Diabetes is caused by the inability of electrically coupled, functionally heterogeneous β-cells within the pancreatic islet to provide adequate insulin secretion. Functional networks have been used to represent synchronized oscillatory [Ca2+] dynamics and to study β-cell subpopulations, which play an important role in driving islet function. The mechanism by which highly synchronized β-cell subpopulations drive islet function is unclear. We used experimental and computational techniques to investigate the relationship between functional networks, structural (gap junction) networks, and intrinsic β-cell dynamics in slow and fast oscillating islets. Highly synchronized subpopulations in the functional network were differentiated by intrinsic dynamics, including metabolic activity and KATP channel conductance, more than structural coupling. Consistent with this, intrinsic dynamics were more predictive of high synchronization in the islet functional network as compared to high levels of structural coupling. Finally, dysfunction of gap junctions, which can occur in diabetes, caused decreases in the efficiency and clustering of the functional network. These results indicate that intrinsic dynamics rather than structure drive connections in the functional network and highly synchronized subpopulations, but gap junctions are still essential for overall network efficiency. These findings deepen our interpretation of functional networks and the formation of functional subpopulations in dynamic tissues such as the islet.
isletnetwork theorycoordinated oscillationsβ-cell oscillationsResearch organismMousehttp://dx.doi.org/10.13039/100000002National Institutes of HealthR01 DK102950BenningerRichard KPhttp://dx.doi.org/10.13039/100000002National Institutes of HealthR01 DK106412BenningerRichard KPhttp://dx.doi.org/10.13039/100000001National Science FoundationGraduate Research Fellowship DGE-1938058_BriggsBriggsJennifer Khttp://dx.doi.org/10.13039/100008871Juvenile Diabetes Research Foundation United States of America3-PDF-2019-741-A-NKravetsVirahttp://dx.doi.org/10.13039/100008307Beckman Research Institute, City of HopeUC24 DK104162KravetsVirahttp://dx.doi.org/10.13039/100000861Burroughs Wellcome Fund25B1756KravetsVirahttp://dx.doi.org/10.13039/100000002National Institutes of HealthDK126360DwuletJaeAnn Mhttp://dx.doi.org/10.13039/100000002National Institutes of HealthLM012734AlbersDavid Jhttp://dx.doi.org/10.13039/501100000855University of BirminghamDynamic Investment FundMarinelliIsabellaThe funders had no role in study design, data collection and interpretation, or the decision to submit the work for publication.Author impact statementThe functional network of β-cells in the pancreatic islet, which has been used to identify important subpopulations of β-cells implicated in diabetes progression, is indicative of intrinsic dynamics rather than structural communication, implying islet robustness.Introduction
Diabetes mellitus is a global epidemic afflicting >500M adults world-wide (Sun et al., 2022) and is associated with dysfunction or death of insulin-producing β-cells within pancreatic islets. β-cells are electrically coupled through connexin36 (Cx36) gap junctions (Benninger et al., 2008; Serre-Beinier et al., 2000; Ravier et al., 2005), and are functionally heterogeneous. Electrical coupling and heterogeneity are both central to regulating the homeostasis of glucose and other nutrients (Benninger and Kravets, 2022; Pørksen et al., 2002; Matveyenko et al., 2012; Bratusch-Marrain et al., 1986). Gap junction electrical coupling is essential for synchronizing β-cell electrical dynamics and allowing insulin to be released in robust coordinated pulses (Benninger et al., 2008; Benninger and Kravets, 2022; Weitz et al., 2021; Head et al., 2012). Functional heterogeneity allows improved islet robustness (Dwulet et al., 2021) and control of islet pulsatile dynamics (Benninger and Kravets, 2022; Dwulet et al., 2021; Dwulet et al., 2019). In diabetes, there can be a lack of intra-islet β-cell communication (Matveyenko et al., 2012; Satin et al., 2015; Farnsworth and Benninger, 2014) and an insufficient number of adequately functioning β-cells (Matveyenko et al., 2012; Bratusch-Marrain et al., 1986; Weitz et al., 2021; Head et al., 2012). Therefore, the role of islet communication, β-cell heterogeneity, and collective synchronization is fundamental to understanding islet dysfunction in diabetes.
Network analysis is a tool used to quantify relationships between interacting parts of a system (Koutrouli et al., 2020; Strogatz, 2001; Newman, 2003; Watts and Strogatz, 1998; Lynn and Bassett, 2019), represented by its entities (nodes) and their interactions (edges). The islet can be studied using structural or functional networks. In a structural network, edges represent physical conduits for communication (Lynn and Bassett, 2019; Bullmore and Sporns, 2009; Gansterer et al., 2019; Rings et al., 2022), which include gap junctions in the islet. However, islet networks are often studied using a functional network representation, where edges connect β-cell pairs that have highly correlated [Ca2+] dynamics (Lynn and Bassett, 2019; Chialvo, 2010; Stožer et al., 2013; Johnston et al., 2016). β-Cell functional networks have been suggested to follow a scale-free distribution (Stožer et al., 2013) with high clustering (Gosak et al., 2018; Markovič et al., 2015) and ‘small-world-like qualities’ (see Table 1). In small world networks, there exists a subset of highly synchronized nodes (referred to as ‘β-cell hubs’ in the islet; Bullmore and Sporns, 2009; Johnston et al., 2016), which have stronger than average influence over network dynamics. Optogenetic-mediated hyper-polarization of these β-cell hubs has been demonstrated to perturb overall islet coordination, and these hubs are disrupted under diabetogenic conditions (Johnston et al., 2016). Therefore, in theory, preferential removal of these hubs will stop the network from functioning. However, it has been questioned whether this small world topology is possible in the islet, as individual cells (or small subpopulations of cells) do not have enough electrical influence to control the entire islet (Dwulet et al., 2021; Satin et al., 2020; Rutter et al., 2020).
Essential network statistics and types.
Left: Five network statistics used to quantify the network in this paper. Representative networks show an example of the statistic in red and the rest of the network in blue. Right: Five network types referred to in this paper. Regular, small world, and random networks are all made with the same number of nodes and edges, but different configurations of edges. Scale free network shows three “hub” nodes where node size is proportional to degree.
Network essentials
Network statistics
Network types
Degree
Number of edges for a given node
Regular
A network with ordered connections: high clustering coefficient and long average path length
Degree distribution
The distribution of the network’s degrees
Small world
A regular network with a few rewired edges: high clustering coefficient and short average path length
Clustering coefficient
Likelihood of how often neighbors of a node share connections with each other3(#oftriangles)(#ofconnectedtriples)
Random
A network with random connections has low clustering coefficient and short average path length
Shortest path
Shortest distance between any two nodes
Scale-free
A network whose degree distribution follows a power law, such that there are a few very highly connected nodes called hubs
Efficiency
Inverse of the shortest path
Weighted
A network whose edges are weighted by some edge property
Oscillations of [Ca2+] in β-cells are driven by the activity of glucokinase, closure of ATP-sensitive K+ (KATP) channels, and subsequent bursts of action potentials. Therefore, synchronized [Ca2+] dynamics, and the corresponding functional network, may be influenced by both gap junction communication and factors influencing intrinsic cell dynamics such as glucokinase and KATP activity. Because debate exists over the topology of the islet network and whether β-cell subpopulations can dictate islet dynamics, an important question is how the functional network and β-cell hubs are driven by structural communication compared to intrinsic cell dynamics. This question is important in our understanding of islet dynamics in diabetes because both the gap junctions (e.g. the structural network) (St. Clair et al., 2020; Corezola do Amaral et al., 2020; Farnsworth et al., 2022; Hodson et al., 2013; Carvalho et al., 2012; Farnsworth et al., 2016) and synchronization (e.g. the functional network) (Pørksen et al., 2002; Satin et al., 2015; Corezola do Amaral et al., 2020; Hodson et al., 2013; Haefliger et al., 2013) can be impaired in diabetic conditions.
To investigate the relationship between functional networks, structural networks, and the intrinsic dynamics of the β-cells, we framed our study around three questions. Our first question asks, are highly correlated subpopulations (β-cell hubs) within the functional network differentiated with respect to the structural network or by their intrinsic properties? While sometimes assumed, it is unclear whether β-cell hubs show significantly greater structural connections or whether they are differentiated by their intrinsic dynamics. Our second question asks: what does the islet functional network indicate about its underlying structure or the intrinsic dynamics of individual cells? That is, are edges within the functional network representing strong structural connections or pathways between β-cell nodes? Finally, as gap junctions may become dysfunctional in diabetic conditions, our third question asks: how do changes to the structural network influence the islet functional network? By answering these questions, we provide insight into the relationship between structure, function, and intrinsic cell dynamics in the pancreatic islet.
ResultsSimulations of fast Ca<sup>2+</sup> oscillations reveal that β-cell hubs have a more pronounced difference in metabolic activity than gap junction coupling
To investigate the relationships between functional and structural networks, we first analyzed a simulated β-cell network. We used a well-validated multi-cellular coupled ODE model, which describes the electrophysiological properties of the β-cells (Dwulet et al., 2019; Notary et al., 2016; Hraha et al., 2014; Westacott et al., 2017) and results in fast calcium oscillations (<2 min). This model is based on the Cha-Noma model (Cha et al., 2011) and contains detailed kinetics of the main ion channels in the islet that affect membrane potential, including the voltage-gated Ca2+ channels (CaV), KATP channels, or store-operated currents. We include heterogeneity in parameters underlying cellular excitability, such as the rate of glucokinase (kglyc) and maximum KATP conductance (gKATP). We also included heterogeneity in the gap junction electrical coupling (gcoup) (Figure 1a). The model was simulated at 11 mM glucose.
Analysis of parameters underlying functionally connected cells from the Cha-Noma model, representing fast oscillations.
(a) Schematic of 1000 β-cell computational model, with cells false-colored by heterogeneity in rate of glucokinase, kglyc (left) and gap junction conductance, gcoup (right) parameter values. (b) Distribution of functional connections (edges), determined from functional network analysis. Colors show five simulated islets. Hub cells (red outline) are any cell with >60% of maximum number of links. (c) Two-dimensional slice of the simulated islet, with lines (edges) representing functional connections between synchronized cells. Hub cells indicated in red. Slice is taken from middle of islet (see inset). (d) Representative [Ca2+] time course of a hub (red) and non-hub (blue). (e) Average rate of glucose metabolism (kglyc) values compared between hubs and non-hubs in a Cha-Noma simulated islet. Each data point represents the averaged values for a single islet. Effect size was 2.85. (f) As in e for maximum conductance of ATP-sensitive potassium channel (gKATP). Effect size was 0.31. (g) As in e for gap junction conductance (gcoup). Effect size was 0.82. (h) Pearson correlation coefficient between duty cycle and kglyc, gKATP, and gcoup for all cells. Significance in e–g was determined by paired Student’s t-test. *p ≤ 0.05 and ***p ≤ 0.001. In e–g each data point corresponds to the average value over a single simulated islet. Error bars represent s.e.m.
Functional network sensitivity to threshold in Cha-Noma model.
(a) Degree distribution of five simulated islets with five different thresholds. (b) kglyc comparing hubs and non-hubs across five different thresholds. (c) gcoup comparing hubs and non-hubs across five different thresholds. Significance is assessed using Tukey’s multiple comparisons. (d) Islet with representative cell connections calculated by a lower threshold (Rth=0.999, right) and higher threshold (Rth=0.99995, left). Synchronized cell pairs are shown in blue, gap junction pairs are shown in red, and both are shown in black. (e) Probability of a synchronized cell pair given gap junction for five thresholds. (f) Probability of gap junction given synchronization for the five thresholds. (g) Venn diagrams for the probability of synchronization (blue) and gap junction connection (red) with the overlapping area representing cell pairs that were both synchronized and gap junction coupled.
Functional network dependence with alternative hub cell definition in Cha-Noma model.
Hubs can be alternatively defined as the 10% of cells with the highest number of functional connections. (a) kglyc comparing alternatively defined hubs and non-hubs. (b) gKATP comparing alternatively defined hubs and non-hubs. (c) gcoup comparing alternatively defined hubs and non-hubs. Significance was determined by paired Student’s t-test. *p≤0.05 and ***p≤0.001. Each data point corresponds to the average value over a single simulated islet.
Our first question asks whether subpopulations that emerge from the β-cell functional network are driven by the structural network or by intrinsic properties of the cells. We extracted the islet functional network by assigning a node to each cell and an edge between any cell pair for which the intracellular free Ca2+ ([Ca2+]) time course showed a correlation coefficient greater than a threshold (Rth). The degree was defined as the number of edges a given node possesses (Table 1). In order to study similar subpopulations as in previous studies, we set the threshold Rth to match previous studies which show a normalized degree distribution that includes many cells with a low degree and few cells with a high degree (Stožer et al., 2013; Johnston et al., 2016) (see Methods) (Figure 1b and Table 1). We then identified β-cell hubs as cells in which the normalized functional network degree was greater than 60%, in accordance with prior studies that identified β-cell hubs (Johnston et al., 2016; Figure 1b–d). β-Cell hubs made up 5–12% of the coupled Cha-Noma model.
We examined the relationship between functional network-derived β-cell hubs and parameters impacting their intrinsic dynamics. Glucokinase conversion of glucose to glucose 6-phosphate is the rate-limiting step in glycolysis, therefore the rate of glucokinase activity (kglyc) is a proxy for the metabolic activity in the cell. The maximal open-channel KATP conductance (gKATP) is equivalent to the number of KATP channels, which also influences β-cell dynamics. β-Cell hubs had a significantly higher kglyc than non-hubs (mean± SD 0.14±0.0007 ms–1 vs 0.12±0.0017 ms–1, p<0.001) (Figure 1e) but did not differ in gKATP (2.27±0.05 nS vs. 2.31±0.01 nS, p=0.18) (Figure 1f). These results indicate that in the coupled Cha-Noma model, β-cell hubs are determined by their metabolic activity more than solely the number of KATP channels. Interestingly, β-cell hubs had only a slightly higher gcoup than non-hubs (0.74±0.07 pS vs. 0.63±0.01 pS, p=0.032) (Figure 1g). To investigate the influence of these parameters on Ca2+ dynamics in this model, we correlated the parameter values with [Ca2+] oscillation duty cycle for each simulated islet. In the coupled Cha-Noma model, kglyc was more strongly correlated with duty cycle than gKATP (p<0.0001) and gcoup (p<0.0001) (Figure 1h). Therefore, β-cell hubs were most strongly differentiated by kglyc, which in turn was strongly correlated with the [Ca2+] oscillation duty cycle.
To ensure consistency in our findings, we repeated the analysis with several thresholds (Rth) (Figure 1—figure supplement 1a). For all thresholds analyzed, the difference between hubs and non-hubs was greater for kglyc than for gcoup (Figure 1—figure supplement 1b and c), with kglyc being significantly higher in hubs for all five thresholds and gcoup being significantly higher in hubs for only two thresholds. We also used an alternative method to identify β-cell hubs, as the 10% of cells with the highest network degree. With this alternative definition, β-cell hubs still had a significantly higher kglyc, a slightly higher gcoup and similar gKATP as compared to non-hubs (Figure 1—figure supplement 2).
These results indicate that both rate of glucokinase activity and gap junction coupling correlate with high cellular synchronization, where the rate of glucokinase activity has a much stronger correlation.
Simulations of slow Ca<sup>2+</sup> oscillations reveal that K<sub>ATP</sub> channel conductance and gap junction coupling are distinct features of β-cell hubs
In addition to fast oscillations modeled in Figure 1, β-cells exhibit slow Ca2+ oscillations with periods greater than 2 min (Liu et al., 1998). We next used the integrated oscillator model (IOM) (Marinelli et al., 2021; Marinelli et al., 2022) that describes details of the metabolic oscillations that underlie slow Ca2+ oscillations to test whether β-cell hubs emerging from a model of slow oscillations were differentiated by their gap junction coupling or intrinsic dynamics. Unlike the Cha-Noma model (Figure 1), the IOM has not previously been coupled using more than two cells, which is necessary for studying heterogeneity. We formed a coupled IOM, again with heterogeneity in parameters describing the rate of glucokinase activity (kglyc) (referred to as vGK in previous manuscripts; Marinelli et al., 2021; Marinelli et al., 2022), maximum KATP conductance (gKATP), and structural gap junction electrical coupling (gcoup) and simulated at 11 mM glucose. Due to limitations in computational power, we limited the islets to 260 cells (Figure 2a). We again chose a threshold to obtain a ‘scale-free-like’ distribution (Figure 2b and c). β-Cell hubs made up 3–7% of the coupled IOM.
Analysis of parameters underlying functionally connected cells from the integrated oscillator model, representing slow oscillations.
(a) Two-dimensional slice of the simulated islet from slow simulation, with lines (edges) representing functional connections between synchronized cells. Hub cells indicated in red. Slice is taken from middle of islet (see inset). (b) Distribution of functional connections (edges), determined from functional network analysis. Colors show five simulated islets. Hub cells (red outline) are any cell with >60% of maximum number of links. (c) Representative [Ca2+] time course of a hub (red) and non-hub (blue) from slow islet simulation. (d) Average rate of glucokinase (kglyc) values compared between hubs and non-hubs in a slow simulated islet, retrospectively analyzed. Each data point represents the averaged values for a whole islet. Effect size was 0.65. (e) As in d for maximum conductance of ATP-sensitive potassium channel (gKATP). Effect size was 1.27. (f) As in d for gap junction conductance (gcoup). Effect size was 2.94. (g) Pearson correlation coefficient between duty cycle and kglyc, gKATP, and gcoup for all cells. (h) As in d but for fast model with 260 cells. (i) As in e but for fast model with 260 cells. (j) As in f but for fast model with 260 cells. Significance in d–j was determined by paired Student’s t-test. *p≤0.05 and **p≤0.01. In d–j each data point corresponds to the average value over a single simulated islet. Error bars represent s.e.m.
Functional network sensitivity to threshold in integrated oscillator model (IOM).
(a) Average rate of glucose metabolism (kglyc) values compared between hubs and non-hub in IOM simulated islet determined using threshold 0.999999998. There were on average 18 hubs. This threshold is lower than that used in Figure 2. Each data point represents the averaged values for a single islet. (b) As in a for maximum conductance of ATP-sensitive potassium channel (gKATP). (c) As in a for gap junction conductance (gcoup). (d) Average rate of glucose metabolism (kglyc) values compared between hubs and non-hubs in IOM simulated islet determined using threshold 0.9999999995. There were on average 11 hubs. This threshold is higher than that used in Figure 2. Each data point represents the averaged values for a single islet. (e) As in d for maximum conductance of ATP-sensitive potassium channel (gKATP). (f) As in d for gap junction conductance (gcoup).
Functional network dependence with alternative hub cell definition in integrated oscillator model.
Hubs can be alternatively defined as the 10% of cells with the highest number of functional connections. (a) kglyc comparing alternatively defined hubs and non-hubs. (b) gKATP comparing alternatively defined hubs and non hubs. (c) gcoup comparing alternatively defined hubs and non-hubs. Significance was determined by paired Student’s t-test. ***p≤0.001. Each data point corresponds to the average value over a single simulated islet.
β-Cell hubs did not differ in kglyc compared to non-hubs (mean ± SD 3.67±0.47 ms–1 vs 3.36±0.07 ms–1, p=0.19) (Figure 2d). However, β-cell hubs had significantly lower gKATP (20.1±0.3 nS vs. 20.4±0.1 nS, p=0.04) (Figure 2e) and significantly higher gcoup (9.5±1.2 pS vs. 6.0±0.1 pS, p=0.0028) (Figure 2f). These findings were less stable than the Cha-Noma model when the threshold Rth was altered (Figure 2—figure supplement 1). When identifying β-cell hubs as the 10% of cells with the highest network degree, hubs still had significantly higher gcoup but similar gKATP and kglyc compared to non-hubs (Figure 2—figure supplement 2).
To test why hubs in the coupled IOM were determined by gKATP more than kglyc (as in the Cha-Noma model), we again correlated duty cycle with parameter value. As opposed to the coupled Cha-Noma model (Figure 1h), in the coupled IOM, duty cycle was strongly negatively correlated with gKATP and weakly positively correlated with kglyc (gKATP vs. kglyc: p<0.0001 and gcoup vs kglyc: p=0.0002) (Figure 2g). Therefore, independent of oscillation type and computational model, the parameter that was most correlated with duty cycle was also most influential in determining hubs in the functional network.
