A Multicellular Model of Primary Saliva Secretion in the Parotid Gland

Abstract

We construct a three-dimensional anatomically accurate multicellular model of a parotid gland acinus to investigate the influence that the topology of its lumen has on primary fluid secretion. Our model consists of seven individual cells, coupled via a common lumen and intercellular signalling. Each cell is equipped with the intracellular calcium ($\mathrm{Ca}^{2+}$)-signalling model developed by Pages et al, Bull Math Biol 81: 1394–1426, 2019. https://doi.org/10.1007/s11538-018-00563-z and the secretion model constructed by Vera-Sigüenza et al., Bull Math Biol 81: 699–721, 2019. https://doi.org/10.1007/s11538-018-0534-z. The work presented here is a continuation of these studies. While previous mathematical research has proven invaluable, to the best of our knowledge, a multicellular modelling approach has never been implemented. Studies have hypothesised the need for a multiscale model to understand the primary secretion process, as acinar cells do not operate on an individual basis. Instead, they form racemous clusters that form intricate water and protein delivery networks that join the acini with the gland’s ducts-questions regarding the extent to which the acinus topology influences the efficiency of primary fluid secretion to persist. We found that (1) The topology of the acinus has almost no effect on fluid secretion. (2) A multicellular spatial model of secretion is not necessary when modelling fluid flow. Although the inclusion of intercellular signalling introduces vastly more complex dynamics, the total secretory rate remains fundamentally unchanged. (3) To obtain an acinus, or better yet a gland flow rate estimate, one can multiply the output of a well-stirred single-cell model by the total number of cells required.

Appendices

Appendix

Fluxes of the Model

i. Sodium/Potassium ATPases (NaK-ATPases–Basolateral)

$$\begin{aligned} j_{\text {NaK}}&= S_b \alpha _{\text {NaK}} \left( r \frac{[\text {K}^+]_e^2 [\text {Na}^+]_i^3}{[\text {K}^+]_e^2+\alpha [\text {Na}^+]_i^3} \right) . \end{aligned}$$

• Membrane surface density: $\alpha _{\text {NaK}} = 1.6$ (amol/$\upmu \hbox {m}^2$)

• NaK-ATPase rate: $r=0.0016$ (mM$^{-3}\hbox { s}^{-1}$)

ii. Sodium–Potassium–Chloride Cotransporters (Nkcc1 - Basolateral)

$$\begin{aligned} j_{\text {Nkcc1}}&=S_b\alpha _{\text {Nkcc1}} \left( \frac{a_1-a_2 [\text {Na}^+]_i [\text {K}^+]_i [\text {Cl}^-]_i^2}{a_3+a_4 [\text {Na}^+]_i [\text {K}^+]_i [\text {Cl}^-]_i^2}\right) . \end{aligned}$$

• Membrane surface density: $\alpha _{\text {Nkcc1}} = 2.15$ (amol/$\upmu$m$^2$)

• Nkcc1 rate: $a_1 = 157.55$ (s$^{-2}$)

• Nkcc1 rate: $a_2 = 2.009 \times 10^{-5}$ (mM$^{-4}$ s$^{-2}$)

• Nkcc1 rate: $a_3 = 1.0306$ (s$^{-1}$)

• Nkcc1 rate: $a_4 = 1.38 \times 10^{-6}$ (mM$^{-4}$ s$^{-1}$)

iii. Sodium/Proton Antiporter (Nhe1–Basolateral)

$$\begin{aligned} j_{\text {Nhe1}}=S_b G_{\text {Nhe1}} \Bigg [ \Bigg (\frac{[\text {H}^+]_i}{[\text {H}^+]_i+K_\text {H}}\Bigg )^2&\Bigg (\frac{[\mathrm{Na}^+]_e}{[\mathrm{Na}^+]_e+K_{\text {Na}}} \Bigg ) \\ -&\Bigg (\frac{[\text {H}^+]_e}{[\text {H}^+]_e+K_\text {H}}\Bigg )^2 \Bigg (\frac{[\mathrm{Na}^+]_i}{[\mathrm{Na}^+]_i+K_{\text {Na}}} \Bigg ) \Bigg ]. \end{aligned}$$

