Pancreatic Beta-Cell Mathematical Models

Ca-Ca Model

This paper explores calcium-dependent inactivation of L-type calcium channels in beta cells, proposing that inactivation depends on calcium in the micro- or nano-domain near open calcium channels rather than bulk calcium. This should apply to L-type calcium channels in other cell types.

Domain model for Ca²⁺ inactivation of Ca²⁺ channels at low channel density.

Keizer J, Rinzel J, Sherman A.

Biophys. J. (1990) 58:985-995. Abstract/Full Text

Chay-Cook Model

We adapted this model for our analysis of classification of bursting mechanisms because it can flexibly produce burst patterns similar to those seen in beta-cells and neurons.

Topological and phenomenological classification of bursting mechanisms.

Bertram R, Butte M, Kiemel T, Sherman A.

Bull. Math. Biol. (1995) 57:413-439. Abstract/Full Text

Endogenous Bursting Patterns in Excitable Cells.

Adapted from: Chay TR, Cook D.

Math. Biosci. (1988) 90(1-2):139–153. Abstract/Full Text

Chay-Keizer Model

This is the foundational model that launched the field of modeling bursting electrical activity in pancreatic beta cells. Bursting was achieved by appending a calcium-activated potassium channel to voltage-dependent squid axon channels. Many details have changed over the years, but the fundamental mathematical structure has been largely retained.

Minimal model for membrane oscillations in the pancreatic beta-cell.

Chay TR, Keizer J

Biophys J. (1983) 42(2):181-90. Abstract/Full Text

Keizer-Magnus Model

This was the first model to propose that bursting is driven primarily by calcium-dependent inhibition of ATP production in the mitochondria).

Slow voltage inactivation of Ca²⁺ currents and bursting mechanisms for the mouse pancreatic beta-cell.

Keizer J, Magnus G.

J. Membr. Biol. (1989) 127(1):9 – 19. Abstract/Full Text

Keizer-Smolen Model

In this model bursting is driven by inactivation of L-type calcium channels.

Bursting electrical activity in pancreatic beta cells caused by Ca²⁺- and voltage-inactivated ^Ca2+ channels.

Keizer J, Smolen P.

(1991) 88(9):3897–3901. Abstract/Full Text

S Model

This is a simplified bursting model convenient for mathematical analysis. Bursting is driven by a slow, voltage-dependent potassium current. Electrical coupling between two such cells is also included.

Rhythmogenic effects of weak electrotonic coupling in neuronal models.

Rinzel J, Sherman A.

Proc. Natl. Acad. Sci. USA (1992) 89:2471-2474. Abstract/Full Text

Anti-phase, asymmetric, and aperiodic oscillations in excitable cells I. Coupled bursters.

Sherman A.

Bull. Math. Biol. (1994) 56:811-835. Abstract/Full Text

SRK Model

A Chay-Keizer-like model with bursting driven by calcium-activated potassium channels but ionic currents based more closely on measurements in beta cells rather than neurons.

Emergence of organized bursting in clusters of pancreatic beta-cells by channel sharing.

Keizer J, Sherman A, Rinzel J.

Biophys. J. (1988) 54:411-425. Abstract/Full Text

Beta Cell Resetting

This model accounts for the puzzling observation that when islets are reset from active phase to silent phase with a strong electrical impulse, the duration of the induced phase is independent of when the stimulus was applied in the original phase.

Phase-independent resetting in relaxation and burst oscillators.

Sherman A, Smolen P.

J. Theor. Biol. (1994) 169:339-348. Abstract/Full Text

Phantom Burster I

This model introduced the concept of phantom bursting, i.e., that the process that drives bursting may not be a single channel but a composite of two or more channels. This is a way of accounting for the wide range of burst periods observed in beta cells that may be more plausible than a single mechanism with an elastic time constant.

The phantom burster model for pancreatic beta-cells.

Bertram R, Kinard TA, Previte J, Satin LS, Sherman A.

Biophys. J. (2000) 79:2880-2892. Abstract/Full Text

Phantom Burster II

This is a more biophysical implementation of the phantom bursting model for pancreatic beta-cells. The role of s1 is played by cytosolic calcium (c) and the role of s2 is played by ER calcium (cer) or the ADP/ATP ratio (a), or both.

The phantom burster model for pancreatic beta-cells.

Bertram R, Sherman A.

Bull. Math. Biol. (2004) 66:1313-1344 2004. Abstract/Full Text

Calcium Subspace Model

In this model, the calcium-activated potassium channel is governed by calcium released from the ER into a small subspace between the ER and the plasma membrane rather than the usual bulk cytoplasmic calcium. This accounts for the paradoxical observation that emptying the ER with thapsigargin increases K(Ca) current.

Calcium-activated K+ channels of mouse beta-cells are controlled by both store and cytoplasmic Ca2+: experimental and theoretical studies.

