A Model for the Coexistence of Competing Mechanisms for Ca 2 + Oscillations in T-lymphocytes.

Ca 2 + is a ubiquitous signaling mechanism across different cell types. In T-cells, it is associated with cytokine production and immune function. Benson et al. have shown the coexistence of competing Ca 2 + oscillations during antigen stimulation of T-cell receptors, depending on the presence of extracellular Ca 2 + influx through the Ca 2 + release-activated Ca 2 + channel (Benson in J Biol Chem 29:105310, 2023). In this paper, we construct a mathematical model consisting of five ordinary differential equations and analyze the relationship between the competing oscillatory mechanisms.. We perform bifurcation analysis on two versions of our model, corresponding to the two oscillatory types, to find the defining characteristics of these two families.

Non-Markovian processes on heteroclinic networks.

Sets of saddle equilibria connected by trajectories are known as heteroclinic networks. Trajectories near a heteroclinic network typically spend a long period of time near one of the saddles before rapidly transitioning to the neighborhood of a different saddle. The sequence of saddles visited by a trajectory can be considered a stochastic sequence of states. In the presence of small-amplitude noise, this sequence may be either Markovian or non-Markovian, depending on the appearance of a phenomenon called lift-off at one or more saddles of the network. In this paper, we investigate how lift-off occurring at one saddle affects the dynamics near the next saddle visited, how we might determine the order of the associated Markov chain of states, and how we might calculate the transition probabilities of that Markov chain. We first review methods developed by Bakhtin to determine the map describing the dynamics near a linear saddle in the presence of noise and extend the results to include three different initial probability distributions. Using Bakhtin's map, we determine conditions under which the effect of lift-off persists as the trajectory moves past a subsequent saddle. We then propose a method for finding a lower bound for the order of this Markov chain. Many of the theoretical results in this paper are only valid in the limit of small noise, and we numerically investigate how close simulated results get to the theoretical predictions over a range of noise amplitudes and parameter values.

Emergence of broad cytosolic Ca2+ oscillations in the absence of CRAC channels: A model for CRAC-mediated negative feedback on PLC and Ca2+ oscillations through PKC.

The role of Ca2+ release-activated Ca2+ (CRAC) channels mediated by ORAI isoforms in calcium signalling has been extensively investigated. It has been shown that the presence or absence of different isoforms has a significant effect on store-operated calcium entry (SOCE). Yoast et al. (2020) showed that, in addition to the reported narrow-spike oscillations (whereby cytosolic calcium decreases quickly after a sharp increase), ORAI1 knockout HEK293 cells were able to oscillate with broad-spike oscillations (whereby cytosolic calcium decreases in a prolonged manner after a sharp increase) when stimulated with a muscarinic agonist. This suggests that Ca2+ influx through ORAI-mediated CRAC channels negatively regulates the duration of Ca2+ oscillations. We hypothesise that, through the activation of protein kinase C (PKC), ORAI1 negatively regulates phospholipase C (PLC) activity to decrease inositol 1,4,5-trisphosphate (IP3) production and limit the duration of agonist-evoked Ca2+ oscillations. Based on this hypothesis, we construct a new mathematical model, which shows that the formation of broad-spike oscillations is highly dependent on the absence of ORAI1. Predictions of this model are consistent with the experimental results.

A multiple-oscillator mechanism underlies antigen-induced Ca2+ oscillations in Jurkat T-cells.

T-cell receptor stimulation triggers cytosolic Ca2+ signaling by inositol-1,4,5-trisphosphate (IP3)-mediated Ca2+ release from the endoplasmic reticulum (ER) and Ca2+ entry through Ca2+ release-activated Ca2+ (CRAC) channels gated by ER-located stromal-interacting molecules (STIM1/2). Physiologically, cytosolic Ca2+ signaling manifests as regenerative Ca2+ oscillations, which are critical for nuclear factor of activated T-cells-mediated transcription. In most cells, Ca2+ oscillations are thought to originate from IP3 receptor-mediated Ca2+ release, with CRAC channels indirectly sustaining them through ER refilling. Here, experimental and computational evidence support a multiple-oscillator mechanism in Jurkat T-cells whereby both IP3 receptor and CRAC channel activities oscillate and directly fuel antigen-evoked Ca2+ oscillations, with the CRAC channel being the major contributor. KO of either STIM1 or STIM2 significantly reduces CRAC channel activity. As such, STIM1 and STIM2 synergize for optimal Ca2+ oscillations and activation of nuclear factor of activated T-cells 1 and are essential for ER refilling. The loss of both STIM proteins abrogates CRAC channel activity, drastically reduces ER Ca2+ content, severely hampers cell proliferation and enhances cell death. These results clarify the mechanism and the contribution of STIM proteins to Ca2+ oscillations in T-cells.

