Nonlinear compartmental modeling to monitor ovarian follicle population dynamics on the whole lifespan.

In this work, we introduce a compartmental model of ovarian follicle development all along lifespan, based on ordinary differential equations. The model predicts the changes in the follicle numbers in different maturation stages with aging. Ovarian follicles may either move forward to the next compartment (unidirectional migration) or degenerate and disappear (death). The migration from the first follicle compartment corresponds to the activation of quiescent follicles, which is responsible for the progressive exhaustion of the follicle reserve (ovarian aging) until cessation of reproductive activity. The model consists of a data-driven layer embedded into a more comprehensive, knowledge-driven layer encompassing the earliest events in follicle development. The data-driven layer is designed according to the most densely sampled experimental dataset available on follicle numbers in the mouse. Its salient feature is the nonlinear formulation of the activation rate, whose formulation includes a feedback term from growing follicles. The knowledge-based, coating layer accounts for cutting-edge studies on the initiation of follicle development around birth. Its salient feature is the co-existence of two follicle subpopulations of different embryonic origins. We then setup a complete estimation strategy, including the study of structural identifiability, the elaboration of a relevant optimization criterion combining different sources of data (the initial dataset on follicle numbers, together with data in conditions of perturbed activation, and data discriminating the subpopulations) with appropriate error models, and a model selection step. We finally illustrate the model potential for experimental design (suggestion of targeted new data acquisition) and in silico experiments.

A Lifshitz-Slyozov type model for adipocyte size dynamics: limit from Becker-Döring system and numerical simulation.

Biological data show that the size distribution of adipose cells follows a bimodal distribution. In this work, we introduce a Lifshitz-Slyozov type model, based on a transport partial differential equation, for the dynamics of the size distribution of adipose cells. We prove a new convergence result from the related Becker-Döring model, a system composed of several ordinary differential equations, toward mild solutions of the Lifshitz-Slyozov model using distribution tail techniques. Then, this result allows us to propose a new advective-diffusive model, the second-order diffusive Lifshitz-Slyozov model, which is expected to better fit the experimental data. Numerical simulations of the solutions to the diffusive Lifshitz-Slyozov model are performed using a well-balanced scheme and compared to solutions to the transport model. Those simulations show that both bimodal and unimodal profiles can be reached asymptotically depending on several parameters. We put in evidence that the asymptotic profile for the second-order system does not depend on initial conditions, unlike for the transport Lifshitz-Slyozov model.

Mathematical modeling of adipocyte size distributions: Identifiability and parameter estimation from rat data.

Fat cells, called adipocytes, are designed to regulate energy homeostasis by storing energy in the form of lipids. Adipocyte size distribution is assumed to play a role in the development of obesity-related diseases. These cells that do not have a characteristic size, indeed a bimodal size distribution is observed in adipose tissue. We propose a model based on a partial differential equation to describe adipocyte size distribution. The model includes a description of the lipid fluxes and the cell size fluctuations and using a formulation of a stationary solution fast computation of bimodal distribution is achieved. We investigate the parameter identifiability and estimate parameter values with CMA-ES algorithm. We first validate the procedure on synthetic data, then we estimate parameter values with experimental data of 32 rats. We discuss the estimated parameter values and their variability within the population, as well as the relation between estimated values and their biological significance. Finally, a sensitivity analysis is performed to specify the influence of parameters on cell size distribution and explain the differences between the model and the measurements. The proposed framework enables the characterization of adipocyte size distribution with four parameters and can be easily adapted to measurements of cell size distribution in different health conditions.

An overview of deep learning applications in precocious puberty and thyroid dysfunction.

In the last decade, deep learning methods have garnered a great deal of attention in endocrinology research. In this article, we provide a summary of current deep learning applications in endocrine disorders caused by either precocious onset of adult hormone or abnormal amount of hormone production. To give access to the broader audience, we start with a gentle introduction to deep learning and its most commonly used architectures, and then we focus on the research trends of deep learning applications in thyroid dysfunction classification and precocious puberty diagnosis. We highlight the strengths and weaknesses of various approaches and discuss potential solutions to different challenges. We also go through the practical considerations useful for choosing (and building) the deep learning model, as well as for understanding the thought process behind different decisions made by these models. Finally, we give concluding remarks and future directions.

Pharmacological Characterization of Low Molecular Weight Biased Agonists at the Follicle Stimulating Hormone Receptor.

