Research paperAssessing the origin and velocity of Ca2+ waves in three-dimensional tissue: Insights from a mathematical model and confocal imaging in mouse pancreas tissue slices
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.
Section snippets
Numerical model of wave propagation in a homogeneous excitable lattice
For numerical simulations of excitation wave propagation in a homogeneous lattice of coupled cells, we utilized the two-dimensional Rulkov iterative map, a phenomenological model of excitable cellular oscillators. It was first proposed by Rulkov [86] for modelling of spiking-bursting neural behavior but it lends itself well for modeling of other types of excitable cells, including beta cells [72]. It consists of two iterative equations:
Theoretical considerations
3D excitation waves in pancreatic islets can be experimentally assessed only in 2D cross-sections, but they can be triggered anywhere outside this focal plane. Here we examined the shape of wave velocity profiles in one plane as a function of the wave origin location. We derived a simple theoretical model of 3D wave propagation using the following assumptions:
- i)
The origin of the wave is directly beneath the first detected activity in the observed (focal) plane. With this we assume that the tissue
Excitation wave propagation in a 3D lattice of homogeneous excitable cells
Firstly, we simulated the excitation wave propagation in a 3D lattice of excitable cells to examine how the wave behavior matches theoretical considerations. The coordinates of the grid element (i,j,k) were uniformly placed within a unit cube, with the lattice distance 1/17. The arrangement is visualized in Fig. 2A. The dynamics of the (i,j,k) grid element was governed by the Rulkov map model, as described in the Methods section. Pacemaker cells were placed at the bottom of the system in the
Discussion
The combination of a theoretical model, numerical simulations, and experimental data analysis enabled us to quantitatively assess the difference between real Ca2+ wave velocities and velocities estimated from signals recorded in an optical plane during fMCI in mouse islets of Langerhans from acute pancreatic tissue slices. The first main finding is that the wave velocities determined from cells at the beginning and end of a wave in the optical plane overestimated more precisely computed values
CRediT authorship contribution statement
Marko Šterk: Conceptualization, Software, Formal analysis, Writing - original draft, Visualization. Jurij Dolenšek: Conceptualization, Methodology, Writing - review & editing. Lidija Križančić Bombek: Methodology, Writing - review & editing. Rene Markovič: Software, Writing - review & editing, Visualization. Darko Zakelšek: Software, Writing - review & editing. Matjaž Perc: Software, Writing - review & editing, Funding acquisition. Viljem Pohorec: Methodology, Writing - review & editing. Andraž
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We dedicate this paper to the memory of the late Roger Yonchien Tsien and Michael John Berridge, two pioneers of calcium signaling. We thank Maruša Rošer and Rudi Mlakar for their excellent technical assistance.
Funding
The work presented in this study was financially supported by the Slovenian Research Agency (research core funding nos. P3-0396, P1-0403 and I0-0029, as well as research projects nos. J3-9289, J4-9302, J1-9112, N3-0048, and N3-0133).
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