The correspondence between mathematical structures and experimental systems is the basis of the generalizability of results found with specific systems and is the basis of the predictive power of theoretical physics. While physicists have confidence in this correspondence, it is less recognized in cellular biophysics. On the one hand, the complex organization of cellular dynamics involving a plethora of interacting molecules and the basic observation of cell variability seem to question its possibility. The practical difficulties of deriving the equations describing cellular behaviour from first principles support these doubts. On the other hand, ignoring such a correspondence would severely limit the possibility of predictive quantitative theory in biophysics. Additionally, the existence of functional modules (like pathways) across cell types suggests also the existence of mathematical structures with comparable universality. Only a few cellular systems have been sufficiently investigated in a variety of cell types to follow up these basic questions. IP 3 induced Ca 2þ signalling is one of them, and the mathematical structure corresponding to it is subject of ongoing discussion. We review the system's general properties observed in a variety of cell types. They are captured by a reaction diffusion system. We discuss the phase space structure of its local dynamics. The spiking regime corresponds to noisy excitability. Models focussing on different aspects can be derived starting from this phase space structure. We discuss how the initial assumptions on the set of stochastic variables and phase space structure shape the predictions of parameter dependencies of the mathematical models resulting from the derivation. V

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.

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A note on versions:The version presented here may differ from the published version or from the version of record. If you wish to cite this item you are advised to consult the publisher's version. Please see the repository url above for details on accessing the published version and note that access may require a subscription. Main points:1) Transient stimuli trigger astrocyte calcium waves in a probabilistic manner.2) Response probability can be modulated by other signalling pathways.3) Wave initiation follows synchronous calcium release from three or more subcellular "puff" sites.Keywords: Astrocyte; calcium puffs; crosstalk; ATP; glutamate. 2 AbstractAstrocyte calcium signals can range in size from subcellular microdomains, to waves that spread through the whole cell (and into connected cells). The differential roles of such local or global calcium signalling are under intense investigation, but the mechanisms by which local signals evolve into global signals in astrocytes are not well understood, nor are the computational rules by which physiological stimuli are transduced into a global signal. To investigate these questions, we transiently applied receptor agonists linked to calcium signalling to primary cultures of cerebellar astrocytes. Astrocytes repetitively tested with the same stimulus responded with global signals intermittently, indicating that each stimulus had a defined probability for triggering a response. The response probability varied between agonists, increased with agonist concentration, and could be positively and negatively modulated by crosstalk with other signalling pathways.To better understand the processes determining the evolution of a global signal, we recorded subcellular calcium "puffs" throughout the whole cell during stimulation. The key requirement for puffs to trigger a global calcium wave following receptor activation appeared to be the synchronous release of calcium from three or more sites, rather than an increasing calcium load accumulating in the cytosol due to increased puff size, amplitude or frequency. These results suggest that the concentration of transient stimuli will be encoded into a probability of generating a global calcium response, determined by the likelihood of synchronous release from multiple subcellular sites.3

