Biological Oscillators Research Articles

An Unstable Singularity Underlies Stochastic Phasing of the Circadian Clock in Individual Cyanobacterial Cells

The endogenous circadian clock synchronizes with environmental time by appropriately resetting its phase in response to external cues. Of note, some resetting stimuli induce attenuated oscillations of clock output, which has been observed at the population-level in several organisms and in studies of individual humans. To investigate what is happening in individual cellular clocks, we studied the unicellular cyanobacterium S.elongatus. By measuring its phase-resetting responses to temperature changes, we found that population-level arrhythmicity occurs when certain perturbations cause stochastic phases of oscillations in individual cells. Combining modeling with experiments, we related stochastic phasing to the dynamical structure of the cyanobacterial clock as an oscillator and explored the physiological relevance of the oscillator structure for accurately timed rhythmicity in changing environmental conditions. Our findings and approach can be applied to other biological oscillators.

The Trade-Off Mechanism in Mammalian Circadian Clock Model with Two Time Delays

Circadian clock is an autonomous oscillator which orchestrates the daily rhythms of physiology and behaviors. This study is devoted to explore how a positive feedback loop affects the dynamics of mammalian circadian clock. We simplify an experimentally validated mathematical model in our previous work, to a nonlinear differential equation with two time delays. This simplified mathematical model incorporates the pacemaker of mammalian circadian clock, a negative primary feedback loop, and a critical positive auxiliary feedback loop, [Formula: see text] loop. We perform analytical studies of the system. Delay-dependent conditions for the asymptotic stability of the nontrivial positive steady state of the model are investigated. We also prove the existence of Hopf bifurcation, which leads to self-sustained oscillation of mammalian circadian clock. Our theoretical analyses show that the oscillatory regime is reduced upon the participation of the delayed positive auxiliary loop. However, further simulations reveal that the auxiliary loop can enable the circadian clock gain widely adjustable amplitudes and robust period. Thus, the positive auxiliary feedback loop may provide a trade-off mechanism, to use the small loss in the robustness of oscillation in exchange for adaptable flexibility in mammalian circadian clock. The results obtained from the model may gain new insights into the dynamics of biological oscillators with interlocked feedback loops.

Long-time analytic approximation of large stochastic oscillators: Simulation, analysis and inference.

In order to analyse large complex stochastic dynamical models such as those studied in systems biology there is currently a great need for both analytical tools and also algorithms for accurate and fast simulation and estimation. We present a new stochastic approximation of biological oscillators that addresses these needs. Our method, called phase-corrected LNA (pcLNA) overcomes the main limitations of the standard Linear Noise Approximation (LNA) to remain uniformly accurate for long times, still maintaining the speed and analytically tractability of the LNA. As part of this, we develop analytical expressions for key probability distributions and associated quantities, such as the Fisher Information Matrix and Kullback-Leibler divergence and we introduce a new approach to system-global sensitivity analysis. We also present algorithms for statistical inference and for long-term simulation of oscillating systems that are shown to be as accurate but much faster than leaping algorithms and algorithms for integration of diffusion equations. Stochastic versions of published models of the circadian clock and NF-κB system are used to illustrate our results.

The cyanobacterial circadian clock follows midday in vivo and in vitro.

Circadian rhythms are biological oscillations that schedule daily changes in physiology. Outside the laboratory, circadian clocks do not generally free-run but are driven by daily cues whose timing varies with the seasons. The principles that determine how circadian clocks align to these external cycles are not well understood. Here, we report experimental platforms for driving the cyanobacterial circadian clock both in vivo and in vitro. We find that the phase of the circadian rhythm follows a simple scaling law in light-dark cycles, tracking midday across conditions with variable day length. The core biochemical oscillator comprised of the Kai proteins behaves similarly when driven by metabolic pulses in vitro, indicating that such dynamics are intrinsic to these proteins. We develop a general mathematical framework based on instantaneous transformation of the clock cycle by external cues, which successfully predicts clock behavior under many cycling environments.

Incoherent Inputs Enhance the Robustness of Biological Oscillators

Robust biological oscillators retain the critical ability to function in the presence of environmental perturbations. Although central architectures that support robust oscillations have been extensively studied, networks containing the same core vary drastically in their potential to oscillate, and it remains elusive what peripheral modifications to the core contribute to this functional variation. Here, we have generated a complete atlas of two- and three-node oscillators computationally, then systematically analyzed the association between network structure and robustness. We found that, while certain core topologies are essential for producing a robust oscillator, local structures can substantially modulate the robustness of oscillations. Notably, local nodes receiving incoherent or coherent inputs respectively promote or attenuate the overall network robustness in an additive manner. We validated these relationships in larger-scale networks reflective of real biological oscillators. Our findings provide an explanation for why auxiliary structures not required for oscillation are evolutionarily conserved and suggest simple ways to evolve or design robust oscillators.

