A primer on modelling pancreatic islets: from models of coupled β-cells to multicellular islet models

ABSTRACT

Pancreatic islets are mini-organs composed of hundreds or thousands of ɑ, β and δ-cells, which, respectively, secrete glucagon, insulin and somatostatin, key hormones for the regulation of blood glucose. In pancreatic islets, hormone secretion is tightly regulated by both internal and external mechanisms, including electrical communication and paracrine signaling between islet cells. Given its complexity, the experimental study of pancreatic islets has been complemented with computational modeling as a tool to gain a better understanding about how all the mechanisms involved at different levels of organization interact. In this review, we describe how multicellular models of pancreatic cells have evolved from the early models of electrically coupled β-cells to models in which experimentally derived architectures and both electrical and paracrine signals have been considered.

GRAPHICAL ABSTRACT

Introduction

Pancreatic islets, also known as islets of Langerhans, are mainly composed of ɑ, β and δ-cells, endocrine cells that secrete glucagon, insulin and somatostatin, respectively, critical hormones for the regulation of blood glucose. Insulin, the only hormone capable of reducing glucose levels directly, is secreted when blood glucose rises, producing its hypoglycemic effect by promoting the uptake of glucose by hepatic, muscular and adipose tissues. On the contrary, glucagon is secreted when blood glucose decreases, promoting the release of the stored glucose (i.e., glycogen) mainly from the liver to restore the normal glucose levels. Finally, somatostatin, secreted by δ-cells, is not involved in the regulation of blood glucose directly, although it has a relevant indirect role by inhibiting the secretion of both glucagon and insulin from ɑ and β-cells, respectively.¹

Pancreatic ɑ, β and δ-cells share a common secretion mechanism involving glucose transport, metabolism and electrical activity, which, by driving the entry of Ca²⁺ to the intracellular space, ultimately promotes exocytosis. Despite this, ɑ, β and δ-cells have particularities that make them secrete their respective hormone in a finely tuned fashion following the body’s needs (for details see refs.^2–4). In addition, glucagon, insulin and somatostatin also serve as paracrine and autocrine signals, forming regulatory loops that shape the secretory behavior of the islet in response to the dynamic changes in blood glucose.^5,6 On the other hand, it is also well known that β-cells in close contact can communicate via gap-junctions, thus allowing the flow of ions and small molecules between the connected cells.⁷ Moreover, it has been demonstrated that gap-junctions also connect neighbor β and δ-cells.^8,9 All these communication pathways within the islets, in addition to the external endocrine, nutritional and neural inputs, conform an extremely complex dynamical system and as such has been widely studied both experimentally and theoretically.

Experimentally, the study of isolated pancreatic cells and pancreatic islets has grown in complexity following the technological advances. For instance, electrophysiological measurements of the electrical activity of single β-cells was first performed using the intracellular recording technique.^10–12 Then, the patch clamp techniques¹³ allowed researchers to characterize the properties of the ionic channels involved and their role on the electrical response of the cells (see for instance refs.^14–19). The recordings of the electrical activity of β-cells were eventually complemented with measurements of intracellular Ca^{2+ 20,21} and metabolism,^22–25 thus making it possible to delineate the relationship between cellular mechanisms leading to the exocytosis of insulin, which has been described extensively in more recent years owing to the use of advanced microscopy techniques.^26–28 Afterward, the simultaneous recording of two neighbor β-cells (separated by 20–50 μm) suggested the possibility of electrical coupling between β-cells in intact islets.^29,30 Moreover, fluorescence and electrophysiological techniques, along with the use of specifical chemical blockers and genetic knockout of gap-junctions have allowed to elucidate how electrical signals propagate throughout intact islets.^31,32 On the other hand, paracrine interactions between islet cells have been studied experimentally,^{33–37,37–40} thus allowing us to gain a better understanding of how glucose is regulated at the islet level considering the hormonal crosstalk. Similarly, optogenetic techniques have been adopted to evaluate whether a small population of specialized hub β-cells could act as pacemakers within the islets.⁴¹ Recently, RNA sequencing has been combined with electrophysiology to identify genes associated with cell function in health and disease.^42,43

As a complement, theoretical models of the electrical activity of ɑ,^44,45 β^46–48 and δ-cells⁴⁹ have been extensively used to explain the experimental observations at the single-cell level. Similarly, mathematical and computational models of clusters of communicating β-cells and recently, of pancreatic islets including ɑ, β and δ-cells, have been proposed with the objective of evaluating plausible hypotheses to explain the collective behavior observed experimentally.

In this review, we describe how multicellular models of pancreatic cells have evolved along the years, briefly explaining how they are constructed in terms of its components, their geometrical configuration, the modeling approaches adopted and the underlying physiological hypotheses. The characteristics of all the multicellular models of pancreatic cells reviewed are summarized in the table provided as Supplementary Material.

Modeling strategies in multicellular models of pancreatic cells

The presence of gap-junctions between neighbor β-cells was demonstrated by Orci et al.⁵⁰ in 1973 using electron microscopy images of rat islets. A few years later, functional electrophysiological evidence obtained by the simultaneous measurement of the electrical activity of two neighbor β-cells in mouse islets by Eddlestone et al.,²⁹ Meissner³⁰ and Meda et al.⁵¹ demonstrated that they are electrically coupled, as indicated by the synchronized electrical response following a glucose stimulus. After the proposal of the paradigmatic mathematical model of the electrical activity of the pancreatic β-cell by Chay and Keizer,⁵² it was a matter of time before theoretical studies of electrical coupling between β-cells were developed, recently followed by models of paracrine signaling between islet cells, as growing evidence has emerged about its importance for the regulation of the secretion of pancreatic hormones. Throughout the years, several multicellular models of clusters of β-cells and pancreatic islets have been proposed using different modeling strategies, briefly described in the following.

Dynamical models of electrically coupled cells and paracrine signaling

As the great majority of single-cell models (for a review see refs.^46–48), most of the multicellular models of pancreatic cells are dynamical models, in which all the variables of interest (e.g. membrane potential, intracellular Ca²⁺, etc.) change over time and therefore are represented by a system of coupled differential equations.

