A Mathematical Model Supports a Key Role for Ae4 (Slc4a9) in Salivary Gland Secretion

Abstract

We develop a mathematical model of a salivary gland acinar cell with the objective of investigating the role of two $\mathrm{Cl}^-/\mathrm{HCO}_3^-$ exchangers from the solute carrier family 4 (Slc4), Ae2 (Slc4a2) and Ae4 (Slc4a9), in fluid secretion. Water transport in this type of cell is predominantly driven by $\mathrm{Cl}^-$ movement. Here, a basolateral $\mathrm{Na}^+/ \mathrm{K}^+$ adenosine triphosphatase pump (NaK-ATPase) and a $\mathrm{Na}^+$–$\mathrm{K}^+$–$2 \mathrm{Cl}^-$ cotransporter (Nkcc1) are primarily responsible for concentrating the intracellular space with $\mathrm{Cl}^-$ well above its equilibrium potential. Gustatory and olfactory stimuli induce the release of $\mathrm{Ca}^{2+}$ ions from the internal stores of acinar cells, which triggers saliva secretion. $\mathrm{Ca}^{2+}$-dependent $\mathrm{Cl}^-$ and $\mathrm{K}^+$ channels promote ion secretion into the luminal space, thus creating an osmotic gradient that promotes water movement in the secretory direction. The current model for saliva secretion proposes that $\mathrm{Cl}^-/ \mathrm{HCO}_3^-$ anion exchangers (Ae), coupled with a basolateral $\mathrm{Na}^+/\hbox {proton}$ ($\hbox {H}^+$) (Nhe1) antiporter, regulate intracellular pH and act as a secondary $\mathrm{Cl}^-$ uptake mechanism (Nauntofte in Am J Physiol Gastrointest Liver Physiol 263(6):G823–G837, 1992; Melvin et al. in Annu Rev Physiol 67:445–469, 2005. https://doi.org/10.1146/annurev.physiol.67.041703.084745). Recent studies demonstrated that Ae4 deficient mice exhibit an approximate $30\%$ decrease in gland salivation (Peña-Münzenmayer et al. in J Biol Chem 290(17):10677–10688, 2015). Surprisingly, the same study revealed that absence of Ae2 does not impair salivation, as previously suggested. These results seem to indicate that the Ae4 may be responsible for the majority of the secondary $\mathrm{Cl}^-$ uptake and thus a key mechanism for saliva secretion. Here, by using ‘in-silico’ Ae2 and Ae4 knockout simulations, we produced mathematical support for such controversial findings. Our results suggest that the exchanger’s cotransport of monovalent cations is likely to be important in establishing the osmotic gradient necessary for optimal transepithelial fluid movement.

Appendices

Appendix

$\hbox {Na}^+$–$\hbox {K}^+$–$2\hbox {Cl}^-$ Cotransporter (Nkcc1)

The Nkcc1 cotransporter expression is ubiquitous in nearly all cells and secretory epithelia (Wang et al. 2003). The Nkcc1-mediated-$\hbox {Cl}^-$ uptake mechanism is a secondary active transport, i.e. the energy required for its activity comes indirectly from ATP hydrolysis. The Nkcc1 model we use was first constructed by Benjamin and Johnson (1997). The model assumes equilibrium ion binding, binding symmetry and identity of $\hbox {Cl}^-$ binding sites. This results in a 10 state model. Palk et al. (2010) and Gin et al. (2007) simplified it to a two-state model that assumes simultaneous binding and unbinding of $\hbox {Cl}^-$, $\hbox {K}^+$ and $\hbox {Na}^+$. The reaction occurs as follows:

$$\begin{aligned} O + \text {K}^+_e \text {Na}^+_e (\text {Cl}_e)^2 \underset{k_1^-}{{\mathop {\rightleftharpoons }\limits ^{k_1^+}}} I \underset{k_2^-}{{\mathop {\rightleftharpoons }\limits ^{k_2^+}}} O + \text {K}^+_i \text {Na}^+_i (\text {Cl}_i^-)^2. \end{aligned}$$

The steady-state flux is given by

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

(17)

where $\alpha _{\mathrm{Nkcc1}}$ is the density of the cotransporter.