A Tale of two receptors.

Calcium (Ca2+) oscillations in hepatocytes have a wide dynamic range. In particular, recent experimental evidence shows that agonist stimulation of the P2Y family of receptors leads to qualitatively diverse Ca2+ oscillations. We present a new model of Ca2+ oscillations in hepatocytes based on these experiments to investigate the mechanisms controlling P2Y-activated Ca2+ oscillations. The model accounts for Ca2+ regulation of the IP3 receptor (IP3R), the positive feedback from Ca2+ on phospholipase C (PLC) and the P2Y receptor phosphorylation by protein kinase C (PKC). Furthermore, PKC is shown to control multiple cellular substrates. Utilising the model, we suggest the activity and intensity of PLC and PKC necessary to explain the qualitatively diverse Ca2+ oscillations in response to P2Y receptor activation.

Dual mechanisms of Ca2+ oscillations in hepatocytes.

Calcium (Ca2+) oscillations in hepatocytes control many critical cellular functions, including glucose metabolism and bile secretion. The mechanisms underlying repetitive Ca2+ oscillations and how these mechanisms regulate these oscillations is not fully understood. Recent experimental evidence has shown that both Ca2+ regulation of the inositol 1,4,5-trisphosphate (IP3) receptor and IP3 metabolism generate Ca2+ oscillations and co-exist in hepatocytes. To investigate the effects of these feedback mechanisms on the Ca2+ response, we construct a mathematical model of the Ca2+ signalling network in hepatocytes. The model accounts for the biphasic regulation of Ca2+ on the IP3 receptor (IP3R) and the positive feedback from Ca2+ on IP3 metabolism, via activation of phospholipase C (PLC) by agonist and Ca2+. Model simulations show that Ca2+ oscillations exist for both constant [IP3] and for [IP3] changing dynamically. We show, both experimentally and in the model, that as agonist concentration increases, Ca2+ oscillations transition between simple narrow-spike oscillations and complex broad-spike oscillations. The model predicts that narrow-spike oscillations persist when Ca2+ transport across the plasma membrane is blocked. This prediction has been experimentally validated. In contrast, broad-spike oscillations are terminated when plasma membrane transport is blocked. We conclude that multiple feedback mechanisms participate in regulating Ca2+ oscillations in hepatocytes.

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.

On the dynamical structure of calcium oscillations.

Oscillations in the concentration of free cytosolic Ca2+ are an important and ubiquitous control mechanism in many cell types. It is thus correspondingly important to understand the mechanisms that underlie the control of these oscillations and how their period is determined. We show that Class I Ca2+ oscillations (i.e., oscillations that can occur at a constant concentration of inositol trisphosphate) have a common dynamical structure, irrespective of the oscillation period. This commonality allows the construction of a simple canonical model that incorporates this underlying dynamical behavior. Predictions from the model are tested, and confirmed, in three different cell types, with oscillation periods ranging over an order of magnitude. The model also predicts that Ca2+ oscillation period can be controlled by modulation of the rate of activation by Ca2+ of the inositol trisphosphate receptor. Preliminary experimental evidence consistent with this hypothesis is presented. Our canonical model has a structure similar to, but not identical to, the classic FitzHugh-Nagumo model. The characterization of variables by speed of evolution, as either fast or slow variables, changes over the course of a typical oscillation, leading to a model without globally defined fast and slow variables.

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.

Effects of quasi-steady-state reduction on biophysical models with oscillations.

Many biophysical models have the property that some variables in the model evolve much faster than others. A common step in the analysis of such systems is to simplify the model by assuming that the fastest variables equilibrate instantaneously, an approach that is known as quasi-steady state reduction (QSSR). QSSR is intuitively satisfying but is not always mathematically justified, with problems known to arise, for instance, in some cases in which the full model has oscillatory solutions; in this case, the simplified version of the model may have significantly different dynamics to the full model. This paper focusses on the effect of QSSR on models in which oscillatory solutions arise via one or more Hopf bifurcations. We first illustrate the problems that can arise by applying QSSR to a selection of well-known models. We then categorize Hopf bifurcations according to whether they involve fast variables, slow variables or a mixture of both, and show that Hopf bifurcations that involve only slow variables are not affected by QSSR, Hopf bifurcations that involve fast and slow variables (i.e., singular Hopf bifurcations) are generically preserved under QSSR so long as a fast variable is kept in the simplified system, and Hopf bifurcations that primarily involve fast variables may be eliminated by QSSR. Finally, we present some guidelines for the application of QSSR if one wishes to use the method while minimising the risk of inadvertently destroying essential features of the original model.

