A mathematical model of fluid secretion from a parotid acinar cell

Tissue hydraulics: Physics of lumen formation and interaction

Lumen formation plays an essential role in the morphogenesis of tissues during development. Here we review the physical principles that play a role in the growth and coarsening of lumens. Solute pumping by the cell, hydraulic flows driven by differences of osmotic and hydrostatic pressures, balance of forces between extracellular fluids and cell-generated cytoskeletal forces, and electro-osmotic effects have been implicated in determining the dynamics and steady-state of lumens. We use the framework of linear irreversible thermodynamics to discuss the relevant force, time and length scales involved in these processes. We focus on order of magnitude estimates of physical parameters controlling lumen formation and coarsening.

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.

Modeling of Ca2+ transients initiated by GPCR agonists in mesenchymal stromal cells

The integrative study that included experimentation and mathematical modeling was carried out to analyze dynamic aspects of transient Ca²⁺ signaling induced by brief pulses of GPCR agonists in mesenchymal stromal cells from the human adipose tissue (AD-MSCs). The experimental findings argued for IP₃/Ca²⁺-regulated Ca²⁺ release via IP₃ receptors (IP₃Rs) as a key mechanism mediating agonist-dependent Ca²⁺ transients. The consistent signaling circuit was proposed to formalize coupling of agonist binding to Ca²⁺ mobilization for mathematical modeling. The model properly simulated the basic phenomenology of agonist transduction in AD-MSCs, which mostly produced single Ca²⁺ spikes upon brief stimulation. The spike-like responses were almost invariantly shaped at different agonist doses above a threshold, while response lag markedly decreased with stimulus strength. In AD-MSCs, agonists and IP₃ uncaging elicited similar Ca²⁺ transients but IP₃ pulses released Ca²⁺ without pronounced delay. This suggested that IP₃ production was rate-limiting in agonist transduction. In a subpopulation of AD-MSCs, brief agonist pulses elicited Ca²⁺ bursts crowned by damped oscillations. With properly adjusted parameters of IP₃R inhibition by cytosolic Ca²⁺, the model reproduced such oscillatory Ca²⁺ responses as well. GEM-GECO1 and R-CEPIA1er, the genetically encoded sensors of cytosolic and reticular Ca²⁺, respectively, were co-expressed in HEK-293 cells that also responded to agonists in an "all-or-nothing" manner. The experimentally observed Ca²⁺ signals triggered by ACh in both compartments were properly simulated with the suggested signaling circuit. Thus, the performed modeling of the transduction process provides sufficient theoretical basis for deeper interpretation of experimental findings on agonist-induced Ca²⁺ signaling in AD-MSCs.

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.

Fluid and solute transport across the retinal pigment epithelium: a theoretical model

The retina is composed of two main layers-the neuroretina and the retinal pigment epithelium (RPE)-that are separated by a potential gap termed the sub-retinal space (SRS). Accumulation of fluid in the SRS may result in a retinal detachment. A key function of the RPE is to prevent fluid accumulation in the SRS by actively pumping fluid from this space to the choroid. We have developed a mathematical model of this process that incorporates the transport of seven chemical species: Na⁺, K⁺, Cl^-, HCO3-, H⁺, CO₂ and H₂CO₃. This allows us to estimate solute and water fluxes and to understand the role of the different membrane ion channels. We have performed a global sensitivity analysis using the extended Fourier amplitude sensitivity test to investigate the relative importance of parameters in generating the model outputs. The model predicts that flow across the RPE is driven by an osmotic gradient in the cleft gap between adjacent cells. Moreover, the model estimates how water flux is modified in response to inhibition of membrane ion channels and carbonic anhydrase (CA). It provides a possible explanation for how CA inhibitors, which are used clinically to prevent fluid accumulation in the SRS, may be acting.

Wet-tip versus dry-tip regimes of osmotically driven fluid flow

The secretion of osmolytes into a lumen and thereby caused osmotic water inflow can drive fluid flows in organs without a mechanical pump. Such fluids include saliva, sweat, pancreatic juice and bile. The effects of elevated fluid pressure and the associated mechanical limitations of organ function remain largely unknown since fluid pressure is difficult to measure inside tiny secretory channels in vivo. We consider the pressure profile of the coupled osmolyte-flow problem in a secretory channel with a closed tip and an open outlet. Importantly, the entire lateral boundary acts as a dynamic fluid source, the strength of which self-organizes through feedback from the emergent pressure solution itself. We derive analytical solutions and compare them to numerical simulations of the problem in three-dimensional space. The theoretical results reveal a phase boundary in a four-dimensional parameter space separating the commonly considered regime with steady flow all along the channel, here termed “wet-tip” regime, from a “dry-tip” regime suffering ceased flow downstream from the closed tip. We propose a relation between the predicted phase boundary and the onset of cholestasis, a pathological liver condition with reduced bile outflow. The phase boundary also sets an intrinsic length scale for the channel which could act as a length sensor during organ growth.

