Model for synchronization of pancreatic beta-cells by gap junction coupling.

Pancreatic beta-cells coupled by gap junctions in sufficiently large clusters exhibit regular electrical bursting activity, which is described by the Chay-Keizer model and its variants. According to most reports, however, isolated cells exhibit disorganized spiking. We have previously (Sherman, A. J. Rinzel, and J. Keizer, 1988. Biophys. J. 54:411-425) modeled these behaviors by hypothesizing that stochastic channel fluctuations disrupt the bursts. We showed that when cells are coupled by infinite conductance gap junctions, so that the cluster is isopotential and may be viewed as a single "supercell," the fluctuations are shared over a larger membrane area and hence dampened. Bursting emerges when there are more than approximately 50 cells in the cluster. In the model the temporal organization of spikes into bursts increases the amplitude of intracellular calcium oscillations, which may be relevant for insulin secretion. We now extend the previous work by considering the case of a true "multicell" model with finite gap junctional conductance. Whereas the previous study assumed that the cells were synchronized, we can now study the process of synchronization itself. We show that, for sufficiently large clusters, the cells both synchronize and begin to burst with moderate, physiologically reasonable gap junctional conductance. An unexpected finding is that the burst period is longer, and calcium amplitude greater, than when coupling is infinitely strong, with an optimum in the range of 150-250 pS. Our model is in good agreement with recent experimental data of Perez-Armendariz, M., D. C. Spray, and M. V. L. Bennett. (1991. Biophys. J. 59:76-92) showing extensive gap junctions in beta-cell pairs with mean interfacial conductance of 213 +/- 113 pS. The optimality property of our model is noteworthy because simple slow-wave models without spikes do not show the same behavior.     

From spikers to bursters via coupling: Help from heterogeneity

It has been shown previously that identical spiking cells, incapable of bursting by themselves, may burst under weak diffusive coupling conditions. With stronger coupling, the coupled system reverts to bursting can be recovered by introducing heterogeneity in the model parameters. For a two-cell system, we explain the phenomenon with bifurcation analysis. For larger clusters of coupled cells, we demonstrate by numerical simulation that the phenomenon carries over. In addition, we use a perturbation analysis to deduce the dependence of the heterogeneity parameter used in the bifurcation analysis on the original heterogeneity parameters and the coupling strength. Implications of the phenomenon of emergent bursting are discussed in the context of electrical activity in pancreatic beta cells.     

Channel sharing in pancreatic beta -cells revisited: enhancement of emergent bursting by noise

Secretion of insulin by electrically coupled populations of pancreatic beta -cells is governed by bursting electrical activity. Isolated beta -cells, however, exhibit atypical bursting or continuous spike activity. We study bursting as an emergent property of the population, focussing on interactions among the subclass of spiking cells. These are modelled by equipping the fast subsystem with a saddle-node-loop bifurcation, which makes it monostable. Such cells can only spike tonically or remain silent when isolated, but can be induced to burst with weak diffusive coupling. With stronger coupling, the cells revert to tonic spiking. We demonstrate that the addition of noise dramatically increases, via a phenomenon like stochastic resonance, the coupling range over which bursting is seen.     

Gap junction coupling and calcium waves in the pancreatic islet

The pancreatic islet is a highly coupled, multicellular system that exhibits complex spatiotemporal electrical activity in response to elevated glucose levels. The emergent properties of islets, which differ from those arising in isolated islet cells, are believed to arise in part by gap junctional coupling, but the mechanisms through which this coupling occurs are poorly understood. To uncover these mechanisms, we have used both high-speed imaging and theoretical modeling of the electrical activity in pancreatic islets under a reduction in the gap junction mediated electrical coupling. Utilizing islets from a gap junction protein connexin 36 knockout mouse model together with chemical inhibitors, we can modulate the electrical coupling in the islet in a precise manner and quantify this modulation by electrophysiology measurements. We find that after a reduction in electrical coupling, calcium waves are slowed as well as disrupted, and the number of cells showing synchronous calcium oscillations is reduced. This behavior can be reproduced by computational modeling of a heterogeneous population of β-cells with heterogeneous levels of electrical coupling. The resulting quantitative agreement between the data and analytical models of islet connectivity, using only a single free parameter, reveals the mechanistic underpinnings of the multicellular behavior of the islet.     

