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

Transcription

1 Konra-Zuse-Zentrum für Informationstechnik Berlin Takustraße 7 D Berlin-Dahlem Germany H.M.T.BOER, C.STÖTZEL, S.RÖBLITZ, P.DEUFLHARD, R.F.VEERKAMP, H.WOELDERS A simple mathematical moel of the bovine estrous cycle: follicle evelopment an enocrine interactions ZIB-Report (Mai 010)

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3 A simple mathematical moel of the bovine estrous cycle: follicle evelopment an enocrine interactions H.M.T.Boer C.Stötzel S.Röblitz P.Deuflhar R.F.Veerkamp H.Woelers May 7, 010 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 1 ays, with three waves of follicle growth per cycle. The moel contains 1 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 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. AMS MSC 000: 9C4, 9C30, 90C31, 65L09 Keywors: cow, reprouction, hormone patterns, ifferential equations, systems biology Animal Breeing an Genomics Center, Wageningen UR Livestock Research, 800 AB Lelysta, The Netherlans Aaptation Physiology Group, Department of Animal Sciences, Wageningen University, 6700 AH Wageningen, The Netherlans Zuse Institute Berlin (ZIB), Takustraße 7, Berlin, Germany Corresponing author. susanna.roeblitz@zib.e 1

4 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers Contents 1 Introuction 3 Biological backgroun 4.1 The bovine estrous cycle Follicle evelopment Follicular waves Corpus luteum Dynamics of ovarian an uterine hormones Estraiol Inhibin Progesterone Prostaglanin Fα Dynamics of hypothalamic an pituitary hormones Gonaotropin releasing hormone Luteinizing hormone Follicle stimulating hormone Mathematical formulation Hill functions Moel equations Parameter ientification an sensitivity analysis Simulation results 17 5 Discussion an Outlook 0 A List of Hill functions B List of parameters 3 C List of equations 5

5 A simple mathematical moel of the bovine estrous cycle 3 1 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 [59, 75]). This ecline in fertility is shown by e.g. alterations in hormone patterns uring the estrous cycle, reuce expression of estrous behavior an lower conception rates [79]. 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 [11]. 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 [49, 70]. A number of moels have been evelope for other ruminant species, especially ewes [13, 35], 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 [61]. This moel 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. Therefore, some mechanisms of the human moel in [61] 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 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.

6 4 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers Biological backgroun.1 The bovine estrous cycle The main tissues an organs involve in the regulation of the estrous cycle are the ovaries, the uterus, the hypothalamus an the anterior pituitary. These organs interact via hormones in the bloo (reviewe in [11]). In the bovine, most estrous cycles have two or three waves in which a cohort of follicles start to grow an prouce estraiol (E). The first one or two waves prouce a ominant follicle that oes not ovulate, but unergoes regression uner influence of progesterone (P4). The ominant follicle that evelops in the last wave becomes the ovulatory follicle. The average uration of the bovine estrous cycle is 0 ays for -wave an ays for 3-wave cycles (reviewe in [3]). During pro-estrus, when the corpus luteum (CL) is regresse an the concentration of P4 is ecrease, the ominant follicle, eviate from a cohort of antral follicles, evelops an matures uner the influence of follicle stimulating hormone (FSH) an luteinizing hormone (LH) secrete from the pituitary. FSH plays an important role in the beginning of follicular growth, while LH is important for maturation up to ovulation. During evelopment to preovulatory size, the ominant follicle secretes increasing amounts of estraiol (E), an also inhibin (Inh), which inhibits FSH synthesis an, hence, inhibits the growth of suborinate follicles [10, 7]. E inhibits gonaotropin releasing hormone (GnRH) secretion from the hypothalamus an therefore LH secretion from the pituitary throughout most of the cycle. However, uring pro-estrus, elevate E levels increase the secretion of GnRH, which, together with irect effects of E on the pituitary, triggers the LH surge, which inuces ovulation. Once an oocyte is successfully ovulate, the remains of the follicle form a new P4-proucing CL. P4 maintains the reainess of the enometrium for receiving the embryo. If conception has faile, the CL regresses uner influence of prostaglanin Fα (PGFα) prouce in the enometrium [47], P4 levels ecrease, an the cycle restarts (reviewe in [11]).. Follicle evelopment Follicle evelopment is controlle by changes in hormone levels. In each cycle, one follicle ovulates, while all other growing follicles go into regression at a certain stage of their evelopment. The follicles prouce E an Inh, which are release into peripheral bloo. Different follicles are recruite, growing, an regressing in each cycle an in each wave. However, total E an Inh prouction capacity is moele as a continuous function throughout subsequent waves an cycles, sometimes rising, sometimes falling, representing the total amount of hormone prouction of the follicles present at any moment. Small follicles of an emerging cohort may release very small amounts of E an Inh per follicle, but taken together, this amount is not negligible. Furthermore, there is always a meium-size or large follicle present [36, 80, 81], which results in a basal hormone prouction throughout the cycle. The capacity of follicles to prouce E an Inh is enote as follicular function in the rest of this paper.

