Stochastic binding of Ca2+ ions in the dyadic cleft; continuous versus random walk description of diffusion

T cell Ca2+ microdomains through the lens of computational modeling

Cellular Ca2+ signaling is highly organized in time and space. Locally restricted and short-lived regions of Ca2+ increase, called Ca2+ microdomains, constitute building blocks that are differentially arranged to create cellular Ca2+ signatures controlling physiological responses. Here, we focus on Ca2+ microdomains occurring in restricted cytosolic spaces between the plasma membrane and the endoplasmic reticulum, called endoplasmic reticulum-plasma membrane junctions. In T cells, these microdomains have been finely characterized. Enough quantitative data are thus available to develop detailed computational models of junctional Ca2+ dynamics. Simulations are able to predict the characteristics of Ca2+ increases at the level of single channels and in junctions of different spatial configurations, in response to various signaling molecules. Thanks to the synergy between experimental observations and computational modeling, a unified description of the molecular mechanisms that create Ca2+ microdomains in the first seconds of T cell stimulation is emerging.

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.

Influence of T-Bar on Calcium Concentration Impacting Release Probability

The relation of form and function, namely the impact of the synaptic anatomy on calcium dynamics in the presynaptic bouton, is a major challenge of present (computational) neuroscience at a cellular level. The Drosophila larval neuromuscular junction (NMJ) is a simple model system, which allows studying basic effects in a rather simple way. This synapse harbors several special structures. In particular, in opposite to standard vertebrate synapses, the presynaptic boutons are rather large, and they have several presynaptic zones. In these zones, different types of anatomical structures are present. Some of the zones bear a so-called T-bar, a particular anatomical structure. The geometric form of the T-bar resembles the shape of the letter “T” or a table with one leg. When an action potential arises, calcium influx is triggered. The probability of vesicle docking and neurotransmitter release is superlinearly proportional to the concentration of calcium close to the vesicular release site. It is tempting to assume that the T-bar causes some sort of calcium accumulation and hence triggers a higher release probability and thus enhances neurotransmitter exocytosis. In order to study this influence in a quantitative manner, we constructed a typical T-bar geometry and compared the calcium concentration close to the active zones (AZs). We compared the case of synapses with and without T-bars. Indeed, we found a substantial influence of the T-bar structure on the presynaptic calcium concentrations close to the AZs, indicating that this anatomical structure increases vesicle release probability. Therefore, our study reveals how the T-bar zone implies a strong relation between form and function. Our study answers the question of experimental studies (namely “Wichmann and Sigrist, Journal of neurogenetics 2010”) concerning the sense of the anatomical structure of the T-bar.

Arrhythmogenic Current Generation by Myofilament-Triggered Ca2+ Release and Sarcomere Heterogeneity

Heterogeneous mechanical dyskinesis has been implicated in many arrhythmogenic phenotypes. Strain-dependent perturbations to cardiomyocyte electrophysiology may contribute to this arrhythmogenesis through processes referred to as mechanoelectric feedback. Although the role of stretch-activated ion currents has been investigated using computational models, experimental studies suggest that mechanical strain may also promote arrhythmia by facilitating calcium wave propagation. To investigate whether strain-dependent changes in calcium affinity to the myofilament may promote arrhythmogenic intracellular calcium waves, we modified a mathematical model of rabbit excitation-contraction coupling coupled to a model of myofilament activation and force development. In a one-dimensional compartmental analysis, we bidirectionally coupled 50 sarcomere models in series to model calcium diffusion and stress transfer between adjacent sarcomeres. These considerations enabled the model to capture 1) the effects of mechanical feedback on calcium homeostasis at the sarcomeric level and 2) the combined effects of mechanical and calcium heterogeneities at the cellular level. The results suggest that in conditions of calcium overload, the vulnerable window of stretch-release to trigger suprathreshold delayed afterdepolarizations can be affected by heterogeneity in sarcomere length. Furthermore, stretch and sarcomere heterogeneity may modulate the susceptibility threshold for delayed afterdepolarizations and the aftercontraction wave propagation velocity.