Collective Signal Processing in Cluster Chemotaxis: Roles of Adaptation,Amplification, and Co-attraction in Collective Guidance

Single eukaryotic cells commonly sense and follow chemical gradients, performing chemotaxis. Recent experiments and theories, however, show that even when single cells do not chemotax, clusters of cells may, if their interactions are regulated by the chemoattractant. We study this general mechanism of “collective guidance” computationally with models that integrate stochastic dynamics for individual cells with biochemical reactions within the cells, and diffusion of chemical signals between the cells. We show that if clusters of cells use the well-known local excitation, global inhibition (LEGI) mechanism to sense chemoattractant gradients, the speed of the cell cluster becomes non-monotonic in the cluster’s size—clusters either larger or smaller than an optimal size will have lower speed. We argue that the cell cluster speed is a crucial readout of how the cluster processes chemotactic signals; both amplification and adaptation will alter the behavior of cluster speed as a function of size. We also show that, contrary to the assumptions of earlier theories, collective guidance does not require persistent cell-cell contacts and strong short range adhesion. If cell-cell adhesion is absent, and the cluster cohesion is instead provided by a co-attraction mechanism, e.g. chemotaxis toward a secreted molecule, collective guidance may still function. However, new behaviors, such as cluster rotation, may also appear in this case. Co-attraction and adaptation allow for collective guidance that is robust to varying chemoattractant concentrations while not requiring strong cell-cell adhesion.     

Diffusion of extracellular K+ can synchronize bursting oscillations in a model islet of Langerhans.

Electrical bursting oscillations of mammalian pancreatic beta-cells are synchronous among cells within an islet. While electrical coupling among cells via gap junctions has been demonstrated, its extent and topology are unclear. The beta-cells also share an extracellular compartment in which oscillations of K+ concentration have been measured (Perez-Armendariz and Atwater, 1985). These oscillations (1-2 mM) are synchronous with the burst pattern, and apparently are caused by the oscillating voltage-dependent membrane currents: Extracellular K+ concentration (Ke) rises during the depolarized active (spiking) phase and falls during the hyperpolarized silent phase. Because raising Ke depolarizes the cell membrane by increasing the potassium reversal potential (VK), any cell in the active phase should recruit nonspiking cells into the active phase. The opposite is predicted for the silent phase. This positive feedback system might couple the cells' electrical activity and synchronize bursting. We have explored this possibility using a theoretical model for bursting of beta-cells (Sherman et al., 1988) and K+ diffusion in the extracellular space of an islet. Computer simulations demonstrate that the bursts synchronize very quickly (within one burst) without gap junctional coupling among the cells. The shape and amplitude of computed Ke oscillations resemble those seen in experiments for certain parameter ranges. The model cells synchronize with exterior cells leading, though incorporating heterogeneous cell properties can allow interior cells to lead. The model islet can also be forced to oscillate at both faster and slower frequencies using periodic pulses of higher K+ in the medium surrounding the islet. Phase plane analysis was used to understand the synchronization mechanism. The results of our model suggest that diffusion of extracellular K+ may contribute to coupling and synchronization of electrical oscillations in beta-cells within an islet.     

Role of single-channel stochastic noise on bursting clusters of pancreatic beta-cells.

To study why pancreatic beta-cells prefer to burst as a multi-cellular complex, we have formulated a stochastic model for bursting clusters of excitable cells. Our model incorporated a delayed rectifier K+ channel, a fast voltage-gated Ca2+ channel, and a slow Cai-blockable Ca2+ channel. The fraction of ATP-sensitive K+ channels that may still be active in the bursting regime was included in the model as a leak current. We then developed an efficient method for simulating an ionic current component of an excitable cell that contains several thousands of channels opening simultaneously under unclamped voltage. Single channel open-close stochastic events were incorporated into the model by use of binomially distributed random numbers. Our simulations revealed that in an isolated beta-cell [Ca2+]i oscillates with a small amplitude about a low [Ca2+]i. However, in a large cluster of tightly coupled cells, stable bursts develop, and [Ca2+]i oscillates with a larger amplitude about a higher [Ca2+]i. This may explain why single beta-cells do not burst and also do not release insulin.     