• Membrane surface density: $G_{\text {Nhe1}} = 0.9$ (amol/$\upmu$m$^2$)

• Half maximal $\mathrm{Na}^+$ concentration: K$_{\text {Na}}=15$ (mM)

• Half maximal $\mathrm{H}^+$ concentration: K$_{\text {H}}=4.5 \times 10^{-4}$ (mM)

iv. Sodium-Bicarbonate/Chloride Exchanger (Ae4–Basolateral)

$$\begin{aligned} j_{\text {Ae4}}&= S_b G_{Ae4} \Bigg [ \Bigg (\frac{[\mathrm{Cl}^-]_e}{[\mathrm{Cl}^-]_e + K_{\text {Cl}}}\Bigg ) \Bigg (\frac{[\mathrm{Na}^+]_i}{[\mathrm{Na}^+]_i+ K_{\text {Na}}}\Bigg ) \Bigg (\frac{[\mathrm{HCO}_3^-]_i}{[\mathrm{HCO}_3^-]_i+ K_{\text {B}}}\Bigg )^2 \\&-\, \Bigg (\frac{[\mathrm{Cl}^-]_i}{[\mathrm{Cl}^-]_i + K_{\text {Cl}}}\Bigg ) \Bigg (\frac{[\mathrm{Na}^+]_e}{[\mathrm{Na}^+]_e + K_{\text {Na}}}\Bigg ) \Bigg (\frac{[\mathrm{HCO}_3^-]_e}{[\mathrm{HCO}_3^-]_e + K_{\text {B}}}\Bigg )^2\Bigg ]. \end{aligned}$$

• Membrane surface density: $G_{\text {Ae4}} = 1.3$ (amol/$\upmu \hbox {m}^2$)

• Half maximal $\mathrm{Na}^+$ concentration: $\hbox {K}_{\text {Na}}=15$ (mM)

• Half maximal $\mathrm{H}^+$ concentration: $\hbox {K}_{\text {B}}=1\times 10^{4}$ (mM)

• Half maximal $\mathrm{H}^+$ concentration: $\hbox {K}_{\text {Cl}}=5.6$ (mM)

v. Intracellular Bicarbonate Buffering (CA-IV)

$$\begin{aligned} j_{\text {Buffer}}&= \omega _i \Bigg (k_1[\text {CO}_2]_i-k_{-1}[\text {H}^+]_i[\text {HCO}_3^-]_i\Bigg ). \end{aligned}$$

• Buffering association rate: $k_\mathrm{p}=312$ (s$^{-1}$)

• Buffering dissociation rate: $k_n=0.132$ (s$^{-1}$)

vi. Basolateral Calcium-Activated Potassium Channels (K.Ca1 - BK/IK)

$$\begin{aligned}&j_{K}= \frac{G_{K}}{F} \Bigg [ V_{b} - \frac{R T}{F z^{K}} \ln \Bigg (\frac{[\mathrm{K}^+]_l}{[\mathrm{K}^+]_i} \Bigg ) \Bigg ], \\&G_{K} = \Bigg (\frac{g_{K}}{S_b^{\{c\}}} \Bigg ) \sum _{k=1}^K \int _{\partial \varOmega _{k}^{b\{c\}} } \Bigg (\frac{\tilde{[\mathrm{Ca}^{2+}]_i}_k^{\eta _2}}{K_{KCa}^{\eta _2}+ \tilde{[\mathrm{Ca}^{2+}]_i}_k^{\eta _2}}\Bigg ) \ dS_k. \end{aligned}$$