7 A simple mathematical moel of the bovine estrous cycle 5.3 Follicular waves 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 [6]. A normal bovine estrous cycle inclues two or three wave-like patterns of follicle evelopment. The ovaries contain a pool of primorial follicles, consisting of an immature oocyte surroune by a single layer of epithelial cells. Follicle growth starts with the transformation of primorial to primary follicles. Uner influence of FSH, a cohort of 8-41 growing follicles emerge [3]. One follicle becomes the ominant follicle an continues to grow [3]. Growing follicles prouce increasing amounts of Inh. Inh suppresses FSH secretion, an above a certain Inh prouction this will cause a ecline in FSH serum levels. Inh prouction eclines with ovulation or regression of the ominant follicle, thereby allowing FSH to rise again. This FSH rise inuces the emergence of the next follicular wave [7]. Follicle growth is associate with increase mrna expression for LH an FSH receptors an steroiogenic enzymes, which regulates the follicles ability to prouce steroi hormones [6]. Approximately two ays after cohort recruitment, one follicle of 8-9 mm in iameter is selecte to grow to 15 mm [6]. The growing ominant follicle secretes increase amounts of Inh, which suppresses FSH release, thereby inhibiting the growth of suborinate follicles. Because the ominant follicle increases its sensitivity for FSH an LH, it continues to grow, even when FSH levels are ecrease [7, 8]. Follicular stages iffer in expression of steroiogenic enzymes an therefore iffer in E prouction. Enzyme concentrations in follicles epen on LH an FSH concentrations in the bloo an on concentration of activate LH an FSH receptors at the follicle. Deviation of the ominant follicle from the cohort of growing follicles is associate with increase FSH an LH receptor bining, activating the enzymes that catalyze steroiogenesis, resulting in increase E prouction an higher E serum levels. If P4 remains at luteal levels, the ominant follicle goes into regression. Follicle regression is associate with own-regulation in mrna expression of receptors for LH an FSH an steroiogenic enzymes, resulting in ecline E prouction [6]. Therefore, neither the LH surge nor ovulation take place [3, 6]. The ominant follicle that is present at the onset of CL regression becomes the ovulatory follicle [3, 6]. In the preovulatory follicle, the LH peak inuces a shift from E to P4 prouction [18] by changing the expression of steroiogenic enzymes [6, 7]. Follicle evelopment is moele as growth of E an Inh proucing tissue. In the moel, emergence of a follicular wave is inuce when FSH excees a threshol. This threshol for FSH becomes lower when follicles become larger, representing that larger follicles are more sensitive to FSH. Follicle regression is promote by high P4 levels an by the LH surge (Equation 7). The current moel comprises three follicular waves per cycle: two non-ovulatory waves an a thir wave that is ovulatory..4 Corpus luteum The LH surge at the en of the thir wave inuces ovulation of the ominant follicle an ifferentiation of cells of the follicular wall into luteal cells [51]. Continue CL growth is stimulate by autocrine an paracrine mechanisms

