When are cellular oscillators sufficient for sequential segmentation?

Sequential segmentation during embryogenesis involves the generation of a repeated pattern along the embryo, which is concurrently undergoing axial elongation by cell division. Most mathematical models of sequential segmentation involve inherent cellular oscillators, acting as a segmentation clock. The cellular oscillation is assumed to be governed by the cell's physiological age or by its interaction with an external morphogen gradient. Here, we address the issue of when cellular oscillators alone are sufficient for predicting segmentation, and when a morphogen gradient is required. The key to resolving this issue lies in how cells determine positional information in the model - this is directly related to the distribution of cell divisions responsible for axial elongation. Mathematical models demonstrate that if axial elongation occurs through cell divisions restricted to the posterior end of the unsegmented region, a cell can obtain its positional information from its physiological age, and therefore cellular oscillators will suffice. Alternatively, if axial elongation occurs through cell divisions distributed throughout the unsegmented region, then positional information can be obtained through another mechanism, such as a morphogen gradient. Two alternative ways to establish a morphogen gradient in tissue with distributed cell divisions are presented - one with diffusion and the other without diffusion. Our model produces segment polarity and a distribution of segment size from the anterior-to-posterior ends, as observed in some systems. Furthermore, the model predicts segment deletions when there is an interruption in cell division, just as seen in heat shock experiments, as well as the growth and final shrinkage of the presomitic mesoderm during somitogenesis.     

Control of interaction strength in a network of the true slime mold by a microfabricated structure

The plasmodium of the true slime mold, Physarum polycephalum, which shows various nonlinear oscillatory phenomena, for example, in its thickness, protoplasmic streaming and concentration of intracellular chemicals, can be regarded as a collective of nonlinear oscillators. The plasmodial oscillators are interconnected by microscale tubes whose dimensions can be closely related to the strength of interaction between the oscillators. Investigation of the collective behavior of the oscillators under the conditions in which the interaction strength can be systematically controlled gives significant information on the characteristics of the system. In this study, we proposed a living model system of a coupled oscillator system in the Physarum plasmodium. We patterned the geometry and dimensions of the microscale tube structure in the plasmodium by a microfabricated structure (microstructure). As the first step, we constructed a two-oscillator system for the plasmodium that has two wells (oscillator part) and a channel (coupling part). We investigated the oscillation behavior by monitoring the thickness oscillation of the plasmodium in the microstructure with various channel widths. It was found that the oscillation behavior of two oscillators dynamically changed depending on the channel width. Based on the results of measurements of the tube dimensions and the velocity of the protoplasmic streaming in the tube, we discuss how the channel width relates to the interaction strength of the coupled oscillator system.     

Entrainment of two coupled van der pol oscillators by an external oscillation

A system composed of two coupled internal oscillators and an external oscillation is studied as a model of biological control systems. The type of interaction between the internal oscillators is a mutual and dissipative one. Three macroscopic states of the internal oscillators are demonstrated in the absence of the external oscillation. Strict and loose entrainment regions of the internal oscillators by the external oscillation are shown in respect to the intensity of the mutual interaction, the intrinsic frequency difference of the internal oscillations, and the magnitude and frequency of the external oscillation. On the other hand, an idea of holonic system is introduced and the fundamental properties of the model as a holonic system are elucidated.     

An oscillator network exhibiting a long-lasting response to an external signal

This study proposes an oscillator network to model the long-lasting responses observed in neural circuits. The responses of the proposed network model are represented by the temporal synchronization of the oscillators. The response duration does not depend on the natural frequency of the oscillators, which allows the responses to last much longer than the oscillation period of the oscillators. We can control the response duration by tuning the connection strengths between the oscillators and the external signal that triggers the responses. It is possible to break and restart the responses regardless of the way in which the oscillators are connected.     

Modulation analysis of forced non-linear oscillators for biological modelling.

The synchronization of self-sustained oscillations by a forcing oscillation is of interest in a number of biological models. It has been considered for circadian rhythm modelling, heart-rate variability studies and forced breathing experiments. Outside the range of synchronization, conditions of almost-entrainment occur in which changes in amplitude and/or frequency are apparent. It is shown in this paper that such conditions can be analysed as modulation phenomena using the analytical method of harmonic balance. The degree of non-linearity in the self-sustained oscillation affects the nature of modulation, in that increasing distortion gives a trend towards frequency rather than amplitude modulation. The analytical results compare favourably with spectral analysis of simulated oscillators.     