The first theoretical study of a pair of coupled β-cells was proposed by Banghman et al.⁵³ in 1986 using the classical dynamical model of Chay and Keizer,⁵² where it was shown that electrical coupling was capable of synchronizing the electrical activity of the pair of cells. Since the beginning, most of the dynamical models of coupled β-cells included a coupling current $I_{ij}$ between cells i and j, which only depends on the difference between the membrane potentials of the coupled cells ($V_i$ and $V_j$) and the conductance of the gap-junctions $g_{ij}$ (see Figure 1a):

$$I_{ij}=g_{ij}\left(V_i-V_j\right).$$ (1)

Figure 1.

A. Electrical coupling through gap-junctions between three hypothetical β-cells can be modeled using Eq. 2 where $I_{ion,i}$ represents the currents through the ionic channels and the coupling currents $I_{12}$, $I_{13}$ (and $I_{21}$, $I_{31}$) are described by Eq. 1. B. Other models considered the changes in concentration and flow of ions (e.g., Ca²⁺ and K⁺) and metabolites (e.g., FBP, G6P) through gap-junctions (GJ) in addition to the voltage differences between connected cells.

In practice, Eq. 1 means that whenever the membrane potentials of the coupled cells are different, there will be a current flowing between them, either inward or outward, in addition to the currents produced by the populations of ionic channels located in the cell membrane, thus contributing to shape the electrical response of the cells. In mathematical terms, a typical dynamical model of coupled β-cells can be described as:

$$\frac{d{V}_i}{dt}=\frac{-1}{C_{mi}}\left(I_{ion,i}+{\sum }_j^{\hspace{0.17em}}{I}_{ij}\right),$$ (2)

where $V_i$ is the membrane potential of cell i, $C{m}_i$ is the membrane capacitance, which is used in models of excitable cells as an indirect measure of cell size, $I_{ion}$ includes all the currents flowing through the populations of ionic channels in the cell membrane and the last term, $\sum _j^{\hspace{0.17em}}{I}_{ij}$, represents the currents flowing through the gap-junctions connecting cell i with its neighbors, j. In Figure 1a, an example considering three β-cells (i = 1,2,3) where cell 1 is connected to both cells 2 and 3 is shown. In this hypothetical case, assuming that β-cells have populations of Na⁺, K⁺, Ca²⁺ and leak channels, the first term of the right-hand side of Eq. 2 would be $I_{ion}=I_{Na}+I_K+I_{Ca}+I_{Leak}$ for the three cells, while the respective coupling terms $I_{12}$, $I_{13}$, $I_{21}$ and $I_{31}$ would be given by Eq. 1.

A minority of dynamical models (see for instance refs.^54,55) adopted the slightly more complex approach proposed by Kitagawa et al.⁵⁴ to simulate the gap junction currents. Their proposal was based on the Goldman-Hodgkin-Katz approximation and considered the flow of K⁺ and Ca²⁺ ions through the gap-junctions (see Figure 1b). In mathematical terms:

$$I_{ij}=\sum _j\frac{u_K{z}_{K}F{S}_G}{d_G}\left(V_i-V_j\right)\left(\frac{K_i-K_j{e}^{-z_{K}F\left(V_i-V_j\right)/RT}}{1-e^{-z_{K}F\left(V_i-V_j\right)/RT}}\right)+\sum _j\frac{u_{Ca}{z}_{Ca}F{S}_G}{d_G}\left(V_i-V_j\right)\left(\frac{C{a}_i-C{a}_j{e}^{-z_{Ca}F\left(V_i-V_j\right)/RT}}{1-e^{-z_{Ca}F\left(V_i-V_j\right)/RT}}\right)$$ (3)

where $I_{ij}$ is the junctional current between coupled cells (second term of Eq. 2), $u_K$ and $u_{Ca}$ are the mobilities of K⁺ and Ca²⁺, $z_K$ and $z_{Ca}$ are the corresponding valences, $S_G$ and $d_G$ represent the area and length of the gap-junctions, respectively, $K_i$, $K_j$, $C{a}_i$ and $C{a}_j$ are the intracellular concentrations of K⁺ and Ca²⁺ of cells i and j, and F, R and T are the Faraday constant, the gas constant and the absolute temperature, respectively.

This representation of gap-junctions in models of coupled β-cells allowed the simulation of more detailed scenarios, although inevitably increasing the complexity of the models since a mathematical representation of the changes in the intracellular ionic concentrations (K_i and Ca_i in Eq. 3) and several additional parameters are needed in comparison to the basic representation of gap-junctions given by Eq. 1.

In addition to electrical coupling and the diffusion of K⁺ and Ca²⁺ through gap junctions, other models explored the effect of gap-junctional diffusion of metabolites, such as fructose 1–6-biphosphate and glucose 6-phosphate, on the coordination of β-cells^56–58 (see Figure 1b). The diffusion of metabolites through the gap-junctions adopted a similar form than that used to simulate electrical coupling between two cells, i.e. $J_X^{ij}={\sum }_{j\in \mathrm{\Gamma }i}{p}_{c,X}\left(X_i-X_j\right)$, where $J_X^{ij}$ represents the flux of X = Ca²⁺, FBP or G6P through the gap junctions between the cell i and cells j in its neighborhood $\mathrm{\Gamma }i$, $p_c$ is the gap-junctional permeability, and X_i and X_j are the corresponding intracellular concentrations.

The modeling approach described in this section is, by far, the most popular strategy to develop multicellular models of pancreatic cells, since it allows a relatively detailed description of all the variables of interest. However, it must be noted that the computational cost involved in solving a dynamical model composed of hundreds or thousands of cells, each described by tens of coupled differential equations, is considerably high, thus requiring the use of specialized computer clusters or parallel computing.