Studying dyadic structure-function relationships: a review of current modeling approaches and new insights into Ca2+ (mis)handling

Excitation–contraction coupling in cardiac myocytes requires calcium influx through L-type calcium channels in the sarcolemma, which gates calcium release through sarcoplasmic reticulum ryanodine receptors in a process known as calcium-induced calcium release, producing a myoplasmic calcium transient and enabling cardiomyocyte contraction. The spatio-temporal dynamics of calcium release, buffering, and reuptake into the sarcoplasmic reticulum play a central role in excitation–contraction coupling in both normal and diseased cardiac myocytes. However, further quantitative understanding of these cells’ calcium machinery and the study of mechanisms that underlie both normal cardiac function and calcium-dependent etiologies in heart disease requires accurate knowledge of cardiac ultrastructure, protein distribution and subcellular function. As current imaging techniques are limited in spatial resolution, limiting insight into changes in calcium handling, computational models of excitation–contraction coupling have been increasingly employed to probe these structure–function relationships. This review will focus on the development of structural models of cardiac calcium dynamics at the subcellular level, orienting the reader broadly towards the development of models of subcellular calcium handling in cardiomyocytes. Specific focus will be given to progress in recent years in terms of multi-scale modeling employing resolved spatial models of subcellular calcium machinery. A review of the state-of-the-art will be followed by a review of emergent insights into calcium-dependent etiologies in heart disease and, finally, we will offer a perspective on future directions for related computational modeling and simulation efforts.

Hybrid finite element and Brownian dynamics method for charged particles

Diffusion is often the rate-determining step in many biological processes. Currently, the two main computational methods for studying diffusion are stochastic methods, such as Brownian dynamics, and continuum methods, such as the finite element method. A previous study introduced a new hybrid diffusion method that couples the strengths of each of these two methods, but was limited by the lack of interactions among the particles; the force on each particle had to be from an external field. This study further develops the method to allow charged particles. The method is derived for a general multidimensional system and is presented using a basic test case for a one-dimensional linear system with one charged species and a radially symmetric system with three charged species.

Microdomain [Ca(2+)] Fluctuations Alter Temporal Dynamics in Models of Ca(2+)-Dependent Signaling Cascades and Synaptic Vesicle Release

Ca(2+)-dependent signaling is often localized in spatially restricted microdomains and may involve only 1 to 100 Ca(2+) ions. Fluctuations in the microdomain Ca(2+) concentration (Ca(2+)) can arise from a wide range of elementary processes, including diffusion, Ca(2+) influx, and association/dissociation with Ca(2+) binding proteins or buffers. However, it is unclear to what extent these fluctuations alter Ca(2+)-dependent signaling. We construct Markov models of a general Ca(2+)-dependent signaling cascade and Ca(2+)-triggered synaptic vesicle release. We compare the hitting (release) time distribution and statistics for models that account for [Ca(2+)] fluctuations with the corresponding models that neglect these fluctuations. In general, when Ca(2+) fluctuations are much faster than the characteristic time for the signaling event, the hitting time distributions and statistics for the models with and without Ca(2+) fluctuation are similar. However, when the timescale of Ca(2+) fluctuations is on the same order as the signaling cascade or slower, the hitting time mean and variability are typically increased, in particular when the average number of microdomain Ca(2+) ions is small, a consequence of a long-tailed hitting time distribution. In a model of Ca(2+)-triggered synaptic vesicle release, we demonstrate the conditions for which [Ca(2+)] fluctuations do and do not alter the distribution, mean, and variability of release timing. We find that both the release time mean and variability can be increased, demonstrating that Ca(2+) fluctuations are an important aspect of microdomain Ca(2+) signaling and further suggesting that Ca(2+) fluctuations in the presynaptic terminal may contribute to variability in synaptic vesicle release and thus variability in neuronal spiking.

A multiscale computational model of spatially resolved calcium cycling in cardiac myocytes: from detailed cleft dynamics to the whole cell concentration profiles

Mathematical modeling of excitation-contraction coupling (ECC) in ventricular cardiac myocytes is a multiscale problem, and it is therefore difficult to develop spatially detailed simulation tools. ECC involves gradients on the length scale of 100 nm in dyadic spaces and concentration profiles along the 100 μm of the whole cell, as well as the sub-millisecond time scale of local concentration changes and the change of lumenal Ca²⁺ content within tens of seconds. Our concept for a multiscale mathematical model of Ca²⁺ -induced Ca²⁺ release (CICR) and whole cardiomyocyte electrophysiology incorporates stochastic simulation of individual LC- and RyR-channels, spatially detailed concentration dynamics in dyadic clefts, rabbit membrane potential dynamics, and a system of partial differential equations for myoplasmic and lumenal free Ca²⁺ and Ca²⁺-binding molecules in the bulk of the cell. We developed a novel computational approach to resolve the concentration gradients from dyadic space to cell level by using a quasistatic approximation within the dyad and finite element methods for integrating the partial differential equations. We show whole cell Ca²⁺-concentration profiles using three previously published RyR-channel Markov schemes.