8 6 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers [68]. The CL evelops within -3 ays after ovulation, grows until a maximum iameter of about 30 mm, an starts to regress at ay of the cycle [50, 73], although P4 levels may start to ecline earlier [81]. The primary function of the CL is P4 secretion. Maintenance of P4 secretion is prerequisite for pregnancy. If pregnancy oes not occur, regression of the CL is require to allow the next ovulation. CL regression is inuce by pulsatile PGFα secretion from the uterus [51]. In each cycle a new CL evelops, but CL evelopment is moele as a continuous function of P4 proucing tissue. The capacity of the CL to prouce P4 is 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. If the CL reaches a certain size, it continues to grow without further stimulation by LH. A couple of ays after P4 reaches a threshol, PGFα rises an inuces a ecline in CL function an thereby in P4 serum levels (Equation 9)..5 Dynamics of ovarian an uterine hormones The prouction of E, Inh, P4 an PGFα epens largely on follicle an CL stage. The follicle is the main proucer of E an Inh, while the CL mainly prouces P4, an PGFα is prouce in the enometrium. E, P4 an Inh are transporte via peripheral bloo. Transport of PGFα to the ovaries occurs via a counter current mechanism between the uterine vein an the ovarian artery [39], but concentration is assume to be represente by peripheral PGFM, the main PGFα metabolite [43, 8]..5.1 Estraiol E affects LH synthesis an release [31] an FSH release [7, 41]. Furthermore, E plays an important role in the expression of estrous behavior (reviewe in [11]). E concentration in follicular flui increases when follicles grow, an the concentration in flui of ovulatory follicles is twice as high compare to non-ovulatory follicles [3]. Serum peak concentrations measure aroun estrus range from pg/ml [10, 3, 4, 30, 37, 80]. E serum levels are higher in ovulatory compare to non-ovulatory waves [10, 3]. This suggests that the preovulatory follicle has the largest capacity to prouce an release E, although its maximum size is not significantly ifferent from the maximum size of nonovulatory ominant follicles. E prouction an conuctivity of E efflux to bloo is thus not constant throughout the cycle, but appears to change uner influence of ifferential gene expression patterns affecte by LH, FSH, P4 an PGFα [, 1]. Consiering the results in [, 1], where a better vascularity of the ovulatory follicle is reporte, it is reasonable that the ovulatory follicle can secrete more E than non-ovulatory follicles an, consequently, E serum levels are highest at estrus. In our moel, the rate of E prouction an release to the bloo is taken as proportional to follicular function (Equation 11)..5. Inhibin Inh inhibits FSH synthesis an thus release [7]. Inh prouction is increase when follicles grow. Inh secretion by ominant follicles suppresses FSH serum

9 A simple mathematical moel of the bovine estrous cycle 7 levels an thereby reuces the growth of suborinate follicles [48]. The ominant follicle expresses more LH an FSH receptors, which means that the sensitivity of the follicle for LH an FSH is increase, an it can therefore continue to grow even when LH an FSH serum levels are low [5]. FSH inuces the emergence of a new follicular wave. However, when follicles grow, also Inh concentration grows, which suppresses FSH secretion, an from a certain level on thus causes a ecline in FSH serum levels. Inh prouction eclines with ovulation or regression of the ominant follicle, thereby allowing FSH to rise again, an a new follicular wave will emerge [7]. Peak Inh levels in serum are on average 33 pg/ml in nonovulatory an 464 pg/ml in ovulatory waves respectively, an minimum Inh levels are about 180 pg/ml [55]. 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 [7]. In our moel, Inh prouction rate is taken as proportional to follicular function (Equation 1)..5.3 Progesterone Serum P4 concentration etermines if a ominant follicle will be able to evelop to an ovulatory follicle. Ovulation of the ominant follicle of the first two waves of the cycle is inhibite because P4 levels are high, an the follicle will regress. The ominant follicle in the thir wave will not regress, because P4 rops below a certain threshol. Ovulation can then take place because the inhibiting effect of P4 on the LH surge is remove [43]. The CL is the main source of P4. P4 levels ecline as a result of CL regression [3]. P4 inhibits the luteolytic signal [43], an controls the life span of the CL by regulating uterine PGFα secretion [9]. Receptors for E an P4 are highly expresse in the CL in the early luteal phase, which probably stimulates P4 secretion an prevents apoptosis [65]. Serum P4 concentration is near to zero aroun estrus an high uring the luteal phase, with average peak levels ranging from ng/ml [4, 17, 3, 38, 71]. A high correlation between CL iameter an P4 output was reporte in [58, 64, 81]. However, Aams [4] foun a correlation between CL size an P4 serum levels in the early luteal phase only, which coul be ue to changes in CL vascularity uring the cycle, resulting in changes in P4 releasing capacity [46, 50]. In our moel, the rate of P4 release into the bloo is taken as proportional to CL function (Equation 10)..5.4 Prostaglanin Fα Pulsatile PGFα release from the uterus inuces CL regression, an is regulate by oxytocin (OT), P4 an E [67]. In the first half of the cycle, P4 inhibits expression of genes for E synthesis an OT receptors in the enometrium, to prevent a too early release of PGFα pulses. Simultaneously, P4 stimulates synthesis of enzymes require for PGFα prouction. In the secon half of the cycle, expression of P4 receptors is own-regulate an expression of OT receptors is up-regulate, resulting in a graual ecrease in the suppression of PGFα []. As a result of own-regulation of P4 receptors in the hypothalamus, OT secretion increases, which inuces uterine pulses of PGFα [47]. At the en of the luteal phase, PGFα is secrete in 5-8 iscrete pulses with 6-8h interval [67]. A broa range of PGFα levels is reporte in literature, but in general peak levels are 3-fol higher than basal levels. Average basal an peak levels

