Coupling delay controls synchronized oscillation in the segmentation clock.

Individual cellular activities fluctuate but are constantly coordinated at the population level via cell-cell coupling. A notable example is the somite segmentation clock, in which the expression of clock genes (such as Hes7) oscillates in synchrony between the cells that comprise the presomitic mesoderm (PSM)1,2. This synchronization depends on the Notch signalling pathway; inhibiting this pathway desynchronizes oscillations, leading to somite fusion3-7. However, how Notch signalling regulates the synchronicity of HES7 oscillations is unknown. Here we establish a live-imaging system using a new fluorescent reporter (Achilles), which we fuse with HES7 to monitor synchronous oscillations in HES7 expression in the mouse PSM at a single-cell resolution. Wild-type cells can rapidly correct for phase fluctuations in HES7 oscillations, whereas the absence of the Notch modulator gene lunatic fringe (Lfng) leads to a loss of synchrony between PSM cells. Furthermore, HES7 oscillations are severely dampened in individual cells of Lfng-null PSM. However, when Lfng-null PSM cells were completely dissociated, the amplitude and periodicity of HES7 oscillations were almost normal, which suggests that LFNG is involved mostly in cell-cell coupling. Mixed cultures of control and Lfng-null PSM cells, and an optogenetic Notch signalling reporter assay, revealed that LFNG delays the signal-sending process of intercellular Notch signalling transmission. These results-together with mathematical modelling-raised the possibility that Lfng-null PSM cells shorten the coupling delay, thereby approaching a condition known as the oscillation or amplitude death of coupled oscillators8. Indeed, a small compound that lengthens the coupling delay partially rescues the amplitude and synchrony of HES7 oscillations in Lfng-null PSM cells. Our study reveals a delay control mechanism of the oscillatory networks involved in somite segmentation, and indicates that intercellular coupling with the correct delay is essential for synchronized oscillation.

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.

Optogenetic perturbation and bioluminescence imaging to analyze cell-to-cell transfer of oscillatory information.

Cells communicate with each other to coordinate their gene activities at the population level through signaling pathways. It has been shown that many gene activities are oscillatory and that the frequency and phase of oscillatory gene expression encode various types of information. However, whether or how such oscillatory information is transmitted from cell to cell remains unknown. Here, we developed an integrated approach that combines optogenetic perturbations and single-cell bioluminescence imaging to visualize and reconstitute synchronized oscillatory gene expression in signal-sending and signal-receiving processes. We found that intracellular and intercellular periodic inputs of Notch signaling entrain intrinsic oscillations by frequency tuning and phase shifting at the single-cell level. In this way, the oscillation dynamics are transmitted through Notch signaling, thereby synchronizing the population of oscillators. Thus, this approach enabled us to control and monitor dynamic cell-to-cell transfer of oscillatory information to coordinate gene expression patterns at the population level.

Oscillatory control of Delta-like1 in cell interactions regulates dynamic gene expression and tissue morphogenesis.

Notch signaling regulates tissue morphogenesis through cell-cell interactions. The Notch effectors Hes1 and Hes7 are expressed in an oscillatory manner and regulate developmental processes such as neurogenesis and somitogenesis, respectively. Expression of the mRNA for the mouse Notch ligand Delta-like1 (Dll1) is also oscillatory. However, the dynamics of Dll1 protein expression are controversial, and their functional significance is unknown. Here, we developed a live-imaging system and found that Dll1 protein expression oscillated in neural progenitors and presomitic mesoderm cells. Notably, when Dll1 expression was accelerated or delayed by shortening or elongating the Dll1 gene, Dll1 oscillations became severely dampened or quenched at intermediate levels, as modeled mathematically. Under this condition, Hes1 and Hes7 oscillations were also dampened. In the presomitic mesoderm, steady Dll1 expression led to severe fusion of somites and their derivatives, such as vertebrae and ribs. In the developing brain, steady Dll1 expression inhibited proliferation of neural progenitors and accelerated neurogenesis, whereas optogenetic induction of Dll1 oscillation efficiently maintained neural progenitors. These results indicate that the appropriate timing of Dll1 expression is critical for the oscillatory networks and suggest the functional significance of oscillatory cell-cell interactions in tissue morphogenesis.

Synchronization and stability analysis of an exponentially diverging solution in a mathematical model of asymmetrically interacting agents.