Goforth et al.

J. Gen. Physiol. (Sept 2002) 120(3):307-322. Abstract/Full Text

The Ca2+ dynamics of isolated mouse beta-cells and islets: implications for mathematical models.

Zhang et al.

Biophys. J. (2003) 84:2852-2870. Abstract/Full Text

CRAC Model I

This model shows that a Calcium Release Activated Channel (CRAC), activated when ER calcium declines, can account for the effects of acetylcholine to potentiate beta-cell electrical activity and the transient first phase of electrical activity when glucose is elevated from basal.

A role for calcium-release activated current (CRAC) in cholinergic modulation of electrical activity in pancreatic beta-cells.

Atwater I, Bertram R, Martin F, Mears D, Sherman A, Smolen P, Soria B.

Biophys. J. (1995) 68:2323-2332. Abstract/Full Text

CRAC Model II

This model experimentally tested the prediction of CRAC Model I that ER depletion can account for the transient first phase of electrical activity when glucose is elevated from basal.

Evidence that calcium release-activated current mediates transient glucose-induced electrical activity in the pancreatic beta-cell.

Atwater I, Bertram R, Mears D, Rojas E, Sheppard Jr NF, Sherman A.

J. Membr. Bio. (1997) 155:47-59. Abstract/Full Text

Passive ER Model

This model shows that a passive endoplasmic reticulum that takes up calcium when beta cells are depolarized and releases calcium when the cells are silent is sufficient to account for experiments in pancreatic islets. Active calcium-induced calcium release, proposed by others, is not necessary and is in conflict with the data if the effect is dominant.

Filtering of calcium transients by the endoplasmic reticulum in pancreatic beta-cells.

Bertram R, Sherman A.

Biophys. J. (2004) 87:3775-3785. Abstract/Full Text

DOM 1.5

This model is a blend of DOM 1.0 and DOM 2.0 and introduces a major role for consumption of ATP by calcium pumps. It was used to demonstrate that glycolytic oscillations may be dependent on calcium, reinterpreting experimental data in the companion paper below.

Calcium and metabolic oscillations in pancreatic islets: Who's driving the bus?

Bertram R, Fendler B, Merrins MJ, Satin LS, Sherman A, Watts M.

SIAM J. Appl. Dyn. Syst. (2014) 13(2):683-703. Abstract/Full Text

Metabolic oscillations in pancreatic islets depend on the intracellular Ca²⁺ level but not Ca²⁺ oscillations.

Bertram R, Fendler B, Merrins MJ, Satin LS, Sherman A, Zhang M.

Biophys. J. (2010) 99:(1)76-94. Abstract/Full Text

DOM 2.0 (mitochondria)

The dual oscillator model is extended to include details of mitochondrial dynamics, including mitochondrial membrane potential, calcium and NADH.

Interaction of glycolysis and mitochondrial respiration in metabolic oscillations of pancreatic islets.

Bertram R, Luciani DS, Gram Pedersen M, Satin LS, Sherman A.

Biophys. J. (2007) 92(5):1544-55. Abstract/Full Text

Dual Oscillator Model (DOM) 1.0

This model exhibits complex oscillations of calcium and membrane potential based on interactions of glycolytic and ionic oscillatiors. That combination, termed the Dual Oscillator Model (DOM), accounts for the three major oscillation patterns, fast, slow and compound, as well as small amplitude calcium oscillations in sub-threshold glucose.

Calcium and glycolysis mediate multiple bursting modes in pancreatic islets

Bertram R, Satin L, Sherman A, Smolen P, Zhang M.

Biophys. J. (2004) 87:3074-3087. Abstract/Full Text

Glucose Sensing with DOM 1.0

These models show that the DOM 1.0 can account for not only diverse oscillation patterns of pancreatic but also how the oscillations change with glucose. The agreement and confirmed predictions lend strong support to the DOM.

Glucose modulates Ca²⁺ oscillations in pancreatic islets via ionic and glycolytic mechanisms.

Bertram R, Daniel CR, Nunemaker CS, Satin LS, Sherman A, Tsaneva-Atanosova K.

Biophys. J. (2006) 91:2082-2096. Abstract/Full Text

IOM 1.0

In order to account for measurements of oscillations in fructose 1,6 bisphosphate (FBP) in the companion experimental paper, we hypothesized an additional effect of calcium on metabolism, to stimulate uptake of glycolytic substrate into the mitochondria. Reflecting the tighter connection between the electrical and metabolic oscillators, we call this the Integrated Oscillator Model (IOM).

Ca²⁺ Effects on ATP Production and Consumption Have Key Regulatory Roles on Oscillatory Islet Activity

Bertram R, Ha J, McKenna JP, Merrins MJ, Satin LS, Sherman A.

Biophys. J. (2016) 110(3):733-742. Abstract/Full Text

Phase Analysis of Metabolic Oscillations and Membrane Potential in Pancreatic Islet β-cells.