Oscillation of the electric potential of frog skin under the effect of Li+: Experimental approach

Summary When a frog skin is used to separate two compartments, and lithium is added to the external medium, transmembrane electric potential oscillations frequently occur. When no external current is imposed, sustained oscillations, with a period of about 10 min, are maintained for several hours. An oscillation of the Na⁺ influx accompanies the electric oscillation, though the two oscillations are out of phase to a greater or less extent.Theophyllin promotes a significant decrease in the mean electric potential of the skin, but it does not affect very much the characteristics of the oscillation. Important factors influencing the oscillation are temperature, permeability of the external membrane to lithium, and potassium concentration in the internal medium. No correlation can be detected between oscilation characteristics and skin area. This suggests that the oscillation is of a local nature, possibly originating at the cellular level. Occurrence of macroscopic oscillations implies coupling between local oscillators. Coupling between two epithelia has been studied under diverse conditions. The coupling is of an electrical nature: by varying the value of the coupling resistance, it is possible to control synchronization of the oscillations.     

Multiple redundant circadian oscillators within the isolated avian pineal gland

Summary The avian pineal gland contains a circadian pacemaker that oscillates in vitro. Using a flow-through culture system it is possible to measure melatonin production from very small subsections of an individual gland. We have used this technique to attempt to localize the oscillators in the pineal. Progressive tissue reduction did not affect the rhythmicity of cultured pineals. Multiple pieces (up to eight) from a single pineal all were capable of circadian oscillation — establishing directly that a pineal gland contains at least eight oscillators. All pineal pieces were responsive to light, and single light pulses shifted the phase of the melatonin rhythm. Because pieces equivalent to less than one per cent of the whole gland were rhythmic and because the capacity for oscillation was distributed throughout the gland, an individual pineal appears to be composed of a population of circadian oscillators.     

动态神经网络中的同步振荡

目前有一种假设认为同一视觉对象是由一群神经元的同步振荡活动来表征的。这一神经元发放活动的时间特性,是解决视觉信息处理中“结合问题（Ｂｉｎｄｉｎｇｐｒｏｂｌｅｍ）”的可能机制。本文用我们所提出的一种简化现实性神经网络模型［１］所构造的时滞非线性振子网络［２］,模拟生物神经网络的同步振荡活动。并考虑了振子各参数的设置与振荡活动的关系,以及网络振子间耦联对同步活动的影响．     

Sources of Variability in a Synthetic Gene Oscillator

Synthetic gene oscillators are small, engineered genetic circuits that produce periodic variations in target protein expression. Like other gene circuits, synthetic gene oscillators are noisy and exhibit fluctuations in amplitude and period. Understanding the origins of such variability is key to building predictive models that can guide the rational design of synthetic circuits. Here, we developed a method for determining the impact of different sources of noise in genetic oscillators by measuring the variability in oscillation amplitude and correlations between sister cells. We first used a combination of microfluidic devices and time-lapse fluorescence microscopy to track oscillations in cell lineages across many generations. We found that oscillation amplitude exhibited high cell-to-cell variability, while sister cells remained strongly correlated for many minutes after cell division. To understand how such variability arises, we constructed a computational model that identified the impact of various noise sources across the lineage of an initial cell. When each source of noise was appropriately tuned the model reproduced the experimentally observed amplitude variability and correlations, and accurately predicted outcomes under novel experimental conditions. Our combination of computational modeling and time-lapse data analysis provides a general way to examine the sources of variability in dynamic gene circuits.     

Analysis of the gait generation principle by a simulated quadruped model with a CPG incorporating vestibular modulation

This study aims to understand the principles of gait generation in a quadrupedal model. It is difficult to determine the essence of gait generation simply by observation of the movement of complicated animals composed of brains, nerves, muscles, etc. Therefore, we build a planar quadruped model with simplified nervous system and mechanisms, in order to observe its gaits under simulation. The model is equipped with a mathematical central pattern generator (CPG), consisting of four coupled neural oscillators, basically producing a trot pattern. The model also contains sensory feedback to the CPG, measuring the body tilt (vestibular modulation). This spontaneously gives rise to an unprogrammed lateral walk at low speeds, a transverse gallop while running, in addition to trotting at a medium speed. This is because the body oscillation exhibits a double peak per leg frequency at low speeds, no peak (little oscillation) at medium speeds, and a single peak while running. The body oscillation autonomously adjusts the phase differences between the neural oscillators via the feedback. We assume that the oscillations of the four legs produced by the CPG and the body oscillation varying according to the current speed are synchronized along with the varied phase differences to keep balance during locomotion through postural adaptation via the vestibular modulation, resulting in each gait. We succeeded in determining a single simple principle that accounts for gait transition from walking to trotting to galloping, even without brain control, complicated leg mechanisms, or a flexible trunk.     