A comprehensive reduced model of the mammalian cell cycle

The cellular division cycle is an essential process to ensure healthy tissue homeostasis, which can, due to its periodicity, be interpreted as a biological oscillator. This work focuses on identifying the main mechanisms underlying cell cycle rhythms in mammals and propose a mathematical model to describe them. The model is based on post-translational modifications of cyclin B-cdk1, also called mitosis promoting factor (MPF), known to be the essential protein of the mammalian cell cycle, as well as in its degradation by the APC:cdc20 complex. The final result is a two variable reduced model of the mammalian cell cycle that is able to reproduce oscillatory behaviors and properties consistent with observations, namely the period being tunable by an external input of growth factor. We calibrate and validate this model and study its behavior in a simple open-loop control configuration, showing that it can exhibit bistability and oscillations. The model presents an advantage to work with due to its low variable and parameter size.

Integrated Communication and Control for Collective Motion with Limited Communication

Decentralized collective motion coordination is a rapidly developing field, motivated by its extensive applications in mobile sensor networks, cooperative robotics, and unmanned aerial vehicles. In most existing results, the distributed controller for collective motion coordination is designed in the continuous-time domain in order to conform to the continuous kinetic dynamics of vehicle systems, and discretization of the continuous controller is used in implementation since the information can only be exchanged at discrete-time instants among these systems. However, this approach cannot guarantee the original design performance since discretization may harm or even destabilize the closed-loop system. Inspired by pulse-based synchronization among biological pulse-coupled oscillators, we propose an integrated communication and control approach for collective motion coordination which can circumvent discretization and guarantee the design performance in final implementation. Not only can heading control be achieved to coordinate vehicle headings, but also spacing control is achievable for the circular motion. Numerical simulations are provided to confirm the theoretical results.

Commentary: 2016 Epilepsia Basic Science Prize.

Commentary: 2016 Epilepsia Basic Science Prize.

Robustness of synthetic oscillators in growing and dividing cells

Synthetic biology sets out to implement new functions in cells, and to develop a deeper understanding of biological design principles. Elowitz and Leibler [Nature (London) 403, 335 (2000)NATUAS0028-083610.1038/35002125] showed that by rational design of the reaction network, and using existing biological components, they could create a network that exhibits periodic gene expression, dubbed the repressilator. More recently, Stricker etal. [Nature(London) 456, 516 (2008)NATUAS0028-083610.1038/nature07389] presented another synthetic oscillator, called the dual-feedback oscillator, which is more stable. Detailed studies have been carried out to determine how the stability of these oscillators is affected by the intrinsic noise of the interactions between the components and the stochastic expression of their genes. However, as all biological oscillators reside in growing and dividing cells, an important question is how these oscillators are perturbed by the cell cycle. In previous work we showed that the periodic doubling of the gene copy numbers due to DNA replication can couple not only natural, circadian oscillators to the cell cycle [Paijmans et al., Proc. Natl. Acad. Sci. (USA) 113, 4063 (2016)PNASA60027-842410.1073/pnas.1507291113], but also these synthetic oscillators. Here we expand this study. We find that the strength of the locking between oscillators depends not only on the positions of the genes on the chromosome, but also on the noise in the timing of gene replication: noise tends to weaken the coupling. Yet, even in the limit of high levels of noise in the replication times of the genes, both synthetic oscillators show clear signatures of locking to the cell cycle. This work enhances our understanding of the design of robust biological oscillators inside growing and diving cells.

A new link between plant metabolism and circadian rhythms?

A new link between plant metabolism and circadian rhythms?

Entrainment Control of Phase Dynamics

First order phase reduced model is a good approximation of the dynamics of forced nonlinear oscillators near its limit cycle. The phase evolution is determined by the unforced frequency, the forcing term, and the phase response curve (PRC). Such models arise in biological oscillations such as in circadian rhythm, neural signaling, heart beat, etc. This technical note focuses on the phase regulation of the circadian rhythm using light intensity as the input. Though the model is simple, the circle topology of the state space needs to be carefully addressed. The most common entrainment method is to use a periodic input, such as in our daily light-dark cycle. We obtain the complete stable entrainment condition based on the entraiment input and the PRC. Motivated by the jet-lag problem, we also consider the minimum time entrainment control to achieve a specified phase shift. Application of the Pontryagin Minimum Principle leads to an efficient solution strategy for the optimal control, without solving the two-point boundary value problem. The optimal control may be further represented as a feedback control law based on the current and desired phases. Our analysis allows the answer to questions such as: When traveling from New York to Paris, is it faster to use light to shift the phase forward by 6 hours or delay the phase by 18 hours? The answer is somewhat counter-intuitive - delaying by 18 hours requires less time. The general answer depends on the light intensity level and the shape of the PRC. PRCs for human and Drosophila from the literature are used to illustrate the results.

