A Mathematical Model of Salivary Gland Duct Cells.

Saliva is produced in two stages in the salivary glands: the secretion of primary saliva by the acinus and the modification of saliva composition to final saliva by the intercalated and striated ducts. In order to understand the saliva modification process, we develop a mathematical model for the salivary gland duct. The model utilises the realistic 3D structure of the duct reconstructed from an image stack of gland tissue. Immunostaining results show that TMEM16A and aquaporin are expressed in the intercalated duct cells and that ENaC is not. Based on this, the model predicts that the intercalated duct does not absorb Na[Formula: see text] and Cl[Formula: see text] like the striated duct but secretes a small amount of water instead. The input to the duct model is the time-dependent primary saliva generated by an acinar cell model. Our duct model produces final saliva output that agrees with the experimental measurements at various stimulation levels. It also shows realistic biological features such as duct cell volume, cellular concentrations and membrane potentials. Simplification of the model by omission of all detailed 3D structures of the duct makes a negligible difference to the final saliva output. This shows that saliva production is not sensitive to structural variation of the duct.

Calcium Dynamics and Water Transport in Salivary Acinar Cells.

Saliva is secreted from the acinar cells of the salivary glands, using mechanisms that are similar to other types of water-transporting epithelial cells. Using a combination of theoretical and experimental techniques, over the past 20 years we have continually developed and modified a quantitative model of saliva secretion, and how it is controlled by the dynamics of intracellular calcium. However, over approximately the past 5 years there have been significant developments both in our understanding of the underlying mechanisms and in the way these mechanisms should best be modelled. Here, we review the traditional understanding of how saliva is secreted, and describe how our work has suggested important modifications to this traditional view. We end with a brief description of the most recent data from living animals and discuss how this is now contributing to yet another iteration of model construction and experimental investigation.

A Multicellular Model of Primary Saliva Secretion in the Parotid Gland.

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 ([Formula: see text])-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.

A Model of [Formula: see text] Dynamics in an Accurate Reconstruction of Parotid Acinar Cells.

We have constructed a spatiotemporal model of [Formula: see text] dynamics in parotid acinar cells, based on new data about the distribution of inositol trisphophate receptors (IPR). The model is solved numerically on a mesh reconstructed from images of a cluster of parotid acinar cells. In contrast to our earlier model (Sneyd et al. in J Theor Biol 419:383-393. https://doi.org/10.1016/j.jtbi.2016.04.030 , 2017b), which cannot generate realistic [Formula: see text] oscillations with the new data on IPR distribution, our new model reproduces the [Formula: see text] dynamics observed in parotid acinar cells. This model is then coupled with a fluid secretion model described in detail in a companion paper: A mathematical model of fluid transport in an accurate reconstruction of a parotid acinar cell (Vera-Sigüenza et al. in Bull Math Biol. https://doi.org/10.1007/s11538-018-0534-z , 2018b). Based on the new measurements of IPR distribution, we show that Class I models (where [Formula: see text] oscillations can occur at constant [[Formula: see text]]) can produce [Formula: see text] oscillations in parotid acinar cells, whereas Class II models (where [[Formula: see text]] needs to oscillate in order to produce [Formula: see text] oscillations) are unlikely to do so. In addition, we demonstrate that coupling fluid flow secretion with the [Formula: see text] signalling model changes the dynamics of the [Formula: see text] oscillations significantly, which indicates that [Formula: see text] dynamics and fluid flow cannot be accurately modelled independently. Further, we determine that an active propagation mechanism based on calcium-induced calcium release channels is needed to propagate the [Formula: see text] wave from the apical region to the basal region of the acinar cell.

A Mathematical Model of Fluid Transport in an Accurate Reconstruction of Parotid Acinar Cells.