Calcium oscillations in a triplet of pancreatic acinar cells

We use a mathematical model of calcium dynamics in pancreatic acinar cells to investigate calcium oscillations in a ring of three coupled cells. A connected group of cells is modeled in two different ways: 1), as coupled point oscillators, each oscillator being described by a spatially homogeneous model; and 2), as spatially distributed cells coupled along their common boundaries by gap-junctional diffusion of inositol trisphosphate and/or calcium. We show that, although the point-oscillator model gives a reasonably accurate general picture, the behavior of the spatially distributed cells cannot always be predicted from the simpler analysis; spatially distributed diffusion and cell geometry both play important roles in determining behavior. In particular, oscillations in which two cells are in synchrony, with the third phase-locked but not synchronous, appears to be more dominant in the spatially distributed model than in the point-oscillator model. In both types of model, intercellular coupling leads to a variety of synchronous, phase-locked, or asynchronous behaviors. For some parameter values there are multiple, simultaneous stable types of oscillation. We predict 1), that intercellular calcium diffusion is necessary and sufficient to coordinate the responses in neighboring cells; 2), that the function of intercellular inositol trisphosphate diffusion is to smooth out any concentration differences between the cells, thus making it easier for the diffusion of calcium to synchronize the oscillations; 3), that groups of coupled cells will tend to respond in a clumped manner, with groups of synchronized cells, rather than with regular phase-locked periodic intercellular waves; and 4), that enzyme secretion is maximized by the presence of a pacemaker cell in each cluster which drives the other cells at a frequency greater than their intrinsic frequency.     

Molecular dynamics simulations of carbon nanotube oscillators deformed by encapsulated copper nanowires

Pure carbon nanotube (CNT) oscillators are compared to the corresponding CNT oscillators encapsulating copper nanowires (Cu@CNTs) by molecular dynamics simulations. The classical oscillation theory provides a fairly good estimate of the mass dependence of the operating frequency when the CNT surface is not deformed by the Cu nanowire. The structural deformations of the CNT induced by the encapsulated copper nanowire have a greater effect on the oscillation frequency than the mass of the copper nanowire. The excess forces of the Cu@CNT oscillator are slightly higher than those of the CNT oscillator and the excess van der Waals forces induced by the inter-wall interactions are 17 times higher than the excess forces induced by the Cu nanowire–CNT interactions.     

Effects of adenylates on the circadian interaction of KaiB with the KaiC complex in the reconstituted cyanobacterial Kai protein oscillator

Cyanobacteria are photosynthetic prokaryotes that possess circadian oscillators. Clock proteins, KaiA, KaiB, KaiC compose the central circadian oscillator, which can be reconstituted in vitro in the presence of ATP. KaiC has ATPase, autokinase, and autophosphatase enzymatic activities. These activities are modulated by protein–protein interactions among the Kai proteins. The interaction of KaiB with the KaiC complex shows a circadian rhythm in the reconstituted system. We previously developed a quantitative, real-time monitoring system for the dynamic behavior of the complex using fluorescence correlation spectroscopy. Here, we examined the effects of ATP and ADP on the rhythmic interaction of KaiB. We show that increased concentration of ATP or ADP shortened period length. Adding ADP to the Kai protein oscillation shifted its phase in a phase-dependent manner. These results provide insight into how circadian oscillation entrainment mechanism is linked to cellular metabolism.     

Cellular oscillators in animal segmentation

Kinetic modeling of developmental dynamics requires detailed knowledge about genetic and metabolic networks that underlie developmental processes. However, such knowledge is not available for a vast majority of developmental processes. Here, we present an coarse-grained, phenomenological model of periodic pattern formation in multicellular organisms based on cellular oscillators (CO) that can be applied to systems for which little or no molecular data is available. An oscillatory process within cells serves as a developmental clock whose period is tightly regulated by cell-autonomous and non-autonomous mechanisms. A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillators freezing into spatial order as the clocks slow down and stop at different times or phases in their cycles. When applied to vertebrate somitogenesis, the CO model can reproduce the dynamics of periodic gene expression patterns observed in the presomitic mesoderm. Different somite lengths can be generated by altering the period of the oscillation. There is evidence that a CO-type mechanism might also underlie segment formation in certain invertebrates, such as annelids and short germ insects. This suggests that the dynamical principles of sequential segmentation might be equivalent throughout the animal kingdom although most of the genes involved in segment determination differ between distant phyla.     

On partial contraction analysis for coupled nonlinear oscillators

We describe a simple yet general method to analyze networks of coupled identical nonlinear oscillators and study applications to fast synchronization, locomotion, and schooling. Specifically, we use nonlinear contraction theory to derive exact and global (rather than linearized) results on synchronization, antisynchronization, and oscillator death. The method can be applied to coupled networks of various structures and arbitrary size. For oscillators with positive definite diffusion coupling, it can be shown that synchronization always occurs globally for strong enough coupling strengths, and an explicit upper bound on the corresponding threshold can be computed through eigenvalue analysis. The discussion also extends to the case when network structure varies abruptly and asynchronously, as in flocks of oscillators or dynamic elements.     