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.

Cell volume changes contribute to epithelial morphogenesis in zebrafish Kupffer's vesicle

How epithelial cell behaviors are coordinately regulated to sculpt tissue architecture is a fundamental question in biology. Kupffer’s vesicle (KV), a transient organ with a fluid-filled lumen, provides a simple system to investigate the interplay between intrinsic cellular mechanisms and external forces during epithelial morphogenesis. Using 3-dimensional (3D) analyses of single cells we identify asymmetric cell volume changes along the anteroposterior axis of KV that coincide with asymmetric cell shape changes. Blocking ion flux prevents these cell volume changes and cell shape changes. Vertex simulations suggest cell shape changes do not depend on lumen expansion. Consistent with this prediction, asymmetric changes in KV cell volume and shape occur normally when KV lumen growth fails due to leaky cell adhesions. These results indicate ion flux mediates cell volume changes that contribute to asymmetric cell shape changes in KV, and that these changes in epithelial morphology are separable from lumen-generated forces.

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

We develop a mathematical model of a salivary gland acinar cell with the objective of investigating the role of two [Formula: see text] 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 [Formula: see text] movement. Here, a basolateral [Formula: see text] adenosine triphosphatase pump (NaK-ATPase) and a [Formula: see text]-[Formula: see text]-[Formula: see text] cotransporter (Nkcc1) are primarily responsible for concentrating the intracellular space with [Formula: see text] well above its equilibrium potential. Gustatory and olfactory stimuli induce the release of [Formula: see text] ions from the internal stores of acinar cells, which triggers saliva secretion. [Formula: see text]-dependent [Formula: see text] and [Formula: see text] 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 [Formula: see text] anion exchangers (Ae), coupled with a basolateral [Formula: see text] ([Formula: see text]) (Nhe1) antiporter, regulate intracellular pH and act as a secondary [Formula: see text] 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 [Formula: see text] 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 [Formula: see text] 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.

Computational modeling of epithelial fluid and ion transport in the parotid duct after transfection of human aquaporin-1

Previous studies have shown that localized delivery of the aquaporin-1 (AQP1) gene to the parotid duct can restore saliva flow in minipigs following irradiation-induced salivary hypofunction. The resulting flow rate and electrochemistry of secreted saliva contradicts current understanding of ductal fluid transport. We hypothesized that changes in expression of ion transport proteins have occurred following AQP1 transfection. We use a mathematical model of ion and fluid transport across the parotid duct epithelial cells to predict the expression profile of ion transporters that are consistent with the experimental measurements of saliva composition and secretion rates. Using a baseline set of parameters, the model reproduces the data for the irradiated, non-AQP1-transfected case. We propose three scenarios which may have occurred after transfection, which differ in the location of the AQP1 gene. The first scenario places AQP1 within nonsecretory cells, and requires that epithelial sodium channel (ENaC) expression is greatly reduced (1.3% of baseline), and ductal bicarbonate concentration is increased from 40.6 to 137.0 mM, to drive water secretion into the duct. The second scenario introduces the AQP1 gene into all ductal cells. The final scenario has AQP1 primarily in the proximal duct cells which secrete water under baseline conditions. We find the change in the remaining cells includes a 95.8% reduction in ENaC expression, enabling us to reproduce all experimental ionic concentrations within 9 mM. These findings provide a mechanistic basis for the observations and will guide the further development of gene transfer therapy for salivary hypofunction.

NEW & NOTEWORTHY

Following transfection of aquaporin into the parotid ducts of minipigs with salivary hypofunction, the resulting increase in salivary flow rates contradicts current understanding of ductal fluid transport. We show that the change in saliva electrochemistry and flow rate can be explained by changes in expression of ion transporters in the ductal cell membranes, using a mathematical model replicating a single parotid duct.