This study deals with an existing mathematical model of asymmetrically interacting agents. We analyze the following two previously unfocused features of the model: (i) synchronization of growth rates and (ii) initial value dependence of damped oscillation. By applying the techniques of variable transformation and timescale separation, we perform the stability analysis of a diverging solution. We find that (i) all growth rates synchronize to the same value that is as small as the smallest growth rate and (ii) oscillatory dynamics appear if the initial value of the slowest-growing agent is sufficiently small. Furthermore, our analytical method proposes a way to apply stability analysis to an exponentially diverging solution, which we believe is also a contribution of this study. Although the employed model is originally proposed as a model of infectious disease, we do not discuss its biological relevance but merely focus on the technical aspects.

Phase and frequency linear response theory for hyperbolic chaotic oscillators.

We formulate a linear phase and frequency response theory for hyperbolic flows, which generalizes phase response theory for autonomous limit cycle oscillators to hyperbolic chaotic dynamics. The theory is based on a shadowing conjecture, stating the existence of a perturbed trajectory shadowing every unperturbed trajectory on the system attractor for any small enough perturbation of arbitrary duration and a corresponding unique time isomorphism, which we identify as phase such that phase shifts between the unperturbed trajectory and its perturbed shadow are well defined. The phase sensitivity function is the solution of an adjoint linear equation and can be used to estimate the average change of phase velocity to small time dependent or independent perturbations. These changes in frequency are experimentally accessible, giving a convenient way to define and measure phase response curves for chaotic oscillators. The shadowing trajectory and the phase can be constructed explicitly in the tangent space of an unperturbed trajectory using co-variant Lyapunov vectors. It can also be used to identify the limits of the regime of linear response.

A Liposomal Gemcitabine, FF-10832, Improves Plasma Stability, Tumor Targeting, and Antitumor Efficacy of Gemcitabine in Pancreatic Cancer Xenograft Models.

PURPOSE: The clinical application of gemcitabine (GEM) is limited by its pharmacokinetic properties. The aim of this study was to characterize the stability in circulating plasma, tumor targeting, and payload release of liposome-encapsulated GEM, FF-10832. METHODS: Antitumor activity was assessed in xenograft mouse models of human pancreatic cancer. The pharmacokinetics of GEM and its active metabolite dFdCTP were also evaluated. RESULTS: In mice with Capan-1 tumors, the dose-normalized areas under the curve (AUCs) after FF-10832 administration in plasma and tumor were 672 and 1047 times higher, respectively, than after using unencapsulated GEM. The tumor-to-bone marrow AUC ratio of dFdCTP was approximately eight times higher after FF-10832 administration than after GEM administration. These results indicated that liposomal encapsulation produced long-term stability in circulating plasma and tumor-selective targeting of GEM. In mice with Capan-1, SUIT-2, and BxPC-3 tumors, FF-10832 had better antitumor activity and tolerability than GEM. Internalization of FF-10832 in tumor-associated macrophages (TAMs) was revealed by flow cytometry and confocal laser scanning microscopy, and GEM was efficiently released from isolated macrophages of mice treated with FF-10832. These results suggest that TAMs are one of the potential reservoirs of GEM in tumors. CONCLUSION: This study found that FF-10832 had favorable pharmacokinetic properties. The liposomal formulation was more effective and tolerable than unencapsulated GEM in mouse xenograft tumor models. Hence, FF-10832 is a promising candidate for the treatment of pancreatic cancer.

Low temperature nullifies the circadian clock in cyanobacteria through Hopf bifurcation.

Cold temperatures lead to nullification of circadian rhythms in many organisms. Two typical scenarios explain the disappearance of rhythmicity: the first is oscillation death, which is the transition from self-sustained oscillation to damped oscillation that occurs at a critical temperature. The second scenario is oscillation arrest, in which oscillation terminates at a certain phase. In the field of nonlinear dynamics, these mechanisms are called the Hopf bifurcation and the saddle-node on an invariant circle bifurcation, respectively. Although these mechanisms lead to distinct dynamical properties near the critical temperature, it is unclear to which scenario the circadian clock belongs. Here we reduced the temperature to dampen the reconstituted circadian rhythm of phosphorylation of the recombinant cyanobacterial clock protein KaiC. The data led us to conclude that Hopf bifurcation occurred at ∼19 °C. Below this critical temperature, the self-sustained rhythms of KaiC phosphorylation transformed to damped oscillations, which are predicted by the Hopf bifurcation theory. Moreover, we detected resonant oscillations below the critical temperature when temperature was periodically varied, which was reproduced by numerical simulations. Our findings suggest that the transition to a damped oscillation through Hopf bifurcation contributes to maintaining the circadian rhythm of cyanobacteria through resonance at cold temperatures.