Discrete models based on percolation theory

Discrete models of communicating β-cells, in contrast to the complex dynamical models, do not require a detailed description of all the cellular mechanisms included, which reduces the computational power required to perform simulations significantly, with the drawback that physiological interpretability, in terms of the subcellular processes involved, is completely lost.

In short, according to this approach, coupled β-cells form a network where each cell is a node and linked nodes or cells are assumed to be functionally coupled (e.g., if a cell is active, all the other cells coupled to it are also active). Coupling between cells is determined using a probabilistic approach, where p is the probability of being open for coupling and the complement, 1 – p, is the probability of not being coupled. According to percolation theory, there is a critical probability value p_c that determines if the network is formed by disconnected clusters (p < p_c) or a large predominant cluster is formed (p > p_c). An additional parameter S_p, representing the percentage of excitable and non-excitable cells, can be included in the model to simulate, using the same probabilistic approach, how coupling (given by the parameter p) and the proportion of the different subpopulations included in the model would influence the collective behavior of the cluster of cells. An example of this can be seen in Figure 2c-e, where both the connectivity between cells (represented by the gray links) and the proportions of active and inactive cells (green and black cells, respectively) have been determined probabilistically.

Figure 2.

A. One-dimensional arrangement of clusters of cells including a coupled pair of cells and an open-ended line composed of 10 coupled cells. According to the continuous approximation, the propagation of electrical signals through a linear arrangement of cells can be approximated by Eq. 5 where it is assumed that each cell contributes with ∆x to the length of the continuous line (see main text for details). B. Bi-dimensional arrangement of cells including the simple and hexagonal cubic packing (SCP and HCP, respectively). C-F. Three-dimensional structures used in multicellular models of pancreatic cells. In C-E, green and black cells represent active and inactive cells, respectively, as modeled using the discrete probabilistic approach based on percolation theory. In F, red, green and blue cells represent the ɑ, β and δ-cells, respectively.

As will be described in more detail below (see Section Applications of multicellular models of pancreatic cells), discrete models have been used to study how gap-junction coupling affects synchronization and wave propagation,³¹ to explain the effects of the cluster architecture (size and shape) on the functionality of both 2D and 3D cell aggregates,⁵⁹ to predict the effect of the proportion of excitable and non-excitable cells and coupling on the cluster activity,⁶⁰ and to determine the critical value of β-cell loss where an islet is not functional,⁶¹ which proves the usefulness of this modeling approach in spite of its relative simplicity.

Model of coupled oscillators

Another simplified approach that does not consider explicitly the cellular mechanisms involved in electrical activity, electrical coupling and paracrine signaling was adopted by Hoang et al.,⁶² who reinterpreted the Kuramoto model of coupled oscillators^63,64 to represent the pulsatile secretory behavior of ɑ, β and δ-cells and the corresponding paracrine interactions (see Figure 4c). This model is based on the following differential equation describing the oscillation phase (θ_i) of cell i, assumed to interact with cells j in its neighborhood ${\mathrm{\Lambda }}_i$:

$$\frac{d{\theta }_i}{dt}={\omega }_i+\sum _{j\in {\mathrm{\Lambda }}_i}{K}_{{\sigma }_i{\sigma }_j}sin\left({\theta }_j-{\theta }_i\right)$$ (4)

Figure 4.

A. Diagram of the communication signals included in the multicellular model of Briant et al.⁸ When active, β-cells promote the activity of δ-cells due to the electrical communication between them via gap-junctions. Somatostatin secretion then inhibits the secretion of glucagon from ɑ-cells. B. The model by Watts et al.¹²⁵ includes paracrine signals between the three types of cells. C. Model of coupled oscillators used by Hoang et al.⁶² to simulate the paracrine interactions between the ɑ, β and δ-cells. Note that only one cell of each type is shown while the model included hundreds or thousands of cells. ATP: adenosine triphosphate, KATP: ATP-dependent K⁺ channels, VDCC: voltage-dependent Ca²⁺ channels, GJ: gap-junctions, GIRK: G-protein coupled inwardly-rectifying K⁺ channel.

where ${\omega }_i$ represents the intrinsic frequency of cell i, ${\sigma }_i$ and ${\sigma }_j$ indicate if cells i and j are either ɑ, β or δ, and $K_{{\sigma }_i{\sigma }_j}$ is an interaction parameter that determines the strength of the interaction between communicating cells. In simple terms, each cell oscillates at a given frequency ${\omega }_i$ with an oscillation phase influenced by the phases of the neighbor cells (see Figure 4c). Although relatively simple, by varying the sign and value of the interaction parameter $K_{{\sigma }_i{\sigma }_j}$ in Eq. 4, this type of model is suitable for simulating both excitatory or inhibitory effects between cells and to evaluate the degree of synchronization of the islet. On the other hand, the main disadvantage of this approach is that cellular mechanisms are not considered explicitly, thus limiting the interpretation of the simulation results from a physiological viewpoint.

Continuous propagation models

In other works, in order to study the propagation of excitation waves through clusters of coupled β-cells,^65–68 the system of equations for a one-dimensional cluster of β-cells was simplified to a single equation by treating the cluster as a continuous medium formed by the connected cells. For instance, for a cluster of β-cells in 1-D, the continuous approximation can be described mathematically as:

$$\frac{\mathrm{\partial }V}{\mathrm{\partial }t}=D\frac{{\mathrm{\partial }}^{2}V}{\mathrm{\partial }{x}^2}-\frac{1}{C_m}{I}_{ion},\left(0\le x\le L\right)$$ (5)

where V is the propagating voltage through the line of N cells located along the x coordinate, each located at $x=i\mathrm{\Delta }x$(i = 0 … N), $L=N\mathrm{\Delta }x$ being the length of the continuous medium (see the bottom panel of Figure 2a), C_m the cell capacitance and $I_{ion}$ includes all the membrane currents as in the dynamical models described previously (e.g. $I_{ion}=I_{Na}+I_K+I_{Ca}+I_{Leak}$ as in the example used above). This approach was also used to simulate the propagation of waves across a bi-dimensional array of cells in rectangular coordinates,⁶⁷ as well as in spherical coordinates, assuming that an islet can be morphologically represented by a sphere.⁶⁶

Since a continuous approximation implies a reduction of the mathematical complexity of the model, the computational power needed to solve it is considerably low when compared to that required to solve a multicellular dynamical model, with the drawback that the resolution at the cellular and subcellular levels is lost.