肽类抗生素生物合成基因簇在Escherichia coli中的异源表达

微生物能够产生众多结构和生物活性多样的次生代谢产物,而其生物合成基因簇的挖掘和异源表达是药物创新和产量提高的必要前提. 在过去20年里,大量重要天然产物的生物合成基因簇在微生物中被不断的发现. 在这些被挖掘的基因簇中,肽类抗生素的生物合成基因簇占了很大比重.肽类抗生素因具有抗菌、抗肿瘤、抗病毒等多种生物学活性而备受化学家和药物学家的重视. 如能了解它们的生物合成机制,实现其基因簇的异源表达,将使合理化遗传修饰生物合成通路获取结构类似物（药物开发）和提高产量成为可能. 大肠杆菌作为最广泛、最成功的表达体系,常用来表达外源基因,但一般只能表达一个或几个基因,却很少有用它来表达整个生物合成基因簇. 2001年,Khosla和Cane在E.coli中成功异源表达了一个复杂聚酮天然产物（红霉素苷原6dEB）基因簇. 这是首个有关在E.coli中异源表达天然产物生物合成基因簇的研究. 至此之后,大肠杆菌开始作为生物合成基因簇的异源表达宿主,越来越受到相关领域的重视. 紧接着核糖体肽和非核糖体肽生物合成基因簇也相继在大肠杆菌中成功异源表达. 本文对肽类抗生素生物合成基因簇在E.coli中的异源表达进行了综述.     

Investigating the role of islet cytoarchitecture in its oscillation using a new beta-cell cluster model

The oscillatory insulin release is fundamental to normal glycemic control. The basis of the oscillation is the intercellular coupling and bursting synchronization of beta cells in each islet. The functional role of islet beta cell mass organization with respect to its oscillatory bursting is not well understood. This is of special interest in view of the recent finding of islet cytoarchitectural differences between human and animal models. In this study we developed a new hexagonal closest packing (HCP) cell cluster model. The model captures more accurately the real islet cell organization than the simple cubic packing (SCP) cluster that is conventionally used. Using our new model we investigated the functional characteristics of beta-cell clusters, including the fraction of cells able to burst f(b), the synchronization index lambda of the bursting beta cells, the bursting period T(b), the plateau fraction p(f), and the amplitude of intracellular calcium oscillation [Ca]. We determined their dependence on cluster architectural parameters including number of cells n(beta), number of inter-beta cell couplings of each beta cell n(c), and the coupling strength g(c). We found that at low values of n(beta), n(c) and g(c), the oscillation regularity improves with their increasing values. This functional gain plateaus around their physiological values in real islets, at n(beta) approximately 100, n(c) approximately 6 and g(c) approximately 200 pS. In addition, normal beta-cell clusters are robust against significant perturbation to their architecture, including the presence of non-beta cells or dead beta cells. In clusters with n(beta)> approximately 100, coordinated beta-cell bursting can be maintained at up to 70% of beta-cell loss, which is consistent with laboratory and clinical findings of islets. Our results suggest that the bursting characteristics of a beta-cell cluster depend quantitatively on its architecture in a non-linear fashion. These findings are important to understand the islet bursting phenomenon and the regulation of insulin secretion, under both physiological and pathological conditions.     

Innate synchronous oscillations in freely-organized small neuronal circuits

Background

Information processing in neuronal networks relies on the network''s ability to generate temporal patterns of action potentials. Although the nature of neuronal network activity has been intensively investigated in the past several decades at the individual neuron level, the underlying principles of the collective network activity, such as the synchronization and coordination between neurons, are largely unknown. Here we focus on isolated neuronal clusters in culture and address the following simple, yet fundamental questions: What is the minimal number of cells needed to exhibit collective dynamics? What are the internal temporal characteristics of such dynamics and how do the temporal features of network activity alternate upon crossover from minimal networks to large networks?

