The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment.

The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

Fitness advantage of Bacteroides thetaiotaomicron capsular polysaccharide in the mouse gut depends on the resident microbiota.

Many microbiota-based therapeutics rely on our ability to introduce a microbe of choice into an already-colonized intestine. In this study, we used genetically barcoded Bacteroides thetaiotaomicron (B. theta) strains to quantify population bottlenecks experienced by a B. theta population during colonization of the mouse gut. As expected, this reveals an inverse relationship between microbiota complexity and the probability that an individual wildtype B. theta clone will colonize the gut. The polysaccharide capsule of B. theta is important for resistance against attacks from other bacteria, phage, and the host immune system, and correspondingly acapsular B. theta loses in competitive colonization against the wildtype strain. Surprisingly, the acapsular strain did not show a colonization defect in mice with a low-complexity microbiota, as we found that acapsular strains have an indistinguishable colonization probability to the wildtype strain on single-strain colonization. This discrepancy could be resolved by tracking in vivo growth dynamics of both strains: acapsular B.theta shows a longer lag phase in the gut lumen as well as a slightly slower net growth rate. Therefore, as long as there is no niche competitor for the acapsular strain, this has only a small influence on colonization probability. However, the presence of a strong niche competitor (i.e., wildtype B. theta, SPF microbiota) rapidly excludes the acapsular strain during competitive colonization. Correspondingly, the acapsular strain shows a similarly low colonization probability in the context of a co-colonization with the wildtype strain or a complete microbiota. In summary, neutral tagging and detailed analysis of bacterial growth kinetics can therefore quantify the mechanisms of colonization resistance in differently-colonized animals.

Circadian Misalignment and Metabolic Disorders: A Story of Twisted Clocks.

Biological clocks are cell-autonomous oscillators that can be entrained by periodic environmental cues. This allows organisms to anticipate predictable daily environmental changes and, thereby, to partition physiological processes into appropriate phases with respect to these changing external conditions. Nowadays our 24/7 society challenges this delicate equilibrium. Indeed, many studies suggest that perturbations such as chronic jet lag, ill-timed eating patterns, or shift work increase the susceptibility to cardiometabolic disorders, diabetes, and cancers. However the underlying mechanisms are still poorly understood. A deeper understanding of this complex, dynamic system requires a global holistic approach for which mathematical modeling can be highly beneficial. In this review, we summarize several experimental works pertaining to the effect of adverse conditions on clock gene expression and on physiology, and we show how computational models can bring interesting insights into the links between circadian misalignment and metabolic diseases.

Modeling clock-related metabolic syndrome due to conflicting light and food cues.

Most organisms possess a light- and food- entrainable circadian clock system enabling their adaptation to daily environmental changes in sunlight and food availability. The mammalian circadian system is composed of multiple clocks throughout the body. These local clocks are entrained by nutrient, neural, endocrine and temperature cues and drive diverse physiological functions including metabolism. In particular, the clock of the pancreatic ß cell rhythmically regulates the transcription of genes involved in glucose-stimulated insulin secretion. Perturbations of this fine-tuned oscillatory network increase the susceptibility to diseases. Besides chronic jet lag and shift work, common perturbations are ill-timed eating patterns which can lead to metabolic troubles (such as hypoinsulinemia). We have built a mathematical model describing the clock-dependent pancreatic regulation of glucose homeostasis in rodents. After calibrating the model using experimental data, we have investigated the effect of restricting food access to the normal rest phase. Our simulations show that the conflict between the light-dark cycle and the feeding-fasting cycle creates a differential phase shift in the expression of core clock genes (consistent with experimental observations). Our model further predicts that this induces a non-concomitance between nutrient cues and clock-controlled cues driving metabolic outputs which results in hypoinsulinemia, hyperglycemia as well as in a loss of food anticipation.

Modeling-Based Investigation of the Effect of Noise in Cellular Systems.

Noise is pervasive in cellular biology and inevitably affects the dynamics of cellular processes. Biological systems have developed regulatory mechanisms to ensure robustness with respect to noise or to take advantage of stochasticity. We review here, through a couple of selected examples, some insights on possible robustness factors and constructive roles of noise provided by computational modeling. In particular, we focus on (1) factors that likely contribute to the robustness of oscillatory processes such as the circadian clocks and the cell cycle, (2) how reliable coding/decoding of calcium-mediated signaling could be achieved in presence of noise and, in some cases, enhanced through stochastic resonance, and (3) how embryonic cell differentiation processes can exploit stochasticity to create heterogeneity in a population of identical cells.

A Mathematical Model of the Liver Circadian Clock Linking Feeding and Fasting Cycles to Clock Function.

To maintain energy homeostasis despite variable energy supply and consumption along the diurnal cycle, the liver relies on a circadian clock synchronized to food timing. Perturbed feeding and fasting cycles have been associated with clock disruption and metabolic diseases; however, the mechanisms are unclear. To address this question, we have constructed a mathematical model of the mammalian circadian clock, incorporating the metabolic sensors SIRT1 and AMPK. The clock response to various temporal patterns of AMPK activation was simulated numerically, mimicking the effects of a normal diet, fasting, and a high-fat diet. The model reproduces the dampened clock gene expression and NAD+ rhythms reported for mice on a high-fat diet and predicts that this effect may be pharmacologically rescued by timed REV-ERB agonist administration. Our model thus identifies altered AMPK signaling as a mechanism leading to clock disruption and its associated metabolic effects and suggests a pharmacological approach to resetting the clock in obesity.

The bird circadian clock: insights from a computational model.

The circadian timekeeping system appears more complex in birds than in mammals. In mammals, the main pacemaker is centralized in the suprachiasmatic nuclei, whereas in birds, the pacemaker involves the interplay between the pineal and hypothalamic oscillators. In order to investigate the consequence of this complex mechanism, we propose here a mathematical model for the bird circadian clock. The model is based on the internal resonance between the pineal and hypothalamic oscillators, each described by Goodwin-like equations. We show that, consistently with experimental observations, self-sustained oscillations can be generated by mutual inhibitory coupling of the 2 clocks, even if individual oscillators present damped oscillations. We study the effect of constant and periodic administrations of melatonin, which, in intact birds, acts as the coupling variable between the pineal and the hypothalamus, and compare the prediction of the model with the experiments performed in pinealectomized birds. We also assess the entrainment properties when the system is subject to light-dark cycles. Analyses of the entrainment range, resynchronization time after jet lag, and entrainment phase with respect to the photoperiod lead us to formulate hypotheses about the physiological advantage of the particular architecture of the avian circadian clock. Although minimal, our model opens promising perspectives in modeling and understanding the bird circadian clock.