10 8 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers are 0-40 pg/ml an pg/ml, respectively [, 44, 66], but higher values (300 an 700 pg/ml for basal an peak levels respectively) were reporte in [5]. Exposure to substantial amounts of P4 must last for a couple of ays to inuce PGFα pulses [45]. In the moel, PGFα increases a couple of ays after P4 levels reach a threshol. P4 with another elay inuces the ecline in PGFα a few ays later (Equation 8)..6 Dynamics of hypothalamic an pituitary hormones Effects of P4 an E on LH an FSH synthesis an release occur irectly on the anterior pituitary, but also via pulsatile GnRH release. GnRH is release from the hypothalamus to the anterior pituitary via the portal bloo system. LH is require for normal luteal function an follicle growth, an FSH is important uring early follicular evelopment..6.1 Gonaotropin releasing hormone Pulsatile signaling of GnRH regulates LH an FSH release, an controls LH secretion [56]. Because GnRH inuces the LH surge, it inirectly inuces ovulation [74]. The GnRH pulse generator is locate in the hypothalamus an is moulate by P4 an E [3]. During the luteal phase, both P4 an E suppress the activity of the GnRH pulse generator. During pro-estrus however, elevate E inuces a GnRH surge [31, 3]. GnRH is prouce an store in hypothalamic neurons an release in pulses when these neurons are stimulate by high E concentrations. GnRH is release from the hypothalamus into the portal circulation of the pituitary an bins to GnRH receptors of the anterior pituitary [76]. Low P4 levels, combine with increase E serum levels an pulsatile GnRH release up-regulates expression of GnRH receptors [3, 76], which will make the pituitary more sensitive to GnRH. GnRH pulsatility is important because prolonge constant GnRH exposure leas to own-regulation of GnRH receptor [76, 77]. GnRH pulse frequency affects height of the LH peak [76], an also treatment with a single GnRH injection can increase LH peak height [14]. Aministration of GnRH inuces an LH surge within 30 minutes [54]. In the moel, GnRH stimulates LH, 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 E levels are above a threshol. GnRH release switches when P4 levels are low an E 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 E levels, representing that E up-regulates expression of GnRH receptors (Equation )..6. Luteinizing hormone LH is require for normal CL function an follicle evelopment [76]. The LH surge at the ay before ovulation inuces ovulation of the ovulatory follicle an formation of the CL. The LH surge is cause by continue increase in serum E uring the ovulatory wave. Besies irect effects of E on the pituitary, E stimulates the LH surge because it stimulates GnRH [3]. The LH surge will

11 A simple mathematical moel of the bovine estrous cycle 9 shut own E an Inh prouction capacity of the ovulatory follicle [1, 81] an inuces a shift from E to P4 prouction [19]. LH level in the bloo eclines because the LH content of the anterior pituitary is reuce an further LH synthesis therein an release to the bloo is ecrease because E is ecrease. Experiments in ewes showe that LH content of the anterior pituitary is highest at the ay before ovulation an is ecline by 89% two ays after ovulation [69]. High P4 levels prevent ovulation in the first two follicular waves, because P4 suppresses the release of LH via the inhibition of the GnRH pulse generator [8]. Aitionally, high P4 levels ecrease pituitary sensitivity to E, thereby increasing the amount of E require to inuce an LH surge above physiological levels [3]. Basal LH levels are about 1- ng/ml, an average peak levels of the LH surge range from ng/ml [10, 3, 40], an even higher peak values were foun in e.g. [18]. In the moel, LH synthesis is stimulate by E 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)..6.3 Follicle stimulating hormone Each follicular wave is initiate by an increase in FSH release from the anterior pituitary [9]. Growth of small follicles is epenent on FSH [48]. FSH synthesis is inhibite by Inh prouce by growing follicles [7]. P4 an E moulate FSH release via effects on the anterior pituitary an on the GnRH pulse generator in the hypothalamus. Some stuies in cows show an FSH peak concurrently with the LH surge [38, 40], while other stuies fail to show a clear peak [4, 10, 4]. Basal FSH levels are about 5-10 ng/ml, an peak levels about 15 ng/ml [10, 4]. In our 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 E (Equation 3b). 3 Mathematical formulation The mathematical approach use for the bovine moel is comparable to the approach use for the moel of the human menstrual cycle [60], which has been evelope at the Zuse Institute. 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. The complete mechanisms are shown in Figure. Base on these mechanisms, 1 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 [33], which has been esigne for the solution of stiff elay ifferential equations.

