Simplification and analysis of models of calcium dynamics based on IP3-sensitive calcium channel kinetics

Optogenetic Calcium Ion Influx in Myoblasts and Myotubes by Near-Infrared Light Using Upconversion Nanoparticles

Bioactuators made of cultured skeletal muscle cells are generally driven by electrical or visible light stimuli. Among these, the technology to control skeletal muscle consisting of myoblasts genetically engineered to express photoreceptor proteins with visible light is very promising, as there is no risk of cell contamination by electrodes, and the skeletal muscle bioactuator can be operated remotely. However, due to the low biopermeability of visible light, it can only be applied to thin skeletal muscle films, making it difficult to realize high-power bioactuators consisting of thick skeletal muscle. To solve this problem, it is desirable to realize thick skeletal muscle bioactuators that can be driven by near-infrared (NIR) light, to which living tissue is highly permeable. In this study, as a promising first step, upconversion nanoparticles (UCNPs) capable of converting NIR light into blue light were bound to C2C12 myoblasts expressing the photoreceptor protein channelrhodopsin-2 (ChR2), and the myoblasts calcium ion (Ca2+) influx was remotely manipulated by NIR light exposure. UCNP-bound myoblasts and UCNP-bound differentiated myotubes were exposed to NIR light, and the intracellular Ca2+ concentrations were measured and compared to myoblasts exposed to blue light. Exposure of the UCNP-bound cells to NIR light was found to be more efficient than exposure to blue light in terms of stimulating Ca2+ influx.

Modeling IP3-induced Ca2+ signaling based on its interspike interval statistics

Inositol 1,4,5-trisphosphate (IP3)-induced Ca2+ signaling is a second messenger system used by almost all eukaryotic cells. Recent research demonstrated randomness of Ca2+ signaling on all structural levels. We compile eight general properties of Ca2+ spiking common to all cell types investigated and suggest a theory of Ca2+ spiking starting from the random behavior of IP3 receptor channel clusters mediating the release of Ca2+ from the endoplasmic reticulum capturing all general properties and pathway-specific behavior. Spike generation begins after the absolute refractory period of the previous spike. According to its hierarchical spreading from initiating channel openings to cell level, we describe it as a first passage process from none to all clusters open while the cell recovers from the inhibition which terminated the previous spike. Our theory reproduces the exponential stimulation response relation of the average interspike interval Tav and its robustness properties, random spike timing with a linear moment relation between Tav and the interspike interval SD and its robustness properties, sensitive dependency of Tav on diffusion properties, and nonoscillatory local dynamics. We explain large cell variability of Tav observed in experiments by variability of channel cluster coupling by Ca2+-induced Ca2+ release, the number of clusters, and IP3 pathway component expression levels. We predict the relation between puff probability and agonist concentration and [IP3] and agonist concentration. Differences of spike behavior between cell types and stimulating agonists are explained by the different types of negative feedback terminating spikes. In summary, the hierarchical random character of spike generation explains all of the identified general properties.

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.

Effect of delay in transportation of extracellular glucose into cardiomyocytes under diabetic condition: a study through mathematical model

A four-dimensional model was built to mimic the cross-talk among plasma glucose, plasma insulin, intracellular glucose and cytoplasmic calcium of a cardiomyocyte. A time delay was considered to represent the time required for performing various cellular mechanisms between activation of insulin receptor and subsequent glucose entry from extracellular region into intracellular region of a cardiac cell. We analysed the delay-induced model and deciphered conditions for stability and bifurcation. Extensive numerical computations were performed to validate the analytical results and give further insights. Sensitivity study of the system parameters using LHS-PRCC method reveals that some rate parameters, which represent the input of plasma glucose, absorption of glucose by noncardiac cells and insulin production, are sensitive and may cause significant change in the system dynamics. It was observed that the time taken for transportation of extracellular glucose into the cell through GLUT4 plays an important role in maintaining physiological oscillations of the state variables. Parameter recalibration exercise showed that reduced input rate of glucose in the blood plasma or an alteration in transportation delay may be used for therapeutic targets in diabetic-like condition for maintaining normal cardiac function.

Lessons from the Endoplasmic Reticulum Ca2+ Transporters-A Cancer Connection

Ca2+ is an integral mediator of intracellular signaling, impacting almost every aspect of cellular life. The Ca2+-conducting transporters located on the endoplasmic reticulum (ER) membrane shoulder the responsibility of constructing the global Ca2+ signaling landscape. These transporters gate the ER Ca2+ release and uptake, sculpt signaling duration and intensity, and compose the Ca2+ signaling rhythm to accommodate a plethora of biological activities. In this review, we explore the mechanisms of activation and functional regulation of ER Ca2+ transporters in the establishment of Ca2+ homeostasis. We also contextualize the aberrant alterations of these transporters in carcinogenesis, presenting Ca2+-based therapeutic interventions as a means to tackle malignancies.

A Statistical View on Calcium Oscillations

Transient rises and falls of the intracellular calcium concentration have been observed in numerous cell types and under a plethora of conditions. There is now a growing body of evidence that these whole-cell calcium oscillations are stochastic, which poses a significant challenge for modelling. In this review, we take a closer look at recently developed statistical approaches to calcium oscillations. These models describe the timing of whole-cell calcium spikes, yet their parametrisations reflect subcellular processes. We show how non-stationary calcium spike sequences, which e.g. occur during slow depletion of intracellular calcium stores or in the presence of time-dependent stimulation, can be analysed with the help of so-called intensity functions. By utilising Bayesian concepts, we demonstrate how values of key parameters of the statistical model can be inferred from single cell calcium spike sequences and illustrate what information whole-cell statistical models can provide about the subcellular mechanistic processes that drive calcium oscillations. In particular, we find that the interspike interval distribution of HEK293 cells under constant stimulation is captured by a Gamma distribution.

Attenuation of calmodulin regulation evokes Ca2+ oscillations: evidence for the involvement of intracellular arachidonate-activated channels and connexons

Intracellular Са²⁺ controls its own level by regulation of Ca²⁺ transport across the plasma and organellar membranes, often acting via calmodulin (CaM). Drugs antagonizing CaM action induce an increase in cytosolic Ca²⁺ concentration in different cells. We have found persistent Са²⁺ oscillations in cultured white adipocytes in response to calmidazolium (CMZ). They appeared at [CMZ] > 1 μM as repetitive sharp spikes mainly superimposed on a transient or elevated baseline. Similar oscillations were observed when we used trifluoperazine. Oscillations evoked by 5 μM CMZ resulted from the release of stored Ca²⁺ and were supported by Са²⁺ entry. Inhibition of store-operated channels by YM-58483 or 2-APB did not change the responses. Phospholipase A₂ inhibited by AACOCF₃ was responsible for initial Ca²⁺ mobilization, but not for subsequent oscillations, whereas inhibition of iPLA₂ by BEL had no effect. Phospholipase C was partially involved in both stages as revealed with U73122. Intracellular Са²⁺ stores engaged by CMZ were entirely dependent on thapsigargin. The oscillations existed in the presence of inhibitors of ryanodine or inositol 1,4,5-trisphosphate receptors, or antagonists of Ca²⁺ transport by lysosome-like acidic stores. Carbenoxolone or octanol, blockers of hemichannels (connexons), when applied for two hours, prevented oscillations but did not affect the initial Са²⁺ release. Incubation with La³⁺ for 2 or 24 h inhibited all responses to CMZ, retaining the thapsigargin-induced Ca²⁺ rise. These results suggest that Ca²⁺-CaM regulation suppresses La³⁺-sensitive channels in non-acidic organelles, of which arachidonate-activated channels initiate Ca²⁺ oscillations, and connexons are intimately implicated in their generation mechanism.