The influence of Ca²⁺ buffers on free [Ca²⁺] fluctuations and the effective volume of Ca²⁺ microdomains

Intracellular calcium (Ca(2+)) plays a significant role in many cell signaling pathways, some of which are localized to spatially restricted microdomains. Ca(2+) binding proteins (Ca(2+) buffers) play an important role in regulating Ca(2+) concentration ([Ca(2+)]). Buffers typically slow [Ca(2+)] temporal dynamics and increase the effective volume of Ca(2+) domains. Because fluctuations in [Ca(2+)] decrease in proportion to the square-root of a domain's physical volume, one might conjecture that buffers decrease [Ca(2+)] fluctuations and, consequently, mitigate the significance of small domain volume concerning Ca(2+) signaling. We test this hypothesis through mathematical and computational analysis of idealized buffer-containing domains and their stochastic dynamics during free Ca(2+) influx with passive exchange of both Ca(2+) and buffer with bulk concentrations. We derive Langevin equations for the fluctuating dynamics of Ca(2+) and buffer and use these stochastic differential equations to determine the magnitude of [Ca(2+)] fluctuations for different buffer parameters (e.g., dissociation constant and concentration). In marked contrast to expectations based on a naive application of the principle of effective volume as employed in deterministic models of Ca(2+) signaling, we find that mobile and rapid buffers typically increase the magnitude of domain [Ca(2+)] fluctuations during periods of Ca(2+) influx, whereas stationary (immobile) Ca(2+) buffers do not. Also contrary to expectations, we find that in the absence of Ca(2+) influx, buffers influence the temporal characteristics, but not the magnitude, of [Ca(2+)] fluctuations. We derive an analytical formula describing the influence of rapid Ca(2+) buffers on [Ca(2+)] fluctuations and, importantly, identify the stochastic analog of (deterministic) effective domain volume. Our results demonstrate that Ca(2+) buffers alter the dynamics of [Ca(2+)] fluctuations in a nonintuitive manner. The finding that Ca(2+) buffers do not suppress intrinsic domain [Ca(2+)] fluctuations raises the intriguing question of whether or not [Ca(2+)] fluctuations are a physiologically significant aspect of local Ca(2+) signaling.

Microdomain calcium fluctuations as a colored noise process

Calcium ions play a key role in subcellular signaling as localized transients of the intracellular calcium concentration modify the activity of ion channels, enzymes and transcription factors, among others. The intracellular calcium concentration is inherently noisy, as diffusion, the transient binding to and dissociation from buffer molecules and stochastically gating calcium channels contribute to the fluctuations of the local copy number of Ca²⁺ ions. We study the properties of the fluctuating calcium concentration in sub-femtoliter volumes using an exact stochastic simulation algorithm and approximations to the exact stochastic solution. It is shown that the time course of the local calcium concentration represents a colored noise process whose autocorrelation time is a function of buffer kinetics and diffusion constants. Using the chemical Langevin description and the excess buffer approximation of the process, fast approximative algorithms and theoretical connections to the Ornstein-Uhlenbeck process are obtained. In a generic example, we show how calcium noise can couple to the dynamics of a single variable moving in a double-well potential, leading to a colored noise induced transition. Our work shows how a multitude of intracellular signaling pathways may be influenced by the inherent stochasticity of calcium signals, a key messenger in virtually any cell type, and how the calcium signal can be implemented efficiently in cellular signaling models.

Cardiac sodium transport and excitation-contraction coupling

The excitation-contraction coupling (EC-coupling) links membrane depolarization with contraction in cardiomyocytes. Ca(2+) induced opening of ryanodine receptors (RyRs) leads to Ca(2+) induced Ca(2+) release (CICR) from the sarcoplasmic reticulum (SR) into the dyadic cleft between the t-tubules and SR. Ca(2+) is removed from the cytosol by the SR Ca(2+) ATPase (SERCA2) and the Na,Ca-exchanger (NCX). The NCX connects cardiac Ca(2+) and Na(+)-transport, leading to Na(+)-dependent regulation of EC-coupling by several mechanisms of which some still lack firm experimental evidence. Firstly, NCX might contribute to CICR during an action potential (AP) as Na(+)-accumulation at the intracellular site together with depolarization will trigger reverse mode exchange bringing Ca(2+) into the dyadic cleft. The controversial issue is the nature of the compartment in which Na(+) accumulates. It seems not to be the bulk cytosol, but is it part of a widespread subsarcolemmal space, a localized microdomain ("fuzzy space"), or as we propose, a more localized "spot" to which only a few membrane proteins have shared access (nanodomains)? Also, there seems to be spots where the Na,K-pump (NKA) will cause local Na(+) depletion. Secondly, Na(+) determines the rate of cytosolic Ca(2+) removal and SR Ca(2+) load by regulating the SERCA2/NCX-balance during the decay of the Ca(2+) transient. The aim of this review is to describe available data and current concepts of Na(+)-mediated regulation of cardiac EC-coupling, with special focus on subcellular microdomains and the potential roles of Na(+) transport proteins in regulating CICR and Ca(2+) extrusion in cardiomyocytes. We propose that voltage gated Na(+) channels, NCX and the NKA α2-isoform all regulate cardiac EC-coupling through control of the "Na(+) concentration in specific subcellular nanodomains in cardiomyocytes. This article is part of a Special Issue entitled "Na(+) Regulation in Cardiac Myocytes."

