Integrated multiscale mathematical modeling of insulin secretion reveals the role of islet network integrity for proper oscillatory glucose-dose response

Both electrical and metabolic coupling shape the collective multimodal activity and functional connectivity patterns in beta cell collectives: A computational model perspective

Pancreatic beta cells are coupled excitable oscillators that synchronize their activity via different communication pathways. Their oscillatory activity manifests itself on multiple timescales and consists of bursting electrical activity, subsequent oscillations in the intracellular Ca^{2+}, as well as oscillations in metabolism and exocytosis. The coordination of the intricate activity on the multicellular level plays a key role in the regulation of physiological pulsatile insulin secretion and is incompletely understood. In this paper, we investigate theoretically the principles that give rise to the synchronized activity of beta cell populations by building up a phenomenological multicellular model that incorporates the basic features of beta cell dynamics. Specifically, the model is composed of coupled slow and fast oscillatory units that reflect metabolic processes and electrical activity, respectively. Using a realistic description of the intercellular interactions, we study how the combination of electrical and metabolic coupling generates collective rhythmicity and shapes functional beta cell networks. It turns out that while electrical coupling solely can synchronize the responses, the addition of metabolic interactions further enhances coordination, the spatial range of interactions increases the number of connections in the functional beta cell networks, and ensures a better consistency with experimental findings. Moreover, our computational results provide additional insights into the relationship between beta cell heterogeneity, their activity profiles, and functional connectivity, supplementing thereby recent experimental results on endocrine networks.

Elementary intracellular Ca signals approximated as a transition of release channel system from a metastable state

Cardiac muscle contraction is initiated by an elementary Ca signal (called Ca spark) which is achieved by collective action of Ca release channels in a cluster. The mechanism of this synchronization remains uncertain. We approached Ca spark activation as an emergent phenomenon of an interactive system of release channels. We constructed a weakly lumped Markov chain that applies an Ising model formalism to such release channel clusters and probable open channel configurations and demonstrated that spark activation is described as a system transition from a metastable to an absorbing state, analogous to the pressure required to overcome surface tension in bubble formation. This yielded quantitative estimates of the spark generation probability as a function of various system parameters. We performed numerical simulations to find spark probabilities as a function of sarcoplasmic reticulum Ca concentration, obtaining similar values for spark activation threshold as our analytic model, as well as those reported in experimental studies. Our parametric sensitivity analyses also showed that the spark activation threshold decreased as Ca sensitivity of RyR activation and RyR cluster size increased.

Glucose-dependent activation, activity, and deactivation of beta cell networks in acute mouse pancreas tissue slices

Many details of glucose-stimulated intracellular calcium changes in β cells during activation, activity, and deactivation, as well as their concentration-dependence, remain to be analyzed. Classical physiological experiments indicated that in islets, functional differences between individual cells are largely attenuated, but recent findings suggest considerable intercellular heterogeneity, with some cells possibly coordinating the collective responses. To address the above with an emphasis on heterogeneity and describing the relations between classical physiological and functional network properties, we performed functional multicellular calcium imaging in mouse pancreas tissue slices over a wide range of glucose concentrations. During activation, delays to activation of cells and any-cell-to-first-responder delays are shortened, and the sizes of simultaneously responding clusters increased with increasing glucose concentrations. Exactly the opposite characterized deactivation. The frequency of fast calcium oscillations during activity increased with increasing glucose up to 12 mM glucose concentration, beyond which oscillation duration became longer, resulting in a homogenous increase in active time. In terms of functional connectivity, islets progressed from a very segregated network to a single large functional unit with increasing glucose concentration. A comparison between classical physiological and network parameters revealed that the first-responders during activation had longer active times during plateau and the most active cells during the plateau tended to deactivate later. Cells with the most functional connections tended to activate sooner, have longer active times, and deactivate later. Our findings provide a common ground for recent differing views on β cell heterogeneity and an important baseline for future studies of stimulus-secretion and intercellular coupling.NEW & NOTEWORTHY We assessed concentration-dependence in coupled β cells, degree of functional heterogeneity, and uncovered possible specialized subpopulations during the different phases of the response to glucose at the level of many individual cells. To this aim, we combined acute mouse pancreas tissue slices with functional multicellular calcium imaging over a wide range from threshold (7 mM) and physiological (8 and 9 mM) to supraphysiological (12 and 16 mM) glucose concentrations, classical physiological, and advanced network analyses.

Hubs, diversity, and synchronization in FitzHugh-Nagumo oscillator networks: Resonance effects and biophysical implications

Using the FitzHugh-Nagumo equations to represent the oscillatory electrical behavior of β-cells, we develop a coupled oscillator network model with cubic lattice topology, showing that the emergence of pacemakers or hubs in the system can be viewed as a natural consequence of oscillator population diversity. The optimal hub to nonhub ratio is determined by the position of the diversity-induced resonance maximum for a given set of FitzHugh-Nagumo equation parameters and is predicted by the model to be in a range that is fully consistent with experimental observations. The model also suggests that hubs in a β-cell network should have the ability to "switch on" and "off" their pacemaker function. As a consequence, their relative amount in the population can vary in order to ensure an optimal oscillatory performance of the network in response to environmental changes, such as variations of an external stimulus.