Follicle-stimulating hormone receptor (FSHR) plays a key role in reproduction through the activation of multiple signaling pathways. Low molecular weight (LMW) ligands composed of biased agonist properties are highly valuable tools to decipher complex signaling mechanisms as they allow selective activation of discrete signaling cascades. However, available LMW FSHR ligands have not been fully characterized yet. In this context, we explored the pharmacological diversity of three benzamide and two thiazolidinone derivatives compared to FSH. Concentration/activity curves were generated for Gαs, Gαq, Gαi, ß-arrestin 2 recruitment, and cAMP production, using BRET assays in living cells. ERK phosphorylation was analyzed by Western blotting, and CRE-dependent transcription was assessed using a luciferase reporter assay. All assays were done in either wild-type, Gαs or ß-arrestin 1/2 CRISPR knockout HEK293 cells. Bias factors were calculated for each pair of read-outs by using the operational model. Our results show that each ligand presented a discrete pharmacological efficacy compared to FSH, ranging from super-agonist for ß-arrestin 2 recruitment to pure Gαs bias. Interestingly, LMW ligands generated kinetic profiles distinct from FSH (i.e., faster, slower or transient, depending on the ligand) and correlated with CRE-dependent transcription. In addition, clear system biases were observed in cells depleted of either Gαs or ß-arrestin genes. Such LMW properties are useful pharmacological tools to better dissect the multiple signaling pathways activated by FSHR and assess their relative contributions at the cellular and physio-pathological levels.

Stochastic nonlinear model for somatic cell population dynamics during ovarian follicle activation.

In mammals, female germ cells are sheltered within somatic structures called ovarian follicles, which remain in a quiescent state until they get activated, all along reproductive life. We investigate the sequence of somatic cell events occurring just after follicle activation, starting by the awakening of precursor somatic cells, and their transformation into proliferative cells. We introduce a nonlinear stochastic model accounting for the joint dynamics of the two cell types, and allowing us to investigate the potential impact of a feedback from proliferative cells onto precursor cells. To tackle the key issue of whether cell proliferation is concomitant or posterior to cell awakening, we assess both the time needed for all precursor cells to awake, and the corresponding increase in the total cell number with respect to the initial cell number. Using the probabilistic theory of first passage times, we design a numerical scheme based on a rigorous finite state projection and coupling techniques to compute the mean extinction time and the cell number at extinction time. We find that the feedback term clearly lowers the number of proliferative cells at the extinction time. We calibrate the model parameters using an exact likelihood approach. We carry out a comprehensive comparison between the initial model and a series of submodels, which helps to select the critical cell events taking place during activation, and suggests that awakening is prominent over proliferation.

Mathematical modeling approaches of cellular endocrinology within the hypothalamo-pituitary-gonadal axis.

The reproductive neuroendocrine axis, or hypothalamo-pituitary-gonadal (HPG) axis, is a paragon of complex biological system involving numerous cell types, spread over several anatomical levels communicating through entangled endocrine feedback loops. The HPG axis exhibits remarkable dynamic behaviors on multiple time and space scales, which are an inexhaustible source of studies for mathematical and computational biology. In this review, we will describe a variety of modeling approaches of the HPG axis from a cellular endocrinology viewpoint. We will in particular investigate the questions raised by some of the most striking features of the HPG axis: (i) the pulsatile secretion of hypothalamic and pituitary hormones, and its counterpart, the cell signaling induced by frequency-encoded hormonal signals, and (ii) the dual, gametogenic and glandular function of the gonads, which relies on the tight control of the somatic cell populations ensuring the proper maturation and timely release of the germ cells.

Biased Signaling and Allosteric Modulation at the FSHR.

Knowledge on G protein-coupled receptor (GPCRs) structure and mechanism of activation has profoundly evolved over the past years. The way drugs targeting this family of receptors are discovered and used has also changed. Ligands appear to bind a growing number of GPCRs in a competitive or allosteric manner to elicit balanced signaling or biased signaling (i.e., differential efficacy in activating or inhibiting selective signaling pathway(s) compared to the reference ligand). These novel concepts and developments transform our understanding of the follicle-stimulating hormone (FSH) receptor (FSHR) biology and the way it could be pharmacologically modulated in the future. The FSHR is expressed in somatic cells of the gonads and plays a major role in reproduction. When compared to classical GPCRs, the FSHR exhibits intrinsic peculiarities, such as a very large NH2-terminal extracellular domain that binds a naturally heterogeneous, large heterodimeric glycoprotein, namely FSH. Once activated, the FSHR couples to Gαs and, in some instances, to other Gα subunits. G protein-coupled receptor kinases and ß-arrestins are also recruited to this receptor and account for its desensitization, trafficking, and intracellular signaling. Different classes of pharmacological tools capable of biasing FSHR signaling have been reported and open promising prospects both in basic research and for therapeutic applications. Here we provide an updated review of the most salient peculiarities of FSHR signaling and its selective modulation.