Bertram R, Ha J, McKenna JP, Merrins MJ, Poudel C, Satin LS, Sherman A.

Biophys. J. (2016) 110(3):691-699. Abstract/Full Text

Closing in on the mechanisms of pulsatile insulin secretion.

Bertram R, Satin LS, Sherman A.

Diabetes. (2018) 67:351-359. Abstract/Full Text

IOM 1.5

The IOM has been refined to account for a wider range of oscillation modes. The two papers below provide mathematical analysis and further predictions to be tested experimentally.

Transitions Between Bursting Modes in the Integrated Oscillator Model for Pancreatic β-Cells.

Bertram R, Gerardo-Giorda L, Marinelli I, Vo T.

Journal of Theoretical Biology. (2018) 454:310-319. Abstract/Full Text

Fast-Slow Analysis of the Integrated Oscillator Model for Pancreatic β-Cells.

Bertram R, McKenna JP.

Journal of Theoretical Biology. (2018) 457:152-162. Abstract/Full Text

IOM 1.7

IOM 1.5 has been extended to include secretion of insulin and applied to illustrate the hypothesized transition from glycolytic oscillations in sub-threshold (basal) glucose to calcium-driven metabolic oscillations in super-threshold (post-prandial) glucose.

Pulsatile Basal Insulin Secretion Is Driven by Glycolytic Oscillations

Fletcher PA, Marinelli I, Bertram R, Satin LS, Sherman AS.

Physiology (Bethesda). (2022) 37(4):0. Abstract/Full Text

IOM 2.0

Mitochondrial variables have been added to the Integrated Oscillator Model to make IOM2.0. It has been used in the following publications:

Slow oscillations persist in pancreatic beta cells lacking phosphofructokinase M

Marinelli I, Parekh V, Fletcher P, Thompson B, Ren J, Tang X, Saunders TL, Ha J, Sherman A, Bertram R, Satin LS.

Biophysical Journal. (2022) 121(5):692-704. Abstract/Full Text

Oscillations in K(ATP) conductance drive slow calcium oscillations in pancreatic β-cells

Marinelli I, Thompson BM, Parekh VS, Fletcher PA, Gerardo-Giorda L, Sherman AS, Satin LS, Bertram R.

Biophysical Journal. (2022) 121(8):1449-1464. Abstract/Full Text

Pitchfork

These models show that oscillations in the DOM can be abolished by a pitchfork bifurcation when cells are coupled through the slow negative feedback variable, which is calcium, for a simple electrical oscillator, and G6P, for a glycolytic oscillator.

Diffusion of calcium and metabolites in pancreatic islets: Killing oscillat ions with a pitchfork.

Bertram R, Sherman A, Tsaneva-Atanasova K, Zimliki CL.

Biophys. J. (2006) 90:3434-3446. Abstract/Full Text

Ramp IVs

Here, a ramp IV protocol was used to measure conductance of K(ATP) channels during slow oscillations in islets. The time course of the conductance was consistent with glycolytic oscillations but not metabolic oscillations driven by calcium.

Slow oscillations of KATP conductance in mouse pancreatic islets provide support for electrical bursting driven by metabolic oscillations.

Bertram R, Goforth PB, Nunemaker C, Ren J, Satin LS, Sherman A, Waters C.

Am. J. Physiol. (Endocrinol. and Metab.) (2013) 305(7):E805-17. Abstract/Full Text

Smolen-Keizer Model

In this model bursting is driven primarily by calcium-dependent inhibition of ATP production in the mitochondria, as in Keizer-Magnus, but more detail on calcium-channel inactivation is incorporated.

Slow voltage inactivation of Ca2+ currents and bursting mechanisms for the mouse pancreatic beta-cell.

Keizer J, Smolen P.

J. Membr. Biol. (1992) 127(1):9–19. Abstract/Full Text

cAMP Oscillations

cAMP in beta cells is increased by calcium, but some experiments indicate that cAMP can oscillate in the absence of calcium oscillations. We show that oscillations of adenosine monophosphate (AMP), which can occur independent of calcium by the DOM, can drive cAMP oscillations.

Modeling of glucose-induced cAMP oscillations in pancreatic β-cells: cAMP rocks when metabolism rolls.

Bertram R, Peercy BE, Sherman AS.

Biophys. J. (2015) 109(2):439-49. Abstract/Full Text

PKA Diffusion

Part of the way beta cells adapt to chronic hyperglycemia is mediated by cAMP entering the nucleus. This model offers an explanation for why short-term elevation of cAMP does not lead to nuclear entry but long-term elevation does.

How pancreatic beta-cells distinguish long- and short-time scale cAMP signals

Peercy BE, Sherman AS.

Biophys. J. (2010) 99(2):398-406. Abstract/Full Text