The Role of Mammalian Glial Cells in Circadian Rhythm Regulation.

Circadian rhythms are biological oscillations with a period of about 24 hours. These rhythms are maintained by an innate genetically determined time-keeping system called the circadian clock. A large number of the proteins involved in the regulation of this clock are transcription factors controlling rhythmic transcription of so-called clock-controlled genes, which participate in a plethora of physiological functions in the organism. In the brain, several areas, besides the suprachiasmatic nucleus, harbor functional clocks characterized by a well-defined time pattern of clock gene expression. This expression rhythm is not restricted to neurons but is also present in glia, suggesting that these cells are involved in circadian rhythmicity. However, only certain glial cells fulfill the criteria to be called glial clocks, namely, to display molecular oscillators based on the canonical clock protein PERIOD, which depends on the suprachiasmatic nucleus for their synchronization. In this contribution, we summarize the current information about activity of the clock genes in glial cells, their potential role as oscillators as well as clinical implications.

Efficient Computation of Multiscale Entropy over Short Biomedical Time Series Based on Linear State-Space Models

The most common approach to assess the dynamical complexity of a time series across multiple temporal scales makes use of the multiscale entropy (MSE) and refined MSE (RMSE) measures. In spite of their popularity, MSE and RMSE lack an analytical framework allowing their calculation for known dynamic processes and cannot be reliably computed over short time series. To overcome these limitations, we propose a method to assess RMSE for autoregressive (AR) stochastic processes. The method makes use of linear state-space (SS) models to provide the multiscale parametric representation of an AR process observed at different time scales and exploits the SS parameters to quantify analytically the complexity of the process. The resulting linear MSE (LMSE) measure is first tested in simulations, both theoretically to relate the multiscale complexity of AR processes to their dynamical properties and over short process realizations to assess its computational reliability in comparison with RMSE. Then, it is applied to the time series of heart period, arterial pressure, and respiration measured for healthy subjects monitored in resting conditions and during physiological stress. This application to short-term cardiovascular variability documents that LMSE can describe better than RMSE the activity of physiological mechanisms producing biological oscillations at different temporal scales.

Design Principles of Biological Oscillators through Optimization: Forward and Reverse Analysis.

From cyanobacteria to human, sustained oscillations coordinate important biological functions. Although much has been learned concerning the sophisticated molecular mechanisms underlying biological oscillators, design principles linking structure and functional behavior are not yet fully understood. Here we explore design principles of biological oscillators from a multiobjective optimization perspective, taking into account the trade-offs between conflicting performance goals or demands. We develop a comprehensive tool for automated design of oscillators, based on multicriteria global optimization that allows two modes: (i) the automatic design (forward problem) and (ii) the inference of design principles (reverse analysis problem). From the perspective of synthetic biology, the forward mode allows the solution of design problems that mimic some of the desirable properties appearing in natural oscillators. The reverse analysis mode facilitates a systematic exploration of the design space based on Pareto optimality concepts. The method is illustrated with two case studies: the automatic design of synthetic oscillators from a library of biological parts, and the exploration of design principles in 3-gene oscillatory systems.

Viewpoint oscillation improves the perception of distance travelled based on optic flow.

When static observers are presented with a visual simulation of forward self-motion, they generally misestimate distance travelled relative to a previously seen distant target: It has been suggested that this finding can be accounted for by a "leaky path integration" model. In the present study, using a similar experimental procedure, this result was confirmed. It was also established that combining the translational optical flow with simulated head oscillations (similar to those during natural walking) improved the subjects' perception of the distance travelled in comparison with a purely translational flow. This improvement may be attributable to the fact that an optic flow pattern resembling that associated with walking enhances the path integration process. In a subsequent experiment, we investigated whether it was the biological or the rhythmical characteristics of the simulation that enhanced the subjects' estimates of the distance travelled. The results obtained confirm that adding rhythmic components to the optic flow pattern improved the accuracy of subjects' perception of the distance travelled. However, no significant differences between biological and rhythmical oscillations were detected. These results relate to recent studies on the effects of smooth and jittering optic flows on vection onset and strength. One possible conclusion is that oscillations may increase the global retinal motion and thus improve the vection and path integration processes. Another possibility is that the nonmonotonous pattern of retinal motion induced by oscillatory inputs may maintain optimum sensitivity to the optic flow over time and thus improve the accuracy of subjects' perception of the distance travelled.

Elements of biological oscillations in time and space.

