A bifurcation analysis of two coupled calcium oscillators

In many cell types, asynchronous or synchronous oscillations in the concentration of intracellular free calcium occur in adjacent cells that are coupled by gap junctions. Such oscillations are believed to underlie oscillatory intercellular calcium waves in some cell types, and thus it is important to understand how they occur and are modified by intercellular coupling. Using a previous model of intracellular calcium oscillations in pancreatic acinar cells, this article explores the effects of coupling two cells with a simple linear diffusion term. Depending on the concentration of a signal molecule, inositol (1,4,5)-trisphosphate, coupling two identical cells by diffusion can give rise to synchronized in-phase oscillations, as well as different-amplitude in-phase oscillations and same-amplitude antiphase oscillations. Coupling two nonidentical cells leads to more complex behaviors such as cascades of period doubling and multiply periodic solutions. This study is a first step towards understanding the role and significance of the diffusion of calcium through gap junctions in the coordination of oscillatory calcium waves in a variety of cell types. © 2001 American Institute of Physics.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/70869/2/CHAOEH-11-1-237-1.pd

Depolarization-induced Calcium Responses in Sympathetic Neurons: Relative Contributions from Ca2+ Entry, Extrusion, ER/Mitochondrial Ca2+ Uptake and Release, and Ca2+ Buffering

Many models have been developed to account for stimulus-evoked [Ca2+] responses, but few address how responses elicited in specific cell types are defined by the Ca2+ transport and buffering systems that operate in the same cells. In this study, we extend previous modeling studies by linking the time course of stimulus-evoked [Ca2+] responses to the underlying Ca2+ transport and buffering systems. Depolarization-evoked [Ca2+]i responses were studied in sympathetic neurons under voltage clamp, asking how response kinetics are defined by the Ca2+ handling systems expressed in these cells. We investigated five cases of increasing complexity, comparing observed and calculated responses deduced from measured Ca2+ handling properties. In Case 1, [Ca2+]i responses were elicited by small Ca2+ currents while Ca2+ transport by internal stores was inhibited, leaving plasma membrane Ca2+ extrusion intact. In Case 2, responses to the same stimuli were measured while mitochondrial Ca2+ uptake was active. In Case 3, responses were elicited as in Case 2 but with larger Ca2+ currents that produce larger and faster [Ca2+]i elevations. Case 4 included the mitochondrial Na/Ca exchanger. Finally, Case 5 included ER Ca2+ uptake and release pathways. We found that [Ca2+]i responses elicited by weak stimuli (Cases 1 and 2) could be quantitatively reconstructed using a spatially uniform model incorporating the measured properties of Ca2+ entry, removal, and buffering. Responses to strong depolarization (Case 3) could not be described by this model, but were consistent with a diffusion model incorporating the same Ca2+ transport and buffering descriptions, as long as endogenous buffers have low mobility, leading to steep radial [Ca2+]i gradients and spatially nonuniform Ca2+ loading by mitochondria. When extended to include mitochondrial Ca2+ release (Case 4) and ER Ca2+ transport (Case 5), the diffusion model could also account for previous measurements of stimulus-evoked changes in total mitochondrial and ER Ca concentration

A deterministic model predicts the properties of stochastic calcium oscillations in airway smooth muscle cells

The inositol trisphosphate receptor ([Formula: see text]) is one of the most important cellular components responsible for oscillations in the cytoplasmic calcium concentration. Over the past decade, two major questions about the [Formula: see text] have arisen. Firstly, how best should the [Formula: see text] be modeled? In other words, what fundamental properties of the [Formula: see text] allow it to perform its function, and what are their quantitative properties? Secondly, although calcium oscillations are caused by the stochastic opening and closing of small numbers of [Formula: see text], is it possible for a deterministic model to be a reliable predictor of calcium behavior? Here, we answer these two questions, using airway smooth muscle cells (ASMC) as a specific example. Firstly, we show that periodic calcium waves in ASMC, as well as the statistics of calcium puffs in other cell types, can be quantitatively reproduced by a two-state model of the [Formula: see text], and thus the behavior of the [Formula: see text] is essentially determined by its modal structure. The structure within each mode is irrelevant for function. Secondly, we show that, although calcium waves in ASMC are generated by a stochastic mechanism, [Formula: see text] stochasticity is not essential for a qualitative prediction of how oscillation frequency depends on model parameters, and thus deterministic [Formula: see text] models demonstrate the same level of predictive capability as do stochastic models. We conclude that, firstly, calcium dynamics can be accurately modeled using simplified [Formula: see text] models, and, secondly, to obtain qualitative predictions of how oscillation frequency depends on parameters it is sufficient to use a deterministic model

A Multi-Scale Approach to Airway Hyperresponsiveness: From Molecule to Organ

Airway hyperresponsiveness (AHR), a characteristic of asthma that involves an excessive reduction in airway caliber, is a complex mechanism reflecting multiple processes that manifest over a large range of length and time scales. At one extreme, molecular interactions determine the force generated by airway smooth muscle (ASM). At the other, the spatially distributed constriction of the branching airways leads to breathing difficulties. Similarly, asthma therapies act at the molecular scale while clinical outcomes are determined by lung function. These extremes are linked by events operating over intermediate scales of length and time. Thus, AHR is an emergent phenomenon that limits our understanding of asthma and confounds the interpretation of studies that address physiological mechanisms over a limited range of scales. A solution is a modular computational model that integrates experimental and mathematical data from multiple scales. This includes, at the molecular scale, kinetics, and force production of actin-myosin contractile proteins during cross-bridge and latch-state cycling; at the cellular scale, Ca2+ signaling mechanisms that regulate ASM force production; at the tissue scale, forces acting between contracting ASM and opposing viscoelastic tissue that determine airway narrowing; at the organ scale, the topographic distribution of ASM contraction dynamics that determine mechanical impedance of the lung. At each scale, models are constructed with iterations between theory and experimentation to identify the parameters that link adjacent scales. This modular model establishes algorithms for modeling over a wide range of scales and provides a framework for the inclusion of other responses such as inflammation or therapeutic regimes. The goal is to develop this lung model so that it can make predictions about bronchoconstriction and identify the pathophysiologic mechanisms having the greatest impact on AHR and its therapy

Modelling the effects of calcium waves and oscillations on saliva secretion

An understanding of Ca2+Ca2+ signalling in saliva-secreting acinar cells is important, as Ca2+Ca2+ is the second messenger linking stimulation of cells to production of saliva. Ca2+Ca2+ signals affect secretion via the ion channels located both apically and basolaterally in the cell. By approximating Ca2+Ca2+ 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 Ca2+Ca2+ concentration is found to be the most significant property in signalling secretion. Wave speed was found to encode a range of secretion rates. Ca2+Ca2+ oscillation frequency and amplitude had little effect on fluid secretion