Modelling the transition from simple to complex Ca²⁺oscillations in pancreatic acinar cells

Abstract

A mathematical model is proposed which systematically investigates complex calcium oscillations in pancreatic acinar cells. This model is based on calcium-induced calcium release via inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR) and includes calcium modulation of inositol (1,4,5) trisphosphate (IP₃) levels through feedback regulation of degradation and production. In our model, the apical and the basal regions are separated by a region containing mitochondria, which is capable of restricting Ca²⁺ responses to the apical region. We were able to reproduce the observed oscillatory patterns, from baseline spikes to sinusoidal oscillations. The model predicts that calcium-dependent production and degradation of IP₃ is a key mechanism for complex calcium oscillations in pancreatic acinar cells. A partial bifurcation analysis is performed which explores the dynamic behaviour of the model in both apical and basal regions.

Appendix A

$$ \begin{array}{c}\hfill \frac{\partial \left[C{a}^{2+}\right]}{\partial t}={D}_c\frac{\partial^2\left[C{a}^{2+}\right]}{\partial {x}^2}+\left({J}_{IPR}+{J}_{RyR}+{J}_{ER}\right)\hfill \\ {}\hfill \left(-{J}_{SERCA}\right)-{J}_{MITO}+\delta \left({J}_{IN}-{J}_{PM}\right)\hfill \end{array} $$

(A1)

Note that JMITO is nonzero only in the mitochondrial region.

$$ \frac{1}{\gamma}\frac{d{\left[C{a}^{2+}\right]}_{ER}}{ dt}=\left(-\left({J}_{IPR}+{J}_{RyR}+{J}_{ER}\right)+{J}_{SERCA}\right) $$

(A2)

$$ \frac{d\left[I{P}_3\right]}{ dt}=\frac{J_0}{\tau_p}\left(\begin{array}{l}\left(v\frac{\left[C{a}^{2+}\right]+\left(1-\alpha \right){k}_5}{\left[C{a}^{2+}\right]+{k}_5}\right)-\\ {}\left({k}_{5p}+{k}_{3k}\frac{{\left[C{a}^{2+}\right]}^2}{{\left[C{a}^{2+}\right]}^2+{k}_{\deg}^2}\right)\left[I{P}_3\right]\end{array}\right) $$

(A3)

$$ \frac{ dR}{ dt}={\phi}_{-2}O-{\phi}_2\left[I{P}_3\right]R+\left({k}_{-1}+{l}_{-2}\right){l}_1-{\phi}_1R $$

(A4)

$$ \frac{ dO}{ dt}={\phi}_2\left[I{P}_3\right]R-\left({\phi}_{-2}+{\phi}_4+{\phi}_3\right)O+{\phi}_{-4}A+{k}_{{}_{-3}}S $$

(A5)

$$ \frac{ dA}{ dt}={\phi}_4O-{\phi}_{-4}A-{\phi}_5A+\left({k}_{-1}+{l}_{-2}\right){l}_2 $$

(A6)

$$ \frac{d{I}_1}{ dt}={\phi}_1R-\left({k}_{-1}+{l}_{-2}\right){l}_1 $$

(A7)

$$ \frac{d{I}_2}{ dt}={\phi}_5A-\left({k}_{-1}+{l}_{-2}\right){I}_2 $$

(A8)

$$ \frac{ dw}{ dt}=\frac{k_c^{-}\left({w}^{\infty}\left[C{a}^{2+}\right]-w\right)}{w^{\infty}\left[C{a}^{2+}\right]} $$

(A9)

$$ {P}_{IPR}=\left(0.1\ast O+0.9\ast A\right),\kern0.5em \mathrm{IPR}\kern0.5em \mathrm{open}\kern0.5em \mathrm{probability} $$

(A10)

$$ {J}_{IPR}={k}_{IPR}{P}_{IPR}\left({\left[C{a}^{2+}\right]}_{ER}-\left[C{a}^{2+}\right]\right) $$

(A11)

$$ {P}_{RyR}=\frac{w\left(1+{\left(\left[C{a}^{2+}\right]/{K}_b\right)}^3\right)}{\left(1+{\left(\left[C{a}^{2+}\right]\right)}^4+{\left(\left[C{a}^{2+}\right]/{K}_b\right)}^3\right)},\kern0.5em \mathrm{RyR}\kern0.5em \mathrm{open}\kern0.5em \mathrm{probability} $$

(A12)

$$ {J}_{RyR}={k}_{RyR}{P}_{RyR}\left({\left[c{a}^{2+}\right]}_{ER}-\left[c{a}^{2+}\right]\right) $$

(A13)

$$ {J}_{PM}={V}_{PM}\frac{{\left[C{a}^{2+}\right]}^2}{K_{PM}^2+{\left[C{a}^{2+}\right]}^2} $$

(A14)

$$ {J}_{SERCA}={V}_{SERCA}\frac{\left[C{a}^{2+}\right]}{K_{SERCA}+\left[C{a}^{2+}\right]}\times \frac{1}{{\left[C{a}^{2+}\right]}_{er}} $$

(A15)

$$ {J}_{MITO}={v}_{MITO}\frac{\left[C{a}^{2+}\right]}{1+{\left(1/\left[C{a}^{2+}\right]\right)}^2} $$

(A16)

$$ {J}_{IN}={\alpha}_1+{\alpha}_2v $$

(A17)

$$ {J}_{ER}=0.002{s}^{-1} $$

(A18)

All ϕ are saturating (non-mass action) binding rates and are function of [Ca²⁺] derived by Sneyd and Dufour.

$$ {\phi}_1\left[C{a}^{2+}\right]=\frac{\left({k}_1{L}_1+{l}_2\right)\left[C{a}^{2+}\right]}{L_1+\left[C{a}^{2+}\right]\left(1+\frac{L_2}{L_3}\right)} $$

(A19)

$$ {\phi}_2\left[C{a}^{2+}\right]=\frac{k_2{L}_3+{l}_4\left[C{a}^{2+}\right]}{L_3+\left[C{a}^{2+}\right]\left(1+\frac{L_3}{L_1}\right)} $$

(A20)

$$ {\phi}_{-2}\left[C{a}^{2+}\right]=\frac{k_{-2}+{l}_{-4}\left[C{a}^{2+}\right]}{\left(1+\frac{\left[C{a}^{2+}\right]}{L_5}\right)} $$

(A21)

$$ {\phi}_3\left[C{a}^{2+}\right]=\frac{k_3{L}_5}{L_5+\left[C{a}^{2+}\right]} $$

(A22)

$$ {\phi}_4\left[C{a}^{2+}\right]=\frac{\left({k}_4{L}_5+{l}_6\right)\left[C{a}^{2+}\right]}{L_5+\left[C{a}^{2+}\right]} $$

(A23)

$$ {\phi}_{-4}\left[C{a}^{2+}\right]=\frac{L_1\left({k}_{-4}+{l}_{-6}\right)}{L_1+\left[C{a}^{2+}\right]} $$

(A24)

$$ {\phi}_5\left[C{a}^{2+}\right]=\frac{\left({k}_1{L}_1+{l}_2\right)\left[C{a}^{2+}\right]}{L_1+\left[C{a}^{2+}\right]} $$

(A25)