Fundamental properties of Ca2+ signals

BAPTA-based potentiometric polymer sensor: towards sensing inflammations and infections

Potentiometric ion sensors represent a significant subgroup of electrochemical sensors. In this study, we have developed a potentiometric sensor using an electrically conductive copolymer of 2,2'-bithiophene (BT) and 1,2-bis(o-aminophenoxy)ethane-N,N,N',N'-tetraacetic acid (BAPTA) for the selective detection of Ca2+ ions in extracellular interstitial fluids. The integration of BAPTA with its highly selective calcium chelating properties into a polymer matrix via electrochemical polymerization results in a sensitive conductive polymer layer that effectively detects the presence of calcium ions. This sensor aims at the early detection of inflammation or infection around implants because local calcium concentration is strongly elevated in interstitial fluid in such pathologies. The potentiometric study proves the incorporation of BAPTA into the polymer matrix was successful and its potential decreased upon calcium binding demonstrating the Nernstian behavior with a slope of approximately 20 ± 0.3 mV per decade in the concentration range from 0.1 mM to 1 mM. Moreover, the selectivity coefficient of -0.4 was measured by SSM and calculated from the Nicolsky-Eisenmann equation, which indicates selectivity towards Ca2+ ions with respect to Mg2+ ions.

On the role of calcium diffusion and its rapid buffering in intraflagellar signaling

We have considered the realistic mechanism of rapid Ca2+ (calcium ion) buffering within the wave of calcium ions progressing along the flagellar axoneme. This buffering is an essential part of the Ca2+ signaling pathway aimed at controlling the bending dynamics of flagella. It is primarily achieved by the mobile region of calmodulin molecules and by stationary calaxin, as well as by the part of calmodulin bound to calcium/calmodulin-dependent kinase II and kinase C. We derived and elaborated a model of Ca2+ diffusion within a signaling wave in the presence of these molecules which rapidly buffer Ca2+. This approach has led to a single nonlinear transport equation for the Ca2+ wave that contains the effects brought about by both as necessary buffers for signaling. The presence of mobile buffer calmodulin gives rise to a transport equation that is not strictly diffusive but also exhibits a sink-like effect. We solved straightforwardly the final transport equation in an analytical framework and obtained the implied function of calcium concentration. The effective diffusion coefficient depends on local Ca2+ concentration. It is plausible that these buffers' presence can impact Ca2+ wave speed and shape, which are essential for decoding Ca2+ signaling in flagella. We present the solution of the transport equation for a few specified cases with physiologically reasonable sets of parameters involved.

A Review of Potato Salt Tolerance

Potato is the world's fourth largest food crop. Due to limited arable land and an ever-increasing demand for food from a growing population, it is critical to increase crop yields on existing acreage. Soil salinization is an increasing problem that dramatically impacts crop yields and restricts the growing area of potato. One possible solution to this problem is the development of salt-tolerant transgenic potato cultivars. In this work, we review the current potato planting distribution and the ways in which it overlaps with salinized land, in addition to covering the development and utilization of potato salt-tolerant cultivars. We also provide an overview of the current progress toward identifying potato salt tolerance genes and how they may be deployed to overcome the current challenges facing potato growers.

Close agreement between deterministic versus stochastic modeling of first-passage time to vesicle fusion

Ca2+-dependent cell processes, such as neurotransmitter or endocrine vesicle fusion, are inherently stochastic due to large fluctuations in Ca2+ channel gating, Ca2+ diffusion, and Ca2+ binding to buffers and target sensors. However, previous studies revealed closer-than-expected agreement between deterministic and stochastic simulations of Ca2+ diffusion, buffering, and sensing if Ca2+ channel gating is not Ca2+ dependent. To understand this result more fully, we present a comparative study complementing previous work, focusing on Ca2+ dynamics downstream of Ca2+ channel gating. Specifically, we compare deterministic (mean-field/mass-action) and stochastic simulations of vesicle exocytosis latency, quantified by the probability density of the first-passage time (FPT) to the Ca2+-bound state of a vesicle fusion sensor, following a brief Ca2+ current pulse. We show that under physiological constraints, the discrepancy between FPT densities obtained using the two approaches remains small even if as few as ∼50 Ca2+ ions enter per single channel-vesicle release unit. Using a reduced two-compartment model for ease of analysis, we illustrate how this close agreement arises from the smallness of correlations between fluctuations of the reactant molecule numbers, despite the large magnitude of fluctuation amplitudes. This holds if all relevant reactions are heteroreaction between molecules of different species, as is the case for bimolecular Ca2+ binding to buffers and downstream sensor targets. In this case, diffusion and buffering effectively decorrelate the state of the Ca2+ sensor from local Ca2+ fluctuations. Thus, fluctuations in the Ca2+ sensor's state underlying the FPT distribution are only weakly affected by the fluctuations in the local Ca2+ concentration around its average, deterministically computable value.

