Conceptual Models of Entrainment, Jet Lag, and Seasonality

Differences in phenology across three trophic levels between two Afrotropical sites separated by four degrees latitude

Birds time their life cycle events to favourable windows in environmental conditions. In tropical environments, where photoperiod variation is small, birds show high variability in the timing of life cycle stages, yet these species have been severely underrepresented in phenology research. Here, we investigated temporal patterns in bird life cycles and resource availability in two sites in tropical Africa: Weppa (Nigeria, 7° N) and Elat (Cameroon, 3° N). In these sites we captured common bulbuls (Pycnonotus barbatus), a widespread generalist, and recorded breeding and moult over a 12-month period. Simultaneously, we surveyed fruiting tree and arthropod abundance. Our aim was to quantify seasonal patterns in moult and breeding in bulbuls at both sites, and link them to fluctuations in local fruit and arthropod abundance and precipitation. Moult was more seasonal than breeding in both sites, and seasonality of both life cycle events was stronger in Nigeria than Cameroon. The peak timing for moult was 1.5 months earlier in Nigeria than Cameroon. Seasonal variation in abundance of fruiting trees and arthropods was different between sites, as were the associations with breeding and moulting. In Nigeria, we found a positive association between moult and arthropod abundance, and a negative one with fruiting tree abundance. In contrast, in Cameroon moult was associated with higher precipitation, while breeding occurred at times with higher fruit abundance. Our results provide evidence that, even in similar habitats separated by four degrees in latitude, seasonal patterns across three trophic levels are variable. Understanding links between environmental conditions and life cycle events can reveal potential vulnerabilities of tropical species, and guide conservation efforts.

Are circadian amplitudes and periods correlated? A new twist in the story

Three parameters are important to characterize a circadian and in general any biological clock: period, phase and amplitude. While circadian periods have been shown to correlate with entrainment phases, and clock amplitude influences the phase response of an oscillator to pulse-like zeitgeber signals, the co-modulations of amplitude and periods, which we term twist, have not been studied in detail. In this paper we define two concepts: parametric twist refers to amplitude-period correlations arising in ensembles of self-sustained, limit cycle clocks in the absence of external inputs, and phase space twist refers to the co-modulation of an individual clock's amplitude and period in response to external zeitgebers. Our findings show that twist influences the interaction of oscillators with the environment, facilitating entrainment, speeding upfastening recovery to pulse-like perturbations or modifying the response of an individual clock to coupling. This theoretical framework might be applied to understand the emerging properties of other oscillating systems.

Contribution of membrane-associated oscillators to biological timing at different timescales

Environmental rhythms such as the daily light-dark cycle selected for endogenous clocks. These clocks predict regular environmental changes and provide the basis for well-timed adaptive homeostasis in physiology and behavior of organisms. Endogenous clocks are oscillators that are based on positive feedforward and negative feedback loops. They generate stable rhythms even under constant conditions. Since even weak interactions between oscillators allow for autonomous synchronization, coupling/synchronization of oscillators provides the basis of self-organized physiological timing. Amongst the most thoroughly researched clocks are the endogenous circadian clock neurons in mammals and insects. They comprise nuclear clockworks of transcriptional/translational feedback loops (TTFL) that generate ∼24 h rhythms in clock gene expression entrained to the environmental day-night cycle. It is generally assumed that this TTFL clockwork drives all circadian oscillations within and between clock cells, being the basis of any circadian rhythm in physiology and behavior of organisms. Instead of the current gene-based hierarchical clock model we provide here a systems view of timing. We suggest that a coupled system of autonomous TTFL and posttranslational feedback loop (PTFL) oscillators/clocks that run at multiple timescales governs adaptive, dynamic homeostasis of physiology and behavior. We focus on mammalian and insect neurons as endogenous oscillators at multiple timescales. We suggest that neuronal plasma membrane-associated signalosomes constitute specific autonomous PTFL clocks that generate localized but interlinked oscillations of membrane potential and intracellular messengers with specific endogenous frequencies. In each clock neuron multiscale interactions of TTFL and PTFL oscillators/clocks form a temporally structured oscillatory network with a common complex frequency-band comprising superimposed multiscale oscillations. Coupling between oscillator/clock neurons provides the next level of complexity of an oscillatory network. This systemic dynamic network of molecular and cellular oscillators/clocks is suggested to form the basis of any physiological homeostasis that cycles through dynamic homeostatic setpoints with a characteristic frequency-band as hallmark. We propose that mechanisms of homeostatic plasticity maintain the stability of these dynamic setpoints, whereas Hebbian plasticity enables switching between setpoints via coupling factors, like biogenic amines and/or neuropeptides. They reprogram the network to a new common frequency, a new dynamic setpoint. Our novel hypothesis is up for experimental challenge.

