Adaptation of pancreatic islet cyto-architecture during development

Human pancreatic tissues were obtained from the University of Chicago Human Tissue Resource Center with an exemption from the Institutional Review Board. The two-dimensional sections were stained for insulin, glucagon, somatostatin and nuclei. Cells were labeled based on the hormone concentration surrounding each nucleus determining the cellular composition of each islet.

Graphs consist of vertices and edges. The vertices represent endocrine cells found in each individual islet (a representative islet is shown in figure 1(a)). To define edges between vertices, it is necessary to use the measured spatial separation between two cells to decide if the cells can interact, either by surface interactions or by intercellular signaling. If they can, based on a given neighborhood radius, an edge is added to the graph connecting the vertices corresponding to the two cells. The graphs of interest are (1) the β-cell cluster (ββ) graph (figure 1(b)) consisting of vertices representing β cells and edges between neighboring β cells, (2) the αδ graph (figure 1(c)) where vertices represent α and δ cells and edges are between nearby pairs of αα, αδ, and δδ cells, and (3) the αδ -β cell cluster interaction (αδ-β) graph (figure 1(d)) where vertices represent α, β and δ cells and edges are between αβ and βδ cell pairs.

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Knowing the range of intercellular distances that is sufficient and necessary for two cells to interact is critical for an accurate graph representation of surface interactions. Since this range is unknown in pancreatic islets, we determine the critical neighborhood radius based on the data. The pair correlation function, g(r), normalizes the number of cell pairs that are a given distance apart by the density of cells in the islet. For a given r₀, if g(r₀) is greater (less) than 1 then there are more (less) cell pairs within an intercell distance of r₀ than would be expected if the cells were randomly distributed. Because the graphs here do not represent all cell interactions but rather inter- and/or intra- cell-type interactions, we use a slightly modified version of g(r), given by

$$\begin{eqnarray*}&&g(r)dr=\displaystyle \frac{1}{{N}_{Y}}\displaystyle \frac{Area}{{N}_{X}}\displaystyle \sum _{i\in Y}\displaystyle \sum _{j\in X}\displaystyle \frac{\delta (r-{r}_{ij})}{2\pi r}\end{eqnarray*}$$

for two sets, X and Y, where $\delta (r-{r}_{ij})=\left\{\begin{array}{cc}1 & \mathrm{if}\;r={r}_{ij}\\ 0 & \mathrm{otherwise}\end{array}\right.,$ r_ij is the distance between vertices i and j, and N_X and N_Y are the number of vertices in sets X and Y, respectively, was used. For the ββ graph which captures interactions between β cells, X and Y both represent the β-cell vertices and N_X and N_Y are the number of β cells. Since the αδ graph captures interactions between α and δ cells, X consists of the α- and δ-cell vertices and Y is the same. N_X is the number of α and δ cells and N_Y is the same. The αδ-β graph represents interactions between the α and δ cells and β-cell clusters. Here X is the set of α- and δ-cell vertices, Y is the set of β-cell vertices, N_X is the number of α and δ cells and N_Y is the number of β cells.

The degree of a vertex (figure 1(e)) in a graph is the number of edges incident on the vertex. A component (figure 1(f)) is a subgraph of the graph such that each vertex shares an edge with at least one other vertex in the subgraph and this subgraph cannot be decomposed into further subgraphs. Components can be either singular (S), i.e. containing only one vertex, or nonsingular (NS). We used these two measures to quantify islet architecture as viewed through the graphs we defined above. We used the degree averaged over all vertices in a given islet to calculate the mean degree. Component results are shown with respect to the mean number of components (NS + S and NS only) per islet, and the mean number of vertices per component (NS + S and NS only). Computational code was written in C++ using the Boost Graph Library [25].

Since the calculated measures are discrete values the Mann-Whitney test, a nonparametric test for the significance of the difference between the distributions of two independent samples, with a Bonferroni corrections were used to test for significance. The standard deviations of each data point (given in tables S1–S9) are large due to the fact that these values are discrete. The degree, number of components per islet and number of cells per component counts are given in tables S1–9.

ββ graph measures change across developmental stages. Here, we model how the observed architectural alterations occur by taking the graphs corresponding to islets at an initial time and apply parameterized processes to these graphs to determine which process produces the architecture observed in the next developmental stage. The set of processes applied to the initial graphs should encompass (1) any rearrangements known to occur in islets, such as translocation [26–28], replication [29], and death [29], and (2) account for the increase in β cells per islet observed from developmental stage to developmental stage (figure 2(a)).