Nkcc1	Description	Value	Units
$\alpha _{\mathrm{Nkcc1}}^{\tiny {\dagger }}$	Membrane density	2.15	$\hbox {amol}/\upmu \hbox {m}^3$
$a_1$	Rate	157.5	$\hbox {s}^{-1}$
$a_2$	Rate	$2.0096\times 10^7$	$\hbox {mM}^{-4}$ $\hbox {s}^{-1}$
$a_3$	Rate	1.0306	$\hbox {s}^{-1}$
$a_4$	Rate	$1.3852\times 10^6$	$\hbox {mM}^{-4}\ \hbox {s}^{-1}$

1. $^{\dagger }$Parameter value determined from the model
2. Parameter values taken from Palk et al. (2010)

$\hbox {Na}^+/\hbox {K}^+$ ATPase Pump (NaK)

The $\hbox {Na}^+/\hbox {K}^+$ ATPase pump extrudes 3 $\hbox {Na}^+$ ions while introducing 2 $\hbox {K}^+$ ions against their electrochemical gradients at the expense of hydrolysing an ATP molecule per cycle. The net reaction for the pump cycle is

$$\begin{aligned} \text {ATP} + 3 \text {Na}_i^+ + 2 \text {K}_e^+ \leftrightarrow \text {ADP} + \text {P}_i + 3\text {Na}_e^+ +\text {K}_i^+, \end{aligned}$$

where $\text {P}_i$ represents the intracellular concentration of phosphate (a result of ATP conversion to ADP). Smith and Crampin (2004) constructed a mathematical model of the NaK-ATPase pump intended to be used as a component in whole-cell myocyte modelling with the objective to predict pump function and whole myocyte behaviour when cellular metabolism is compromised. Palk et al. (2010) reduced the model to two states:

$$\begin{aligned} \text {O} + 2\text {K}_e \underset{k_1^-}{{\mathop {\rightleftharpoons }\limits ^{k_1^+}}} \text {I}+ 3\text {Na}_e,\\ \text {I} + 3\text {Na}_i \underset{k_2^-}{{\mathop {\rightleftharpoons }\limits ^{k_2^+}}} \text {O} + 2\text {K}_i. \end{aligned}$$

where I refers to an ‘Inside’ state and O to an ‘Outside’ state. The simplification assumes that external $\hbox {Na}^+$ and internal $\hbox {K}^+$ ions simultaneously bind and unbind and are supplied at a constant rate. In addition, the forward reaction rates are higher than the reverse and that the steady-state flux through the pump is given by

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

(18)

where $\alpha _{\mathrm{NaK}}$ is the density of the pump.

NaK-ATPase	Description	Value	Units
$\alpha _{\mathrm{NaK}}^{\dagger }$	Membrane density	4.84	$\hbox {amol}/\upmu \hbox {m}^3$
r	Rate	$1.305\times 10^6$	$\hbox {mM}^{-3}\ \hbox {s}^{-1}$
$\alpha _1$	Half saturation	0.641	$\hbox {mM}^{-1}$

1. $^{\dagger }$Parameter value determined from the model
2. Parameter values taken from Palk et al. (2010)

$\hbox {Ca}^{2+}$-Activated-$\hbox {K}^+$ Channel (CaKC)

Takahata et al. (2003) characterised the biophysical and pharmacological properties of native TEA-insensitive $\hbox {Ca}^{2+}$ activated currents in bovine parotid acinar cells. In their study, they developed a mathematical model that describes the basolateral CaKC current. The open probability of the channel is given by

$$\begin{aligned} P_{\mathrm{CaKC}}=\left( \frac{[\text {Ca}^{2+}]_i}{[\text {Ca}^{2+}]_i+K_{\mathrm{CaKC}}}\right) ^{\eta _2}, \end{aligned}$$