Slow Ca²⁺ sparks de-synchronize Ca²⁺ release in failing cardiomyocytes: evidence for altered configuration of Ca²⁺ release units?

In heart failure, cardiomyocytes exhibit slowing of the rising phase of the Ca(2+) transient which contributes to the impaired contractility observed in this condition. We investigated whether alterations in ryanodine receptor function promote slowing of Ca(2+) release in a murine model of congestive heart failure (CHF). Myocardial infarction was induced by left coronary artery ligation. When chronic CHF had developed (10 weeks post-infarction), cardiomyocytes were isolated from viable regions of the septum. Septal myocytes from SHAM-operated mice served as controls. Ca(2+) transients rose markedly slower in CHF than SHAM myocytes with longer time to peak (CHF=152 ± 12% of SHAM, P<0.05). The rise time of Ca(2+) sparks was also increased in CHF (SHAM=9.6 ± 0.6 ms, CHF=13.2 ± 0.7 ms, P<0.05), due to a sub-population of sparks (≈20%) with markedly slowed kinetics. Regions of the cell associated with these slow spontaneous sparks also exhibited slowed Ca(2+) release during the action potential. Thus, greater variability in spark kinetics in CHF promoted less uniform Ca(2+) release across the cell. Dyssynchronous Ca(2+) transients in CHF additionally resulted from T-tubule disorganization, as indicated by fast Fourier transforms, but slow sparks were not associated with orphaned ryanodine receptors. Rather, mathematical modeling suggested that slow sparks could result from an altered composition of Ca(2+) release units, including a reduction in ryanodine receptor density and/or distribution of ryanodine receptors into sub-clusters. In conclusion, our findings indicate that slowed, dyssynchronous Ca(2+) transients in CHF result from alterations in Ca(2+) sparks, consistent with rearrangement of ryanodine receptors within Ca(2+) release units.

Hybrid finite element and Brownian dynamics method for diffusion-controlled reactions

Diffusion is often the rate determining step in many biological processes. Currently, the two main computational methods for studying diffusion are stochastic methods, such as Brownian dynamics, and continuum methods, such as the finite element method. This paper proposes a new hybrid diffusion method that couples the strengths of each of these two methods. The method is derived for a general multidimensional system, and is presented using a basic test case for 1D linear and radially symmetric diffusion systems.

Exact and approximate stochastic simulation of intracellular calcium dynamics

In simulations of chemical systems, the main task is to find an exact or approximate solution of the chemical master equation (CME) that satisfies certain constraints with respect to computation time and accuracy. While Brownian motion simulations of single molecules are often too time consuming to represent the mesoscopic level, the classical Gillespie algorithm is a stochastically exact algorithm that provides satisfying results in the representation of calcium microdomains. Gillespie's algorithm can be approximated via the tau-leap method and the chemical Langevin equation (CLE). Both methods lead to a substantial acceleration in computation time and a relatively small decrease in accuracy. Elimination of the noise terms leads to the classical, deterministic reaction rate equations (RRE). For complex multiscale systems, hybrid simulations are increasingly proposed to combine the advantages of stochastic and deterministic algorithms. An often used exemplary cell type in this context are striated muscle cells (e.g., cardiac and skeletal muscle cells). The properties of these cells are well described and they express many common calcium-dependent signaling pathways. The purpose of the present paper is to provide an overview of the aforementioned simulation approaches and their mutual relationships in the spectrum ranging from stochastic to deterministic algorithms.

Cardiac myocytes and local signaling in nano-domains

It is well known that calcium-induced calcium-release in cardiac myocytes takes place in spatially restricted regions known as dyads, where discrete patches of junctional sarcoplasmic reticulum tightly associate with the t-tubule membrane. The dimensions of a dyad are so small that it contains only a few Ca²⁺ ions at any given time. Ca²⁺ signaling in the dyad is therefore noisy, and dominated by the Brownian motion of Ca²⁺ ions in a potential field. Remarkably, from this complexity emerges the integrated behavior of the myocyte in which, under normal conditions, precise control of Ca²⁺ release and muscle contraction is maintained over the life of the cell. This is but one example of how signal processing within the cardiac myocyte and other cells often occurs in small "nano-domains" where proteins and protein complexes interact at spatial dimensions on the order of ∼1-10 nm and at time-scales on the order of nanoseconds to perform the functions of the cell. In this article, we will review several examples of local signaling in nano-domains, how it contributes to the integrative behavior of the cardiac myocyte, and present computational methods for modeling signal processing within these domains across differing spatio-temporal scales.