A mathematical and computational model of the calcium dynamics in Caenorhabditis elegans ASH sensory neuron

We propose a mathematical and computational model that captures the stimulus-generated Ca²⁺ transients in the C. elegans ASH sensory neuron. The rationale is to develop a tool that will enable a cross-talk between modeling and experiments, using modeling results to guide targeted experimental efforts. The model is built based on biophysical events and molecular cascades known to unfold as part of neurons' Ca²⁺ homeostasis mechanism, as well as on Ca²⁺ signaling events. The state of ion channels is described by their probability of being activated or inactivated, and the remaining molecular states are based on biochemically defined kinetic equations or known biochemical motifs. We estimate the parameters of the model using experimental data of hyperosmotic stimulus-evoked Ca²⁺ transients detected with a FRET sensor in young and aged worms, unstressed and exposed to oxidative stress. We use a hybrid optimization method composed of a multi-objective genetic algorithm and nonlinear least-squares to estimate the model parameters. We first obtain the model parameters for young unstressed worms. Next, we use these values of the parameters as a starting point to identify the model parameters for stressed and aged worms. We show that the model, in combination with experimental data, corroborates literature results. In addition, we demonstrate that our model can be used to predict ASH response to complex combinations of stimulation pulses. The proposed model includes for the first time the ASH Ca²⁺ dynamics observed during both "on" and "off" responses. This mathematical and computational effort is the first to propose a dynamic model of the Ca²⁺ transients' mechanism in C. elegans neurons, based on biochemical pathways of the cell's Ca²⁺ homeostasis machinery. We believe that the proposed model can be used to further elucidate the Ca²⁺ dynamics of a key C. elegans neuron, to guide future experiments on C. elegans neurobiology, and to pave the way for the development of more mathematical models for neuronal Ca²⁺ dynamics.

Coupling calcium dynamics and mitochondrial bioenergetic: an in silico study to simulate cardiomyocyte dysfunction

The coupling of intracellular Ca(2+) dynamics with mitochondrial bioenergetic is crucial for the functioning of cardiomyocytes both in healthy and disease conditions. The pathophysiological signature of the Cardiomyocyte Dysfunction (CD) is commonly related to decreased ATP production due to mitochondrial functional impairment and to an increased mitochondrial calcium content ([Ca(2+)]m). These features advanced the therapeutic approaches which aim to reduce [Ca(2+)]m. But whether [Ca(2+)]m overload is the pathological trigger for CD or a physiological consequence, remained controversial. We addressed this issue in silico and showed that [Ca(2+)]m might not directly cause CD. Through model parameter recalibration, we demonstrated how mitochondria cope up with functionally impaired processes and consequently accumulate calcium. A strong coupling of the [Ca(2+)]m oscillations with the ATP synthesis rate ensures robust calcium cycling and avoids CD. We suggested a cardioprotective role of the mitochondrial calcium uniporter and predicted that a mitochondrial sodium calcium exchanger could be a potential therapeutic target to restore the normal functioning of the cardiomyocyte.

An allosteric model of the inositol trisphosphate receptor with nonequilibrium binding

The inositol trisphosphate receptor (IPR) is a crucial ion channel that regulates the Ca(2+) influx from the endoplasmic reticulum (ER) to the cytoplasm. A thorough study of the IPR channel contributes to a better understanding of calcium oscillations and waves. It has long been observed that the IPR channel is a typical biological system which performs adaptation. However, recent advances on the physical essence of adaptation show that adaptation systems with a negative feedback mechanism, such as the IPR channel, must break detailed balance and always operate out of equilibrium with energy dissipation. Almost all previous IPR models are equilibrium models assuming detailed balance and thus violate the dissipative nature of adaptation. In this article, we constructed a nonequilibrium allosteric model of single IPR channels based on the patch-clamp experimental data obtained from the IPR in the outer membranes of isolated nuclei of the Xenopus oocyte. It turns out that our model reproduces the patch-clamp experimental data reasonably well and produces both the correct steady-state and dynamic properties of the channel. Particularly, our model successfully describes the complicated bimodal [Ca(2+)] dependence of the mean open duration at high [IP3], a steady-state behavior which fails to be correctly described in previous IPR models. Finally, we used the patch-clamp experimental data to validate that the IPR channel indeed breaks detailed balance and thus is a nonequilibrium system which consumes energy.

Activation of store-operated calcium entry in airway smooth muscle cells: insight from a mathematical model

Intracellular dynamics of airway smooth muscle cells (ASMC) mediate ASMC contraction and proliferation, and thus play a key role in airway hyper-responsiveness (AHR) and remodelling in asthma. We evaluate the importance of store-operated entry (SOCE) in these dynamics by constructing a mathematical model of ASMC signaling based on experimental data from lung slices. The model confirms that SOCE is elicited upon sufficient depletion of the sarcoplasmic reticulum (SR), while receptor-operated entry (ROCE) is inhibited in such conditions. It also shows that SOCE can sustain agonist-induced oscillations in the absence of other influx. SOCE up-regulation may thus contribute to AHR by increasing the oscillation frequency that in turn regulates ASMC contraction. The model also provides an explanation for the failure of the SERCA pump blocker CPA to clamp the cytosolic of ASMC in lung slices, by showing that CPA is unable to maintain the SR empty of . This prediction is confirmed by experimental data from mouse lung slices, and strongly suggests that CPA only partially inhibits SERCA in ASMC.

Excitation and adaptation in bacteria-a model signal transduction system that controls taxis and spatial pattern formation

The machinery for transduction of chemotactic stimuli in the bacterium E. coli is one of the most completely characterized signal transduction systems, and because of its relative simplicity, quantitative analysis of this system is possible. Here we discuss models which reproduce many of the important behaviors of the system. The important characteristics of the signal transduction system are excitation and adaptation, and the latter implies that the transduction system can function as a "derivative sensor" with respect to the ligand concentration in that the DC component of a signal is ultimately ignored if it is not too large. This temporal sensing mechanism provides the bacterium with a memory of its passage through spatially- or temporally-varying signal fields, and adaptation is essential for successful chemotaxis. We also discuss some of the spatial patterns observed in populations and indicate how cell-level behavior can be embedded in population-level descriptions.