A mathematical model of calcium dynamics in HSY cells

Saliva is an essential part of activities such as speaking, masticating and swallowing. Enzymes in salivary fluid protect teeth and gums from infectious diseases, and also initiate the digestion process. Intracellular calcium (Ca2+) plays a critical role in saliva secretion and regulation. Experimental measurements of Ca2+ and inositol trisphosphate (IP3) concentrations in HSY cells, a human salivary duct cell line, show that when the cells are stimulated with adenosine triphosphate (ATP) or carbachol (CCh), they exhibit coupled oscillations with Ca2+ spike peaks preceding IP3 spike peaks. Based on these data, we construct a mathematical model of coupled Ca2+ and IP3 oscillations in HSY cells and perform model simulations of three different experimental settings to forecast Ca2+ responses. The model predicts that when Ca2+ influx from the extracellular space is removed, oscillations gradually slow down until they stop. The model simulation of applying a pulse of IP3 predicts that photolysis of caged IP3 causes a transient increase in the frequency of the Ca2+ oscillations. Lastly, when Ca2+-dependent activation of PLC is inhibited, we see an increase in the oscillation frequency and a decrease in the amplitude. These model predictions are confirmed by experimental data. We conclude that, although concentrations of Ca2+ and IP3 oscillate, Ca2+ oscillations in HSY cells are the result of modulation of the IP3 receptor by intracellular Ca2+, and that the period is modulated by the accompanying IP3 oscillations.

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.

Mitotic cell rounding and epithelial thinning regulate lumen growth and shape

Many organ functions rely on epithelial cavities with particular shapes. Morphogenetic anomalies in these cavities lead to kidney, brain or inner ear diseases. Despite their relevance, the mechanisms regulating lumen dimensions are poorly understood. Here, we perform live imaging of zebrafish inner ear development and quantitatively analyse the dynamics of lumen growth in 3D. Using genetic, chemical and mechanical interferences, we identify two new morphogenetic mechanisms underlying anisotropic lumen growth. The first mechanism involves thinning of the epithelium as the cells change their shape and lose fluids in concert with expansion of the cavity, suggesting an intra-organ fluid redistribution process. In the second mechanism, revealed by laser microsurgery experiments, mitotic rounding cells apicobasally contract the epithelium and mechanically contribute to expansion of the lumen. Since these mechanisms are axis specific, they not only regulate lumen growth but also the shape of the cavity.

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.

Modelling the transition from simple to complex Ca²⁺ oscillations in pancreatic acinar cells

A mathematical model is proposed which systematically investigates complex calcium oscillations in pancreatic acinar cells. This model is based on calcium-induced calcium release via inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR) and includes calcium modulation of inositol (1,4,5) trisphosphate (IP3) levels through feedback regulation of degradation and production. In our model, the apical and the basal regions are separated by a region containing mitochondria, which is capable of restricting Ca2+ responses to the apical region. We were able to reproduce the observed oscillatory patterns, from baseline spikes to sinusoidal oscillations. The model predicts that calcium-dependent production and degradation of IP3 is a key mechanism for complex calcium oscillations in pancreatic acinar cells. A partial bifurcation analysis is performed which explores the dynamic behaviour of the model in both apical and basal regions.

Model discrimination in dynamic molecular systems: application to parotid de-differentiation network

In modern systems biology the modeling of longitudinal data, such as changes in mRNA concentrations, is often of interest. Fully parametric, ordinary differential equations (ODE)-based models are typically developed for the purpose, but their lack of fit in some examples indicates that more flexible Bayesian models may be beneficial, particularly when there are relatively few data points available. However, under such sparse data scenarios it is often difficult to identify the most suitable model. The process of falsifying inappropriate candidate models is called model discrimination. We propose here a formal method of discrimination between competing Bayesian mixture-type longitudinal models that is both sensitive and sufficiently flexible to account for the complex variability of the longitudinal molecular data. The ideas from the field of Bayesian analysis of computer model validation are applied, along with modern Markov Chain Monte Carlo (MCMC) algorithms, in order to derive an appropriate Bayes discriminant rule. We restrict attention to the two-model comparison problem and present the application of the proposed rule to the mRNA data in the de-differentiation network of three mRNA concentrations in mammalian salivary glands as well as to a large synthetic dataset derived from the model used in the recent DREAM6 competition.

What do aquaporin knockout studies tell us about fluid transport in epithelia?