A solvable model for the diffusion and reaction of neurotransmitters in a synaptic junction

BACKGROUND

The diffusion and reaction of the transmitter acetylcholine in neuromuscular junctions and the diffusion and binding of Ca2+ in the dyadic clefts of ventricular myocytes have been extensively modeled by Monte Carlo simulations and by finite-difference and finite-element solutions. However, an analytical solution that can serve as a benchmark for testing these numerical methods has been lacking.

RESULT

Here we present an analytical solution to a model for the diffusion and reaction of acetylcholine in a neuromuscular junction and for the diffusion and binding of Ca2+ in a dyadic cleft. Our model is similar to those previously solved numerically and our results are also qualitatively similar.

CONCLUSION

The analytical solution provides a unique benchmark for testing numerical methods and potentially provides a new avenue for modeling biochemical transport.

Control of Ca2+ release by action potential configuration in normal and failing murine cardiomyocytes

Cardiomyocytes from failing hearts exhibit spatially nonuniform or dyssynchronous sarcoplasmic reticulum (SR) Ca(2+) release. We investigated the contribution of action potential (AP) prolongation in mice with congestive heart failure (CHF) after myocardial infarction. AP recordings from CHF and control myocytes were included in a computational model of the dyad, which predicted more dyssynchronous ryanodine receptor opening during stimulation with the CHF AP. This prediction was confirmed in cardiomyocyte experiments, when cells were alternately stimulated by control and CHF AP voltage-clamp waveforms. However, when a train of like APs was used as the voltage stimulus, the control and CHF AP produced a similar Ca(2+) release pattern. In this steady-state condition, greater integrated Ca(2+) entry during the CHF AP lead to increased SR Ca(2+) content. A resulting increase in ryanodine receptor sensitivity synchronized SR Ca(2+) release in the mathematical model, thus offsetting the desynchronizing effects of reduced driving force for Ca(2+) entry. A modest nondyssynchronous prolongation of Ca(2+) release was nevertheless observed during the steady-state CHF AP, which contributed to increased time-to-peak measurements for Ca(2+) transients in failing cells. Thus, dyssynchronous Ca(2+) release in failing mouse myocytes does not result from electrical remodeling, but rather other alterations such as T-tubule reorganization.

A comparison of deterministic and stochastic simulations of neuronal vesicle release models

We study the calcium-induced vesicle release into the synaptic cleft using a deterministic algorithm and MCell, a Monte Carlo algorithm that tracks individual molecules. We compare the average vesicle release probability obtained using both algorithms and investigate the effect of the three main sources of noise: diffusion, sensor kinetics and fluctuations from the voltage-dependent calcium channels (VDCCs). We find that the stochastic opening kinetics of the VDCCs are the main contributors to differences in the release probability. Our results show that the deterministic calculations lead to reliable results, with an error of less than 20%, when the sensor is located at least 50 nm from the VDCCs, corresponding to microdomain signaling. For smaller distances, i.e. nanodomain signaling, the error becomes larger and a stochastic algorithm is necessary.

A univariate model of calcium release in the dyadic cleft of cardiac myocytes

Local calcium sparks in the dyadic cleft of cardiac myocytes are triggered by calcium influxes via L-type calcium channels (LCCs) located on the transverse tubule (TT) membrane, and subsequently controlled by the regeneration of ryanodine receptors (RyRs) on the sarcoplasmic reticulum (SR). Calcium released from SR channels is known to be responsible for the sparks. Therefore, the activities of RyRs provide straightforward indication to the calcium concentration alteration. A method to study calcium signaling by analyzing RyR-gating statistics is described in the present study. Here we propose a univariate model with a simplified geometry of the dyadic cleft, which specifies the spatial localization of LCCs and RyRs to monitor the activity changes of RyRs. This model is used to explore two crucial aspects of local calcium signaling: the first is to disclose the tight control of calcium influxes via LCCs, and the second is to reveal the interactional impact of the self-regenerative RyRs. Patterns of active RyRs are rendered through numerous computational simulation experiments, manipulating the state initialization and the spatial localization of LCCs and RyRs to observe gating transition of RyRs.