The SCN-HPA-Periphery Circadian Timing System: Mathematical Modeling of Clock Synchronization and the Effects of Photoperiod on Jetlag Adaptation

Synchronizing the circadian timing system (CTS) to external light/dark cycles is crucial for homeostasis maintenance and environmental adaptation. The CTS is organized hierarchically, with the central pacemaker located in the suprachiasmatic nuclei (SCN) generating coherent oscillations that are entrained to light/dark cycles. These oscillations regulate the release of glucocorticoids by the hypothalamus-pituitary-adrenal (HPA) axis, which acts as a systemic entrainer of peripheral clocks throughout the body. The SCN adjusts its network plasticity in response to variations in photoperiod, leading to changes in the rhythmic release of glucocorticoids and ultimately impacting peripheral clocks. However, the effects of photoperiod-induced variations of glucocorticoids on the synchronization of peripheral clocks are not fully understood, and the interaction between jetlag adaption and photoperiod changes is unclear. This study presents a semi-mechanistic mathematical model to investigate how the CTS responds to changes in photoperiod. Specifically, the study focuses on the entrainment properties of a system composed of the SCN, HPA axis, and peripheral clocks. The results show that high-amplitude glucocorticoid rhythms lead to a more coherent phase distribution in the periphery. In addition, our study investigates the effect of photoperiod exposure on jetlag recovery time and phase shift, proposing different interventional strategies for eastward and westward jetlag. The findings suggest that decreasing photic exposure before jetlag during eastward traveling and after jetlag during westward traveling can accelerate jetlag readaptation. The study provides insights into the mechanisms of CTS organization and potential recovery strategies for transitions between time zones and lighting zones.

Optimal phase-selective entrainment of heterogeneous oscillator ensembles

We develop a framework to design optimal entrainment signals that entrain an ensemble of heterogeneous nonlinear oscillators, described by phase models, at desired phases. We explicitly take into account heterogeneity in both oscillation frequency and the type of oscillators characterized by different Phase Response Curves. The central idea is to leverage the Fourier series representation of periodic functions to decode a phase-selective entrainment task into a quadratic program. We demonstrate our approach using a variety of phase models, where we entrain the oscillators into distinct phase patterns. Also, we show how the generalizability gained from our formulation enables us to meet a wide range of design objectives and constraints, such as minimum-power, fast entrainment, and charge-balanced controls.

Noise facilitates entrainment of a population of uncoupled limit cycle oscillators

Many biological oscillators share two properties: they are subject to stochastic fluctuations (noise) and they must reliably adjust their period to changing environmental conditions (entrainment). While noise seems to distort the ability of single oscillators to entrain, in populations of uncoupled oscillators noise allows population-level entrainment for a wider range of input amplitudes and periods. Here, we investigate how this effect depends on the noise intensity and the number of oscillators in the population. We have found that, if a population consists of a sufficient number of oscillators, increasing noise intensity leads to faster entrainment after a phase change of the input signal (jet lag) and increases sensitivity to low-amplitude input signals.

Coupling Between Subregional Oscillators Within the Suprachiasmatic Nucleus Determines Free-Running Period in the Rat

Rats housed in a 22-h light-dark cycle (11:11, T22) exhibit 2 distinct circadian locomotor activity (LMA) bouts simultaneously: one is entrained to the LD cycle and a second dissociated bout maintains a period greater than 24 h. These 2 activity bouts are associated with independent clock gene oscillations in the ventrolateral (vl-) and dorsomedial (dm-) suprachiasmatic nucleus (SCN), respectively. Previous results in our laboratory have shown that the vl- and dm-SCN oscillators are weakly coupled under T22 and that the period of the dissociated bout depends on coupling between the 2 subdivisions. Here, we sought to study the behavior of the T22 SCN pacemaker upon release into free-running conditions and compare it to the behavior of the system upon release from typical 24-h (12:12, T24) entrainment. T22-desynchronized rats or T24-entrained rats were released into constant darkness (DD). Activity rhythms in T22 rats rapidly resynchronized upon release into DD, and the free-running period (FRP) of the fused rhythm was longer than the FRP of T24 rats. We then asked whether the in vivo period changes were also present in the ex vivo SCN. Per1-luc rats were desynchronized in T22 for assessment of SCN Per1-luc ex vivo. Similar to behavioral FRP, the period of ex vivo SCN explanted from T22 rats was longer than that for T24 animals. Mathematical models supported the observed behavior of the dual oscillator system as the result of mutual coupling between the vl- and dm-SCN oscillators. This bidirectionally coupled model predicted both the FRP of the T22 system and its phase-shifting response to light. Together, these data support a model of pacemaker organization in which a light-sensitive vl-SCN oscillator is mutually coupled with a light-insensitive dm-SCN oscillator, and together they determine the period of the coupled system as a whole and its response to light pulses.