Geometry of clusters of β-cells and islet architectures

Including electrical and paracrine communication between islet cells in multicellular models implies that a geometrical representation of the cluster of cells or islet must be defined in advance in order to determine how cells communicate with each other. Along the years, the geometry of the multicellular models of islet cells has increased in complexity, aiming to adopt the most realistic representation of the actual islet architecture.

As explained before, the first model of coupled cells only considered the interaction between two β-cells⁵³ (Figure 2a), a basic geometrical configuration that, in spite of its simplicity, has been continuously used to describe how a pair of coupled cells interact before exploring a more complex multicellular system (see for instance refs.^53,56,69,70). As an example of this, even very recently, Ren et al.⁷¹ proposed a model with only two cells to simulate the interactions between α and β cells where, instead of gap-junctions, communication between the pair of cells relies on paracrine mechanisms.

Next in complexity are the 1-D arrays of more than two β-cells, ranging in size from triplets⁷² to open-ended lines^67,68 or closed rings⁷³ including tens or hundreds of cells (see Figure 2a). The main drawback of the 1D models is that each cell has only two neighbors, and, therefore, the number of interactions between cells is considerably limited.

Bi-dimensional clusters of β-cells have adopted squared (2D-Simple Cubic Packing, SCP),^{54,59,67,74,75} hexagonal⁷⁶ (2D Hexagonal Closest Packing, HCP) and rectangular geometries⁷⁷ thus increasing the number of possible interactions between cells to four in the case of the 2D SCP and rectangular structures and six for the case of the 2D HCP geometry (see Figure 2b).

Finally, three-dimensional models have adopted a variety of geometries, including the 3D SCP and HCP structures (see Figure 2c, d), as well as more complex three-dimensional structures generated using sphere-packing algorithms (Figure 2e). Other recent models have used real islet architectures obtained via confocal microscopy and immunofluorescence techniques, aiming to develop more realistic models in terms of the composition and organization of the simulated islets (see Figure 2f). Regarding the connectivity between islet cells, 3D SCP and HCP structures have a pre-defined maximum number of interactions of 6 and 12, respectively. On the other hand, the architectures based on sphere packing (Figure 2e) and reconstructed architectures (Figure 2f) have relied on a threshold-based criteria to determine the intercellular contacts in the simulated islets, using the Euclidean distance between the coordinates of the center of the cell as the main factor to determine the intercellular interactions taking place in the simulations. A minority of models have adopted a probabilistic approach based on percolation theory to determine the connectivity of clusters of β-cells (briefly explained above; see also Figures 2c, d).⁶⁰ While these are reasonable assumptions, there are some aspects that are worth being considered. Firstly, the electrical interactions between β-cells via gap-junctions require that the membranes of the interacting cells are in close contact, and, therefore, it could be easy to overestimate the number of cells interacting locally if a large distance threshold is used. On the other hand, paracrine crosstalk within the islet might involve interactions between more distant cells, which could be also wrongly implemented if an incorrect threshold is selected. Recently, an optimization algorithm was proposed to reconstruct three-dimensional islets composed of non-overlapping ɑ, β and δ spherical cells with varying sizes assigned randomly in accordance with the capacitance distributions obtained via electrophysiological experiments.^78–80 Along with the mechanisms considered in the multicellular model at the single-cell level, the geometrical configuration of the multicellular model, and therefore its connectivity, is a key aspect with important functional implications.

Applications of multicellular models of pancreatic cells

Role of noise and heterogeneity in the electrical activity of coupled β-cells

The origin of mathematical models of clusters of β-cells is closely related to the need for a plausible explanation of the experimental observations of the heterogeneous electrical behavior of isolated β-cells and β-cells in intact islets; that is, irregular spiking or bursting in single β-cells,⁸¹ and robust synchronized bursting in intact islets.^29,51 The main goal of early models of clusters of β-cells was to elucidate whether bursting is produced intrinsically at the cellular level or is a property emerging as a result of cell coupling, a hypothesis based on the simultaneous measurement of both the electrical activity^21,29 and intracellular Ca²⁺ of neighbor cells,^82–85 which showed an apparent perfect synchronization.

One of the main hypotheses proposed to explain these observations was the channel sharing hypothesis, originally proposed by Atwater et al.,⁸⁶ postulating that the stochastic gating of “shared” ionic channels due to cell-to-cell coupling via gap-junctions, prevents single β-cells from bursting, which can be recovered when cells are electrically coupled^87–89 (see Figure 3a). Representative models of the channel sharing hypothesis are those proposed by Sherman et al.⁸⁸ and Chay and Kang,⁸⁷ who considered the stochastic nature of the gating of the ionic channels as the source of the irregular and noisy electrical activity observed in isolated β-cells, which was gradually counteracted as the number of electrically coupled cells increased, thus leading to an organized bursting activity throughout the islet. In Figure 3a, simulations of the electrical activity of single and clusters of increasing number of cells are shown. When the number of cells in the cluster increases, and therefore the number of shared channels, robust bursting is observed (top panel of Figure 3A), which contrasts with the electrical behavior observed in isolated cells and small clusters (middle and lower panels in Figure 3a). The main assumption of these models, that is, an infinite gap-junction conductance (or zero resistance), produced that the whole ensemble of cells behaves as a single cell with a higher number of channels (hence these models received the name of “supercell” models). Models by Carroll and Sherman et al.^90,91 modified the previous supercell model by Sherman et al.⁸⁸ to introduce a finite coupling conductance based on the experimental observations by Pérez-Armendariz et al.⁹² in isolated pairs of β-cells from mice. The updated multicellular model⁹⁰ showed that only moderate coupling (not infinite as suggested by the previous models) was required to synchronize the bursts of action potentials. Other models considered different sources of noise and stochasticity. For instance, Kang et al.⁹³ and Jo et al.⁹⁴ modified Eq. 2 by adding a random (additive) noise term to represent the effect of thermal fluctuations on membrane voltage; that is:

$$\frac{d{V}_i}{dt}=\frac{-1}{C_{mi}}\left(I_{ion}+{\xi }_i+\sum _j^{\hspace{0.17em}}{I}_{ij}\right)$$ (6)

Figure 3.

