A dynamic model of saliva secretion

Abstract

We construct a mathematical model of the parotid acinar cell with the aim of investigating how the distribution of K(+) and Cl(-) channels affects saliva production. Secretion of fluid is initiated by Ca(2+) signals acting on Ca(2+) dependent K(+) and Cl(-) channels. The opening of these channels facilitates the movement of Cl(-) ions into the lumen which water follows by osmosis. We use recent results into both the release of Ca(2+) from internal stores via the inositol (1,4,5)-trisphosphate receptor (IP(3)R) and IP(3) dynamics to create a physiologically realistic Ca(2+) model which is able to recreate important experimentally observed behaviours seen in parotid acinar cells. We formulate an equivalent electrical circuit diagram for the movement of ions responsible for water flow which enables us to calculate and include distinct apical and basal membrane potentials to the model. We show that maximum saliva production occurs when a small amount of K(+) conductance is located at the apical membrane, with the majority in the basal membrane. The maximum fluid output is found to coincide with a minimum in the apical membrane potential. The traditional model whereby all Cl(-) channels are located in the apical membrane is shown to be the most efficient Cl(-) channel distribution.

Figure 1

Mechanisms underlying fluid flow. The basal membrane separates the cytoplasm from the interstitium, with the apical membrane at the other pole separating the cytoplasm from the lumen. Cl⁻ moves into the lumen through the apical membrane and water follows by osmosis. Paracellular movement of cations through the tight junctions balances the movement of negative Cl⁻ ions into lumen. The model allows for the possibility of apical as well as basal K⁺ channels. We also allow for possible Cl⁻ efflux through the basal membrane.

Figure 2

Schematic of the calcium model. Ca²⁺ fluxes are shown with solid black arrows. IP₃ and Ca²⁺ feedback on the IP₃R and IP₃ degradation are shown with dashed arrows. Agonist concentration stimulates IP₃ production via PLC. IP₃ then degrades by dephosphorylation at rate k_5p and phosphorylation at a Ca²⁺-dependent rate k_3K (grey arrows). Increases in Ca²⁺ and IP₃ concentration raise the open probability of the IP₃R, releasing Ca²⁺ from the ER.

Figure 3

Four state model of the IP₃ receptor

Figure 4

Circuit model of two membranes in series

Figure 5

Ca²⁺ traces for three agonist concentrations with the model result in (a) and the experimental trace (b) reproduced from original figure in Gin et al. (2007) with permission from the authors. Parameter ν is zero except when indicated by a horizontal bar with its non-zero value written above. In the experimental trace, F340/F380 represents the fluorescence of fura-2 and gives a measure of Ca²⁺ concentration.

Figure 6

Effect of blocking the Ryanodine receptor in the model (a) by setting k_RyR = 0 and (b) experimentally by applying a large (500 µM) Ryanodine concentration. Ca²⁺ oscillations are initiated in the model by setting ν = 5100 nM/s, and experimentally by adding 300 nM carbachol at the times shown with the horizontal bars. With the exception of the period when k_RyR = 0 all the model parameters are those found in Table 3. Experimental ‘ratio unit’ represents the fluorescence of fura-2 and gives a measure for Ca²⁺ concentration. Experimental trace reproduced from Bruce et al. (2002) with permission from the authors.

Figure 7

The effect of stopping Ca²⁺ entry from the interstitium which is seen to damp oscillations, experimental result shown in (b), experiment reproduced in the model by setting J_in = 0, seen in (a). Ca²⁺ oscillations are initiated in the model by setting ν = 5100 nM/s, and experimentally by adding 100 nM carbachol at the times shown with the horizontal bars. Experimental ‘ratio unit’ represents the fluorescence of fura-2 and gives a measure for Ca²⁺ concentration. Experimental trace reproduced from Bruce et al. (2002) with permission from the authors.

Figure 8

Model ionic and potential changes with simulated saliva production. Parameter ν = 5100 nM/s for the duration to represent continued stimulation with agonist.

Figure 9

Model volume (red trace) change with Ca²⁺ oscillations (black trace) upon stimulation with agonist. 9b shows the final 50 seconds of 9a and shows clearly the simultaneous Ca²⁺ and volume oscillations.

Figure 10

Dependence of fluid flow and apical membrane potential on location of K⁺ conductance at three different IP₃ productions rates.

Figure 11

Change in normalised water flow and apical membrane potential with tight junctional resistance

Figure G.12

A comparison between a model simulation run with (a) the simplified NKCC and NaK fluxes found in Appendix E, Appendix F and (b) run with more complicated 4-state models found in Appendix C and Appendix D. ν = 5100 nM/s for duration to simulate continuous stimulation with agonist.

Figure H.13

Change in normalised water flow with distribution of Cl⁻ channels, in (a) with varying IP₃ production rates and in (b) with varying K⁺ distributions.