Methodology/Principal Findings

We used network engineering techniques to induce self-organization of cultured networks into neuronal clusters of different sizes. We found that small clusters made of as few as 40 cells already exhibit spontaneous collective events characterized by innate synchronous network oscillations in the range of 25 to 100 Hz. The oscillation frequency of each network appeared to be independent of cluster size. The duration and rate of the network events scale with cluster size but converge to that of large uniform networks. Finally, the investigation of two coupled clusters revealed clear activity propagation with master/slave asymmetry.

Conclusions/Significance

The nature of the activity patterns observed in small networks, namely the consistent emergence of similar activity across networks of different size and morphology, suggests that neuronal clusters self-regulate their activity to sustain network bursts with internal oscillatory features. We therefore suggest that clusters of as few as tens of cells can serve as a minimal but sufficient functional network, capable of sustaining oscillatory activity. Interestingly, the frequencies of these oscillations are similar those observed in vivo.     

Dispersion and time delay effects in synchronized spike–burst networks

We study spike–burst neural activity and investigate its transitions to synchronized states under electrical coupling. Our reported results include the following: (1) Synchronization of spike–burst activity is a multi-time scale phenomenon and burst synchrony is easier to achieve than spike synchrony. (2) Synchrony of networks with time-delayed connections can be achieved at lower coupling strengths than within the same network with instantaneous couplings. (3) The introduction of parameter dispersion into the network destroys the existence of synchrony in the strict sense, but the network dynamics in major regimes of the parameter space can still be effectively captured by a mean field approach if the couplings are excitatory. Our results on synchronization of spiking networks are general of nature and will aid in the development of minimal models of neuronal populations. The latter are the building blocks of large scale brain networks relevant for cognitive processing.     

Network dynamics and synchronous activity in cultured cortical neurons

Neurons extracted from specific areas of the Central Nervous System (CNS), such as the hippocampus, the cortex and the spinal cord, can be cultured in vitro and coupled with a micro-electrode array (MEA) for months. After a few days, neurons connect each other with functionally active synapses, forming a random network and displaying spontaneous electrophysiological activity. In spite of their simplified level of organization, they represent an useful framework to study general information processing properties and specific basic learning mechanisms in the nervous system. These experimental preparations show patterns of collective rhythmic activity characterized by burst and spike firing. The patterns of electrophysiological activity may change as a consequence of external stimulation (i.e., chemical and/or electrical inputs) and by partly modifying the "randomness" of the network architecture (i.e., confining neuronal sub-populations in clusters with micro-machined barriers). In particular we investigated how the spontaneous rhythmic and synchronous activity can be modulated or drastically changed by focal electrical stimulation, pharmacological manipulation and network segregation. Our results show that burst firing and global synchronization can be enhanced or reduced; and that the degree of synchronous activity in the network can be characterized by simple parameters such as cross-correlation on burst events.     

Short inversions and conserved gene cluster

MOTIVATION: Two independent sets of recent observations on newly sequenced microbial genomes pertain to the prevalence of short inversion as a gene order rearrangement process and to the lack of conservation of gene order within conserved gene clusters. We propose a model of inversion where the key parameter is the length of the inverted fragment. RESULTS: We show that there is a qualitative difference in the pattern of evolution when the inversion length is small with respect to the cluster size and when it is large. This suggests an explanation of the lack of parallel gene order in conserved clusters and raises questions about the statistical validity of putative functionally selected gene clusters if these have only been tested against inappropriate null hypotheses.     

Changing patterns of ganglion cell coupling and connexin expression during chick retinal development

We have used dye injection and immunolabeling to investigate the relationship between connexin (Cx) expression and dye coupling between ganglion cells (GCs) and other cells of the embryonic chick retina between embryonic days 5 and 14 (E5-14). At E5, GCs were usually coupled, via soma-somatic or dendro-somatic contacts, to only one or two other cells. Coupling increased with time until E11 when GCs were often coupled to more than a dozen other cells with somata in the ganglion cell layer (GCL) or inner nuclear layer (INL). These coupled clusters occupied large areas of the retina and coupling was via dendro-dendritic contacts. By E14, after the onset of synaptogenesis and at a time of marked cell death, dye coupling was markedly decreased with GCs coupled to three or four partners. At this time, coupling was usually to cells of the same morphology, whereas earlier coupling was heterogeneous. Between E5 and E11, GCs were sometimes coupled to cells of neuroepithelial morphology that spanned the thickness of the retina. The expression of Cx 26, 32, and 43 differed and their distribution changed during the period studied, showing correlation with events such as proliferation, migration, and synaptogenesis. These results suggest specific roles for gap junctions and Cx's during retinal development.     