Simulation of Calcium Dynamics in Realistic Three-Dimensional Domains

The cytosolic concentration of free calcium ions ([Ca2+]) is an important intracellular messenger in most cell types, and the spatial distribution of [Ca2+] is often critical. In a salivary gland acinar cell, a polarised epithelial cell, whose principal function is to transport water and thus secrete saliva, [Ca2+] controls the secretion of primary saliva, but increases in [Ca2+] are localised to the apical regions of the cell. Hence, any quantitative explanation of how [Ca2+] controls saliva secretion must take into careful account the spatial distribution of the various Ca2+ sources, Ca2+ sinks, and Ca2+-sensitive ion channels. Based on optical slices, we have previously constructed anatomically accurate three-dimensional models of seven salivary gland acinar cells, and thus shown that a model in which Ca2+ responses are confined to the apical regions of the cell is sufficient to provide a quantitative and predictive explanation of primary saliva secretion. However, reconstruction of such anatomically accurate cells is extremely time consuming and inefficient. Here, we present an alternative, mostly automated method of constructing three-dimensional cells that are approximately anatomically accurate and show that the new construction preserves the quantitative accuracy of the model.

Nonequilibrium thermodynamics of glycolytic traveling wave: Benjamin-Feir instability

Evolution of the nonequilibrium thermodynamic entities corresponding to dynamics of the Hopf instabilities and traveling waves at a nonequilibrium steady state of a spatially extended glycolysis model is assessed here by implementing an analytically tractable scheme incorporating a complex Ginzburg-Landau equation (CGLE). In the presence of self and cross diffusion, a more general amplitude equation exploiting the multiscale Krylov-Bogoliubov averaging method serves as an essential tool to reveal the various dynamical instability criteria, especially Benjamin-Feir (BF) instability, to estimate the corresponding nonlinear dispersion relation of the traveling wave pattern. The critical control parameter, wave-number selection criteria, and magnitude of the complex amplitude for traveling waves are modified by self- and cross-diffusion coefficients within the oscillatory regime, and their variabilities are exhibited against the amplitude equation. Unlike the traveling waves, a low-amplitude broad region appears for the Hopf instability in the concentration dynamics as the system phase passes through minima during its variation with the control parameter. The total entropy production rate of the uniform Hopf oscillation and glycolysis wave not only qualitatively reflects the global dynamics of concentrations of intermediate species but almost quantitatively. Despite the crucial role of diffusion in generating and shaping the traveling waves, the diffusive part of the entropy production rate has a negligible contribution to the system's total entropy production rate. The Hopf instability shows a more complex and colossal change in the energy profile of the open nonlinear system than in the traveling waves. A detailed analysis of BF instability shows a contrary nature of the semigrand Gibbs free energy with discrete and continuous wave numbers for the traveling wave. We hope the Hopf and traveling wave pattern around the BF instability in terms of energetics and dissipation will open up new applications of such dynamical phenomena.

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

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

Stationary Ca2+ nanodomains in the presence of buffers with two binding sites

We examine closed-form approximations for the equilibrium Ca²⁺ and buffer concentrations near a point Ca²⁺ source representing a Ca²⁺ channel, in the presence of a mobile buffer with two Ca²⁺ binding sites activated sequentially and possessing distinct binding affinities and kinetics. This allows us to model the impact on Ca²⁺ nanodomains of realistic endogenous Ca²⁺ buffers characterized by cooperative Ca²⁺ binding, such as calretinin. The approximations we present involve a combination or rational and exponential functions, whose parameters are constrained using the series interpolation method that we recently introduced for the case of simpler Ca²⁺ buffers with a single Ca²⁺ binding site. We conduct extensive parameter sensitivity analysis and show that the obtained closed-form approximations achieve reasonable qualitative accuracy for a wide range of buffer's Ca²⁺ binding properties and other relevant model parameters. In particular, the accuracy of the derived approximants exceeds that of the rapid buffering approximation in large portions of the relevant parameter space.

The Oscillation Amplitude, Not the Frequency of Cytosolic Calcium, Regulates Apoptosis Induction

Although a rising concentration of cytosolic Ca²⁺ has long been recognized as an essential signal for apoptosis, the dynamical mechanisms by which Ca²⁺ regulates apoptosis are not clear yet. To address this, we constructed a computational model that integrates known biochemical reactions and can reproduce the dynamical behaviors of Ca²⁺-induced apoptosis as observed in experiments. Model analysis shows that oscillating Ca²⁺ signals first convert into gradual signals and eventually transform into a switch-like apoptotic response. Via the two processes, the apoptotic signaling pathway filters the frequency of Ca²⁺ oscillations effectively but instead responds acutely to their amplitude. Collectively, our results suggest that Ca²⁺ regulates apoptosis mainly via oscillation amplitude, rather than frequency, modulation. This study not only provides a comprehensive understanding of how oscillatory Ca²⁺ dynamically regulates the complex apoptotic signaling network but also presents a typical example of how Ca²⁺ controls cellular responses through amplitude modulation.

Efficient Approximations for Stationary Single-Channel Ca2+ Nanodomains across Length Scales

We consider the stationary solution for the Ca²⁺ concentration near a point Ca²⁺ source describing a single-channel Ca²⁺ nanodomain in the presence of a single mobile Ca²⁺ buffer with 1:1 Ca²⁺ binding. We present computationally efficient approximants that estimate stationary single-channel Ca²⁺ nanodomains with great accuracy in broad regions of parameter space. The presented approximants have a functional form that combines rational and exponential functions, which is similar to that of the well-known excess buffer approximation and the linear approximation but with parameters estimated using two novel, to our knowledge, methods. One of the methods involves interpolation between the short-range Taylor series of the free buffer concentration and its long-range asymptotic series in inverse powers of distance from the channel. Although this method has already been used to find Padé (rational-function) approximants to single-channel Ca²⁺ and buffer concentrations, extending this method to interpolants combining exponential and rational functions improves accuracy in a significant fraction of the relevant parameter space. A second method is based on the variational approach and involves a global minimization of an appropriate functional with respect to parameters of the chosen approximations. An extensive parameter-sensitivity analysis is presented, comparing these two methods with previously developed approximants. Apart from increased accuracy, the strength of these approximants is that they can be extended to more realistic buffers with multiple binding sites characterized by cooperative Ca²⁺ binding, such as calmodulin and calretinin.