To enable direct comparisons between the coupled Cha-Noma and coupled IOM, we re-simulated the coupled Cha-Noma model with similar numbers of cells as the IOM. In the smaller coupled Cha-Noma model, the influence of kglyc on β-cell hubs became more similar to the influence of gcoup (Figure 2h and j). There was still no significant difference in gKATP for β-cell hubs and non-hubs (Figure 2i). These results suggest that as the islet becomes smaller, the influence of gap junction coupling in determining β-cell hubs grows to be comparable to the intrinsic dynamics (kglyc, gKATP) that drive Ca2+ oscillation duty cycle.
Elevated metabolic activity, but not elevated gap junction permeability, is observed experimentally in highly synchronized cells
We next applied experimental approaches to investigate the β-cell functional network and the properties of highly synchronized cells. We performed time-lapse imaging of [Ca2+] within islets isolated from Mip-CreER;Rosa-LSL-GCamP6s mice that express GCamP6s in β-cells (β-GCamP6s mice, see Methods). We identified β-cells that showed synchronous slow oscillations in [Ca2+] (Figure 3a and b). We extracted the functional network from [Ca2+] oscillations in the second phase using a threshold of Rth = 0.9 for all islets analyzed (Figure 3c), where this threshold was chosen to reflect a scale-free-like distribution for most islets (Figure 3d). Our results showed no correlation between network degree and mean GCamP6 fluorescence intensity (p>0.8). β-Cell hubs (>60% of max degree) made up 3–40% of islets.
Experimental comparison between the functional network from slow β-cell oscillations, NAD(P)H activity, and coupling conductance.
(a) Mouse pancreatic islet expressing fluorescent calcium indicator GCaMP6s in β-cells. Glucose level 11 mM. (b) GCamp6s time traces recorded at 2 and 11 mM glucose. Red curves represent dynamics of the most coordinated cells. These cells had highest number of edges, i.e., normalized degree >0.9. Green curves represent dynamics of the least coordinated cells, i.e., normalized degree <0.1, the rest of the cells are shown in gray. Only second phase (shown in black box) was used for functional network analysis. (c) Ca2+-based functional network of the islet shown in (e). Red dots represent most coordinated cells, and green dots – least coordinated cells which had at least 1 edge. (d) Degree distribution of all 11 analyzed islets. Threshold of Rth = 0.9 was used to filter the Pearson-based coordination matrix and obtain the functional network. (e) Left: Mouse pancreatic islet expressing fluorescent calcium indicator GCaMP6s. Glucose level 11 mM. Middle: NAD(P)H autofluorescence of the islet and the same cell layer, recorded at 2 mM glucose. Right: NAD(P)H autofluorescence recorded at 11 mM glucose. (f) Change of the NAD(P)H levels in each cell in response to glucose elevation, with respect to the islet average change. Metabolic activity here is compared between β-cell hubs and non-hubs. (g) Change in NAD(P)H levels in response to glucose elevation, with respect to the islet average change: here comparison is made for the most coordinated cells (normalized degree >0.9) with the less coordinated cells (normalized degree >0.8, >0.7, >0.6, …,<0.2, <0.1). Linear regression resulted in a trending relationship between normalized degree and NAD(P)H with slope of –0.126 (p=0.079). (f, g) represent n=5 islets with a total of 131 cells. Each dot represents the average NAD(P)H level for cells with respective degrees in an islet. (h) Rhodamine-123 fluorescence of the islet and the same cell layer as in (a), recorded immediately before photobleaching of the top half of each islet (left, red), immediately after photobleaching (middle, yellow) and 360 s after the photobleaching (right, green). (i) Fluorescence recovery after photobleaching (FRAP) recovery rate (s–1) between β-cell hubs and non-hubs. (j) FRAP recovery rate (s–1) in each of the photobleached cells: here the comparison is made for the most coordinated cells (normalized degree >90%) with the less coordinated cells (normalized degree >80, >70, >60, …<20,<10%). Linear regression resulted in a non-significant relationship between normalized degree and recovery rate (p=0.29). (i, j) shows results from 9 islets. Data points represent the average recovery rate for cells with respective degrees in an islet. **p≤0.01. In all panels, error bars represent s.e.m.
Experimental results for fast and mixed oscillations reveal no relationship between gap junction conductance and functional degree.
(a) Recovery rate vs. normalized degree for functional network synchronization threshold 0.85. (b) Recovery rate vs. normalized degree for functional network synchronization threshold 0.95. (c) Example time course of calcium oscillations in fast oscillations. (d) Fluorescence recovery after photobleaching (FRAP) recovery rate separated by hubs and non-hubs. (e) FRAP recovery rate (s–1) in each of the photobleached cells: here the comparison is made for the most coordinated cells (normalized degree >90%) with the less coordinated cells (normalized degree >80, >70, >60, …<20, <10%). (d, e) show results from 4 islets. (f) As in c for mixed oscillations. (g) As in d for mixed oscillations. (h) As in e for mixed oscillations. (f, h) show results from 4 islets. Significance in d, g was determined by unpaired two-tailed t-test. Significance and slope in d, g were determined by linear regression.
Experimental results with alternative hub cell definition.
Hubs can be alternatively defined as the 10% of cells with the highest number of functional connections. (a) Change of the NAD(P)H levels compared between β-cell hubs and non-hubs. Change in NAD(P)H is calculated in each cell in response to glucose elevation, and expressed with respect to the islet average change. (b) Fluorescence recovery after photobleaching (FRAP) recovery rate (s–1) between β-cell hubs and non-hubs. Data points represent the average recovery rate for cells with respective degrees in an islet. Significance was determined by unpaired two-tailed t-test. *p≤0.05. One identified outlier in panel a (value = –0.23, Grubbs’ test) was excluded from the comparison.
To test whether metabolic activity was correlated with the functional network degree, we measured two-photon excited NAD(P)H autofluorescence in conjunction with time-lapse imaging of [Ca2+] dynamics at 2 mM and 11 mM glucose (Figure 3e). The metabolic response of individual β-cells in the islet was calculated as the difference between NAD(P)H autofluorescence at high glucose compared to low glucose (NAD(P)H11-2), with the islet average subtracted to account for inter-islet variability (NAD(P)H11-2 -NAD(P)H 11av-2av). This metric was 0 if the NAD(P)H response of a cell was the same as the islet average, >0 if a cell showed an elevated metabolic response, and <0 if a cell showed a reduced metabolic response. β-Cell hubs (>60% of max degree) had a significantly higher NAD(P)H autofluorescence than in non-hubs (<60% of max degree) (p=0.0061) (Figure 3f). Similarly, the NAD(P)H response trended toward a relationship with cell’s normalized degree (p=0.079) (Figure 3g). When identifying β-cell hubs as the 10% of cells with the highest network degree, hubs (top 10% by degree) also had a significantly higher NAD(P)H autofluorescence than in non-hubs (bottom 90% by degree) (Figure 3—figure supplement 2a). This indicates that highly synchronized cells are more metabolically active than less synchronized cells, in agreement with findings from simulated islets in the coupled Cha-Noma model (Figure 1).
The relationship between gap junction coupling and the functional network is not known. Fluorescence recovery after photobleaching (FRAP) measurements of dye transfer kinetics can quantify gap junction permeability (Farnsworth et al., 2014). We performed FRAP measurements in conjunction with time-lapse imaging of [Ca2+] dynamics, to map cellular gap junction connections in the same cell layer as the functional network (Figure 3h). There was no difference in the rate of recovery for β-cell hubs and non-hubs (p=0.15) (Figure 3i). Similarly, the rate of recovery, which assesses gap junction permeability, was not statistically related to the normalized degree (p=0.37) (Figure 3j). The lack of relationship between rate of recovery and normalized degree was maintained over different functional network synchronization thresholds (Figure 3—figure supplement 1a and b). We also repeated these analyses in islets that showed fast oscillations or mixed oscillations. Irrespective of oscillation type, there was no difference between the rate of recovery for β-cell hubs and non-hubs, nor was there an association between functional network degree and the rate of recovery (Figure 3—figure supplement 1c–h). When identifying β-cell hubs as the 10% of cells with the highest network degree, there was also no difference between the rate of recovery for β-cell hubs and non-hubs (Figure 3—figure supplement 2b). These results indicate that cell synchronization is not correlated with gap junction permeability, as measured by FRAP. Therefore, the functional network is not strongly related to the structural network.
Synchronization between a cell pair more likely indicates shared intrinsic properties than elevated gap junction coupling in simulated islets
We next used the coupled Cha-Noma simulated islet to investigate the second question: what does the islet functional network indicate about its underlying structure or intrinsic properties, on an individual cell basis? Irrespective of the functional network degree of a cell, it was rare for a cell pair to be connected in both the functional and structural networks (Figure 4a). The probability that two cells were synchronized in the functional network, given that they shared a gap junction in the structural network, was Pr(Sync│GJ)=0.39 (Figure 4b). Therefore, the presence of a structural edge does not imply a functional edge. Importantly, when the threshold Rth was decreased, the probability of synchronization given gap junction increased to almost 1.0 (Figure 1—figure supplement 1d and e). This indicates, as expected, that gap junctions are required for synchronizing cell pairs. However, additional similarities are required for a cell pair to be synchronized enough to surpass a threshold large enough to cause the functional network to appear ‘scale-free-like’ that is demonstrated in prior studies (Stožer et al., 2013; Johnston et al., 2016).
Comparison of functional connections, gap junctions, and glucose metabolism in the simulated islet.
(a) Functional connections and structural connections for a representative highly connected β-cell hub, average cell, and low connected cell. Blue cells and edges indicate cells determined as ‘synchronized’ via functional network analysis. Red cells are GJ coupled. Black cells are both GJ coupled and synchronized. (b) Probability that a cell is synchronized given it is gap junction coupled, or vice versa, that a cell is gap junction coupled given it is synchronized. (c) Venn diagram showing overlap between the synchronized cells and gap junction coupled cells within the simulated islet. (d) Functional connections and metabolic connections for a β-cell hub and a cell with averaged synchronization. Blue cells and edges indicate cells determined as ‘synchronized’ via functional network analysis. Green cells and edges indicate cells that have similar kglyc and are within 15% islet from each other. Black cells and edges indicate both synchronized and similar kglyc. (e) Probability that a cell pair is synchronized given it shares a gap junction has similar kglyc. (f) Probability that the cell pair is gap junction coupled has similar kglyc and within 15% of islet distance (Pr=0.0052) , respectively, given the pair is synchronized. (g) Venn diagram showing overlap between the synchronized cells and metabolically similar cells. Shaded area in c and g is proportional to indicated probability. Significance in e and f was determined by a repeated measures one-way ANOVA with Tukey’s multiple comparisons **p≤0.01. In b, e, f each data point corresponds to the average value over a single simulated islet.
The alternative probability of a structural edge given a functional edge was Pr(GJ |Sync)=0.04 (Figure 4b), indicating that very few functionally coupled cells are connected by gap junctions. These findings indicate that structurally connected cells and functionally connected cells are two distinct groups with little overlap (Figure 4c). We analyzed the sensitivity of these measures to the synchronization threshold Rth. Tautologically, as the threshold was increased, the number of functionally connected cells decreased (Figure 1—figure supplement 1d), causing Pr(GJ│Sync) to increase (Figure 1—figure supplement 1f). However, the ‘overlap’ between the functional network and structural network did not increase (Figure 1—figure supplement 1g). Our findings that the functional and structural networks are two distinct groups with little overlap is consistent across thresholds.
These data corroborate with initial findings showing the functional network is not strongly correlated with the structural network. To investigate whether kglyc was associated with edges in the functional network, we created a new network where edges were drawn between cells with similar kglyc rates (see Methods) (Figure 4d). Parameters were chosen such that there was no difference between Pr(Sync│Similar kglyc) and Pr(Sync│GJ) (Figure 4e), allowing for a direct comparison between probabilities. The alternative probability Pr(Similar kglyc|Sync) was significantly higher than Pr(GJ│Sync) (Figure 4f). If two cells have synchronized Ca2+ oscillations, they are more likely to contain shared rate of glucokinase activity than to be gap junction coupled. This is further indicated by the increased overlap between the [Ca2+] synchronization-derived functional network and the intrinsic rate of glucokinase activity-derived network (Figure 4g). These results further indicate that intrinsic cell dynamics is a greater driving factor than gap junction connections for cells to show high [Ca2+] synchronization and influence the functional network.
Elevated metabolic activity is a greater driver of long-range functional connections in simulated islets
In accordance with the islet cytoarchitecture, gap junction connections only exist between highly proximal cells. However, there is no distance constraint on [Ca2+] oscillation synchronization (Figure 5a). To determine whether this spatial constraint was responsible for the low correspondence between the functional and structural networks, we asked whether a series of highly conductive gap junction connections could predict long-range functional connections. We weighted the structural network by gcoup and calculated the path which maximized conductance (see Methods) (Figure 5b). Not all synchronized cell pairs had chains of larger total gcoup than non-synchronized cell pairs (Figure 5b). The probability distributions of total gcoup for synchronized cells highly overlapped with those for non-synchronized cells, irrespective of distance (Figure 5c and d, and Figure 5—figure supplement 1a). However, the total conductance normalized by separation distance was significantly less for non-synchronized cells than for synchronized cells (Figure 5e). Thus, on average, long-range functional connections traverse cells that are connected by higher gap junction conductance.
Comparison of long-range functional synchronization, gap junction network, and glucose metabolism in the simulated islet.
(a) Functional connections and gap junction connections for a representative cell (black). Synchronized cells in blue, gap junction coupled cells in red. Inset shows the entire structural network (gray) with functional connections for the same cell shown in blue. (b) Two representative cells (black) and the shortest path to a synchronized cell (blue) and a non-synchronized cell (green). Path is weighted (shown by edge thickness) by gcoup. For the cell on the top panel, the synchronized path has a higher cumulative gap junction conductivity (0.68 nS) than the non-synchronized path (0.43 nS). For the cell on the bottom panel, the synchronized path has a lower cumulative gap junction conductance (0.74 nS) than the non-synchronized path (1.02 nS). (c) Probability distribution of total gcoup for synchronized cells (blue) and non-synchronized cells (green) that are directly connected by gap junctions. (d) As in c for cells pairs that are 7 cells apart. (e) Comparison of the total gcoup, normalized by distance for synchronized cells (blue) and non-synchronized cells (green). Each dot indicates the average resistance for a single simulated islet. (f) As in c but with connections weighted by rate of glucokinase activity (kglyc). (g) As in d but with connections weighted by kglyc. (h) As in e but with connections weighted by kglyc. (i) Total gcoup for synchronized and non-synchronized cells organized by cell distance. (j) As in i but for the network weighted by kglyc. Significance in e, h was assessed by a two-tailed paired t-test, with p-value indicated. Significance in i and j was assessed by paired t-tests with Bonferroni correction. Reported p-values are adjusted. *p≤0.05, **p≤0.01, ***p≤0.001. In (e, h, i, j) each data point corresponds to the average value for synchronized/non-synchronized cell pairs (e, h) or respective number of cells apart over a single simulated islet.
Paths that maximized gap junction conductivity.
(a) Cumulative distribution functions of total conductance between two cells from 1 to 12 cells apart. Synchronized CDF is shown in blue, non-synchronized CDF is shown in green. (b) Bar charts comparing the average conductance for synchronized and non-synchronized cells. p-Values are not corrected, but significance (denoted by asterisks) is determined using a Bonferroni-corrected paired t-test. Since there were 15 cell distances, significance threshold was set at α=0.003.
Paths that maximized metabolic rate.
(a) Cumulative distribution functions of total kglyc between two cells from 1 to 12 cells apart. Synchronized CDF is shown in blue, non-synchronized CDF is shown in green. (b) Bar charts comparing the average conductance for synchronized and non-synchronized cells. p-Values are not corrected, but significance (denoted by asterisks) is determined using a Bonferroni-corrected paired t-test. Since there were 15 cell distances, significance threshold was set at α=0.003.
To examine how the intrinsic rate of glucokinase activity influences long-range functional connections, we repeated this procedure but weighted graph edges by kglyc rather than gcoup. Again, the probability distributions of the total kglyc for synchronized cells overlapped with those for non-synchronized cells, irrespective of distance (Figure 5f and g and Figure 5—figure supplement 2). The total rate of glucokinase activity, normalized by the separation distance, was significantly less for non-synchronized cells than for synchronized cells (Figure 5h). Thus, on average, long-range functional connections also traverse cells with higher metabolic activity.
To test whether there was a spatial relationship for rate of glucokinase activity or gap junction conductance-controlled synchronization, we analyzed the total conductance or total rate of glucose metabolism for different separation distances and between synchronized or non-synchronized cell pairs. Total gcoup was significantly higher for synchronized cells 4–7 cells apart (Figure 5i). Total kglyc was significantly higher for synchronized cells 1–5 cells apart, with the significance larger than that of total gcoup (Figure 5j). Thus, cell pairs with similar kglyc can strongly influence functional connections up to 5 cells apart, while high gcoup is necessary for influencing functional connections over longer distances (5–7 cells apart).
Decreasing gap junction conductance in the structural network decreases the average network degree in the functional network constructed based on the experimental and simulated data
Our third question asks how changes to the structural network influence the islet functional network. Reducing Cx36 gap junction coupling via genetic or pharmacological means reduces overall islet Ca2+ oscillation synchronization (Benninger et al., 2008; Farnsworth and Benninger, 2014). We performed functional network analysis on islets from wild-type (WT) mice (Cx36+/+), heterozygous Cx36 knockout mice with ~50% reduced gap junction coupling (Cx36+/-), and homozygous Cx36 knockout mice with no gap junction coupling (Cx36-/-) (Benninger et al., 2008; Figure 6a–b). We used a single synchronization threshold Rth across all genotypes so the average degree distribution of WT islets roughly followed a scale-free-like distribution (Stožer et al., 2013) and had an average of between 5 and 15 connections (see Methods) (Figure 6—figure supplement 1). Cx36+/- islets demonstrated decreased average network degree compared to WT Cx36+/+ islets (Figure 6c), and homozygous Cx36-/- demonstrated further decreased degree compared to both WT Cx36+/+ islets and Cx36+/- islets (Figure 6c). Therefore, the functional network becomes highly sparse as edges in the structural network decrease.
Experimental islet functional networks are influenced by changes in the structural network.
(a) Representative images of Cx36+/+ islet (left), heterozygous Cx36+/- islet (middle), and homozygous islet (right), overlaid with a synchronization network. Dot signifies a cell, blue lines represent functional network edges that connect synchronized cells. Red cells are β-cell hubs. (b) Oscillations in [Ca2+] of corresponding islet in a. Gray lines represent time course for each cell, colored line represents mean islet time course. (c) Normalized degree of functional network between cell pairs for each islet in the Cx36+/+, Cx36+/-, Cx36-/- mice. (d) Clustering coefficient of the functional network for each islet in the Cx36+/+, Cx36+/-, Cx36-/- mice. Overlaid is the box and whisker plot of 1000 random networks per islet. (e) As in d for global efficiency. (f) Average distance between synchronized cells normalized to average distance between all cells in islet. Dashed line shows average distance between cells. Adjusted p-values: *p≤0.05, **p≤0.01, ***p≤0.001 For c–d, each dot represents an islet.
Alternative network metrics for calcium islets from Cx36KO (Cx36 knockout) mice.
(a) Averaged degree distribution histograms for all islets analyzed. Error bars indicate standard deviation. Purple (top) histogram shows the distribution of connections for wild-type (Cx36+/+) islets. Orange (middle) shows the distribution of connections for heterozygous Cx36 knockout (Cx36+/-) islets. Blue (right) shows the distribution of connections for homozygous Cx36 knockout (Cx36-/-) islets. (b) Average Pearson cross-correlation coefficient, between all islet in the islet. (c) Number of cells in each islet. Significance or lack thereof is determined by an ordinary one-way ANOVA and Tukey’s multiple comparison test. *p≤0.05, **p≤0.01.