Comparison of models for IP3 receptor kinetics using stochastic simulations

Inositol 1,4,5-trisphosphate receptor (IP₃R) is a ubiquitous intracellular calcium (Ca²⁺) channel which has a major role in controlling Ca²⁺ levels in neurons. A variety of computational models have been developed to describe the kinetic function of IP₃R under different conditions. In the field of computational neuroscience, it is of great interest to apply the existing models of IP₃R when modeling local Ca²⁺ transients in dendrites or overall Ca²⁺ dynamics in large neuronal models. The goal of this study was to evaluate existing IP₃R models, based on electrophysiological data. This was done in order to be able to suggest suitable models for neuronal modeling. Altogether four models (Othmer and Tang, 1993; Dawson etal., 2003; Fraiman and Dawson, 2004; Doi etal., 2005) were selected for a more detailed comparison. The selection was based on the computational efficiency of the models and the type of experimental data that was used in developing the model. The kinetics of all four models were simulated by stochastic means, using the simulation software STEPS, which implements the Gillespie stochastic simulation algorithm. The results show major differences in the statistical properties of model functionality. Of the four compared models, the one by Fraiman and Dawson (2004) proved most satisfactory in producing the specific features of experimental findings reported in literature. To our knowledge, the present study is the first detailed evaluation of IP₃R models using stochastic simulation methods, thus providing an important setting for constructing a new, realistic model of IP₃R channel kinetics for compartmental modeling of neuronal functions. We conclude that the kinetics of IP₃R with different concentrations of Ca²⁺ and IP₃ should be more carefully addressed when new models for IP₃R are developed.

A data-driven model of a modal gated ion channel: the inositol 1,4,5-trisphosphate receptor in insect Sf9 cells

The inositol 1,4,5-trisphosphate (IP(3)) receptor (IP(3)R) channel is crucial for the generation and modulation of intracellular Ca(2+) signals in animal cells. To gain insight into the complicated ligand regulation of this ubiquitous channel, we constructed a simple quantitative continuous-time Markov-chain model from the data. Our model accounts for most experimentally observed gating behaviors of single native IP(3)R channels from insect Sf9 cells. Ligand (Ca(2+) and IP(3)) dependencies of channel activity established six main ligand-bound channel complexes, where a complex consists of one or more states with the same ligand stoichiometry and open or closed conformation. Channel gating in three distinct modes added one complex and indicated that three complexes gate in multiple modes. This also restricted the connectivity between channel complexes. Finally, latencies of channel responses to abrupt ligand concentration changes defined a model with specific network topology between 9 closed and 3 open states. The model with 28 parameters can closely reproduce the equilibrium gating statistics for all three gating modes over a broad range of ligand concentrations. It also captures the major features of channel response latency distributions. The model can generate falsifiable predictions of IP(3)R channel gating behaviors and provide insights to both guide future experiment development and improve IP(3)R channel gating analysis. Maximum likelihood estimates of the model parameters and of the parameters in the De Young-Keizer model yield strong statistical evidence in favor of our model. Our method is simple and easily applicable to the dynamics of other ion channels and molecules.

Stochastic simulations of pattern formation in excitable media

We present a method for mesoscopic, dynamic Monte Carlo simulations of pattern formation in excitable reaction-diffusion systems. Using a two-level parallelization approach, our simulations cover the whole range of the parameter space, from the noise-dominated low-particle number regime to the quasi-deterministic high-particle number limit. Three qualitatively different case studies are performed that stand exemplary for the wide variety of excitable systems. We present mesoscopic stochastic simulations of the Gray-Scott model, of a simplified model for intracellular Ca2+ oscillations and, for the first time, of the Oregonator model. We achieve simulations with up to 10(10) particles. The software and the model files are freely available and researchers can use the models to reproduce our results or adapt and refine them for further exploration.

TEMPERATURE EFFECTS ON THE STOCHASTIC GATING OF THE IP3R CALCIUM RELEASE CHANNEL: A NUMERICAL SIMULATION STUDY

The importance of the kinetic study of endoplasmatic calcium ion channels in different intracellular processes is known today. Although there are few experimental reports on the temperature dependency of IP3R channel functions, we did not find any detailed theoretical study on this subject. For this purpose, we used a modified Gillespie algorithm to investigate the effect of temperature on the conditions affecting the open state of a single subunit of the De Young-Keizer (DYK) model. Population of the states was considered as the subject of fluctuation. Key features of the channel, such as bell-shaped dependency of open probability to the Calcium concentrations were modeled at different temperatures, too. The range of temperature variation was selected by regarding the experimental data on IP3R channel.

By increasing the temperature, we had the very slow time domains (t: 10-1 s ) and the much slower time domains (t: 100 s ) in addition to other time domains, which could be seen as new time categories in InsP3R studies, and so the results were reported in these time domains, as well.

We found out that increase in temperature declined the open probability in some concentrations of Ca 2+ and/or IP3. Also, by introducing the intensity graphs, broadening of the range of fluctuations and lowering of the order of frequency of fluctuations for the population of each state were observed due to the temperature increments.

The temperature effects on the activation and inactivation states of the channel were studied in the framework of the reaction paths. We did not find similar paths at different time domains; several paths observed which were totally different all together. These time-dependent reaction paths are also depending on the Ca 2+ and/or the IP3 concentrations. So, one can predict the most probable reaction paths at different concentrations and temperatures and also determine which kind of the path it is; a path for closing the channel or a path to open it.

Finally, the temperature effects on the calcium inhibited states were studied. We found out that calcium ion inhibitions were shifted to lower calcium concentration by increasing the temperature. The results suggests that inhibiting role of calcium is not only [ Ca 2+] and/or [IP3] dependent, but also temperature dependent.

Exploring phospholipase C-coupled Ca(2+) signalling networks using Boolean modelling

In this study, the authors explored the utility of a descriptive and predictive bionetwork model for phospholipase C-coupled calcium signalling pathways, built with non-kinetic experimental information. Boolean models generated from these data yield oscillatory activity patterns for both the endoplasmic reticulum resident inositol-1,4,5-trisphosphate receptor (IP(3)R) and the plasma-membrane resident canonical transient receptor potential channel 3 (TRPC3). These results are specific as randomisation of the Boolean operators ablates oscillatory pattern formation. Furthermore, knock-out simulations of the IP(3)R, TRPC3 and multiple other proteins recapitulate experimentally derived results. The potential of this approach can be observed by its ability to predict previously undescribed cellular phenotypes using in vitro experimental data. Indeed, our cellular analysis of the developmental and calcium-regulatory protein, DANGER1a, confirms the counter-intuitive predictions from our Boolean models in two highly relevant cellular models. Based on these results, the authors theorise that with sufficient legacy knowledge and/or computational biology predictions, Boolean networks can provide a robust method for predictive modelling of any biological system. [Includes supplementary material].

