Modeling of the gap junction of pancreatic β-cells and the robustness of insulin secretion

Abstract

Pancreatic β-cells are interconnected by gap junctions, which allow small molecules to pass from cell to cell. In spite of the importance of the gap junctions in cellular communication, modeling studies have been limited by the complexity of the system. Here, we propose a mathematical gap junction model that properly takes into account biological functions, and apply this model to the study of the β-cell cluster. We consider both electrical and metabolic features of the system. Then, we find that when a fraction of the ATP-sensitive K⁺ channels are damaged, robust insulin secretion can only be achieved by gap junctions. Our finding is consistent with recent experiments conducted by Rocheleau et al. Our study also suggests that the free passage of potassium ions through gap junctions plays an important role in achieving metabolic synchronization between β-cells.

Synchronization of rhythms via receptors plays important roles in a variety of cellular processes. Cellular slime mold1, in which rich quantitative experimental data are available, has played a significant role in studies of rhythm synchronization via cAMP receptors. A synchronization scheme developed for the cellular slime mold2 has been generalized as a universal mathematical scheme3–5. Another important player in cellular synchronization is the gap junction. Most cells in animal tissues including pancreatic β-cells are in communication with their neighbors both electrically and metabolically via gap junctions, which allow the passage of small water-soluble ions and molecules smaller than ^∼ 1000 daltons. The phenomenon of synchronization via gap junctions in the pancreatic β-cells has been extensively investigated6,7. Nonetheless, the study of gap junctions is still in a challenging stage in large part because of the complex relationship between structure and function.

Pancreatic β-cells, located in the islets of Langerhans, secrete insulin in response to the level of glucose in the blood allowing the glucose level in the blood to be maintained within a small range around 5.6 mM. Insulin secretion is oscillatory with a period of ≤1 min (fast oscillatory mode), 2–7 min (slow oscillatory mode), or a combination of the two8–10.

It is generally believed that Ca²⁺ feedback to ATP production is responsible for the fast oscillatory mode of insulin secretion, whereas glycolysis is responsible for the slow oscillatory mode. The glucose-phosphorylating enzyme glucokinase act as a metabolic glucose sensor in liver and β-cells11. The electrical activity of β-cells has a characteristic behavior known as bursting. A burst consists of an active phase of spiking followed by a silent phase of hyperpolarization. During the silent phase, Ca²⁺ is cleared by Ca²⁺ ATPases.

The essential processes in insulin secretion are as follows (see Fig. 1): ATP-sensitive K⁺ channels in the plasma membrane are activated by ADP and inactivated by ATP; thus the ratio of these nucleotides determines the fraction of open K(ATP) channels. When the ATP/ADP ratio is elevated, there is a reduction in the number of open K(ATP) channels. This results in membrane depolarization, causing voltage-dependent Ca²⁺ channels to open. The resulting Ca²⁺ influx evokes insulin secretion. The difference among various mathematical models may originate in the extent to which common cellular processes beyond the essential processes are taken into account.

Figure 1.

Essential processes of the insulin secretion6. Glucose is taken into the β-cell by GLUT-2 transporters, and broken down during glycolysis. Glycolytic product pyruvate is taken into the mitochondria in order to produce ATP. The ATP-sensitive K⁺ channel regulates membrane potential, and Ca²⁺ flow into the cell, and the insulin release into the blood is prompted by the elevated cytosolic Ca²⁺ concentration.

Comparisons among experimental observations have been performed mostly in the context of mathematical models of single β-cells. However, experimental observations are conducted in β-cell clusters, where β-cells are connected with gap junctions. In this letter, we propose a gap junction model that clarifies the relationship between the synchronization phenomena and the cellular processes via gap junctions. From among many proposed single pancreatic β-cell models6,10,12–20, here we adopt one by Pedersen et al.6, and generalize their model from the single-cell level to a multicellular system. The model by Pedersen et al. (see Fig. 2) is not complete as a whole β-cell model, but its relative simplicity facilitates generalization to the multicellular system. We expect that the simplicity of the model can help our intuitive understanding of the numerical investigations. Naturally, this also gives some limitations to our study as discussed in Appendix A.