Workflow Description to Dynamically Model ß-Arrestin Signaling Networks.

Dynamic models of signaling networks allow the formulation of hypotheses on the topology and kinetic rate laws characterizing a given molecular network, in-depth exploration, and confrontation with kinetic biological data. Despite its standardization, dynamic modeling of signaling networks still requires successive technical steps that need to be carefully performed. Here, we detail these steps by going through the mathematical and statistical framework. We explain how it can be applied to the understanding of ß-arrestin-dependent signaling networks. We illustrate our methodology through the modeling of ß-arrestin recruitment kinetics at the follicle-stimulating hormone (FSH) receptor supported by in-house bioluminescence resonance energy transfer (BRET) data.

Advances in computational modeling approaches of pituitary gonadotropin signaling.

INTRODUCTION: Pituitary gonadotropins play an essential and pivotal role in the control of human and animal reproduction within the hypothalamic-pituitary-gonadal (HPG) axis. The computational modeling of pituitary gonadotropin signaling encompasses phenomena of different natures such as the dynamic encoding of gonadotropin secretion, and the intracellular cascades triggered by gonadotropin binding to their cognate receptors, resulting in a variety of biological outcomes. Areas covered: The authors provide an overview of the historical and ongoing issues in modeling and data analysis related to gonadotropin secretion in the field of both physiology and neuroendocrinology. They mention the different mathematical formalisms involved, their interest and limits. They also discuss open statistical questions in signal analysis associated with key endocrine issues and review recent advances in the modeling of the intracellular pathways activated by gonadotropins, which yields promising development for innovative approaches in drug discovery. Expert opinion: The greatest challenge to be tackled in computational modeling of pituitary gonadotropin signaling is the embedding of gonadotropin signaling within its natural multi-scale environment, from the single cell level, to the organic and whole HPG level. The development of modeling approaches of G protein-coupled receptor signaling, together with multicellular systems biology may lead to unexampled mechanistic understanding with critical expected fallouts in the therapeutic management of reproduction.

Follicle-Stimulating Hormone Receptor: Advances and Remaining Challenges.

Follicle-stimulating hormone (FSH) is produced in the pituitary and is essential for reproduction. It specifically binds to a membrane receptor (FSHR) expressed in somatic cells of the gonads. The FSH/FSHR system presents many peculiarities compared to classical G protein-coupled receptors (GPCRs). FSH is a large naturally heterogeneous heterodimeric glycoprotein. The FSHR is characterized by a very large NH2-terminal extracellular domain, which binds FSH and participates to the activation/inactivation switch of the receptor. Once activated, the FSHR couples to Gαs and, in some instances, to other Gα-subunits. GPCR kinases and ß-arrestins are also recruited to the FSHR and account for its desensitization, the control of its trafficking and its intracellular signaling. Of note, the FSHR internalization and recycling are very fast and involve very early endosomes (EE) instead of EE. All the transduction mechanisms triggered upon FSH stimulation lead to the activation of a complex signaling network that controls gene expression by acting at multiple levels. The integration of these mechanisms not only leads to context-adapted responses from the target gonadal cells but also indirectly affects the fate of germ cells. Depending on the physiological/developmental stage, FSH elicits proliferation, differentiation, or apoptosis in order to maintain the homeostasis of the reproductive system. Pharmacological tools targeting FSHR recently came to the fore and open promising prospects both for basic research and therapeutic applications. This chapter provides an updated review of the most salient aspects and peculiarities of FSHR biology and pharmacology.

Human Luteinizing Hormone and Chorionic Gonadotropin Display Biased Agonism at the LH and LH/CG Receptors.

