Mathematical models of electrical activity of the pancreatic β-cell: a physiological review

Abstract

Mathematical modeling of the electrical activity of the pancreatic β-cell has been extremely important for understanding the cellular mechanisms involved in glucose-stimulated insulin secretion. Several models have been proposed over the last 30 y, growing in complexity as experimental evidence of the cellular mechanisms involved has become available. Almost all the models have been developed based on experimental data from rodents. However, given the many important differences between species, models of human β-cells have recently been developed. This review summarizes how modeling of β-cells has evolved, highlighting the proposed physiological mechanisms underlying β-cell electrical activity.

Keywords: ADP, adenosine diphosphate; ATP, adenosine triphosphate; CK, Chay-Keizer; CRAC, calcium release-activated current; Ca2+, calcium ions; DOM, dual oscillator model; ER, endoplasmic reticulum; F6P, fructose-6-phosphate; FBP, fructose-1,6-bisphosphate; GLUT, glucose transporter; GSIS, glucose-stimulated insulin secretion; HERG, human eter à-go-go related gene; IP3R, inositol-1,4,5-trisphosphate receptors; KATP, ATP-sensitive K+ channels; KCa, Ca2+-dependent K+ channels; Kv, voltage-dependent K+ channels; MCU, mitochondrial Ca2+ uniporter; NCX, Na+/Ca2+ exchanger; PFK, phosphofructokinase; PMCA, plasma membrane Ca2+-ATPase; ROS, reactive oxygen species; RyR, ryanodine receptors; SERCA, sarco-endoplasmic reticulum Ca2+-ATPase; T2D, Type 2 Diabetes; TCA, trycarboxylic acid cycle; TRP, transient receptor potential; VDCC, voltage-dependent Ca2+ channels; Vm, membrane potential; [ATP]i, cytosolic ATP; [Ca2+]i, intracellular calcium concentration; [Ca2+]m, mitochondrial calcium; [Na+], Na+ concentration; action potentials; bursting; cAMP, cyclic AMP; calcium; electrical activity; ion channels; mNCX, mitochondrial Na+/Ca2+ exchanger; mathematical model; β-cell.

Figure 1.

Glucose-stimulated insulin secretion (GSIS). After glucose is transported into the cell by the GLUT transporters, it is metabolized, potentiating the production of ATP and the closure of the ATP-sensitive K⁺ channels (K_ATP). The membrane is depolarized and voltage-dependent Ca²⁺ channels (VDCCs) are activated, allowing the influx of Ca²⁺. The increase of the intracellular Ca²⁺ concentration ([Ca²⁺]_i) stimulates Ca²⁺-dependent insulin secretion.

Figure 2.

Electrical activity patterns in pancreatic β-cells. (A) Simulated fast (top), slow (middle), and compound (bottom) bursting behavior in rodent cells (simulations made with the Dual Oscillator Model¹²⁰). (B) Simulations of action potential firing (top), fast (middle), and slow (bottom) bursting in human cells (simulated with the model of the human β-cell of Riz et al.²⁸).

Figure 3.

Minimal model of Chay and Keizer (CK model). A. Scheme of the CK model. The active phase (1) is sustained by the VDCCs and K_v channels, slowly increasing [Ca²⁺]_i. The K_Ca channels are activated, eventually repolarizing the membrane (2). During the silent phase (3), the VDCCs and K_v channels are closed and Ca²⁺ is extruded from the cell, inhibiting the activity of the K_Ca channels. The slow depolarization eventually activates the VDCCs and K_v channels, initiating a new burst. B. Fast bursting simulated with the CK model. Top: Membrane potential (black curve) and intracellular Ca²⁺ concentration ([Ca²⁺]_i, yellow curve). Bottom: Ca²⁺-dependent K⁺ (K_Ca) current.

Figure 4.

Oscillations in ATP regulate the conductance of the K_ATP channels. (A) During the active phase (1), sustained by the VDCC and the K_v channels, [Ca²⁺]_i increases, exerting a negative effect on the production of ATP, reflected in the increase in ADP and the corresponding decrease in the ATP/ADP ratio. The K_ATP channels are slowly opened, eventually repolarizing the membrane (2). During the silent phase, VDCCs are inhibited, and the influx of Ca²⁺ is ceased as Ca²⁺ is also extruded from the cell. As [Ca²⁺]_i decreases, the production of ATP is potentiated, closing the K_ATP channels and initiating slow depolarization (3). (B) Simulations with the model of Smolen-Keizer. Top: V_m (black curve) and [ADP]_i (purple curve). Bottom: the fast dynamics of [Ca²⁺]_i resembles the experimental observations.

