Multi-scale data-driven modeling and observation of calcium puffs

Luminal transport through intact endoplasmic reticulum limits the magnitude of localized Ca2+ signals

The endoplasmic reticulum (ER) forms an interconnected network of tubules stretching throughout the cell. Understanding how ER functionality relies on its structural organization is crucial for elucidating cellular vulnerability to ER perturbations, which have been implicated in several neuronal pathologies. One of the key functions of the ER is enabling Ca[Formula: see text] signaling by storing large quantities of this ion and releasing it into the cytoplasm in a spatiotemporally controlled manner. Through a combination of physical modeling and live-cell imaging, we demonstrate that alterations in ER shape significantly impact its ability to support efficient local Ca[Formula: see text] releases, due to hindered transport of luminal content within the ER. Our model reveals that rapid Ca[Formula: see text] release necessitates mobile luminal buffer proteins with moderate binding strength, moving through a well-connected network of ER tubules. These findings provide insight into the functional advantages of normal ER architecture, emphasizing its importance as a kinetically efficient intracellular Ca[Formula: see text] delivery system.

Elementary intracellular Ca signals approximated as a transition of release channel system from a metastable state

Cardiac muscle contraction is initiated by an elementary Ca signal (called Ca spark) which is achieved by collective action of Ca release channels in a cluster. The mechanism of this synchronization remains uncertain. We approached Ca spark activation as an emergent phenomenon of an interactive system of release channels. We constructed a weakly lumped Markov chain that applies an Ising model formalism to such release channel clusters and probable open channel configurations and demonstrated that spark activation is described as a system transition from a metastable to an absorbing state, analogous to the pressure required to overcome surface tension in bubble formation. This yielded quantitative estimates of the spark generation probability as a function of various system parameters. We performed numerical simulations to find spark probabilities as a function of sarcoplasmic reticulum Ca concentration, obtaining similar values for spark activation threshold as our analytic model, as well as those reported in experimental studies. Our parametric sensitivity analyses also showed that the spark activation threshold decreased as Ca sensitivity of RyR activation and RyR cluster size increased.

Intracellular Injection of Brain Extracts from Alzheimer's Disease Patients Triggers Unregulated Ca2+ Release from Intracellular Stores That Hinders Cellular Bioenergetics

Strong evidence indicates that amyloid beta (Aβ) inflicts its toxicity in Alzheimer's disease (AD) by promoting uncontrolled elevation of cytosolic Ca²⁺ in neurons. We have previously shown that synthetic Aβ42 oligomers stimulate abnormal intracellular Ca²⁺ release from the endoplasmic reticulum stores, suggesting that a similar mechanism of Ca²⁺ toxicity may be common to the endogenous Aβs oligomers. Here, we use human postmortem brain extracts from AD-affected patients and test their ability to trigger Ca²⁺ fluxes when injected intracellularly into Xenopus oocytes. Immunological characterization of the samples revealed the elevated content of soluble Aβ oligomers only in samples from AD patients. Intracellular injection of brain extracts from control patients failed to trigger detectable changes in intracellular Ca²⁺. Conversely, brain extracts from AD patients triggered Ca²⁺ events consisting of local and global Ca²⁺ fluorescent transients. Pre-incubation with either the conformation-specific OC antiserum or caffeine completely suppressed the brain extract's ability to trigger cytosolic Ca²⁺ events. Computational modeling suggests that these Ca²⁺ fluxes may impair cells bioenergetic by affecting ATP and ROS production. These results support the hypothesis that Aβ oligomers contained in neurons of AD-affected brains may represent the toxic agents responsible for neuronal malfunctioning and death associated with the disruption of Ca²⁺ homeostasis.