Human luteinizing hormone (LH) and chorionic gonadotropin (hCG) have been considered biologically equivalent because of their structural similarities and their binding to the same receptor; the LH/CGR. However, accumulating evidence suggest that LH/CGR differentially responds to the two hormones triggering differential intracellular signaling and steroidogenesis. The mechanistic basis of such differential responses remains mostly unknown. Here, we compared the abilities of recombinant rhLH and rhCG to elicit cAMP, ß-arrestin 2 activation, and steroidogenesis in HEK293 cells and mouse Leydig tumor cells (mLTC-1). For this, BRET and FRET technologies were used allowing quantitative analyses of hormone activities in real-time and in living cells. Our data indicate that rhLH and rhCG differentially promote cell responses mediated by LH/CGR revealing interesting divergences in their potencies, efficacies and kinetics: rhCG was more potent than rhLH in both HEK293 and mLTC-1 cells. Interestingly, partial effects of rhLH were found on ß-arrestin recruitment and on progesterone production compared to rhCG. Such a link was further supported by knockdown experiments. These pharmacological differences demonstrate that rhLH and rhCG act as natural biased agonists. The discovery of novel mechanisms associated with gonadotropin-specific action may ultimately help improve and personalize assisted reproduction technologies.

ß-arrestin signalling and bias in hormone-responsive GPCRs.

G protein-coupled receptors (GPCRs) play crucial roles in the ability of target organs to respond to hormonal cues. GPCRs' activation mechanisms have long been considered as a two-state process connecting the agonist-bound receptor to heterotrimeric G proteins. This view is now challenged as mounting evidence point to GPCRs being connected to large arrays of transduction mechanisms involving heterotrimeric G proteins as well as other players. Amongst the G protein-independent transduction mechanisms, those elicited by ß-arrestins upon their recruitment to the active receptors are by far the best characterized and apply to most GPCRs. These concepts, in conjunction with remarkable advances made in the field of GPCR structural biology and biophysics, have supported the notion of ligand-selective signalling also known as pharmacological bias. Interestingly, recent reports have opened intriguing prospects to the way ß-arrestins control GPCR-mediated signalling in space and time within the cells. In the present paper, we review the existing evidence linking endocrine-related GPCRs to ß-arrestin recruitement, signalling, pathophysiological implications and selective activation by biased ligands and/or receptor modifications. Emerging concepts surrounding ß-arrestin-mediated transduction are discussed in the light of the peculiarities of endocrine systems.

Profiling of FSHR negative allosteric modulators on LH/CGR reveals biased antagonism with implications in steroidogenesis.

Biased signaling has recently emerged as an interesting means to modulate the function of many G protein-coupled receptors (GPCRs). Previous studies reported two negative allosteric modulators (NAMs) of follicle-stimulating hormone receptor (FSHR), ADX68692 and ADX68693, with differential effects on FSHR-mediated steroidogenesis and ovulation. In this study, we attempted to pharmacologically profile these NAMs on the closely related luteinizing hormone/chorionic gonadotropin hormone receptor (LH/CGR) with regards to its canonical Gs/cAMP pathway as well as to ß-arrestin recruitment in HEK293 cells. The NAMs' effects on cAMP, progesterone and testosterone production were also assessed in murine Leydig tumor cell line (mLTC-1) as well as rat primary Leydig cells. We found that both NAMs strongly antagonized LH/CGR signaling in the different cell models used with ADX68693 being more potent than ADX68692 to inhibit hCG-induced cAMP production in HEK293, mLTC-1 and rat primary Leydig cells as well as ß-arrestin 2 recruitment in HEK293 cells. Interestingly, differential antagonism of the two NAMs on hCG-promoted steroidogenesis in mLTC-1 and rat primary Leydig cells was observed. Indeed, a significant inhibition of testosterone production by the two NAMs was observed in both cell types, whereas progesterone production was only inhibited by ADX68693 in rat primary Leydig cells. In addition, while ADX68693 totally abolished testosterone production, ADX68692 had only a partial effect in both mLTC-1 and rat primary Leydig cells. These observations suggest biased effects of the two NAMs on LH/CGR-dependent pathways controlling steroidogenesis. Interestingly, the pharmacological profiles of the two NAMs with respect to steroidogenesis were found to differ from that previously shown on FSHR. This illustrates the complexity of signaling pathways controlling FSHR- and LH/CGR-mediated steroidogenesis, suggesting differential implication of cAMP and ß-arrestins mediated by FSHR and LH/CGR. Together, our data demonstrate that ADX68692 and ADX68693 are biased NAMs at the LH/CGR in addition to the FSHR. These pharmacological characteristics are important to consider for potential contraceptive and therapeutic applications based on such compounds.