A. According to the channel sharing and heterogeneity hypothesis, robust bursting observed in β-cells of intact islets (top panel) is a result of coupling of a large number of cells, while in small clusters (middle panel) and isolated cells (bottom panel), irregular bursting is regularly observed. B. A simulated islet composed of heterogeneous β-cells (i.e., slow and fast bursting cells) show intermediate bursting as a result of cell coupling. C. The origin and propagation of waves in clusters of β-cells has been an aspect extensively studied using computational models. D. Multicellular models have been used to evaluate the existence and role of hubs, leader and first responder β-cells. E. Following a glucose challenge, first responder cells participate during the first phase of oscillations while hub and leader cells are thought to be relevant during the second phase.

where ${\xi }_i$ is the noise term randomly added at each time step of the simulation. As an alternative, they also explored adding a voltage-dependent (multiplicative) noise term to Eq. 2 to simulate channel gating stochasticity:

$$\frac{d{V}_i}{dt}=\frac{-1}{C_{mi}}\left(I_{ion}+{\eta }_i\left(V_i-V_{eq}\right)+\sum _j^{\hspace{0.17em}}{I}_{ij}\right)$$ (7)

where ${\eta }_i$ represents, for instance, the randomly varying conductance of the open channels and $\left(V_i-V_{eq}\right)$ is the driving force of the stochastic current depending on the membrane voltage of cell i ($V_i$ and the reversal potential ($V_{eq}$) of the permeating ion. Another example of stochasticity in ionic channels is the model by Meyer-Hermann and Benninger,⁹⁵ who included, in addition to stochastic variations in the gap-junction currents, stochasticity in the expression of ATP-dependent K⁺ channels (KATP) aiming to explain the origin of active and inactive regions observed experimentally by Rocheleau et al.⁹⁶ in an islet subject to a glucose gradient.

The heterogeneity hypothesis, on the other hand, postulates that a population of electrically coupled β-cells, with heterogeneous properties such as ionic channel conductance (or channel density), membrane capacitance (cell size), gap-junction conductance, metabolism, etc., is capable of producing robust bursting in spite of the variety of electrical behaviors produced individually (i.e., continuous spiking, bursting with different periodicities, etc.). In contrast to the channel sharing hypothesis, based on the inclusion of stochastic terms in the models, heterogeneity is achieved by assigning different values to certain model parameters to generate different electrical behaviors among the simulated cells (e.g., fast or slow bursting) without the need of stochastic terms. A representative example of this hypothesis is the model by Zimliki et al.⁹⁷ who used the phantom burster model of the pancreatic β-cell proposed by Bertram et al.,^98,99 which is based on the idea that the heterogeneous periodicity of the electrical behavior of β-cells is a result of the interaction of slow and fast oscillatory variables, as shown schematically in Figure 3b. Following this proposal, Zimliki et al.⁹⁷ simulated heterogeneous clusters of cells composed of fast and slow β-cells with the main objective of explaining why β-cells in intact islets generally show bursting with an intermediate period between 10 and 60 s, while isolated cells showed a wide range of bursting periods, from <10 s (fast) or >60 s (slow). From their simulations, they concluded that bursting of intermediate periodicity could be obtained if islets are composed either by all fast or slow cells or by a combination of the two.

An alternative hypothesis explored by early models of clusters of β-cells was based on the experiments performed by Pérez-Armendariz et al.,^100,101 who proposed a role for the intercellular K⁺ concentration in the synchronization of bursting among islet cells via depolarization of adjacent cells due to the efflux of K⁺ from neighbor cells in conjunction with a “long distance” effect caused by the diffusion of K⁺ throughout the intercellular space. This hypothesis was explored theoretically by Stokes and Rinzel¹⁰² in an islet model composed of concentric shells of uncoupled β-cells surrounded by non-β-cells, thus resembling the organization of rodent islets. A model of K⁺ diffusion was also incorporated into their model to calculate the local K⁺ concentrations affecting the K⁺ currents through the effect on the K⁺ reversal potential. As a result, in spite of the lack of direct electrical communication between adjacent β-cells (i.e. gap-junctions), their simulations showed that β-cells could synchronize by the solely effect of K⁺ accumulation and diffusion in the intercellular space, as proposed by Pérez-Armendariz et al.^100,101 although this mechanism alone was not capable of reproducing experiments in which current injection in one cell produced a corresponding voltage change in an impaled adjacent cell.^29,48 The authors concluded that both coupling through gap junctions as well as diffusion of ions in the intercellular space could be participating in the synchronized response of islet cells. Additional theoretical support to this hypothesis was provided by Rinzel et al.¹⁰³ An interesting result obtained by Stokes and Rinzel¹⁰² when heterogeneity was considered in their model, is the idea of groups of β-cells leading the electrical behavior of the islet, which can be related to recent observations suggesting the existence of a subpopulation of leader β-cells^41,104 (see Section Hubs, leaders and first responders below).

Since the first experimental reports of cellular heterogeneity in pancreatic islets published at the time by D. Pipeleers,¹⁰⁵ it has been an aspect included in practically all the multicellular models developed throughout the years. In contrast, noise (in any of its forms) and concentration gradients have not been widely considered, although early models have shown the possibility that both mechanisms could partially dictate the behavior of islet cells as members of interacting cell populations.