• Half maximal concentration: $K_{K_{\text {Ca}}}=0.3$ ($\upmu$M)

• Hill coefficient: $\eta _2=4.7$

• $\mathrm{K}^+$ channel conductance: $G_{\text {K}}=14.6$ (nS)

vii. Tight Junctional Ionic Fluxes (Paracellular)

$$\begin{aligned}&j_{t_{\mathrm{Na}}} = \frac{G_{t_{\mathrm{Na}^+}}}{F} \Bigg [V_\mathrm{a} - V_b - \frac{RT}{F} \ln \Bigg ( \frac{[\mathrm{Na}^+]_l}{[\mathrm{Na}^+]_e}\Bigg )\Bigg ],\\&j_{t_{\mathrm{K}}} = \frac{G_{t_{\mathrm{K}^+}}}{F} \Bigg [V_\mathrm{a} - V_b - \frac{RT}{F} \ln \Bigg ( \frac{[\mathrm{K}^+]_l}{[\mathrm{K}^+]_e}\Bigg )\Bigg ]. \end{aligned}$$

• Temperature: $T=310$ (Kelvin)

• Universal Gas Constant: $R=8.314$ (J mol$^{-1}$ K$^{-1}$)

• Faraday’s Constant: $F=6485.332$ (C)

• Tight Junctional $\mathrm{Na}^+$ conductance: $G^t_{\text {Na}}=0.2$ (nS)

• Tight Junctional $\mathrm{K}^+$ conductance: $G^t_{\text {K}}=0.16$ (nS)

viii. Apical Calcium-Activated Chloride Channels (TMeM16a - KCa)

$$\begin{aligned}&j_{Cl}= \frac{G_{Cl}}{F} \Bigg [ V_\mathrm{a} - \frac{R T}{F z^{Cl}} \ln \Bigg (\frac{[\mathrm{Cl}^-]_l}{[\mathrm{Cl}^-]_i}\Bigg ) \Bigg ], \\&G_{Cl} = \Bigg (\frac{g_{Cl}}{S_\mathrm{a}^{\{c,(c,n)\}}} \Bigg ) \sum _{k=1}^K \int _{\partial \varOmega _{k}^{a\{c,(c,n)\}} } \Bigg (\frac{\tilde{[\mathrm{Ca}^{2+}]_i}_k^{\eta }}{K_{CaCl}^{\eta }+ \tilde{[\mathrm{Ca}^{2+}]_i}_k^{\eta }}\Bigg ) \ dS_k. \end{aligned}$$

• Half maximal concentration: $K_{Cl_{\text {Ca}}}=0.2$ ($\upmu$M)

• Hill coefficient: $\eta _1=4.49$

• $\mathrm{K}^+$ channel conductance: $G_{\text {Cl}}=18.2$ (nS)