We next quantified how the removal of edges in the structural network influences the functional network topology (Table 1). Network topology statistics can provide insight into how the network functions, and how it changes with specific perturbations. Clustering coefficient (Cavg) quantifies the proportion of nodes connected to a given node that are also connected with each other (Table 1). A high clustering coefficient indicates that cells tend to synchronize with other cells that share some property, such as proximity or intrinsic dynamics. The clustering coefficient significantly decreased as gap junction coupling decreased (Figure 6d). To quantify whether the functional network topology was a result of random edge generation, we compared each metric with that determined from 1000 Erdos-Renyi random networks (Watts and Strogatz, 1998; Gansterer et al., 2019; Stožer et al., 2013; Table 1). If gap junctions are the property that gives the functional network a high clustering coefficient, then reducing gap junction coupling should result in a clustering coefficient that shows greater overlap with the random graph. For all levels of Cx36, the clustering coefficient of the islet functional network was greater than that of a random network (Figure 6d, box and whisker plot). Similar to our previous findings, this further suggests that something other than the structural network contributes to the functional network topology. Global efficiency is defined as the inverse of the average smallest number of nodes required to traverse between any cell pair (Table 1). High global efficiency can occur in a random network, or in a regular network (Table 1) with a few edges randomly moved (Watts and Strogatz, 1998). The global efficiency significantly decreased as gap junction coupling decreased (Figure 6e). For Cx36+/+ islets, the global efficiency was less than that of a random network. As gap junction coupling decreased, the global efficiency showed a larger intersection with that of a random network. For Cx36-/- islets all measurements intersected with and thus could be explained by a random network. For each genotype, the distance between synchronized cells was less than the average distance between all cells (Figure 6f). These results suggest that the gap junction structural network contributes to whole islet global properties (global efficiency) of the functional network topology, but not immediate local properties (clustering), similar to our previous findings (Figure 5).
We next asked how changes to the structural network influence the islet functional network in the coupled Cha-Noma model. We decreased coupling conductance (gcoup) (Figure 7a). Similar to experimental findings, as average gap junction conductance decreased, the average degree and average correlation also decreased (Figure 7c, Figure 7—figure supplement 1b). Pr(Sync│GJ) also decreased significantly as gap junction conductance was decreased (Figure 7b and d). This indicates that as the gap junction conductance decreases, the probability that two structurally connected cells are also functionally connected decreases. We then quantified the functional network topology in the coupled Cha-Noma model upon reduced gap junction conductance. The clustering coefficient (Cavg) progressively decreased with decreasing gap junction conductance and was always greater than that determined from a random network (Figure 7e). Global efficiency also progressively decreased with decreased gap junction conductance (Figure 7f) and was always less than that of a random network. These trends are similar to our experimental observations.
Simulated islet functional networks are influenced by changes in the structural network.
(a) Functional connections and structural connections in simulated islets of hub cells for high and low gap junction conductance. Blue cells and edges indicate cells determined as ‘synchronized’ via functional network analysis. Red cells are GJ coupled. Black cells are both GJ coupled and synchronized. (b) Venn diagrams of the probability of synchronization and probability of gap junction with the probability of both synchronization and gap junction as the overlapped area. (c) Average degree in the functional network. (d) Probability that two cells are synchronized given they share a gap junction connection. (e) Clustering coefficient for 0.12nS, 0.06nS, 0.03nS , for each simulated islet. Overlaid is the box and whisker plot of 1000 random networks per simulated islet. (f) As in e for global efficiency for 0.12nS, 0.06nS, 0.03nS, for each simulated islet. Adjusted p-values: *p≤0.05 , **p≤0.01, ***p≤0.001. For c–f, each dot represents a simulated islet.
Alternative network metrics for simulated islet with gap junction coupling change.
(a) Averaged degree distribution histograms for all islets analyzed. Error bars indicate standard deviation. Purple (top) histogram shows the distribution of connections for fully coupled (0.12 nS) islets. Orange (middle) shows the distribution of connections for half coupled (0.06 nS) islets. Blue (right) shows the distribution of connections for homozygous quarter coupled (0.03 nS) islets. (b) Average Pearson cross-correlation coefficient between all islets in the islet. Significance or lack thereof is determined by an ordinary one-way ANOVA and Tukey’s multiple comparison tests. *p≤0.05, **p≤0.01.
Overall, our findings from experimental and simulated islets both indicate that the gap junction structural network influences overall islet synchronization, as expected. However, while it influences the global functional network topology, it has less influence on the local functional network topology.
Discussion
The islet of Langerhans is central to glucose homeostasis and dysfunction of the islet underlies diabetes. Heterogeneity among β-cells in the islet plays a key role in islet function (Benninger and Kravets, 2022; Johnston et al., 2016; Salem et al., 2019; Kravets et al., 2022). Functional network analysis enables the investigation of β-cell heterogeneity and how subpopulations of β-cells influence islet dynamics, over which there is currently a debate (Dwulet et al., 2021; Satin et al., 2020; Rutter et al., 2020; Peercy and Sherman, 2022). To correctly interpret the β-cell functional network and its implications in diabetes, we must understand what drives edges and subpopulations such as β-cell hubs. While it is known that gap junctions are essential for whole islet synchronization, it is unclear how the functional network relates to the gap junction structural network. Our aim was to investigate the relationship between functional networks, structural networks, and underlying β-cell dynamics in the pancreatic islets. We framed our study around three questions. (1) Are highly correlated subpopulations (β-cell hubs) that emerge from the β-cell functional network differentiated by their unique qualities with respect to the structural network or by intrinsic properties of the cells? (2) What does the islet functional network indicate about its underlying structure or intrinsic cell properties on an individual cell basis? (3) How do changes in the structural network affect the islet’s functional network? Figure 8 shows a graphical representation of the answers to these questions. By understanding the formation of islet functional networks, we can understand how β-cell subpopulations drive islet function. Knowledge about the role of β-cell subpopulations will also be useful when developing new treatments, such as identifying and preserving functional subpopulations in diabetes and generating stem cell-derived insulin-secreting cells for restoring insulin secretion.
Graphical summary of results.
Graphical summary of results from data. Inset shows how structural gap junctions are translated to structural network and synchronized calcium oscillations are translated to functional network. To answer question 1 we show that subpopulations of β-cells in the functional network are driven by intrinsic properties more than gap junction coupling (top). To answer question 2, we show that intrinsic properties of the β-cells drive synchronization in the functional network (right). To answer question 3, we show that removal of gap junctions in the structural network decreases global communication efficiency and local connectivity (bottom).
Highly connected β-cells are driven by intrinsic cell dynamics
Our computational and experimental results show a subpopulation of highly synchronized cells, known as β-cell hubs (Johnston et al., 2016; Benninger and Hodson, 2018). In the coupled Cha-Noma model of fast β-cell oscillations, the elevated rate of glucokinase kglyc (which indicates metabolic activity), was most strongly associated with β-cell hubs and duty cycle of the [Ca2+] oscillations (Figure 1). gKATP showed little association with duty cycle, which is consistent with the greater influence of ATP production (primarily controlled by kglyc) and ATP-driven KATP closure on membrane potential, compared to KATP open-channel conductance (Notary et al., 2016). In the coupled IOM of slow β-cell oscillations, β-cell hubs were associated with both gKATP and gap junction conductance (Figure 2). In this case, the gKATP parameter showed a high association with oscillation duty cycle, as shown previously in uncoupled β-cell simulations (Marinelli et al., 2022). Thus, results from two independent models show that the cell intrinsic properties that influence oscillation duty cycle are associated with driving β-cell hubs, regardless of β-cell oscillation type. Notably an association between network degree and oscillation duty cycle has recently been demonstrated (Šterk et al., 2023; Stožer et al., 2021; Gosak et al., 2022).
A key difference between the two coupled models is the strong association between β-cell hubs and gap junction coupling in the coupled IOM and the weak association in the coupled Cha-Noma model. While there was an ~15% difference in gcoup between hubs and non-hubs in the Cha-Noma model (compared to ~10% difference in kglyc), this difference was subject to high variability and not observed for all thresholds used. This study represents one of the first uses of a coupled IOM to study β-cell heterogeneity. The Cha-Noma model, which has been used extensively to study β-cell heterogeneity (Dwulet et al., 2021; Dwulet et al., 2019; Notary et al., 2016; Westacott et al., 2017; Silva et al., 2014), is simulated with 1000 β-cells to roughly match experimental observations (Pisania et al., 2010; Steiner et al., 2010). However, we simulated the coupled IOM as a smaller islet model due to model stiffness and computational limits. When the coupled Cha-Noma model was simulated with a similarly small number of cells, gap junction conductance had a higher association with β-cell hubs (Figure 2). Therefore, we attribute the increased contribution of gap junction coupling to β-cell hubs in the coupled IOM to be partially influenced by islet size. We predict if the coupled IOM could be simulated with a larger number of cells, β-cell hubs would become more strongly associated with gKATP than gap junction coupling.
Our experimental measurements showed that β-cell hubs were associated with elevated metabolic activity, quantified by NAD(P)H fluorescence. In contrast, there was no association between β-cell hubs and gap junction permeability (Figure 3). Previous studies have shown that the functional network topology of β-cells is highly dependent on whether the β-cell oscillation is fast, slow, or plateau (Zmazek et al., 2021; Stožer et al., 2022). However, the relationship between β-cell hubs and gap junction permeability was not influenced by [Ca2+] oscillations type: slow, fast, and mixed. These results further illustrate how subpopulations of β-cell hubs in the functional network are likely emerge due to their intrinsic properties such as metabolic activity, rather than gap junction conductance. Importantly, we are unable to spatially resolve KATP channel conductance or channel number, thus we cannot also exclude that β-cell hubs may show decreased KATP conductance. Further, the variability that we observe comparing metabolic activity between hubs and non-hubs in both experiments and simulated islets suggests other factors (including KATP conductance) may potentially play some additional role in determining β-cell hubs.
Our findings also agree with previous evidence that β-cell hubs have elevated metabolic activity. Semi-quantitative immunofluorescence demonstrated elevated glucokinase protein in β-cell hubs, suggesting elevated glycolytic activity and increased accumulation of tetramethylrhodamine ethyl ester in β-cell hubs was indicative of increased oxidative phosphorylation (Johnston et al., 2016). However, evidence for elevated gap junction coupling in β-cell hubs was argued either on the basis of [Ca2+] synchronization analysis which we show is not necessarily indicative of gap junction coupling (Figures 1—5) or a global knockdown of GJD2 (which codes for Cx36) (Johnston et al., 2016) which will influence both β-cell hubs and non-hubs, as we also observe (Figures 6–7). Further supporting our findings here, prior experiments have shown that cells that are preferentially activated by optogenetic ChR2 stimulation, which indicates function, were more metabolically active, but not preferentially gap junction coupled (Westacott et al., 2017).
In answering our first question, we conclude that gap junctions are not the primary distinguishing factor in determining variations in oscillation synchronization between cells. Rather cell intrinsic properties including metabolic activity distinguish variations in synchronization between cells because of their influence on [Ca2+] oscillation duty cycle.
Functional network edges primarily reflect variations in intrinsic cellular properties
To compare the entire islet functional network with its structural network (instead of hubs and non-hubs in question 1), we analyzed the coupled Cha-Noma model, and generated conditional probabilities that quantify the relationship between structural and functional networks at an individual cell basis (Figure 4). We found the probability of a gap junction connection given a synchronized cell pair (Pr(GJ|Sync)) was <5%, indicating that high [Ca2+] synchronization rarely implies a gap junction connection. However, the probability that two synchronized cells had similar metabolic activity was significantly larger (~20%). Therefore, the functional network primarily reflects variations in cellular metabolic activity rather than structural gap junction connections.
Importantly the overlap in cellular metabolic activity with the functional network refers to cells sharing some intrinsic metabolic properties (kglyc) that drives duty cycle and functional network edges. It has been suggested that β-cells may be metabolically coupled, where metabolites diffuse between cells via gap junctions (Rao and Rizzo, 2020; Tsaneva-Atanasova et al., 2006). However, our findings regarding metabolic activity or intrinsic dynamics driving the functional network are not referring to metabolic coupling, but rather the fact that intrinsically similar cells will appear more synchronized, and therefore will be more likely to be connected in the functional network.
Surprisingly, we found that gap junctions do not guarantee that [Ca2+] synchronization will be higher than the functional network threshold (Rth), as previously assumed (Markovič et al., 2015). The probability of high synchronization given gap junction connections (Pr(Sync|GJ)) was only ~40% (Figure 4). When threshold is reduced, Pr(Sync|GJ) increases to almost 100% (Figure 1—figure supplement 1e) but the functional degree distribution no longer appears scale-free-like, as indicated by previous studies (Stožer et al., 2013; Johnston et al., 2016). Gap junctions facilitate ion passage between β-cells and therefore play a primary role in synchronizing electrical oscillations across the islet (Figure 5). However, they are not the only factor contributing to the high synchronization seen between some β-cell pairs. The cell membrane potential is determined by cell metabolic activity, KATP conductance, and other ion channels. The membrane potential of 2 cells determines the electrochemical gradient along the gap junction, and thus the coupling current which drives synchronization (Dwulet et al., 2021; Plonsey et al., 2007). As such, high metabolic activity can drive a high coupling current. Furthermore, cells with similar intrinsic characteristics, such as metabolic activity, require less electrical coupling to be highly synchronized. Thus, while gap junctions are important for the overall synchronization of islet [Ca2+] dynamics, additional intrinsic β-cell factors are necessary for very strong synchronization between proximal β-cells.
Perturbing the gap junction structural network altered the islet functional network topology
Changes in gap junction coupling and [Ca2+] oscillation synchronization are observed in conditions associated with diabetes. Impaired [Ca2+] oscillation synchronization (representing a disrupted functional network) and disrupted pulsatile insulin secretion have been observed in islets from rodent models of type 2 diabetes (Pørksen et al., 2002; Satin et al., 2015; Corezola do Amaral et al., 2020), type 1 diabetes (Pørksen et al., 2002; O’Meara et al., 1995), human subjects with type 2 diabetes (St. Clair et al., 2020) or obesity (Hodson et al., 2013), and when exposed to glucolipotoxicity, or pro-inflammatory cytokines (Hodson et al., 2013; Farnsworth et al., 2016). Similarly, altered gap junction coupling (representing a disrupted structural network) has been observed in islets from mouse models of type 2 diabetes (St. Clair et al., 2020; Corezola do Amaral et al., 2020), type 1 diabetes (Farnsworth et al., 2022), prediabetes (Carvalho et al., 2012), and exposure to hyperglycemia or pro-inflammatory cytokines (Farnsworth et al., 2016; Haefliger et al., 2013). When gap junction coupling and [Ca2+] synchronization are perturbed upon a genetic deletion of Cx36, first-phase insulin secretion and pulsatile second-phase secretion are diminished, causing glucose intolerance. Therefore, it is important to understand the relationship between altered [Ca2+] oscillation synchronization and gap junctions coupling.
Our computational and experimental results show that as structural edges were removed by reducing gap junction conductance, the functional network degree, efficiency, and clustering decreased (6,7Figures 6 and 7). This implies that the islet loses efficient signal propagation and the ability to synchronize. This in part explains decreased islet function in diabetes. However, clustering could not be explained by randomness (Figures 6 and 7). If gap junctions were solely responsible for variations in islet synchronization, a decrease in gap junction coupling would cause the functional network to appear more random. Instead, our findings indicate that β-cells are preferentially, not randomly, synchronized even in the presence of low gap junction coupling. This is consistent with synchronization between cell pairs being driven by intrinsic cellular dynamics, such as metabolic activity. Thus, while removal of gap junctions is detrimental to overall islet function, changes in synchronization are not directly indicative of only changes in structural connections. Perturbations in cellular metabolic activity, which occurs in diabetes (Ostenson et al., 1993), will also significantly influence islet synchronization. Additionally, our results indicate that the relationship between the structural and functional network is not strongly influenced by β-cell oscillation type (Figure 3—figure supplement 1), therefore, fast, slow, and mixed oscillations should show similar changes with decreased gap junction coupling.
Potential limitations
Our computational models are based on careful physiological recordings and have been previously validated (Dwulet et al., 2019; Hraha et al., 2014; Westacott et al., 2017; Bertram et al., 2018). Nevertheless, they will still be limited in describing β-cell dynamics. Because our results are internally derived, where the functional and structural networks were obtained from the same model, they are not dependent on the models’ ability to reflect islet dynamics. This fact also allows our methods to translate to any oscillatory network with similar characteristics – this is reflected by our use of two independent coupled models. The choice of threshold (Rth) is influential in creating the functional network and should be carefully considered. We used thresholds to match that of the prior experimental results (Stožer et al., 2013; Johnston et al., 2016; Korošak et al., 2021). The computational thresholds were higher in these prior experiments because the computational model did not include noise, so that synchronization was more acute. However, the trends in our results for the Cha-Noma model and experiment were consistent across thresholds and hub definition (Figure 1—figure supplements 1 and 2, Figure 3—figure supplements 1 and 2). Within the coupled IOM, a very high threshold was required to generate a degree distribution that matched the coupled Cha-Noma model results. This was likely due to the small islet size, which was limited by model instability. Nevertheless, consistent results were found in terms of parameters that influence duty cycle associating with β-cell hubs.
FRAP measurements of gap junction conductance are likely less sensitive than gold-standard patch clamping, but it allows measurements in multiple cells in situ and subsequent calcium imaging, which was central to our study. Additionally, we cannot exclude that the Cx36 global deletion may influence β-cell specification. Thus, generating effective inducible and conditional Cx36 knockout models are needed. However, studies have shown no significant change in NAD(P)H response and islet architecture in Cx36 knockout islets (Benninger et al., 2011). Dissociated β-cells of both Cx36 knockout and WT islets also show similar insulin secretion responses, indicating a Cx36 knockout islets has no significant defect in the intrinsic ability of a β-cell to release insulin.
Dynamic changes in the distribution of gap junction coupling can occur over time (Miranda et al., 2022a), which is not accounted for in our computational models. However, since our experimental results indicate that β-cell hubs are more influenced by intrinsic activity, we do not predict that changes in the distribution of gap junction coupling over time would substantially impact β-cell hubs.
Due to propagating Ca2+ waves across the islet, there is a phase lag between [Ca2+] oscillation that is small for β-cells in close proximity and greater for β-cells far apart. A large phase lag will decrease the apparent [Ca2+] oscillation synchronization, thus influencing the functional network. Future work will be needed to examine ‘wave initiator’ like cells that lie at the origin of propagating Ca2+ waves, or cells located at the wave end, and determine their position and influence on the islet functional network. Indeed, prior work has shown that cells at the wave end likely influence the overall [Ca2+] oscillation dynamics (Dwulet et al., 2021).
The islet contains α-cells, δ-cells, and other cell types which can influence β-cell dynamics via paracrine mechanisms (Henquin, 2021; Moede et al., 2020; van der Meulen et al., 2015). Our study only analyzes β-cells, as performed in other network studies which we compare (Stožer et al., 2013; Johnston et al., 2016; Lei et al., 2018). However, we cannot exclude long-range structural projections from δ-cells (Arrojo E Drigo et al., 2019; Briant et al., 2018), and that δ-cells may be gap junction coupled (Miranda et al., 2022b). Additionally, β-cell oscillation types (slow, fast, or mixed) may be directly related to the number of α-cells present in the islet (Ren et al., 2022). Future analysis should include other cell types, which may provide valuable insight on how their interactions can shape the overall islet response. This is particularly important when translating our findings to human islets because α-cells, δ-cells, and β-cells are known to be more mixed in human islets (Kim et al., 2009), and there is experimental evidence that human islet functional networks may differ from mouse functional networks (Gosak et al., 2022).
Overall summary
While it is well known that gap junctions allow for whole islet synchronization (Benninger et al., 2008), it is not clear whether the functional network directly reflects gap junction structural network or if the functional network is influenced by intrinsic β-cell characteristics. We investigated the relationship between the synchronization-based functional network, the gap junction structural network, and β-cell intrinsic properties in the islets of Langerhans. We concluded with three major points. First, highly connected subpopulations of β-cells, including β-cell hubs, are defined by intrinsic properties more than gap junctions. Second, the functional network primarily reflects variations in intrinsic cellular properties. Third, removal of edges in the structural network upon decreasing gap junction conductance decreases global communication efficiency and local connectivity and causes the functional network to become sparser. Our results also show that forming conclusions about structure using the functional network should be done with caution. Broadly, these results provide insight into the relationship between function and structure in biological networks and the understanding that synchronization can give insight into cellular properties that drive excitability of the cell can be useful in many systems.
