Synchronization of Oscillations in Pancreatic Islets

By Nicole Bruce, I-An Wei, Michael Roper, and Richard Bertram

Beta cells, which are located in micro-organs of the pancreas called pancreatic islets, secrete insulin to effectively regulate blood glucose levels. Individual islets release insulin in pulses that stem from oscillations in intracellular \(\textrm{Ca}^{2+}\) [5]. Blood insulin measurements reflect this insulin pulsatility, indicating that insulin release is synchronized among the millions of islets in the pancreas [6]. While the exact method of this synchronization is unknown, feedback interactions between the pancreas and liver serve as one possible mechanism (see Figure 1). Liver cells lower blood glucose in response to insulin, and all islets feel this change in glucose after a time delay \(\tau_d\) that accounts for glucose circulation in the blood. This glucose adjustment provides negative feedback on insulin secretion, and the islets coordinate their activity by all responding to the same signal. 

Previous research successfully utilized the liver feedback mechanism to synchronize a small number of mouse islets in vitro without a time delay [2]. We sought to examine the effects of a time delay in the negative feedback for both model and mouse islets in vitro. We found that the liver feedback mechanism successfully synchronized islets in the presence of a time delay; it was also capable of producing a second, slower oscillation pattern.

We studied the effect of a time delay on model islets using an existing model for islet activity called the integrated oscillator model [3]. We modified the IOM to include a representation of liver feedback, following from a previous model [4]. First, we added an equation for insulin secretion:

\[\frac{dI}{dt}=\frac{I_\infty-I}{\tau_I},\tag1\]

where \(\tau_I\) is a time constant and \(I_\infty\) is an increasing linear function of \(\textrm{Ca}^{2+}\) concentration past a threshold \(\textrm{Ca}_\textrm{thr}\):

\[I_\infty = \left\{ \begin{array}{l}
I_{\textrm{slope}}\left( \textrm{Ca} - \textrm{Ca}_\textrm{thr} \right) &\textrm{for } \: \textrm{Ca} \ge \textrm{Ca}_\textrm{thr}\\
0 & \textrm{for } \: \textrm{Ca} < \textrm{Ca}_\textrm{thr}
\end{array} \right. .\tag2\]

We averaged the insulin secretion over all islets \((I_\textrm{avg})\) and used it to negatively regulate the extracellular glucose concentration, as is done by the liver: 

\[\frac{dG_e}{dt}=\frac{G_\infty-G_e}{\tau_G}.\tag3\]

Here, \(\tau_G\) is a time constant and \(G_\infty\)—the glucose response function—is a decreasing sigmoidal function of \(I_\textrm{avg}\) with a time delay \(\tau_G\): 

\[G_\infty=G_\textrm{min}+\frac{G_\textrm{max}-G_\textrm{min}}{1+\exp{\left(\frac{I_\textrm{avg}\left(t-\tau_d\right)-\hat{I}}{S_G}\right)}}.\tag4\]

\(G_\textrm{min}\) and \(G_\textrm{max}\) are the respective minimum and maximum glucose levels, and \(\hat{I}\) corresponds to the halfway point between those values. The parameter \(S_G\) determines the steepness of the response function. 

We modeled five heterogeneous islets with various natural periods for time delay values \(\tau_d =\) 0, 1, 2, 3, 4, 5, 6, and 7 minutes. Figure 2 illustrates results for \(\tau_d =\) 3 minutes. Since glucose concentration was initially held constant, the average \(\textrm{Ca}^{2+}\) trace \((\textrm{Ca}_\textrm{avg})\) was initially disorganized due to the individual islets that were bursting out of phase. After 20 minutes, glucose was allowed to vary according to the liver feedback model and the islets synchronized to a 4-minute period (see Figure 2a). This synchronization is evidenced by the \(\textrm{Ca}_\textrm{avg}\) trace’s transition to more organized and periodic, with high-amplitude pulses. The individual islet traces during feedback also show that the islets are oscillating in phase. 

However, we observed additional behavior besides the islets’ synchronization to a 4-minute period; with a time delay, the islets were also able to generate slow oscillation rhythms (see Figure 2b). Beginning with different initial conditions, the same group of islets with the same time delay (\(\tau_d =\) 3 minutes) now produce an average calcium trace with a slow oscillation period of 11.5 minutes. From the individual islet calcium traces, we see that these slow oscillations arise from the organization of individual fast oscillations into longer episodes.