Model Summary

1.1 Fluid Flow Model

For any cell $c = 1,\ldots ,7$, and its neighbours n,

$$\begin{aligned} \frac{\mathrm{d} \omega _i^{\{c \}}}{\mathrm{d}t} = j^{\{c\}}_{w_b} - \sum _n j^{\{c,(c,n)\}}_{w_\mathrm{a}},\\ \frac{\mathrm{d}\Big ([\mathrm{Na}^+]_i^{\{c\}} \omega ^{\{c\}}_i\Big )}{\mathrm{d}t}&= S_b^{\{c\}} \Bigg ( j^{\{c\}}_{\text {Nkcc1}} - 3 j^{\{c\}}_{\text {NaK}} + j^{\{c\}}_{\text {Nhe1}} - j^{\{c\}}_{\text {Ae4}} \Bigg ),\\ \frac{\mathrm{d}\Big ([\mathrm{K}^+]_i^{\{c\}} \omega ^{\{c\}}_i \Big )}{\mathrm{d}t}&= S_b^{\{c\}} \Bigg ( j^{\{c\}}_{\text {Nkcc1}} + 2 j^{\{c\}}_{\text {NaK}} \Bigg ) - \Bigg (\int _{\partial \varOmega _b^{\{c\}}} j^{\{c\}}_{\mathrm{Cl}} \ \text {d}S \Bigg ),\\ \frac{\mathrm{d} \Big ([\mathrm{Cl}^-]_i^{\{c\}} \omega ^{\{c\}}_i\Big )}{\mathrm{d}t}&= S_b^{\{c\}} \Bigg ( 2j^{\{c\}}_{\text {Nkcc1}} + j^{\{c\}}_{\text {Ae4}} \Bigg ) - \sum _n \Bigg (\int _{\partial \varOmega _\mathrm{a}^{\{c,(c,n)\}}} j^{\{c,(c,n)\}}_{\mathrm{Cl}} \ \text {d}S \Bigg ),\\ \frac{\mathrm{d} \Big ([\mathrm{HCO}_3^-]_i^{\{c\}} \omega ^{\{c\}}_i\Big )}{\mathrm{d}t}&= j^{\{c\}}_{\text {Buffer}} - S_b^{\{c\}} 2j^{\{c\}}_{\text {Ae4}}, \\ \frac{\mathrm{d} \Big ([\text {H}^+]^{\{c\}}_i \omega ^{\{c\}}_i\Big )}{\mathrm{d}t}&= j^{\{c\}}_{\text {Buffer}} - S_b^{\{c\}} j_{\text {Nhe1}}, \\ \frac{C_\mathrm{m}}{F} \frac{\mathrm{d} V^{\{c\}}_\mathrm{a}}{\mathrm{d} t}&= - \sum _n \Bigg (\int _{\partial \varOmega _\mathrm{a}^{\{c,(c,n)\}}} j^{\{c,(c,n)\}}_{\mathrm{Cl}} \ \text {d}S \Bigg ) + \sum _n \Bigg ( j^{t\{c,(c,n)\}}_\mathrm{Na} + j^{t\{c,(c,n)\}}_\mathrm{K} \Bigg ),\\ \frac{C_\mathrm{m}}{F} \frac{\mathrm{d} V^{\{c\}}_b}{\mathrm{d} t}&= - S_b^{\{c\}} j^{\{c\}}_\mathrm{NaK} - \Bigg (\int _{\partial \varOmega _b^{\{c\}}} j^{\{c\}}_{\mathrm{Cl}} \ \text {d}S \Bigg ) + \sum _n \Bigg ( j^{t\{c,(c,n)\}}_\mathrm{Na} + j^{t\{c,(c,n)\}}_\mathrm{K} \Bigg ), \\ \frac{\mathrm{d} \Bigg ([\mathrm{Na}^+]_l^{(c,n)} \omega ^{(c,n)}_l \Bigg )}{\mathrm{d}t}&= \sum _* j^*_w [\mathrm{Na}^+]_l^* + \Bigg (j^{\{c,(c,n)\}}_{t_{Na}} + j^{\{n,(c,n)\}}_{t_{Na}} \Bigg ) \\&\quad - \Bigg ( \sum _* j^*_w + j^{\{c,(c,n)\}}_{w} + j^{\{n,(c,n)\}}_{w} \Bigg ) [\mathrm{Na}^+]_l^{(c,n)},\\ \frac{\mathrm{d} \Bigg ( [\mathrm{K}^+]_l^{(c,n)} \omega ^{(c,n)}_l \Bigg )}{\mathrm{d}t}&= \sum _* j^*_w [\mathrm{K}^+]_l^* +\Bigg ( j^{\{c,(c,n)\}}_{t_K} + j^{\{n,(c,n)\}}_{t_K} \Bigg ) \\&\quad - \Bigg ( \sum _* j^*_w + j^{\{c,(c,n)\}}_{w} + j^{\{n,(c,n)\}}_{w} \Bigg ) [\mathrm{K}^+]_l^{(c,n)},\\ \frac{\mathrm{d} \Bigg ([\mathrm{Cl}^-]_l^{(c,n)} \omega ^{(c,n)}_l\Bigg )}{\mathrm{d}t}&= \sum _* j^*_w [\mathrm{Cl}^-]_l^* + \Bigg (\int _{\partial \varOmega _\mathrm{a}^{\{c,(c,n)\}}} j^{\{c,(c,n)\}}_{\mathrm{Cl}} \ \text {d}S \\&\quad + \int _{\partial \varOmega _\mathrm{a}^{\{n,(c,n)\}}} j^{\{n,(c,n)\}}_{\mathrm{Cl}} \ \text {d}S\Bigg )\\&\quad - \Bigg ( \sum _* j^*_w + j^{\{c,(c,n)\}}_{w} + j^{\{n,(c,n)\}}_{w} \Bigg ) [\mathrm{Cl}^-]_l^{(c,n)}. \end{aligned}$$