Strong current response to slow modulation: A metabolic case-study

We study the current response to periodic driving of a crucial biochemical reaction network, namely, substrate inhibition. We focus on the conversion rate of substrate into product under time-varying metabolic conditions, modeled by a periodic modulation of the product concentration. We find that the system exhibits a strong nonlinear response to small driving frequencies both for the mean time-averaged current and for the fluctuations. For the first, we obtain an analytic formula by coarse-graining the original model to a solvable one. The result is nonperturbative in the modulation amplitude and frequency. We then refine the picture by studying the stochastic dynamics of the full system using a large deviation approach that allows us to show the resonant effect at the level of the time-averaged variance and signal-to-noise ratio. Finally, we discuss how this nonequilibrium effect may play a role in metabolic and synthetic networks.

Energetic and entropic cost due to overlapping of Turing-Hopf instabilities in the presence of cross diffusion

A systematic introduction to nonequilibrium thermodynamics of dynamical instabilities are considered for an open nonlinear system beyond conventional Turing pattern in presence of cross diffusion. An altered condition of Turing instability in presence of cross diffusion is best reflected through a critical control parameter and wave number containing both the self- and cross-diffusion coefficients. Our main focus is on entropic and energetic cost of Turing-Hopf interplay in stationary pattern formation. Depending on the relative dispositions of Turing-Hopf codimensional instabilities from the reaction-diffusion equation it clarifies two aspects: energy cost of pattern formation, especially how Hopf instability can be utilized to dictate a stationary concentration profile, and the possibility of revealing nonequilibrium phase transition. In the Brusselator model, to understand these phenomena, we have analyzed through the relevant complex Ginzberg-Landau equation using multiscale Krylov-Bogolyubov averaging method. Due to Hopf instability it is observed that the cross-diffusion parameters can be a source of huge change in free-energy and concentration profiles.

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.

Extension of Rapid Buffering Approximation to Ca2+ Buffers with Two Binding Sites

Fundamental cell processes such as synaptic neurotransmitter release, endocrine hormone secretion, and myocyte contraction are controlled by highly localized calcium (Ca²⁺) signals resulting from brief openings of trans-membrane Ca²⁺ channels. On short temporal and spatial scales, the corresponding local Ca²⁺ nanodomains formed in the vicinity of a single or several open Ca²⁺ channels can be effectively approximated by quasi-stationary solutions. The rapid buffering approximation (RBA) is one of the most powerful of such approximations, and is based on the assumption of instantaneous equilibration of the bimolecular Ca²⁺ buffering reaction, combined with the conservation condition for the total Ca²⁺ and buffer molecule numbers. Previously, RBA has been generalized to an arbitrary arrangement of Ca²⁺ channels on a flat membrane, in the presence of any number of simple Ca²⁺ buffers with one-to-one Ca²⁺ binding stoichiometry. However, many biological buffers have multiple binding sites. For example, buffers and sensors phylogenetically related to calmodulin consist of two Ca²⁺-binding domains (lobes), with each domain binding two Ca²⁺ ions in a cooperative manner. Here we consider an extension of RBA to such buffers with two interdependent Ca²⁺ binding sites. We show that in the presence of such buffers, RBA solution is given by the solution to a cubic equation, analogous to the quadratic equation describing RBA in the case of a simple, one-to-one Ca²⁺ buffer. We examine in detail the dependence of RBA accuracy on buffering parameters, to reveal conditions under which RBA provides sufficient precision.

Information Thermodynamics of Turing Patterns

We set up a rigorous thermodynamic description of reaction-diffusion systems driven out of equilibrium by time-dependent space-distributed chemostats. Building on the assumption of local equilibrium, nonequilibrium thermodynamic potentials are constructed exploiting the symmetries of the chemical network topology. It is shown that the canonical (resp. semigrand canonical) nonequilibrium free energy works as a Lyapunov function in the relaxation to equilibrium of a closed (resp. open) system, and its variation provides the minimum amount of work needed to manipulate the species concentrations. The theory is used to study analytically the Turing pattern formation in a prototypical reaction-diffusion system, the one-dimensional Brusselator model, and to classify it as a genuine thermodynamic nonequilibrium phase transition.