Computational modeling approaches in gonadotropin signaling.

Follicle-stimulating hormone and LH play essential roles in animal reproduction. They exert their function through binding to their cognate receptors, which belong to the large family of G protein-coupled receptors. This recognition at the plasma membrane triggers a plethora of cellular events, whose processing and integration ultimately lead to an adapted biological response. Understanding the nature and the kinetics of these events is essential for innovative approaches in drug discovery. The study and manipulation of such complex systems requires the use of computational modeling approaches combined with robust in vitro functional assays for calibration and validation. Modeling brings a detailed understanding of the system and can also be used to understand why existing drugs do not work as well as expected, and how to design more efficient ones.

First passage times in homogeneous nucleation: Dependence on the total number of particles.

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present an extension to a thorough analysis of the stochastic self-assembly of a fixed number of identical particles in a finite volume. We study the statistics of times required for maximal clusters to be completed, starting from a pure-monomeric particle configuration. For finite volumes, we extend previous analytical approaches to the case of arbitrary size-dependent aggregation and fragmentation kinetic rates. For larger volumes, we develop a scaling framework to study the first assembly time behavior as a function of the total quantity of particles. We find that the mean time to first completion of a maximum-sized cluster may have a surprisingly weak dependence on the total number of particles. We highlight how higher statistics (variance, distribution) of the first passage time may nevertheless help to infer key parameters, such as the size of the maximum cluster. Finally, we present a framework to quantify formation of macroscopic sized clusters, which are (asymptotically) very unlikely and occur as a large deviation phenomenon from the mean-field limit. We argue that this framework is suitable to describe phase transition phenomena, as inherent infrequent stochastic processes, in contrast to classical nucleation theory.

Adiabatic reduction of a model of stochastic gene expression with jump Markov process.

This paper considers adiabatic reduction in a model of stochastic gene expression with bursting transcription considered as a jump Markov process. In this model, the process of gene expression with auto-regulation is described by fast/slow dynamics. The production of mRNA is assumed to follow a compound Poisson process occurring at a rate depending on protein levels (the phenomena called bursting in molecular biology) and the production of protein is a linear function of mRNA numbers. When the dynamics of mRNA is assumed to be a fast process (due to faster mRNA degradation than that of protein) we prove that, with appropriate scalings in the burst rate, jump size or translational rate, the bursting phenomena can be transmitted to the slow variable. We show that, depending on the scaling, the reduced equation is either a stochastic differential equation with a jump Poisson process or a deterministic ordinary differential equation. These results are significant because adiabatic reduction techniques seem to have not been rigorously justified for a stochastic differential system containing a jump Markov process. We expect that the results can be generalized to adiabatic methods in more general stochastic hybrid systems.

First passage times in homogeneous nucleation and self-assembly.

Motivated by nucleation and molecular aggregation in physical, chemical, and biological settings, we present a thorough analysis of the general problem of stochastic self-assembly of a fixed number of identical particles in a finite volume. We derive the backward Kolmogorov equation (BKE) for the cluster probability distribution. From the BKE, we study the distribution of times it takes for a single maximal cluster to be completed, starting from any initial particle configuration. In the limits of slow and fast self-assembly, we develop analytical approaches to calculate the mean cluster formation time and to estimate the first assembly time distribution. We find, both analytically and numerically, that faster detachment can lead to a shorter mean time to first completion of a maximum-sized cluster. This unexpected effect arises from a redistribution of trajectory weights such that upon increasing the detachment rate, paths that take a shorter time to complete a cluster become more likely.

Molecular distributions in gene regulatory dynamics.

Extending the work of Friedman et al. (2006), we study the stationary density of the distribution of molecular constituents in the presence of noise arising from either bursting transcription or translation, or noise in degradation rates. We examine both the global stability of the stationary density as well as its bifurcation structure. We have compared our results with an analysis of the same model systems (either inducible or repressible operons) in the absence of any stochastic effects, and shown the correspondence between behaviour in the deterministic system and the stochastic analogs. We have identified key dimensionless parameters that control the appearance of one or two stable steady states in the deterministic case, or unimodal and bimodal densities in the stochastic systems, and detailed the analytic requirements for the occurrence of different behaviours. This approach provides, in some situations, an alternative to computationally intensive stochastic simulations. Our results indicate that, within the context of the simple models we have examined, bursting and degradation noise cannot be distinguished analytically when present alone.