Here, the expression: $\sum _* j^*_w [\mathrm ion]^*$ represents the sum of all neighbouring luminal convective water fluxes into the luminal segment shared between cells c and n. Similarly, $\sum _* j^*_w$ represents the sum of all neighbouring luminal water fluxes in and out to the luminal segment shared between cells c and n.

1.2 Calcium Dynamics Model

i. Inositol-1,4,5-Trisphosphate Receptors (InsP$_3$R)

$$\begin{aligned} m(c)&= \frac{[\mathrm{Ca}^{2+}]_i^4}{K_c^4+[\mathrm{Ca}^{2+}]_i^4},\\ 1-A([\mathrm{InsP}_3]_i)&= B([\mathrm{InsP}_3]_i) = \frac{[\mathrm{InsP}_3]_i^2}{K_\mathrm{p}^2 + [\mathrm{InsP}_3]_i^2},\\ h_\infty ([\mathrm{Ca}^{2+}]_i)&= \frac{K_h^4}{K_h^4 + [\mathrm{Ca}^{2+}]_i^4},\\ \tau _h([\mathrm{Ca}^{2+}]_i)&= \tau _{\max } \frac{K_{\tau }^4}{K_{\tau }^4+[\mathrm{Ca}^{2+}]_i^4},\\ \alpha&= A([\mathrm{InsP}_3]_i)(1-m([\mathrm{Ca}^{2+}]_i)h_\infty ([\mathrm{Ca}^{2+}]_i)), \\ \beta&= B([\mathrm{InsP}_3]_i)m([\mathrm{Ca}^{2+}]_i)h([\mathrm{Ca}^{2+}]_i), \\ P_0&= \frac{\beta }{\beta +k_\beta (\beta +\alpha )} \\ J_\mathrm{IPR}&= k_\mathrm{IPR}P_0, \end{aligned}$$

ii. Sarcoplasmic/Endoplasmic Ca2+ ATPase Pumps (SERCA)

$$\begin{aligned}&J_{SERCA}= V_\mathrm{p}\frac{[\mathrm{Ca}^{2+}]_i^2-\bar{K}[\mathrm{Ca}^{2+}]_e^2}{[\mathrm{Ca}^{2+}]_i^2+K_s^2} \end{aligned}$$

iii. Inositol-1,4,5-Trisphosphate (InsP$_3$ Dynamics)

$$\begin{aligned}&V_{\text {PLC}} = \mu (x) \frac{[\mathrm{Ca}^{2+}]_i^2}{[\mathrm{Ca}^{2+}]_i^2 +K_{\text {PLC}}^2},\\&V_\mathrm{deg} = \left( V_{5K}+V_{3K}\frac{[\mathrm{Ca}^{2+}]_i^2}{[\mathrm{Ca}^{2+}]_i^2 +K_{3K}^2}\right) [\mathrm{InsP}_3]_i,\\&\mu (x) = {\left\{ \begin{array}{ll} \begin{array}{rll} k_\mathrm{PLC} &{}\text {if }&{} d_b(x)<d_\mathrm{PLC} \text { and } d_\mathrm{a}(x)>dl_\mathrm{PLC},\\ &{} 0 \qquad \text {else}, \\ \end{array} \end{array}\right. } \end{aligned}$$