The Function of Mitochondrial Calcium Uniporter at the Whole-Cell and Single Mitochondrion Levels in WT, MICU1 KO, and MICU2 KO Cells

Mitochondrial Ca2+ ([Ca2+]M) uptake through its Ca2+ uniporter (MCU) is central to many cell functions such as bioenergetics, spatiotemporal organization of Ca2+ signals, and apoptosis. MCU activity is regulated by several intrinsic proteins including MICU1, MICU2, and EMRE. While significant details about the role of MICU1, MICU2, and EMRE in MCU function have emerged recently, a key challenge for the future experiments is to investigate how these regulatory proteins modulate mitochondrial Ca2+ influx through MCU in intact cells under pathophysiological conditions. This is further complicated by the fact that several variables affecting MCU function change dynamically as cell functions. To overcome this void, we develop a data-driven model that closely replicates the behavior of MCU under a wide range of cytosolic Ca2+ ([Ca2+]C), [Ca2+]M, and mitochondrial membrane potential values in WT, MICU1 knockout (KO), and MICU2 KO cells at the single mitochondrion and whole-cell levels. The model is extended to investigate how MICU1 or MICU2 KO affect mitochondrial function. Moreover, we show how Ca2+ buffering proteins, the separation between mitochondrion and Ca2+-releasing stores, and the duration of opening of Ca2+-releasing channels affect mitochondrial function under different conditions. Finally, we demonstrate an easy extension of the model to single channel function of MCU.

TraceSpecks: A Software for Automated Idealization of Noisy Patch-Clamp and Imaging Data

Experimental records of single molecules or ion channels from fluorescence microscopy and patch-clamp electrophysiology often include high-frequency noise and baseline fluctuations that are not generated by the system under investigation and have to be removed. Moreover, multiple channels or conductance levels can be present at a time in the data that need to be quantified to accurately understand the behavior of the system. Manual procedures for removing these fluctuations and extracting conducting states or multiple channels are laborious, prone to subjective bias, and likely to hinder the processing of often very large data sets. We introduce a maximal likelihood formalism for separating signal from a noisy and drifting background such as fluorescence traces from imaging of elementary Ca²⁺ release events called puffs arising from clusters of channels, and patch-clamp recordings of ion channels. Parameters such as the number of open channels or conducting states, noise level, and background signal can all be optimized using the expectation-maximization algorithm. We implement our algorithm following the Baum-Welch approach to expectation-maximization in the portable Java language with a user-friendly graphical interface and test the algorithm on both synthetic and experimental data from the patch-clamp electrophysiology of Ca²⁺ channels and fluorescence microscopy of a cluster of Ca²⁺ channels and Ca²⁺ channels with multiple conductance levels. The resulting software is accurate, fast, and provides detailed information usually not available through manual analysis. Options for visual inspection of the raw and processed data with key parameters are provided, in addition to a range of statistics such as the mean open probabilities, mean open times, mean close times, dwell-time distributions for different number of channels open or conductance levels, amplitude distribution of all opening events, and number of transitions between different number of open channels or conducting levels in asci format with a single click.

Intracellular calcium stores mediate metaplasticity at hippocampal dendritic spines

Key points

Calcium (Ca²⁺) entry mediated by NMDA receptors is considered central to the induction of activity‐dependent synaptic plasticity in hippocampal area CA1; this description does not, however, take into account the potential contribution of endoplasmic reticulum (ER) Ca²⁺ stores.

The ER has a heterogeneous distribution in CA1 dendritic spines, and may introduce localized functional differences in Ca²⁺ signalling between synapses, as suggested by experiments on metabotropic receptor‐dependent long‐term depression.

A physiologically detailed computational model of Ca²⁺ dynamics at a CA3–CA1 excitatory synapse characterizes the contribution of spine ER via metabotropic signalling during plasticity induction protocols.

ER Ca²⁺ release via IP₃ receptors modulates NMDA receptor‐dependent plasticity in a graded manner, to selectively promote synaptic depression with relatively diminished effect on LTP induction; this may temper further strengthening at the stronger synapses which are preferentially associated with ER‐containing spines.

Acquisition of spine ER may thus represent a local, biophysically plausible ‘metaplastic switch’ at potentiated CA1 synapses, contributing to the plasticity–stability balance in neural circuits.