Analysis of developmentally homogeneous neural crest cell populations in vitro: I. Formation,morphology and differentiative behavior

When early embryonic quail neural tubes are dissected free from surrounding tissues and placed in culture, small stellate neural crest cells usually migrate from the explant onto the substratum. This outgrowth has been reported to consist of a mixture of cells, some of which undergo melanogenesis, while the rest remain unpigmented. We have, in contrast to earlier observations, obtained a spatial separation of the two phenotypes. In these cultures the primary outgrowth of migrating cells remained almost free of pigment-forming cells, whereas small spherical clusters containing several hundred pigment-forming cells appeared on the explanted neural tubes. Whether the clusters remained with the tube explants or were subcultured, all cluster cells differentiated into melanocytes. Prior to melanogenesis, the appearance of the cultured cells from a cluster was indistinguishable from the cells in the outgrowth. The clusters provide a source of neural crest cells, that (1) can be easily obtained in comparatively large numbers, (2) is not contaminated with any other cell type, (3) can be isolated before the onset of differentiation, and (4) is developmentally homogeneous. Thus, the cluster population is well suited for many types of experiments, such as the identification of specific environmental factors that might control neural crest cell differentiation.     

Fast subsystem bifurcations in strongly coupled heterogeneous collections of excitable cells

A continuum model for a heterogeneous collection of excitable cells electrically coupled through gap junctions is introduced and analysed using spatial averaging, asymptotic and numerical techniques. Heterogeneity is modelled by imposing a spatial dependence on parameters which define the single cell model and a diffusion term is used to model the gap junction coupling. For different parameter values, single cell models can exhibit bursting, beating and a myriad of other complex oscillations. A procedure for finding asymptotic estimates of the thresholds between these (synchronous) behaviors in the cellular aggregates is described for the heterogeneous case where the coupling strength is strong. This procedure is tested on a model of a strongly coupled heterogeneous collection of bursting and beating cells. Since isolated pancreatic β-cells have been observed to both burst and beat, this test of the spatial averaging techniques provides a possible explanation to measured discrepancies between the electrical activities of isolated β-cells and coupled collections (islets) of β-cells. This work was supported by the National Science Foundation Grant DMS-97-04-966.     

Biophysical properties of gap junctions between freshly dispersed pairs of mouse pancreatic beta cells.

Coupling between beta cells through gap junctions has been postulated as a principal mechanism of electrical synchronization of glucose-induced activity throughout the islet of Langerhans. We characterized junctional conductance between isolated pairs of mouse pancreatic beta cells by whole-cell recording with two independent patch-clamp circuits. Most pairs were coupled (67%, n = 155), although the mean junctional conductance (gj) (215 +/- 110 pS) was lower than reported in other tissues. Coupling could be recorded for long periods, up to 40 min. Voltage imposed across the junctional or nonjunctional membranes had no effect on gj. Up to several hours of treatment to increase intracellular cAMP levels did not affect gj. Electrically coupled pairs did not show transfer of the dye Lucifer yellow. Octanol (2 mM) reversibly decreased gj. Lower concentrations of octanol (0.5 mM) and heptanol (0.5 mM) than required to uncouple beta cells decreased voltage-dependent K+ and Ca2+ currents in nonjunctional membranes. Although gj recorded in these experiments would be expected to be provided by current flowing through only a few channels of the unitary conductance previously reported for other gap junctions, no unitary junctional currents were observed even during reversible suppression of gj by octanol. This result suggests either that the single channel conductance of gap junction channels between beta cells is smaller than in other tissues (less than 20 pS) or that the small mean conductance is due to transitions between open and closed states that are too rapid or too slow to be resolved.     