Calcium oscillations

Calcium signaling results from a complex interplay between activation and inactivation of intracellular and extracellular calcium permeable channels. This complexity is obvious from the pattern of calcium signals observed with modest, physiological concentrations of calcium-mobilizing agonists, which typically present as sequential regenerative discharges of stored calcium, a process referred to as calcium oscillations. In this review, we discuss recent advances in understanding the underlying mechanism of calcium oscillations through the power of mathematical modeling. We also summarize recent findings on the role of calcium entry through store-operated channels in sustaining calcium oscillations and in the mechanism by which calcium oscillations couple to downstream effectors.

Unitary Ca(2+) current through recombinant type 3 InsP(3) receptor channels under physiological ionic conditions

The ubiquitous inositol 1,4,5-trisphosphate (InsP(3)) receptor (InsP(3)R) channel, localized primarily in the endoplasmic reticulum (ER) membrane, releases Ca(2+) into the cytoplasm upon binding InsP(3), generating and modulating intracellular Ca(2+) signals that regulate numerous physiological processes. Together with the number of channels activated and the open probability of the active channels, the size of the unitary Ca(2+) current (i(Ca)) passing through an open InsP(3)R channel determines the amount of Ca(2+) released from the ER store, and thus the amplitude and the spatial and temporal nature of Ca(2+) signals generated in response to extracellular stimuli. Despite its significance, i(Ca) for InsP(3)R channels in physiological ionic conditions has not been directly measured. Here, we report the first measurement of i(Ca) through an InsP(3)R channel in its native membrane environment under physiological ionic conditions. Nuclear patch clamp electrophysiology with rapid perfusion solution exchanges was used to study the conductance properties of recombinant homotetrameric rat type 3 InsP(3)R channels. Within physiological ranges of free Ca(2+) concentrations in the ER lumen ([Ca(2+)](ER)), free cytoplasmic [Ca(2+)] ([Ca(2+)](i)), and symmetric free [Mg(2+)] ([Mg(2+)](f)), the i(Ca)-[Ca(2+)](ER) relation was linear, with no detectable dependence on [Mg(2+)](f). i(Ca) was 0.15 +/- 0.01 pA for a filled ER store with 500 microM [Ca(2+)](ER). The i(Ca)-[Ca(2+)](ER) relation suggests that Ca(2+) released by an InsP(3)R channel raises [Ca(2+)](i) near the open channel to approximately 13-70 microM, depending on [Ca(2+)](ER). These measurements have implications for the activities of nearby InsP(3)-liganded InsP(3)R channels, and they confirm that Ca(2+) released by an open InsP(3)R channel is sufficient to activate neighboring channels at appropriate distances away, promoting Ca(2+)-induced Ca(2+) release.

Reaction-diffusion systems in intracellular molecular transport and control

Chemical reactions make cells work only if the participating chemicals are delivered to desired locations in a timely and precise fashion. Most research to date has focused on active-transport mechanisms, although passive diffusion is often equally rapid and energetically less costly. Capitalizing on these advantages, cells have developed sophisticated reaction-diffusion (RD) systems that control a wide range of cellular functions-from chemotaxis and cell division, through signaling cascades and oscillations, to cell motility. These apparently diverse systems share many common features and are "wired" according to "generic" motifs such as nonlinear kinetics, autocatalysis, and feedback loops. Understanding the operation of these complex (bio)chemical systems requires the analysis of pertinent transport-kinetic equations or, at least on a qualitative level, of the characteristic times of the constituent subprocesses. Therefore, in reviewing the manifestations of cellular RD, we also describe basic theory of reaction-diffusion phenomena.

Spatiotemporal organization of Ca dynamics: a modeling-based approach

Calcium is a ubiquitous second messenger that mediates vital physiological responses such as fertilization, secretion, gene expression, or apoptosis. Given this variety of processes mediated by Ca(2+), these signals are highly organized both in time and space to ensure reliability and specificity. This review deals with the spatiotemporal organization of the Ca(2+) signaling pathway in electrically nonexcitable cells in which InsP(3) receptors are by far the most important Ca(2+) channels. We focus on the aspects of this highly regulated dynamical system for which an interplay between experiments and modeling is particularly fruitful. In particular, the importance of the relative densities of the different InsP(3) receptor subtypes will be discussed on the basis of a modeling approach linking the steady-state behaviors of these channels in electrophysiological experiments with their behavior in a cellular environment. Also, the interplay between InsP(3) metabolism and Ca(2+) oscillations will be considered. Finally, we discuss the relationships between stochastic openings of the Ca(2+) releasing channels at the microscopic level and the coordinated, regular behavior observed at the whole cell level on the basis of a combined experimental and modeling approach.

Modeling and analysis of the molecular basis of pain in sensory neurons

Intracellular calcium dynamics are critical to cellular functions like pain transmission. Extracellular ATP plays an important role in modulating intracellular calcium levels by interacting with the P2 family of surface receptors. In this study, we developed a mechanistic mathematical model of ATP-induced P2 mediated calcium signaling in archetype sensory neurons. The model architecture, which described 90 species connected by 162 interactions, was formulated by aggregating disparate molecular modules from literature. Unlike previous models, only mass action kinetics were used to describe the rate of molecular interactions. Thus, the majority of the 252 unknown model parameters were either association, dissociation or catalytic rate constants. Model parameters were estimated from nine independent data sets taken from multiple laboratories. The training data consisted of both dynamic and steady-state measurements. However, because of the complexity of the calcium network, we were unable to estimate unique model parameters. Instead, we estimated a family or ensemble of probable parameter sets using a multi-objective thermal ensemble method. Each member of the ensemble met an error criterion and was located along or near the optimal trade-off surface between the individual training data sets. The model quantitatively reproduced experimental measurements from dorsal root ganglion neurons as a function of extracellular ATP forcing. Hypothesized architecture linking phosphoinositide regulation with P2X receptor activity explained the inhibition of P2X-mediated current flow by activated metabotropic P2Y receptors. Sensitivity analysis using individual and the whole system outputs suggested which molecular subsystems were most important following P2 activation. Taken together, modeling and analysis of ATP-induced P2 mediated calcium signaling generated qualitative insight into the critical interactions controlling ATP induced calcium dynamics. Understanding these critical interactions may prove useful for the design of the next generation of molecular pain management strategies.

What can we learn from the irregularity of Ca2+ oscillations?