Regulation of electrical bursting in a spatiotemporal model of a GnRH neuron.

Gonadotropin-releasing hormone (GnRH) neurons are hypothalamic neurons that control the pulsatile release of GnRH that governs fertility and reproduction in mammals. The mechanisms underlying the pulsatile release of GnRH are not well understood. Some mathematical models have been developed previously to explain different aspects of these activities, such as the properties of burst action potential firing and their associated Ca(2+) transients. These previous studies were based on experimental recordings taken from the soma of GnRH neurons. However, some research groups have shown that the dendrites of GnRH neurons play very important roles. In particular, it is now known that the site of action potential initiation in these neurons is often in the dendrite, over 100 µm from the soma. This raises an important question. Since some of the mechanisms for controlling the burst length and interburst interval are located in the soma, how can electrical bursting be controlled when initiated at a site located some distance from these controlling mechanisms? In order to answer this question, we construct a spatio-temporal mathematical model that includes both the soma and the dendrite. Our model shows that the diffusion coefficient for the spread of electrical potentials in the dendrite is large enough to coordinate burst firing of action potentials when the initiation site is located at some distance from the soma.

Changes in the criticality of Hopf bifurcations due to certain model reduction techniques in systems with multiple timescales.

A major obstacle in the analysis of many physiological models is the issue of model simplification. Various methods have been used for simplifying such models, with one common technique being to eliminate certain 'fast' variables using a quasi-steady-state assumption. In this article, we show when such a physiological model reduction technique in a slow-fast system is mathematically justified. We provide counterexamples showing that this technique can give erroneous results near the onset of oscillatory behaviour which is, practically, the region of most importance in a model. In addition, we show that the singular limit of the first Lyapunov coefficient of a Hopf bifurcation in a slow-fast system is, in general, not equal to the first Lyapunov coefficient of the Hopf bifurcation in the corresponding layer problem, a seemingly counterintuitive result. Consequently, one cannot deduce, in general, the criticality of a Hopf bifurcation in a slow-fast system from the lower-dimensional layer problem.

Understanding anomalous delays in a model of intracellular calcium dynamics.

In many cell types, oscillations in the concentration of free intracellular calcium ions are used to control a variety of cellular functions. It has been suggested [J. Sneyd et al., "A method for determining the dependence of calcium oscillations on inositol trisphosphate oscillations," Proc. Natl. Acad. Sci. U.S.A. 103, 1675-1680 (2006)] that the mechanisms underlying the generation and control of such oscillations can be determined by means of a simple experiment, whereby a single exogenous pulse of inositol trisphosphate (IP(3)) is applied to the cell. However, more detailed mathematical investigations [M. Domijan et al., "Dynamical probing of the mechanisms underlying calcium oscillations," J. Nonlinear Sci. 16, 483-506 (2006)] have shown that this is not necessarily always true, and that the experimental data are more difficult to interpret than first thought. Here, we use geometric singular perturbation techniques to study the dynamics of models that make different assumptions about the mechanisms underlying the calcium oscillations. In particular, we show how recently developed canard theory for singularly perturbed systems with three or more slow variables [M. Wechselberger, "A propos de canards (Apropos canards)," Preprint, 2010] applies to these calcium models and how the presence of a curve of folded singularities and corresponding canards can result in anomalous delays in the response of these models to a pulse of IP(3).

A bifurcation analysis of calcium buffering.

Many mathematical models of calcium oscillations model buffering implicitly by using a rapid buffering approximation. This approximation assumes that separate time scales can be distinguished, with the buffer reactions occurring on a faster time scale than the other calcium fluxes. The rapid buffering approximation is convenient as it reduces the model to a single transport equation for calcium, but buffering is not always so fast. We investigated what happens if such an assumption is made for slower buffers for parameter values typical of both endogenous and exogenous buffers. We found that no qualitative differences are introduced to the bifurcation structure, i.e. there are no anomalous behaviour or artifacts in the dynamics when a rapid buffering approximation is used compared with including buffering explicitly in the model. We found that there exist distinct buffer parameter regions in which either the rapid buffering approximation or an assumption of no buffering could be used. Separating the two regions was a small transition region of buffer parameters for which care needs to be taken in modelling buffers. However, overall, the qualitative behaviour in all three regions was similar.