Size distribution of mouse Langerhans islets

Abstract

Pancreatic beta-cells are clustered in islets of Langerhans, which are typically a few hundred micrometers in a variety of mammals. In this study, we propose a theoretical model for the growth of pancreatic islets and derive the islet size distribution, based on two recent observations: First, the neogenesis of new islets becomes negligible after some developmental stage. Second, islets grow via a random process, where any cell in an islet proliferates with the same rate regardless of the present size of the islet. Our model predicts either log-normal or Weibull distributions of the islet sizes, depending on whether cells in an islet proliferate coherently or independently. To confirm this, we also measure the islet size by selectively staining islets, which are exposed from exocrine tissues in mice after enzymatic treatment. Indeed revealed are skewed distributions with the peak size of approximately 100 cells, which fit well to the theoretically derived ones. Interestingly, most islets turned out to be bigger than the expected minimal size (approximately 10 or so cells) necessary for stable synchronization of beta-cells through electrical gap-junction coupling. The collaborative behavior among cells is known to facilitate synchronized insulin secretion and tends to saturate beyond the critical (saturation) size of approximately 100 cells. We further probe how the islets change as normal mice grow from young (6 weeks) to adult (5 months) stages. It is found that islets may not grow too large to maintain appropriate ratios between cells of different types. Our results implicate that growing of mouse islets may be regulated by several physical constraints such as the minimal size required for stable cell-to-cell coupling and the upper limit to keep the ratios between cell types. Within the lower and upper limits the observed size distributions of islets can be faithfully regenerated by assuming random and uncoordinated proliferation of each beta-cell at appropriate rates.

FIGURE 1

Coherent cell proliferation in an islet. At each replication time, all cells together replicate or not according to the proliferation rate λ. The growth process is illustrated until the second replications and the growth status of each islet is described by the proliferation stage k.

FIGURE 2

Independent cell proliferation in an islet. At each replication time, every cell replicates with the proliferation rate λ, independently of each other. The growth process is again illustrated until the second replications and the growth status of each islet is described by the total number n of cells in it.

FIGURE 3

Evolution of the islet size distribution in the case of independent cell proliferation. Starting from the initial configuration of a single cell, the islet evolves with the cell proliferation rate λ = 0.5. Open squares and circles represent the islet size distribution at time t = 5 and 6, respectively, obtained from MC simulations on 10⁶ samples. The data are fitted to the Weibull distribution, plotted with solid and dashed lines. The inset shows the shape parameter and the scale parameter versus the replication time t, where the solid line corresponds to the function a(1 + λ)^t with a = 1.14.

FIGURE 4

Images of Langerhans islets: (A) Langerhans islets in a low magnification; (B) medium, (C) small, and (D) big islets at the magnification four times higher than panel A. The islets are stained red and attached to the white acinar cells. The size of scale bars is 200 μm.

FIGURE 5

Schematic diagram of a prolate spheroid with the equatorial radius a and the polar radius b.

FIGURE 6

Islet size distribution obtained from eight mice of (A) 6-weeks old and (B) 5-months old. Dotted lines represent the log-normal distribution with A (μ, σ) = (1.89, 0.68) and (B) (μ, σ) = (2.10, 0.64). Solid lines represent the two-parameter Weibull distribution with (A) (γ, η) = (1.64, 8.89) and (B) (γ, η) = (1.70, 10.15).

FIGURE 7

Radius ratio κ ≡ a/b between the equatorial radius a and the polar radius b versus the islet size s ≡ n^1/3. The dotted line represents the linear regression equation given by κ = −0.013 s + 1.00.

FIGURE 8

Islet size distribution obtained from three different methods. The dotted line, plotting the result for ob/ + mice, from the stereological method (ST), corresponds to the Weibull distribution with γ = 0.91 and η = 3.36, reconstructed in accord with the number-weighted mean islet volume V_N and the volume-weighted mean islet volume V_V of the data in Bock et al. (30) (see Table 2). The solid and open circles plot the data for 5-month-old BALB/c mice, obtained from the mild enzymatic treatment (EN) and from the complete isolation of islets (IS), respectively. They are fitted to the solid line and the dashed line, which represent the Weibull distributions with (γ, η) = (1.70, 10.15) and (3.57, 16.60), respectively.