In most cells, Ca(2+) increases in response to external stimulation are organized in the form of oscillations and waves that sometimes propagate from one cell to another. Numerous experimental and theoretical studies reveal that this spatiotemporal organization contains a non-negligible level of stochasticity. In this study, we extend the previous work based on a statistical analysis of experimental Ca(2+) traces in isolated, hormone-stimulated hepatocytes and on stochastic simulations of Ca(2+) oscillations based on the Gillespie's algorithm. Comparison of the coefficients of variation in the periods of experimental and simulated Ca(2+) spikes provides information about the clustering and the specific subtypes of the Ca(2+) channels. In hepatocytes coupled by gap junctions, the global perfusion with a hormone leads to successive Ca(2+) responses, giving the appearance of an intercellular wave. Statistical analysis of experimental Ca(2+) oscillations in coupled hepatocytes confirms that this coordinated Ca(2+) spiking corresponds to a phase wave but suggests the existence of an additional coupling mechanism.

Toward a predictive model of Ca2+ puffs

We investigate the key characteristics of Ca(2+) puffs in deterministic and stochastic frameworks that all incorporate the cellular morphology of IP(3) receptor channel clusters. In the first step, we numerically study the Ca(2+) liberation in a three-dimensional representation of a cluster environment with reaction-diffusion dynamics in both the cytosol and the lumen. These simulations reveal that Ca(2+) concentrations at a releasing cluster range from 80 to 170 microM and equilibrate almost instantaneously on the time scale of the release duration. These highly elevated Ca(2+) concentrations eliminate Ca(2+) oscillations in a deterministic model of an IP(3)R channel cluster at physiological parameter values as revealed by a linear stability analysis. The reason lies in the saturation of all feedback processes in the IP(3)R gating dynamics, so that only fluctuations can restore experimentally observed Ca(2+) oscillations. In this spirit, we derive master equations that allow us to analytically quantify the onset of Ca(2+) puffs and hence the stochastic time scale of intracellular Ca(2+) dynamics. Moving up the spatial scale, we suggest to formulate cellular dynamics in terms of waiting time distribution functions. This approach prevents the state space explosion that is typical for the description of cellular dynamics based on channel states and still contains information on molecular fluctuations. We illustrate this method by studying global Ca(2+) oscillations.

Ca2+ spiral waves in a spatially discrete and random medium

It is well known that the spatial distribution of the calcium ion channels in the endoplasmic reticulum is discrete. We study the Ca(2+) spiral pattern formation based on a model in which ion channels are discretely and randomly distributed. Numerical simulations are performed on different types of media with the Ca(2+) release sites uniformly distributed, discretely and uniformly arranged, or discretely and randomly arranged. The comparisons among the different media show that random distribution is necessary for spontaneous initiation of Ca(2+) spiral waves, and the discrete and random distribution is of significance for spiral waves under physiologically reasonable conditions. The period and velocity of spiral waves are also calculated, and they are not prominently changed by varying the type of medium.

Dynamic simulation of the effect of calcium-release activated calcium channel on cytoplasmic Ca2+ oscillation

A mathematical model is proposed to illustrate the activation of STIM1 (stromal interaction molecule 1) protein, the assembly and activation of calcium-release activated calcium (CRAC) channels in T cells. In combination with De Young-Keizer-Li-Rinzel model, we successfully reproduce a sustained Ca(2+) oscillation in cytoplasm. Our results reveal that Ca(2+) oscillation dynamics in cytoplasm can be significantly affected by the way how the Orai1 CRAC channel are assembled and activated. A low sustained Ca(2+) influx is observed through the CRAC channels across the plasma membrane. In particular, our model shows that a tetrameric channel complex can effectively regulate the total quantity of the channels and the ratio of the active channels to the total channels, and a period of Ca(2+) oscillation about 29 s is in agreement with published experimental data. The bifurcation analyses illustrate the different dynamic properties between our mixed Ca(2+) feedback model and the single positive or negative feedback models.

Calcium-dependent inactivation and the dynamics of calcium puffs and sparks

Localized intracellular Ca(2+) elevations known as puffs and sparks arise from the cooperative activity of inositol 1,4,5-trisphosphate receptor Ca(2+) channels (IP(3)Rs) and ryanodine receptor Ca(2+) channels (RyRs) clustered at Ca(2+) release sites on the surface of the endoplasmic reticulum or sarcoplasmic reticulum. When Markov chain models of these intracellular Ca(2+)-regulated Ca(2+) channels are coupled via a mathematical representation of a Ca(2+) microdomain, simulated Ca(2+) release sites may exhibit the phenomenon of "stochastic Ca(2+) excitability" reminiscent of Ca(2+) puffs and sparks where channels open and close in a concerted fashion. To clarify the role of Ca(2+) inactivation of IP(3)Rs and RyRs in the dynamics of puffs and sparks, we formulate and analyze Markov chain models of Ca(2+) release sites composed of 10-40 three-state intracellular Ca(2+) channels that are inactivated as well as activated by Ca(2+). We study how the statistics of simulated puffs and sparks depend on the kinetics and dissociation constant of Ca(2+) inactivation and find that puffs and sparks are often less sensitive to variations in the number of channels at release sites and strength of coupling via local [Ca(2+)] when the average fraction of inactivated channels is significant. Interestingly, we observe that the single channel kinetics of Ca(2+) inactivation influences the thermodynamic entropy production rate of Markov chain models of puffs and sparks. While excessively fast Ca(2+) inactivation can preclude puffs and sparks, moderately fast Ca(2+) inactivation often leads to time-irreversible puffs and sparks whose termination is facilitated by the recruitment of inactivated channels throughout the duration of the puff/spark event. On the other hand, Ca(2+) inactivation may be an important negative feedback mechanism even when its time constant is much greater than the duration of puffs and sparks. In fact, slow Ca(2+) inactivation can lead to release sites with a substantial fraction of inactivated channels that exhibit puffs and sparks that are nearly time-reversible and terminate without additional recruitment of inactivated channels.

Calcium oscillations

Changes in cellular Ca2+ concentration control a wide range of physiological processes, from the subsecond release of synaptic neurotransmitters, to the regulation of gene expression over months or years. Ca2+ can also trigger cell death through both apoptosis and necrosis, and so the regulation of cellular Ca2+ concentration must be tightly controlled through the concerted action of pumps, channels and buffers that transport Ca2+ into and out of the cell cytoplasm. A hallmark of cellular Ca2+ signalling is its spatiotemporal complexity: stimulation of cells by a hormone or neurotransmitter leads to oscillations in cytoplasmic Ca2+ concentration that can vary markedly in time course, amplitude, frequency, and spatial range. In this chapter we review some of the biological roles of Ca2+, the experimental characterisation of complex dynamic changes in Ca2+ concentration, and attempts to explain this complexity using computational models. We consider the 'toolkit' of cellular proteins which influence Ca2+ concentrarion, describe mechanistic models of key elements of the toolkit, and fit these into the framework of whole cell models of Ca2+ oscillations and waves. Finally, we will touch on recent efforts to use stochastic modelling to elucidate elementary Ca2+ signal events, and how these may evolve into global signals.