A qualitative mathematical model of the ontogeny of a circadian rhythm in crayfish

Based on experimental work on the ontogeny of the electroretinogram circadian rhythm in crayfish, we present a mathematical model simulating changes in both frequency and amplitude of the electroretinogram oscillation during several developmental stages until shortly before the adult age. Simultaneously, we propose a hypothetical oscillation in the hormonal release whose frequency is imposed on the electroretinogram oscillation. The model consists of two coupled nonlinear oscillators in which a dynamical response is obtained mainly through an Andronov-Hopf bifurcation. Through the construction of the model, a biological hypothesis about the essential elements underlying the ERG circadian rhythm and their interrelations is formulated and discussed.     

Design principles for robust oscillatory behavior

Oscillatory responses are ubiquitous in regulatory networks of living organisms, a fact that has led to extensive efforts to study and replicate the circuits involved. However, to date, design principles that underlie the robustness of natural oscillators are not completely known. Here we study a three-component enzymatic network model in order to determine the topological requirements for robust oscillation. First, by simulating every possible topological arrangement and varying their parameter values, we demonstrate that robust oscillators can be obtained by augmenting the number of both negative feedback loops and positive autoregulations while maintaining an appropriate balance of positive and negative interactions. We then identify network motifs, whose presence in more complex topologies is a necessary condition for obtaining oscillatory responses. Finally, we pinpoint a series of simple architectural patterns that progressively render more robust oscillators. Together, these findings can help in the design of more reliable synthetic biomolecular networks and may also have implications in the understanding of other oscillatory systems.

Electronic supplementary material

The online version of this article (doi:10.1007/s11693-015-9178-6) contains supplementary material, which is available to authorized users.     

Periodic activity of the cortical cytoskeleton of the lymphoblast: modelling by a reaction-diffusion system

The cytoplasm of cultured human lymphoblasts has a particular organization. A ring formed by actin and myosin delimits a region of the cell where an active membrane veil forms. Depolymerization of the microtubular cytoskeleton releases this ring, which then oscillates between the two poles of the cell. This periodic movement has been studied and described with great precision. We have used these experimental results to model the oscillation of the ring. We have constructed a system of reaction—diffusion equations designed to simulate the behaviour of the actin and have considered three variables representing, respectively, the proteins involved in actin nucléation, F-actin bound to the plasma membrane and free F-actin, pan of which is assumed to constitute the ring, together with myosin. There are two types of results. First, from a theoretical point of view, we have succeeded in simulating a perfect back-and-forth movement of a wave. Second, this model suggests that actin alone, by virtue of its intrinsic properties, can generate an oscillatory movement within the cell, without the need for other oscillators.     

Dendritic synchrony and transient dynamics in a coupled oscillator model of the dopaminergic neuron

Transient increases in spontaneous firing rate of mesencephalic dopaminergic neurons have been suggested to act as a reward prediction error signal. A mechanism previously proposed involves subthreshold calcium-dependent oscillations in all parts of the neuron. In that mechanism, the natural frequency of oscillation varies with diameter of cell processes, so there is a wide variation of natural frequencies on the cell, but strong voltage coupling enforces a single frequency of oscillation under resting conditions. In previous work, mathematical analysis of a simpler system of oscillators showed that the chain of oscillators could produce transient dynamics in which the frequency of the coupled system increased temporarily, as seen in a biophysical model of the dopaminergic neuron. The transient dynamics was shown to be consequence of a slow drift along an invariant subset of phase space, with rate of drift given by a Lyapunov function. In this paper, we show that the same mathematical structure exists for the full biophysical model, giving physiological meaning to the slow drift and the Lyapunov function, which is shown to describe differences in intracellular calcium concentration in different parts of the cell. The duration of transients was long, being comparable to the time constant of calcium disposition. These results indicate that brief changes in input to the dopaminergic neuron can produce long lasting firing rate transients whose form is determined by intrinsic cell properties.     

Unidirectional waves on rings: Models for chiral preference of circumnutating plants

Twining plants exhibit a striking oscillation of their stems in their quest for a support. The oscillations, called circumnutation, have periods generally of 1–5 hr, and virtually all species have a preferred direction of twining. I seek to explain these chiral asymmetries in plant behavior by hypothesizing a chiral asymmetry in plant anatomy. Such asymmetries already exist, for example, in phyllotaxis. I explore wave phenomena on asymmetric but isotropic rings, and seek systems which will only support (stable) waves in one direction around the ring, and not in the other. Simulations indicate that (1) oscillatory reaction-diffusion systems do not support unidirectional waves on rings; (2) excitable reaction-diffusion systems do support unidirectional waves on rings; and (3) unidirectional phase-locking (discrete unidirectional waves) occurs in rings of coupled oscillators. Thus, chiral asymmetries of circumnutating plants cannot be explained by continuum oscillator phenomena, but can be explained by general discrete oscillators, or excitable phenomena on the continuum.