iv. Wave Propagation Model

$$\begin{aligned}&{\left\{ \begin{array}{ll} \begin{array}{rcl} &{}J_\mathrm{Wav} = V_\mathrm{Wav}(x)\frac{[\mathrm{Ca}^{2+}]_i^4}{[\mathrm{Ca}^{2+}]_i^4+K_\mathrm{Wav1}^4}g,\\ &{} \frac{\text {d}g}{\text {d}t} = \left( g_{\infty } - g\right) /\tau _\mathrm{Wav},\\ \end{array} \end{array}\right. }\\&g_{\infty } = \frac{K_{\mathrm{hWav}}^2}{K_{\mathrm{hWav}}^2+[\mathrm{Ca}^{2+}]_i^2}, \\&V_\mathrm{Wav}(x) = {\left\{ \begin{array}{ll} \begin{array}{rll} V_\mathrm{Wav}^\mathrm{max} \quad \frac{d_\mathrm{a}(x)}{d_\mathrm{Wav}} &{}\text {if }&{} d_\mathrm{a}(x)<d_\mathrm{Wav},\\ V_\mathrm{Wav}^\mathrm{max} \quad \text {else}. \end{array} \end{array}\right. } \end{aligned}$$

v. Equations of the Ca$^{2+}$ Model

$$\begin{aligned} \frac{\partial \Big ( [\mathrm{Ca}^{2+}]_i\omega ^{\{c\}}_i \Big )}{\partial t}&= D_c \nabla ^2 [\mathrm{Ca}^{2+}]_i^{\{c\}} \ldots \\&\quad + {\omega ^{\{c\}}_{i_0}} \Bigg (J_\mathrm{Wav}([\mathrm{Ca}^{2+}]^{\{c\}}_{ER}-[\mathrm{Ca}^{2+}]_i^{\{c\}}) -J_{SERCA}\Bigg ),\\ \frac{\partial [\mathrm{Ca}^{2+}]^{\{c\}}_{ER}}{\partial t}&= D_c \nabla ^2 [\mathrm{Ca}^{2+}]^{\{c\}}_{ER} \ldots \\&\quad - \frac{1}{\gamma \Big ( \omega ^{\{c\}}_i \Big )}\Bigg (J_\mathrm{Wav}\Big ([\mathrm{Ca}^{2+}]^{\{c\}}_{ER}-[\mathrm{Ca}^{2+}]_i^{\{c\}}\Big )-J_{SERCA}\Bigg ), \\ \frac{\partial \Bigg ([\mathrm{InsP}_3]_i^{\{c\}} \omega ^{\{c\}}_i \Bigg )}{\partial t}&= D_\mathrm{p} \nabla ^2 [\mathrm{InsP}_3]_i^{\{c\}} + {\omega _{i_0}^{\{c\}} } \Bigg (V_\mathrm{PLC}-V_\mathrm{deg}\Bigg ), \\ \tau _h([\mathrm{Ca}^{2+}]_i)\frac{\text {d}h}{\text {d}t}&= h_\infty ([\mathrm{Ca}^{2+}]_i)-h,\\ \displaystyle \frac{\mathrm{d} g}{\mathrm{d} t}&= \displaystyle (g_\infty - g\omega ^{\{c\}}_i)/\tau ,\\ \frac{\mathrm{d} h^{\{c\}}}{\mathrm{d}t}&= \frac{1}{\tau ([\mathrm{Ca}^{2+}]_i)}(h_{\infty } - h), \end{aligned}$$