Figure 2.

A scheme for insulin secretion adopted by Pedersen et al.6 G_e is the extracellular glucose concentration; G_i is the intracellular glucose concentration; G6P is glucose 6-phosphate concentration; FBP is fructose 1-6-bisphosphate concentration; and SERCA means the SERCA pumps.

To model gap junctions, the electrical coupling between cells has often been handled by adding the following linear coupling term6, −g_c(v_i − v_j), where g_c is the coupling strength between neighbor cells i and j. Such a linear coupling approximation has recently been extended to Ca²⁺ and glycolytic metabolites7. In these works, the coupling strength of gap junctions has been treated as a free parameter while the relationship between the adopted coupling parameters and the structure and the function of gap junctions has been left as an unanswered question. Furthermore, the electrical coupling and the ion currents between β-cells must be studied in a consistent manner since ions such as K⁺ and Ca²⁺ can freely move through gap junctions. Here, we propose a gap junction model based on the Goldman, Hogkin, and Katz (GHK) approximation21 to answer such a fundamental question. Our proposed scheme can handle both the membrane ion currents and the gap currents in a consistent manner.

The structure of this paper is as follows. Following this brief introduction (Section I), we present our mathematical modeling of gap junctions in Section II, and describe our numerical investigations in Section III. Comparison of the prediction of our model to the recent experimental results of Rocheleau et al.22 is presented in Section IV. Limitations of the linear coupling scheme are discussed in Section V. Finally, we discuss our results in Section VI.

Modeling of gap junctions of β-cells

We generalize the single β-cell model proposed by Pedersen et al to a multicellular system as follows. Our equation for the membrane potential (v) of the i-th β-cell is given by

$$C_m\frac{d{v}_i}{dt}+I_{K(v)}^{(i)}+I_{\mathit{Ca}(v)}^{(i)}+I_{K(\mathit{Ca})}^{(i)}+I_{K(\mathit{ATP})}^{(i)}=I_G^{(i)},$$ (1)

where C_m is the membrane capacitance, $I_{K(v)}^{(i)}$ is the v-dependent K⁺ current, $I_{\mathit{Ca}(v)}^{(i)}$ is the v-dependent Ca²⁺ current, $I_{K(\mathit{Ca})}^{(i)}$ is the calcium-activated K⁺ current, $I_{K(\mathit{ATP})}^{(i)}$ is the ATP-sensitive K⁺ current, and $I_G^{(i)}$ is the gap current. Figure 3 shows the general structure of gap junctions. We also assume that the flow of ions through gap junctions is driven not only by the electric field but also by the concentration gradient. Thus, we obtain the Nernst-Planck equation for the flow of ions21

$$J_{G(A)}=-S_G{D}_A\left(\frac{d{c}_A}{dx}+\frac{Z_{A}F{c}_A}{\mathit{RT}}\frac{d\varphi }{dx}\right),$$ (2)

where ϕ is the electrical potential, c_A is the concentration of type A ions, F is Faraday’s constant, z is the valence of ions, D_A is the diffusion constant of type A ions, S_G is the cross-sectional area of the gap junction, and RT is the gas constant multiplied by the absolute temperature. Then, the gap current I_G is

$$I_G=\sum _A{z}_{A}F{J}_{G(A)}.$$

We also use $S_G=\pi r_C^2{N}_C$, where r_C ≅ 0.75 nm is the radius of a single channel and N_C is number of channels in a gap junction. It is known that N_C is a few to many thousands23. When the length of the gap junctions is d (see Fig. 3b), the potential difference in the gap junction can be approximated (Goldman, Hogkin, and Katz21) as

$$\frac{d\varphi }{dx}\cong \frac{\left({\varphi }^{(i)}-{\varphi }^{(j)}\right)}{d}=\frac{v_i-v_j}{d}.$$

