## Section: New Results

### Modeling, observation and control in biosciences: ovulation control

#### Foliation-based parameter tuning in a model of the GnRH pulse and surge generator

Participants : Frédérique Clément, Alexandre Vidal.

We investigate a model of the GnRH pulse and surge generator [6] , with the definite aim of constraining the model GnRH output with respect to a physiologically relevant list of specifications [30] . The alternating pulse and surge pattern of secretion results from the interaction between a GnRH secreting system and a regulating system exhibiting slow-fast dynamics. The mechanisms underlying the behavior of the model are reviewed from the study of the Boundary-Layer System according to the "dissection method" principle. Using singular perturbation theory, we describe the sequence of bifurcations undergone by the regulating (FitzHugh-Nagumo) system, encompassing the rarely investigated case of homoclinic connection. Basing on pure dynamical considerations, we restrict the space of parameter search for the regulating system and describe a foliation of this restricted space, whose leaves define constant duration ratios between the surge and the pulsatility phase in the whole system. We propose an algorithm to fix the parameter values to also meet the other prescribed ratios dealing with amplitude and frequency features of the secretion signal. We finally apply these results to illustrate the dynamics of GnRH secretion in the ovine species and the rhesus monkey.

#### Endogenous circannual rhythm in LH secretion: insight from signal analysis coupled to mathematical modelling

Participants : Frédérique Clément, Claire Médigue, Alexandre Vidal.

In sheep, as in many vertebrates, the seasonal pattern of reproduction is timed by the annual photoperiodic cycle, characterized by seasonal changes in the day length. The photoperiodic information is translated into a circadian profile of melatonin secretion. After multiple neuronal relays (within the hypothalamus), melatonin affects gonadotrophin-releasing hormone (GnRH) secretion, which in turn controls ovarian cyclicity. The pattern of GnRH secretion is mirrored by that of luteinizing hormone (LH) secretion, whose plasmatic level can be easily measured. We addressed the question of whether there exists an endogenous circannual rhythm in a tropical sheep (Blackbelly) population that exhibits clear seasonal ovarian activity when ewes are subject to temperate latitudes [42] . We based our analysis on LH time series collected in the course of 3 years from ewes subject to a constant photoperiodic regime. Owing to intra- and interanimal variability and unequal sampling times, the existence of an endogenous rhythm is not straightforward. We have used time–frequency signal processing methods, and especially the smooth pseudo-Wigner–Ville distribution, to extract possible hidden rhythms from the data. To further investigate the low-frequency (LF) and high-frequency (HF) components of the signals, we have designed a simple mathematical model of the LH plasmatic level accounting for the effect of experimental sampling times. The model enables us to (i) confirm the existence of an endogenous circannual rhythm as detected by the LF signal component, (ii) investigate the action mechanism of the photoperiod on the pulsatile pattern of LH secretion (control of the interpulse interval), and (iii) conclude that the HF component is mainly due to the experimental sampling protocol.

#### Towards a systems biology approach of G protein-coupled receptors' signalling: challenges and expectations

Participant : Frédérique Clément.

Collaboration with François Fages (contraintes ), Domitille Heitzler and Eric Reiter (UMR CNRS-INRA 6175)

G protein-coupled receptors (GPCRs) control all the main physiological functions and are targeted by more than 50% of therapeutics. Our perception of GPCRs signalling has grown increasingly complex since it is now accepted that they activate large signalling networks which are integrating the information fluxes into appropriate biological responses. These concepts lead the way to the development of pathway-selective agonists (or antagonists) with fewer side effects. Systems biology approaches focused on GPCR-mediated signalling would help dealing with the huge complexity of these mechanisms therefore speeding-up the discovery of new drug classes. In this review [34] , we present the various technical and conceptual possibilities allowing a systems approach of GPCR-mediated signalling. The main remaining limitations are also discussed.