Noise stability of synchronization and optimal network structures.

We provide a theoretical framework for quantifying the expected level of synchronization in a network of noisy oscillators. Through linearization around the synchronized state, we derive the following quantities as functions of the eigenvalues and eigenfunctions of the network Laplacian using a standard technique for dealing with multivariate Ornstein-Uhlenbeck processes: the magnitude of the fluctuations around a synchronized state and the disturbance coefficients αi that represent how strongly node i disturbs the synchronization. With this approach, we can quantify the effect of individual nodes and links on synchronization. Our theory can thus be utilized to find the optimal network structure for accomplishing the best synchronization. Furthermore, when the noise levels of the oscillators are heterogeneous, we can also find optimal oscillator configurations, i.e., where to place oscillators in a given network depending on their noise levels. We apply our theory to several example networks to elucidate optimal network structures and oscillator configurations.

The effects of coating culture dishes with collagen on fibroblast cell shape and swirling pattern formation.

Motile human-skin fibroblasts form macroscopic swirling patterns when grown to confluence on a culture dish. In this paper, we investigate the effect of coating the culture-dish surface with collagen on the resulting pattern, using human-skin fibroblast NB1RGB cells as the model system. The presence of the collagen coating is expected to enhance the adherence of the fibroblasts to the dish surface, and thereby also enhance the traction that the fibroblasts have as they move. We find that, contrary to our initial expectation, the coating does not significantly affect the motility of the fibroblasts. Their eventual number density at confluence is also unaffected. However, the coherence length of cell orientation in the swirling pattern is diminished. We also find that the fibroblasts cultured in collagen-coated dishes are rounder in shape and shorter in perimeter, compared with those cultured in uncoated polystyrene or glass culture dishes. We hypothesise that the rounder cell-shape which weakens the cell-cell nematic contact interaction is responsible for the change in coherence length. A simple mathematical model of the migrating fibroblasts is constructed, which demonstrates that constant motility with weaker nematic interaction strength does indeed lead to the shortening of the coherence length.

Partial synchronization of relaxation oscillators with repulsive coupling in autocatalytic integrate-and-fire model and electrochemical experiments.

Experiments and supporting theoretical analysis are presented to describe the synchronization patterns that can be observed with a population of globally coupled electrochemical oscillators close to a homoclinic, saddle-loop bifurcation, where the coupling is repulsive in the electrode potential. While attractive coupling generates phase clusters and desynchronized states, repulsive coupling results in synchronized oscillations. The experiments are interpreted with a phenomenological model that captures the waveform of the oscillations (exponential increase) followed by a refractory period. The globally coupled autocatalytic integrate-and-fire model predicts the development of partially synchronized states that occur through attracting heteroclinic cycles between out-of-phase two-cluster states. Similar behavior can be expected in many other systems where the oscillations occur close to a saddle-loop bifurcation, e.g., with Morris-Lecar neurons.

Delayed feedback induced multirhythmicity in the oscillatory electrodissolution of copper.

Occurrence of bi- and trirhythmicities (coexistence of two or three stable limit cycles, respectively, with distinctly different periods) has been studied experimentally by applying delayed feedback control to the copper-phosphoric acid electrochemical system oscillating close to a Hopf bifurcation point under potentiostatic condition. The oscillating electrode potential is delayed by τ and the difference between the present and delayed values is fed back to the circuit potential with a feedback gain K. The experiments were performed by determining the period of current oscillations T as a function of (both increasing and decreasing) τ at several fixed values of K. With small delay times, the period exhibits a sinusoidal type dependence on τ. However, with relatively large delays (typically τ ≫ T) for each feedback gain K, there exists a critical delay τcrit above which birhythmicity emerges. The experiments show that for weak feedback, Kτcrit is approximately constant. At very large delays, the dynamics becomes even more complex, and trirhythmicity could be observed. Results of numerical simulations based on a general kinetic model for metal electrodissolution were consistent with the experimental observations. The experimental and numerical results are also interpreted by using a phase model; the model parameters can be obtained from experimental data measured at small delay times. Analytical solutions to the phase model quantitatively predict the parameter regions for the appearance of birhythmicity in the experiments, and explain the almost constant value of Kτcrit for weak feedback.