Salivary gland acinar cells use the calcium ([Formula: see text]) ion as a signalling messenger to regulate a diverse range of intracellular processes, including the secretion of primary saliva. Although the underlying mechanisms responsible for saliva secretion are reasonably well understood, the precise role played by spatially heterogeneous intracellular [Formula: see text] signalling in these cells remains uncertain. In this study, we use a mathematical model, based on new and unpublished experimental data from parotid acinar cells (measured in excised lobules of mouse parotid gland), to investigate how the structure of the cell and the spatio-temporal properties of [Formula: see text] signalling influence the production of primary saliva. We combine a new [Formula: see text] signalling model [described in detail in a companion paper: Pages et al. in Bull Math Biol 2018, submitted] with an existing secretion model (Vera-Sigüenza et al. in Bull Math Biol 80:255-282, 2018. https://doi.org/10.1007/s11538-017-0370-6 ) and solve the resultant model in an anatomically accurate three-dimensional cell. Our study yields three principal results. Firstly, we show that spatial heterogeneities of [Formula: see text] concentration in either the apical or basal regions of the cell have no significant effect on the rate of primary saliva secretion. Secondly, in agreement with previous work (Palk et al., in J Theor Biol 305:45-53, 2012. https://doi.org/10.1016/j.jtbi.2012.04.009 ) we show that the frequency of [Formula: see text] oscillation has no significant effect on the rate of primary saliva secretion, which is determined almost entirely by the mean (over time) of the apical and basal [Formula: see text]. Thirdly, it is possible to model the rate of primary saliva secretion as a quasi-steady-state function of the cytosolic [Formula: see text] averaged over the entire cell when modelling the flow rate is the only interest, thus ignoring all the dynamic complexity not only of the fluid secretion mechanism but also of the intracellular heterogeneity of [Formula: see text]. Taken together, our results demonstrate that an accurate multiscale model of primary saliva secretion from a single acinar cell can be constructed by ignoring the vast majority of the spatial and temporal complexity of the underlying mechanisms.

New saliva secretion model based on the expression of Na+-K+ pump and K+ channels in the apical membrane of parotid acinar cells.

The plasma membrane of parotid acinar cells is functionally divided into apical and basolateral regions. According to the current model, fluid secretion is driven by transepithelial ion gradient, which facilitates water movement by osmosis into the acinar lumen from the interstitium. The osmotic gradient is created by the apical Cl- efflux and the subsequent paracellular Na+ transport. In this model, the Na+-K+ pump is located exclusively in the basolateral membrane and has essential role in salivary secretion, since the driving force for Cl- transport via basolateral Na+-K+-2Cl- cotransport is generated by the Na+-K+ pump. In addition, the continuous electrochemical gradient for Cl- flow during acinar cell stimulation is maintained by the basolateral K+ efflux. However, using a combination of single-cell electrophysiology and Ca2+-imaging, we demonstrate that photolysis of Ca2+ close to the apical membrane of parotid acinar cells triggered significant K+ current, indicating that a substantial amount of K+ is secreted into the lumen during stimulation. Nevertheless, the K+ content of the primary saliva is relatively low, suggesting that K+ might be reabsorbed through the apical membrane. Therefore, we investigated the localization of Na+-K+ pumps in acinar cells. We show that the pumps appear evenly distributed throughout the whole plasma membrane, including the apical pole of the cell. Based on these results, a new mathematical model of salivary fluid secretion is presented, where the pump reabsorbs K+ from and secretes Na+ to the lumen, which can partially supplement the paracellular Na+ pathway.

Modeling calcium waves in an anatomically accurate three-dimensional parotid acinar cell.

We construct a model of calcium waves in a three-dimensional anatomically accurate parotid acinar cell, constructed from experimental data. Gradients of inositol trisphosphate receptor (IPR) density are imposed, with the IPR density being greater closer to the lumen, which has a branched structure, and inositol trisphosphate (IP3) is produced only at the basal membrane. We show (1) that IP3 equilibrates so quickly across the cell that it can be assumed to be spatially homogeneous; (2) spatial separation of the sites of IP3 action and IP3 production does not preclude the formation of stable oscillatory Ca2+ waves. However, these waves are not waves in the mathematical sense of a traveling wave with fixed profile. They result instead from a time delay between the Ca2+ rise in the apical and basal regions; (3) the ryanodine receptors serve to reinforce the Ca2+ wave, but are not necessary for the wave to exist; (4) a spatially independent model is not sufficient to study saliva secretion, although a one-dimensional model might be sufficient. Our results here form the first stages of the construction of a multiscale and multicellular model of saliva secretion in an entire acinus.