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The rearrangement processes were discussed in detail in [24]. In short, these processes assume the deletion of a vertex or addition of a nearby vertex is dependent on the vertex's degree (an alternate criterion, the size of a vertex's associated component, was also tried with comparable results; results not shown). It is natural to assume these processes can take place in islets since connectivity and cluster size play a role in optimal insulin production. There is, at this time, no model system where such rearrangements can be observed in vivo. These simulation processes add and delete one vertex per graph during each iteration keeping the number of vertices constant per graph. We consider a number of possible models for the determinants of these rearrangements. In each model, the vertices that are chosen for either deletion or to have a vertex added nearby are chosen by a Monte Carlo approach based on their degree and a relative likelihood function. The first relative likelihood function is given by

$$\begin{eqnarray*}&&R{L}_{-}=\mathrm{0.5\; -\; 0.5}\;\mathrm{tanh}(x-rl{p}_{*})\;\end{eqnarray*}$$

where x is the degree of the vertex and rlp_* is the given add or delete parameter for the process (see figure 2(b)). For this function higher-degreed vertices are less likely to be picked. The second relative likelihood function, given by

$$\begin{eqnarray*}&&R{L}_{+}=\mathrm{0.5\; +\; 0.5}\;\mathrm{tanh}(x-rl{p}_{*}),\end{eqnarray*}$$

shows that higher-degreed vertices are more likely to be picked (see figure 2(c)). The relative likelihood parameter shifts the range of what is considered a higher-degreed vertex as shown in figures 2(b) and (c). There are two processes, addition and deletion, and two relative likelihood functions, RL₋ and RL₊, resulting in four combinations of (Add, Delete), namely (RL₊, RL₊), (RL₊, RL₋), (RL₋, RL₊) and (RL₋, RL₋) which will be referred to as PP (plus,plus), PM (plus,minus), MP (minus,plus), and MM (minus,minus), respectively. Thus, the models we consider are defined by these relative likelihood functions and the parameters implicit in these functions.

Thus far we have addressed the rearrangement processes that leave the number of vertices unchanged. However, there is an increase in the number of β cells per islet observed between developmental stages, for which the stochastic process must account. The number of vertices to be added per iteration were calculated so that the resulting mean number of cells per islet matched that of the next developmental stage. For example, there are 199 197 total cells (38 281 total islets) and 176 463 total cells (29 174 total islets) in the gestational and 1–35 weeks datasets (tables 1–3), respectively. During the transition from gestation to 1–35 weeks, the number of cells per islet must increase from 5.20 to 6.05, resulting in the addition of 32 403 cells over a total of 500 iterations. Therefore there are 65 cells added per iteration. The placement of each cell in a chosen islet follows the given relative likelihood function.

Choosing which islet graphs will have additional vertices is not straightforward. We reasoned that the observed differences in the islet size distributions between developmental stages (figures 3(a)–(c)) should determine the islets gaining cells. For each vertex added, the islet size distribution (Dist(s), where s is the number of vertices per islet) is calculated. The bin-size for the distribution is 1 vertex. Note that figure 3 is shown with respect to log (base 2) of the number of vertices per islet. The difference between this distribution and that of the next developmental stage is given by $Diff(s)=Dis{t}_{{t}_{0}}(s)-Dis{t}_{{t}_{1}}(s)$ (figures 3(d)–(f)). This function gives a positive value for excess islets of a given islet size found in the simulated graphs. If an islet of a given size (s_m) is chosen, Dist(s_m) will decrease and Dist(s_m+1) will increase when recalculated. To reduce islet sizes that have excess and to avoid adding to islet sizes that have excess the relative likelihood function $RL(s)=Diff(s)-Diff(s+1)$ is used for choosing which islet receives an additional vertex. Notice this relative likelihood function is different from those discussed for vertex placement and is updated after each vertex addition. Figures 3(g)–(i) illustrates the size of islets chosen for vertex additions for each transition. The size of the islets chosen correlates with the observed distribution differences for each transition.

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In summary, the simulation process consists of 500 iterations, with each iteration consisting of steps where: (1) a vertex is removed from each islet chosen based on the given relative likelihood deletion function and parameter, (2) a vertex is added near a chosen vertex based on the given relative likelihood addition function and parameter, and (3) a determined number of vertices are added to the graph where the islet graph is chosen based on the difference in its islet size distribution and that of the next developmental stage and the placement of the vertex in the chosen islet is based on the given relative likelihood addition function. There are 500 simulations run for each combination of (Add, Delete) relative likelihood functions with relative likelihood addition (rlp_a) parameter values ranging from 0 to 7 and relative likelihood death (rlp_d) parameter values ranging from 0 to 7, resulting in a total of 128 000 simulations. As a test of convergence, the change in mean and standard deviation for each additional simulation was calculated.