Weakly nonlinear analysis on synchronization and oscillation quenching of coupled mechanical oscillators.

Various oscillatory phenomena occur in the world. Because some are associated with abnormal states (e.g. epilepsy), it is important to establish ways to terminate oscillations by external stimuli. However, despite the prior development of techniques for stabilizing unstable oscillations, relatively few studies address the transition from oscillatory to resting state in nonlinear dynamics. This study mainly analyzes the oscillation-quenching of metronomes on a platform as an example of such transitions. To facilitate the analysis, we describe the impulsive force (escapement mechanism) of a metronome by a fifth-order polynomial. By performing both averaging approximation and numerical simulation, we obtain a phase diagram for synchronization and oscillation quenching. We find that quenching occurs when the feedback to the oscillator increases, which will help explore the general principle regarding the state transition from oscillatory to resting state. We also numerically investigate the bifurcation of out-of-phase synchronization and beat-like solution. Despite the simplicity, our model successfully reproduces essential phenomena in interacting mechanical clocks, such as the bistability of in-phase and anti-phase synchrony and oscillation quenching occurring for a large mass ratio between the oscillator and the platform. We believe that our simple model will contribute to future analyses of other dynamics of mechanical clocks.

Diffusive coupling facilitates and impedes noise-induced escape in interacting bistable elements.

Diverse complex systems often undergo sudden changes in their states, such as epileptic seizures, climate changes, and social uprisings. Such behavior has been modeled by noise-induced escape of bistable elements, which is the escape from an attracting state driven by a fluctuation in the system's state. We consider a system of interacting bistable elements and investigate the effect of diffusive coupling among elements on the process of noise-induced escape. We focus on the influence of the coupling strength over the escape time, which is the time it takes for noise-induced escape to occur. We performed numerical simulations and observed that weak coupling reduced the mean escape time, whereas strong coupling impeded escape. We argue that, although diffusive coupling both facilitates and impedes escape, the facilitating effect is dominant when coupling is weak. For weak coupling cases, we develop an approximate theory that can predict the mean and variance of escape times. In contrast, strong coupling reduces the effective noise intensity to impede escape. Our results suggest that diffusive coupling among multistable elements contributes to regulating the rate of transitions among attracting states.

Partial synchronization and community switching in phase-oscillator networks and its analysis based on a bidirectional, weighted chain of three oscillators.

Complex networks often possess communities defined based on network connectivity. When dynamics undergo in a network, one can also consider dynamical communities, i.e., a group of nodes displaying a similar dynamical process. We have investigated both analytically and numerically the development of a dynamical community structure, where the community is referred to as a group of nodes synchronized in frequency, in networks of phase oscillators. We first demonstrate that using a few example networks, the community structure changes when network connectivity or interaction strength is varied. In particular, we found that community switching, i.e., a portion of oscillators change the group to which they synchronize, occurs for a range of parameters. We then propose a three-oscillator model: a bidirectional, weighted chain of three Kuramoto phase oscillators, as a theoretical framework for understanding the community formation and its variation. Our analysis demonstrates that the model shows a variety of partially synchronized patterns: oscillators with similar natural frequencies tend to synchronize for weak coupling, while tightly connected oscillators tend to synchronize for strong coupling. We obtain approximate expressions for the critical coupling strengths by employing a perturbative approach in a weak coupling regime and a geometric approach in strong coupling regimes. Moreover, we elucidate the bifurcation types of transitions between different patterns. Our theory might be useful for understanding the development of partially synchronized patterns in a wider class of complex networks than community structured networks.

An extended Hilbert transform method for reconstructing the phase from an oscillatory signal.