12 10 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers Figure 1: Schematic representation of the compartments in the moel of the bovine estrous cycle. Figure : Complete mechanisms of the bovine moel. Boxes represent the 1 key components of the system. Differential equations are erive for these 1 components. Arrows enote functional epenencies. Stimulating an inhibiting effects are inicate by + an - respectively.

13 A simple mathematical moel of the bovine estrous cycle 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) n (S(t); 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. H S t,t,n m n 1 n n 5 n 10 Figure 3: Scale positive Hill functions with ifferent steepness. 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 =, 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 A, an parameter values can be foun in B. 3. 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

14 1 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers moele as logistic growth, Syn GnRH (t) = c GnRH,1 As long as GnRH is far below its maximum, the factor 1 ( 1 GnRH ) Hypo(t) GnRH max. (1a) Hypo GnRH Hypo(t) GnRH max Hypo has only a small impact. The release of GnRH from the hypothalamus to the pituitary is epenent on its current level in the hypothalamus. E inhibits GnRH release uring the luteal phase, i.e. if P4 an E are high at the same time, escribe by H 1 (P 4&E ). H 1 (P 4&E ) enotes the sum of two Hill functions minus their prouct, an inhibits GnRH release only if both substrates are above their threshol. Aitionally, the release of GnRH is inhibite by P4 only, Rel GnRH (t) = (H 1 (P 4&E ) + H (P 4)) 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 E. E 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, GnRH P it (t), in which c GnRH, is a constant, t GnRH Pit(t) = Rel GnRH (t) H + 3 (E ) c GnRH, GnRH Pit (t). () 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 Harris (001). 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 E, Rel FSH = (H + 5 (P 4) + H 6 (E ) + H + 7 (GnRH Pit)) FSH Pit (t). (3b) 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, release into the bloo an clearance thereof, t LH Pit(t) = Syn LH (t) Rel LH (t). (5)

15 A simple mathematical moel of the bovine estrous cycle 13 LH synthesis in the pituitary is stimulate by E an inhibite by P4, Syn LH (t) = H + 8 (E ) + H 9 (P 4). (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). Summarizing, LH in the bloo is obtaine as (5b) 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+ 1 (P 4) + 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 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), an the Hill function for the effect of FSH on follicular function becomes H 11 + (FSH ) := mfoll FSH h + Foll (FSH Bloo (t); T FSH (t), ). (7a) PGFα initiates the functional regression of the CL, an thereby the ecrease in P4 levels. After a large time elay, PGFα synthesis is stimulate by elevate P4 levels above a specifie threshol value. The PGFα level eclines a couple of ays after its rise, which is inclue as a elaye positive effect of P4 on the ecay of PGFα, t PGF α(t) = H+ 14 (P 4,τ ) H + 15 (P 4,τ ) PGF 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 PGFα is low. The CL starts to regress when PGFα levels rise above a threshol, t CL(t) = H+ 16 (LH τ ) + H 17 + (CL) H+ 18 (PGF α) CL(t). (9) The prouction of P4 is assume to be proportional to CL function, an the prouction of E an Inh is assume to be proportional to follicular function, t P 4(t) = c P4 CL CL(t) c P 4 P 4 (t), (10) t E (t) = c E Foll Foll(t) c E E (t), (11) Inh(t) t = cinh Foll Foll(t) c Inh Inh(t). (1) The parameters c P4, c E an c Inh enote the respective clearance rate constants. Figure gives an overview of all mechanisms escribe by the moel equations. Detaile notations for the Hill functions, parameters, an equations are given in A, B, an C respectively.