This dataset is comprised of 139 subjects ranging in age from gestation to 28 months and older. There are ∼150 000 islets consisting of over 2,000,000 cells. The number of cells, average number of cells per islet and islet cell fraction are shown in table 1 for large and small islets. Over time the average number of β cells increases whereas the average number of δ cells decreases per islet.

The pair correlation function was calculated for the three graphs described in Methods. In addition, it was calculated for pairs of the same cell type to see how cells are distributed with respect to the same cell type (supplementary figures S1(a)–(c)), for cell pairs of differing cell types to see how cells are distributed with respect to other cell types (supplementary figures S1(d)–(f)), and for all cell types together (supplementary figure S1(g)).

The pair correlation for the ββ graph (figure 4(a)) is greater than one (more intercell distance pairs than expected if the vertices were randomly distributed) between 7 and 27 microns for the gestation and 7 and 28 microns for the 1–35 weeks groups. There is a decrease in range observed in the 12–24 months and 28 months+ groups which had a range of 7–22 and 7–24 microns, respectively (figure 4(a)). However, all groups show a peak around 10 microns. Similar results are seen between same cell type pairs, such as αα and δδ (supplementary figures S1(a)–(c)). When comparing the pair correlation of β cells with respect to either α or δ cells, there is no peak which means that β cells are randomly distributed with respect to α and δ cells (supplementary figures S1(d) and (f)). However, when the pair correlation of α cells with respect to δ cells is calculated, there is a range of values greater than 1, less pronounced than that observed in the ββ graph (supplementary figure S1(e)).

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The pair correlation for the αδ graph shows a non-random range of values with a peak around 10 microns (figure 4(b)), which is expected since it contains the cell pairs αα, αδ, and δδ, each of which individually showed correlation values greater than one and a peak around 10 microns. In contrast, the αδ-β graph does not have a peak (figure 4(c)) which is consistent with the correlation values seen for the cell pairs αβ and αδ.

Since most pair correlation functions calculated here had a peak around 10 microns, this neighborhood radius was used to create all graphs for comparing measure results. The measure values themselves are strongly dependent on the neighborhood radius used (supplementary figures S2–S6). However, the statistical significance of the measure differences between developmental stages is relatively constant for neighborhood radii around 10 microns (supplementary figures S2–S6). Because of this, the focus here is not on the measure values per se but the statistically significant differences observed between developmental stages.

3.3.1. Mean degree

For the ββ graphs there is an increase in mean degree observed between 1–35 weeks and 12–24 months (figure 5(a)). The mean degree transitions between gestation and 1–35 weeks and between 12–24 months and 28 months+ remain rather constant, especially in the large islets between gestation and 1–35 weeks and the small islets between 12–24 months and 28 months+. The αδ graphs show mean degree differences between all developmental stages for all islet sizes (figure 6(a)). However, there is a pronounced decrease in mean degree between 12–24 months and 28 months+. For the αδ-β graphs the mean degree decreases for all islet sizes until 12–24 months (figure 6(d)). This is followed by a further decrease in mean degree in small islets and an increase in mean degree for large islets.

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3.3.2. Number of components

3.3.3. Number of vertices per component

The first simulated transition (T1) is between gestation and 1–35 weeks. The simulation process was as described in Methods with the initial and next developmental stage datasets given by the gestation and 1–35 weeks datasets, respectively. For each (rlp_a, rlp_d) combination (denoted by points on the difference plots in figure 7), the simulations were run and the measures of the resulting graphs were calculated. The resulting measures were subtracted from the measures observed in the next developmental stage, in this case 1–35 weeks, so that measure equilibrium can be easily detected (when difference values are zero) as shown in figure 7. The mean degree difference is shown in figure 7. The number of components per islet and the number of vertices per component differences can be found in supplementary figure S7. For PM (figure 7(b)) and MM (figure 7(d)), no measure equilibria are found. For MP (figure 7(c)), all rlp_a values produce a measure equilibrium when rlp_d was approximately 1.65 (intercept values range from 1.63–1.68). For PP (figure 7(a)), all rlp_a produce a measure equilibrium but the rlp_d value have a larger range, from 1.7 to 2.65.