What Is Required for Neuronal Calcium Waves? A Numerical Parameter Study

Neuronal calcium signals propagating by simple diffusion and reaction with mobile and stationary buffers are limited to cellular microdomains. The distance intracellular calcium signals can travel may be significantly increased by means of calcium-induced calcium release from internal calcium stores, notably the endoplasmic reticulum. The organelle, which can be thought of as a cell-within-a-cell, is able to sequester large amounts of cytosolic calcium ions via SERCA pumps and selectively release them into the cytosol through ryanodine receptor channels leading to the formation of calcium waves. In this study, we set out to investigate the basic properties of such dendritic calcium waves and how they depend on the three parameters dendrite radius, ER radius and ryanodine receptor density in the endoplasmic membrane. We demonstrate that there are stable and abortive regimes for calcium waves, depending on the above morphological and physiological parameters. In stable regimes, calcium waves can travel across long dendritic distances, similar to electrical action potentials. We further observe that abortive regimes exist, which could be relevant for spike-timing dependent plasticity, as travel distances and wave velocities vary with changing intracellular architecture. For some of these regimes, analytic functions could be derived that fit the simulation data. In parameter spaces, that are non-trivially influenced by the three-dimensional calcium concentration profile, we were not able to derive such a functional description, demonstrating the mathematical requirement to model and simulate biochemical signaling in three-dimensional space.

Optogenetic Inhibition of Gαq Protein Signaling Reduces Calcium Oscillation Stochasticity

As fast terminators of G-protein coupled receptor (GPCR) signaling, regulators of G-protein signaling (RGS) serve critical roles in fine-tuning second messenger levels and, consequently, cellular responses to external stimuli. Here, we report the creation of an optogenetic RGS2 (opto-RGS2) that suppresses agonist-evoked calcium oscillations by the inactivation of Gαq protein. In this system, cryptochrome-mediated heterodimerization of the catalytic RGS2-box with its N-terminal amphipathic helix reconstitutes a functional membrane-localized complex that can dynamically suppress store-operated release of calcium. Engineered opto-RGS2 cell lines were used to establish the role of RGS2 as a key inhibitory feedback regulator of the stochasticity of the Gαq-mediated calcium spike timing. RGS2 reduced the stochasticity of carbachol-stimulated calcium oscillations, and the feedback inhibition was coupled to the global calcium elevation by calmodulin/RGS2 interactions. The identification of a critical negative feedback circuit exemplifies the utility of optogenetic approaches for interrogating RGS/GPCR biology and calcium encoding principles through temporally precise molecular gain-of-function.

Analysis of network motifs in cellular regulation: Structural similarities, input-output relations and signal integration

Much of the complexity of regulatory networks derives from the necessity to integrate multiple signals and to avoid malfunction due to cross-talk or harmful perturbations. Hence, one may expect that the input-output behavior of larger networks is not necessarily more complex than that of smaller network motifs which suggests that both can, under certain conditions, be described by similar equations. In this review, we illustrate this approach by discussing the similarities that exist in the steady state descriptions of a simple bimolecular reaction, covalent modification cycles and bacterial two-component systems. Interestingly, in all three systems fundamental input-output characteristics such as thresholds, ultrasensitivity or concentration robustness are described by structurally similar equations. Depending on the system the meaning of the parameters can differ ranging from protein concentrations and affinity constants to complex parameter combinations which allows for a quantitative understanding of signal integration in these systems. We argue that this approach may also be extended to larger regulatory networks.

A Bayesian approach to modelling heterogeneous calcium responses in cell populations

Calcium responses have been observed as spikes of the whole-cell calcium concentration in numerous cell types and are essential for translating extracellular stimuli into cellular responses. While there are several suggestions for how this encoding is achieved, we still lack a comprehensive theory. To achieve this goal it is necessary to reliably predict the temporal evolution of calcium spike sequences for a given stimulus. Here, we propose a modelling framework that allows us to quantitatively describe the timing of calcium spikes. Using a Bayesian approach, we show that Gaussian processes model calcium spike rates with high fidelity and perform better than standard tools such as peri-stimulus time histograms and kernel smoothing. We employ our modelling concept to analyse calcium spike sequences from dynamically-stimulated HEK293T cells. Under these conditions, different cells often experience diverse stimulus time courses, which is a situation likely to occur in vivo. This single cell variability and the concomitant small number of calcium spikes per cell pose a significant modelling challenge, but we demonstrate that Gaussian processes can successfully describe calcium spike rates in these circumstances. Our results therefore pave the way towards a statistical description of heterogeneous calcium oscillations in a dynamic environment.

Upon stimulation a large number of cell types respond with transient increases of the intracellular calcium concentration, which often take the form of repetitive spikes. It is therefore believed that calcium spikes play a central role in cellular signal transduction. A critical feature of these calcium spikes is that they occur randomly, which raises the question of how we can predict the timing of calcium spikes. We here show that by using Bayesian ideas and concepts from stochastic processes, we can quantitatively compute the calcium spike rate for a given stimulus. Our analysis also demonstrates that traditional methods for spike rate estimation perform less favourably compared to a Bayesian approach when small numbers of cells are investigated. To test our methodology under conditions that closely mimic those experienced in vivo we challenged cells with agonist concentrations that vary both in space and time. We find that cells that experience similar stimulus profiles are described by similar calcium spike rates. This suggests that calcium spike rates may constitute a quantitative description of whole-cell calcium spiking that reflects both the randomness and the spatiotemporal organisation of the calcium signalling machinery.