16 14 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers 3.3 Parameter ientification an sensitivity analysis Parameter ientification is a mathematical challenge on its own. Many of the parameters are not measurable, sometimes the range of values is known, an some are completely unknown. Uner those circumstances, estimating all 54 parameters simultaneously is impossible. To obtain goo initial guesses of the parameter values for the optimization proceure, we use a moel ecomposition approach an successively enlarge the set of estimate parameters. Figure 4: The first steps in generating goo initial guesses for parameter optimization an enlarging the set of estimate parameters. The first step in parameter estimation is to efine input curves representing the evelopment of Inh, P4, an E levels in the bloo over time. Composition of these input curves is base on publishe ata for enocrine profiles of cows with a normal estrous cycle, see for example [57], ( ) ( ) (t + 8) (t + 1) E (t) = exp exp 3 ( ) ( ) (t 6) (t + 9) exp exp, ( ) ( ) (t 9.5) (t ) P 4 (t) = exp + exp, ( ) ( ) (t + 8) (t + 1) Inh(t) = exp exp ( ) ( ) (t 6) (t + 9) exp exp. Moreover, we use the ata for FSH, LH an GnRH as liste in Table 1. These are han-mae ata chosen in such a way that the ata points cover the qualitative behavior of the hormones as escribe in the literature. Therefore, we refer to these ata as artificial ata. They are normalize to be between zero an one in orer to simplify parameter ientification. However, as soon as we want to use measurement ata with certain imensions, the moel components can be scale easily. Note that the ata points are chosen such that the LH peak occurs on ay zero.

17 A simple mathematical moel of the bovine estrous cycle 15 Table 1: Artificial ata values. ay LH GnRH FSH GnRH 0.5 FSH ays ays (a) GnRH (b) FSH 1 1 LH 0.5 P ays ays (c) LH () P4 1 1 E 0.5 Inh ays ays (e) E (f) Inh Figure 5: Simulate curves of the close moel together with the artificial ata points use for parameter optimization. Day zero correspons to the ay of LH peak.

18 16 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers Column Norms p1 p p3 p4 p5 p6 p7 p8 p9 p10 p11 p1 p13 p14 p15 p16 p17 p18 p19 p0 p1 p p3 p4 p5 p6 p7 p8 p9 p30 p31 p3 p33 p34 p35 p36 p37 p38 p39 p40 p41 p4 p43 p44 p45 p46 p47 p48 p49 p50 p51 p5 p53 p54 Figure 6: The column norms of the sensitivity matrix. Following the approach in [34], we first take a small part of the moel: the influence of P4 an E on GnRH. For this purpose, P4 an E are replace by the corresponing input curves, an a subset of 10 parameters is optimize such that the simulate GnRH profile fits to the ata values from Table 1. Thereafter, we consier the influence of P4, E, an GnRH on LH, an optimize 8 more parameters to obtain a goo fit for the LH bloo level. Subsequently, Inh is also replace by its input curve, an 9 aitional parameters are now estimate such that the resulting FSH bloo profile fits our artificial ata. These first steps are visualize in Figure 4. Afterwars, the profiles of follicular function, PGFα an CL function are fitte to be in line with empirical knowlege. Finally, the input curves for P4, E, an Inh are replace by their original ODE/DDE escription to obtain a close network. The number of estimate parameters increases to 54 for the complete moel. To estimate the remaining parameters, we generate artificial ata for E, P4, an Inh by evaluating the input functions at 1 ays with t = { 10, 9,..., 10}. An overview of the parameters is given in B. Parameter estimation is one subsequently while generating the components of the moel an enlarging the set of parameters. A parameter value obtaine by the optimization proceure can be overwritten throughout the moeling process; the previous value then serves as starting value for the optimization proceure in the next moelling step. The final parameter values are liste in Table, an the corresponing simulation results are illustrate in Figure 5. Throughout the moel evelopment, parameter ientification is performe with the software package NLSCON [53], which has been evelope an improve at the Zuse Institute over many years [5, 16]. This program takes into account parameter sensitivities an linear epenencies, an it inclues a number of optimization methos such as, for example, affine covariant Gauss-Newton methos [15]. Figures 6 an 7 illustrate the results of a sensitivity analysis for the complete moel. The column norms of the sensitivity matrix contain the information

