Understanding broad-spike oscillations in a model of intracellular calcium dynamics

TL;DR

An ordinary differential equation model of calcium oscillations in hepatocytes is constructed in an attempt to understand the origin of two distinct types of oscillation observed in experiments, and ideas from geometric singular perturbation theory are used to investigate the origin of broad-spike solutions in this model.

Abstract

Oscillations of free intracellular calcium concentration are thought to be important in the control of a wide variety of physiological phenomena, and there is long-standing interest in understanding these oscillations via the investigation of suitable mathematical models. Many of these models have the feature that different variables or terms in the model evolve on very different time-scales, which often results in the accompanying oscillations being temporally complex. Cloete et al [5] constructed an ordinary differential equation model of calcium oscillations in hepatocytes in an attempt to understand the origin of two distinct types of oscillation observed in experiments: narrow-spike oscillations in which rapid spikes of calcium concentration alternate with relatively long periods of quiescence, and broad-spike oscillations in which there is a fast rise in calcium levels followed by a slower decline then a period of quiescence. These two types of oscillation can be observed in the model if a single system parameter is varied but the mathematical mechanisms underlying the different types of oscillations were not explored in detail in [5]. We use ideas from geometric singular perturbation theory to investigate the origin of broad-spike solutions in this model. We find that the analysis is intractable in the full model, but are able to uncover structure in particular singular limits of a related model that point to the origin of the broad-spike solutions.

Figures (11)

• Figure 1: Experimental results from Cloete1, showing that changes in applied agonist can cause changes in the qualitative behaviour of calcium oscillations in hepatocytes. Both panels show the Fura-2 ratio, which is proportional to the concentration of free Ca$^{2+}$ ions in the cytosol of a cell, as a function of time. Panel (a) shows narrow-spike oscillations (excluding the first oscillation) occurring when 1-10 $\mu$M of the agonist ADP is applied and panel (b) shows broad-spike oscillations induced by application of 1-10 $\mu$M of the agonist UTP.
• Figure 2: Time series for $c$ for the attracting solution found at two values of $p$ in equations (\ref{['eq4']}) with parameter values as in Table \ref{['table:one']}. Panel (a) shows periodic narrow-spike oscillations for $p=0.02 \ \mu$M and panel (b) shows one of a periodic sequence of broad spikes for $p=0.09 \ \mu$M.
• Figure 3: Normalised Hill functions from equations (\ref{['eq6']}), illustrating the relative speeds of the switching transitions and the values of $c$ where the transitions occur.
• Figure 4: Singular geometry for the layer problem \ref{['eq:layer_R1']} in (R1) with the parameter values from Table \ref{['table:two']}. Panel (a) shows the critical manifolds $S_1$ and $S_2$ in red and purple, respectively. $S_2$ is non-hyperbolic, while $S_1$ is composed of three segments: $S_1^{\textup{a}}$ (solid line, attracting equilibria of the layer problem), $S_1^{\textup{r}}$ (dotted line, repelling equilibria of the layer problem), and $S_1^{\textup{s}}$ (dashed line, saddle-type equilibria of the layer problem). The global slow variable $c_t$ can be interpreted as a bifurcation parameter, in which case we find regular fold and subcritical Hopf points of the layer problem on $S_1$ at $c_t \approx 0.23$ and $c_t \approx 0.35$, resp., indicated by black disks and denoted by $p_{\rm f}$ and $p_{\rm h}$, resp. A branch of unstable periodic orbits (blue surface) is born in the Hopf bifurcation and terminates at a homoclinic bifurcation of the saddle equilibrium at $p_{\rm s} \in S_1^s$ for $c_t \approx 0.46$. (b) Phase portrait for $c_t=0.34$. (c) Phase portrait for $c_t=0.40$. (d) Phase portrait for $c_t=0.50$. In panels (b)-(d), red dots mark the location of the two equilibria of the layer problem ($q_f$ (resp. $q_s$) is the equilibrium on the upper (resp. lower) branch of $S_1$), and the grey (resp. purple) curve is the unstable (resp. stable) manifold of $q_s$.
• Figure 5: Reduced flow on $S_1$. The point $p_*$ is the equilibrium of the reduced system \ref{['eq:reduced_R1']} and the direction of flow is indicated by the grey arrows. To aid comparison with Figure \ref{['fig:singgeom']}, the locations of points $p_\mathrm{f}$ and $p_\mathrm{h}$ are shown.

Theorems & Definitions (13)

• Remark 2.1
• Remark 2.2
• Remark 2.3
• Remark 2.4
• Remark 2.5
• Remark 2.6
• Remark 3.1
• Remark 3.2
• Remark 3.3
• Remark 3.4