A spatio-temporal model to reveal oscillator phenotypes in molecular clocks: Parameter estimation elucidates circadian gene transcription dynamics in single-cells

We propose a stochastic distributed delay model together with a Markov random field prior and a measurement model for bioluminescence-reporting to analyse spatio-temporal gene expression in intact networks of cells. The model describes the oscillating time evolution of molecular mRNA counts through a negative transcriptional-translational feedback loop encoded in a chemical Langevin equation with a probabilistic delay distribution. The model is extended spatially by means of a multiplicative random effects model with a first order Markov random field prior distribution. Our methodology effectively separates intrinsic molecular noise, measurement noise, and extrinsic noise and phenotypic variation driving cell heterogeneity, while being amenable to parameter identification and inference. Based on the single-cell model we propose a novel computational stability analysis that allows us to infer two key characteristics, namely the robustness of the oscillations, i.e. whether the reaction network exhibits sustained or damped oscillations, and the profile of the regulation, i.e. whether the inhibition occurs over time in a more distributed versus a more direct manner, which affects the cells' ability to phase-shift to new schedules. We show how insight into the spatio-temporal characteristics of the circadian feedback loop in the suprachiasmatic nucleus (SCN) can be gained by applying the methodology to bioluminescence-reported expression of the circadian core clock gene Cry1 across mouse SCN tissue. We find that while (almost) all SCN neurons exhibit robust cell-autonomous oscillations, the parameters that are associated with the regulatory transcription profile give rise to a spatial division of the tissue between the central region whose oscillations are resilient to perturbation in the sense that they maintain a high degree of synchronicity, and the dorsal region which appears to phase shift in a more diversified way as a response to large perturbations and thus could be more amenable to entrainment.

Delays are Self-enhancing: An Explanation of the East-West Asymmetry in Recovery from Jetlag

There is evidence in mammals that recovering from jetlag after westward travel is faster than after eastward travel. To understand why, mathematical models have been used, along with theories of entrainment rooted in experimental evidence. The most complete understanding relies on detailed mathematical modeling, so it is helpful to develop an intuition about why there is an east-west asymmetry. One such intuition is that humans have long periods and therefore recover better when they can delay. Although this is part of the reason, it does not explain why short-period mice also recover from westward travel faster. Our goal is to provide a simple intuition consistent with detailed mathematical theories, but which does not require mathematical expertise to follow. Here, we present the intuition that westward travel is easier to recover from because of a simple principle: delays are self-enhancing.

A Two-Step Model of Human Entrainment: A Quantitative Study of Circadian Period and Phase of Entrainment

One of the essential characteristics of an authentic circadian clock is that the free-running period sustains an approximately 24-hour cycle. When organisms are exposed to an external stimulus, the endogenous oscillators synchronize to the cycling environment signal in a process known as entrainment. These environmental cues perform an important role in resetting the phase and period of the circadian clock. A "generalized assumption" states that when an organism has a short period, it will experience a phase advance, while an organism with a long period experiences a phase delay. Despite widespread use, this positive relationship relating period to the phase of entrainment does not describe all known experimental data. We developed a two-step entrainment model to explain a broader range of results as well as provide more quantitative analysis. We prove existence and stability of periodic orbits and given analytical solutions of the range of entrainment, fit the phase trajectory over the entire entrainment process to data from a published study for 12 subjects in extended day cycles, i.e., longer than 24 h. Our simulations closely replicated the phase data and predicted correctly the phase of entrainment. We investigate the factors related to the rate of entrainment (ROE) and present the three-dimensional parameter spaces, illustrating the various behaviors of the phase of entrainment and ROE. Our findings can be applied to diagnostics and treatments for patients with sleep disorders caused by shift work or jet lag.

A functional context for heterogeneity of the circadian clock in cells

Characterization of circadian systems at the organism level—a top-down approach—has led to definition of unifying properties, a hallmark of the science of chronobiology. The next challenge is to use a bottom-up approach to show how the molecular workings of the cellular circadian clock work as building blocks of those properties. We review new studies, including a recently published PLOS Biology paper by Nikhil and colleagues, that show how programmed but also stochastic generation of variation in cellular circadian period explain important adaptive features of entrained circadian phase.

A recent PLOS Biology paper shows that clonal cell populations are themselves a collection heterogeneous cellular circadian clocks; this Primer explores the implications, proposing that the phase of entrainment of biological clocks (to time of day or to season) is granular, built from the contributions of individual cells.