Rapid ligand-regulated gating kinetics of single inositol 1,4,5-trisphosphate receptor Ca2+ release channels

The ubiquitous inositol 1,4,5-trisphosphate receptor (InsP(3)R) intracellular Ca(2+) release channel is engaged by thousands of plasma membrane receptors to generate Ca(2+) signals in all cells. Understanding how complex Ca(2+) signals are generated has been hindered by a lack of information on the kinetic responses of the channel to its primary ligands, InsP(3) and Ca(2+), which activate and inhibit channel gating. Here, we describe the kinetic responses of single InsP(3)R channels in native endoplasmic reticulum membrane to rapid ligand concentration changes with millisecond resolution, using a new patch-clamp configuration. The kinetics of channel activation and deactivation showed novel Ca(2+) regulation and unexpected ligand cooperativity. The kinetics of Ca(2+)-mediated channel inhibition showed the single-channel bases for fundamental Ca(2+) release events and Ca(2+) release refractory periods. These results provide new insights into the channel regulatory mechanisms that contribute to complex spatial and temporal features of intracellular Ca(2+) signals.

Complex intracellular calcium oscillations. A theoretical exploration of possible mechanisms

Intracellular Ca(2+) oscillations are commonly observed in a large number of cell types in response to stimulation by an extracellular agonist. In most cell types the mechanism of regular spiking is well understood and models based on Ca(2+)-induced Ca(2+) release (CICR) can account for many experimental observations. However, cells do not always exhibit simple Ca(2+) oscillations. In response to given agonists, some cells show more complex behaviour in the form of bursting, i.e. trains of Ca(2+) spikes separated by silent phases. Here we develop several theoretical models, based on physiologically plausible assumptions, that could account for complex intracellular Ca(2+) oscillations. The models are all based on one- or two-pool models based on CICR. We extend these models by (i) considering the inhibition of the Ca(2+)-release channel on a unique intracellular store at high cytosolic Ca(2+) concentrations, (ii) taking into account the Ca(2+)-activated degradation of inositol 1,4,5-trisphosphate (IP(3)), or (iii) considering explicity the evolution of the Ca(2+) concentration in two different pools, one sensitive and the other one insensitive to IP(3). Besides simple periodic oscillations, these three models can all account for more complex oscillatory behaviour in the form of bursting. Moreover, the model that takes the kinetics of IP(3) into account shows chaotic behaviour.

Simulations of intracellular calcium release dynamics in response to a high-intensity, ultrashort electric pulse

Numerical simulations for electrically induced, intracellular calcium release from the endoplasmic reticulum are reported. A two-step model is used for self-consistency. Distributed electrical circuit representation coupled with the Smoluchowski equation yields the ER membrane nanoporation for calcium outflow based on a numerical simulation. This is combined with the continuum Li-Rinzel model and drift diffusion for calcium dynamics. Our results are shown to be in agreement with reported calcium release data. A modest increase (rough doubling) of the cellular calcium is predicted in the absence of extra-cellular calcium. In particular, the applied field of 15 kV/cm with 60 ns pulse duration makes for a strong comparison. No oscillations are predicted and the net recovery period of about 5 min are both in agreement with published experimental results. A quantitative explanation for the lack of such oscillatory behavior, based on the density dependent calcium fluxes, is also provided.

A kinetic Monte Carlo simulation study of inositol 1,4,5-trisphosphate receptor (IP3R) calcium release channel

Most of the previously theoretical studies about the stochastic nature of the IP3R calcium release channel gating use the chemical master equation (CME) approach. Because of the limitations of this approach we have used a stochastic simulation algorithm (SSA) presented by Gillespie. A single subunit of De Young-Keizer (DYK) model was simulated using Gillespie algorithm. The model has been considered in its complete form with eight states. We investigate the conditions which affect the open state of the model. Calcium concentrations were the subject of fluctuation in the previous works while in this study the population of the states is the subject of stochastic fluctuations. We found out that decreasing open probability is a function of Ca(2+) concentration in fast time domain, while in slow time domain it is a function of IP3 concentration. Studying the population of each state shows a time dependent reaction pattern in fast and medium time domains (10(-4) and 10(-3)s). In this pattern the state of X(010) has a determinative role in selecting the open state path. Also, intensity and frequency of fluctuations and Ca(2+) inhibitions have been studied. The results indicate that Gillespie algorithm can be a better choice for studying such systems, without using any approximation or elimination while having acceptable accuracy. In comparison with the chemical master equation, Gillespie algorithm is also provides a wide area for studying biological systems from other points of view.

The dynamics of luminal depletion and the stochastic gating of Ca2+-activated Ca2+ channels and release sites

Single channel models of intracellular calcium (Ca(2+)) channels such as the 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca(2+)-dependent transitions are mediated by constant background cytosolic [Ca(2+)]. This assumption neglects the fact that Ca(2+) released by open channels may influence subsequent gating through the processes of Ca(2+)-activation or inactivation. Similarly, the influence of the dynamics of luminal depletion on the stochastic gating of intracellular Ca(2+) channels is often neglected, in spite of the fact that the sarco/endoplasmic reticulum [Ca(2+)] near the luminal face of intracellular Ca(2+) channels influences the driving force for Ca(2+), the rate of Ca(2+) release, and the magnitude and time course of the consequent increase in cytosolic domain [Ca(2+)]. Here we analyze how the steady-state open probability of several minimal Ca(2+)-regulated Ca(2+) channel models depends on the conductance of the channel and the time constants for the relaxation of elevated cytosolic [Ca(2+)] and depleted luminal [Ca(2+)] to the bulk [Ca(2+)] of both compartments. Our approach includes Monte Carlo simulation as well as numerical solution of a system of advection-reaction equations for the multivariate probability density of elevated cytosolic [Ca(2+)] and depleted luminal [Ca(2+)] conditioned on each state of the stochastically gating channel. Both methods are subsequently used to study the role of luminal depletion in the dynamics of Ca(2+) puff/spark termination in release sites composed of Ca(2+) channels that are activated, but not inactivated, by cytosolic Ca(2+). The probability density approach shows that such minimal Ca(2+) release site models may exhibit puff/spark-like dynamics in either of two distinct parameter regimes. In one case, puffs/spark termination is due to the process of stochastic attrition and facilitated by rapid Ca(2+) domain collapse [cf. DeRemigio, H., Smith, G., 2005. The dynamics of stochastic attrition viewed as an absorption time on a terminating Markov chain. Cell Calcium 38, 73-86]. In the second case, puff/spark termination is promoted by the local depletion of luminal Ca(2+).