Next, we obtain the following current equation of type A ions for the gap junctions

$$\begin{array}{l}{I}_{G(A)}^{i\leftarrow j}=\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}-\frac{D_A}{d}\frac{z_A^2{F}^2}{\mathit{RT}}{S}_G\left(v_i-v_j\right)\frac{c_A^{(i)}-c_A^{(j)}\text{exp}\hspace{0.17em}[-z_{A}F\left(v_i-v_j\right)[RT]}{1-\text{exp}\hspace{0.17em}[-z_{A}F\left(v_i-v_j\right)[\mathit{RT}]}\hfill \end{array}$$ (3)

$$=-\frac{u_A}{d}{z}_{A}F{S}_G\left(v_i-v_j\right)\frac{c_A^{(i)}-c_A^{(j)}\hspace{0.17em}\text{exp}\hspace{0.17em}[-z_{A}F\left(v_i-v_j\right)[\mathit{RT}]}{1-\text{exp}\hspace{0.17em}[-z_{A}F\left(v_i-v_j\right)[\mathit{RT}]},$$ (4)

where the Einstein relation between the diffusion coeffcient D_A and the ionic mobility u_A, namely D_A = u_ART/z_AF is used. Thus, the gap current becomes

$$I_G^{(i)}=\sum _{j,A}{I}_{G(A)}^{i\leftarrow j},$$ (5)

where j is the nearest-neighbor cell of cell i.

Figure 3.

The general structure of gap junctions23. Within a gap junction, there are a few to many thousands channels of diameter 1.5 nm. The length of the cell gap is about 2∼4 nm. The thickness of the cell membrane is about 5 nm. Thus, the length of the gap junctions d is nearly 13 nm. The channels allow inorganic ions and molecules with a mass of less than 1 kDa to pass into the other cell.

Using the gap current, the equation of conservation for K⁺ ions within the i-th cell becomes

$$V_{\mathit{cyt}}\frac{d{K}^{(i)}}{dt}=\frac{1}{z_{K}F}[-I_{K(v)}^{(i)}-I_{K(\mathit{Ca})}^{(i)}-I_{K(\mathit{ATP})}^{(i)}+I_{K(in)}^{(i)}+I_{G(K)}^{(i)}],$$ (6)

where V_cyt(= 1150(μm)³) is the cytosolic volume of a single cell, $I_{K(in)}^{(i)}$ is the inward potassium current $I_{K(in)}^{(i)}$ (see Appendix A for its detailed discussion), and $I_{G(K)}^{(i)}$ is the gap current of K⁺ ions. Summing the length of the cell gap (2∼4 nm) and the thickness of the two neighboring cell membranes (∼5 nm), we obtain d=3 nm+5 nm * 2=13 nm as an approximation here (see Fig. 3(b)). Such an approximation is necessary because the intracellular potential difference must vanish when the coupled system is in equilibrium.

Similarly, the equation of conservation for the free cytosolic Ca²⁺ concentration is

$$\begin{array}{l}{V}_{\mathit{cyt}}\frac{dC{a}^{(i)}}{dt}=\hfill \\ \hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}\hspace{0.17em}{f}_{\mathit{cyt}}\left[-{\tilde{k}}_{\mathit{PMCA}}C{a}^{(i)}+J_{er}^{(i)}+\frac{1}{z_{\mathit{Ca}}F}\left(-I_{\mathit{Ca}(v)}^{(i)}+I_{G(\mathit{Ca})}^{(i)}\right)\right],\hfill \end{array}$$ (7)

and the equation for the free Ca²⁺ concentration in the endoplasmic reticulum is

$$V_{er}\frac{d{\mathit{Ca}}_{er}^{(i)}}{dt}=-f_{er}{J}_{er}^{(i)},$$ (8)

where

$$J_{er}^{(i)}={\tilde{p}}_{\mathit{leak}}\left({\mathit{Ca}}_{er}^{(i)}-{\mathit{Ca}}^{(i)}\right)-{\tilde{k}}_{\mathit{SERCA}}\hspace{0.17em}{\mathit{Ca}}^{(i)}$$

is the Ca²⁺ flux out of the endoplasmic reticulum, $I_{G(\mathit{Ca})}^{(i)}$ is the gap current of Ca²⁺, f_cyt is the fraction of free to total cytosolic Ca²⁺, and k_PMCA ≡ k_PMCAV_cyt, p˜_leak ≡ p_leakV_cyt, k˜_SERCA ≡ k_SERCAV_cyt. Where k_PMCA, k_SERCA, and p_leak are the PMCA pump rate, SERCA pump rate, and the Ca²⁺ leakage rate from ER, f_cyt and f_er are the fraction of free to total cytosolic and the fraction of free to total ER Ca²⁺, respectively, V_er is the volume of the ER compartment. Thus, we can handle gap junctions consistently.