Wave propagation

Simultaneous recordings of the electrical activity of neighbor β-cells in intact islets showed that the electrical signals of both cells were similar both in shape and periodicity, which led to believe that the connectivity between β-cells via gap-junctions would produce a perfect synchronized response of the islet β-cells.^21,29 Similarly, the first experimental measurements of the oscillations of intracellular Ca²⁺ in rodent islets suggested that β-cells synchronized perfectly (probably due to the low time resolution of the recordings).^82–84 Other groups, based on Ca²⁺ measurements and electrophysiological measurements, suggested the possibility of wave propagation throughout the islets^85,106,107 as a synchronizing mechanism in response to glucose stimulation (described schematically in Figure 3c). For instance, Palti et al.⁸⁵ concluded, based on external measurements of the electrical signal produced by intact islets, that waves were originated from pacemaker cells with special properties such as high excitability and/or high firing frequency. On the other hand, Bertuzzi et al.,^106,107 using fluorescence techniques to measure the changes in cytosolic Ca²⁺ in isolated islets from both mouse and pigs, found that the propagation of Ca²⁺ signals involved both electrical communication and extracellular mechanisms such as the diffusion of molecules throughout the intercellular space, conclusion reached after observing that Ca²⁺ waves elicited by a mechanical stimulus rely on functional gap-junctions, while waves produced by glucose stimulation propagate independently of gap-junction communication, as indicated by the minor effect of a specific inhibitor of gap-junctions (AGA, ɑ-glycyrrhetinic acid) on the propagation of the oscillatory Ca²⁺ signals, as well as the fact that depletion of extracellular Ca²⁺ produced the blockade of the Ca²⁺ oscillations.¹⁰⁶

Following these experimental observations, theoretical models were developed to elucidate how the connectivity and properties of the coupled β-cells could be related to both the origin and propagation of electrical and Ca²⁺ waves. The first model to investigate wave propagation was proposed by Aslanidi et al.,⁶⁵ who demonstrated that Ca²⁺ oscillations are mostly asynchronous, thus appearing as periodic propagating waves. Their simulations of a linear cluster of β-cells supported the hypothesis postulating that the Ca²⁺ waves observed experimentally are produced by the propagation of electric impulses through the cluster of electrically coupled cells. From the modeling perspective, the authors proposed the continuous approximation of the propagation of Ca²⁺ waves described before (Eq. 5), later adopted by Pedersen⁶⁷ to model 1D and 2D clusters of β-cells to show that the wave propagation speed decreases as the heterogeneity of the coupling conductance increases. In a later work, Aslanidi et al.⁶⁶ reformulated their continuous model using spherical coordinates and complemented it with a diffusion model to account for the extracellular distribution of glucose, as proposed previously by a model by Bertram and Pernarowski¹⁰⁸ based on the experiments of Bennett et al.¹⁰⁹ who measured the metabolic activity (i.e. NAD(P)H) elicited by the penetration of glucose into an intact islet. Their results indicated that extracellular glucose gradients could initiate the Ca²⁺ waves observed experimentally; that is, waves initiated at the periphery, then spreading throughout the islet as a result of the diffusion of glucose into the islet.

Experiments by Rocheleau et al.⁹⁶ showed that Ca²⁺ waves elicited by a glucose gradient in mice islets, propagated only through islet regions where the glucose concentration was above the activation threshold and was halted in regions below that threshold. In contrast to the proposal of pacemaker cells initiating the propagation of waves, their results led them to hypothesize that Ca²⁺ waves were propagated as a result of the strength of coupling and the activity of KATP channels. Based on these experiments, Pedersen and Sørensen⁶⁸ showed, using both discrete and continuous models, that this phenomena could be in fact produced by the spatial variation of the ATP-dependent K⁺ channels, as hypothesized by Rocheleau et al.⁹⁶ In a different model, Meyer-Hermann and Benninger⁹⁵ proposed that a stochastic variation of gap-junction currents and the expression of KATP channels could be behind the observations by Rocheleau et al.⁹⁶

In a detailed study involving both experiments and computational modeling, Benninger et al.³¹ measured the velocity of the Ca²⁺ waves in mouse islets and evaluated the effects of reducing the electrical coupling between β-cells. They found that chemical blocking of gap-junctions slowed the propagation of the Ca²⁺ waves, and that gap-junction knockout completely disrupted both the propagation of Ca²⁺ waves, as well as the synchronization of Ca²⁺ oscillations. They reproduced these experimental observations using a computational model of a cubic cluster of heterogeneous β-cells, concluding that cellular heterogeneity is sufficient to produce propagating Ca²⁺ waves. In a follow-up study by the same group,⁵⁵ they demonstrated experimentally that in an islet subject to a glucose gradient, waves originate in regions where cells are highly excitable in comparison with the average excitability of the islet β-cells, being the direction of propagation determined by the intrinsic heterogeneity of the islet β-cells, while the extent of propagation across the islet is regulated by the glucose-dependent gap junction coupling. The authors concluded that including the electrochemical gradients in the junctional currents (as in Eq. 3) was needed to reproduce the experimental observations. From these results, it was suggested that islets could have a pacemaker region randomly determined by the location of highly excitable cells, although they did not discard a possible role for islet development in the distribution of the pacemaker region.

A different approach based on network theory was adopted by Stozer et al.¹¹⁰ to analyze the functional connectivity of β-cells in slices of mouse islets based on the correlation between Ca²⁺ signals recorded experimentally at different locations of the islet slices. They found that functional networks with small-world properties (i.e. high local coupling with shortcuts to distant cells) were formed (see the works by Stozer et al. and Markovic et al.^110–112 for details). Since the mechanism of communication between distant regions of the islets is still unknown, following these observations, Cappon and Pedersen¹¹³ developed a three-dimensional model aiming to elucidate how the functional networks observed experimentally could be formed in pancreatic islets. Their simulations showed that the small-world networks in pancreatic islets are not necessarily produced as a result of a long-range mechanism or signal communicating distant cells, but they can arise as a result of heterogeneity and wave propagation throughout the islet via gap-junction coupling, without discarding a potential contribution of intra-islet neurons or paracrine signals.

