t) p 3 (h (y 10 (t); p 5, 2) + h (y 11 (t); p 4, 2) p 1 ) t y 2(t) = h (y 10 (t); p 5, 2) h (y 11 (t); p 4, 2)) + p 6 h (y 10 (t); p 7, 2) ( p 3 (h (y 10 (t); p 5, 2) + h (y 11 (t), p 4, 2) h (y 10 (t); p 5, 2) h (y 11 (t); p 4, 2) ) ) +p 6 h (y 10 (t); p 7, 2) y 1 (t) p 8 h + (y 11 (t); p 9, 5) p 10 y 2 (t) ( t y 3(t) = p 12 h (y 12 (t p 11 ); p 13, 2) p 14 h + (y 10 (t); p 15, 2) ) +p 16 h (y 11 (t); p 17, 2) + p 18 h + (y 2 (t); p 19, 1) y 3 (t) ( t y 4(t) = p 14 h + (y 10 (t); p 15, 2) + p 16 h (y 11 (t); p 17, 2) ) +p 18 h + (y 2 (t); p 19, 1) y 3 (t) p 20 y 4 (t) t y 5(t) = p 21 h + (y 11 (t); p 22, 2) + p 23 h (y 10 (t); p 24, 2) ( p 27 + p 25 h + (y 2 (t); p 26, 2) ) y 5 (t) t y 6(t) = (p 27 + p 25 h + (y 2 (t); p 26, 2)) y 5 (t) p 28 y 6 (t) t y 7(t) = p 29 h + (y 4 (t); p 30 h (y 7 (t); p 31, 1), 2) ( p 32 h + (y 10 (t); p 33, 2) + p 34 h + (y 6 (t); p 35, 2) ) y 7 (t) y 1 (t) t y 8(t) = p 37 h + (y 10 (t p 36 ); p 38, 2) p 40 h + (y 10 (t p 39 ); p 41, 10) y 8 (t) t y 9(t) = p 43 h + (y 6 (t p 42 ); p 44, 2) + p 45 h + (y 9 (t); p 46, 1) p 47 h + (y 8 (t); p 48, 2) t y 10(t) = p 49 y 9 (t) p 50 y 10 (t) t y 11(t) = p 51 y 7 (t) p 52 y 11 (t) t y 12(t) = p 53 y 7 (t) p 54 y 12 (t) 36

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