Assessing the origin and velocity of Ca²⁺ waves in three-dimensional tissue: Insights from a mathematical model and confocal imaging in mouse pancreas tissue slices

Highlights

• Intercellular Ca²⁺ waves coordinate the function of multicellular systems.

• Determining the origin and velocity of Ca²⁺ waves in 3D tissue is challenging.

• We combined Ca²⁺ imaging with models to study Ca²⁺ waves in beta cell networks.

• The origin and velocity of Ca²⁺ waves in pancreatic islets are revealed.

• Our theoretical and numerical results are of particular value to experimentalists.

Abstract

Many tissues are gap-junction-coupled syncytia that support cell-to-cell communication via propagating calcium waves. This also holds true for pancreatic islets of Langerhans, where several thousand beta cells work in synchrony to ensure proper insulin secretion. Two emerging functional parameters of islet function are the location of wave initiator regions and the velocity of spreading calcium waves. High-frequency confocal laser-scanning imaging in tissue slices is one of the best available methods to determine these markers, but it is limited to two-dimensional cross-sections of an otherwise three-dimensional islet. Here we show how mathematical modeling can significantly improve this limitation. Firstly, we analytically determine the shape of velocity profiles of spherical excitation waves in the focal plane of a homogeneous three-dimensional space. Secondly, we introduce a mathematical model consisting of coupled excitable cells that considers cellular heterogeneities to approach more realistic conditions by means of numerical simulations. We demonstrate the effectiveness of our approach on experimentally recorded waves from an islet that was stimulated with 9 mM glucose. Furthermore, we show that calcium waves were primarily triggered by a specific region located 30 µm bellow the focal plane at the periphery of the islet. Additionally, we show that the velocity of the calcium wave was around 80 µm/s. We discuss the importance of our approach for the correct determination of the origin and velocity of calcium waves from experimental data, as well as the pitfalls that are due to improper procedural simplifications.

Introduction

Already in 1883, Ca²⁺ ions were identified as an essential and ubiquitous intracellular messenger [1]. More than a century later, the search for molecular and cellular mechanism of Ca²⁺ signaling are still ongoing. A major breakthrough in the quest was the development of Ca²⁺-sensitive dyes and imaging techniques that helped to elucidate the role of Ca²⁺ ions as a most versatile secondary messenger regulating processes as diverse and universal as fertilization, cell cleavage, development, contraction, secretion, and cell death [2], [3], [4], [5]. Since Ca²⁺ is a simple ion with binding sites on a plethora of target molecules, it cannot serve as a signal merely by its presence or specific binding. Instead, the information it carries is encoded in spatial and temporal patterns, which take the form of intracellular oscillations and waves [6], [7]. Intercellular Ca²⁺ waves of different shapes, directions and frequencies, that are mediated by different mechanisms are crucial for the coordinated function of coupled cells in a number of different tissues [7], [8], [9].

In the strict sense, the term intercellular Ca²⁺ waves is reserved for non-excitable cells where the waves occur on a temporal scale of several seconds, and spread between a few, tens, or hundreds of cells with a velocity of around 10-30 µm/s, consistent with the diffusion of messengers [10]. In the broader sense, a number of electrically excitable cells are coupled via gap junctions and depolarization waves spreading between cells are often followed by Ca²⁺ increases in these cells, giving the appearance of Ca²⁺ waves. Typically, these waves occur on a temporal scale of 1 second or less, encompass up to thousands of cells and are typically faster (> 30 µm/s), do not show delays, and rely mechanistically on depolarizing electrical currents [6], [7], [9].