Mapping Interpuff Interval Distribution to the Properties of Inositol Trisphosphate Receptors

Tightly clustered inositol trisphosphate receptors (IP₃Rs) control localized Ca²⁺ liberation from the endoplasmic reticulum to generate repetitive Ca²⁺ puffs. Distributions of the interpuff interval (IPI), i.e., the waiting time between successive puffs, are found to be well characterized by a probability density function involving only two parameters, λ and ξ, which represent the basal rate of puff generation and the recovery rate from refractoriness, respectively. However, how the two parameters depend on the kinetic parameters of single IP₃Rs in a cluster is still unclear. In this article, using a stochastic puff model and a single-channel data-based IP₃R model, we establish the dependencies of λ and ξ on two important IP₃R model parameters, IP₃ concentration ([IP₃]) and the recovery rate from Ca²⁺ inhibition (r_low). By varying [IP₃] and r_low in physiologically plausible ranges, we find that the ξ-λ plane is comprised of only two disjoint regions, a biologically impermissible region and a region where each parameter set (ξ, λ) can be caused by using two different combinations of [IP₃] and r_low. The two combinations utilize very different mechanisms to maintain the same IPI distribution, and the mechanistic difference provides a way of identifying IP₃R kinetic parameters by observing properties of the IPI.

Padé Approximation of a Stationary Single-Channel Ca2+ Nanodomain

We consider the stationary solution for the Ca²⁺ concentration near a point Ca²⁺ source describing a single-channel Ca²⁺ nanodomain, in the presence of a single mobile buffer with one-to-one Ca²⁺ binding stoichiometry. Previously, a number of Ca²⁺ nanodomains approximations have been developed, for instance the excess buffer approximation (EBA), the rapid buffering approximation (RBA), and the linear approximation (LIN), each valid for appropriate buffering conditions. Apart from providing a simple method of estimating Ca²⁺ and buffer concentrations without resorting to computationally expensive numerical solution of reaction-diffusion equations, such approximations proved useful in revealing the dependence of nanodomain Ca²⁺ distribution on crucial parameters such as buffer mobility and its Ca²⁺ binding properties. In this study, we present a different form of analytic approximation, which is based on matching the short-range Taylor series of the nanodomain concentration with the long-range asymptotic series expressed in inverse powers of distance from channel location. Namely, we use a "dual" Padé rational function approximation to simultaneously match terms in the short- and the long-range series, and we show that this provides an accurate approximation to the nanodomain Ca²⁺ and buffer concentrations. We compare this approximation with the previously obtained approximations and show that it yields a better estimate of the free buffer concentration for a wide range of buffering conditions. The drawback of our method is that it has a complex algebraic form for any order higher than the lowest bilinear order, and cannot be readily extended to multiple Ca²⁺ channels. However, it may be possible to extend the Padé method to estimate Ca²⁺ nanodomains in the presence of cooperative Ca²⁺ buffers with two Ca²⁺ binding sites, the case that existing methods do not address.

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.

Role of time delay on intracellular calcium dynamics driven by non-Gaussian noises

Effect of time delay (τ) on intracellular calcium dynamics with non-Gaussian noises in transmission processes of intracellular Ca²⁺ is studied by means of second-order stochastic Runge-Kutta type algorithm. By simulating and analyzing time series, normalized autocorrelation function, and characteristic correlation time of cytosolic and calcium store’s Ca²⁺ concentration, the results exhibit: (i) intracellular calcium dynamics’s time coherence disappears and stability strengthens as τ → 0.1s; (ii) for the case of τ < 0.1s, the normalized autocorrelation functions of cytosolic and calcium store’s Ca²⁺ concentration show damped motion when τ is very short, but they trend to a level line as τ → 0.1s, and for the case of τ > 0.1s, they show different variation as τ increases, the former changes from underdamped motion to a level line, but the latter changes from damped motion to underdamped motion; and (iii) at the moderate value of time delay, reverse resonance occurs both in cytosol and calcium store.

Lung Beractant Increases Free Cytosolic Levels of Ca2+ in Human Lung Fibroblasts

Beractant, a natural surfactant, induces an antifibrogenic phenotype and apoptosis in normal human lung fibroblasts (NHLF). As intracellular Ca2+ signalling has been related to programmed cell death, we aimed to assess the effect of beractant on intracellular Ca2+ concentration ([Ca2+]i) in NHLF in vitro. Cultured NHLF were loaded with Fura-2 AM (3 μM) and Ca2+ signals were recorded by microfluorimetric techniques. Beractant causes a concentration-dependent increase in [Ca2+]i with a EC50 of 0.82 μg/ml. The application of beractant, at a concentration of 500 μg/ml, which has been shown to exert an apoptotic effect in human fibroblasts, elicited different patterns of Ca2+ signals in NHLF: a) a single Ca2+ spike which could be followed by b) Ca2+ oscillations, c) a sustained Ca2+ plateau or d) a sustained plateau overlapped by Ca2+ oscillations. The amplitude and pattern of Ca2+ transients evoked by beractant were dependent on the resting [Ca2+]i. Pharmacological manipulation revealed that beractant activates a Ca2+ signal through Ca2+ release from intracellular stores mediated by phospholipase Cβ (PLCβ), Ca2+ release from inositol 1,4,5-trisphosphate receptors (IP3Rs) and Ca2+ influx via a store-operated pathway. Moreover, beractant-induced Ca2+ release was abolished by preventing membrane depolarization upon removal of extracellular Na+ and Ca2+. Finally, the inhibition of store-operated channels prevented beractant-induced NHLF apoptosis and downregulation of α1(I) procollagen expression. Therefore, beractant utilizes SOCE to exert its pro-apoptotic and antifibrinogenic effect on NHLF.