Loss of STIM1 and STIM2 in salivary glands disrupts ANO1 function but does not induce Sjogren's disease.

Sjogren's disease (SjD) is an autoimmune disease characterized by xerostomia (dry mouth), lymphocytic infiltration into salivary glands and the presence of SSA and SSB autoantibodies. Xerostomia is caused by hypofunction of the salivary glands and has been involved in the development of SjD. Saliva production is regulated by parasympathetic input into the glands initiating intracellular Ca 2+ signals that activate the store operated Ca 2+ entry (SOCE) pathway eliciting sustained Ca 2+ influx. SOCE is mediated by the STIM1 and STIM2 proteins and the ORAI1 Ca 2+ channel. However, there are no studies on the effects of lack of STIM1/2 function in salivary acini in animal models and its impact on SjD. Here we report that male and female mice lacking Stim1 and Stim2 ( Stim1/2 K14Cre ) in salivary glands showed reduced intracellular Ca 2+ levels via SOCE in parotid acini and hyposalivate upon pilocarpine stimulation. Bulk RNASeq of the parotid glands of Stim1/2 K14Cre mice showed a decrease in the expression of Stim1/2 but no other Ca 2+ associated genes mediating saliva fluid secretion. SOCE was however functionally required for the activation of the Ca 2+ activated chloride channel ANO1. Despite hyposalivation, ageing Stim1/2 K14Cre mice showed no evidence of lymphocytic infiltration in the glands or elevated levels of SSA or SSB autoantibodies in the serum, which may be linked to the downregulation of the toll-like receptor 8 ( Tlr8 ). By contrast, salivary gland biopsies of SjD patients showed increased STIM1 and TLR8 expression, and induction of SOCE in a salivary gland cell line increased the expression of TLR8 . Our data demonstrate that SOCE is an important activator of ANO1 function and saliva fluid secretion in salivary glands. They also provide a novel link between SOCE and TLR8 signaling which may explain why loss of SOCE does not result in SjD.

A quantitative analysis of electrolyte exchange in the salivary duct.

A healthy salivary gland secretes saliva in two stages. First, acinar cells generate primary saliva, a plasma-like, isotonic fluid high in Na(+) and Cl(-). In the second stage, the ducts exchange Na(+) and Cl(-) for K(+) and HCO(3)(-), producing a hypotonic final saliva with no apparent loss in volume. We have developed a tool that aims to understand how the ducts achieve this electrolyte exchange while maintaining the same volume. This tool is part of a larger multiscale model of the salivary gland and can be used at the duct or gland level to investigate the effects of genetic and chemical alterations. In this study, we construct a radially symmetric mathematical model of the mouse salivary gland duct, representing the lumen, the cell, and the interstitium. For a given flow and primary saliva composition, we predict the potential differences and the luminal and cytosolic concentrations along a duct. Our model accounts well for experimental data obtained in wild-type animals as well as knockouts and chemical inhibitors. Additionally, the luminal membrane potential of the duct cells is predicted to be very depolarized compared with acinar cells. We investigate the effects of an electrogenic vs. electroneutral anion exchanger in the luminal membrane on concentration and the potential difference across the luminal membrane as well as how impairing the cystic fibrosis transmembrane conductance regulator channel affects other ion transporting mechanisms. Our model suggests the electrogenicity of the anion exchanger has little effect in the submandibular duct.

Modelling the effects of calcium waves and oscillations on saliva secretion.