Abstract

Long‐term plasticity mediated by NMDA receptors supports input‐specific, Hebbian forms of learning at excitatory CA3–CA1 connections in the hippocampus. There exists an additional layer of stabilizing mechanisms that act globally as well as locally over multiple time scales to ensure that plasticity occurs in a constrained manner. Here, we investigated the role of calcium (Ca²⁺) stores associated with the endoplasmic reticulum (ER) in the local regulation of plasticity at individual CA1 synapses. Our study was spurred by (1) the curious observation that ER is sparsely distributed in dendritic spines, but over‐represented in larger spines that are likely to have undergone activity‐dependent strengthening, and (2) evidence suggesting that ER motility at synapses can be rapid, and accompany activity‐regulated spine remodelling. We constructed a physiologically realistic computational model of an ER‐bearing CA1 spine, and examined how IP₃‐sensitive Ca²⁺ stores affect spine Ca²⁺ dynamics during activity patterns mimicking the induction of long‐term potentiation and long‐term depression (LTD). Our results suggest that the presence of ER modulates NMDA receptor‐dependent plasticity in a graded manner that selectively enhances LTD induction. We propose that ER may locally tune Ca²⁺‐based plasticity, providing a braking mechanism to mitigate runaway strengthening at potentiated synapses. Our study provides a biophysically accurate description of postsynaptic Ca²⁺ regulation, and suggests that ER in the spine may promote the re‐use of hippocampal synapses with saturated strengths.

Calcium (Ca²⁺) entry mediated by NMDA receptors is considered central to the induction of activity‐dependent synaptic plasticity in hippocampal area CA1; this description does not, however, take into account the potential contribution of endoplasmic reticulum (ER) Ca²⁺ stores.

The ER has a heterogeneous distribution in CA1 dendritic spines, and may introduce localized functional differences in Ca²⁺ signalling between synapses, as suggested by experiments on metabotropic receptor‐dependent long‐term depression.

A physiologically detailed computational model of Ca²⁺ dynamics at a CA3–CA1 excitatory synapse characterizes the contribution of spine ER via metabotropic signalling during plasticity induction protocols.

ER Ca²⁺ release via IP₃ receptors modulates NMDA receptor‐dependent plasticity in a graded manner, to selectively promote synaptic depression with relatively diminished effect on LTP induction; this may temper further strengthening at the stronger synapses which are preferentially associated with ER‐containing spines.

Acquisition of spine ER may thus represent a local, biophysically plausible ‘metaplastic switch’ at potentiated CA1 synapses, contributing to the plasticity–stability balance in neural circuits.

Data-driven modeling of mitochondrial dysfunction in Alzheimer's disease

Intracellular accumulation of oligomeric forms of β amyloid (Aβ) are now believed to play a key role in the earliest phase of Alzheimer's disease (AD) as their rise correlates well with the early symptoms of the disease. Extensive evidence points to impaired neuronal Ca²⁺ homeostasis as a direct consequence of the intracellular Aβ oligomers. However, little is known about the downstream effects of the resulting Ca²⁺ rise on the many intracellular Ca²⁺-dependent pathways. Here we use multiscale modeling in conjunction with patch-clamp electrophysiology of single inositol 1,4,5-trisphosphate (IP₃) receptor (IP₃R) and fluorescence imaging of whole-cell Ca²⁺ response, induced by exogenously applied intracellular Aβ₄₂ oligomers to show that Aβ₄₂ inflicts cytotoxicity by impairing mitochondrial function. Driven by patch-clamp experiments, we first model the kinetics of IP₃R, which is then extended to build a model for the whole-cell Ca²⁺ signals. The whole-cell model is then fitted to fluorescence signals to quantify the overall Ca²⁺ release from the endoplasmic reticulum by intracellular Aβ₄₂ oligomers through G-protein-mediated stimulation of IP₃ production. The estimated IP₃ concentration as a function of intracellular Aβ₄₂ content together with the whole-cell model allows us to show that Aβ₄₂ oligomers impair mitochondrial function through pathological Ca²⁺ uptake and the resulting reduced mitochondrial inner membrane potential, leading to an overall lower ATP and increased production of reactive oxygen species and H₂O₂. We further show that mitochondrial function can be restored by the addition of Ca²⁺ buffer EGTA, in accordance with the observed abrogation of Aβ₄₂ cytotoxicity by EGTA in our live cells experiments.