MethodsCalcium imaging, FRAP, and NAD(P)HAnimal care
Male and female mice were used under protocols approved by the University of Colorado Institutional Animal Care and Use Committee. β-Cell-specific GCaMP6s expression (β-GCaMP6s) was achieved through crossing a MIP-CreER (The Jackson Laboratory) and a GCaMP6s line (The Jackson Laboratory) (Kravets et al., 2022). Genotype was verified through qPCR (Transetyx, Memphis, TN, USA). Mice were held in a temperature-controlled environment with a 12 hr light/dark cycle and given continuous access to food and water. CreER-mediated recombination was induced by 5 daily doses of tamoxifen (50 mg/kg bw in corn oil) delivered IP.
Islet isolation and culture
Islets were isolated from mice under ketamine/xylazine anesthesia (80 and 16 mg/kg) by collagenase delivery into the pancreas via injection into the bile duct. The collagenase-inflated pancreas was surgically removed and digested. Islets were handpicked and planted into the glass-bottom dishes (MatTek) using CellTak cell tissue adhesive (Sigma-Aldrich). Islets were cultured in RPMI medium (Corning, Tewksbury, MA, USA) containing 10% fetal bovine serum, 100 U/mL penicillin, and 100 mg/mL streptomycin. Islets were incubated at 37°C, 5% CO2 for 24–72 hr before imaging.
Imaging
An hour prior to imaging nutrition media from the isolated islets was replaced by an imaging solution (125 mM NaCl, 5.7 mM KCl, 2.5 mM CaCl2, 1.2 mM MgCl2, 10 mM HEPES, and 0.1% BSA, pH 7.4) containing 2 mM glucose. During imaging the glucose level was raised to 11 mM. Islets were imaged using either an LSM780 system (Carl Zeiss, Oberkochen, Germany) with a 40×1.2 NA objective or with an LSM800 system (Carl Zeiss) with 20×0.8 NA PlanApochromat objective, or a 40×1.2 NA objective, with samples held at 37°C.
For [Ca2+] measurements GCaMP6s fluorescence was excited using a 488 nm laser. Images were acquired at 1 frame/s at 10–20 µm depth from the surface of the islet. Glucose was elevated 3 min after the start of recording, unless stated otherwise.
NAD(P)H autofluorescence and [Ca2+] dynamics were performed in the same z-position within the islet. NADH(P)H autofluorescence was imaged under two-photon excitation using a tunable mode-locked Tisapphire laser (Chameleon; Coherent, Santa Clara, CA, USA) set to 710 nm. Fluorescence emission was detected at 400–450 nm using the internal detector. Z-stacks of 6–7 images were acquired spanning a depth of 5 μm. First NAD(P)H autofluorescence (1 z-stack) was recorded at 2 mM glucose, then the [Ca2+] dynamics was recorded at 2 mM and during transition to 11 mM glucose (~30 min). After measuring the oscillatory [Ca2+] time course, NAD(P)H autofluorescence (1 z-stack) was recorded at 11 mM glucose.
Cx36 gap junction permeability and [Ca2+] dynamics were performed in the same z-position within the islet, with gap junction permeability measured using FRAP, as previously described. After [Ca2+] imaging, islets were loaded with 12 mM Rhodamine-123 for 30 min at 37°C in imaging solution. Islets were then washed and FRAP performed at 11 mM glucose at room temperature. Room temperature was used because at this temperature the Rhodamine-123 travels between the cells only through the Cx36 gap junctions, versus at 37°C it can permeate a cell membrane. Rhodamine-123 was excited using a 488 nm laser line, and fluorescence emission was detected at 500–580 nm. Three baseline intensity images were initially recorded. A region of interest was then photobleached achieving, on average, a 50% decrease in fluorescence, and images were then acquired every 5–15 s for 15–30 min.
Analysis of [Ca<sup>2+</sup>] dynamics
Pearson-product-based network analysis presented in Figure 2 was performed as previously reported (Stožer et al., 2013). [Ca2+] time courses were analyzed during the second-phase [Ca2+] response when the slow calcium wave was established. For fast oscillations, 400–500 s of [Ca2+] response was analyzed, matching the simulation time. For slow oscillations, 741–1000 s of [Ca2+] response was analyzed. This time period was chosen so at least 3 oscillations were completed. In mixed oscillations, 858–1000 s of [Ca2+] response was analyzed.
The Pearson product for each cell pair islet was calculated over each time point, and the time-average values were computed to construct a correlation matrix. An adjacency matrix was calculated by applying a threshold to the correlation matrix. The same threshold of 0.9 was applied to all islets. All cell pairs with a non-zero values in the adjacency matrix were considered to have a functional edge. The percent of edges was calculated with respect to the maximum number of edges per cell in each individual islet. For example, if a most connected cell possessed max = 10 edges, and other cells had 1, 3, …, 7 edges – then the % were: 10%, 30%, …, 70%.
Prior calcium imaging first presented in (<xref ref-type="bibr" rid="bib2">Benninger et al., 2008</xref>)Islet isolation
Islets were isolated as described in Scharp et al., 1973, and Stefan et al., 1987, and maintained in Roswell Park Memorial. Institute medium containing 10% fetal bovine serum, 11 mM glucose at 37°C under humidified 5% CO2 for 24–48 hr before imaging.
Imaging islets
Isolated islets were stained with 4 mM Fluo-4 AM (Invitrogen, Carlsbad, CA, USA) in imaging medium (125 mM NaCl, 5.7 mM KCl, 2.5 CaCl2, 1.2 mM MgCl2, 10 mM HEPES, 2 mM glucose, 0.1% bovine serum albumin, pH 7.4) at room temperature for 1–3 hr before imaging. Islets were imaged in a polydimethylsiloxane microfluidic device, the fabrication of which has been previously described in Rocheleau et al. which holds the islet stable for imaging and allows rapid reagent change, such as varying glucose stimulation or adding gap junction inhibitors. Fluo-4 fluorescence is imaged 15 min after a step increase in glucose from low (2 mM) to high (11 mM). High-speed imaging is performed on an LSM5Live with a 203 0.8 NA Fluar Objective (Zeiss, Jena, Germany) using a 488 nm diode laser for excitation and a 495 nm long-pass filter to detect fluorescence emission. The microfluidic device is held on the microscope stage in a humidified temperature-controlled chamber, maintained at 37°C. Images were acquired at a rate of 4–6 frames/s, with average powers at the sample being minimized to 200 mW/cm2.
Analysis of Ca<sup>2+</sup> imaging data
We present data from 11 islets from 6 WT mice, 11 islets from 11 heterozygous Cx36+/− knockout mice, and 14 islets from 3 homozygous Cx36-/- knockout mice. We extracted cell calcium dynamics by visually identifying and circling all cells islet. We assumed that the pixels within a cell should be well coordinated, so we removed any pixels whose dynamics were not within 5–10 STD of the average. This usually resulted in the removal of 1–5 pixels on the edge of the cell boundary.
Threshold
Previous studies have shown that WT islets should have a degree distribution that is roughly linear when plotted on a log-log plot and average degree of either 8 or between 5 and 15 (Stožer et al., 2013; Johnston et al., 2016; Zmazek et al., 2021; Korošak et al., 2021). To determine Rth that best satisfied both of these findings, we utilized constrained optimization MATLAB (MathWorks Inc, Natick, MA, USA) algorithm fminsearchbnd (D’Errico, 2023) to find the optimal Rth that maximized the goodness of fit to a power law distribution, while forcing 5 ≤ kavg ≤ 15 for each WT islet constrained optimization. The average optimal threshold (Rth = 0.90).
Average distance between connected cells
The average distance between connected cells calculated the total number of pixels between the center of two connected cells. The average distance was expressed as the normalized distance between connected cells and the average distance between all cells in the islet to control for image and islet size.
Computational modelCoupled Cha-Noma model
This ordinary differential equation model has been described previously (Dwulet et al., 2021) and validated on experimental studies (Dwulet et al., 2019; Notary et al., 2016; Hraha et al., 2014; Westacott et al., 2017). This model has been shown to accurately describe experimental findings concerning spatial-temporal dynamics (Westacott et al., 2017), the relationship between electrical coupling and metabolic heterogeneity (Dwulet et al., 2019), the relationship between electrical heterogeneity and electrical activity (Hraha et al., 2014), and the influence of excitability parameters such as rate of glucose metabolism (kglyc) and KATP channel opening kinetics on diabetes mutations (Dwulet et al., 2019; Notary et al., 2016). It is based on a single β-cell electrophysiological model (Cha et al., 2011), in which the membrane potential (Vi) for β-cellsi is solved for each time step using Equation 1a. We created a 1000 β-cell network and electrically coupled any cell pairs within a small distance from each other. We chose 1000 β-cell because most species (including human and mouse) contain on average 1000 β-cells in an islet (Pisania et al., 2010) and it is the number which has been validated experimentally. The coupling current, Icoup, is determined by the difference in voltage between the cell pair and the average coupling conductance (gcoup) between them (Equation 1b). All code was written in C++ and run on the SUMMIT supercomputer (University of Colorado Boulder). All simulations are run at 11 mMol glucose. The membrane potential (Vi) for β-cell i is solved for using the ODE:CmdVidt=ICav+ITRPM+ISOC+IbNSC+IKDr+IKCa(SK)+IKATP+INaCa+IPMCA+INaCa+IcoupIcoup=∑igcoupij(Vi−Vj)
The number of gap junction connections was given by a normal distribution mean 5.25, standard deviation 1.6, with the maximum of 12 and a minimum of 1 gap junction connections per cell. Cellular heterogeneity was introduced by assigning randomized metabolic and electrical parameter values to each cell based on their distributions previously determined by experimental studies. For example, glucokinase (kglyc), the rate-limiting step in glycolysis, was assigned using a normal distribution with mean 1.26*10–04 s–1 and standard deviation 3.15*10–05 s–1. Conductance of the KATP channel (gKATP) was given by a normal distribution with mean 2.31 pA/mV and standard deviation 0.57 pA/mV. The coupling conductance parameter of a cell (gcoupj) was assigned using a gamma distribution with k=4, θ=4 and then shifted so the islet average was gcoupj =0.12 nS. This resulted in average σcoup =0.12 nS and standard deviation of 0.06 nS. We ran the simulation for 500 s and only the second phase was analyzed, in accordance with Johnston et al., 2016. All the results presented are based on five different islets, including all 1000 cells, with randomized parameter values. To explore the effects of coupling conductance on the functional network, we altered islet average coupling conductance of a single gap junction pair to gcoupj =0.06 nS and gcoupij =0.03 nS, and then assigned each cell a randomized gcoupj based on the new target average. The total gap junction conductance (gcoup) for any cell was calculated as the sum of the average conductance between the cell of interest and any cells it shares a gap junction connection with:gcoup=∑i=1m12(gcoupi+gcoupj)
The coupled IOM
The ordinary differential equations for the IOM have been extensively described in previous works (Marinelli et al., 2021; Marinelli et al., 2022). The model simulates the dynamics within a single β-cell and it has been shown to be able to account for key experimental findings (Bertram et al., 2018; Marinelli et al., 2018), including both slow and fast bursting regimes. Allowing this achievement is the interaction of the three modules that make up the model: one module to describe the cellular electrical activity, a second to describe the glycolysis, and a third to describe the mitochondrial metabolism. The change in the membrane potential (Vi) for β-celli is described by Equation 3.CmdVidt=ICa+IK(Ca)+IK(ATP)+IK
Due to limitations of computational power and model stability, we created a 260 β-cell network (instead of the 1000 cells simulated with the coupled Cha-Noma model) where cells we coupled based on their proximity and through the coupling conductance, Icoup , defined in Equation 1b. The differential equations were integrated numerically using MATLAB and all simulations are run at 11 mMol glucose.
In this simulation, the average number of gap junction connections per cell was 6.01 with a standard deviation of 2.66, and with a maximum of 14 and a minimum of 1 gap junction connection per cell. Similar to the computation with the Cha-Noma model, cellular heterogeneity was achieved by allowing key metabolic and electrical parameters to range within a distribution. More precisely, the coupling conductance (gcoup) is assumed to be normally distributed with mean 1 pS and a standard deviation 0.5. We then assign the maximum rate through the glucokinase reaction (kglyc, referred to previously as vGK; Marinelli et al., 2021; Marinelli et al., 2022) using a normal distribution with a mean 0.0037 µM/ms and standard deviation 0.0015 µM/ms, and extract the values for the maximum conductance of the KATP channel (gKATP) from a normal distribution with mean 19,700 pS and standard deviation 3940 pS. Once again, the total gap junction conductance (gcoup) for any cell was calculated as described in Equation 2.
Network analysisCreating functional network from Cha-Noma simulated islet
The methodology was based on that previously defined in Stožer et al., 2013. First, the correlation coefficient between each cell was calculated using corr() function in MATLAB, which follows the equation:Rij=∑[xi(t)−xi¯][xj(t)−xj¯]∑[xi(t)−xi¯]2∑[xi(t)−xi¯]2
Next, a threshold (Rth) was defined to determine whether each cell pair is ‘synchronized’ or ‘not synchronized’. For computational experiments, the threshold was chosen such that the network roughly followed a power law distribution, as predicted in Stožer et al., 2013; Johnston et al., 2016. Unless otherwise noted, Rth = 0.9995 for computational analysis.
Creating functional network from IOM simulated islet
To create the functional network from the IOM simulated islet, we repeated steps for the Cha-Noma model. However, due to small islet size and model stiffness, we were unable to achieve strong heterogeneity in the calcium oscillations (Figure 2a), requiring a threshold of Rth = 0.999999999 to obtain a scale-free-like distribution (Figure 2b and c).
Creating the metabolic network
Because gap junctions enforce localization onto the analysis, we looked at cell pairs whose Pythagorean distance was ≤15% within the islet. Within this sample space, we looked at cell pairs whose average kglyc was more similar than the islet average (Figure 4b).
We chose these parameters such that the Pr(Sync|GJ)=Pr(Sync|Met), which acts as a control allowing us to directly compare Pr(GJ|Sync) to Pr(Met|Sync).
Average degree
The degree ki was calculated by counting the number of connections for cell i and averaging k over all cells in the islet, then normalized to the islet size to remove any size dependence (kavg / n, where n = islet size).
Degree distribution
The degree distribution was calculated by first calculating the degree of each cell by taking the column sum of the adjacency matrix. Then each cell degree was normalized by dividing by the maximum degree of the islet. The histogram was then calculated using GraphPad Prism.
Hub identification
In accordance with Johnston et al., 2016, any cell with more than 60% islet, calculated by the degree distribution, was considered a hub cell.
Probabilities
We first created the functional network and structural (GJ-related probabilities) or metabolic network (metabolism). We then calculated the probability that two cells were synchronized by:Prsync=msyncn-1*n2
where msync = number of edges in the synchronized network, and n = number of nodes. Similarly, we calculated the probability that two cells were either gap junction coupled or metabolically related usingPr(GJorkglyc)=mGJorkglyc(n−1)∗n2
where mGJorkglyc is the number of edges in the gap junction or metabolic network.
To find the probability that a cell pair was both synchronized and GJ or metabolically connected, we calculated using equation:Prboth=mbothn-1*n2
where mboth is any edge that exists in both matrices.
Finally, we calculated conditional probabilities by:PrsyncGJ=PrbothPrGJPrGJsync=PrbothPrsync
These quantities were calculated separately for each simulated islet and then averaged.
Duty cycle
We defined duty cycle as the proportion of time [Ca2+] was elevated above 50% of its maximum value.
The gap junction network was weighted using the inverse of gcoup or kglyc between cell i and cell j : Wij=112(gcoupi+gcoupj) or Wij=112(kglyci+kglycj) . This is done because the shortest path length algorithm views weights as ‘resistance’ and finds the path of least resistance. The shortest weighted path between every cell was calculated using Johnson’s algorithm (Johnson, 1977). Cell pairs were then categorized as synchronized or not synchronized if their correlation coefficient was <Rth . The average for synchronized and non-synchronized cells was calculated over each islet for each distance. To normalize over distance, each data point was divided by the average of the non-synchronized and synchronized islets for the given distance.
Clustering coefficient
The clustering coefficient represents the ‘cliquishness’ of the network. This is defined by Stožer et al., 2013, as the ‘number of existing connections between all neighbors of a node, divided by the number of all possible connections between them’. This was calculated by making a subgraph of each cell’s connections and counting the number of connections between those cells. For example, if A is connected to B, C, and D, and B and C are connected but D is not connected to any other cell (see matrix). Then the clustering for cell A is 23*2=13 . Each node is assigned a clustering coefficient C such that:
Subgraph – A
B
C
D
B
1
0
C
1
0
D
0
0
The average clustering coefficient is Cavg = 1n∑inCi .
Average shortest path length
Shortest path length was calculated with MATLAB function graphallshortestpaths(). This function uses the Johnson’s algorithm (Johnson, 1977) to find the shortest path between every pair of cell islet. For example, the path length between cell i and cell j is 1 if they are directly synchronized, or 2 if cell i is not synchronized with cell j but each is synchronized with cell k . To compensate for highly sparse network, any non-connected node was given a characteristic path length of n+1. Finally, the characteristic path length (L), or average path length, was expressed as the sum of all path lengths normalized to total possible connections (size*(size-1)).
The normalized average shortest path length (Supplemental 2d) is therefore calculated as Lavg=1n1n∑i=1n∑j=1nLij. To compensate for any non-connected cell, whose path length Lix=∞, where x is any cell islet, we set the path length for non-connected cells to Lix→n+1, where n is the number of cells islet.
Global efficiency
The global efficiency is related to the inverse of global path length (Latora and Marchiori, 2001).Eglobal=1Isletsize∗(Isletsize−1)∑jk1Ljk
length using Eglobal=1nn-1∑i=1n∑j=1n1Lij . Because Eglobal=0 for a non-connected cell, the disconnected network is naturally compensated for.
Random networks
Random networks were created using an Erodos Renyi approach (Lakens, 2013). For each islet, we created 1000 random networks with equal number of nodes, edges, and average degree as the islet of interest. The probability that two cells were connected was given by p=kavgn. For each cell pair within the islet, a random number generator created a number from 0 to 1. If that number was below p, whose cells were connected by an edge.
Significance testing
Effect sizes were calculated using Cohen’s dz=t/√n which is appropriate for paired samples (Lakens, 2013), where n=5.
Significance tests are done in Prism (GraphPad). α values are set to 0.5 unless otherwise mentioned.
Figure 1e–g, Figure 2d–f and h–j, Figure 4e, f and Figure 5e, h4,5 are paired two-tailed t-tests.
Figure 1h, Figure 2g, Figure 6c–f, and Figure 7c–f are repeated measures paired one-way ANOVA with multiple comparison using Tukey’s multiple comparison.
Figure 3f, i are unpaired two-tailed t-tests.
Figure 3g and j are linear regressions.
Figure 1—figure supplement 1b is repeated measures paired one-way ANOVA with multiple comparison using Tukey’s multiple comparison.
Figure 5—figure supplement 1b, Figure 5—figure supplement 2b are multiple paired t-tests with Bonferroni-Dunn adjustment. Since there were 15 cell distances, we set the significance threshold α=0.003. For convenience, we present asterisks next to significant p-values.
Additional informationCompeting interests
No competing interests declared
Author contributions
Conceptualization, Data curation, Software, Formal analysis, Funding acquisition, Validation, Investigation, Visualization, Methodology, Writing – original draft, Writing – review and editing
Investigation, Visualization, Writing – review and editing
Investigation, Methodology, Writing – review and editing
Software, Methodology
Writing – review and editing
Data curation, Formal analysis, Methodology, Writing – review and editing
Conceptualization, Supervision, Funding acquisition, Investigation, Methodology, Writing – original draft, Project administration, Writing – review and editing
Ethics
All animal procedures were performed in accordance with guidelines established by the Institutional Animal Care and Use Committee of the University of Colorado Anschutz Medical campus (protocol 000024). All surgeries were performed under ketamine/xylazine anesthesia, with minimal discomfort to the animals.
Additional filesData availability
Raw microscopy imaging data is available on the EMBL-EBI-supported BioImage Archive. Analysis code, model code, and simulated data is available via GitHub at https://github.com/jenniferkbriggs/Functional_and_Structural_Networks (copy archived at Briggs, 2023).