Synchronization of beta-cells via gap junctions

To clarify the role of gap junctions in the cellular processes, we have conducted extensive numerical investigations. From many numerical experiments, first, we show the result for the two β-cell system in Fig. 4, where we adopt ionic mobilities in water at 298 K24, namely, u_K = 7.62×10⁻⁸ m²s⁻¹V⁻¹, u_Ca = 6.17 × 10⁻⁸ m²s⁻¹V⁻¹ as an approximation since specific values for the ionic mobilities in the β-cells are not currently available to our knowledge. In Fig. 4(a), both K⁺ ions and Ca²⁺ ions are allowed to pass through gap junctions. In this case, synchronization is achieved immediately although very different initial parameter values are adopted for the different cells. On the other hand, in Fig. 4(b), only Ca²⁺ ions are allowed to pass through gap junctions. It is clear that without free flow of K⁺ ions through the gap junction, synchronization cannot be achieved. The result is reasonable since the K⁺ ion concentration is much higher than the Ca²⁺ ion concentration within cells. Although we have also tested the case when only K⁺ ions are allowed to pass through gap junctions, we do not show it here because the obtained figure is nearly identical to Fig. 4(a). For our next example, we show the result for the 100 β-cell system in Fig. 5, where the 10×10 square lattice with the periodic boundary condition is used. Once again, we observed synchronous bursting. We do not provide any figure of channel number dependence because no qualitative change was observed when N_c is changed from 1 to 100.

Figure 4.

Synchronization is achieved immediately when both potassium ions and calcium ions are allowed to pass through gap junctions between two β-cells (a), but is not achieved when only calcium ions are allowed to pass through gap junctions (b). Here, ḡ_Ca =1000 pS, and ḡ_K(ATP) =40000 pS, and the extracellular glucose concentration is 7 mM. The solid line and dotted line indicate different β-cells.

Figure 5.

Synchronization of 10×10 β-cell square lattice with periodic boundary conditions. Here, ḡ_Ca =1000 pS, and ḡ_K(ATP) = 40000 pS.

Robustness of the insulin secretion by gap junctions

Recently, Rocheleau et al.22 conducted an interesting experiment by blocking the ATP-sensitive K⁺ channels and gap junctions. They used transgenic mice and showed that the robust control of the insulin secretion can be achieved by the gap junctions. Following their experiment, we conducted a numerical experiment in which G_e = 2 mM. To simulate their experiment, we reduced the conductance of the ATP-sensitive K⁺ channels by decreasing the value of ḡ_K(ATP) from the original value of 40000 pS to 10000 pS in a single β-cell. In this case, the insulin secretion is prompted as shown in Fig. 15. This corresponds to the case where the gap junction is disconnected (case in Fig. 6(c)). If the ATP-sensitive K⁺ channels are fully functional, no insulin secretion should be observed when G_e = 2 mM as shown in Fig. 13. However, when the gap junction is opened (case in Fig. 6(b)), insulin secretion ceases, as shown in Fig. 7(b). This robustness of the insulin secretion is consistent with the experiments of Rocheleau et al. To clarify the detailed mechanism of this robustness, we show the ATP-sensitive potassium current I_K(ATP) and the gap currents I_G(K) and I_K(Ca) in Fig. 8. When the ATP-sensitive potassium channels of both cells are not blocked, the gap currents are negligibly small. However, when one of two cells is blocked, I_G(K) and I_K(ATP) of the normal cell $\left(\equiv I_{K\left(\mathit{ATP}\right)}^{\text{normal}}\right)$ are significantly increased (see Figs. 8(b), 8(c)). Figure 8(d) shows that $I_{K\left(\mathit{ATP}\right)}^{\text{normal}}$ minus I_G(K) is nearly equal to the I_K(ATP) of the blocked cell $\left(\equiv I_{K\left(\mathit{ATP}\right)}^{\text{blocked}}\right)$ plus I_G(K). Besides, both values are roughly equal to I_K(ATP) in equilibrium state in Fig. 8(a). This means that a significant increase of I_K(ATP) of the normal cell is brought about mostly by the outward gap current I_G(K) from the blocked cell. This increase of I_G(K) brings the intracellular potassium density of the blocked cell back to the normal level, and the insulin secretion ceases. Namely, the intercellular collaboration of ATP-sensitive potassium channels via gap junctions is a key of this robustness. Here the positive direction of I_K(ATP) is outward from the cell.