Calcium dynamics on a stochastic reaction-diffusion lattice model

We study a stochastic reaction-diffusion lattice model for describing the calcium dynamics in the endoplasmic reticulum (ER) membrane. Calcium channels and calcium ions are placed in two interpenetrating square lattices which are connected by calcium release and diffusion. Calcium ions are released from the ER through the channels and they can both remain in the membrane or spontaneously leave the membrane into the cytosol. The state of the channel is modulated by calcium ions: a channel can be open, closed, or inactive. The model is studied by numerical simulations and mean field theory and exhibits a phase transition from an active state to an absorbing state which is the result of the catalytic calcium release. The critical behavior of the model is in the directed percolation universality class.

A minimal model for decoding of time-limited Ca2+ oscillations

Calcium oscillations regulate several cellular processes by activating particular proteins. Most theoretical studies focused on the idealized situation of infinitely long oscillations. Here we analyze information transfer by time-limited calcium spike trains. We show that proteins can be selectively activated in a resonance-like manner by time-limited spike trains of different frequencies, while infinitely long oscillations do not show this resonance phenomenon. We found that proteins are activated more specifically by shorter oscillatory signals with narrower spikes.

Models of the inositol trisphosphate receptor

The inositol (1,4,5)-trisphosphate receptor (IPR) plays a crucial role in calcium dynamics in a wide range of cell types, and is often a central feature in quantitative models of calcium oscillations and waves. We review deterministic and stochastic mathematical models of the IPR, from the earliest ones of the 1970s and 1980s, to the most recent. The effects of IPR stochasticity on Ca2+ dynamics are briefly discussed.

Modeling and analysis of calcium signaling events leading to long-term depression in cerebellar Purkinje cells

Modeling and simulation of the calcium signaling events that precede long-term depression of synaptic activity in cerebellar Purkinje cells are performed using the Virtual Cell biological modeling framework. It is found that the unusually high density and low sensitivity of inositol-1,4,5-trisphosphate receptors (IP3R) are critical to the ability of the cell to generate and localize a calcium spike in a single dendritic spine. The results also demonstrate the model's capability to simulate the supralinear calcium spike observed experimentally during coincident activation of the parallel and climbing fibers. The sensitivity of the calcium spikes to certain biological and geometrical effects is investigated as well as the mechanisms that underlie the cell's ability to generate the supralinear spike. The sensitivity of calcium release rates from the IP3R to calcium concentrations, as well as IP3 concentrations, allows the calcium spike to form. The diffusion barrier caused by the small radius of the spine neck is shown to be important, as a threshold radius is observed above which a spike cannot be formed. Additionally, the calcium buffer capacity and diffusion rates from the spine are found to be important parameters in shaping the calcium spike.

The effect of residual on the stochastic gating of -regulated channel models

Single-channel models of intracellular Ca(2+) channels such as the inositol 1,4,5-trisphosphate receptor and ryanodine receptor often assume that Ca(2+)-dependent transitions are mediated by a constant background [Ca(2+)] as opposed to a dynamic [Ca(2+)] representing the formation and collapse of a localized Ca(2+) domain. This assumption neglects the fact that Ca(2+) released by open intracellular Ca(2+) channels may influence subsequent gating through the processes of Ca(2+)-activation or -inactivation. We study the effect of such "residual Ca(2+)" from previous channel opening on the stochastic gating of minimal and realistic single-channel models coupled to a restricted cytoplasmic compartment. Using Monte Carlo simulation as well as analytical and numerical solution of a system of advection-reaction equations for the probability density of the domain [Ca(2+)] conditioned on the state of the channel, we determine how the steady-state open probability (p(open)) of single-channel models of Ca(2+)-regulated Ca(2+) channels depends on the time constant for Ca(2+) domain formation and collapse. As expected, p(open) for a minimal model including Ca(2+) activation increases as the domain time constant becomes large compared to the open and closed dwell times of the channel, that is, on average the channel is activated by residual Ca(2+) from previous openings. Interestingly, p(open) for a channel model that is inactivated by Ca(2+) also increases as a function of the domain time constant when the maximum domain [Ca(2+)] is fixed, because slow formation of the Ca(2+) domain attenuates Ca(2+)-mediated inactivation. Conversely, when the source amplitude of the channel is fixed, increasing the domain time constant leads to elevated domain [Ca(2+)] and decreased open probability. Consistent with these observations, a realistic De Young-Keizer-like IP(3)R model responds to residual Ca(2+) with a steady-state open probability that is a monotonic function of the domain time constant, though minimal models that include both Ca(2+)-activation and -inactivation show more complex behavior. We show how the probability density approach described here can be generalized for arbitrarily complex channel models and for any value of the domain time constant. In addition, we present a comparatively simple numerical procedure for estimating p(open) for models of Ca(2+)-regulated Ca(2+) channels in the limit of a very fast or very slow Ca(2+) domain. When the ordinary differential equation for the [Ca(2+)] in a restricted cytoplasmic compartment is replaced by a partial differential equation for the buffered diffusion of intracellular Ca(2+) in a homogeneous isotropic cytosol, we find the dependence of p(open) on the buffer time constant is qualitatively similar to the above-mentioned results.

Modelling of calcium handling in airway myocytes

Airway myocytes are the primary effectors of airway reactivity which modulates airway resistance and hence ventilation. Stimulation of airway myocytes results in an increase in the cytosolic Ca(2+) concentration ([Ca(2+)](i)) and the subsequent activation of the contractile apparatus. Many contractile agonists, including acetylcholine, induce [Ca(2+)](i) increase via Ca(2+) release from the sarcoplasmic reticulum through InsP(3) receptors. Several models have been developed to explain the characteristics of InsP(3)-induced [Ca(2+)](i) responses, in particular Ca(2+) oscillations. The article reviews the modelling of the major structures implicated in intracellular Ca(2+) handling, i.e., InsP(3) receptors, SERCAs, mitochondria and Ca(2+)-binding cytosolic proteins. We developed theoretical models specifically dedicated to the airway myocyte which include the major mechanisms responsible for intracellular Ca(2+) handling identified in these cells. These biocomputations pointed out the importance of the relative proportion of InsP(3) receptor isoforms and the respective role of the different mechanisms responsible for cytosolic Ca(2+) clearance in the pattern of [Ca(2+)](i) variations. We have developed a theoretical model of membrane conductances that predicts the variations in membrane potential and extracellular Ca(2+) influx. Stimulation of this model by simulated increase in [Ca(2+)](i) predicts membrane depolarisation, but not great enough to trigger a significant opening of voltage-dependant Ca(2+) channels. This may explain why airway contraction induced by cholinergic stimulation does not greatly depend on extracellular calcium. The development of such models of airway myocytes is important for the understanding of the cellular mechanisms of airway reactivity and their possible modulation by pharmacological agents.

Lattice model for calcium dynamics

We present a simplified lattice model to study calcium dynamics in the endoplasmic reticulum membrane. Calcium channels and calcium ions are placed in two interpenetrating square lattices which are connected in two ways: (i) via calcium release and (ii) because transitions between channel states are calcium dependent. The opening or closing of a channel is a stochastic process controlled by two functions which depend on the calcium density on the channel neighborhood. The model is studied through mean field calculations and simulations. We show that the critical behavior of the model changes drastically depending on the opening/closing functions. For certain choices of these functions, all channels are closed at very low and high calcium densities and the model presents one absorbing state.