The following dataset was generated:
BenningerR2024Beta-cell intrinsic dynamics rather than gap junction structure dictates subpopulations in the islet functional networkEBI BioImage ArchiveS-BIAD1024
Acknowledgements
Richard KP Benninger (University of Colorado) is the guarantor of this work and, as such, had full access to all the data in the study and takes responsibility for the integrity of the data and the accuracy of the data analysis. We thank David W Piston (Washington University St Louis) with whom data presented in Figure 6 was previously acquired. We thank Aaron Clauset and Mark Husing for helpful conversations and advice. All authors acknowledge that no conflict of interest exists. National Institutes of Health (NIH) grant R01 DK102950 (RKPB) National Institutes of Health (NIH) grant R01 DK106412 (RKPB) National Science Foundation (NSF) Graduate Research Fellowship DGE-1938058_Briggs (JKB) Juvenile Diabetes Research Foundation (JDRF) grant 3-PDF-2019-741-A-N (VK) Beckman Research Institute-City of Hope (HIRN) grant UC24 DK104162 (VK) Burroughs Wellcome Fund – Careers at Scientific Interfaces Project Number 25B1756 (VK) National Institutes of Health (NIH) grant F31 DK126360 (JMD) National Institutes of Health (NIH) grant LM012734 (DJA) University of Birmingham Dynamic Investment Fund (IM). The authors are grateful for the utilization of the SUMMIT supercomputer from the University of Colorado Boulder Research Computing Group, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the University of Colorado Boulder, and Colorado State University. Microscopy use was supported in part by NIH grant P30 DK116073 and the University of Colorado Neurotechnology center.
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The paper uses both computational and laboratory approaches to test the hypothesis that connectivity in β cells within the islet is due to metabolic rather than gap junctional coupling efficacy. This will be an important advance for understanding the role of heterogeneous β cell populations in driving synchronized oscillations by islets and by extension the oscillatory insulin secretion observed in vivo. There will be implications of the work for understanding the mechanisms of type 2 diabetics and β cell function in general.
10.7554/eLife.83147.sa1Decision letterSatinLeslie SReviewing Editorhttps://ror.org/00jmfr291University of MichiganUnited StatesHodsonDavid JReviewerhttps://ror.org/052gg0110University of OxfordUnited KingdomStozerAndrazReviewerSalemVictoriaReviewerhttps://ror.org/041kmwe10Imperial College LondonUnited Kingdom
Our editorial process produces two outputs: i) public reviews designed to be posted alongside the preprint for the benefit of readers; ii) feedback on the manuscript for the authors, including requests for revisions, shown below. We also include an acceptance summary that explains what the editors found interesting or important about the work.
Decision letter after peer review:
Thank you for submitting your article "Β-cell Metabolic Activity Rather than Gap Junction Structure Dictates Subpopulations in the Islet Functional Network" for consideration by eLife. Your article has been reviewed by 5 peer reviewers, and the evaluation has been overseen by a Reviewing Editor and Naama Barkai as the Senior Editor. The following individuals involved in the review of your submission have agreed to reveal their identity: David J Hodson (Reviewer #1); Andraz Stozer (Reviewer #2); Victoria Salem (Reviewer #3).
The reviewers have discussed their reviews with one another, and the Reviewing Editor has drafted this to help you prepare a revised submission.
Essential revisions:
1) The modeling, based on the Cha et al. model is clearly of fast oscillations but the data shown include both fast and slow oscillations. The model will therefore need to be revised to simulate the slow data more closely either using the Cha model in a different mode, or a different model.
2) There are questions about some of the analyses and plots only showing a small number of data points, which must be addressed and also the choice of R-value as discussed.
3) There was a general concern about the small-worldness of the data and this must be dealt with.
4) It is recommended that some of the key conclusions made by the authors be more tempered as some of the reviewers commented that perhaps some key conclusions were overstated based on the data and analysis.
5). It would help if the authors more clearly indicated which findings were truly novel and which were confirmation of observations made previously by their lab and others.
6). The manuscript is quite long and rather dense and thorough editing must be done to improve its readability (especially by non-experts); this would help the paper.
Reviewer #1 (Recommendations for the authors):
I have the following constructive suggestions to further improve the studies:
– "In diabetes the islet shows disrupted [Ca2+] synchronization (representing a disrupted functional network) and diminished gap junction coupling (representing a disrupted structural network)." A schematic would be helpful here to visually explain the functional and structural networks that are the focus of the studies.
– The resolution of the FRAP studies is impressive, with single-cell definition. Nonetheless, FRAP only looks at cationic dye transfer, does not report conductance, and is unlikely to be as sensitive as patch clamp measures. Conversely, patch clamp is inherently biased, is spatially limited, and likely selects for certain cell subpopulations (i.e. cells that do not form seals or respond to voltage ramps are disposed of). Caveats are needed here, since there is probably no perfect measure of GJ conductance across a cell population, which is the basis for the modelling here.
– "Prior work suggested that cells with highly synchronized Ca2+ oscillations possess both elevated metabolic activity (in agreement with our observations, Figure 2) and high levels of gap junction conductance (unlike what we observe experimentally, Figure 2)". Johnston et al. showed that Gjd2 knockdown disrupted synchronization, functional connectivity, and hub number, implying a role of GJ communication in hub function. However, this knockdown was not hub-specific so any inferences about hub GJ coupling are indiirect.
– "However, as the functional network becomes sparser, the uncertainty in the random network-based approach increases (Supp. 4d, seen more clearly in Supp. 6d)." Maybe I am not following here, but would a sparser functional network not be expected to decrease power-law slope/fit and hence the definition of small-worldness? Some more explanation would be helpful.
– "Semi-quantitative immunofluorescence previously demonstrated elevated glucokinase protein in β-cell hubs, suggesting elevated metabolic activity." The authors also looked at TMRE accumulation in hubs, showing more hyper-polarized mitochondria indicative of increased OXPHOS (hence fitting with the NAD(P)H).
– It is well established that gap junction coupling is decreased in animal models of diabetes, as well as aged/high BMI human islets. However, are functional and structural networks mutually exclusive? That is, do changes in cell states drive changes in GJ coupling or vice versa? Or is there no relationship between these parameters?
– Electrical coupling is assumed to be a static phenomenon within the islet. However, connexins are dynamic proteins, their gating can be influenced by cAMP/PKC signals (shown by this group), and expression levels differ across β cells. How would the authors predict such heterogeneity to influence their model? How might this be built into analyses going forward?
– A comment on the translation of the results to human islets would be appreciated since there are known differences in coordination (more regional) and cell interactions compared to rodents.
– "Finally, the islet also contains α-cells, δ-cells, and other cell types which can influence β-cell dynamics via paracrine mechanisms". It would be worth citing recent studies from the Chen and Tang groups (DOI s41467-022-31373-6).
– The authors note some study limitations. It would be worthwhile also discuss limitations with the models of GJ disruption. For example, GJD2 KO is global and occurs early in development, so any effects may be confounded by β cell de-differentiation/immaturity (and metabolic changes therein). I understand that conditional Cx36 models have been historically unavailable, but conditional-ready KOMP mice could be informative in the future.
Reviewer #2 (Recommendations for the authors):
Introduction
– I suggest a reference for the epidemiological data, i.e., for the number of people affected by diabetes. The IDF Diabetes Atlas is a valuable resource and typically accompanied by citable publications, such as PMID: 34879977
The first part of the Results: Cellular metabolism, but not elevated gap junction coupling, is observed in highly synchronized cells in a simulated β-cell network, the corresponding Figures, and Methods
– Rth was chosen at the value of Rth = 0.9995 – this is a rather high value compared with values typically chosen for experimental calcium traces (e.g., Stožer PLoS Comput Biol 2013, Front Endocrinol 2022), which indicates that the model did not produce large temporal differences/delays between cells. Could this be improved in the model to more closely mimic the experimental situation where delays between cells are on the order of magnitude that is the same as the order of magnitude of burst/fast oscillation durations? If there are objective reasons that this cannot be done easily with existing models (by for instance destabilizing the model with increasing heterogeneity), I suggest that the authors point out this difference between experiments and model and more explicitly address the nature of this discrepancy.
– From the inset in Figure 1d, it is also not clearly visible what the temporal delay between traces in different cells was. Please, provide a more detailed inset/zoom-in.
– In addition to the mean parameter values, I recommend that the authors also provide the range of values more precisely quantify heterogeneity. Since the statistics in Figure 1 are based on a rather small number of simulated islets (N=5), I suggest that effect sizes be reported as well.
– In the model, only 500 seconds were simulated, and then the so-called fast calcium oscillations present during the second (plateau) phase of the calcium response (and brought about by bursts of electrical activity) were analyzed. In the simulated traces, they seem to be around 15-25 seconds long (Figure 1d). From the Methods section, it is not entirely clear what period of calcium traces obtained by calcium imaging in isolated islets was used for the network analysis. The authors state that "[Ca2+] time courses were analyzed during the second-phase [Ca2+] response when the slow calcium wave was established", but Figure 2 and other parts of the manuscript do not provide enough information to be sure whether the interval used for network analyses included the whole traces beyond approximately 500 seconds (Figure 2b) or only smaller parts of these traces that would correspond in duration to intervals used on simulated traces. I think it is critical that the authors address and resolve this question in detail for the following reasons.
– First, simulated traces reflect only fast oscillations, whereas in experimental traces (See Figure 2b), there are a few slow oscillations that last approximately 300 seconds each, and superimposed on them fast oscillations (approximately 10 seconds long), corresponding to oscillations analyzed on simulated traces. Since there may be fundamental mechanistic differences between fast and slow oscillations, the first being more importantly determined by ion channel properties and electrical coupling and the second more importantly by metabolism, the networks constructed from experimental traces may contain and provide information that is different from the information provided by simulated traces and networks based on them. If the authors included in their analyses on experimental traces only one part of a slow wave (e.g. 200-300 seconds) containing a few fast oscillations (e.g. 10 fast oscillations), then the simulated traces and networks can be compared with experimental traces and networks. If not, i.e., if they included longer periods (e.g. 1000 seconds) containing a few slow oscillations and thus many more fast oscillations (e.g., 50 fast oscillations), then the Pearson correlation between traces may heavily depend on the slow trends defined by slow oscillations (as addressed recently in Zmazek et al. Front Physiol 2021 and Stozer et al. Front Endocrinol 2022), and thus the network metrics convey entangled information on both fast and slow properties. Most importantly, the main conclusion from the experimental part that the rate of metabolism may be more important than other factors in determining functional network properties may be due to the prevailing influence of metabolic oscillations on networks based on experimental recordings. In this case, I suggest two possible solutions. The authors could extend the simulations to produce a mixed pattern of oscillations (which should be possible, given the existing work by Bertram, Sherman, Satin, Nunemaker, and Benninger) and then systematically analyze the impact of the fast and slow components on network properties and the importance of different parameters (kglyc, gKATP, gcoup) and compare the findings with experimental data where they could also separately study the two components. Alternatively, the authors could extract the fast component from the experimental traces (by using an appropriate filter) and limit their analyses and discussion to the fast oscillations only.
– Second, resolving the above point is not only important for this manuscript, but also for resolving the (apparent) differences between studies (and the Rutter-Rorsman dispute).
– Finally, resolving the first above point is also particularly important for extending the findings to Cx36-/- mice where fast and slow oscillations may be affected to a different extent by the absence of gap junctions, with desynchronized fast oscillations and possibly less desynchronized slow oscillations.
The other parts of the manuscript
– My comments and suggestions to other parts of the manuscript depend on how the above fast-slow oscillation issue will be resolved since all of the following Results and corresponding Figures strongly depend on the way how networks are constructed. I will be more than happy to do so in a possible next round of revision.
Reviewer #3 (Recommendations for the authors):
Briggs et al. present a strong set of both computational and experimental approaches to investigate some current controversies in the functional relevance of β cell heterogeneity and pancreatic islet function. They conclude that β cell connectivity is driven by metabolic rather than gap-function-mediated structural coupling. This lays the ground for future studies understanding how metabolic coupling relates to the identity of "hub cells" and to what extent this can be targeted in the treatment of diabetes. Overall this is a very strong paper that is of interest to the readership of a journal like eLife. It is extremely well written. I would suggest that more experimental data for the NADPH and FRAP experiments might build up confidence in the finding that connectivity is defined by metabolic coupling – at present, the datasets are convincing that metabolic coupling is absent in cells that lack calcium connectivity but are not convincing enough in the opposite direction.
Here are my general comments.
The authors state that "Functional networks represent the emergent system behavior". Diabetes ensues from the autoimmune destruction of β cells (T1) or the functional demise (a mix of environmental insult with a genetic predisposition – T2). To what extent can network theory really tell us anything about the pathophysiology of these disease states or aid the development of new treatments? Or is this simply another tool, like GSIS, for assessing islet dysfunction? I think it would be useful to have this sort of oversight mentioned in the discussion, to persuade readers of the real relevance of your work.
"cell hubs can exert a strong influence over islet dynamics21,24,25 53 and are preferentially disrupted in diabetes23" Did the Johnston paper really conclude that diabetes is causally related to the preferential loss of hubs over followers? Or is it more likely that in a dysregulated islet loss of "hubs" becomes more apparent? I personally would have prefaced this paper by summarising the mathematical and experimental evidence for hubs that exhibit small-world properties.
When you say "structural location" what do you mean – you didn't for example look at a position relative to blood vessels, α cells, nerve endings, and δ cell projections. As a general comment, I'm not really comfortable with you equating conductance (ie the physical number of gap junction interfaces between β cells) as synonymous with their structural topography. There are so many unknowns here still – for example the contribution of other endocrine cells, nerves, and "humoral factors". Your computational model essentially randomly assigns heterogeneity in terms of metabolic rates, for example, across the 1000 "β" cells. In fact, this may not reflect the fact that such heterogeneity is indeed based on a cell's position relative to other aspects of the islet microenvironment (not just other β cells).
Results section 1
Briggs et al. start with data extracted from a model. Whilst the thresholds selected are generally well justified and the testing of various thresholds adds robustness, it is, in the end, just a model. It has been well validated by electrophysiologists, but the audience of this paper is wider and I think it is worthwhile reminding us in a couple of sentences of what the major inputs are into this model and how that results in oscillatory behaviour (I know you do this in supplemental anyway). Presumably, this is being run for a certain ambient glucose level ie parameters change between low and high glucose. It's important that you point out there is no accounting for other endocrine cells etc – which I think you do in the
Discussion.
Was such a high R-value necessary to get the requisite power-law type distribution a surprise to you? Were all 1000 cells included in your readouts? Why did you choose the 60% connection cut-off to define hubs? Is this done by others? In your supplementary data as you move (in very tiny increments) across R thresholds the number of connections on your axis changes apparently randomly (the scale goes from 0-200 to 0-800). Why?
The differences in Kglyc between hubs and followers are statistically significant but the difference in absolute terms seems very small – is this relevant?
Results section 2
This is the experimental dataset which is very interesting.
"We extracted the functional network (Figure 2c) and again generated a normalized degree distribution which reflects a scale free-like distribution."
– ok but there is no mention of actual R values here – presumably much less than the modelled ones previously and the cut off for defining hubs on normalised degree of "edginess" is also different from the above (and others?). Conversely, you then introduce the concept of a "low degree" cell without defining that.
Just out of interest, the cross-section shows that some β cells are much more fluorescent than others, presumably reflecting variation in GcAMP expression. Do you think this is an issue for your calcium trace analysis?
In the Methods section please can you explain in a bit more detail how you extracted your NADPH data – over what time period/resolution etc.
"Furthermore, the NAD(P)H response trended lower in cells that were functionally disconnected (Ca2+ 128 oscillations lacking any synchronization), compared to connected cells (Figure 2g)."
Presumably, you only looked at calcium oscillating β cells as inactive ones will obviously likely have no other How did you define an oscillation? Your time course of over 30 minutes looks long. The association between low NADPH signal and low connectivity seems much more robust than that between hubs and high NADPH signal. Would repeat experiments firm this up?
Is the FRAP experiment powered? I don't have a feel for the sensitivity of this method to pick apart quantifiable differences in gap junction connections but the numbers here seem low – only 4 islets.
Results section 3
The next section poses the question what does the islet functional network indicate about its underlying structure or intrinsic properties on an individual cell basis?
The authors appear to have returned to their simulated data here which initially confused me so should be headlined at the outset. Given that the EPists know that GJ coupling cannot explain connectivity across more than a few cells, I think it's important to state in the main text that you enforced spatial limits on your structural connectivity analyses.
Some of the surprising findings e.g.
"The probability that two cells were synchronized in the functional network, given that they shared a gap junction in the structural network, was = 0.39" has been discussed well later (eg where they don't tally with prior experimental measures.
On the whole, I find the ending statement "These results further indicate that metabolic activity, not gap junction connections, is a greater driving factor for cells to show high [Ca2+] synchronization and thus influence the functional network." to be robust.
The next section looks at long-range functional connections which traverse cells, quite dense. I thought the experiment that modelled GJs of higher conductance to be rather extraneous and could have gone into supplemental but I don't feel strongly about this.
I think there needs to be some unpacking of the term kglyc as a measure of "glucose metabolism" – how does the sameness of kglyc translate into closer coordination of calcium oscillations? Is this simply the speed with which glucose sensing and insulin release are being cycled? Is metabolic "coupling" simply a coincidence – is this all simply an epiphenomenon?
Finally, the authors look at experimental data from connexin KOs (het and homs vs WTs). I found this data (even though we have seen this before) to be rather surprising given the previous narrative. In essence, what it shows is that a 50% reduction in GJs has a marked effect, and a total loss results in a near-complete loss of hubs. If connections, and therefore presumably networks that deliver hubs, can exist just based on random metabolic homogeneity in distant β cells alone, why does this happen? Is there any NADPH data from the connexin KO islets? With the loss of gap junctions, I can see why the amplitude of the calcium oscillations in β cells might become smaller, but why do other metrics of the oscillations eg frequency seem to change?
Figures
Figure 1 – Kglyc and g coup need to be defined/explained so the figure stands alone.
Figure 1b – the word seed is confusing/comes out of the blue. Do you actually mean "modelled islet"?
Figure 1 f/g – this may display my incomplete understanding of the different populations you defined and measured, but why was a paired t-test used to compare them and if these readings are truly paired were they really normally distributed?
Figure 2 – the defined cellular ROIs in a is not the same as the ones shown on your map in c – I was expecting them to be!
Again, here we have the notion of Kmax mentioned which needs defining in the legend.
Reviewer #4 (Recommendations for the authors):
1. The paper is very dense and complex, and thus is difficult to follow, and as a consequence will only be understandable to a very specific and likely small group of readers. The methods, the analysis, and the models used, while largely based on prior work require careful reading, and a true understanding of their implications will be mostly lost on most readers. The paper would therefore be improved by reducing its size and complexity and removing excessive verbiage. Some of the more mathematical aspects could be relegated to either previously published papers by the group or placed in the Supplementary Material. It might be worth considering breaking the paper into two different but complementary ones, one emphasizing theory and the other measurements but I realize this would make reading the two simultaneously very difficult.
2. The paper would also be improved by very clearly highlighting what is truly new in the paper and deemphasizing what is a restatement of what has been known already (e.g. discussing how the islet can be considered a small world network, etc).
Reviewer #5 (Recommendations for the authors):
I have a major concern about how cells related by cell metabolism and cells related by gap junctional coupling are treated as independent. There is no mechanism mentioned (?) for metabolism to synchronize islet cells independent of gap junctions. Consequently, I have concerns about the conclusions of the results.
This work seems to suggest a disregard of the necessity of gap junctional connections for islet synchronization, and suggest that functional connectivity – based on statistical correlation of traces – somehow predominate…up until Figure 6 where knocking down gap junctions is acknowledged as important.
I think identifying subpopulations is reasonable, but when a 60% functional connection threshold is what defines "hub" cells and that produces between 50 and 200 cells, "hub" seems to lose its meaning, especially when these "hubs" seems to be all grouped together rather than dispersed as might be expected in a small world network. The small network-ness is questionable. The sensitivity to the correlation threshold suggests a certain lack of robustness. Perhaps I am misunderstanding the number of cells labeled as "hub cells."
The excitability of cells via metabolism, for example, must be communicated through some (often structural) mechanisms. I appreciate that this article is attempting to get at how that division breaks down and I think much of the calculation and simulation is useful, but the language expressing the certainty rather than the more appropriate equivocation is not completely appropriate.
[Editors’ note: further revisions were suggested prior to acceptance, as described below.]
Thank you for resubmitting your work entitled "Β-cell Intrinsic Dynamics Rather than Gap Junction Structure Dictates Subpopulations in the Islet Functional Network" for further consideration by eLife.