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The second simulated transition (T2) is from 1–35 weeks to 12–24 months. The PM (figure 7(f)) and MM (figure 7(h)) processes produced no measure equilibria. For MP (figure 7(g)), all rlp_a values produced a measure equilibrium when rlp_d was approximately 1.08 (values ranged from 1.01 to 1.12)- a noticeable shift from the first transition's rlp_d values. The PP processes (figure 7(e)) produced measure equilibria for each rlp_a with rlp_d values ranging from 1.22 to 2.05. Similar results for the number of components per islet and number of vertices per component differences can be found in supplementary figure S8.

The third simulated transition (T3) is from 12–24 months to 28 months+. Again, the PM (figure 7(j)) and MM (figure 7(l)) processes produced no measure equilibria. For MP (figure 7(k)), all rlp_a values produced a measure equilibrium when rlp_d was approximately 1.61 (values ranged from 1.56 to 1.63), similar to what was observed in the first transition. For PP (figure 7(i)), the rlp_d values that produced measure equilibria ranged from 1.7 to 2.67, also similar to what was observed in the first transition. Similar results for the number of components per islet and the number of vertices per component differences can be found in supplementary figure S9.

Bounds on the change in mean and standard deviation for each (rlp_a, rlp_d) combination can be found in supplementary figures S10–12.

The measures calculated for the αδ and αδ-β graphs (figure 6) quantified key transitions in cyto-architecture over developmental stages. For instance, the mean degree of the αδ graphs decreased after 12–24 months probably due to the decrease in δ cells per islet. However, the αδ-β graph degree increased slightly after this developmental stage. Thus, surprisingly, even with the decrease in δ cells, the connectivity between β cells and the α and δ cells was maintained and actually increased slightly after 12–24 months.

For large islets, the mean number of nonsingular components per islet slightly increased (2.86 to 2.88) for the αδ graphs after 12–24 months, whereas for the αδ-β graphs the values increased from 2.10 to 2.70 after this developmental stage. The number of vertices per components decreased in both sets of graphs after 12–24 months, but there was a greater decrease observed in the αδ graphs (2.76 to 2.59) as compared to the αδ-β graphs (2.38–2.34). Therefore even with the decrease in number of components per islet and number of vertices per component found in the αδ graphs, only small differences in the αδ-β graphs are observed after 12–24 months.

The connectivity between β cells increases between 1–35 weeks and 12–24 months (figure 5). This is corroborated quantitatively by the increase in mean degree and the number of vertices per component (S + NS and NS only) for all islet sizes observed between the two developmental stages. The number of components (S + NS and NS only) per islet also increases over time.

4.3.1. Which add/delete process?

4.3.2. Transitional shift

There is a significant shift in the equilibrium-producing rlp_d value from 1.65 for T1 (figure 7(c)) to 1.08 for T2 (figure 7(g)) and then back to 1.61 for T3 (figure 7(k)) in the MP case. There is also a collective shift in the range of rlp_d values producing measure equilibrium for PP processes for each transition. In figure 8, the associated relative likelihood functions for the addition process (rlp_a = 3 is shown, however all rlp_a values produced an equilibrium) and deletion processes for the MP case are shown for each transition. The T1 and T3 deletion parameter values are similar, which is interesting considering that there is little difference in the calculated mean degree and number of vertices per component between gestation and 1–35 weeks and between 12–24 months and 28 months+. Based on this, we suggest that the T1 and T3 rlp_d values maintain a given initial architecture. Between 1–35 weeks and 12–24 months the mean degree and number of vertices per component increase. From the shift observed from T1 to T2, it appears this change in measures occurs from an increase in cells with lower connectivity, represented by lower-degree vertices, being chosen for deletion.

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Chronic hyperglycemia induces morphological changes in islet structure, including large islet β-cell mass loss [9], hypertrophy in β cells [32] and islet amyloid plaque formations [33, 34]. Recently, Brereton et al [35] demonstrated that morphological changes observed in the islet structure are reversible once normoglycemia is attained. During development, humans have changing energy needs and available nutrients. The transitional equilibrium shifts observed between developmental stages possibly demonstrates how islet structure adapts to fluctuations in glycemia.

The simulations shown here are solely based on interactions between β cells. However, β cells may not be the sole determinants of islet architecture. Increased α-cell mass has been reported and distorted islet architectures have been observed in patients with T1D [36] and T2D [5]. Also, an increase in the proportion of α-δ and δ-δ cell contacts was observed in T2D subjects [9]. These results illustrate the importance of the number and placement of α and δ cells to islet function. An interesting extension of this work would be to include α and δ cells to the simulated graphs and determine whether β cell placement is dependent on the location of α and/or δ cells. This would further the understanding of the role of α and δ cells in islet architecture.