Analyzing optical imaging of Ca2+ signals via TIRF microscopy: The limits on resolution due to chemical rates and depth of the channels

High resolution total internal reflection (TIRF) microscopy (TIRFM) together with detailed computational modeling provides a powerful approach towards the understanding of a wide range of Ca²⁺ signals mediated by the ubiquitous inositol 1,4,5-trisphosphate (IP₃) receptor (IP₃R) channel. Exploiting this fruitful collaboration further requires close agreement between the models and observations. However, elementary Ca²⁺ release events, puffs, imaged through TIRFM do not show the rapid single-channel openings and closings during and between puffs as are present in simulated puffs using data-driven single channel models. TIRFM also shows a rapid equilibration of 10ms after a channel opens or closes which is not achievable in simulation using standard Ca²⁺ diffusion coefficients and reaction rates between indicator dye and Ca²⁺. Furthermore, TIRFM imaging cannot decipher the depth of the channel with respect to the microscope, which will affect the change in fluorescence that the microscope detects, thereby affecting its sensitivity to fast single-channel activity. Using the widely used Ca²⁺ diffusion coefficients and reaction rates, our simulations show equilibration rates that are eight times slower than TIRFM imaging. We show that to get equilibrium rates consistent with observed values, the diffusion coefficients and reaction rates have to be significantly higher than the values reported in the literature, and predict the channel depth to be 200-250nm. Finally, we show that with the addition of noise, short events due to 1-2ms opening and closing of channels that are observed in computational models can be missed in TIRFM.

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.

Clusters of calcium release channels harness the Ising phase transition to confine their elementary intracellular signals

Intracellular Ca signals represent a universal mechanism of cell function. Messages carried by Ca are local, rapid, and powerful enough to be delivered over the thermal noise. A higher signal-to-noise ratio is achieved by a cooperative action of Ca release channels such as IP3 receptors or ryanodine receptors arranged in clusters (release units) containing a few to several hundred release channels. The channels synchronize their openings via Ca-induced Ca release, generating high-amplitude local Ca signals known as puffs in neurons and sparks in muscle cells. Despite the positive feedback nature of the activation, Ca signals are strictly confined in time and space by an unexplained termination mechanism. Here we show that the collective transition of release channels from an open to a closed state is identical to the phase transition associated with the reversal of magnetic field in an Ising ferromagnet. Our simple quantitative criterion closely predicts the Ca store depletion level required for spark termination for each cluster size. We further formulate exact requirements that a cluster of release channels should satisfy in any cell type for our mapping to the Ising model and the associated formula to remain valid. Thus, we describe deterministically the behavior of a system on a coarser scale (release unit) that is random on a finer scale (release channels), bridging the gap between scales. Our results provide exact mapping of a nanoscale biological signaling model to an interacting particle system in statistical physics, making the extensive mathematical apparatus available to quantitative biology.

Calcium waves in a grid of clustered channels with synchronous IP3 binding and unbinding