Figure 6.

Two β-cells connected with gap junctions. (a) Wild type cells connected with gap junctions. (b) The K(ATP) channel in one of two β-cells is partially blocked. (c) The K(ATP) channel in one β-cell is blocked, and additionally, the gap junction is non-functional.

Figure 7.

Partial blocking of the ATP-sensitive K⁺ channels of the two β-cell system when G_e= 2 mM. (a) ḡ_K(ATP) =40000 pS (Cell 2), 40000 pS (Cell 1), (b) ḡ_K(ATP) =40000 pS (Cell 2), 10000 pS (Cell 1). The above figures correspond to the cases of Figs. 6(a) and 6(b), respectively. In the case of Fig. 6(c), the result becomes the simple sum of Fig. 14 and Fig. 15 since the gap junction is closed. This means that the insulin secretion cannot be stopped, even though the glucose level is low.

Figure 8.

ATP-sensitive K⁺ current and gap currents of the cases in Fig. 7, where G_e= 2 mM. (a) I_K(ATP) of the case of Fig. 7(a), (b) I_K(ATP) of the case of Fig. 7(b), (c) gap currents I_G(K) and I_G(Ca) of the cases of Figs. 7(a) and 7(b). When the ATP-sensitive potassium channels of both cells are fully functional, the gap currents are negligibly small. However, when one of them is blocked, the outward gap currents from the blocked cell and I_K(ATP) of the normal cell are significantly increased, (d) a comparison of $I_{K(\mathit{ATP})}^{\text{blocked}}+I_{G(K)}$ and $I_{K(\mathit{ATP})}^{\text{normal}}-I_{G(K)}$, where $I_{K(\mathit{ATP})}^{\text{blocked}}$ and $I_{K(\mathit{ATP})}^{\text{normal}}$ are ATP-sensitive potassium current of the blocked cell and that of the normal cell, respectively. Note that $I_{K(\mathit{ATP})}^{\text{blocked}}+I_{G(K)}$ and $I_{K(\mathit{ATP})}^{\text{normal}}-I_{G(K)}$ in Fig. 8(d) are roughly equal to I_K(ATP) in Fig. 8(a). An increase of I_G(K) brings the intracellular potassium density of the blocked cell back to the normal level, and ultimately the insulin secretion is ceased. The intercellular collaboration of ATP-sensitive potassium channels via gap junctions is a key of this robustness. Here the positive direction of I_K(ATP) is outward from the cell.

For the further confirmation of the above analysis, we have increased the extracellular glucose concentration G_e from 2 mM to 7 mM as shown in Fig. 9. Since we have already studied the case where both cells are normal as presented in Fig. 4(a), only the blocked case is shown in Fig. 9(e) except gap currents and I_K(ATP).When G_e= 7 mM, the insulin is constantly secreted, but the role of gap junction for the metabolic synchronization is still important when some cell is damaged.

Figure 9.