v. Boundary Conditions of the Ca$^{2+}$ Model

$$\begin{aligned}&\nabla [\mathrm{Ca}^{2+}]_i^{\{c\}} \cdot \nabla \mathbf{n } = J_\mathrm{IPR} ([\mathrm{Ca}^{2+}]_{ER}-[\mathrm{Ca}^{2+}]_i) \\&\nabla [\mathrm{Ca}^{2+}]^{\{c\}}_{ER} \cdot \nabla \mathbf{n } = - \Bigg (\frac{1}{\gamma (\omega _i)}\Bigg )J_\mathrm{IPR}([\mathrm{Ca}^{2+}]_{ER}-[\mathrm{Ca}^{2+}]_i),\\&\nabla [\mathrm{InsP}_3]^{\{c\}}_i \cdot \nabla \mathbf{n } = {\mathrm{F}_\mathrm{Ip}} \Big ( [\mathrm{InsP}_3]_i^{\{n\}} - [\mathrm{InsP}_3]_i^{\{c\}} \Big ). \end{aligned}$$

Parameter	Description	Value	Units
$V_{\mathrm{Wav}}^{\mathrm{max}}$	Maximal rate of the wave propagation model	0.025	$\upmu$M s$^{-1}$
$K_{\mathrm{wav1}}$	Half maximal concentration of calcium for positive feedback on wave propagation	0.2	$\upmu$M
$d_{\mathrm{Wav}}$	Critical distance for the expression of the wave propagation	1	$\upmu$m
$K_{\mathrm{hWav}}$	Half maximal concentration of calcium for negative feedback on wave propagation	0.15	$\upmu$M
$\tau _{\mathrm{wav}}$	Time scaling for negative feedback of calcium on the wave propagation	10	s
$k_\mathrm{IPR}$	IPR rate	30	$\upmu \,\hbox {M}\,\mathrm{m}\,\mathrm{s}^{-1}$
$k_{\beta }$	–	0.4	–
$K_c$	Half maximal concentration of calcium for positive feedback on IPR	0.2	$\upmu$M
$K_p$	Half maximal concentration of $\mathrm{InsP}_3$ for feedback on the IPR	0.2	$\upmu$M
$K_h$	Half maximal concentration of calcium for negative feedback on IPR	0.08	$\upmu$M
$K_{\tau }$	Half maximal concentration of calcium for feedback on the time scaling factor	0.1	$\upmu$M
$\tau _{\max }$	Time scaling for negative feedback of calcium on the IPR	1000	s
$K_\mathrm{PLC}$	Half maximal concentration of $\mathrm{Ca}^{2+}$ for feedback on PLC	0.07	$\upmu$M
$K_{3K}$	Half maximal concentration of $\mathrm{Ca}^{2+}$ for feedback on Insp-3-kinase	0.4	$\upmu$M
$V_{3K}$	$\mathrm{InsP}_3$ metabolisation rate by InsP-3-kinase	0.05	s$^{-1}$
$V_{5K}$	$\mathrm{InsP}_3$ degradation rate by InsP-5-phosphatase	0.05	s$^{-1}$
$\hbox {d}_\mathrm{PLC}$	Critical distance from the basal membrane for the expression of PLC	0.8	$\upmu$m
$\hbox {dl}_\mathrm{PLC}$	Minimum distance from the apical membrane for the expression of PLC	0.6	$\upmu$m
$k_\mathrm{PLC}$	Maximum PLC rate	0.6	$\upmu$M s$^{-1}$
$d_b(x)$	Distance to the basal region of the point x	–	$\upmu$m
$d_\mathrm{a}(x)$	Distance to the basal region of the point x	–	$\upmu$m
$D_c$	Calcium diffusion	5	$\upmu$m$^2$ s$^{-1}$
$D_p$	IP$_3$ diffusion coefficient	285	$\upmu$m$^2$ s$^{-1}$
$\gamma$	Ratio ER volume/cytosolic volume	0.185	–
$K_{K_{\text {Ca}}}$	Half maximal concentration	0.3	$\upmu$M
$\eta _2$	Hill coefficient	4.7	–