The investigation of near-isosmotic water transport in epithelia goes back over 100 years; however, debates over mechanism and pathway remain. Aquaporin (AQP) knockouts have been used by various research groups to test the hypothesis of an osmotic mechanism as well as to explore the paracellular versus transcellular pathway debate. Nonproportional reductions in the water permeability of a water-transporting epithelial cell (e.g., a reduction of around 80-90 %) compared to the reduction in overall water transport rate in the knockout animal (e.g., a reduction of 50-60 %) are commonly found. This nonproportionality has led to controversy over whether AQP knockout studies support or contradict the osmotic mechanism. Arguments raised for and against an interpretation supporting the osmotic mechanism typically have partially specified, implicit, or incorrect assumptions. We present a simple mathematical model of the osmotic mechanism with clear assumptions and, for models based on this mechanism, establish a baseline prediction of AQP knockout studies. We allow for deviations from isotonic/isosmotic conditions and utilize dimensional analysis to reduce the number of parameters that must be considered independently. This enables a single prediction curve to be used for multiple epithelial systems. We find that a simple, transcellular-only osmotic mechanism sufficiently predicts the results of knockout studies and find criticisms of this mechanism to be overstated. We note, however, that AQP knockout studies do not give sufficient information to definitively rule out an additional paracellular pathway.

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.

Apical Ca2+-activated potassium channels in mouse parotid acinar cells

Ca²⁺ activation of Cl and K channels is a key event underlying stimulated fluid secretion from parotid salivary glands. Cl channels are exclusively present on the apical plasma membrane (PM), whereas the localization of K channels has not been established. Mathematical models have suggested that localization of some K channels to the apical PM is optimum for fluid secretion. A combination of whole cell electrophysiology and temporally resolved digital imaging with local manipulation of intracellular [Ca²⁺] was used to investigate if Ca²⁺-activated K channels are present in the apical PM of parotid acinar cells. Initial experiments established Ca²⁺-buffering conditions that produced brief, localized increases in [Ca²⁺] after focal laser photolysis of caged Ca²⁺. Conditions were used to isolate K⁺ and Cl⁻ conductances. Photolysis at the apical PM resulted in a robust increase in K⁺ and Cl⁻ currents. A localized reduction in [Ca²⁺] at the apical PM after photolysis of Diazo-2, a caged Ca²⁺ chelator, resulted in a decrease in both K⁺ and Cl⁻ currents. The K⁺ currents evoked by apical photolysis were partially blocked by both paxilline and TRAM-34, specific blockers of large-conductance “maxi-K” (BK) and intermediate K (IK), respectively, and almost abolished by incubation with both antagonists. Apical TRAM-34–sensitive K⁺ currents were also observed in BK-null parotid acini. In contrast, when the [Ca²⁺] was increased at the basal or lateral PM, no increase in either K⁺ or Cl⁻ currents was evoked. These data provide strong evidence that K and Cl channels are similarly distributed in the apical PM. Furthermore, both IK and BK channels are present in this domain, and the density of these channels appears higher in the apical versus basolateral PM. Collectively, this study provides support for a model in which fluid secretion is optimized after expression of K channels specifically in the apical PM.

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.

Efficiency of primary saliva secretion: an analysis of parameter dependence in dynamic single-cell and acinus models, with application to aquaporin knockout studies

Secretion from the salivary glands is driven by osmosis following the establishment of osmotic gradients between the lumen, the cell and the interstitium by active ion transport. We consider a dynamic model of osmotically driven primary saliva secretion and use singular perturbation approaches and scaling assumptions to reduce the model. Our analysis shows that isosmotic secretion is the most efficient secretion regime and that this holds for single isolated cells and for multiple cells assembled into an acinus. For typical parameter variations, we rule out any significant synergistic effect on total water secretion of an acinar arrangement of cells about a single shared lumen. Conditions for the attainment of isosmotic secretion are considered, and we derive an expression for how the concentration gradient between the interstitium and the lumen scales with water- and chloride-transport parameters. Aquaporin knockout studies are interpreted in the context of our analysis and further investigated using simulations of transport efficiency with different membrane water permeabilities. We conclude that recent claims that aquaporin knockout studies can be interpreted as evidence against a simple osmotic mechanism are not supported by our work. Many of the results that we obtain are independent of specific transporter details, and our analysis can be easily extended to apply to models that use other proposed ionic mechanisms of saliva secretion.

Systems analysis of salivary gland development and disease

Branching morphogenesis is a crucial developmental process in which vertebrate organs generate extensive epithelial surface area while retaining a compact size. In the vertebrate submandibular salivary gland, branching morphogenesis is crucial for the generation of the large surface area necessary to produce sufficient saliva. However, in many salivary gland diseases, saliva-producing acinar cells are destroyed, resulting in dry mouth and secondary health conditions. Systems-based approaches can provide insights into understanding salivary gland development, function, and disease. The traditional approach to understanding these processes is the identification of molecular signals using reductionist approaches; we review current progress with such methods in understanding salivary gland development. Taking a more global approach, multiple groups are currently profiling the transcriptome, the proteome, and other 'omes' in both developing mouse tissues and in human patient samples. Computational methods have been successful in deciphering large data sets, and mathematical models are starting to make predictions regarding the contribution of molecules to the physical processes of morphogenesis and cellular function. A challenge for the future will be to establish comprehensive, publicly accessible salivary gland databases spanning the full range of genes and proteins; plans are underway to provide these resources to researchers in centralized repositories. The greatest challenge for the future will be to develop realistic models that integrate multiple types of data to both describe and predict embryonic development and disease pathogenesis.