Unifying principles of calcium wave propagation - Insights from a three-dimensional model for atrial myocytes

Atrial myocytes in a number of species lack transverse tubules. As a consequence the intracellular calcium signals occurring during each heartbeat exhibit complex spatio-temporal dynamics. These calcium patterns arise from saltatory calcium waves that propagate via successive rounds of diffusion and calcium-induced calcium release. The many parameters that impinge on calcium-induced calcium release and calcium signal propagation make it difficult to know a priori whether calcium waves will successfully travel, or be extinguished. In this study, we describe in detail a mathematical model of calcium signalling that allows the effect of such parameters to be independently assessed. A key aspect of the model is to follow the triggering and evolution of calcium signals within a realistic three-dimensional cellular volume of an atrial myocyte, but with low computational costs. This is achieved by solving the linear transport equation for calcium analytically between calcium release events and by expressing the onset of calcium liberation as a threshold process. The model makes non-intuitive predictions about calcium signal propagation. For example, our modelling illustrates that the boundary of a cell produces a wave-guiding effect that enables calcium ions to propagate further and for longer, and can subtly alter the pattern of calcium wave movement. The high spatial resolution of the modelling framework allows the study of any arrangement of calcium release sites. We demonstrate that even small variations in randomly positioned release sites cause highly heterogeneous cellular responses. This article is part of a Special Issue entitled: 13th European Symposium on Calcium.

Modulation of elementary calcium release mediates a transition from puffs to waves in an IP3R cluster model

The oscillating concentration of intracellular calcium is one of the most important examples for collective dynamics in cell biology. Localized releases of calcium through clusters of inositol 1,4,5-trisphosphate receptor channels constitute elementary signals called calcium puffs. Coupling by diffusing calcium leads to global releases and waves, but the exact mechanism of inter-cluster coupling and triggering of waves is unknown. To elucidate the relation of puffs and waves, we here model a cluster of IP3R channels using a gating scheme with variable non-equilibrium IP3 binding. Hybrid stochastic and deterministic simulations show that puffs are not stereotyped events of constant duration but are sensitive to stimulation strength and residual calcium. For increasing IP3 concentration, the release events become modulated at a timescale of minutes, with repetitive wave-like releases interspersed with several puffs. This modulation is consistent with experimental observations we present, including refractoriness and increase of puff frequency during the inter-wave interval. Our results suggest that waves are established by a random but time-modulated appearance of sustained release events, which have a high potential to trigger and synchronize activity throughout the cell.

May the remodeling of the Ca²⁺ toolkit in endothelial progenitor cells derived from cancer patients suggest alternative targets for anti-angiogenic treatment?

Endothelial progenitor cells (EPCs) may be recruited from bone marrow to sustain the metastatic switch in a number of solid cancers, including breast cancer (BC) and renal cellular carcinoma (RCC). Preventing EPC mobilization causes tumor shrinkage. Novel anti-angiogenic treatments have been introduced in therapy to inhibit VEGFR-2 signaling; unfortunately, these drugs blocked tumor angiogenesis in pre-clinical murine models, but resulted far less effective in human patients. Understanding the molecular mechanisms driving EPC proliferation and tubulogenesis in cancer patients could outline novel targets for alternative anti-angiogenic treatments. Store-operated Ca²⁺ entry (SOCE) regulates the growth of human EPCs, and it is mediated by the interaction between the endoplasmic reticulum Ca²⁺-sensor, Stim1, and the plasmalemmal Ca²⁺ channels, Orai1 and TRPC1. EPCs do not belong to the neoplastic clone: thus, unlike tumor endothelium and neoplastic cells, they should not remodel their Ca²⁺ toolkit in response to tumor microenvironment. However, our recent work demonstrated that EPCs isolated from naïve RCC patients (RCC-EPCs) undergo a dramatic remodeling of their Ca²⁺ toolkit by displaying a remarkable drop in the endoplasmic reticulum Ca²⁺ content, by down-regulating the expression of inositol-1,4,5-receptors (InsP3Rs), and by up-regulating Stim1, Orai1 and TRPC1. Moreover, EPCs are dramatically less sensitive to VEGF stimulation both in terms of Ca²⁺ signaling and of gene expression when isolated from tumor patients. Conversely, the pharmacological abolition of SOCE suppresses proliferation in these cells. These results question the suitability of VEGFR-2 as a therapeutically relevant target for anti-angiogenic treatments and hint at Orai1 and TRPC1 as more promising alternatives. This article is part of a Special Issue entitled: 13th European Symposium on Calcium.

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

The inositol trisphosphate receptor () 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 have arisen. Firstly, how best should the be modeled? In other words, what fundamental properties of the 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 , 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 , and thus the behavior of the 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, stochasticity is not essential for a qualitative prediction of how oscillation frequency depends on model parameters, and thus deterministic models demonstrate the same level of predictive capability as do stochastic models. We conclude that, firstly, calcium dynamics can be accurately modeled using simplified models, and, secondly, to obtain qualitative predictions of how oscillation frequency depends on parameters it is sufficient to use a deterministic model.