(19)

where $\eta _2=2.54$ is the Hill coefficient and $K_{\mathrm{CaKC}}$ the dissociation constant (a function of the potential difference across the membrane). We used the value found by Palk et al. (2010) of $0.182\ \upmu \hbox {M}$, as we require a small open probability at steady-state $\hbox {Ca}^{2+}$ concentrations. The flux is defined as

$$\begin{aligned} J_{\mathrm{CaKC}}=\frac{G_{\mathrm{CaKC}}P_{\mathrm{CaKC}}}{F}\left( V_b-V_{\mathrm{CaKC}} \right) , \end{aligned}$$

(20)

with Nernst potential

$$\begin{aligned} V_{\mathrm{CaKC}}=\frac{RT}{F z^\text {K}}\ln \left( \frac{[\text {K}^+]_e}{[\text {K}^+]_i} \right) , \end{aligned}$$

(21)

where $z^\text {K}=+1$, the ion’s valence.

$\hbox {Ca}^{2+}$-Activated-$\hbox {Cl}^-$ Channel (CaCC)

Frizzell and Hanrahan (2012) demonstrated that the apical CaCC channels are activated at low $\hbox {Ca}^{2+}$ concentrations by membrane depolarisation and when $\hbox {Ca}^{2+}$ reaches micromolar concentrations. As a simplification, we used a model similar to that of Takahata et al. (2003). Previous mathematical models for the acinar cell have used the model of Arreola et al. (2002). We found that there is no qualitative difference between the models. The CaCC model predicts a large maximum single channel conductance. With a Hill coefficient of $\eta _1=1.46$ and a dissociation constant of $K_{\mathrm{CaCC}}=0.26 \,\upmu \hbox {M}$. The open probability is given by

$$\begin{aligned} P_{\mathrm{CaCC}}=\left( \frac{[\text {Ca}^{2+}]_i}{[\text {Ca}^{2+}]_i+K_{\mathrm{CaCC}}}\right) ^{\eta _1}. \end{aligned}$$

(22)

In this way, the flux is defined as

$$\begin{aligned} J_{\mathrm{CaCC}}=\frac{G_{\mathrm{CaCC}}P_{\mathrm{CaCC}}}{F} \left( V_a-V_{\mathrm{CaCC}} \right) , \end{aligned}$$

(23)

with Nernst potential

$$\begin{aligned} V_{\mathrm{CaCC}}=\frac{RT}{z^{\text {Cl}}F}\ln \left( \frac{[\text {Cl}^-]_l}{[\text {Cl}^-]_i}\right) , \end{aligned}$$

(24)

where $z^{Cl}=-1$, the ion’s valence.

SLC4A2 Anion Exchanger (Ae2)

We use a model created by Falkenberg and Jakobsson (2010). It relies on a concentration gradient, i.e. it uses the $\hbox {HCO}_3^-$ gradient to pump $\hbox {Cl}^-$ into the cell. The energy required for this exchange is derived from the NaK-ATPase activity (Roussa et al. 2001). Its flux depends on the ionic concentrations in the cytoplasm and interstitium, the number of binding sites and the half saturation constants. The model evaluates the product of two terms and the exchanger’s conductance which is proportional to the density of active membrane proteins. The contribution from the concentrations and binding site properties represented by the Michaelis–Menten terms. Additionally as a simplification, we include the difference of the transporter in reverse which renders the exchanger bidirectional.

$$\begin{aligned} J_{\mathrm{Ae2}}&=G_{\mathrm{Ae2}}\Bigg [ \left( \frac{[\text {Cl}^-]_e}{[\text {Cl}^-]_e+K_{\mathrm{Cl}}}\right) \left( \frac{[\text {HCO}_3^-]_i}{[\text {HCO}_3^-]_i+K_\mathrm{B}}\right) \nonumber \\&\quad -\left( \frac{[\text {Cl}^-]_i}{[\text {Cl}^-]_i+K_{\mathrm{Cl}}}\right) \left( \frac{[\text {HCO}_3^-]_e}{[\text {HCO}_3^-]_e+K_\mathrm{B}} \right) \Bigg ]. \end{aligned}$$