Oscillations in time and space are ubiquitous in nature and play critical roles in dynamic cellular processes. Although the molecular mechanisms underlying the generation of the dynamics are diverse, several distinct regulatory elements have been recognized as being critical in producing and modulating oscillatory dynamics. These include negative and positive feedback, time delay, nonlinearity in regulation, and random fluctuations ('noise'). Here we discuss the specific roles of these five elements in promoting or attenuating oscillatory dynamics, by drawing on insights from quantitative analyses of natural or synthetic biological networks.

Generalized synchrony of coupled stochastic processes with multiplicative noise.

We study the effect of multiplicative noise in dynamical flows arising from the coupling of stochastic processes with intrinsic noise. Situations wherein such systems arise naturally are in chemical or biological oscillators that are coupled to each other in a drive-response configuration. Above a coupling threshold we find that there is a strong correlation between the drive and the response: This is a stochastic analog of the phenomenon of generalised synchronization. Since the dynamical fluctuations are large when there is intrinsic noise, it is necessary to employ measures that are sensitive to correlations between the variables of drive and the response, the permutation entropy, or the mutual information in order to detect the transition to generalized synchrony in such systems.

Unmasking ultradian rhythms in gene expression.

Biological oscillations with an ultradian time scale of 1 to several hours include cycles in behavioral arousal, episodic glucocorticoid release, and gene expression. Ultradian rhythms are thought to have an extrinsic origin because of a perceived absence of ultradian rhythmicity in vitro and a lack of known molecular ultradian oscillators. We designed a novel, non–spectral-analysis method of separating ultradian from circadian components and applied it to a published gene expression dataset with an ultradian sampling resolution. Ultradian rhythms in mouse hepatocytes in vivo have been published, and we validated our approach using this control by confirming 175 of 323 ultradian genes identified in a prior study and found 862 additional ultradian genes. For the first time, we now report ultradian expression of >900 genes in vitro. Sixty genes exhibited ultradian transcriptional rhythmicity, both in vivo and in vitro, including 5 genes involved in the cell cycle. Within these 60 genes, we identified significant enrichment of specific DNA motifs in the 1000 bp proximal promotor, some of which associate with known transcriptional factors. These findings are in strong support of instrinsically driven ultradian rhythms and expose potential molecular mechanisms and functions underlying ultradian rhythms that remain unknown.—Van der Veen, D. R., Gerkema, M. P. Unmasking ultradian rhythms in gene expression.

Phase-Amplitude Coupling in Spontaneous Mouse Behavior.

The level of activity of many animals including humans rises and falls with a period of ~ 24 hours. The intrinsic biological oscillator that gives rise to this circadian oscillation is driven by a molecular feedback loop with an approximately 24 hour cycle period and is influenced by the environment, most notably the light:dark cycle. In addition to the circadian oscillations, behavior of many animals is influenced by multiple oscillations occurring at faster—ultradian—time scales. These ultradian oscillations are also thought to be driven by feedback loops. While many studies have focused on identifying such ultradian oscillations, less is known about how the ultradian behavioral oscillations interact with each other and with the circadian oscillation. Decoding the coupling among the various physiological oscillators may be important for understanding how they conspire together to regulate the normal activity levels, as well in disease states in which such rhythmic fluctuations in behavior may be disrupted. Here, we use a wavelet-based cross-frequency analysis to show that different oscillations identified in spontaneous mouse behavior are coupled such that the amplitude of oscillations occurring at higher frequencies are modulated by the phase of the slower oscillations. The patterns of these interactions are different among different individuals. Yet this variability is not random. Differences in the pattern of interactions are confined to a low dimensional subspace where different patterns of interactions form clusters. These clusters expose the differences among individuals—males and females are preferentially segregated into different clusters. These sex-specific features of spontaneous behavior were not apparent in the spectra. Thus, our methodology reveals novel aspects of the structure of spontaneous animal behavior that are not observable using conventional methodology.

Plant shoots exhibit synchronized oscillatory motions

ABSTRACTIn animals, the ability to move has evolved as an important means of protection from predators and for enhancing nutrient uptake. In the animal kingdom, an individual's movements may become coordinated with those of other individuals that belong to the same group, which leads, for example, to the beautiful collective patterns that are observed in flocks of birds and schools of fish or in animal migration. Land plants, however, are fixed to the ground, which limits their movement and, apparently, their interactions and collective behaviors. We show that emergent maize plants grown in a group exhibit synchronized oscillatory motions that may be in-phase or anti-phase. These oscillations occur in short bursts and appear when the leaves rupture from the coleoptile tip. The appearance of these oscillations indicates an abrupt increase in the plant growth rate, which may be associated with a sudden change in the energy uptake for photosynthesis. Our results suggest that plant shoots behave as a complex network of biological oscillators, interacting through biophysical links, e.g. chemical substances or electric signals.