Essential requirement of complex number for oscillatory phenomenon in intracellular trafficking process

Intracellular protein trafficking processes consisting of three intracellular states are described by three differential equations. To solve the equations, a quadratic equation is required, and its roots are generally real or complex. The purpose of the present study is to clarify the meanings of roots of real and complex numbers. To clarify the point, we define that: 1) ‘kI’ is the insertion rate from an insertion state trafficking to the plasma membrane state; 2) ‘kE’, the endocytotic rate from the plasma membrane state trafficking to a recycling state; 3) ‘kR’, the recycling rate from the recycling state trafficking to the insertion state. Amounts of proteins in three states are expressed as αelt+βemt+γ with α,β,γ = constant and l and m are roots of a quadratic equation, r2+kI+kE+kRr+kIkE+kIkR+kEkR=0. When l and m are real kI2+kE2+kR2>2kIkE+kEkR+kRkI, amounts of proteins in three states shows no oscillatory change but a monotonic change after a transient increase (or decrease); when l and m are complex kI2+kE2+kR2<2kIkE+kEkR+kRkI, amounts of proteins in three states are expressed as αelt+βemt+γ=2g2+h2sinbt+σeat+γ (α, β, l, m = complex and γ,a,b,g,h,σ = real: α,β = conjugate each other; l,m = conjugate each other), showing an oscillatory change with time. The frequency of oscillatory change appearance is evaluated to be 60% at random combinations of three trafficking rates, kI, kE and kR. The present study indicates that complex numbers have an essentially important meaning in appearance of oscillatory phenomena in bodily and cellular function.

Assessing Different Temporal Scales of Calcium Dynamics in Networks of Beta Cell Populations

Beta cells within the pancreatic islets of Langerhans respond to stimulation with coherent oscillations of membrane potential and intracellular calcium concentration that presumably drive the pulsatile exocytosis of insulin. Their rhythmic activity is multimodal, resulting from networked feedback interactions of various oscillatory subsystems, such as the glycolytic, mitochondrial, and electrical/calcium components. How these oscillatory modules interact and affect the collective cellular activity, which is a prerequisite for proper hormone release, is incompletely understood. In the present work, we combined advanced confocal Ca2+ imaging in fresh mouse pancreas tissue slices with time series analysis and network science approaches to unveil the glucose-dependent characteristics of different oscillatory components on both the intra- and inter-cellular level. Our results reveal an interrelationship between the metabolically driven low-frequency component and the electrically driven high-frequency component, with the latter exhibiting the highest bursting rates around the peaks of the slow component and the lowest around the nadirs. Moreover, the activity, as well as the average synchronicity of the fast component, considerably increased with increasing stimulatory glucose concentration, whereas the stimulation level did not affect any of these parameters in the slow component domain. Remarkably, in both dynamical components, the average correlation decreased similarly with intercellular distance, which implies that intercellular communication affects the synchronicity of both types of oscillations. To explore the intra-islet synchronization patterns in more detail, we constructed functional connectivity maps. The subsequent comparison of network characteristics of different oscillatory components showed more locally clustered and segregated networks of fast oscillatory activity, while the slow oscillations were more global, resulting in several long-range connections and a more cohesive structure. Besides the structural differences, we found a relatively weak relationship between the fast and slow network layer, which suggests that different synchronization mechanisms shape the collective cellular activity in islets, a finding which has to be kept in mind in future studies employing different oscillations for constructing networks.

Heterogeneity and Delayed Activation as Hallmarks of Self-Organization and Criticality in Excitable Tissue

Self-organized critical dynamics is assumed to be an attractive mode of functioning for several real-life systems and entails an emergent activity in which the extent of observables follows a power-law distribution. The hallmarks of criticality have recently been observed in a plethora of biological systems, including beta cell populations within pancreatic islets of Langerhans. In the present study, we systematically explored the mechanisms that drive the critical and supercritical behavior in networks of coupled beta cells under different circumstances by means of experimental and computational approaches. Experimentally, we employed high-speed functional multicellular calcium imaging of fluorescently labeled acute mouse pancreas tissue slices to record calcium signals in a large number of beta cells simultaneously, and with a high spatiotemporal resolution. Our experimental results revealed that the cellular responses to stimulation with glucose are biphasic and glucose-dependent. Under physiological as well as under supraphysiological levels of stimulation, an initial activation phase was followed by a supercritical plateau phase with a high number of global intercellular calcium waves. However, the activation phase displayed fingerprints of critical behavior under lower stimulation levels, with a progressive recruitment of cells and a power-law distribution of calcium wave sizes. On the other hand, the activation phase provoked by pathophysiologically high glucose concentrations, differed considerably and was more rapid, less continuous, and supercritical. To gain a deeper insight into the experimentally observed complex dynamical patterns, we built up a phenomenological model of coupled excitable cells and explored empirically the model’s necessities that ensured a good overlap between computational and experimental results. It turned out that such a good agreement between experimental and computational findings was attained when both heterogeneous and stimulus-dependent time lags, variability in excitability levels, as well as a heterogeneous cell-cell coupling were included into the model. Most importantly, since our phenomenological approach involved only a few parameters, it naturally lends itself not only for determining key mechanisms of self-organized criticality at the tissue level, but also points out various features for comprehensive and realistic modeling of different excitable systems in nature.