Nevertheless, irrespective of the cell type intercellular waves have the same crucial task, i.e., to help cells overcome differences in their sensitivity and produce a synchronized response [7]. One of the most fascinating aspects of intercellular Ca²⁺ dynamics is that local activity in a population of heterogeneous cells embedded in a changing and noisy environment can be organized into complex and rather regular spatiotemporal patterns, which facilitate the transfer of information. Not only are these structures physiologically important, they are also theoretically appealing and challenging to understand. As a result, numerus computational models have been developed to support experimental endeavors and to elucidate the underlying mechanisms. For instance, previous studies have utilized models and simulations to identify the key intercellular messengers [11], to investigate the role of intercellular topology [12], [13], to explore dynamical transitions and synchronization behavior [14], [15], assess the velocity of signal propagation [16], to evaluate the effect of noise and cellular heterogeneity [17], [18], [19], and, of course, to investigate how they are regulated by mechanical and biochemical factors to maintain essential physiological functions [20]. Models have been developed for networks of non-excitable cells, such as endothelial [12] and epithelial [11] cells, keratinocytes [21], acinar cells [22], hepatocytes [23], and astrocytes [24], to name only a few examples. From the perspective of complexity, numerical simulations of coupled excitable cells are an even more challenging task. Computational studies of Ca²⁺ waves have plenty in common with the study of wave propagation in other excitable models, such as the Hodgkin-Huxley or FitzHugh-Nagumo models, and can predict very complex spatiotemporal patterns [25], [26], [27]. Until today, cell-specific models to study the collective Ca²⁺ activity have been mostly designed for various types of smooth muscle cells [28], [29], [30], [31], [32] and insulin-secreting beta cells from pancreatic islets of Langerhans [33], [34], [35], [36], [37], [38].

The latter is a particularly interesting system to study intra- and intercellular Ca²⁺ signaling. A typical mouse islet is around 100 µm in diameter and consists of around 1000 cells of at least five different types, with the insulin secreting beta cells being the most prevailing in the islet core and representing around 60-80 % of all cells. In terms of composition and size, islets are highly heterogeneous, with some comprising only a few cells and some having diameters well above 500 µm [39]. Cells communicate with each other through different mechanisms; with gap-junctional coupling probably being the main synchronizing mechanism for depolarization and Ca²⁺ waves [40], [41]. These microorgans are biologically highly complex [42], [43], accessible to advanced modelling, experimental, and analytical approaches [44], [45], [46], and their disruption leads to type 2 diabetes mellitus, which is increasingly becoming an important public health problem [47], [48]. When exposed to stimulatory glucose concentrations, mouse beta cells in isolated islets and pancreatic tissue slices exhibit a multimodal oscillatory pattern. The slow oscillatory component with a period of several minutes is believed to reflect glycolytic oscillations, whereas the superimposed fast component resembles a glucose-dependent electrical activity with a frequency of several oscillations per minute [49]. Both slow and fast oscillations are well synchronized between different beta cells of the same islet, predominantly by gap-junctional coupling through Connexin36, even though other means of intercellular communication probably contribute to intercellular Ca²⁺ wave generation as well [38], [42], [43], [50], [51], [52]. The collective activity of slow oscillations has not yet been resolved conclusively and is probably driven by diffusion of the metabolic intermediate glucose-6-phosphate [53], [54]. Synchronous behavior of fast oscillations is receiving much more attention from the scientific community and is governed by electrical depolarization, resulting in the appearance of Ca²⁺ waves that propagate across the islets [55], [56], [57], [58], [59].

Motivated by the fact that well-organized collective activity is a prerequisite for proper hormone secretion, in silico approaches are increasingly becoming an integral part in islet research [60], [61], [62]. Multicellular models incorporating either single cellular type (beta cells) or three cellular types (alpha, beta, and delta cells) are of paramount importance for exploring their synchronous behavior, Ca²⁺ wave propagation, and heterologous cell communication within the islet tissue [34], [63], [64], [65], [66], [67], [68]. In recent years, particular emphasis has been given to the incorporation of the well-known beta cell heterogeneity into beta cell models, as it has been shown to profoundly affect dynamical transitions and intercellular activity [14], [69], [70]. Studies utilizing a combination of experimental findings, advanced analyses, and computational models have revealed that Ca²⁺ waves originate from specific and rather randomly distributed subregions with elevated excitability [36], [65], [71]. Moreover, recent research suggests that the multifaceted heterogeneity also includes cell-to-cell variability and a heterogeneous nature of intercellular interactions. Both must be included in computational models, in order to firmly reproduce experimentally observed collective beta cell behavior [36], [37], [72].