Calcium signals in cells occur at multiple spatial scales and variable temporal duration. However, a physical explanation for transitions between long-lasting global oscillations and localized short-term elevations (puffs) of cytoplasmic Ca²⁺ is still lacking. Here we introduce a phenomenological, coarse-grained model for the calcium variable, which is represented by ordinary differential equations. Due to its small number of parameters, and its simplicity, this model allows us to numerically study the interplay of multi-scale calcium concentrations with stochastic ion channel gating dynamics even in larger systems. We apply this model to a single cluster of inositol trisphosphate (IP ₃) receptor channels and find further evidence for the results presented in earlier work: a single cluster may be capable of producing different calcium release types, where long-lasting events are accompanied by unbinding of IP ₃ from the receptor (Rückl et al., PLoS Comput. Biol. 11, e1003965 (2015)). Finally, we show the practicability of the model in a grid of 64 clusters which is computationally intractable with previous high-resolution models. Here long-lasting events can lead to synchronized oscillations and waves, while short events stay localized. The frequency of calcium releases as well as their coherence can thereby be regulated by the amplitude of IP ₃ stimulation. Finally the model allows for a new explanation of oscillating [IP ₃], which is not based on metabolic production and degradation of IP ₃.

Mode switching of Inositol 1,4,5-trisphosphate receptor channel shapes the Spatiotemporal scales of Ca2+ signals

The inositol 1,4,5-trisphosphate (InsP3) receptor (InsP3R) channel is crucial for the generation and modulation of highly specific intracellular Ca2+ signals performing numerous functions in animal cells. However, the single channel behavior during Ca2+ signals of different spatiotemporal scales is not well understood. To elucidate the correlation between the gating dynamics of single InsP3Rs and spatiotemporal Ca2+ patterns, we simulate a cluster of InsP3Rs under varying ligand concentrations and extract comprehensive gating statistics of all channels during events of different sizes and durations. Our results show that channels gating predominantly in the low activity mode with negligible occupancy of intermediate and high modes leads to single channel Ca2+ release event blips. Increasing occupancies of intermediate and high modes results in events with increasing size. When the channel has more than 50% probability of gating in the intermediate and high modes, the cluster generates very large puffs that would most likely result in global Ca2+ signals. The size, duration and frequency of Ca2+ signals all increase linearly with the total probability of channel gating in the intermediate and high modes. To our knowledge, this is the first study that quantitatively relates the modal characteristics of InsP3R to the shaping of different spatiotemporal scales of Ca2+ signals.

Emergence of ion channel modal gating from independent subunit kinetics

Many ion channels exhibit a slow stochastic switching between distinct modes of gating activity. This feature of channel behavior has pronounced implications for the dynamics of ionic currents and the signaling pathways that they regulate. A canonical example is the inositol 1,4,5-trisphosphate receptor (IP3R) channel, whose regulation of intracellular Ca(2+) concentration is essential for numerous cellular processes. However, the underlying biophysical mechanisms that give rise to modal gating in this and most other channels remain unknown. Although ion channels are composed of protein subunits, previous mathematical models of modal gating are coarse grained at the level of whole-channel states, limiting further dialogue between theory and experiment. Here we propose an origin for modal gating, by modeling the kinetics of ligand binding and conformational change in the IP3R at the subunit level. We find good agreement with experimental data over a wide range of ligand concentrations, accounting for equilibrium channel properties, transient responses to changing ligand conditions, and modal gating statistics. We show how this can be understood within a simple analytical framework and confirm our results with stochastic simulations. The model assumes that channel subunits are independent, demonstrating that cooperative binding or concerted conformational changes are not required for modal gating. Moreover, the model embodies a generally applicable principle: If a timescale separation exists in the kinetics of individual subunits, then modal gating can arise as an emergent property of channel behavior.

Suppressing effect of Ca^{2+} blips on puff amplitudes by inhibiting channels to prevent recovery