The effect of intracellular calcium oscillations on fluid secretion in airway epithelium

Airway epithelium has been shown to elicit fluid secretion after a rise in intracellular calcium. This rise in intracellular calcium has been shown to display complex oscillations in many species after the binding of particular agonists to extracellular receptors. Fluid secreted by the airway epithelium is used to maintain the depth of the periciliary liquid (PCL) above the apical membrane of the epithelial cells lining the bronchial airways. Previous mathematical models have been published which separately consider the electrophysiology involved in regulating periciliary liquid depth, and the transmission of intracellular calcium waves in airway epithelial tissue. In this paper we present a mathematical model that combines these previous models and allows the effect of oscillations in intracellular calcium on fluid secretion by airway epithelial cells to be investigated. We show that an oscillatory calcium response produces different fluid secretion properties to that elicited by a tonic rise in intracellular calcium. These differences are shown to be due to saturation of the Ca(2+) activated ion channels.

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.

Mathematical modelling of calcium wave propagation in mammalian airway epithelium: evidence for regenerative ATP release

Airway epithelium has been shown to exhibit intracellular calcium waves after mechanical stimulation. Two classes of mechanism have been proposed to explain calcium wave propagation: diffusion through gap junctions of the intracellular messenger inositol 1,4,5-trisphosphate (IP3), and diffusion of paracrine extracellular messengers such as ATP. We have used single cell recordings of airway epithelium to parameterize a model of an airway epithelial cell. This was then incorporated into a spatial model of a cell culture where both mechanisms for calcium wave propagation are possible. It is shown that a decreasing return on the radius of Ca2+ wave propagation is achieved as the amount of ATP released from the stimulated cell increases. It is therefore shown that for a Ca2+ wave to propagate large distances, a significant fraction of the intracellular ATP pool would be required to be released. Further to this, the radial distribution of maximal calcium response from the stimulated cell does not produce the same flat profile of maximal calcium response seen in experiential studies. This suggests that an additional mechanism is important in Ca2+ wave propagation, such as regenerative release of ATP from cells downstream of the stimulated cell.

Electrogenic NBCe1 (SLC4A4), but not electroneutral NBCn1 (SLC4A7), cotransporter undergoes cholinergic-stimulated endocytosis in salivary ParC5 cells

Cholinergic agonists are major stimuli for fluid secretion in parotid acinar cells. Saliva bicarbonate is essential for maintaining oral health. Electrogenic and electroneutral Na(+)-HCO(3)(-) cotransporters (NBCe1 and NBCn1) are abundant in parotid glands. We previously reported that angiotensin regulates NBCe1 by endocytosis in Xenopus oocytes. Here, we studied cholinergic regulation of NBCe1 and NBCn1 membrane trafficking by confocal fluorescent microscopy and surface biotinylation in parotid epithelial cells. NBCe1 and NBCn1 colocalized with E-cadherin monoclonal antibody at the basolateral membrane (BLM) in polarized ParC5 cells. Inhibition of constitutive recycling with the carboxylic ionophore monensin or the calmodulin antagonist W-13 caused NBCe1 to accumulate in early endosomes with a parallel loss from the BLM, suggesting that NBCe1 is constitutively endocytosed. Carbachol and PMA likewise caused redistribution of NBCe1 from BLM to early endosomes. The PKC inhibitor, GF-109203X, blocked this redistribution, indicating a role for PKC. In contrast, BLM NBCn1 was not downregulated in parotid acinar cells treated with constitutive recycling inhibitors, cholinergic stimulators, or PMA. We likewise demonstrate striking differences in regulation of membrane trafficking of NBCe1 vs. NBCn1 in resting and stimulated cells. We speculate that endocytosis of NBCe1, which coincides with the transition to a steady-state phase of stimulated fluid secretion, could be a part of acinar cell adjustment to a continuous secretory response. Stable association of NBCn1 at the membrane may facilitate constitutive uptake of HCO(3)(-) across the BLM, thus supporting HCO(3)(-) luminal secretion and/or maintaining acid-base homeostasis in stimulated cells.