ATP-sensitive K⁺ current and gap currents, where G_e = 7 mM. Similar to Fig. 8, the blocked or damaged cell significantly increases the outward gap currents I_G(K) and I_G(Ca) from the cell. (a) I_K(ATP) of the case of Fig. 4(a), (b) I_K(ATP) of the case of Fig. 9(e), (c) gap currents I_G(K) and I_G(Ca) of the cases of Figs. 4(a) and 9(e). (d) a comparison of $I_{K(\mathit{ATP})}^{\text{blocked}}+I_{G(K)}$ and $I_{K(\mathit{ATP})}^{\text{normal}}-I_{G(K)}$. They are nearly identical and in the same range of value of I_K(ATP) in equilibrium state (2000 fA∼10000 fA) in Fig. 9(a).

To investigate the role of gap junctions further, we studies the activity of the voltagegated Ca²⁺ channel by changing ḡ_Ca values. The original ḡ_Ca value is 1000 pS, and Fig. 14 shows the model calculations. We then changed the value of ḡ_Ca from 1000 pS to 900 pS and 1100 pS. Their corresponding results are shown in Figs. 16 and 17, respectively. Two coupled β-cells are successfully synchronized via the gap junction, and a very similar result to that of the ḡ_Ca =1000 pS case is produced (see Fig. 11(a) or Fig. 14) although original individual patterns are very different each other as shown in Figs. 16 and 17. This kind of character has been observed consistently in other tested cases including the three coupled β-cell system.

Figure 11.

Results of the linear coupling scheme. Here, two β-cells with the same and different ḡ_Ca values when G_e = 7 mM. (a) ḡ_Ca =1000 pS (Cell 2), 1000 pS (Cell 1), and (b) ḡ_Ca =1100 pS (Cell 2), 900 pS (Cell 1).

The linear coupling scheme and its limitations

Because of its mathematical simplicity, the linear coupling scheme is often adopted for handling gap junctions without detailed biological discussions. Here we derive it from Eq. (4). By using the condition, $c_A^{(i)}=c_A^{(j)}$ (i, j are neighbor cell numbers), we can obtain

$$\begin{array}{ll}{I}_G^{(i)}\hfill & =-\sum _A\frac{u_A}{d}{z}_{A}F{c}_A^{(i)}{S}_G\sum _j\left(v_i-v_j\right)\hfill \\ \hfill & =-g_c^{(i)}\sum _j\left(v_i-v_j\right),\hfill \end{array}$$ (9)

where $g_c^{(i)}=F{S}_G[d{\sum }_A{u}_A{z}_Z{c}_A^{(i)}$. Here, we assume that only Ca²⁺ and K⁺ ions go through gap junctions. Then,

$$\begin{array}{ll}{I}_G^{(i)}\hfill & =I_{G(K)}^{(i)}+I_{G(\mathit{Ca})}^{(i)}\hfill \\ \hfill & =-g_{c(K)}^{(i)}\sum _j\left(v_i-v_j\right)-g_{c\left(\mathit{Ca}\right)}^{(i)}\sum _j\left(v_i-v_j\right).\hfill \end{array}$$ (10)

Since c_K/c_Ca ≫ 1 within cells, it is clear that K⁺ ions play a much more important role than Ca²⁺ ions in the coupling of β-cells by gap junctions. If we use c_K = 95 mM, c_Ca = 5.5×10⁻² μM, and T =310 K, we obtain g_c ≅ g_c(K) = 95N_cpS and g_c(Ca) = 1.1×10⁻⁵N_cpS.

To clarify the limitation of the linear coupling scheme, we have conducted the following numerical experiments. The results are shown in Figs. 11 and 12, in which the number of channels is N_c = 10 as in previous investigations. By comparing Figs. 11(a) and 4(a), Figs. 11(b) and 10, and Figs. 12 and 7, it becomes clear that the linear coupling scheme works very well except when the ḡ_K(ATP) values of individual β-cells are different as shown in Fig. 12(b). The variation of ḡ_Ca values does not provide any problem as an approximation. This difference originates in how the intra-cellular potassium concentration is a ected by the parameter value variation. As shown in Fig. 15, only ḡ_K(ATP) variation has a ected the intracellular potassium concentration significantly. As a consequence, the precondition of $c_K^{(i)}=c_K^{(j)}$ for this scheme is lost.