As local signals, calcium puffs arise from the concerted opening of a few nearby inositol 1,4,5-trisphospate receptor channels to release Ca^{2+} ions from the endoplasmic reticulum. Although Ca^{2+} puffs have been well studied, little is known about the modulation of cytosolic basal Ca^{2+} concentration ([Ca^{2+}]_{Basal}) on puff dynamics. In this paper we consider a puff model to study how the statistical properties of puffs are modulated by [Ca^{2+}]_{Basal}. The puff frequency and lifetime trivially increase with the increasing [Ca^{2+}]_{Basal}, but an unexpected result is that the puff amplitude and the maximum open-channel number of the puff show decreasing relationship with the increasing [Ca^{2+}]_{Basal}. The underlying dynamics is related not only to the increasing puff frequency which gives a shorter recovery time, but also to the increasing frequency of blips with only one channel open. We indicate that Ca^{2+} blips cause the channels to be inhibited and prevent their recovery during interpuff intervals, resulting in the suppressing effect on puff amplitudes. With increasing [Ca^{2+}]_{Basal}, more blips occur to cause more channels to be inhibited, leaving fewer channels available for puff events. This study shows that the blips may play relevant functions in global Ca^{2+} waves through modulating puff dynamics.

Analyzing and Quantifying the Gain-of-Function Enhancement of IP3 Receptor Gating by Familial Alzheimer's Disease-Causing Mutants in Presenilins

Familial Alzheimer’s disease (FAD)-causing mutant presenilins (PS) interact with inositol 1,4,5-trisphosphate (IP₃) receptor (IP₃R) Ca²⁺ release channels resulting in enhanced IP₃R channel gating in an amyloid beta (Aβ) production-independent manner. This gain-of-function enhancement of IP₃R activity is considered to be the main reason behind the upregulation of intracellular Ca²⁺ signaling in the presence of optimal and suboptimal stimuli and spontaneous Ca²⁺ signals observed in cells expressing mutant PS. In this paper, we employed computational modeling of single IP₃R channel activity records obtained under optimal Ca²⁺ and multiple IP₃ concentrations to gain deeper insights into the enhancement of IP₃R function. We found that in addition to the high occupancy of the high-activity (H) mode and the low occupancy of the low-activity (L) mode, IP₃R in FAD-causing mutant PS-expressing cells exhibits significantly longer mean life-time for the H mode and shorter life-time for the L mode, leading to shorter mean close-time and hence high open probability of the channel in comparison to IP₃R in cells expressing wild-type PS. The model is then used to extrapolate the behavior of the channel to a wide range of IP₃ and Ca²⁺ concentrations and quantify the sensitivity of IP₃R to its two ligands. We show that the gain-of-function enhancement is sensitive to both IP₃ and Ca²⁺ and that very small amount of IP₃ is required to stimulate IP₃R channels in the presence of FAD-causing mutant PS to the same level of activity as channels in control cells stimulated by significantly higher IP₃ concentrations. We further demonstrate with simulations that the relatively longer time spent by IP₃R in the H mode leads to the observed higher frequency of local Ca²⁺ signals, which can account for the more frequent global Ca²⁺ signals observed, while the enhanced activity of the channel at extremely low ligand concentrations will lead to spontaneous Ca²⁺ signals in cells expressing FAD-causing mutant PS.

Aberrant Ca²⁺ signaling caused by IP₃R gating dysregulation is implicated in many neurodegenerative diseases such as Alzheimer’s, Huntington’s, Spinocerebellar ataxias, and endoplasmic reticulum stress-induced brain damage. Thus understanding IP₃R dysfunction is important for the etiology of these diseases. It was previously shown that FAD-causing mutant PS interacts with the IP₃R, leading to its gain-of-function enhancement in optimal Ca²⁺ and sub-saturating IP₃ concentrations. Here, we use data-driven modeling to provide deeper insights into the upregulation of IP₃R gating in a wide range of ligand concentrations and quantify the sensitivity of the channel to its ligands in the presence of mutant PS. Our simulations demonstrate that these changes can alter the statistics of local Ca²⁺ events and we speculate that they lead to Ca²⁺ signaling dysregulations at the whole cell level observed in FAD cells. These models will provide the foundation for future data-driven computational framework for local and global Ca²⁺ signals that will be used to judiciously isolate the primary pathways causing Ca²⁺ dysregulation in FAD from those that are downstream, and to study the effects of upregulation of IP₃R activity on cell function.