Figure 5.

(A) Diagram of the models including ER as a second Ca²⁺ compartment and a non-specific calcium release-activated current (CRAC). During the silent phase (1), Ca²⁺ is released from the ER to the cytoplasm and is simultaneously extruded from the cell. This results in the activation of the CRAC current and the Ca²⁺-inactivated Ca²⁺ current, driving slow depolarization and initiation of a burst of action potentials (2). As [Ca²⁺]_i increases and Ca²⁺ is captured by the ER during the active phase, both the CRAC and the Ca²⁺-inactivating Ca²⁺ currents are inhibited, resulting in membrane repolarization (3). (B and C) Simulations using the model of Chay including ER. Fast (B) and slow (C) bursting is produced by modifying the release rate of Ca²⁺ from the ER. In both cases, V_m (top, black curve), [Ca²⁺]_i, and [Ca²⁺]_ER (bottom, yellow and purple curves, respectively) are shown.

Figure 6.

[Na⁺]_i as a pacemaker variable. (A) The model of Fridlyand et al. is shown schematically. Entry of Ca²⁺ during the active phase activates the Na⁺/Ca²⁺ exchanger, inducing an increase of [Na⁺]_i (1). This promotes the activity of an outward current through the Na⁺/K⁺ pump, eventually repolarizing the membrane (2). In the silent phase, Ca²⁺ influx is inhibited, resulting in a reduction in both the activity of the NCX exchanger and the Na⁺/K⁺ pump, promoting slow depolarization (3). (B) Simulation of slow electrical activity. Top: V_m (black curve) and [Ca²⁺]_i (yellow curve). Middle: Current through the NCX exchanger (INaCa, light purple) and [Na⁺]_i (dark purple). Bottom: Current through the Na⁺/K⁺ pump (INa⁺/K⁺, red curve).

Figure 7.

Intrinsic metabolic oscillations (DOM model). (A) Diagram of the DOM model. The interactions between glycolytic, metabolic, and electrical components drive different electrical behaviors (simulations shown in B–D) depending on the regime of the glycolytic and electrical components. Glucose is metabolized by the glycolytic and metabolic components controlling the production of ATP, which mediate the changes in the conductance of the K_ATP channels, depolarization, and Ca²⁺ influx. The 3 compartments (glycolytic, electrical, and metabolic) are affected by the changes in [Ca²⁺]_i. (B) Slow bursting is produced entirely by oscillatory glycolysis. (C) Fast bursting produced by the electrical component. (D) The combination of glycolytic and electrical components produces compound bursting activity. (B–D) Top: V_m (black curve) and the state of glycolysis (represented by F6P, orange curve). Bottom: [Ca²⁺]_i (yellow curve) and [ATP]_i (green curve).

Figure 8.

(A) Diagram of the mechanisms included in the model of Riz et al. of human β-cells. Channels included in the model: ATP-dependent K⁺ channels (K_ATP), big and small conductance Ca²⁺-dependent K⁺ channels (K_BK and K_SK), voltage-dependent K⁺ channels (K_v), HERG-K⁺ channels (K_ERG), voltage-dependent Na⁺ channels (Na_v), L, T and P/Q-type Ca²⁺ channels (Ca_L, Ca_T, Ca_PQ, respectively), Cl⁻ channels (representing the current mediated by the neurotransmitter γ-aminobutiric acid, GABA). The Ca²⁺ dynamics included a cytoplasmic and a submembrane compartment and the plasma membrane Ca²⁺-ATPase (PMCA) and Na⁺/Ca²⁺ exchanger (NCX). (B–D) Simulations of V_m (black curve), submembrane Ca²⁺ (pink curve), intracellular Ca²⁺ (yellow curve), and glycolysis (FBP, orange curve) are shown. (B) Action potential firing. (C) Fast bursting. (D) Slow bursting.