19 A simple mathematical moel of the bovine estrous cycle Subconitions p36 p11 p0 p39 p3 p51 p8 p47 p5 p9 p44 p1 p9 p6 p34 p50 p4 p53 p30 p33 p13 p10 p14 p41 p45 p1 p31 p54 p15 p35 p38 p p7 p40 p3 p18 p7 p16 p49 p43 p4 p48 p4 p5 p19 p8 p46 p1 p37 p3 p17 p6 p p5 Figure 7: The subconitions of the parameters. about the (normalize) sensitivities of the solution components with respect to the parameters at the artificial ata points. A reorering of the parameters accoring to growing subconition tells us which parameters can be estimate together. The preictability of parameter values ecreases with increasing subconition. It turns out that among the four most sensitive an best preictable parameters, there are three elays. This is not surprising since the elays have a large influence on the cycle length. 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, 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. Each estrous cycle contains three waves of follicular growth (Figure 8). 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 follicles grow, Inh increases an thus FSH ecreases. Therefore, the support for the growth of the follicles eclines, while the negative effect of P4 is still in place, leaing to the ecline of these follicles. 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 E levels is a result of follicular function (Figures 8 an 9). The thir wave of follicular growth takes place when P4 levels are

20 18 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers Foll CL ays (a) Foll CL ays (b) Figure 8: Output curves of follicular function (Foll) an CL function (CL) over time for one cycle (a) an in consecutive cycles (b). 1.5 GnRH LH E ays GnRH LH E ays Figure 9: Output curves of serum concentrations of E an LH, an portal concentration of GnRH over time for one cycle (a) an in consecutive cycles (b).

21 A simple mathematical moel of the bovine estrous cycle FSH Inh ays (a) FSH Inh ays (b) Figure 10: Output curves of serum concentrations of Inh an FSH over time for one cycle (a) an in consecutive cycles (b). low, resulting in increase E levels. These increase E 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 E 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 10). Notice that FSH peak levels in the thir wave of the cycle iffer in consecutive cycles because of corresponing ifferences in height of the GnRH surge (Figures 9 an 10). 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 8). Notice that a lower LH peak height results in a less steep P4 increase an lower levels of P4 in the following cycle (Figures 9 an 11). Increase P4 levels inuce a rise in PGFα after a couple of ays, which causes CL regression an eclining P4.

22 0 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers PGFα P ays (a) PGFα P ays (b) Figure 11: Output curves of serum concentrations of P4 an PGFα over time for one cycle (a) an in consecutive cycles (b). 5 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, as well as the associate hormone level changes in consecutive cycles. The current moel comprises 1 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. Although the current moel is base on artificial ata escribing the mechanisms of an iealize cow, its fitting to measurement ata of an iniviual cow is in principle possible an woul represent the next step in the moeling approach. Because empirical ata are usually noisy, parameter optimization must then also take into account measurement errors. In future work, we want 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. 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

23 A simple mathematical moel of the bovine estrous cycle 1 energy balance, which can cause fertility problems in high proucing airy cows in negative energy balance (for reviews see [0, 6]). Changes in reprouctive performance that are associate with high milk prouction may in part be explaine by elevate P4 an E clearance rates, as escribe in the physiological moel of [79]. 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 E prouction by the ominant follicle [1, 78]. Further, a normal enocrinological cycle is prerequisite for appropriate expression of estrous behavior. The relationships foun between P4, E an intensity of estrous behavior show that hormones involve in regulation of the estrous cycle also affect the expression of estrous behavior [4, 63]. 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.

24 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers A List of Hill functions The Hill functions liste below are the full notations of the Hill functions mentione in Section 3. an represent the mechanisms shown in Figure. ( H1 (P 4&E ) := m P4&E h (P 4 (t); T h GnRH,1 (P 4 (t); T H (P GnRH, 4) := mp 4 h GnRH, (P 4 (t), TP 4, ) GnRH, GnRH,1 P 4, ) + h (E (t), T GnRH,1 E, ) ) P 4, ) h GnRH,1 (E (t), TE, ) GnRH, H 3 + (E ) := me h + (E (t), TE, 5) H4 (Inh τ ) := m FSH Inh h (Inh(t τ Inh ), TInh FSH, ) H 5 + (P 4) := m FSH P 4 H6 (E ) := m FSH E h + (P 4 (t); T FSH P 4, ) h (E (t); T FSH E, ) H + 7 (GnRH Pit) := m FSH GnRH h (GnRH Pit (t); T FSH GnRH, 1) H 8 + (E ) := m LH E H9 (P 4) := m LH P 4 h + (E (t); T LH E, ) h (P 4 (t); TP LH 4, ) H 10 + (GnRH Pit) := m LH GnRH h + (GnRH Pit (t); T LH H 11 + (FSH ) := mfoll FSH h + (FSH Bloo (t); H + 13 GnRH, ) Foll T FSH (t), ), T Foll FSH (t) := T Foll FSH h (Foll(t); T FSH Foll, 1) H 1 + (P 4) := m Foll P 4 h + (P 4 (t); TP Foll 4, ) (LH ) := mfoll LH h (LH Bloo (t); TLH Foll, ) H 14 + (P PGF α,1 4,τ ) := m H + 15 (P 4,τ ) := m P 4 PGF α, P 4 h + (P 4 (t τ P4,1), T PGF α P 4, ) h + (P 4 (t τ P4,), T PGF α P 4, 10) H + 16 (LH τ ) := m CL LH h + (LH Bloo (t τ LH ); T CL LH, ) H 17 + (CL) := mcl h + (CL(t); TCL CL, ) H 18 + (PGF α) := mcl PGF α h + (PGF α(t); TPGF CL α, 1)