(25)

Ae2	Description	Value	Units
$G_{\mathrm{Ae2}}^{\tiny {\dagger }}$	Ae2 activity	0.01807	$\text {fmol}/\text {s}$
$K_{\mathrm{Cl}}$	Half saturation	5.6	mM
$K_\mathrm{B}$	Half saturation	$10^{4}$	mM

1. $^{\dagger }$Parameter derived from the model
2. Parameters from Falkenberg and Jakobsson (2010)

$\hbox {Na}^+/\hbox {H}^+$ Exchanger (Nhe1)

Similarly to the Ae2 exchanger, our model for the Nhe1 exchanger is based on Falkenberg and Jakobsson (2010). Its flux is given by

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

(26)

Nhe1	Description	Value	Units
$G_{\mathrm{Nhe1}}^{\tiny {\dagger }}$	Nhe1 activity	0.0305	$\text {fmol}/\text {s}$
$K_\mathrm{H}$	Half saturation	$4.5\times 10^{-4}$	mM
$K_{\mathrm{Na}}$	Half saturation	15	mM

1. $^{\dagger }$Parameter derived from the model
2. Parameters from Falkenberg and Jakobsson (2010)

SLC4A9 Anion Exchanger (Ae4)

The anion exchanger 4 transports interstitial $\hbox {Cl}^-$ into the cell while extruding 2 $\hbox {HCO}_3^-$ ions and $\hbox {Na}^+$-like monovalent cations (per cycle). In our model, the only monovalent cation, other than $\hbox {Na}^+$, is $\hbox {K}^+$. However, it has been suggested that $\hbox {Cs}^+$, $\hbox {Li}^+$ and $\hbox {Rb}^+$ are also extruded through the Ae4 (Peña-Münzenmayer et al. 2016).

Fig. 10

Schematic diagram of the Markov state model of a bidirectional Ae4 exchanger

We modelled the Ae4 exchanger as a Markov state model for a single exchanger (Dupont et al. 2016). We assume 2 conformational states: (1) the exchanger working forwards and the backward reaction with the $\hbox {HCO}_3^-$ binding site exposed in the interior ($A_i$ and $B_i$) and (2) with the $\hbox {HCO}_3^-$ binding site exposed on the exterior ($A_e$ and $B_e$), respectively (Fig. 10). The lower-case notation $\hbox {cl}$, $\hbox {hco}_3^-$, $\hbox {na}^+$ and $\hbox {k}^+$ denotes the concentration of each ion, respectively. As a simplification, we have assumed simultaneous binding and unbinding of ions to the exchanger. The equations that describe the exchanger under this model are

$$\begin{aligned}&\frac{\mathrm{d}A_i}{\mathrm{d}t}=k_{-1} \hbox {cl}_i B_i+ k_{4} A_e - A_i( k_1 \beta _i + k_{-4}), \end{aligned}$$

(27)

$$\begin{aligned}&\frac{\mathrm{d}B_i}{\mathrm{d}t}=k_1 \beta _i A_i + k_{-2} B_e - B_i(k_{-1} \hbox {cl}_i + k_2), \end{aligned}$$

(28)

$$\begin{aligned}&\frac{\mathrm{d}A_e}{\mathrm{d}t}=k_3 \hbox {cl}_i B_i + k_{-4}A_i - A_e (k_{-3} \beta _e + k_{4}), \end{aligned}$$

