Follicular competition in cows: the selection of dominant follicles as a synergistic effect

Abstract

The reproductive cycle of mono-ovulatory species such as cows or humans is known to show two or more waves of follicular growth and decline between two successive ovulations. Within each wave, there is one dominant follicle escorted by subordinate follicles of varying number. Under the surge of the luteinizing hormone a growing dominant follicle ovulates. Rarely the number of ovulating follicles exceeds one. In the biological literature, the change of hormonal concentrations and individually varying numbers of follicular receptors are made responsible for the selection of exactly one dominant follicle, yet a clear cause has not been identified. In this paper, we suggest a synergistic explanation based on competition, formulated by a parsimoniously defined system of ordinary differential equations (ODEs) that quantifies the time evolution of multiple follicles and their competitive interaction during one wave. Not discriminating between follicles, growth and decline are given by fixed rates. Competition is introduced via a growth-suppressing term, equally supported by all follicles. We prove that the number of dominant follicles is determined exclusively by the ratio of follicular growth and competition. This number turns out to be independent of the number of subordinate follicles. The asymptotic behavior of the corresponding dynamical system is investigated rigorously, where we demonstrate that the $\omega$-limit set only contains fixed points. When also including follicular decline, our ODEs perfectly resemble ultrasound data of bovine follicles. Implications for the involved but not explicitly modeled hormones are discussed.

Appendix

Similar to follicle data and approximating curves of cow number 451 in Figs. 6 and 7 , follicle diameters of another cow (no. 3674) studied by Cummins et al. (2012) are shown in Fig. 8. Parameters and initial values of the ODEs, utilized to approximate follicle diameters over time (Figs. 6, 7, and 8), are provided in Tables 2 and 3 . The following constraints have been implemented: productive dimension $\nu \le 3$, largest initial diameter $x_1(t_0)\le 3$, and maximal diameter $\xi \le 22$, if the corresponding wave is coupled to subsequent waves (where $\lambda >0$), or $\xi \le 25$ otherwise (where $\lambda =0$).

Fig. 6

Follicular kinetics with competition versus data. Follicle diameters of one cow (the same as on the l.h.s. panels in Fig. 3) during the first (l.h.s.) and second follicular wave (r.h.s.) are approximated by the two competitive growth models: logistic decline (14) in Panel (a) and simple decline (15) in Panel (b). The initial time for the second wave has been set to day 10. Six parameters ($\eta ,\gamma ,\kappa ,\nu ,\rho ,\xi$; cf. Fig. 5) as well as three (l.h.s.) and four initial values (r.h.s.) have been determined to achieve the best possible least square fit, locally (cf. Tables 2 and 3 in the Appendix); RMSD indicates the root-mean-squared deviation. The good fitting result for the first wave with simple decline is due to the high $\xi$ value, which (attained at the boundary of 25 mm) is higher than the possible follicle diameter. For the second wave, a better fit would be achievable as well if $\xi$ was not restricted by 25 mm (cf. Appendix)

Fig. 7

Follicular kinetics with competition and interaction between waves. Follicle diameters of one cow (the same as in Fig. 6) during the first (l.h.s.) and second follicular wave (r.h.s.) are approximated by the two competitive growth models including competitive coupling with subsequent waves (16): logistic decline (14) in Panel (a) and simple decline (15) in Panel (b). The initial time for the second wave has been set to day 10. Seven parameters ($\eta ,\gamma ,\kappa ,\lambda ,\nu ,\rho ,\xi$) as well as three (l.h.s.) and four initial values (r.h.s.) have been determined to achieve the best possible least square fit, locally (cf. Tables 2 and 3 in the Appendix); RMSD indicates the root-mean-squared deviation

Fig. 8

Follicular kinetics. The eight panels show data and approximating curves of follicle diameters of cow no. 3674 studied by Cummins et al. (2012). Similar to Fig. 6, Panels (a) and (b) only include follicular interaction within one wave. Similar to Fig. 7, Panels (c) and (d) also incorporate interaction between waves. Panels (a) and (c) model logistic decline, Panels (b) and (d) simple decline. The l.h.s./r.h.s. panels illustrate follicles of the first/second wave, resp

Table 2 Parameter values

Table 3 Initial values