For each nonzero time delay that we incorporated into the negative feedback, islets were able to synchronize to fast oscillations and produce coordinated slow oscillations. The fast oscillation period was always roughly 4 minutes long and unaffected by the time delay, while the slow oscillation period increased linearly with the time delay — resulting in a 20-minute period for our longest time delay (\(\tau_d =\) 7 minutes). The fact that the same group of islets can produce different oscillation behaviors for the same time delay based only on the initial conditions indicates that the model with delayed negative feedback is bistable.

We tested the effects of a time delay with negative feedback on experimental islets by using a microfluidic device that precisely varied glucose concentrations according to the feedback model in \((3)\) and \((4)\). As with the model results, we found that experimental islets with a nonzero time delay either synchronized to a fast oscillation period or produced a slow oscillation with a period that had an increasing linear relationship with the time delay. Yet unlike the model, each experiment used different groups of islets, meaning that we could not conclude that the biological islets were also bistable from this outcome alone. We therefore set out to confirm the model prediction for bistability in biological islets. 

To test for bistability in the feedback system, we first turned on the feedback with a 3-minute time delay and allowed the islets to settle into either a fast or slow oscillation period. We then turned off the feedback and held glucose constant again, after which we turned on the feedback for a second time. We sought to push the islets into the basin of attraction for the opposite oscillation period by turning off the feedback and thus resetting the system’s initial conditions. Figure 3 depicts the results for biological islets. During the first feedback period, the islets produced slow oscillations with a period of 9.8 minutes. When the glucose was subsequently clamped, the islets drifted out of this coherent oscillatory pattern. When we turned on the feedback for a second time, the biological islets instead produced fast synchronized oscillations with a period of 3.5 minutes. This switch from slow to fast oscillations via glucose clamping confirms the model prediction that the islets are bistable. 

Our results support the liver feedback mechanism as a method of synchronizing islet activity. Even with a time delay of up to 7 minutes, feedback successfully synchronized both model and biological islets, proving that the system is robust. The time delay also introduced bistability into the system and was capable of coordinating islet activity into much longer oscillation periods. 

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Nicole Bruce presented this research during a minisymposium presentation at the 2022 SIAM Conference on the Life Sciences (LS22), which took place concurrently with the 2022 SIAM Annual Meeting in Pittsburgh, Pa., in July 2022. She received funding to attend LS22 through a SIAM Student Travel Award. To learn more about Student Travel Awards and submit an application, visit the online page. 

Acknowledgments: This work was supported in part by grant number R01 DK 080714 from the National Institutes of Health and grant number DMS 1853342 from the National Science Foundation. 

References
[1] Bruce, N., Wei, I.-A., Leng, W., Oh, Y., Chiu, Y.C., Roper, M.G., & Bertram, R. (2022). Coordination of pancreatic islet rhythmic activity by delayed negative feedback. Am. J. Physiol. Endocrinol. Metab., epub ahead of print.
[2] Dhumpa, R., Truong, T.M., Wang, X., Bertram, R., & Roper, M.G. (2014). Negative feedback synchronizes islets of Langerhans. Biophys. J., 106(10), 2275-2282.
[3] Marinelli, I., Vo, T., Gerardo-Giorda, L., & Bertram, R. (2018). Transitions between bursting modes in the integrated oscillator model for pancreatic β-cells. J. Theor. Biol., 454, 310-319.
[4] Pedersen, M.G., Bertram, R., & Sherman, A. (2005). Intra- and inter-islet synchronization of metabolically driven insulin secretion. Biophys. J., 89(1), 107-119.
[5] Rorsman, P., & Ashcroft, F.M. (2018). Pancreatic β-cell electrical activity and insulin secretion: Of mice and men. Physiol. Rev., 98(1), 117-214.
[6] Song, S.H., McIntyre, S.S., Shah, H., Veldhuis, J.D., Hayes, P.C., & Butler, P.C. (2000). Direct measurement of pulsatile insulin secretion from the portal vein in human subjects. J. Clin. Endocrinol. Metab., 85(12), 4491-4499.

	Nicole Bruce is a Ph.D. student in the Department of Mathematics at Florida State University. Her research involves the use of mathematical models to investigate potential methods of pancreatic islet synchronization.
	I-An Wei is a Ph.D. student in the Department of Chemistry and Biochemistry at Florida State University.
	Michael Roper is a professor in the Department of Chemistry and Biochemistry at Florida State University.
	Richard Bertram is a professor in the Department of Mathematics and a graduate faculty member in the Molecular Biophysics Program and the Neuroscience Program at Florida State University.