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Geometric singular perturbation analysis of the active metabolic oscillator in pancreatic \b{eta}-cells

Prannath Moolchand, Martin Wechselberger

TL;DR

This work considerably extends the `fast-slow'analysis of glycolytic oscillators and is a stepping stone towards understanding how the slower metabolic system temporally patterns the faster electrical bursting dynamics.

Abstract

Pancreatic \b{eta}-cells secrete insulin in response to blood sugar levels to maintain glucose homeostasis. This vital insulin exocytosis is controlled by the cell's bursting behaviours, which are regulated by tight bidirectional coupling of inherent electrical and metabolic oscillators. The Integrated Oscillator Model suggests that slower metabolic oscillations are mediated either by glycolytic oscillations-through an independent active metabolic oscillator (AMO)-or by Ca2+ effects on ATP consumption via a passive metabolic oscillator (PMO). By clamping the Ca2+ and ATP dynamics, our study focuses on the decoupled AMO which is the driver of pulsatile dynamics. Using appropriate reference scales, we first non-dimensionalise the model to identify small parameters and processes evolving on different timescales. We show that the AMO can be recast as a surrogate relaxation oscillator, a more general class of multiple timescale problems involving oscillation cycles comprising fast and slow segments, which are amenable to rigorous analysis using the machinery of geometric singular perturbation theory. Using the parametrisation method to identify invariant manifolds and blow-up analysis to desingularise degenerate vector fields, we fully characterise the hierarchy of timescales and the complex singular geometry constituting the metabolic oscillations. Our work considerably extends the `fast-slow' analysis of glycolytic oscillators and is a stepping stone towards understanding how the slower metabolic system temporally patterns the faster electrical bursting dynamics.

Geometric singular perturbation analysis of the active metabolic oscillator in pancreatic \b{eta}-cells

TL;DR

This work considerably extends the `fast-slow'analysis of glycolytic oscillators and is a stepping stone towards understanding how the slower metabolic system temporally patterns the faster electrical bursting dynamics.

Abstract

Pancreatic \b{eta}-cells secrete insulin in response to blood sugar levels to maintain glucose homeostasis. This vital insulin exocytosis is controlled by the cell's bursting behaviours, which are regulated by tight bidirectional coupling of inherent electrical and metabolic oscillators. The Integrated Oscillator Model suggests that slower metabolic oscillations are mediated either by glycolytic oscillations-through an independent active metabolic oscillator (AMO)-or by Ca2+ effects on ATP consumption via a passive metabolic oscillator (PMO). By clamping the Ca2+ and ATP dynamics, our study focuses on the decoupled AMO which is the driver of pulsatile dynamics. Using appropriate reference scales, we first non-dimensionalise the model to identify small parameters and processes evolving on different timescales. We show that the AMO can be recast as a surrogate relaxation oscillator, a more general class of multiple timescale problems involving oscillation cycles comprising fast and slow segments, which are amenable to rigorous analysis using the machinery of geometric singular perturbation theory. Using the parametrisation method to identify invariant manifolds and blow-up analysis to desingularise degenerate vector fields, we fully characterise the hierarchy of timescales and the complex singular geometry constituting the metabolic oscillations. Our work considerably extends the `fast-slow' analysis of glycolytic oscillators and is a stepping stone towards understanding how the slower metabolic system temporally patterns the faster electrical bursting dynamics.
Paper Structure (43 sections, 3 theorems, 95 equations, 17 figures, 5 tables)

This paper contains 43 sections, 3 theorems, 95 equations, 17 figures, 5 tables.

Table of Contents

  1. Introduction
  2. The AMO model
  3. The dimensionless model:
  4. Compact domain and confined fold points:
  5. The unstable node:
  6. Comparison with system parameters:
  7. Dealing with the square root:
  8. Scaling regimes analysis:
  9. System X-Z (Regime 1):
  10. The layer problem:
  11. The reduced problem along $\Gamma_2$:
  12. System U-Z (Regime 2):
  13. The layer problem:
  14. The reduced problem:
  15. Summary of Regime 2:
  16. ...and 28 more sections

Key Result

Proposition 4.2

The 3D system eq:syscha1 possesses a local 2-dimensional centre manifold $W^c_{loc}(p_1)$ at $p_1 = (0, \frac{1}{\gamma \sigma_1}, 0)$, tangent to the $(r_1, \varepsilon_1)$-eigenspace. To leading order, the manifold is represented by the graph: The dynamics on $W^c_{loc}(p_1)$ are organised by two invariant 1D sub-manifolds:

Figures (17)

  • Figure 1: Network diagram of the 8D Integrated Oscillator Model (IOM) Marinelli2018, showing the interaction between electrical (blue) and metabolic (green) subsystems.
  • Figure 1: Original Problem, biophysical System $x-y$\ref{['eq:sysxy']}. A: time traces, with dynamic and static phases, Oscillation with a period of $T_{xy} = 4.28e5$ ms = 7.125 min. B: solid black - phase plot of trajectory.
  • Figure 1: The three scaling regimes required for the GSPT analysis, as derived from the asymptotic scalings of the equilibrium and fold points.
  • Figure 1: First Cylindrical Blow-Up
  • Figure 1: Left: Geometric representation of the reduced vector field obtained via the oblique projection $\Pi^S$ onto the tangent bundle $TS$ along the linear fast fiber bundle $\mathcal{N}$. Right: Smooth embedding $\varphi$ from $U_1$ into $U_0$.
  • ...and 12 more figures

Theorems & Definitions (21)

  • Remark 2.1
  • Remark 2.2
  • Remark 3.1
  • Remark 3.2
  • Remark 4.1
  • Proposition 4.2: Existence of the 2D centre manifold
  • Remark 4.3
  • Proposition 4.4: Existence of the 2D centre manifold at $p_3$
  • Remark 4.5
  • Remark 4.6: Resolution of $p_{378}$
  • ...and 11 more