Given the importance of wave propagation as a reflection of the structural and functional properties of the islets, it is key to understand how they originate and propagate in order to determine their functional implications. What is clear is that gap-junction coupling, excitability, expression of KATP channels and intercellular gradients of glucose or other molecules are likely to have a relevant role in the origin and propagation of waves throughout the islets. Pacemaker cells, on the other hand, although still controversial, could participate to some extent, aspect that has also been explored using multicellular models, as described in the next section.

Hubs, leaders and first responders

Throughout the years, different functional subpopulations of β-cells have been identified and characterized by the prolific combination of experiments and computational modeling (Figure 3d, e). Johnston et al.⁴¹ proposed, based on experimental observations, that highly functionally connected β-cells, called hub cells (see Figure 3e), a term firstly used in 2013 by Stozer et al.¹¹⁰ in the context of pancreatic islets, are required to maintain islet oscillations and are key for the coordinated islet response to high glucose and insulin release. According to this proposal, the highly connected hub cells are responsible for coordination of the Ca²⁺ activity of the beta cell network.⁴¹ Although the authors suggested that hub cells could have pacemaker properties, their firing activity, although preceding the activity of the follower cells, also outlasts it, which differs greatly from the typical pacemaker activity (e.g. cardiac pacemaker cells). Hub cells showed no preference regarding their location within the islets and were characterized by a particular metabolic signature (i.e., increased proton pumping in the mitochondria, ATP synthase activity and ATP generation) and a reduced number of insulin granules. On the other hand, Salem et al.¹¹⁴ identified a subpopulation of “leader cells” that currently are known to lead the oscillations during the second phase (see Figure 3e). Although at first, leader cells were thought to be similar to hub cells, as suggested by the high number of functional links, recent experimental studies^115,116 have concluded that they constitute a different subpopulation of β-cells. In addition to hub and leader cells, Kravets et al.¹¹⁵ described a different sub-population of cells denoted as “first responders,” defined as those β-cells showing the fastest response to a glucose stimulus (see Figure 3e), likely to be characterized by a high membrane excitability and lower coupling with its neighbors, as suggested by simulations performed using a multicellular model. In contrast to hub and leader cells, which seem to be relevant for the regulation and coordination of the second phase of secretion, the so-called first responders seem to participate mainly during the first phase of secretion (Figure 3e). Recent studies have confirmed that hubs, leaders and first responders are in fact different subpopulations (see for instance refs.^115,116), thus clarifying the functional organization of β-cells within the islets.

Hogan and Peercy¹¹⁷ implemented a multicellular model to evaluate the feasibility of the hypothesis postulating the existence of hub cells. While they were not able to reproduce the behavior and properties of hub and leader cells as described by Johnston et al.⁴¹ and Salem et al.,¹¹⁴ they found that certain cells, called switch cells, have the ability of switch off the islet activity when they are silenced. Dwulet et al.¹¹⁸ performed computational simulations using a multicellular model to test if small populations of β-cells could have a disproportionate control of the islet Ca²⁺ dynamics. According to their results, there was not a particular type of cell capable of exerting a disproportionate control of the islet Ca²⁺ via direct electrical coupling, which contradicted the hypothesis of hub cells acting as pacemakers. However, as the authors acknowledge, other mechanisms not considered in their model, such as paracrine signaling or the diffusion of intermediate metabolites through gap junctions, cannot be discarded and must be evaluated in future works.

Models of human islets have also been developed aiming to predict how hubs of β-cells could delineate the global islet activity while considering the important differences between rodents and humans, both at the single-cell and islet levels. Loppini and Chiodo¹¹⁹ evaluated the effects of hub cells on the synchronization of human β-cells for the three electrical regimes observed experimentally; that is, action potential spiking and fast and slow bursting.^14,120 Interestingly, their simulations indicated that for the spiking and fast bursting regimes, the presence of hub cells could have a negative effect on the cells’ synchronization. In contrast, for the slow bursting regime, the presence of hub cells enhanced the synchronization of coupled human β-cells. A different model by Lei et al.¹⁰⁴ predicted that the coordinated oscillatory response both in human and mouse islets can be dictated by hub cells, although according to their simulations, human islets could be more susceptible to the inhibition of hubs than mouse islets.

In a recent review, Peercy and Sherman¹²¹ discussed the experimental and theoretical evidence for the existence of hubs of β-cells acting as pacemakers. Although they agree that some β-cells may have a disproportionate influence on the islet behavior, according to their view, the available evidence does not support the existence of pacemaker cells. Other voices (i.e. Korosak et al.¹²² have proposed a reconciled scheme in which hubs of β-cells are generated not by differences in the cellular mechanisms in some “special” β-cells, but as a result of intercellular communication. According to this proposal, if a hub fails at some point, other cells will take over as a new commanding hub as long as communication between β-cells remains possible. Simulations performed by Scialla and Loppini¹²³ supported this proposal by predicting that cells that are not hubs can turn into hubs and vice versa in response to changing stimuli.

Undoubtedly, multicellular models have been useful to identify the characteristics of these subpopulations of β-cells, as well as to evaluate how these subpopulations would influence the islet response, both in humans and rodents. Despite this, further work, both theoretical and experimental, is necessary to detail the mechanisms of action of these populations on the islet function.

Functional implications of the islet architecture

While previous models had focused mainly on synchronization, noise, cellular heterogeneity, wave propagation and hub cells, Nittala et al.⁷⁶ focused on investigating the functional implications of cytoarchitecture (i.e. number of β -cells, connectivity and coupling strength), a relevant aspect not considered in detail in previous models. They proposed the HCP geometry for models of pancreatic islets and evaluated how the cytoarchitecture influenced the fraction of bursting β-cells, the bursting period, plateau fraction and synchronization, as well as the amplitude of intracellular Ca²⁺ oscillations. In a latter work, the same group,¹²⁴ based on extensive simulations, proposed a cytoarchitectural integrity descriptor denominated cluster coupling index as a measure of morphological and structural integrity of the islet.