The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below:
Reviewer #2 (Recommendations for the authors):
I much appreciate the effort that the authors put into addressing my main skepticism about the paper, namely the fact that one type of oscillations were analyzed for experimental data and another type for modelled data, by including an additional model for slow oscillations and additional experimental data.
Given the new results, below, I provide my additional comments and suggestions that should not be too difficult for authors to consider and take into account.
110-197 and Figures 1-S3:
First, the difference between hubs and non-hubs in terms of kglyc is around 0.14 vs 0.13, whereas for gcoup it is around 0.75 vs 0.6. In relative terms, the former is an 8 % difference and the latter a 25 % difference. It is true that the former is more significant statistically, but this is due to the fact that the heterogeneity is larger in the distribution of gcoup for hubs! Therefore, to me, it seems that both kglyc and gcoup seem to define the hub cells to a comparable degree. This conclusion seems to also be more compatible with findings for slow oscillations where the relative difference between hubs and non-hubs for gcoup is again around 20-30 %.
I suggest that authors consider this under the corresponding sections of results and discussion, and perhaps to some extent also in Figure 8.
Second, at a threshold for hub cells of a 60 % normalized degree (i.e., the criterion for the separation line between hubs and non-hubs), different percentages of β cells qualify as hubs for different simulated islets, as evident in Figure 1b. More specifically, the cumulative number of cells qualifying as hubs for islets 3 and 5 is much more than the cumulative number of hubs in simulated islets 1 and 2. This differs from the criterion used in other important recent studies, i.e., in Johnston et al. Cell Metab 2016, hubs typically represent 1-10 % of islet cells, in Lei et al. Islets 2018, hubs represent 10 % of cells, and in Stozer et al. Am J Physiol 2021 and in Sterk et al. Biophys J 2023, hubs represent 1/6th of islet cells. I think it would enhance the comparison between the present and previous research if the authors provided the % of hub cells per islet. Additionally, setting the line of separation between hubs and non-hubs at the given threshold (not changing the threshold and thus the distribution of normalized degree) individually for each islet to achieve a fixed percentage of hub cells per islet close to 10 % of the most connected cells, would be a valuable addition to supplemental figure 1 and could perhaps help detect a larger (or smaller) difference between this more extreme group of cells and the majority of non-hub cells, in terms of kglyc, gkatp, and gcoup. Performing this additional analysis at least for the threshold value used in the main Figure 1 could add much value to the manuscript and make the findings even more directly comparable with the aforementioned studies. The same suggestion could also be taken into account for the analyses of modelled slow oscillations and for the experimental analyses of metabolic activity and coupling, where 10 % of the most connected cells could be considered/classified as hubs as well. Such additional analysis should be easily feasible since it considers a fraction of cells that have already been used in the analyses, but their degree of "hubness" is probably more.
Given that the duty cycle (percentage of active time) so strongly correlates with the role of hub cells, it is somewhat surprising that the authors do not mention in the discussion that this same finding has recently been obtained for both mouse and human islets (Šterk et al. Biophys J 2023, Stožer et al. Am J Physiol 2021, Gosak et al. Diabetes 2022).
Third, I do value the effort of authors in replying to my comment regarding the exceedingly high correlation between simulated traces compared with experimental traces. However, besides noise, one very important aspect is the choice of model and the parameters which determine the observed lags between different cells or the wave speed. For instance, Cappon and Pedersen in their Chaos article built on the model used in Benninger Biophys J 2008 to produce lags between cells that are very realistic compared with experimentally observed values, whereas lags in the present study are just a fraction of the duration of a burst (fast Ca oscillation).
At present, I can only speculate about that, but the rather low conditional probability that two cells sharing gap-junctional conditions also show a functional connection could also be a consequence of the exceedingly short time lags between signals in the model employed in this study. More specifically, in the model by Cappon and Pedersen, there are waves travelling across islets and direct neighbors have shorter phase-lags between their signals compared with more distant cells and therefore on average more similar signals, similar to experimental recordings in isolated islets and tissue slices (the velocity being around 100 um/s, the time lag between direct neighbors is around 0.1 s and the time lag between most distant cells is around 1-2 s). This means that the values of R will be higher for direct neighbors and at any given threshold they will be functionally connected with a higher probability than with more distant cells where lags can be an order of magnitude more.
Therefore, I would suggest that in the future (not in this study), authors also repeat some of the analyses with modeled traces that show larger lags. This would probably also enable them to explore the relationship between different modelled/analyzed parameters and the role of pacemakers, i.e., wave initiators and other populations of cells. Perhaps, if authors consider the above suggestion as useful, they could include it in the discussion as a possible drawback of the present model and as a suggestion for future studies.
Reviewer #5 (Recommendations for the authors):
The authors have thoroughly responded to my (and others') reviews. I think their modifications are reasonable and have tempered the language appropriately. The results are somewhat interesting and the techniques employed to measure and analyze the islets including adding an additional model system are extensive. They have addressed my points.
One point: We discussed (Reviewer #5 question/response 5b) the potential outlier at 2.35 of Figure 1f (gkatp data – Β Cell Hubs). That data point no longer exists on the graph (that I can see?). Since they argue convincingly that point should in fact not be treated as an outlier, I expect it is an oversight of it not being there. This should be fixed (or explained) prior to moving forward with publication.
10.7554/eLife.83147.sa2Author response
Essential revisions:
1) The modeling, based on the Cha et al. model is clearly of fast oscillations but the data shown include both fast and slow oscillations. The model will therefore need to be revised to simulate the slow data more closely either using the Cha model in a different mode, or a different model.
In addition to the Cha et al. based model, we created a coupled simulation of the Integrated Oscillator Model (IOM) which was derived to simulate both fast and slow oscillations. Results from this coupled IOM are described in Figure 2, and support our overall conclusions. These essential revisions are included in Figure 2, Results page 6 lines 128-156.
We also conducted additional experimental measurements of calcium imaging and fluorescence recovery after photobleaching (FRAP) for which we separately analyzed slow, fast, and mixed [Ca2+] oscillations. For all three oscillation types, we found no relationship between functional network degree and gap junction permeability. (Figure 3i,j and Figure 3 Supplemental 1c-h). These essential revisions are included in Figure 3i,j, Figure 3 Supplemental 1c-h, and Results page 7 lines 184-186 and lines 188-191.
2) There are questions about some of the analyses and plots only showing a small number of data points, which must be addressed and also the choice of R-value as discussed.
We have clarified what each data point represents in both the captions and the results text, as well as in the response to reviewers below. Briefly, in simulations, each dot corresponds to the averaged value in a simulated islet (Figures 1, 2, 4, 5, 7). We conducted five independent simulations for both the Cha-Noma and IOM model. In experimental results, each data point also corresponds to the average value over an islet (Figures 3, 6). These essential revisions are included in all figure captions.
Importantly we have conducted additional experiments to address any concerns about the small number of experimental data points. These essential revisions are included in Figure 3j,k.
We have responded to the reviewers regarding the choice of threshold used in the simulations and experiments. The simulation threshold required to match past experimental results is high, primarily because the simulations lack stochastic noise, which strongly influences the correlation threshold. There are also additional computational and model limitations with respect to the Integrated Oscillator Model which reduce the amount of heterogeneity in the simulated [Ca2+] oscillations. We describe these limitations in our discussion and include three supplemental figures showing the effect of threshold on simulation (Figure 1 Supp 1, Figure 2 Supp 1) and experimental (Figure 3 Supp 1) results. With the exception of the coupled Integrated Oscillator model (which we further discuss), the threshold does not influence our conclusions. These essential revisions are included in Supplemental Figures 1-3, Discussion page 14 lines 448-455.
3) There was a general concern about the small-worldness of the data and this must be dealt with.
We agree that we did not conduct a comprehensive small-worldness analysis of our data. This was not a primary focus of the manuscript as small-worldness of the functional network has been shown in Stozer 2013[1]. In efforts to address essential revisions 5 and 6 and simplify the manuscript, we removed all small world analyses from the previous figures 5, 6 and associated supplemental figures. These essential revisions are addressed by the removal of figures and associated text.
4) It is recommended that some of the key conclusions made by the authors be more tempered as some of the reviewers commented that perhaps some key conclusions were overstated based on the data and analysis.
We agree that our initial manuscript miscommunicated some of our interpretations of the results. We have tempered key conclusions and statements throughout the manuscript. One important miscommunication in our initial manuscript is that we are not claiming that metabolic coupling drives the functional network instead of electrical coupling. Rather, intrinsic properties of β-cells drive variations in the functional network rather than structural gap junctions. Gap junctions are still critical to synchronize oscillations across the whole islet. We have taken extensive efforts to remove this miscommunication from the manuscript. This includes a sentence stating this in the discussion (Page 12, Lines 394-397), changing the title to “Β-cell Intrinsic Dynamics Rather than Gap Junction Structure Dictates Subpopulations in the Islet Functional Network”, and altering writing throughout the manuscript to be clearer (particularly in the Results and Discussion). We also highlight data that show the importance of gap junctions so we do not incorrectly communicate that structural coupling is unnecessary for islet synchronization. These essential revisions are addressed in the Introduction (page 3 lines 45-47) by a change in the title, by highlighting data in Figure 6 and 7 and in the Results (page 7 lines 203-205, page 8, lines 241-242, page 9 lines 256 and 293-294, page 10 lines 308-309) and Discussion (page 11 line 318-319, page 13, lines 434-435).
5). It would help if the authors more clearly indicated which findings were truly novel and which were confirmation of observations made previously by their lab and others.
We have substantially rewritten the introduction and Discussion sections to address this comment.
6). The manuscript is quite long and rather dense and thorough editing must be done to improve its readability (especially by non-experts); this would help the paper.
In response to comments 5 and 6, we have significantly restructured the paper –the Introduction, Results and Discussion – with careful emphasis on highlighting which of our findings are novel and which have been shown in previous papers. We also were intentional about making the manuscript less dense and more accessible. To be specific, we reduced question 3 by removing any network metrics that we felt were tangential to the goals of the study and removing additional peripheral data. This includes comments about small world-ness and local efficiency and simulated data correlating metabolism and gap junction coupling. Despite including an additional figure, other additional data, and additional discussion to address comments by all 5 reviewers, the word count remains similar. These essential revisions are addressed throughout the Introduction, Results and Discussion, in Figures 6 and 7, and through removing Supplemental Figures.
Reviewer #1 (Recommendations for the authors):
I have the following constructive suggestions to further improve the studies:
– "In diabetes the islet shows disrupted [Ca2+] synchronization (representing a disrupted functional network) and diminished gap junction coupling (representing a disrupted structural network)." A schematic would be helpful here to visually explain the functional and structural networks that are the focus of the studies.
We thank the reviewers for a useful suggestion on how to make the paper clearer. We have added an inset into Figure 8 that shows the structural and functional networks.
– The resolution of the FRAP studies is impressive, with single-cell definition. Nonetheless, FRAP only looks at cationic dye transfer, does not report conductance, and is unlikely to be as sensitive as patch clamp measures. Conversely, patch clamp is inherently biased, is spatially limited, and likely selects for certain cell subpopulations (i.e. cells that do not form seals or respond to voltage ramps are disposed of). Caveats are needed here, since there is probably no perfect measure of GJ conductance across a cell population, which is the basis for the modelling here.
We thank the reviewer for their enthusiasm for FRAP measurements, which has single cell resolution as we previously demonstrated by Farnsworth et al. [2]. We agree FRAP will not be as sensitive as the gold standard patch clamp measurements, but it does allow multiple cells to be assessed and compared in the same islet. We do note Cx36 is cationic specific so is suited to FRAP here (FRAP of anionic dyes in the islet reveals minimal dye transfer). We now mention caveats associated with FRAP within the discussion. See Discussion page 14 lines 457-464.
– "Prior work suggested that cells with highly synchronized Ca2+ oscillations possess both elevated metabolic activity (in agreement with our observations, Figure 2) and high levels of gap junction conductance (unlike what we observe experimentally, Figure 2)". Johnston et al. showed that Gjd2 knockdown disrupted synchronization, functional connectivity, and hub number, implying a role of GJ communication in hub function. However, this knockdown was not hub-specific so any inferences about hub GJ coupling are indirect.
Thank you for pointing out this need for clarification. The prior work were referring to was in Lei et al. 2017 [3] who conducted a computational study of the functional network in human islets and concluded that hub cells must have stronger than average gap junction conductance in order to silence the islet, as first demonstrated in Johnston et al[4]. We therefore repeated our analysis on simulated islets whose gap junction conductance was proportional to metabolic activity. However, we note that in seeking to make the manuscript shorter and easier to read we have removed this Figure and associated Results text. Nevertheless, when examining Cx36+/- islets we do see a reduction in the network degree consistent with the results the reviewer refers to. However as suggested this is likely due to genetic knockout not being hub specific. We now mention this – See Discussion page 12 lines 372-374.
– "However, as the functional network becomes sparser, the uncertainty in the random network-based approach increases (Supp. 4d, seen more clearly in Supp. 6d)." Maybe I am not following here, but would a sparser functional network not be expected to decrease power-law slope/fit and hence the definition of small-worldness? Some more explanation would be helpful.
We thank the reviewer for this thoughtful comment. Indeed, in a sparse network the power-law fit will decrease exactly as the reviewer says. In the comment above, we refer to the increase in the uncertainty of small worldness in the random networks. In order to establish whether the functional network topology comes from interesting natural behavior or is a result of randomness, we compared the network metrics (such a “small-worldness”) to random graphs. The random graphs are akin to a null hypothesis. This technique is well established in network theory. The random graphs are formed by taking the same number of edges and nodes as in the network of interest and randomly exchanging the edges with some probability. This was described in Table 1 and Methods Network Topology Analysis: Random Networks. The comment above refers to the analysis of these random graphs. As the functional network becomes sparser, the configuration of edges becomes more variable and small changes in the random graph have a larger impact on the small-worldness parameter. Therefore, the uncertainty in this small-worldness increases.
However, we note again that in seeking to make the manuscript shorter and easier to read we have removed results relating to small worldness, and rather present the clustering and global efficiency data (Figure 6, 7).
– "Semi-quantitative immunofluorescence previously demonstrated elevated glucokinase protein in β-cell hubs, suggesting elevated metabolic activity." The authors also looked at TMRE accumulation in hubs, showing more hyper-polarized mitochondria indicative of increased OXPHOS (hence fitting with the NAD(P)H).
Thank you for pointing out these additional measurements- we now refer to the TMRE measurements which are consistent with the NAD(P)H measurements. See Discussion page 12 lines 368-370.
– It is well established that gap junction coupling is decreased in animal models of diabetes, as well as aged/high BMI human islets. However, are functional and structural networks mutually exclusive? That is, do changes in cell states drive changes in GJ coupling or vice versa? Or is there no relationship between these parameters?
We thank the reviewer for this question. In Figure 5 and 6 (original version, now figure 6 and 7), we analyzed the effect of changes in structural networks on functional networks. As the reviewer indicates, in diabetes and other pathological conditions, gap junction coupling is reduced. Therefore, we analyzed how the functional network changes as gap junction coupling (e.g., the structural network is decreased). Our data suggest that functional and structural networks are not mutually exclusive. Decreasing edges in the structural network do lead to decreased oscillation synchronization and, therefore, average degree in the functional network. We also found that the topology of the functional network is affected by the structural network. Data in figure 5 and Figures 6-7 indicate that gap junction coupling is influencing longer range synchronization across the islet, which explains the minimal overlap between functional and structural network on a node or edge scale, but that the networks are not mutually exclusive. We now describe this further. See Discussion page 13 lines 426-427.
– Electrical coupling is assumed to be a static phenomenon within the islet. However, connexins are dynamic proteins, their gating can be influenced by cAMP/PKC signals (shown by this group), and expression levels differ across β cells. How would the authors predict such heterogeneity to influence their model? How might this be built into analyses going forward?
The reviewer is correct and raises an important point that connexin trafficking is dynamic and Cx36 gap junctions can gate, in response to cAMP signals; and also show expression changes in response to Ca2+ – CaN – NFAT signaling. Further, Cx36 gap junction permeability varies between cells (as we include in the model). This variation would be expected to be on the time scale of minutes (cAMP-gating) to hours (cAMP-trafficking) to hours-days (NFAT-transcription) which is much slower than electrical changes that we analyze. Further Cx36 gap junctions are only weekly voltage gated and therefore we can consider gap junction conductance to be static in relation to electrical oscillations.
Against our initial prediction, the synchronization between cells is not substantially influenced by variations in the local gap junction conductance (Question 1). As such we would conclude that if the strength of gap junction coupling varies between cells over time, providing the islet-average conductance remains unchanged, the positioning of hub cells should not change significantly. Thus, if positioning of hub cells does change over time, this would be predicted to result from changes in cell-intrinsic properties. We briefly discuss this – See Discussion page 14 lines 465-468.
Variations in gap junction conductance do seem to influence another cell sub-population – first responder cells (see Kravets et al. PLOS Biology 2022[5]) and the ability of a cell to transition between quiescence and activity upon nutrient stimulation. However, while interesting we avoid further complicating the manuscript by introducing this additional topic.
– A comment on the translation of the results to human islets would be appreciated since there are known differences in coordination (more regional) and cell interactions compared to rodents.
Thank you for this note. We have added the following sentence to the discussion:
“This [referring to future analysis in the influence of other cell types] is particularly important when translating our findings to human islets because α-cells, δ-cells, and β-cells are known to be more mixed in human islets”. See Discussion page 14 lines 475-477.
– "Finally, the islet also contains α-cells, δ-cells, and other cell types which can influence β-cell dynamics via paracrine mechanisms". It would be worth citing recent studies from the Chen and Tang groups (DOI s41467-022-31373-6).
Thank you for the suggestion. We agree that the Chen and Tang paper is very intriguing. We added the citation and comment: “Additionally, β-cell oscillation types (slow, fast, or mixed) may be directly related to the number of α-cells present in the islet[6]. ” See Discussion page 14 lines 472-473.
– The authors note some study limitations. It would be worthwhile also discuss limitations with the models of GJ disruption. For example, GJD2 KO is global and occurs early in development, so any effects may be confounded by β cell de-differentiation/immaturity (and metabolic changes therein). I understand that conditional Cx36 models have been historically unavailable, but conditional-ready KOMP mice could be informative in the future.
We do appreciate the Cx36 ko model has caveats being a global knockout. While changes in β cell specification in such models have not been reported we understand this cannot be excluded. Similarly, while unperturbed floxed models have not been available, these should be available in the future. However, we do note that dissociated β cells from both wild-type and Cx36ko islets show similar insulin secretion responses. In response to this comment we mention potential caveats associated with the Cx36 global ko model. See Discussion page 14 lines 459-464.
Reviewer #2 (Recommendations for the authors):
Introduction
– I suggest a reference for the epidemiological data, i.e., for the number of people affected by diabetes. The IDF Diabetes Atlas is a valuable resource and typically accompanied by citable publications, such as PMID: 34879977
Thank you for the reference, this has been added to the paper. See Introduction page 3 lines 42.
The first part of the Results: Cellular metabolism, but not elevated gap junction coupling, is observed in highly synchronized cells in a simulated β-cell network, the corresponding Figures, and Methods
– Rth was chosen at the value of Rth = 0.9995 – this is a rather high value compared with values typically chosen for experimental calcium traces (e.g., Stožer PLoS Comput Biol 2013, Front Endocrinol 2022), which indicates that the model did not produce large temporal differences/delays between cells. Could this be improved in the model to more closely mimic the experimental situation where delays between cells are on the order of magnitude that is the same as the order of magnitude of burst/fast oscillation durations? If there are objective reasons that this cannot be done easily with existing models (by for instance destabilizing the model with increasing heterogeneity), I suggest that the authors point out this difference between experiments and model and more explicitly address the nature of this discrepancy.
As we also responded to reviewer 3: It was not necessarily surprising to us that the computational model required a high threshold to get the power-law type distribution. Correlation coefficients are strongly decreased with stochastic noise. To illustrate this, in Author response image 1, we plot the correlation coefficient between two off phase sine waves. The first has no noise and a correlation of 0.997 and the second has 10% added noise and a correlation of 0.82. There is no noise in the computational model (because it is simulated data) while there is noise in experimental imaging data. We anticipate that this noise difference is the reason for the difference in correlation threshold. We now expand our comment on this: See Discussion page 14 lines 448-455.