The inositol trisphosphate receptor () is one of the most important cellular components responsible for calcium oscillations. Over the past decade, two major questions about the have arisen. Firstly, what fundamental properties of the allow it to perform its function? Secondly, although calcium oscillations are caused by the stochastic properties of small numbers of is it possible for a deterministic model to be a reliable predictor of calcium dynamics? Using airway smooth muscle cells as an example, we show that calcium dynamics can be accurately modeled using simplified models, and, secondly, that deterministic models are qualitatively accurate predictors of calcium dynamics. These results are important for the study of calcium dynamics in many cell types.

Estimating the biophysical properties of neurons with intracellular calcium dynamics

We investigate the dynamics of a conductance-based neuron model coupled to a model of intracellular calcium uptake and release by the endoplasmic reticulum. The intracellular calcium dynamics occur on a time scale that is orders of magnitude slower than voltage spiking behavior. Coupling these mechanisms sets the stage for the appearance of chaotic dynamics, which we observe within certain ranges of model parameter values. We then explore the question of whether one can, using observed voltage data alone, estimate the states and parameters of the voltage plus calcium (V+Ca) dynamics model. We find the answer is negative. Indeed, we show that voltage plus another observed quantity must be known to allow the estimation to be accurate. We show that observing both the voltage time course V(t) and the intracellular Ca time course will permit accurate estimation, and from the estimated model state, accurate prediction after observations are completed. This sets the stage for how one will be able to use a more detailed model of V+Ca dynamics in neuron activity in the analysis of experimental data on individual neurons as well as functional networks in which the nodes (neurons) have these biophysical properties.

Translating intracellular calcium signaling into models

The rich experimental data on intracellular calcium has put theoreticians in an ideal position to derive models of intracellular calcium signaling. Over the last 25 years, a large number of modeling frameworks have been suggested. Here, I will review some of the milestones of intracellular calcium modeling with a special emphasis on calcium-induced calcium release (CICR) through inositol-1,4,5-trisphosphate and ryanodine receptors. I will highlight key features of CICR and how they are represented in models as well as the challenges that theoreticians face when translating our current understanding of calcium signals into equations. The selected examples demonstrate that a successful model provides mechanistic insights into the molecular machinery of the Ca²⁺ signaling toolbox and determines the contribution of local Ca²⁺ release to global Ca²⁺ patterns, which at the moment cannot be resolved experimentally.

Time to blip--stochastic simulation of single channel opening

The stochastic dynamics of the inositol-1,4,5-trisphosphate (IP3) receptor (IP3R) is key to understanding a wide range of observed calcium (Ca²⁺) signals. The stochastic nature results from the constant binding and unbinding of Ca²⁺ and IP3 to and from their respective binding sites and is especially important in the initiation of a Ca²⁺ puff (i.e., the release of Ca²⁺ through a cluster of IP3Rs). Once the first IP3R opens, the Ca²⁺ concentration rises significantly around the ion channel, increasing the open probability for neighboring IP3Rs. This opening may trigger the activation of further receptors, giving rise to a Ca²⁺ puff. In this protocol, we determine the time that it takes for a single IP3R to open from rest. We explicitly take into account the tetrameric structure of the IP3R and the fact that multiple subunits must be active before the channel opens. We develop code for a stochastic simulation of the IP3R and simulate it using the software package MATLAB. This protocol shows the basic form of a stochastic simulation algorithm and may serve as a starting point to investigate more complex gating dynamics.

Exploring oscillations in a point model of the intracellular Ca²⁺ concentration

Intracellular calcium (Ca²⁺) oscillations are a key signaling mechanism in most cell types. A prominent approach to modeling intracellular Ca²⁺ oscillations is the use of ordinary differential equations (ODEs), which treat the intracellular Ca²⁺ concentration as spatially homogenous. Although ODEs cannot account for the interaction of Ca²⁺ microdomains to form cell-wide Ca²⁺ patterns, modelers still choose ODEs because (a) the study of ODEs is computationally cheap, and a large body of techniques is available to investigate ODEs in great detail, or (b) sufficient experimental data are not available to develop a spatially extended model. Irrespective of the reason, analyzing ODEs is a key instrument in the toolbox of modelers. In this protocol, we look at a well-known model for Ca²⁺ oscillations, the De Young-Keizer model, along with the Li-Rinzel approximation of the De Young-Keizer model. The main emphasis of this protocol is the use of the open source software package XPPAUT to numerically study ODEs. The knowledge gained here can be directly transferred to other ODE systems and therefore may serve as a template for future studies.

The many faces of calmodulin in cell proliferation, programmed cell death, autophagy, and cancer

Calmodulin (CaM) is a ubiquitous Ca(2+) receptor protein mediating a large number of signaling processes in all eukaryotic cells. CaM plays a central role in regulating a myriad of cellular functions via interaction with multiple target proteins. This review focuses on the action of CaM and CaM-dependent signaling systems in the control of vertebrate cell proliferation, programmed cell death and autophagy. The significance of CaM and interconnected CaM-regulated systems for the physiology of cancer cells including tumor stem cells, and processes required for tumor progression such as growth, tumor-associated angiogenesis and metastasis are highlighted. Furthermore, the potential targeting of CaM-dependent signaling processes for therapeutic use is discussed.