Several experimental studies have investigated the extent and mechanism of synchronicity between fast Ca²⁺ oscillations in beta cells. In isolated islets, the oscillations from different parts of the islets were found to be phase-shifted for up to 2 seconds [73]. Considering the characteristic size of the islets (100 µm), it could thus be estimated that the velocity of signal transmission was around 50-100 µm/s. Additionally, it was reported that the direction of the waves might change from one oscillation to another [73]. Moreover, similar results were obtained in clusters of beta cells [74], isolated mouse [75] and human [76] islets and in islets in vivo [77], [78], [79], but, remarkably, neither of the studies explicitly mentioned intercellular Ca²⁺ waves. Similar values were also obtained with electrophysiological experiments [80]. A more explicit Ca²⁺ imaging and simulation study by Aslanidi et al. [33] provided first convincing evidence for depolarization front-dependent Ca²⁺ waves, quantified the relationship between gap junctional coupling, membrane conductance, and wave velocity, and concluded that the predicted wave velocities are typically between 30 and 100 µm/s in glucose-stimulated islets. Later, Benninger et al. [34] provided a decisive piece of evidence for the dependence of Ca²⁺ waves on gap junctions and depolarization in isolated islets and that a genetic disruption of connexin 36 slows and disrupts Ca²⁺ waves. The estimated wave velocities were between 20 and 200 µm/s and the observed directions of spreading were not always the same. Interestingly, Cappon and Pedersen [69] also demonstrated numerically that the wave velocity is a function of the average gap junctional conductance, which holds true also in ensembles of heterogeneous networks of beta cells.

We would like to conclude this brief review of previous findings by pointing out some promising recent studies that assessed the velocity of Ca²⁺ waves in vivo in zebrafish islets, islets transplanted into the anterior chamber of the eye, and in exteriorized pancreata, some even in all three dimensions (3D) [79], [81], [82], [83]. However, according to the available information, none of the studies so far primarily addressed the problem of true wave origin and velocity in 3D tissue and the abovementioned studies were limited by rather low temporal resolutions. More precisely, both tissue slices and islets are 3D. Imaging methods to detect Ca²⁺ waves enable one to detect Ca²⁺ changes predominantly in the stronger stained periphery of isolated islets and in the part of the islet closer to the objective in the case of a camera-based system, or in an approximately 5 µm thick focal plane in case of confocal microscopy. Thus, velocities calculated by simply dividing the distances between wave origins and ends by phase lags are probably imprecise estimates. Moreover, true origins and ends of waves could lie outside the recorded area. Finding true wave-initiating regions is especially important in the light of recent findings that pacemakers might exhibit metabolic features different from follower cells and other specialized cells [41], [43], [82], [84]. Typically, pacemakers are found at the periphery of islets, which may be the result of peripheral cells being exposed to glucose sooner than other cells in some experimental settings, a true biological heterogeneity in some metabolic parameters, being coupled to fewer neighboring cells, or due to a combination of these factors [33], [34], [35], [36], [72]. However, sometimes they appear to be located more centrally, but this could also be an experimental artefact. Finally, understanding the spreading of waves in 3D could also help us resolve some previous findings of complete synchronicity between cells as well as the findings that some cells, called hub cells, are strongly synchronized with many other cells [85].

The main goal of this study was therefore to investigate how the Ca²⁺ wave velocity can be computed based on multicellular Ca²⁺ imaging and to determine the distribution of wave origins in 3D space. To this end, we combined theoretical models and simulations with experimental observations. With respect to the latter, we limit ourselves to tissue slices and confocal microscopy, but one shall easily extrapolate our findings to other models and modes of recording. We resorted to theoretical models to gain quantitative insight as our and other experimental setups do not yet make the observation of intracellular Ca²⁺ changes possible in 3D. In particular, we first considered the spread of spherical excitation waves in homogeneous 3D space to determine analytically the shapes of velocity profiles in the focal plane. Then, we built multicellular models of coupled excitable cells to examine this issue by means of numerical simulations, also in the scenario which included realistic physiological determinants, such as cellular heterogeneity. Finally, we used the acquired insights from theoretical and numerical models to analyze experimental data and to determine the Ca²⁺ wave velocity and the spatial distribution of wave initiators within the islet.