Targeted reduction of complex models with time scale hierarchy--a case study

With the increasing flow of biological data there is a growing demand for mathematical tools whereby essential aspects of complex causal dynamic models can be captured and detected by simpler mathematical models without sacrificing too much of the realism provided by the original ones. Given the presence of a time scale hierarchy, singular perturbation techniques represent an elegant method for making such minimised mathematical representations. Any reduction of a complex model by singular perturbation methods is a targeted reduction by the fact that one has to pick certain mechanisms, processes or aspects thought to be essential in a given explanatory context. Here we illustrate how such a targeted reduction of a complex model of melanogenesis in mammals recently developed by the authors provides a way to improve the understanding of how the melanogenic system may behave in a switch-like manner between production of the two major types of melanins. The reduced model is shown by numerical means to be in good quantitative agreement with the original model. Furthermore, it is shown how the reduced model discloses hidden robustness features of the full model, and how the making of a reduced model represents an efficient analytical sensitivity analysis. In addition to yielding new insights concerning the melanogenic system, the paper provides an illustration of a protocol that could be followed to make validated simplifications of complex biological models possessing time scale hierarchies.

A model of calcium waves in pancreatic and parotid acinar cells

We construct a mathematical model of Ca(2+) wave propagation in pancreatic and parotid acinar cells. Ca(2+) release is via inositol trisphosphate receptors and ryanodine receptors that are distributed heterogeneously through the cell. The apical and basal regions are separated by a region containing the mitochondria. In response to a whole-cell, homogeneous application of inositol trisphosphate (IP(3)), the model predicts that 1), at lower concentrations of IP(3), the intracellular waves in pancreatic cells begin in the apical region and are actively propagated across the basal region by Ca(2+) release through ryanodine receptors; 2), at higher [IP(3)], the waves in pancreatic and parotid cells are not true waves but rather apparent waves, formed as the result of sequential activation of inositol trisphosphate receptors in the apical and basal regions; 3), the differences in wave propagation in pancreatic and parotid cells can be explained in part by differences in inositol trisphosphate receptor density; 4), in pancreatic cells, increased Ca(2+) uptake by the mitochondria is capable of restricting Ca(2+) responses to the apical region, but that this happens only for a relatively narrow range of [IP(3)]; and 5), at higher [IP(3)], the apical and basal regions of the cell act as coupled Ca(2+) oscillators, with the basal region partially entrained to the apical region.

Modeling the dependence of the period of intracellular Ca2+ waves on SERCA expression

Contrary to intuitive expectations, overexpression of sarco-endoplasmic reticulum (ER) Ca(2+) ATPases (SERCAs) in Xenopus oocytes leads to a decrease in the period and an increase in the amplitude of intracellular Ca(2+) waves. Here we examine these experimental findings by modeling Ca(2+) release using a modified Othmer-Tang-model. An increase in the period and a reduction in the amplitude of Ca(2+) wave activity are obtained when increases in SERCA density are simulated while keeping all other parameters of the model constant. However, Ca(2+) wave period can be reduced and the wave amplitude and velocity can be significantly increased when an increase in the luminal ER Ca(2+) concentration due to SERCA overexpression is incorporated into the model. Increased luminal Ca(2+) occurs because increased SERCA activity lowers cytosolic Ca(2+), which is partially replenished by Ca(2+) influx across the plasma membrane. These simulations are supported by experimental data demonstrating higher luminal Ca(2+) levels, decreased periods, increased amplitude, and increased velocity of Ca(2+) waves in response to increased SERCA density.

Wave bifurcation and propagation failure in a model of Ca(2+) release

The De Young Keizer model for intracellular calcium oscillations is based around a detailed description of the dynamics for inositol trisphosphate (IP(3)) receptors. Systematic reductions of the kinetic schemes for IP(3) dynamics have proved especially fruitful in understanding the transition from excitable to oscillatory behaviour. With the inclusion of diffusive transport of calcium ions the model also supports wave propagation. The analysis of waves, even in reduced models, is typically only possible with the use of numerical bifurcation techniques. In this paper we review the travelling wave properties of the biophysical De Young Keizer model and show that much of its behaviour can be reproduced by a much simpler Fire-Diffuse-Fire (FDF) type model. The FDF model includes both a refractory process and an IP(3) dependent threshold. Parameters of the FDF model are constrained using a comprehensive numerical bifurcation analysis of solitary pulses and periodic waves in the De Young Keizer model. The linear stability of numerically constructed solution branches is calculated using pseudospectral techniques. The combination of numerical bifurcation and stability analysis also allows us to highlight the mechanisms that give rise to propagation failure. Moreover, a kinematic theory of wave propagation, based around numerically computed dispersion curves is used to predict waves which connect periodic orbits. Direct numerical simulations of the De Young Keizer model confirm this prediction. Corresponding travelling wave solutions of the FDF model are obtained analytically and are shown to be in good qualitative agreement with those of the De Young Keizer model. Moreover, the FDF model may be naturally extended to include the discrete nature of calcium stores within a cell, without the loss of analytical tractability. By considering calcium stores as idealised point sources we are able to explicitly construct solutions of the FDF model that correspond to saltatory periodic travelling waves.

Simulation of waves in calcium models with 3D spherical geometry

Waves of calcium ions are present in fertilized eggs of many species. Models for pulse and tidal wave propagation have usually been studied in one or two spatial coordinates only. We examine in three spatial coordinates some established models, based on Ca(2+)-induced Ca(2+)-release from both (assumed) continuously or heterogeneously distributed stores of endoplasmic reticulum (ER) through channels activated by inositol triphosphate (IP(3)). With continuous IP(3) distribution decreasing radially towards the interior, we obtain concave pulse shapes for waves penetrating the interior. Concave waves are also recorded in systems with ER confined to distributions of small spheres (microdomains) inside the cell, which we simulate for front waves (tides) in bistable systems.

Long-range signal transmission in autocrine relays

Intracellular signaling induced by peptide growth factors can stimulate secretion of these molecules into the extracellular medium. In autocrine and paracrine networks, this can establish a positive feedback loop between ligand binding and ligand release. When coupled to intercellular communication by autocrine ligands, this positive feedback can generate constant-speed traveling waves. To demonstrate that, we propose a mechanistic model of autocrine relay systems. The model is relevant to the physiology of epithelial layers and to a number of in vitro experimental formats. Using asymptotic and numerical tools, we find that traveling waves in autocrine relays exist and have a number of unusual properties, such as an optimal ligand binding strength necessary for the maximal speed of propagation. We compare our results to recent observations of autocrine and paracrine systems and discuss the steps toward experimental tests of our predictions.