A stochastic model of calcium puffs based on single-channel data

Calcium puffs are local transient Ca(2+) releases from internal Ca(2+) stores such as the endoplasmic reticulum or the sarcoplasmic reticulum. Such release occurs through a cluster of inositol 1,4,5-trisphosphate receptors (IP3Rs). Based on the IP3R model (which is determined by fitting to stationary single-channel data) and nonstationary single-channel data, we construct a new IP3R model that includes time-dependent rates of mode switches. A point-source model of Ca(2+) puffs is then constructed based on the new IP3R model and is solved by a hybrid Gillespie method with adaptive timing. Model results show that a relatively slow recovery of an IP3R from Ca(2+) inhibition is necessary to reproduce most of the experimental outcomes, especially the nonexponential interpuff interval distributions. The number of receptors in a cluster could be severely underestimated when the recovery is sufficiently slow. Furthermore, we find that, as the number of IP3Rs increases, the average duration of puffs initially increases but then becomes saturated, whereas the average decay time keeps increasing linearly. This gives rise to the observed asymmetric puff shape.

Canonical transient receptor potential 3 channel triggers vascular endothelial growth factor-induced intracellular Ca2+ oscillations in endothelial progenitor cells isolated from umbilical cord blood

Endothelial colony-forming cells (ECFCs) are the only endothelial progenitor cells (EPCs) that are capable of acquiring a mature endothelial phenotype. ECFCs are mainly mobilized from bone marrow to promote vascularization and represent a promising tool for cell-based therapy of severe ischemic diseases. Vascular endothelial growth factor (VEGF) stimulates the proliferation of peripheral blood-derived ECFCs (PB-ECFCs) through oscillations in intracellular Ca(2+) concentration ([Ca(2+)]i). VEGF-induced Ca(2+) spikes are driven by the interplay between inositol-1,4,5-trisphosphate (InsP3)-dependent Ca(2+) release and store-operated Ca(2+) entry (SOCE). The therapeutic potential of umbilical cord blood-derived ECFCs (UCB-ECFCs) has also been shown in recent studies. However, VEGF-induced proliferation of UCB-ECFCs is faster compared with their peripheral counterpart. Unlike PB-ECFCs, UCB-ECFCs express canonical transient receptor potential channel 3 (TRPC3) that mediates diacylglycerol-dependent Ca(2+) entry. The present study aimed at investigating whether the higher proliferative potential of UCB-ECFCs was associated to any difference in the molecular underpinnings of their Ca(2+) response to VEGF. We found that VEGF induces oscillations in [Ca(2+)]i that are patterned by the interaction between InsP3-dependent Ca(2+) release and SOCE. Unlike PB-ECFCs, VEGF-evoked Ca(2+) oscillations do not arise in the absence of extracellular Ca(2+) entry and after pharmacological (with Pyr3 and flufenamic acid) and genetic (by employing selective small interference RNA) suppression of TRPC3. VEGF-induced UCB-ECFC proliferation is abrogated on inhibition of the intracellular Ca(2+) spikes. Therefore, the Ca(2+) response to VEGF in UCB-ECFCs is shaped by a different Ca(2+) machinery as compared with PB-ECFCs, and TRPC3 stands out as a promising target in EPC-based treatment of ischemic pathologies.

The timing statistics of spontaneous calcium release in cardiac myocytes

A variety of cardiac arrhythmias are initiated by a focal excitation that disrupts the regular beating of the heart. In some cases it is known that these excitations are due to calcium (Ca) release from the sarcoplasmic reticulum (SR) via propagating subcellular Ca waves. However, it is not understood what are the physiological factors that determine the timing of these excitations at both the subcellular and tissue level. In this paper we apply analytic and numerical approaches to determine the timing statistics of spontaneous Ca release (SCR) in a simplified model of a cardiac myocyte. In particular, we compute the mean first passage time (MFPT) to SCR, in the case where SCR is initiated by spontaneous Ca sparks, and demonstrate that this quantity exhibits either an algebraic or exponential dependence on system parameters. Based on this analysis we identify the necessary requirements so that SCR occurs on a time scale comparable to the cardiac cycle. Finally, we study how SCR is synchronized across many cells in cardiac tissue, and identify a quantitative measure that determines the relative timing of SCR in an ensemble of cells. Using this approach we identify the physiological conditions so that cell-to-cell variations in the timing of SCR is small compared to the typical duration of an SCR event. We argue further that under these conditions inward currents due to SCR can summate and generate arrhythmogenic triggered excitations in cardiac tissue.

Hierarchic stochastic modelling applied to intracellular Ca(2+) signals

Important biological processes like cell signalling and gene expression have noisy components and are very complex at the same time. Mathematical analysis of such systems has often been limited to the study of isolated subsystems, or approximations are used that are difficult to justify. Here we extend a recently published method (Thurley and Falcke, PNAS 2011) which is formulated in observable system configurations instead of molecular transitions. This reduces the number of system states by several orders of magnitude and avoids fitting of kinetic parameters. The method is applied to Ca(2+) signalling. Ca(2+) is a ubiquitous second messenger transmitting information by stochastic sequences of concentration spikes, which arise by coupling of subcellular Ca(2+) release events (puffs). We derive analytical expressions for a mechanistic Ca(2+) model, based on recent data from live cell imaging, and calculate Ca(2+) spike statistics in dependence on cellular parameters like stimulus strength or number of Ca(2+) channels. The new approach substantiates a generic Ca(2+) model, which is a very convenient way to simulate Ca(2+) spike sequences with correct spiking statistics.