Rhythmic activity is ubiquitous in biological systems from the cellular to organism level. Reconstructing the instantaneous phase is the first step in analyzing the essential mechanism leading to a synchronization state from the observed signals. A popular method of phase reconstruction is based on the Hilbert transform, which can only reconstruct the interpretable phase from a limited class of signals, e.g., narrow band signals. To address this issue, we propose an extended Hilbert transform method that accurately reconstructs the phase from various oscillatory signals. The proposed method is developed by analyzing the reconstruction error of the Hilbert transform method with the aid of Bedrosian's theorem. We validate the proposed method using synthetic data and show its systematically improved performance compared with the conventional Hilbert transform method with respect to accurately reconstructing the phase. Finally, we demonstrate that the proposed method is potentially useful for detecting the phase shift in an observed signal. The proposed method is expected to facilitate the study of synchronization phenomena from experimental data.

Structure of cell networks critically determines oscillation regularity.

Biological rhythms are generated by pacemaker organs, such as the heart pacemaker organ (the sinoatrial node) and the master clock of the circadian rhythms (the suprachiasmatic nucleus), which are composed of a network of autonomously oscillatory cells. Such biological rhythms have notable periodicity despite the internal and external noise present in each cell. Previous experimental studies indicate that the regularity of oscillatory dynamics is enhanced when noisy oscillators interact and become synchronized. This effect, called the collective enhancement of temporal precision, has been studied theoretically using particular assumptions. In this study, we propose a general theoretical framework that enables us to understand the dependence of temporal precision on network parameters including size, connectivity, and coupling intensity; this effect has been poorly understood to date. Our framework is based on a phase oscillator model that is applicable to general oscillator networks with any coupling mechanism if coupling and noise are sufficiently weak. In particular, we can manage general directed and weighted networks. We quantify the precision of the activity of a single cell and the mean activity of an arbitrary subset of cells. We find that, in general undirected networks, the standard deviation of cycle-to-cycle periods scales with the system size N as 1/N, but only up to a certain system size N(⁎) that depends on network parameters. Enhancement of temporal precision is ineffective when N>N(⁎). We provide an example in which temporal precision considerably improves with increasing N while the level of synchrony remains almost constant; temporal precision and synchrony are independent dynamical properties. We also reveal the advantage of long-range interactions among cells to temporal precision.

Accuracy of a one-dimensional reduction of dynamical systems on networks.

Resilience is an ability of a system with which the system can adjust its activity to maintain its functionality when it is perturbed. To study resilience of dynamics on networks, Gao et al. [Nature (London) 530, 307 (2016)0028-083610.1038/nature16948] proposed a theoretical framework to reduce dynamical systems on networks, which are high dimensional in general, to one-dimensional dynamical systems. The accuracy of this one-dimensional reduction relies on three approximations in addition to the assumption that the network has a negligible degree correlation. In the present study, we analyze the accuracy of the one-dimensional reduction assuming networks without degree correlation. We do so mainly through examining the validity of the individual assumptions underlying the method. Across five dynamical system models, we find that the accuracy of the one-dimensional reduction hinges on the spread of the equilibrium value of the state variable across the nodes in most cases. Specifically, the one-dimensional reduction tends to be accurate when the dispersion of the node's state is small. We also find that the correlation between the node's state and the node's degree, which is common for various dynamical systems on networks, is unrelated to the accuracy of the one-dimensional reduction.

A light-induced small G-protein gem limits the circadian clock phase-shift magnitude by inhibiting voltage-dependent calcium channels.

Calcium signaling is pivotal to the circadian clockwork in the suprachiasmatic nucleus (SCN), particularly in rhythm entrainment to environmental light-dark cycles. Here, we show that a small G-protein Gem, an endogenous inhibitor of high-voltage-activated voltage-dependent calcium channels (VDCCs), is rapidly induced by light in SCN neurons via the calcium (Ca2+)-mediated CREB/CRE transcriptional pathway. Gem attenuates light-induced calcium signaling through its interaction with VDCCs. The phase-shift magnitude of locomotor activity rhythms by light, at night, increases in Gem-deficient (Gem-/-) mice. Similarly, in SCN slices from Gem-/- mice, depolarizing stimuli induce larger phase shifts of clock gene transcription rhythms that are normalized by the application of an L-type VDCC blocker, nifedipine. Voltage-clamp recordings from SCN neurons reveal that Ca2+ currents through L-type channels increase in Gem-/- mice. Our findings suggest that transcriptionally activated Gem feeds back to suppress excessive light-evoked L-type VDCC activation, adjusting the light-induced phase-shift magnitude to an appropriate level in mammals.