Analyzing and Modeling the Kinetics of Amyloid Beta Pores Associated with Alzheimer's Disease Pathology

Amyloid beta (Aβ) oligomers associated with Alzheimer’s disease (AD) form Ca²⁺-permeable plasma membrane pores, leading to a disruption of the otherwise well-controlled intracellular calcium (Ca²⁺) homeostasis. The resultant up-regulation of intracellular Ca²⁺ concentration has detrimental implications for memory formation and cell survival. The gating kinetics and Ca²⁺ permeability of Aβ pores are not well understood. We have used computational modeling in conjunction with the ability of optical patch-clamping for massively parallel imaging of Ca²⁺ flux through thousands of pores in the cell membrane of Xenopus oocytes to elucidate the kinetic properties of Aβ pores. The fluorescence time-series data from individual pores were idealized and used to develop data-driven Markov chain models for the kinetics of the Aβ pore at different stages of its evolution. Our study provides the first demonstration of developing Markov chain models for ion channel gating that are driven by optical-patch clamp data with the advantage of experiments being performed under close to physiological conditions. Towards the end, we demonstrate the up-regulation of gating of various Ca²⁺ release channels due to Aβ pores and show that the extent and spatial range of such up-regulation increases as Aβ pores with low open probability and Ca²⁺ permeability transition into those with high open probability and Ca²⁺ permeability.

Frequency and relative prevalence of calcium blips and puffs in a model of small IP₃R clusters

In this work, we model the local calcium release from clusters with a few inositol 1,4,5-trisphosphate receptor (IP3R) channels, focusing on the stochastic process in which an open channel either triggers other channels to open (as a puff) or fails to cause any channel to open (as a blip). We show that there are linear relations for the interevent interval (including blips and puffs) and the first event latency against the inverse cluster size. However, nonlinearity is found for the interpuff interval and the first puff latency against the inverse cluster size. Furthermore, the simulations indicate that the blip fraction among all release events and the blip frequency are increasing with larger basal [Ca(2+)], with blips in turn giving a growing contribution to basal [Ca(2+)]. This result suggests that blips are not just lapses to trigger puffs, but they may also possess a biological function to contribute to the initiation of calcium waves by a preceding increase of basal [Ca(2+)] in cells that have small IP3R clusters.

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.

Termination of calcium puffs and coupled closings of inositol trisphosphate receptor channels

Calcium puffs are localized Ca(2+) signals mediated by Ca(2+) release from the endoplasmic reticulum (ER) through clusters of inositol trisphosphate receptor (IP3R) channels. The recruitment of IP3R channels during puffs depends on Ca(2+)-induced Ca(2+) release, a regenerative process that must be terminated to maintain control of cell signaling and prevent Ca(2+) cytotoxicity. Here, we studied puff termination using total internal reflection microscopy to resolve the gating of individual IP3R channels during puffs in intact SH-SY5Y neuroblastoma cells. We find that the kinetics of IP3R channel closing differ from that expected for independent, stochastic gating, in that multiple channels tend to remain open together longer than predicted from their individual open lifetimes and then close in near-synchrony. This behavior cannot readily be explained by previously proposed termination mechanisms, including Ca(2+)-inhibition of IP3Rs and local depletion of Ca(2+) in the ER lumen. Instead, we postulate that the gating of closely adjacent IP3Rs is coupled, possibly via allosteric interactions, suggesting an important mechanism to ensure robust puff termination in addition to Ca(2+)-inactivation.

Excitability in a stochastic differential equation model for calcium puffs

Calcium dynamics are essential to a multitude of cellular processes. For many cell types, localized discharges of calcium through small clusters of intracellular channels are building blocks for all spatially extended calcium signals. Because of the large noise amplitude, the validity of noise-approximating model equations for this system has been questioned. Here we revisit the master equations for local calcium release, examine the multiple scales of calcium concentrations in the cluster domain, and derive adapted stochastic differential equations. We show by comparison of discrete and continuous trajectories that the Langevin equations can be made consistent with the master equations even for very small channel numbers. In its deterministic limit, the model reveals that excitability, a dynamical phenomenon observed in many natural systems, is at the core of calcium puffs. The model also predicts a bifurcation from transient to sustained release which may link local and global calcium signals in cells.