– From the inset in Figure 1d, it is also not clearly visible what the temporal delay between traces in different cells was. Please, provide a more detailed inset/zoom-in.
Thank you for the suggestion, we have changed the figure accordingly. See Figure 1d (and also Figure 2c).
– In addition to the mean parameter values, I recommend that the authors also provide the range of values more precisely quantify heterogeneity. Since the statistics in Figure 1 are based on a rather small number of simulated islets (N=5), I suggest that effect sizes be reported as well.
We have updated the methods to provide the exact distributions used for imposing heterogeneity. The number of gap junction connections was given by a normal distribution mean 5.25, standard deviation 1.6, with the maximum of 12 and a minimum of 1 gap junction connections per cell. Cellular heterogeneity was introduced by assigning randomized metabolic and electrical parameter values to each cell based on their distributions previously determined by experimental studies. For example, coupling conductance (gcoup) was assigned using a γ distribution with k = 4, θ = 4 and then shifted so the islet average was gcoup=0.12nS. Glucokinase (kglyc), the rate-limiting step in glycolysis, was assigned using a normal distribution with mean 1.26x10-4 s-1 and standard deviation 3.15x10-5 s-1. Conductance of the KATP channel (gKATP) was given by a normal distribution with mean 2.41 pA mV-1 and standard deviation 0.57 pA mV-1. These details are all now included: See Methods page 19 lines 594-603 and 623-632.
Effect sizes are now included – See Figure 1 caption (page 34) and figure 2 caption (page 35), and Methods page 22 lines 719-720-753. This further supports the dominant association of kglyc and marginal association of gcoup in hub cells in the Cha-Noma Model:
– In the model, only 500 seconds were simulated, and then the so-called fast calcium oscillations present during the second (plateau) phase of the calcium response (and brought about by bursts of electrical activity) were analyzed. In the simulated traces, they seem to be around 15-25 seconds long (Figure 1d). From the Methods section, it is not entirely clear what period of calcium traces obtained by calcium imaging in isolated islets was used for the network analysis. The authors state that "[Ca2+] time courses were analyzed during the second-phase [Ca2+] response when the slow calcium wave was established", but Figure 2 and other parts of the manuscript do not provide enough information to be sure whether the interval used for network analyses included the whole traces beyond approximately 500 seconds (Figure 2b) or only smaller parts of these traces that would correspond in duration to intervals used on simulated traces. I think it is critical that the authors address and resolve this question in detail for the following reasons.
Thank you for your careful reading and pointing this out. We have now updated the methods with the following information. See Methods page 16 lines 534-537.
“For fast oscillations, 400-500 seconds of [Ca2+] response was analyzed, matching the simulation time. For slow oscillations, 741-1000 seconds of [Ca2+] response was analyzed. This time period was chosen so at least 3 oscillations were completed. In mixed oscillations, 858 – 1000 s of [Ca2+] response was analyzed.”
– First, simulated traces reflect only fast oscillations, whereas in experimental traces (See Figure 2b), there are a few slow oscillations that last approximately 300 seconds each, and superimposed on them fast oscillations (approximately 10 seconds long), corresponding to oscillations analyzed on simulated traces. Since there may be fundamental mechanistic differences between fast and slow oscillations, the first being more importantly determined by ion channel properties and electrical coupling and the second more importantly by metabolism, the networks constructed from experimental traces may contain and provide information that is different from the information provided by simulated traces and networks based on them. If the authors included in their analyses on experimental traces only one part of a slow wave (e.g. 200-300 seconds) containing a few fast oscillations (e.g. 10 fast oscillations), then the simulated traces and networks can be compared with experimental traces and networks. If not, i.e., if they included longer periods (e.g. 1000 seconds) containing a few slow oscillations and thus many more fast oscillations (e.g., 50 fast oscillations), then the Pearson correlation between traces may heavily depend on the slow trends defined by slow oscillations (as addressed recently in Zmazek et al. Front Physiol 2021 and Stozer et al. Front Endocrinol 2022), and thus the network metrics convey entangled information on both fast and slow properties. Most importantly, the main conclusion from the experimental part that the rate of metabolism may be more important than other factors in determining functional network properties may be due to the prevailing influence of metabolic oscillations on networks based on experimental recordings. In this case, I suggest two possible solutions. The authors could extend the simulations to produce a mixed pattern of oscillations (which should be possible, given the existing work by Bertram, Sherman, Satin, Nunemaker, and Benninger) and then systematically analyze the impact of the fast and slow components on network properties and the importance of different parameters (kglyc, gKATP, gcoup) and compare the findings with experimental data where they could also separately study the two components. Alternatively, the authors could extract the fast component from the experimental traces (by using an appropriate filter) and limit their analyses and discussion to the fast oscillations only.
The reviewer raises an important point and caveat associated with our comparison of simulated and experimental data. This point was also raised by reviewer 4. The simulated data (Figure 1) has Ca2+ oscillations of ~20s period (i.e. fast oscillations) whereas the experimental data (Figure 2) has Ca2+ oscillations of ~300s period (i.e. slow oscillations). To address this comment, we have performed several additional experiments and analyses:
1. We collected additional Ca2+ (to identify the functional network and hubs) and FRAP data (to assess gap junction permeability) in islets which show either pure slow, pure fast, or mixed oscillations. We generated networks based on each time scale to compare with FRAP gap junction permeability data. We found that the conclusions of our first draft to be consistent across all oscillation types. There was no relationship between gap junction conductance, as approximated using FRAP, and normalized degree for slow (Figure 3j), fast (Figure 3 Supp 1d,e), or mixed (Figure 3 Supp 1g,h) oscillations. We also include discussion of these conclusions – See Results page 7 lines 184-186 and lines 188-191, Discussion page 12 lines 357-360.
2. We also performed additional simulations with a coupled ‘Integrated Oscillator Model’ which shows slow oscillations because of metabolic oscillations (Figure 2). We compared connectivity with gap junction coupling and underlying cell parameters. In this case, there is an association between functional and structural networks, with highly-connected hub cells showing higher gap junction conductance (Figure 2f) but also low KATP channel conductance (gKATP) (Figure 2e). However, there are some caveats to these findings – given the nature of the IOM model, we were limited to simulating smaller islets (260 cells) and less heterogeneity in the calcium traces was observed. Additional analysis suggests the greater association between functional and structural networks in this model was a result of the smaller islets, and the association was also dependent on threshold (unlike in the Cha-Noma fast oscillator model) robust. These limitations and results are discussed further (Discussion page 11 lines 344-354).
Additionally, in the IOM, the underlying cell dynamics of highly-connected hub cells are differentiated by KATP channel conductance (gKATP), which is different than in the fast oscillator model (differentiated by metabolism, kglyc). However this difference between models can be linked to differences in the way duty cycle is influenced by gKATP and kglyc (Figure 1h, Figure 2g). In each model there was a similar association between duty cycle and highly-connected hub cells. We also discuss these findings (Discussion page 11 lines 334-343).
Overall these results and discussion with respect to the coupled IOM oscillator model can be found in Figure 2, Results page 6 lines 128-156 and Discussion page 11 lines 332-354.
– Second, resolving the above point is not only important for this manuscript, but also for resolving the (apparent) differences between studies (and the Rutter-Rorsman dispute).
This is an important comment. We also hope that our manuscript will help resolve the apparent differences between studies. We were sure to cite the Rutter-Rorsman dispute in the Introduction page 3 line 63-65.
– Finally, resolving the first above point is also particularly important for extending the findings to Cx36-/- mice where fast and slow oscillations may be affected to a different extent by the absence of gap junctions, with desynchronized fast oscillations and possibly less desynchronized slow oscillations.
This is an excellent point. While homozygous Cx36ko islets lack any synchronization, a partial decrease in Cx36 such as in a heterozygous Cx36ko islet may influence fast and slow oscillations differently. Our findings show the association between β cell hubs or the functional network with gap junction coupling structural network is not dependent on oscillation timescale (Figure 3 and Figure 3 Supp 1). Thus, both fast and slow oscillating islets should show similar changes with decreased gap junction coupling. We now mention this – Discussion page 13 lines 438-440.
Reviewer #3 (Recommendations for the authors):
Briggs et al. present a strong set of both computational and experimental approaches to investigate some current controversies in the functional relevance of β cell heterogeneity and pancreatic islet function. They conclude that β cell connectivity is driven by metabolic rather than gap-function-mediated structural coupling. This lays the ground for future studies understanding how metabolic coupling relates to the identity of "hub cells" and to what extent this can be targeted in the treatment of diabetes. Overall this is a very strong paper that is of interest to the readership of a journal like eLife. It is extremely well written. I would suggest that more experimental data for the NADPH and FRAP experiments might build up confidence in the finding that connectivity is defined by metabolic coupling – at present, the datasets are convincing that metabolic coupling is absent in cells that lack calcium connectivity but are not convincing enough in the opposite direction.
We thank the reviewer for their high enthusiasm for our manuscript. As detailed below we have collected and present more experimental data and also include more robust statistical analysis of the experimental data.
Here are my general comments
The authors state that "Functional networks represent the emergent system behavior". Diabetes ensues from the autoimmune destruction of β cells (T1) or the functional demise (a mix of environmental insult with a genetic predisposition – T2). To what extent can network theory really tell us anything about the pathophysiology of these disease states or aid the development of new treatments? Or is this simply another tool, like GSIS, for assessing islet dysfunction? I think it would be useful to have this sort of oversight mentioned in the discussion, to persuade readers of the real relevance of your work.
While the islet network does provide a tool to assess overall islet function, we would suggest it also provides a means to examine individual cell sub-populations and provide insight into how they are driving islet function. We now mention in the discussion the motivation to use network theory in the context of diabetes (below). See Discussion page 10 lines 236-330.
“By understanding the formation of islet functional networks we can understand how β-cell subpopulations drive islet function. Knowledge about the role of β-cell subpopulations will also be useful when developing new treatments, such as identifying and preserving functional sub-populations in diabetes and generating stem cell derived insulin secreting cells for restoring insulin secretion.”
"cell hubs can exert a strong influence over islet dynamics21,24,25 53 and are preferentially disrupted in diabetes23" Did the Johnston paper really conclude that diabetes is causally related to the preferential loss of hubs over followers? Or is it more likely that in a dysregulated islet loss of "hubs" becomes more apparent? I personally would have prefaced this paper by summarising the mathematical and experimental evidence for hubs that exhibit small-world properties.
Thank you for pointing out a very important miscommunication in our introduction. We agree that Johnston did not show any causality and instead showed that diabetic cytokines influenced the existence of β cell hubs but did not show a preferential loss of hubs over followers. We have updated our introduction accordingly. We have also changed our introduction as you recommend to summarize the evidence more clearly for and against small-world properties. See Introduction page 3 lines 57-65.
When you say "structural location" what do you mean – you didn't for example look at a position relative to blood vessels, α cells, nerve endings, and δ cell projections. As a general comment, I'm not really comfortable with you equating conductance (ie the physical number of gap junction interfaces between β cells) as synonymous with their structural topography. There are so many unknowns here still – for example the contribution of other endocrine cells, nerves, and "humoral factors". Your computational model essentially randomly assigns heterogeneity in terms of metabolic rates, for example, across the 1000 "β" cells. In fact, this may not reflect the fact that such heterogeneity is indeed based on a cell's position relative to other aspects of the islet microenvironment (not just other β cells).
We have updated our manuscript to be clearer about this topic and do not use the words location or position for cells within the structural network. For example “are highly correlated subpopulations (β-cell hubs) within the β-cell functional network differentiated with respect to the structural network”. Thank you for the suggestion. See for example Introduction page 3 lines 76-78; Results page 5 lines 99-100 and throughout the manuscript.
Results section 1
Briggs et al. start with data extracted from a model. Whilst the thresholds selected are generally well justified and the testing of various thresholds adds robustness, it is, in the end, just a model. It has been well validated by electrophysiologists, but the audience of this paper is wider and I think it is worthwhile reminding us in a couple of sentences of what the major inputs are into this model and how that results in oscillatory behaviour (I know you do this in supplemental anyway). Presumably, this is being run for a certain ambient glucose level ie parameters change between low and high glucose. It's important that you point out there is no accounting for other endocrine cells etc – which I think you do in the Discussion.
We have included more detailed information about the model in the results – See Results page 5 lines 92-96. We similarly describe the coupled IOM model – See Results page 6 lines 131-138.
Thank you also for pointing out the important limitations of the model that we use – and models in general. We agree and describe these limitations in greater depth within the discussion, in the “Potential Limitations” section – see Discussion page 14 lines 443-446.
Was such a high R-value necessary to get the requisite power-law type distribution a surprise to you?
As we also responded to reviewer 2: It was not necessarily surprising to us that the computational model required a high threshold to get the power-law type distribution. Correlation coefficients are strongly decreased with stochastic noise. In Author response image 1, we plot the correlation coefficient between two off phase sine waves. The first has no noise and a correlation of 0.997 and the second has 10% added noise and a correlation of 0.82. There is no noise in the computational model (because it is simulated data) while there is noise in imaging data. We hypothesize that this noise difference is the reason for the difference in correlation threshold. We now comment on this See Discussion page 14 lines 448-455
Were all 1000 cells included in your readouts?
Yes, we used 1000 cells in all five model runs. We mention this now – see Methods page 18 line 607-608.
Why did you choose the 60% connection cut-off to define hubs? Is this done by others?
The threshold was set to 60% of the max degree to align with Johnston et al.[4], which was the first paper to define hubs. We describe this now in the Methods and in the Results where we first describe the threshold cutoff. See Methods page 20 lines 661-662 and Results page 5 lines 106-108.
In your supplementary data as you move (in very tiny increments) across R thresholds the number of connections on your axis changes apparently randomly (the scale goes from 0-200 to 0-800). Why?
This is a result of the functional islet network having a roughly scale-free distribution. In Figure 1 Supplemental 1, we see that as the threshold decreases, the number of cells that are considered “high degree” will increase in a power-law like manner.
The differences in Kglyc between hubs and followers are statistically significant but the difference in absolute terms seems very small – is this relevant?
In line with the question from reviewer 2 (question 4), we outlined the distributions of the parameter values in the methods to provide more clarify about the spread of the parameter distribution. The average kglyc for non-hubs was the 1.26x10-4 s-1 (which is the average of the distribution because most cells are non hubs) while the average kglyc for hubs was 1.4x10-4 s-1 which is about half of a standard deviation higher. This is further supported all effect sizes being much greater than 0.8 for all significantly different findings (Cha Noma – kglyc: 2.85, gcoup: 0.82) (IOM: gKATP: 1.27, gcoup: 2.94) – We have included these effect sizes in the captions see Figure 1 and 2 captions (pages 34, 35) and Methods page 22 lines 722-723.
Results section 2
This is the experimental dataset which is very interesting.
"We extracted the functional network (Figure 2c) and again generated a normalized degree distribution which reflects a scale free-like distribution."
– ok but there is no mention of actual R values here – presumably much less than the modelled ones previously and the cut off for defining hubs on normalised degree of "edginess" is also different from the above (and others?).
We have updated the sentence to contain the threshold. We feel the network degree vs recovery plot was informative. However, we now also include a comparison between hubs vs non-hubs, where hubs are defined in the same as the computational method and Johnston et al. 2016 (e.g. nodes with >60% of maximal edges in the network). See Results page 6 line 163, page 7 line 175-176 and Figure 3d.
Conversely, you then introduce the concept of a "low degree" cell without defining that.
Following above, we include both network degree vs recovery/NADH and hub vs non-hub comparison. We now do not include any ‘low degree’ comparison as we agree this may cause some confusion.
Just out of interest, the cross-section shows that some β cells are much more fluorescent than others, presumably reflecting variation in GcAMP expression. Do you think this is an issue for your calcium trace analysis?
This is a very insightful comment. We normally see variations in the GCamP6 signal likely due to variations in GCamP6 expression (as also seen in other fluorescent-protein expressing models). However, since the time-courses are normalized to the mean GCamP6 signal we do not anticipate that this has an impact on subsequent analysis. We did analyze the relationship between GCamP6 signal and normalized degree and found no correlation (Author response image 2), which we now mention. See Results page 6 lines 166-167.
In the Methods section please can you explain in a bit more detail how you extracted your NADPH data – over what time period/resolution etc.
NAD(P)H was measured before and after the Ca2+ time course where glucose starts at 2mM and is elevated to 11mM. Thus the 2mM measurement is prior to Ca2+ imaging and the 11mM measurement after the conclusion of Ca2+ imaging. We also updated the methods with these details and exactly how the NAD(P)H response metric was determined – see Methods page 16 lines 517-524.
"Furthermore, the NAD(P)H response trended lower in cells that were functionally disconnected (Ca2+ 128 oscillations lacking any synchronization), compared to connected cells (Figure 2g)."
Presumably, you only looked at calcium oscillating β cells as inactive ones will obviously likely have no other How did you define an oscillation? Your time course of over 30 minutes looks long. The association between low NADPH signal and low connectivity seems much more robust than that between hubs and high NADPH signal. Would repeat experiments firm this up?
We did only examine β cells in which GCamP6 signal was above a background threshold. However, any β-cell with a GCamP6 signal above background will be included in the analysis, irrespective of whether oscillations occurred or not. Inactive cells will likely show little coordination and thus have a low to zero network degree. Thus, we do not need to define an oscillation in our analysis.
We collect Ca2+ data for ~30 min in order to include sufficient numbers of oscillations to generate the functional network – this is because the oscillations are often slow (although for fast osculation we use a narrower time window). The time window used is now described – see Methods page 17 lines 537-540
We do appreciate that low connectivity does indeed appear to show lower NAD(P)H as compared to high connectivity showing higher NAD(P)H. One speculation for this is that given the power-law-like degree distribution, there are more cells with low degree than high degree in each islet. Since each dot represents the average over each islet, the lower degrees are more robust to outliers than the higher degrees. However as described above we now compare either NADH vs network degree or NADH between hub and non-hub cells. In each case we provide more robust statistical analysis and linear regression to show that a relationship between functional network degree and metabolism exists (or is trends toward signficance): for NADH hub vs non-hub (p=0.0061), for NADH vs network degree linear regression (p=0.079). See figure 3 caption (page 36) and Results page 7 lines 184-191.
Is the FRAP experiment powered? I don't have a feel for the sensitivity of this method to pick apart quantifiable differences in gap junction connections but the numbers here seem low – only 4 islets.
We have performed additional Ca2+ and FRAP measurements, and include data from an additional 11 islets. These data are further separated into slow, fast, and mixed oscillations and compared as either linear regression vs network degree or between hub and non-hubs (none of which show a significant relationship between network degree and gap junction permeability). See Figure 3 i,j and Figure 3 Supp 3.
Results section 3
The next section poses the question what does the islet functional network indicate about its underlying structure or intrinsic properties on an individual cell basis?
The authors appear to have returned to their simulated data here which initially confused me so should be headlined at the outset.
We have adjusted the Results section titles and writing to make this clearer. See Results page 7 lines 196, 230.
Given that the EPists know that GJ coupling cannot explain connectivity across more than a few cells, I think it's important to state in the main text that you enforced spatial limits on your structural connectivity analyses.
This is an important fact and we now mention this in the section “Elevated glucose metabolism is a greater driver of longer-range functional connections”. See Results 8 lines 231-233.
“In accordance with the islet cytoarchitecture, gap junction connections only exist between highly proximal cells. However, there is no distance constraint on [Ca2+] oscillation synchronization (Figure 5a). To determine whether this spatial constraint was responsible for the low correspondence between the functional and structural networks…”
Some of the surprising findings e.g.
"The probability that two cells were synchronized in the functional network, given that they shared a gap junction in the structural network, was = 0.39" has been discussed well later (eg where they don't tally with prior experimental measures.
Thank you for your comment, we elaborate on these findings in the Discussion (page 13 lines 400-410).
On the whole, I find the ending statement "These results further indicate that metabolic activity, not gap junction connections, is a greater driving factor for cells to show high [Ca2+] synchronization and thus influence the functional network." to be robust.
Thank you for this comment.
The next section looks at long-range functional connections which traverse cells, quite dense. I thought the experiment that modelled GJs of higher conductance to be rather extraneous and could have gone into supplemental but I don't feel strongly about this.
We have edited the section concerning long range connections so that it is more assessable. Results examining long range connections are restricted to one figure, with additional data in Figures 5 Supp. 1. However, we do feel some of these findings are important to retain as it is quite important as it removes the spatial component of the structural network.