Stability of two cluster solutions in pulse coupled networks of neural oscillators

Phase response curves (PRCs) have been widely used to study synchronization in neural circuits comprised of pacemaking neurons. They describe how the timing of the next spike in a given spontaneously firing neuron is affected by the phase at which an input from another neuron is received. Here we study two reciprocally coupled clusters of pulse coupled oscillatory neurons. The neurons within each cluster are presumed to be identical and identically pulse coupled, but not necessarily identical to those in the other cluster. We investigate a two cluster solution in which all oscillators are synchronized within each cluster, but in which the two clusters are phase locked at nonzero phase with each other. Intuitively, one might expect this solution to be stable only when synchrony within each isolated cluster is stable, but this is not the case. We prove rigorously the stability of the two cluster solution and show how reciprocal coupling can stabilize synchrony within clusters that cannot synchronize in isolation. These stability results for the two cluster solution suggest a mechanism by which reciprocal coupling between brain regions can induce local synchronization via the network feedback loop.     

A Mathematical Model of Collective Cell Migration in a Three-Dimensional,Heterogeneous Environment

Cell migration is essential in animal development, homeostasis, and disease progression, but many questions remain unanswered about how this process is controlled. While many kinds of individual cell movements have been characterized, less effort has been directed towards understanding how clusters of cells migrate collectively through heterogeneous, cellular environments. To explore this, we have focused on the migration of the border cells during Drosophila egg development. In this case, a cluster of different cell types coalesce and traverse as a group between large cells, called nurse cells, in the center of the egg chamber. We have developed a new model for this collective cell migration based on the forces of adhesion, repulsion, migration and stochastic fluctuation to generate the movement of discrete cells. We implement the model using Identical Math Cells, or IMCs. IMCs can each represent one biological cell of the system, or can be aggregated using increased adhesion forces to model the dynamics of larger biological cells. The domain of interest is filled with IMCs, each assigned specific biophysical properties to mimic a diversity of cell types. Using this system, we have successfully simulated the migration of the border cell cluster through an environment filled with larger cells, which represent nurse cells. Interestingly, our simulations suggest that the forces utilized in this model are sufficient to produce behaviors of the cluster that are observed in vivo, such as rotation. Our framework was developed to capture a heterogeneous cell population, and our implementation strategy allows for diverse, but precise, initial position specification over a three- dimensional domain. Therefore, we believe that this model will be useful for not only examining aspects of Drosophila oogenesis, but also for modeling other two or three-dimensional systems that have multiple cell types and where investigating the forces between cells is of interest.     

The Role of Cellular Coupling in the Spontaneous Generation of Electrical Activity in Uterine Tissue

The spontaneous emergence of contraction-inducing electrical activity in the uterus at the beginning of labor remains poorly understood, partly due to the seemingly contradictory observation that isolated uterine cells are not spontaneously active. It is known, however, that the expression of gap junctions increases dramatically in the approach to parturition, by more than one order of magnitude, which results in a significant increase in inter-cellular electrical coupling. In this paper, we build upon previous studies of the activity of electrically excitable smooth muscle cells (myocytes) and investigate the mechanism through which the coupling of these cells to electrically passive cells results in the generation of spontaneous activity in the uterus. Using a recently developed, realistic model of uterine muscle cell dynamics, we investigate a system consisting of a myocyte coupled to passive cells. We then extend our analysis to a simple two-dimensional lattice model of the tissue, with each myocyte being coupled to its neighbors, as well as to a random number of passive cells. We observe that different dynamical regimes can be observed over a range of gap junction conductances: at low coupling strength, corresponding to values measured long before delivery, the activity is confined to cell clusters, while the activity for high coupling, compatible with values measured shortly before delivery, may spread across the entire tissue. Additionally, we find that the system supports the spontaneous generation of spiral wave activity. Our results are both qualitatively and quantitatively consistent with observations from in vitro experiments. In particular, we demonstrate that the increase in inter-cellular electrical coupling observed experimentally strongly facilitates the appearance of spontaneous action potentials that may eventually lead to parturition.