Towards the Physics of Calcium Signalling in Plants

Calcium is an abundant element with a wide variety of important roles within cells. Calcium ions are inter- and intra-cellular messengers that are involved in numerous signalling pathways. Fluctuating compartment-specific calcium ion concentrations can lead to localised and even plant-wide oscillations that can regulate downstream events. Understanding the mechanisms that give rise to these complex patterns that vary both in space and time can be challenging, even in cases for which individual components have been identified. Taking a systems biology approach, mathematical and computational techniques can be employed to produce models that recapitulate experimental observations and capture our current understanding of the system. Useful models make novel predictions that can be investigated and falsified experimentally. This review brings together recent work on the modelling of calcium signalling in plants, from the scale of ion channels through to plant-wide responses to external stimuli. Some in silico results that have informed later experiments are highlighted.

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.

Cytosolic [Ca2+] regulation of InsP3-evoked puffs

InsP3-mediated puffs are fundamental building blocks of cellular Ca2+ signalling, and arise through the concerted opening of clustered InsP3Rs (InsP3 receptors) co-ordinated via Ca2+-induced Ca2+ release. Although the Ca2+ dependency of InsP3Rs has been extensively studied at the single channel level, little is known as to how changes in basal cytosolic [Ca2+] would alter the dynamics of InsP3-evoked Ca2+ signals in intact cells. To explore this question, we expressed Ca2+-permeable channels (nicotinic acetylcholine receptors) in the plasma membrane of voltage-clamped Xenopus oocytes to regulate cytosolic [Ca2+] by changing the electrochemical gradient for extracellular Ca2+ entry, and imaged Ca2+ liberation evoked by photolysis of caged InsP3. Elevation of basal cytosolic [Ca2+] strongly increased the amplitude and shortened the latency of global Ca2+ waves. In oocytes loaded with EGTA to localize Ca2+ signals, the number of sites at which puffs were observed and the frequency and latency of puffs were strongly dependent on cytosolic [Ca2+], whereas puff amplitudes were only weakly affected. The results of the present study indicate that basal cytosolic [Ca2+] strongly affects the triggering of puffs, but has less of an effect on puffs once they have been initiated.

Maternal high-altitude hypoxia and suppression of ryanodine receptor-mediated Ca2+ sparks in fetal sheep pulmonary arterial myocytes

Ca(2+) sparks are fundamental Ca(2+) signaling events arising from ryanodine receptor (RyR) activation, events that relate to contractile and dilatory events in the pulmonary vasculature. Recent studies demonstrate that long-term hypoxia (LTH) can affect pulmonary arterial reactivity in fetal, newborn, and adult animals. Because RyRs are important to pulmonary vascular reactivity and reactivity changes with ontogeny and LTH we tested the hypothesis that RyR-generated Ca(2+) signals are more active before birth and that LTH suppresses these responses. We examined these hypotheses by performing confocal imaging of myocytes in living arteries and by performing wire myography studies. Pulmonary arteries (PA) were isolated from fetal, newborn, or adult sheep that lived at low altitude or from those that were acclimatized to 3,801 m for > 100 days. Confocal imaging demonstrated preservation of the distance between the sarcoplasmic reticulum, nucleus, and plasma membrane in PA myocytes. Maturation increased global Ca(2+) waves and Ca(2+) spark activity, with sparks becoming larger, wider, and slower. LTH preferentially depressed Ca(2+) spark activity in immature pulmonary arterial myocytes, and these sparks were smaller, wider, and slower. LTH also suppressed caffeine-elicited contraction in fetal PA but augmented contraction in the newborn and adult. The influence of both ontogeny and LTH on RyR-dependent cell excitability shed new light on the therapeutic potential of these channels for the treatment of pulmonary vascular disease in newborns as well as adults.