Figure 12.

Results of the linear coupling scheme. Here, two β-cells with the same and different ḡ_K(ATP) values when G_e = 2 mM. (a) ḡ_K(ATP) =40000 pS (Cell 2), 40000 pS (Cell 1), (b) ḡ_K(ATP) =40000 pS (Cell 2), 10000 pS (Cell 1).

Figure 10.

Two β-cells with different ḡ_Ca values, where ḡ_Ca =1100 pS (Cell 2), 900 pS (Cell 1), and G_e = 7 mM.

Discussion

In this study, we have proposed a mathematical model for the gap junction based on the biological functions and the structure of gap junctions, and applied it to the β-cell cluster in the islets of Langerhans. Our model demonstrates that free flow of potassium ions through gap junctions plays an essential role in achieving synchronous bursting. When metabolic synchronization between β-cells is achieved, the net current through gap junctions becomes negligible. However, the ionic flow through gap junctions becomes essential when some of the ATP-sensitive K⁺ channels in the β-cells are damaged. The proved robustness of the insulin secretion is consistent with the experimental observations of Rocheleau et al.22. We offer the following interpretation of our findings. Intracellular calcium ion concentration is relatively insensitive to the variation of the conductance ḡ_Ca so long as the function of the ATP-sensitive K⁺ channels is normal. However, when a fraction of the ATP-sensitive K⁺ channels is damaged, intracellular potassium ion concentration is increased (Fig. 15). This increase also brings about an increase in calcium ion concentration. As a consequence, insulin secretion is stimulated even when the glucose level is low. However, when the damaged cell is connected with the normal cells with gap junctions, mutual membrane potential synchronization is achieved immediately due to the free flow of potassium ions through gap junctions. Thus, the intracellular potassium ion concentration of the damaged cell returns to the normal level, and calcium ion concentration is reduced to the normal level a little later. This stops insulin secretion, and the robust insulin secretion can be achieved (Figs. 7 and 8).

We have also derived a linear coupling scheme from the GHK approximation, assuming that the ion concentrations between neighbor cells are nearly equal. The performance of the linear coupling scheme is excellent, so long as the ATP-sensitive K⁺ channels are normal between neighbor cells because the intracellular potassium density is sensitive to ATP-sensitive K⁺ channels (Fig. 15). One cannot expect nearly equal ion concentrations between neighboring cells, as that would break the basis of the linear coupling scheme.

As discussed in Appendix A, we have introduced I_K(in) =−I_Ca(v) term into the equation of conservation for potassium ions as an approximation. This mathematical (as opposed to biological) approximation is necessary to hold intracellular potassium concentration steady, in the context of the single β-cell model by Pedersen et al. and their original parameter values. This problem does not appear explicitly so far as the constant Nernt potential is adopted in any scheme. This inconsistency may have occurred due to the lack of pumps that are responsible for the inward potassium current in the model equations. Thus, in future models, we would like to introduce such pumps along with the sodium current in order to avoid such a drastic approximation. Until then, we should take our findings as predictions and they must be confirmed by the direct observation of gap currents.

APPENDIX A. Modification of the single b-cell model by Pedersen et al.

In this letter, we adopt the single β-cell model by Pedersen et al., and their fitting parameters6 unless stated otherwise (only typical parameters are shown in Table 1). However, for handling the ion flow through gap junctions, we must modify their model, although their original parameter values are not changed.

Table 1.

Typical variable values. See ref. 6 for other variables

Variables
Cm = 5300 fF	ḡK =2700 pS
ḡCa =1000 pS	ḡK(Ca) =400 pS
ḡK(ATP)=40000 pS	vK = −75 mV
vCa = 25 mV	Vcyt = 1150 µm3
Vcyt/Ver = 31	kPMCA = 0.18 ms−1
kSERCA = 0.4 ms−1	pleak = 0.0002 ms−1
fcyt = 0.01	fer = 0.01