Hraha et al.⁵⁹ studied the role of cytoarchitecture and coupling in the islet function by forming 2D and 3D aggregates of MIN6 β-cells to determine the effect of dimensionality (2D or 3D) and size on the synchronization of excitability. They showed that in 3D aggregates at high glucose, a greater proportion of cells synchronizes, and that the velocity of Ca²⁺ waves is greater in comparison to the 2D aggregates. Interestingly, 2D aggregates showed spontaneous Ca²⁺ transients at low glucose, which were not observed in the 3D aggregates, although large 2D aggregates also showed a quiescent behavior at low glucose. The authors then used a probabilistic model based on percolation theory (previously used by Benninger et al.)³¹ to explain these observations. In a later work,⁶⁰ they evaluated the capacity of a network of coupled β-cells to generate a robust response when there is a subpopulation of non-excitable cells. Their simulations indicated that there could be an excitability threshold where there is a rapid transition between the active and inactive state of the islet, determined by the proportions of excitable and non-excitable cells with a relevant role of gap-junction coupling. These models highlighted the importance of cytoarchitecture for the functionality of clusters of β-cells, an important aspect that must be considered in future models of pancreatic islets.

Models of paracrine signaling

From all the multicellular models developed so far, only the dynamical model by Briant et al.⁸ has considered the electrical communication between neighbor β-cells and β and δ-cells, as well as the paracrine interactions between δ and ɑ-cells (see Figure 4a) in a multicellular model in which actual architectures of human islets were used to determine the structural and connectivity properties of the islets. With their model, Briant et al.⁸ investigated how activation of δ-cells by electrically coupled β-cells could mediate the inhibition of glucagon secretion. In addition to an electrical model mathematically described by Eq. 2, the authors included minimal secretion models to estimate the concentration of glucagon, insulin and somatostatin. The concentration of somatostatin was then used to regulate the activity of ɑ-cells via a somatostatin-dependent ionic current, as depicted in Figure 4a. Previously, Watts et al.¹²⁵ had used a similar approach in a model of the paracrine interactions between islet cells where only one cell of each type (ɑ, β and δ) represented the average behavior of the corresponding populations. In contrast to the multicellular model by Briant et al.,⁸ the model by Watts¹²⁵ included paracrine interactions between the three types of cells, as described graphically in Figure 4b.

A different approach to simulate the paracrine interactions between ɑ, β and δ-cells in real islet architectures was adopted by Hoang et al.⁶² in a model of coupled oscillators, mathematically and graphically described by Eq. 4 and Figure 4c, respectively. In this model, each cell is represented by an oscillating function with an intrinsic frequency and phase (ω and θ in Eq. 4 and Figure 4c), depending on the type of cell (ɑ, β and δ) where paracrine signals are described by positive and negative interaction terms of variable strength (parameter K in Eq. 4 and Figure 4c). As a result, depending on the value and sign of the corresponding interaction term and the phase difference between the interacting cells, oscillation phases are affected between neighbor cells following Eq. 4. In spite of the oversimplified representation of the physiological mechanisms involved, the model of coupled oscillators by Hoang et al.⁶² was useful to evaluate how synchronization of hormone pulses can be controlled at different glucose regimens.

Including paracrine signaling in multicellular models of pancreatic islets, in addition to electrical coupling, is crucial to capture the complexity of the regulatory mechanisms involved in the control of blood glucose. However, building such a complete model is still challenging. Despite this, recent models including paracrine and electrical communication have laid the foundations for the development of increasingly realistic models.

Discussion

As described throughout this article, multicellular models of pancreatic islets are composed of a myriad of elements at different levels of organization. Firstly, a mathematical representation of the physiological behavior at the cellular level must be adopted, which implies selecting 1) a model for each type of cell considered in the model and 2) a modeling strategy (i.e., dynamical, continuous, probabilistic or oscillators). Secondly, a geometrical representation of the model must be selected, including its dimensionality (1, 2 or 3 dimensions) and configuration; that is, open or closed ended for 1-D, cubic, hexagonal (for 2 or 3-D), sphere-packed or reconstructed realistic architecture (3D). Implicitly, this determines the connectivity of the model, of great importance for the global behavior of the cluster or islet model. Thirdly, the communication pathways to be included in the model must be selected; that is, electrical and/or paracrine signaling. Once the components of the model have been determined, researchers can determine if a pathological scenario will be simulated, either by inducing alterations at the cellular level (e.g., ionic channels, metabolism, secretion), at the communication mechanisms such as gap-junctions or paracrine signaling or at the architecture level (e.g., loss of β-cell mass).

By definition, a model is a representation of a real system with a certain degree of approximation. As such, models have limitations that extend from its constituents. In this case, approximations are made practically at every stage of the process described above. For instance, different models of the β-cells have been proposed in the last decades, each based on a different physiological hypothesis (reviewed extensively in ^46–48). Then, when a particular β-cell model is selected as the building block of a multicellular model, the simulation results are delineated by the cellular mechanisms included and the underlying physiological hypothesis. In addition, although important advances have been achieved, there are still open questions about the physiology of the ɑ, β and δ-cells. On the other hand, the selection of the geometrical representation greatly determines the connectivity of the model that could be either under or overestimated. Moreover, if paracrine signals are not included in the multicellular model, as in most of the models described in this review, the simulated scenarios will not reflect the physiological behavior of the islet. In fact, even considering the paracrine interactions, models of islets proposed so far lack a representation of relevant aspects such as the islet vasculature, hormone diffusion through the intercellular space, the irregular shape of δ-cells that, according to some reports, show long neurite-like processes which can greatly influence the connectivity of the islet cells.³

Despite these limitations, in our view, models of pancreatic islets, as challenging as they are, along with all the advanced experimental techniques available to date, will keep contributing greatly to gain a better understanding about how all the cellular mechanisms involved in the regulation of glucose interact at the islet level and importantly, to predict how this coordinated behavior would be affected by functional and structural alterations.

Funding Statement

The author(s) reported that there is no funding associated with the work featured in this article.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Supplementary material

Supplemental data for this article can be accessed online at https://doi.org/10.1080/19382014.2023.2231609