Parameter	Description	Value	Units
$G_{\text {K}}$	$\mathrm{Cl}^-$ channel conductance	14.6	nS
$K_{Cl_{\text {Ca}}}$	Half maximal concentration	0.2	$\upmu$M
$\eta$	Hill coefficient	4.49	–
$G_{\text {Cl}}$	$\mathrm{Cl}^-$ channel conductance	18.2	nS
$V_{\mathrm{Wav}}^{\mathrm{max}}$	Wave propagation rate	0.025	$\upmu$M s$^{-1}$
$K_{\mathrm{wav1}}$	Half maximal concentration of calcium	0.2	$\upmu$M
$d_{\mathrm{Wav}}$	Critical distance of wave propagation	1	$\upmu$m
$K_{\mathrm{hWav}}$	Half maximal concentration of calcium propagation	0.15	$\upmu$M
$\tau _{\mathrm{wav}}$	Time scaling	10	s
$S_b^{\{1\}}$	Cell 1 Basolateral Surface Area	285.0	$\upmu \hbox {m}^2$
$S_b^{\{2\}}$	Cell 2 Basolateral Surface Area	238.6	$\upmu \hbox {m}^2$
$S_b^{\{3\}}$	Cell 3 Basolateral Surface Area	311.9	$\upmu \hbox {m}^2$
$S_b^{\{4\}}$	Cell 4 Basolateral Surface Area	284.8	$\upmu \hbox {m}^2$
$S_b^{\{5\}}$	Cell 5 Basolateral Surface Area	144.03	$\upmu \hbox {m}^2$
$S_b^{\{6\}}$	Cell 6 Basolateral Surface Area	304.05	$\upmu \hbox {m}^2$
$S_b^{\{7\}}$	Cell 7 Basolateral Surface Area	324.8	$\upmu \hbox {m}^2$
$S_\mathrm{a}^{\{1\}}$	Cell 1 Apical Surface Area	92.7	$\upmu \hbox {m}^2$
$S_\mathrm{a}^{\{2\}}$	Cell 2 Apical Surface Area	100.4	$\upmu \hbox {m}^2$
$S_\mathrm{a}^{\{3\}}$	Cell 3 Apical Surface Area	93.9	$\upmu \hbox {m}^2$
$S_\mathrm{a}^{\{4\}}$	Cell 4 Apical Surface Area	95.3	$\upmu \hbox {m}^2$
$S_\mathrm{a}^{\{5\}}$	Cell 5 Apical Surface Area	48.9	$\upmu \hbox {m}^2$
$S_\mathrm{a}^{\{6\}}$	Cell 6 Apical Surface Area	73.9	$\upmu \hbox {m}^2$
$S_\mathrm{a}^{\{7\}}$	Cell 7 Apical Surface Area	54.3	$\upmu \hbox {m}^2$
$\omega _i^{\{1\}}$	Cell 1 Resting Volume	1004.8	$\upmu \hbox {m}^3$
$\omega _i^{\{2\}}$	Cell 2 Resting Volume	1006.9	$\upmu \hbox {m}^3$
$\omega _i^{\{3\}}$	Cell 3 Resting Volume	1084.8	$\upmu \hbox {m}^3$
$\omega _i^{\{4\}}$	Cell 4 Resting Volume	1109.3	$\upmu \hbox {m}^3$
$\omega _i^{\{5\}}$	Cell 5 Resting Volume	495.04	$\upmu \hbox {m}^3$
$\omega _i^{\{6\}}$	Cell 6 Resting Volume	1036.2	$\upmu \hbox {m}^3$
$\omega _i^{\{7\}}$	Cell 7 Resting Volume	904.2	$\upmu \hbox {m}^3$