Mathematical investigation of IP₃-dependent calcium dynamics in astrocytes

Abstract

We study evoked calcium dynamics in astrocytes, a major cell type in the mammalian brain. Experimental evidence has shown that such dynamics are highly variable between different trials, cells, and cell subcompartments. Here we present a qualitative analysis of a recent mathematical model of astrocyte calcium responses. We show how the major response types are generated in the model as a result of the underlying bifurcation structure. By varying key channel parameters, mimicking blockers used by experimentalists, we manipulate this underlying bifurcation structure and predict how the distributions of responses can change. We find that store-operated calcium channels, plasma membrane bound channels with little activity during calcium transients, have a surprisingly strong effect, underscoring the importance of considering these channels in both experiments and mathematical settings. Variation in the maximum flow in different calcium channels is also shown to determine the range of stable oscillations, as well as set the range of frequencies of the oscillations. Further, by conducting a randomized search through the parameter space and recording the resulting calcium responses, we create a database that can be used by experimentalists to help estimate the underlying channel distribution of their cells.

Appendix

1.1 Mathematical Model

The differential equations driving the model are

$$\begin{array}{@{}rcl@{}} \frac{d c}{d t} &=& J_{\text{IP3R}}(c,c_{ER},p)+J_{\text{ER\_leak}}(c,c_{ER})-J_{\text{SERCA}}(c) \\ &&+\delta[J_{\text{ECS\_add}}(c)-J_{\text{PMCA}}(c) +J_{\text{{SOC}}}(c_{ER})], \end{array} $$

(1)

$$\begin{array}{@{}rcl@{}} \frac{dc_{\text{tot}}}{d t} &=& \delta[J_{\text{ECS\_add}}(c)-J_{\text{PMCA}}(c) +J_{\text{{SOC}}}(c_{ER})], \end{array} $$

(2)

$$\begin{array}{@{}rcl@{}} \frac{d h}{d t} &=& \frac{h_{\infty}(c,p) - h}{\tau_{h}(c,p)}, \end{array} $$

(3)

where we denote the calcium concentration in the ER as c _{E R} = (c _tot − c)γ, and IP₃ concentration as p. The J _i ’s are the fluxes found in Fig. 1. Specifically, we use the Li-Rinzel IP₃ receptor model to capture the calcium dynamics through the IP₃R channel (Li and Rinzel 1996), which is governed by the following equations

$$\begin{array}{@{}rcl@{}} J_{\text{IP3R}}(c,c_{ER},p) &=& v_{\text{IP3R}} m_{\infty}(p)^{3} n_{\infty}(c)^{3} h(c,p)^{3} (c_{ER} - c), \\ \frac{d h}{d t} &=& \frac{h_{\infty}(c,p) - h}{\tau_{h}(c,p)}, \end{array} $$

where

$$m_{\infty}(p) = \frac{p}{p + d_{1}},\;\;\; n_{\infty}(c) = \frac{c}{c + d_{5}}, $$

and

$$\begin{array}{@{}rcl@{}} \tau_{h}(p,c) &=&\frac{1}{a_{2}(Q_{2}(p)+c)}, \;\; h_{\infty}(p,c)= \frac{Q_{2}(p)}{Q_{2}(p)+c}, \\ Q_{2}(p) &=& d_{2} \left( \frac{p + d_{1}}{p+d_{3}}\right). \end{array} $$

The SERCA and PMCA pumps are both model as Hill functions, the forms found in Cao et al. (2014) and Croisier et al. (2013) respectively, and are given by the equations,

$$J_{\text{SERCA}}(c) = v_{\text{SERCA}}\frac{c^{1.75}}{c^{1.75} + k_{\text{SERCA}}^{1.75}}, $$

and

$$J_{\text{PMCA}}(c) = v_{\text{PMCA}}\frac{c^{2}}{c^{2} + k_{\text{PMCA}}^{2}}. $$

Similar to the work in Cao et al. (2014), we model SOC channels as the following reverse Hill function

$$J_{\text{SOC}}(c_{ER}) = v_{\text{SOC}}\frac{k_{\text{SOC}}^{2}}{k_{\text{SOC}}^{2}+c_{ER}^{2}}, $$

since it has been shown that they open when calcium is depleted in the ER (Verkhratsky et al. 2012b). The model also includes an IP₃R-independent leak between the cytosol and the ER with the following equation

$$J_{\text{ER\_leak}}(c,c_{ER}) = v_{\text{ER\_leak}}(c_{ER}-c). $$

Further, we account for additional fluxes across the plasma membrane with the equation

$$J_{\text{ECS\_add}}(c) = v_{\text{in}} - k_{\text{out}}c, $$

where v _in captures the constant leak from the extracellular space, and −k _out c accounts for additional calcium extrusion not explicitly model, such as the sodium-calcium exchanger (Höfer et al. 2002; Ullah et al. 2006; Keener and Sneyd 2009; Verkhratsky et al. 2012a). Lastly, the explicit equation for IP₃ is

$$\begin{array}{@{}rcl@{}} p(t) = \left\{\begin{array}{ll} 0 & t < t^{*}\\ s_{\infty}\cdot(1-e^{-r_{\text{rise}}(t-t^{*})}) & t^{*}\le t < t^{*}+d_{\text{rise}} \\ A\cdot e^{-r_{\text{dec}}\cdot(t-[t^{*}+d_{\text{rise}}])} & t^{*}+d_{\text{rise}} \le t \end{array}\right., \end{array} $$

(4)

where

$$s_{\infty}= \frac{A}{1-e^{-r_{\text{rise}}\cdot d_{\text{rise}}}},\;\; r_{\text{dec}}=-\frac{1}{d_{\text{decay}}}\log\left( \frac{0.005}{A}\right), $$

t ^∗ is the time of stimulus, A is the max amplitude, r _rise and r _dec are the rate of rise and decay respectively, and d _rise and d _decay are the duration of the rising and decaying phase. These parameters allow us the flexibility to explore a large distribution of IP₃ responses easily and effectively.

The complete range of IP₃ parameters, as well as the other parameters mentioned in this section, are included in Table 1. Specific values used in the figures are included in the figure captions and text.