An understanding of Ca(2+) signalling in saliva-secreting acinar cells is important, as Ca(2+) is the second messenger linking stimulation of cells to production of saliva. Ca(2+) signals affect secretion via the ion channels located both apically and basolaterally in the cell. By approximating Ca(2+) waves with periodic functions on the apical and basolateral membranes, we isolate individual wave properties and investigate them for their effect on fluid secretion in a mathematical model of the acinar cell. Mean Ca(2+) concentration is found to be the most significant property in signalling secretion. Wave speed was found to encode a range of secretion rates. Ca(2+) oscillation frequency and amplitude had little effect on fluid secretion.

Highly localized intracellular Ca2+ signals promote optimal salivary gland fluid secretion.

Salivary fluid secretion involves an intricate choreography of membrane transporters to result in the trans-epithelial movement of NaCl and water into the acinus lumen. Current models are largely based on experimental observations in enzymatically isolated cells where the Ca2+ signal invariably propagates globally and thus appears ideally suited to activate spatially separated Cl and K channels, present on the apical and basolateral plasma membrane, respectively. We monitored Ca2+ signals and salivary secretion in live mice expressing GCamp6F, following stimulation of the nerves innervating the submandibular gland. Consistent with in vitro studies, Ca2+ signals were initiated in the apical endoplasmic reticulum. In marked contrast to in vitro data, highly localized trains of Ca2+ transients that failed to fully propagate from the apical region were observed. Following stimuli optimum for secretion, large apical-basal gradients were elicited. A new mathematical model, incorporating these data was constructed to probe how salivary secretion can be optimally stimulated by apical Ca2+ signals.

A dynamic model of saliva secretion.

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.

Fluoride exposure alters Ca2+ signaling and mitochondrial function in enamel cells.

Fluoride ions are highly reactive, and their incorporation in forming dental enamel at low concentrations promotes mineralization. In contrast, excessive fluoride intake causes dental fluorosis, visually recognizable enamel defects that can increase the risk of caries. To investigate the molecular bases of dental fluorosis, we analyzed the effects of fluoride exposure in enamel cells to assess its impact on Ca2+ signaling. Primary enamel cells and an enamel cell line (LS8) exposed to fluoride showed decreased internal Ca2+ stores and store-operated Ca2+ entry (SOCE). RNA-sequencing analysis revealed changes in gene expression suggestive of endoplasmic reticulum (ER) stress in fluoride-treated LS8 cells. Fluoride exposure did not alter Ca2+ homeostasis or increase the expression of ER stress-associated genes in HEK-293 cells. In enamel cells, fluoride exposure affected the functioning of the ER-localized Ca2+ channel IP3R and the activity of the sarco-endoplasmic reticulum Ca2+-ATPase (SERCA) pump during Ca2+ refilling of the ER. Fluoride negatively affected mitochondrial respiration, elicited mitochondrial membrane depolarization, and disrupted mitochondrial morphology. Together, these data provide a potential mechanism underlying dental fluorosis.

Evidence That Calcium Entry Into Calcium-Transporting Dental Enamel Cells Is Regulated by Cholecystokinin, Acetylcholine and ATP.

Dental enamel is formed by specialized epithelial cells which handle large quantities of Ca2+ while producing the most highly mineralized tissue. However, the mechanisms used by enamel cells to handle bulk Ca2+ safely remain unclear. Our previous work contradicted the dogma that Ca2+ is ferried through the cytosol of Ca2+-transporting cells and instead suggested an organelle-based route across enamel cells. This new paradigm involves endoplasmic reticulum (ER)-associated Ca2+ stores and their concomitant refilling by store-operated Ca2+ entry (SOCE) mediated by Ca2+ release activated Ca2+ (CRAC) channels. Given that Ca2+ handling is maximal during the enamel-mineralization stage (maturation), we anticipated that SOCE would also be elevated then. Confirmation was obtained here using single-cell recordings of cytosolic Ca2+ concentration ([Ca2+]cyt) in rat ameloblasts. A candidate SOCE agonist, cholecystokinin (CCK), was found to be upregulated during maturation, with Cck transcript abundance reaching 30% of that in brain. CCK-receptor transcripts were also detected and Ca2+ imaging showed that CCK stimulation increased [Ca2+]cyt in a dose-responsive manner that was sensitive to CRAC-channel inhibitors. Similar effects were observed with two other SOCE activators, acetylcholine and ATP, whose receptors were also found in enamel cells. These results provide the first evidence of a potential regulatory system for SOCE in enamel cells and so strengthen the Ca2+ transcytosis paradigm for ER-based transport of bulk Ca2+. Our findings also implicate enamel cells as a new physiological target of CCK and raise the possibility of an auto/paracrine system for regulating Ca2+ transport.