The equation for the membrane potential (v) of a β-cell is and

$$C_m\frac{dv}{dt}+I_{K(v)}+I_{K(\mathit{Ca})}+I_{K(\mathit{ATP})}+I_{\mathit{Ca}(v)}=0,$$ (A1)

the equation for the free cytosolic Ca²⁺ concentration is

$$V_{\mathit{cyt}}\frac{d\mathit{Ca}}{dt}=f_{\mathit{cyt}}\left[-{\tilde{k}}_{\mathit{PMCA}}\mathit{Ca}+J_{er}-\frac{1}{z_{\mathit{Ca}}F}{I}_{\mathit{Ca}(v)}\right].$$ (A2)

For potassium, the approximation of constant K⁺ ion concentration is used. But such an approximation is not appropriate to study the gap junctions. In general, the Nernst potential of the potassium, v_K, assumes the constant K⁺ ion concentrations. A natural replacement of such an approximation is the adoption of the equation for the conservation of K⁺ ions within a cell,

$$V_{\mathit{cyt}}\frac{dK}{dt}=\frac{1}{z_{K}F}\left[-I_{K(v)}-I_{K(\mathit{Ca})}-I_{K(\mathit{ATP})}\right],$$ (A3)

where K ≡ c_K is the intracellular K⁺ ion concentration. However, Eq. (A3) does not guarantee stable K⁺ ion concentration within a cell as shown in Fig. 13. This result is not consistent with the constant Nernst potential, either. This is because pumps that are responsible for the inward potassium current are not included in the scheme to stabilize the intra-cellular K⁺ ion concentration. Then, we rewrite Eq. (A3) as

$$V_{\mathit{cyt}}\frac{dK}{dt}=\frac{1}{z_{K}F}\left[-I_{K(v)}-I_{K(\mathit{Ca})}-I_{K(\mathit{ATP})}+I_{K(in)}\right]$$ (A4)

where I_K(in) is the inward potassium current. But an introduction of the new term I_K(in) should not break the consistency of the whole scheme. In the steady state, dv/dt = dK/dt = 0. Then, we can derive the relationship, I_K(in) = −I_Ca(v), from Eqs. (A1) and (A4). We adopt it as an approximation in the near-equilibrium state. Figure 14 shows that this approximation can successfully stabilize the intracellular K⁺ ion concentration without breaking the self-consistency of the whole scheme. We use c_K = 95 mM25 as the intracellular K⁺ ion concentration at the rest-cell state in the current calculations. An adoption of c_K = 132.4 mM17 instead of c_K = 95 mM only shifts the base values of the K⁺ ion concentration from 95 mM to 132.4 mM in every calculation. The inward potassium current has already been taken into account in the single β-cell models by Meyer-Hermann20 in a different form.

For a wide range of studies of multicellular systems, it is necessary to confirm that our approximation, I_K(in) = −I_Ca(v), does not produce any artificial e ect within the single β-cell system. Otherwise, the reliability of our study of multicellular systems could be lost. Therefore, we have conducted extensive numerical calculations by changing the values of ḡ_K(ATP) and ḡ_Ca. Here, several cases closely related with the current work are shown in Figs. 15–17. In Fig. 14, the ḡ_K(ATP) is changed to 10000 pS from the original value of 40000 pS. ḡ_Ca is changed to 900 pS and 1100 pS from the original value of 1000 pS in Figs. 16 and 17, respectively.

Figure 13.

Single β-cell model calculations of Bedersen et al. together with Eq. (A3). The original model used the approximation of constant potassium ion concentration. Where v is the cell membrane potential, Ca is Ca²⁺ ion concentration, K is K⁺ ion concentration, and FBP is the concentration of fructose 1,6-bisphosphate.

Figure 14.

Modified single β-cell model calculations, where Eq. (A4) was used instead of Eq. (A3).

Figure 15.

Effect of ḡ_K(ATP) variation. Original ḡ_K(ATP) value is 40000 pS. Note that K value is slightly higher than 95 mM here.

Figure 16.

Effect of the ḡ_Ca variation. Original ḡ_Ca value is 1000 pS.

Figure 17.

Effect of the ḡ_Ca variation. Original ḡ_Ca value is 1000 pS.