(29)

$$\begin{aligned}&A_i+A_e+B_i+B_e=1. \end{aligned}$$

(30)

where $\beta _j=(\hbox {hco}_3^-)^2 (\hbox {na}^+ + \hbox {k}^+)$; for $j=i,e$. Eq. (30), is a conservation equation. The steady-state flux is

$$\begin{aligned} J_{Ae4}=k_{4}A_e-k_{-4}A_i=\frac{\beta _i \hbox {cl}_e - K_1K_2K_3K_4 \hbox {cl}_i \beta _e}{\beta _i \gamma _1+\left( \gamma _2+\gamma _3 \beta _i\right) \beta _e+ \left( \gamma _4 + \gamma _5 \beta _e\right) \hbox {cl}_i + \left( \gamma _6 + \gamma _7 \hbox {cl}_i\right) \hbox {cl}_e}, \end{aligned}$$

where $K_i=k_{-i}/k_i$, and $i=1,\ldots ,4$. The values $\gamma _j$, where $j=1,\ldots ,7$, are a condensed form to write the combination of the different parameters that make the equation. After some careful algebraic manipulation, it can be shown that the expression above simplifies to:

$$\begin{aligned} J_{Ae4}=G_{Ae4}\left[ \hbox {k}_+ \hbox {cl}_e (\hbox {hco}_3^-)_i^2 (\hbox {na}^+_i+\hbox {k}^+_i) - \hbox {k}_- \hbox {cl}_i (\hbox {hco}_3^-)_e^2 (\hbox {na}^+_e+\hbox {k}^+_e)\right] , \end{aligned}$$

(31)

where $G_{Ae4}$ (with units of $\text {fmol}$) denotes the density of exchangers. Thus, $J_{Ae4}$ has units of concentration/time. The parameters, $\hbox {k}_+$ and $\hbox {k}_-$, are the association and dissociation rates, respectively. These were found by solving the bicarbonate steady-state equation (Eq. 4).

Ae4	Description	Value	Units
$G_{\mathrm{Ae4}}$	Density of Ae4	0.66	$\hbox {amol}/\upmu \hbox {m}^{3}$
$\hbox {k}_+$	Rate	$1.92\times 10^{-2}$	$\hbox {mM}^{-4}\ \hbox {s}^{-1}$
$\hbox {k}_-$	Rate	$1.3 \times 10^{-5}$	$\hbox {mM}^{-4}\ \hbox {s}^{-1}$

1. Parameter values determined from the model

$\hbox {CO}_2$ Transport and $\hbox {HCO}_3^-$ Buffering

The acinar cell’s membrane is permeable to $\hbox {CO}_2$ which diffuses down its concentration gradient into the cytoplasm where it combines with water to form carbonic acid ($\hbox {H}_2\hbox {CO}_3$). Carbonic anhydrases catalyse the reaction and dissociate the acid quickly into $\hbox {H}^+$ ions and $\hbox {HCO}_3^-$ ions. For simplicity, we assume the reaction occurs as follows:

$$\begin{aligned} \text {CO}_2+\text {H}_2\text {O} \underset{k_{-1}}{{\mathop {\rightleftharpoons }\limits ^{k_{1}}}}\text {H}^+ + \text {HCO}_3^-. \end{aligned}$$

Using the law of mass action, and assuming chemical equilibrium, we have

$$\begin{aligned}&[\text {CO}_2]_l=\frac{k_{-1}}{k_{1}}[\text {HCO}_3^-]_l[\text {H}^+]_l, \\&[\text {CO}_2]_i=\frac{k_{-1}}{k_{1}}[\text {HCO}_3^-]_i[\text {H}^+]_i. \end{aligned}$$

Diffusion across both membranes is modelled as

$$\begin{aligned}&J_{\mathrm{CO}_{2_{a}}}=P_{\mathrm{CO}_2}\left( [\text {CO}_2]_l-[\text {CO}_2]_i \right) ,\\&J_{\mathrm{CO}_{2_{b}}}=P_{\mathrm{CO}_2}\left( [\text {CO}_2]_e-[\text {CO}_2]_i \right) , \\&J_{\mathrm{CO}_2}=P_{\mathrm{CO}_2} \left( 2[\text {CO}_2]_i-[\text {CO}_2]_l-[\text {CO}_2]_e \right) . \end{aligned}$$