Multiscale modelling of saliva secretion.

We review a multiscale model of saliva secretion, describing in brief how the model is constructed and what we have so far learned from it. The model begins at the level of inositol trisphosphate receptors (IPR), and proceeds through the cellular level (with a model of acinar cell calcium dynamics) to the multicellular level (with a model of the acinus), finally to a model of a saliva production unit that includes an acinus and associated duct. The model at the level of the entire salivary gland is not yet completed. Particular results from the model so far include (i) the importance of modal behaviour of IPR, (ii) the relative unimportance of Ca(2+) oscillation frequency as a controller of saliva secretion, (iii) the need for the periodic Ca(2+) waves to be as fast as possible in order to maximise water transport, (iv) the presence of functional K(+) channels in the apical membrane increases saliva secretion, (v) the relative unimportance of acinar spatial structure for isotonic water transport, (vi) the prediction that duct cells are highly depolarised, (vii) the prediction that the secondary saliva takes at least 1mm (from the acinus) to reach ionic equilibrium. We end with a brief discussion of future directions for the model, both in construction and in the study of scientific questions.

A mathematical model of fluid secretion from a parotid acinar cell.

Salivary fluid secretion is crucial for preventing problems such as dryness of mouth, difficulty with mastication and swallowing, as well as oral pain and dental cavities. Fluid flow is driven primarily by the transepithelial movement of chloride and sodium ions into the parotid acinus lumen. The activation of Cl(-) channels is calcium dependent, with the average elevated calcium concentration during calcium oscillations increasing the conductance of the channels, leading to an outflow of Cl(-). The accumulation of NaCl in the lumen drives water flow by osmosis. We construct a mathematical model of the calcium concentration oscillations and couple this to a model for Cl(-) efflux. We also construct a model governing fluid flow in an isolated parotid acinar cell, which includes a description of the rate of change of intracellular ion concentrations, cell volume, membrane potential and water flow rate. We find that [Ca(2+)] oscillations lead to oscillations in fluid flow, and that the rate of fluid flow is regulated by the average calcium concentration and not the frequency of the oscillations.

Akt kinase phosphorylation of inositol 1,4,5-trisphosphate receptors.

A consensus RXRXX(S/T) substrate motif for Akt kinase is conserved in the C-terminal tail of all three inositol 1,4,5-trisphosphate (IP3) receptor (IP3R) isoforms. We have shown that IP3R can be phosphorylated by Akt kinase in vitro and in vivo. Endogenous IP3Rs in Chinese hamster ovary T-cells were phosphorylated in response to Akt activation by insulin. LnCAP cells, a prostate cancer cell line with constitutively active Akt kinase, also showed a constitutive phosphorylation of endogenous type I IP3Rs. In all cases, the IP3R phosphorylation was diminished by the addition of LY294002, an inhibitor of phosphatidylinositol 3-kinase. Mutation of IP3R serine 2681 in the Akt substrate motif to alanine (S2681A) or glutamate (S2681E) prevented IP3R phosphorylation in COS cells transfected with constitutively active Akt kinase. Analysis of the Ca2+ flux properties of these IP3R mutants expressed in COS cell microsomes or in DT40 triple knock-out (TKO) cells did not reveal any modification of channel function. However, staurosporine-induced caspase-3 activation in DT40 TKO cells stably expressing the S2681A mutant was markedly enhanced when compared with wild-type or S2681E IP3Rs. We conclude that IP3 receptors are in vivo substrates for Akt kinase and that phosphorylation of the IP3R may provide one mechanism to restrain the apoptotic effects of calcium.