25 A simple mathematical moel of the bovine estrous cycle 3 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 [E]=pg/ml. t enotes time ; in our moel [t] stans for ays. Table : List of parameters. No. Symbol Value Quantity Explanation 1 GnRH max Hypo 0 c GnRH, m P4&E [GnRH Hypo ] maximum value for GnRH in the hypothalamus [GnRH Hypo] [t] synthesis rate constant of GnRH in the hypothalamus [GnRH Hypo] [t] maximum part of GnRH synthesis rate constant inhibite by E an P4 GnRH,1 4 TE [E] threshol of E to suppress GnRH release GnRH,1 5 TP [P4] threshol of P4 to allow E suppression of GnRH release GnRH, 6 mp /[t] maximum part of GnRH synthesis rate constant inhibite by P4 GnRH, 7 TP [P4] threshol of P4 to inhibit GnRH release irectly GnRH, 8 me 1.5 [GnRH Pit] [GnRH Hypo] maximum scaling of pituitary sensitivity for GnRH GnRH, 9 TE 1.76 [E] threshol of E 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 1 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 P 4 1/[t] maximum part of FSH release rate that is stimulate by P4 15 TP F SH [P4] threshol of P4 to stimulate FSH release 16 m FSH E 0.3 1/[t] maximum part of FSH release rate that is inhibite by E 17 TE FSH.846 [E] threshol of E to inhibit FSH release 18 m FSH GnRH 3 1/[t] maximum part of FSH release rate that is stimulate by GnRH 19 TGnRH FSH 0.4 [GnRH] threshol of GnRH to stimulate FSH release 0 c FSH 0.8 1/[t] FSH clearance rate constant Continue on next page...

26 4 H.M.T.Boer, C.Stötzel, S.Röblitz, P.Deuflhar, R.F.Veerkamp, H.Woelers Table continue from previous page No. Symbol Value Quantity Explanation 1 m LH E 1.5 [LH]/[t] maximum part of LH synthesis that is stimulate by E TE LH 0.1 [E] threshol of E to stimulate LH synthesis 3 m LH P [LH]/[t] maximum part of LH synthesis that is inhibite by P4 4 TP LH [P4] threshol of P4 to inhibit LH synthesis 5 m LH GnRH 4 1/[t] maximum part of LH release rate that is stimulate by GnRH 6 TGnRH LH 4 [GnRH] threshol of GnRH to stimulate LH release 7 b LH /[t] basal LH release rate constant 8 c LH 11 1/[t] LH clearance rate constant 9 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 31 T FSH Foll 0.3 [Foll] threshol of follicular function to ownscale FSH threshol 3 m Foll P 4.5 1/[t] maximum part of follicular ecay stimulate by P4 33 TP Foll [P4] threshol of P4 to stimulate ecrease of follicular function 34 m Foll LH.8 1/[t] maximum part of follicular ecay stimulate by LH 35 T Foll LH 0.55 [LH] threshol of LH to stimulate ecrease of follicular function 36 τ P4,1 1 [t] elay of P4 until stimulating PGFα increase 37 PGF α,1 mp [PGFα]/[t] maximum growth rate of PGFα PGF α,1 38 TP [P4] threshol of P4 to stimulate PGFα increase 39 τ P4, 17 [t] elay of P4 until stimulating PGFα ecrease 40 PGF α, mp 4 11 [PGFα]/[t] maximum ecay rate of PGFα PGF α,1 41 TP [P4] threshol of P4 to stimulate PGFα ecrease 4 τ LH 4.5 [t] elay of LH in CL 43 m CL LH [CL]/[t] maximum increase of CL stimulate by LH 44 T CL LH 1. [LH] threshol of LH to stimulate CL increase 45 m CL CL [CL]/[t] maximum increase of CL stimulate by itself Continue on next page...