$P_{\mathrm{CO}_2}$ is the membrane permeability to $\hbox {CO}_2$ (same for both membranes). From the reaction above, we can derive (using the law of mass action) a term for the production of $\hbox {HCO}_3^-$ and $\hbox {H}^+$ that is proportionally dependent on the influx of $\hbox {CO}_2$,

$$\begin{aligned} J_{\mathrm{Buffer}}=k_1[\text {CO}_2]_i-k_{-1}[\text {H}^+]_i[\text {HCO}_3^-]_i. \end{aligned}$$

$\hbox {HCO}_3^-$ Buffer	Description	Value	Units
$k_{1}$	Rate	11	$\hbox {s}^{-1}$
$k_{-1}$	Rate	$2.6\times 10^4$	$\hbox {s}^{-1}$
$P_{\mathrm{CO}_2}$	Membrane $\hbox {CO}_2$ transport rate	$1.97\times 10^{-13}$	$\hbox {s}^{-1}$

1. Parameter values taken from Sharp et al. (2015)

Electroneutrality

It is a constraint of the model that the interstitium, the cellular media and the acinar lumen must maintain electroneutrality. For instance, in the cellular compartment we keep track of 6 ionic species. However, in reality there are many more. The number of moles of large negatively charged molecules (with valence $z_x \le -1$) that are impermeable to the cellular membrane and thus trapped inside the cell is denoted $x_i$. To find its value, we note

$$\begin{aligned}{}[\text {K}^+]_i+[\text {Na}^+]_i+[\text {H}^+]_i-[\text {Cl}^-]_i -[\text {HCO}_3^-]_i-\frac{x_i}{\omega _i}=0, \end{aligned}$$

(32)

where $\omega _i$ is the volume of the cell. We solve for $x_i$,

$$\begin{aligned} x_i=\omega _i\left( [\text {K}^+]_i+[\text {Na}^+]_i+[\text {H}^+]_i -[\text {Cl}^-]_i-[\text {HCO}_3^-]_i \right) . \end{aligned}$$

(33)

Similarly, in the acinar lumen and the interstitium we must have:

$$\begin{aligned}&[\text {K}^+]_l+[\text {Na}^+]_l +[\text {H}^+]_l-[\text {Cl}^-]_l-[\text {HCO}_3^-]_l=0, \end{aligned}$$

(34)

$$\begin{aligned}&[\text {K}^+]_e+[\text {Na}^+]_e +[\text {H}^+]_e-[\text {Cl}^-]_e-[\text {HCO}_3^-]_e=0. \end{aligned}$$

(35)

Tight Junction

The tight junction currents are given by a linear current–voltage (I–V) relationship:

$$\begin{aligned}&J_\mathrm{K}^t=\frac{g_\mathrm{K}^t}{z^\text {K}F}\left( V_t-V_\mathrm{K}^t \right) , \end{aligned}$$

(36)

$$\begin{aligned}&J_{\mathrm{Na}}^t=\frac{g_{\mathrm{Na}}^t}{z^{\text {Na}}F}\left( V_t-V_{\mathrm{Na}}^t \right) . \end{aligned}$$

(37)

Here $z^{\text {Na}}=+1$ and $z^\text {K}=+1$, correspond to the valence of each ion species, respectively. $V_{Na}^t$ and $V_K^t$ are their respective Nernst potentials:

$$\begin{aligned}&V_{\mathrm{Na}}^t=\frac{RT}{F} \ln \left( \frac{[\text {Na}^+]_l}{[\text {Na}^+]_e}\right) , \\&V_{\mathrm{K}}^t=\frac{RT}{F} \ln \left( \frac{[\text {K}^+]_l}{[\text {K}^+]_e} \right) . \\ \end{aligned}$$

The potential at the tight junction, $V_t$, is given as:

$$\begin{aligned} V_t=V_a-V_b. \end{aligned}$$

Other Parameters of the Model

See Table 2.

Table 2 Parameters of the model