Enhanced precision of circadian rhythm by output system.

Circadian rhythms are biological rhythms with a period of approximately 24 h that persist even under constant conditions without daily environmental cues. The molecular circadian clock machinery generates physiological rhythms, which can be transmitted into the downstream output system. Owing to the stochastic nature of the biochemical reactions and the extracellular environment, the oscillation period of circadian rhythms exhibited by individual organisms or cells is not constant on a daily basis with variations as high as 10%, as reflected by the coefficient of variation. Although the fluctuations in the circadian rhythm are measured through a reporter system such as bioluminescence or fluorescence, which is an example of output systems, experimentally confirming whether the fluctuations found in the reporter system are the same as those in the circadian clock is challenging. This study investigated a coupled system of a circadian clock and its output system numerically and analytically, and then compared the fluctuations in the oscillation period of the two systems. We found that the amount of fluctuations in the output system is smaller than that in the circadian clock, assuming the degradation rate of the molecules responsible for the output system is a typical value for protein degradation. The results indicate that the output system can improve the accuracy of the circadian rhythm without the need for any denoising processes.

Root PRR7 Improves the Accuracy of the Shoot Circadian Clock through Nutrient Transport.

The circadian clock allows plants to anticipate and adapt to periodic environmental changes. Organ- and tissue-specific properties of the circadian clock and shoot-to-root circadian signaling have been reported. While this long-distance signaling is thought to coordinate physiological functions across tissues, little is known about the feedback regulation of the root clock on the shoot clock in the hierarchical circadian network. Here, we show that the plant circadian clock conveys circadian information between shoots and roots through sucrose and K+. We also demonstrate that K+ transport from roots suppresses the variance of period length in shoots and then improves the accuracy of the shoot circadian clock. Sucrose measurements and qPCR showed that root sucrose accumulation was regulated by the circadian clock. Furthermore, root circadian clock genes, including PSEUDO-RESPONSE REGULATOR7 (PRR7), were regulated by sucrose, suggesting the involvement of sucrose from the shoot in the regulation of root clock gene expression. Therefore, we performed time-series measurements of xylem sap and micrografting experiments using prr7 mutants and showed that root PRR7 regulates K+ transport and suppresses variance of period length in the shoot. Our modeling analysis supports the idea that root-to-shoot signaling contributes to the precision of the shoot circadian clock. We performed micrografting experiments that illustrated how root PRR7 plays key roles in maintaining the accuracy of shoot circadian rhythms. We thus present a novel directional signaling pathway for circadian information from roots to shoots and propose that plants modulate physiological events in a timely manner through various timekeeping mechanisms.

Noninvasive inference methods for interaction and noise intensities of coupled oscillators using only spike time data.

Measurements of interaction intensity are generally achieved by observing responses to perturbations. In biological and chemical systems, external stimuli tend to deteriorate their inherent nature, and thus, it is necessary to develop noninvasive inference methods. In this paper, we propose theoretical methods to infer coupling strength and noise intensity simultaneously in two well-synchronized noisy oscillators through observations of spontaneously fluctuating events such as neural spikes. A phase oscillator model is applied to derive formulae relating each of the parameters to spike time statistics. Using these formulae, each parameter is inferred from a specific set of statistics. We verify these methods using the FitzHugh-Nagumo model as well as the phase model. Our methods do not require external perturbations and thus can be applied to various experimental systems.

Synchronization of three electrochemical oscillators: From local to global coupling.

We investigate the formation of synchronization patterns in an oscillatory nickel electrodissolution system in a network obtained by superimposing local and global coupling with three electrodes. We explored the behavior through numerical simulations using kinetic ordinary differential equations, Kuramoto type phase models, and experiments, in which the local to global coupling could be tuned by cross resistances between the three nickel wires. At intermediate coupling strength with predominant global coupling, two of the three oscillators, whose natural frequencies are closer, can synchronize. By adding even a relatively small amount of local coupling (about 9%-25%), a spatially organized partially synchronized state can occur where one of the two synchronized elements is in the center. A formula was derived for predicting the critical coupling strength at which full synchronization will occur independent of the permutation of the natural frequencies of the oscillators over the network. The formula correctly predicts the variation of the critical coupling strength as a function of the global coupling fraction, e.g., with local coupling the critical coupling strength is about twice than that required with global coupling. The results show the importance of the topology of the network on the synchronization properties in a simple three-oscillator setup and could provide guidelines for decrypting coupling topology from identification of synchronization patterns.

Expected Number of Fixed Points in Boolean Networks with Arbitrary Topology.

Boolean network models describe genetic, neural, and social dynamics in complex networks, where the dynamics depend generally on network topology. Fixed points in a genetic regulatory network are typically considered to correspond to cell types in an organism. We prove that the expected number of fixed points in a Boolean network, with Boolean functions drawn from probability distributions that are not required to be uniform or identical, is one, and is independent of network topology if only a feedback arc set satisfies a stochastic neutrality condition. We also demonstrate that the expected number is increased by the predominance of positive feedback in a cycle.

Precision of collective oscillations in complex dynamical systems with noise.

Two kinds of oscillation precision are investigated for complex oscillatory dynamical systems under action of noise. The many-cycle precision determined by the variance of the times needed for a large number of cycles is closely related to diffusion of the global oscillation phase and provides an invariant property of a system. The single-cycle precision given by the variance in durations of single cycles is sensitive to the choice of an output variable and output checkpoint; it can be improved by an appropriate selection of them. A general analysis of the precision properties based on the Floquet perturbation theory is performed and analytical predictions are verified in numerical simulations of a model oscillatory genetic network.

Necessary condition for frequency synchronization in network structures.

We present the necessary condition for complete frequency synchronization of phase-coupled oscillators in network structures. The surface area of a set of sites is defined as the number of links between the sites within the set and those outside the set. The necessary condition is that the surface area of any set of cN (0 < c < 1) oscillators in the N-oscillator system must exceed square root of N in the limit N --> infinity. We also provide the necessary condition for macroscopic frequency synchronization. Thus, we identify networks in which one or both of the above mentioned types of synchronization do not occur.