We have now removed the figure that modelled gap junction conductance correlating with metabolism.
I think there needs to be some unpacking of the term kglyc as a measure of "glucose metabolism" – how does the sameness of kglyc translate into closer coordination of calcium oscillations? Is this simply the speed with which glucose sensing and insulin release are being cycled? Is metabolic "coupling" simply a coincidence – is this all simply an epiphenomenon?
We have more carefully defined kglyc when it is first introduced in the results related to figure 1 (as also with gKATP) – see Results page 5 lines 110-112.
“Glucokinase conversion of glucose to glucose 6-phosphaste is the rate limiting step in glycolysis, therefore the rate of glucokinase activity (kglyc) is a proxy for the metabolic activity in the cell.”
We also clarify within the results and discussion our use of metabolism. We avoid the term “metabolic coupling” but rather describe metabolism as the intrinsic metabolic activity of the cell. We would in fact argue that no evidence for metabolic coupling in the islet has been presented and is not likely given the cationic selectivity of Cx36 gap junctions. See Discussion page 13 lines 395-397.
Finally, the authors look at experimental data from connexin KOs (het and homs vs WTs). I found this data (even though we have seen this before) to be rather surprising given the previous narrative. In essence, what it shows is that a 50% reduction in GJs has a marked effect, and a total loss results in a near-complete loss of hubs. If connections, and therefore presumably networks that deliver hubs, can exist just based on random metabolic homogeneity in distant β cells alone, why does this happen? Is there any NADPH data from the connexin KO islets? With the loss of gap junctions, I can see why the amplitude of the calcium oscillations in β cells might become smaller, but why do other metrics of the oscillations eg frequency seem to change?
It is well accepted that gap junctions are essential for synchronization over the entire islet and that decrease in gap junction changes the total islet behavior. We have rewritten much of the paper to clarify that gap junctions are still essential for overall islet synchronization, but they are not the primary factor driving the variations in synchronization that determines the functional network. For example, while β-cell hubs do not have higher than average gap junction conductance, they still require gap junctions to mediate the effect of their greater metabolic activity to synchronize with other β cells. This can also be seen in figure 5 where paths with higher gap junction conductance do start to show some overlap with long range functional connections. For example, see Results page 9, line 256.
Frequency is known to be affected by gap junctions conductance because as gap junction conductance decreases, the cells’ intrinsic frequency (rather than collective frequency from islet communication) dominates [9], [10].
Figures
Figure 1 – Kglyc and g coup need to be defined/explained so the figure stands alone.
We have adjusted the manuscript accordingly – See Figure 1 and 2 captions (page 34, 35).
Figure 1b – the word seed is confusing/comes out of the blue. Do you actually mean "modelled islet"?
This is a great point – the seed is a technical word for how the random number generator is defined in order to replicate the simulated islet experiments. However it is unnecessary to use this terminology outside of the methods. All references to “seed” are now “simulated islet” (see Figure captions)
Figure 1 f/g – this may display my incomplete understanding of the different populations you defined and measured, but why was a paired t-test used to compare them and if these readings are truly paired were they really normally distributed?
Thank you for this comment. The paired t-test is used because each comparison comes from the same simulated islet. To ensure robustness, I repeated the analysis with unpaired t-test and the interpretation did not change (see Author response table 1). I also tested for normality using Shapiro-wilk and Kolmogorov-Smirnov. All samples fit the threshold of normality except for non-hubs which tested normal under KS but not Shapiro-Wilk.
UnpairedP-value
Pairedp-value
Shapiro-Wilk Normality Test (Yes = normally distributed)(Hubs/Non-hubs)
Figure 2 – the defined cellular ROIs in a is not the same as the ones shown on your map in c – I was expecting them to be!
Thank you so much for your careful observation. The map in c was rotated incorrectly. We have corrected the figure so that the ROIs in Figure 3a (previous figure 2) correspond to the network in Figure 3c.
Again, here we have the notion of Kmax mentioned which needs defining in the legend.
Thank you, we have removed the use of Kmax and redefined the legend as 90% of the max degree. (Figure 3 caption, page 36)
Reviewer #4 (Recommendations for the authors):
1. The paper is very dense and complex, and thus is difficult to follow, and as a consequence will only be understandable to a very specific and likely small group of readers. The methods, the analysis, and the models used, while largely based on prior work require careful reading, and a true understanding of their implications will be mostly lost on most readers. The paper would therefore be improved by reducing its size and complexity and removing excessive verbiage. Some of the more mathematical aspects could be relegated to either previously published papers by the group or placed in the Supplementary Material. It might be worth considering breaking the paper into two different but complementary ones, one emphasizing theory and the other measurements but I realize this would make reading the two simultaneously very difficult.
Thank you for this suggestion. We agree that the manuscript is quite dense. We have made the following changes to make the manuscript more accessible. In consultation with the editor, we decided not to break the paper into two given journal policies; but rather focus on eliminating more peripheral data. Similarly, the mathematical parts of the methods we retain in the methods but are happy to move if the editor suggests this.
1. Significantly shortening of the Introduction and rewriting of the Discussion.
2. Substantially reducing the last two figures from the previous manuscript (figure 5,6) and reducing the associated results text- this includes removing the complex network metrics and kept only clustering coefficient and global efficiency (see Figures 6, 7).
3. Removing simulated islet data and associated Results text where metabolic activity and gap junction coupling were correlated.
2. The paper would also be improved by very clearly highlighting what is truly new in the paper and deemphasizing what is a restatement of what has been known already (e.g. discussing how the islet can be considered a small world network, etc).
Thank you for the suggestion, we have made significant modifications throughout the Introduction, Discussion and Results to be clearer about what is known from previous work and what is newly found in this manuscript.
Reviewer #5 (Recommendations for the authors):
I have a major concern about how cells related by cell metabolism and cells related by gap junctional coupling are treated as independent. There is no mechanism mentioned (?) for metabolism to synchronize islet cells independent of gap junctions. Consequently, I have concerns about the conclusions of the results.
Thank you for this comment and valid point of concern apologize for the confusion. We have tried to make the mechanisms that we are referring to much clearer. Our study is not assessing metabolic coupling or any means of metabolic intermediates actually moving between cells to synchronize them. Rather, we are looking at how to interpret the functional network which shows cells that are highly synchronized. The functional network is based on how synchronized β cell pairs are. Given that β cells are intrinsic oscillators, if two cells have identical intrinsic oscillations (which is primarily driven by the metabolic activity of the cell), then the cells will readily “synchronize” and have an edge in the functional network with minimal gap junction mediated current. Thus two cells can become highly synchronized without necessarily showing high levels of gap junction coupling. This explains why in our partial gap junction knock out experiments (figure 5 and 6), some cells are still highly connected in the functional network. Our aim in this paper was to show that the functional network also reflects the intrinsic cellular characteristics rather than mainly gap junction coupling. As an extreme example, in the complete absence of gap junction coupling a few cells that show identical Ca2+ oscillations and will therefore show a functional edge despite lacking any gap junction connections. We have tried to make this more clear in the paper – see Discussion page 13 lines 395-397.
This work seems to suggest a disregard of the necessity of gap junctional connections for islet synchronization, and suggest that functional connectivity – based on statistical correlation of traces – somehow predominate…up until Figure 6 where knocking down gap junctions is acknowledged as important.
We apologize again for the confusion. We have tried to emphasize the important role of gap junctions in whole islet synchronization to clarify the aim of this paper, which we also attempt to clarify in the reviewer's first comment. Results (page 7 lines 203-205, page 8, lines 241-242, page 9 lines 256 and 293-294, page 10 lines 308-309) and Discussion (page 11 line 318-319, page 13, lines 434-435).
I think identifying subpopulations is reasonable, but when a 60% functional connection threshold is what defines "hub" cells and that produces between 50 and 200 cells, "hub" seems to lose its meaning,
Thank you for the necessary point of clarification. We agree that 60% of functional connections could be large depending on how a hub cell is to be interpreted. In this paper, we chose to use 60% in order to directly match methodology used in Johnston et al[4], which is the first paper to describe β cell hubs. This logic is noted in Methods page 20 lines 661-662 and Results page 5 lines 106-108.
However, given that the degree distribution is roughly logarithmic, 60% of the maximum degree produces a relatively small number of hubs per islet. As you note, the total number of hub cells in a 1000 cell islet was: 52, 47, 109, 88, and 120, which is equivalent to 5%, 4.7%, 10%, 8.8%, and 12% of the total number of cells in the islet.
especially when these "hubs" seems to be all grouped together rather than dispersed as might be expected in a small world network. The small network-ness is questionable.
Thank you for bringing up a very interesting observation – that “hub” cells tend seem to be located towards the center of the islet. We do not have a mechanistic understanding of this observation because the locality of the hubs cells was not a focus of this paper but could be looked at in subsequent work. Indeed prior experimental measurements in Johnston et al[4] suggested that there was not a spatial dependence.
However, we acknowledge the reviewers point that it is strange the cells tend to be grouped together. However, the functional network (from which the small world-ness has been suggested by Stozer et al[1]) is not spatially constrained. Therefore even if the hub cells are physically grouped, they may be more dispersed in the functional network. Additionally, it is a well-known phenomenon in scale-free networks (or roughly scale-free networks) that high degree nodes (hub cells) tend to have be highly connected with other high degree nodes. This comes from the so called “friendship paradox”[15].
While we are happy to further improve our measurement of the small word-ness (beyond the methods we previously used), in order to make the manuscript more accessible and focus on the more important findings of the manuscript, we have de-emphasized the small world-ness results.
The sensitivity to the correlation threshold suggests a certain lack of robustness. Perhaps I am misunderstanding the number of cells labeled as "hub cells."
We completely agree with the reviewer that the sensitivity to the correlation threshold indicates a lack of robustness. This sensitivity would occur in any study which uses functional network analysis, which is one of the reasons that we chose to explicitly analyze the sensitivity. However, in this manuscript, we are not attempting to show that the islet functional network is scale free or small world. These results have been demonstrated in previous papers[1], [4] and our threshold was chosen to match these claims.
Rather, this manuscript is attempting to show that cell metabolic activity is more highly associated with the functional network than is the gap junction structure. We found that in both the Cha-Noma model and experimental measurements, results concerning metabolic activity, and structure were not sensitive to threshold, E.g. hub cells were more differentiated by kglyc (glucokinase rate) than gcoup (gap junction conductance) for all thresholds analyzed – See Figure 1 Supp. 1 and Figure 3 Supp 1.
The excitability of cells via metabolism, for example, must be communicated through some (often structural) mechanisms. I appreciate that this article is attempting to get at how that division breaks down and I think much of the calculation and simulation is useful, but the language expressing the certainty rather than the more appropriate equivocation is not completely appropriate.
We agree that metabolic coupling or communication of metabolites between cells must occur through some type of structural mechanism. Hopefully, our clarification from the reviewers first two questions as well as rewording in the manuscript, clarifies that we are not attempting to describe sharing of metabolism between cells but rather investigating similarities in metabolic rates. We clarify this more explicitly now – For example, see Discussion page 13 lines 395-397.
[Editors’ note: what follows is the authors’ response to the second round of review.]
The manuscript has been improved but there are some remaining issues that need to be addressed, as outlined below:
Reviewer #2 (Recommendations for the authors):
I much appreciate the effort that the authors put into addressing my main skepticism about the paper, namely the fact that one type of oscillations were analyzed for experimental data and another type for modelled data, by including an additional model for slow oscillations and additional experimental data.
Given the new results, below, I provide my additional comments and suggestions that should not be too difficult for authors to consider and take into account.
We thank the reviewer for their positive comments regarding the revisions we made to the manuscript. We appreciate the additional comments and suggestions and have addressed them following the comments below. Please note that the eLife pdf conversion may result in some line and page numbers changing in comparison to the uploaded word version of the manuscript.
110-197 and Figures 1-S3:
First, the difference between hubs and non-hubs in terms of kglyc is around 0.14 vs 0.13, whereas for gcoup it is around 0.75 vs 0.6. In relative terms, the former is an 8 % difference and the latter a 25 % difference. It is true that the former is more significant statistically, but this is due to the fact that the heterogeneity is larger in the distribution of gcoup for hubs! Therefore, to me, it seems that both kglyc and gcoup seem to define the hub cells to a comparable degree. This conclusion seems to also be more compatible with findings for slow oscillations where the relative difference between hubs and non-hubs for gcoup is again around 20-30 %.
I suggest that authors consider this under the corresponding sections of results and discussion, and perhaps to some extent also in Figure 8.
We appreciate the reviewers point with respect to the relative differences in gcoup and kglyc between hubs and non-hubs, and that indeed the greater variability in gcoup than kglyc could perhaps masks how each parameter may define hubs. We now explicitly state the values of kglyc, gKATP, and gcoup between hubs and non-hubs in the respective Results section for figure 1 (where the differences are 11% for kglyc and 15% for gcoup) and for figure 2. However, we do note that the gcoup differences are less robust, in that they do not hold for all Rth values used. We briefly mention this higher difference for gcoup, but has greater variability and is less robust in the Discussion. Given this and that experimental results do not support a difference in gcoup but do with kglyc, we do not feel it warranted to adjust the figure 8 to include a role of gcoup. See Results page 5 line 114-117, page 6 line 146-148 and Discussion page 11 lines 360-362.
Second, at a threshold for hub cells of a 60 % normalized degree (i.e., the criterion for the separation line between hubs and non-hubs), different percentages of β cells qualify as hubs for different simulated islets, as evident in Figure 1b. More specifically, the cumulative number of cells qualifying as hubs for islets 3 and 5 is much more than the cumulative number of hubs in simulated islets 1 and 2. This differs from the criterion used in other important recent studies, i.e., in Johnston et al. Cell Metab 2016, hubs typically represent 1-10 % of islet cells, in Lei et al. Islets 2018, hubs represent 10 % of cells, and in Stozer et al. Am J Physiol 2021 and in Sterk et al. Biophys J 2023, hubs represent 1/6th of islet cells. I think it would enhance the comparison between the present and previous research if the authors provided the % of hub cells per islet.
We thank the reviewer for raising these interesting and important points. By using the same criterion as in Johnston et al. for defining hub cells, we do indeed observe the % of hub cells to vary between 5-12% for fast oscillating simulated islets (Figure 1b); to vary between 3-7% for slow oscillating simulated islets (figure 2b); and to vary between 3%-40% for experimentally measured islets (figure 3d). We now include these values in the Results section to indicate the range of hub cells observed under each condition using this criterion. We do note that in some cases (mainly experimental cases) this range is greater than the previously noted proportions (1-10% or 16%) and this is a good motivation to test whether our findings hold for an alternative definition for hub cells in the next point. See Results page 5 line 108, page 6 line 145 and page 7 line 175.
Additionally, setting the line of separation between hubs and non-hubs at the given threshold (not changing the threshold and thus the distribution of normalized degree) individually for each islet to achieve a fixed percentage of hub cells per islet close to 10 % of the most connected cells, would be a valuable addition to supplemental figure 1 and could perhaps help detect a larger (or smaller) difference between this more extreme group of cells and the majority of non-hub cells, in terms of kglyc, gkatp, and gcoup. Performing this additional analysis at least for the threshold value used in the main Figure 1 could add much value to the manuscript and make the findings even more directly comparable with the aforementioned studies. The same suggestion could also be taken into account for the analyses of modelled slow oscillations and for the experimental analyses of metabolic activity and coupling, where 10 % of the most connected cells could be considered/classified as hubs as well. Such additional analysis should be easily feasible since it considers a fraction of cells that have already been used in the analyses, but their degree of "hubness" is probably more.
We agree another valid way to look at the most connected hub-like cells would be to examine the top 10% most connected cells, especially give that in some cases the range of hubs cells is greater than previously noted proportions. We have performed such analysis for simulated islets presented in figure 1 and figure 2, and compared kglyc, gKATP and gcoup parameters between hubs and non-hubs. We also performed such analysis for experimentally measured islets presented in figure 3 and compared NAD(P)H and gap junction permeability. In each case when comparing the top 10% connected cells with remaining 90% of cells we generally observed similar findings to those presented in figures 1-3. In figure 1 (coupled Cha-Noma) kglyc is significantly different, gcoup is slightly different and gKATP is similar between hubs and non-hubs. In figure 2 (coupled IOM) kglyc is similar, gcoup is significantly different and gKATP is similar between hubs and non-hubs. In figure 3 (experimental) NAD(P)H (metabolic activity) is significantly different between hubs and non-hubs, whereas recovery rate (gap junction coupling) is similar between hubs and non-hubs. We now incorporate these results in new supplemental figures supporting figure 1, figure 2 or figure 3. See Results page 5-6 lines 126-129, page 6 lines 150-151, page 7 lines 186-188 and 202-204; and Figure 1 supplemental 2, Figure 2 supplemental 2, Figure 3 supplemental 2.
Given that the duty cycle (percentage of active time) so strongly correlates with the role of hub cells, it is somewhat surprising that the authors do not mention in the discussion that this same finding has recently been obtained for both mouse and human islets (Šterk et al. Biophys J 2023, Stožer et al. Am J Physiol 2021, Gosak et al. Diabetes 2022).
We apologize for omitting citations to important literature that also observes a relationship between duty cycle and node degree. Our intention was for this relationship to explain how the parameters influence hubs in each model. But this literature is important to cite to provide context that it is a well observed relationship, and we now do so. See Discussion page 11 line 356-357.
Third, I do value the effort of authors in replying to my comment regarding the exceedingly high correlation between simulated traces compared with experimental traces. However, besides noise, one very important aspect is the choice of model and the parameters which determine the observed lags between different cells or the wave speed. For instance, Cappon and Pedersen in their Chaos article built on the model used in Benninger Biophys J 2008 to produce lags between cells that are very realistic compared with experimentally observed values, whereas lags in the present study are just a fraction of the duration of a burst (fast Ca oscillation).
At present, I can only speculate about that, but the rather low conditional probability that two cells sharing gap-junctional conditions also show a functional connection could also be a consequence of the exceedingly short time lags between signals in the model employed in this study. More specifically, in the model by Cappon and Pedersen, there are waves travelling across islets and direct neighbors have shorter phase-lags between their signals compared with more distant cells and therefore on average more similar signals, similar to experimental recordings in isolated islets and tissue slices (the velocity being around 100 um/s, the time lag between direct neighbors is around 0.1 s and the time lag between most distant cells is around 1-2 s). This means that the values of R will be higher for direct neighbors and at any given threshold they will be functionally connected with a higher probability than with more distant cells where lags can be an order of magnitude more.
Therefore, I would suggest that in the future (not in this study), authors also repeat some of the analyses with modeled traces that show larger lags. This would probably also enable them to explore the relationship between different modelled/analyzed parameters and the role of pacemakers, i.e., wave initiators and other populations of cells. Perhaps, if authors consider the above suggestion as useful, they could include it in the discussion as a possible drawback of the present model and as a suggestion for future studies.
We thank the reviewer for their careful explanation of this important point. Indeed, it will be important in future work to compare simulated parameters between wave initiators or other populations of cells. We do note we previously analyzed properties of wave initiators in simulated islets e.g. Dwulet et al. PLOS Comp. Bio. (2021), following the same coupled Cha-Noma model. However, that analysis lacked a comparison with the functional network and thus future work should indeed examine this connection more deeply upon varying the simulated islet, such as to extend the time lags between cells. As such we mention this in the Discussion as a goal for future work. See Discussion page 14 lines 486-492.
Reviewer #5 (Recommendations for the authors):
The authors have thoroughly responded to my (and others') reviews. I think their modifications are reasonable and have tempered the language appropriately. The results are somewhat interesting and the techniques employed to measure and analyze the islets including adding an additional model system are extensive. They have addressed my points.
We thank the reviewer for their positive comments regarding the revisions we made to the manuscript. We appreciate the additional point they make which we have addressed below.
One point: We discussed (Reviewer #5 question/response 5b) the potential outlier at 2.35 of Figure 1f (gkatp data – Β Cell Hubs). That data point no longer exists on the graph (that I can see?). Since they argue convincingly that point should in fact not be treated as an outlier, I expect it is an oversight of it not being there. This should be fixed (or explained) prior to moving forward with publication.
The thank the reviewers for making this point. The data point in question is slightly above 2.35 and therefore as we adjusted the y-axis scale it was mistakenly placed out of the range. We have now corrected this oversight by adjusting the y-axis scale. See Figure 1f.