Calcium (Ca 2+ ) plays a central role in mediating both contractile function and hypertrophic signalling in ventricular cardiomyocytes. L-type Ca 2+ channels trigger release of Ca 2+ from ryanodine receptors (RyRs) for cellular contraction, while signalling downstream of Gq coupled receptors stimulates Ca 2+ release via inositol 1,4,5-trisphosphate receptors (IP 3 Rs), engaging hypertrophic signalling pathways. Modulation of the amplitude, duration, and duty cycle of the cytosolic Ca 2+ contraction signal, and spatial localisation, have all been proposed to encode this hypertrophic signal. Given current knowledge of IP 3 Rs, we develop a model describing the effect of functional interaction (cross-talk) between RyR and IP 3 R channels on the Ca 2+ transient, and examine the sensitivity of the Ca 2+ transient shape to properties of IP 3 R activation. A key result of our study is that IP 3 R activation increases Ca 2+ transient duration for a broad range of IP 3 R properties, but the effect of IP 3 R activation on Ca 2+ transient amplitude is dependent on IP 3 concentration. Furthermore we demonstrate that IP 3 -mediated Ca 2+ release in the cytosol increases the duty cycle of the Ca 2+ transient, the fraction of the cycle for which [Ca 2+ ] is elevated, across a broad range of parameter values and IP 3 concentrations. When coupled to a model of downstream transcription factor (NFAT) activation, we demonstrate that there is a high correspondence between the Ca 2+ transient duty cycle and the proportion of activated NFAT in the nucleus. These findings suggest increased cytosolic Ca 2+ duty cycle as a plausible mechanism for IP 3 -dependent hypertrophic signalling via Ca 2+ -sensitive transcription factors such as NFAT in ventricular cardiomyocytes. SIGNIFICANCE Many studies have identified a role for IP 3 R-mediated Ca 2+ signalling in cardiac hypertrophy, however the mechanism by which this signal is communicated within the cardiomyocyte remains unclear. We present a mathematical model of functional interactions between RyR and IP 3 R channels. We show that IP 3 -mediated Ca 2+ release is capable of providing a modest increase to the duty cycle of the calcium signal, which has been shown experimentally to lead to NFAT activation, and hence hypertrophic signalling. Through a parameter sensitivity analysis we demonstrate that the duty cycle is increased with IP 3 over a broad parameter regime, indicating that this mechanism is robust, and we show that an increase in Ca 2+ duty cycle increases nuclear NFAT activation. These findings suggest a plausible mechanism for IP3R-dependent hypertrophic signalling in cardiomyocytes. INTRODUCTIONCalcium is a universal second messenger that plays a role in controlling many cellular processes across a wide variety of cell types; ranging from fertilisation, cell contraction, and cell growth, to cell death (1, 2). Precisely how Ca 2+ fulfills each of these † These authors contributed equally to the supervision of this work.

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Calcium signaling plays a pivotal role in cardiomyocytes, coupling electrical excitation to mechanical contraction of the heart. Determining locations of active calcium release sites, and how their recruitment changes in response to stimuli and in disease states is therefore of central interest in cardiac physiology. Current algorithms for detecting release sites from live cell imaging data are however not easily validated against a known “ground truth,” which makes interpretation of the output of such algorithms, in particular the degree of confidence in site detection, a challenging task. Computational models are capable of integrating findings from multiple sources into a consistent, predictive framework. In cellular physiology, such models have the potential to reveal structure and function beyond the temporal and spatial resolution limitations of individual experimental measurements. Here, we create a spatially detailed computational model of calcium release in an eight sarcomere section of a ventricular cardiomyocyte, using electron tomography reconstruction of cardiac ultrastructure and confocal imaging of protein localization. This provides a high-resolution model of calcium diffusion from intracellular stores, which can be used as a platform to simulate confocal fluorescence imaging in the context of known ground truth structures from the higher resolution model. We use this capability to evaluate the performance of a recently proposed method for detecting the functional response of calcium release sites in live cells. Model permutations reveal how calcium release site density and mitochondria acting as diffusion barriers impact the detection performance of the algorithm. We demonstrate that site density has the greatest impact on detection precision and recall, in particular affecting the effective detectable depth of sites in confocal data. Our findings provide guidance on how such detection algorithms may best be applied to experimental data and give insights into limitations when using two-dimensional microscopy images to analyse three-dimensional cellular structures.

In cardiomyocytes, the coordinated release of calcium from intracellular stores through 16 ryanodine receptor (RyR) clusters is key to contraction. Recently, a deconvolution algorithm 17 (CaCLEAN) was proposed to detect the functional response of RyR clusters from confocal 18 fluorescence imaging of live cells. However, CaCLEAN cluster detection remained unvalidated 19 without ground truth values. We developed a structurally realistic finite element model of 20 reaction-diffusion of calcium emanating from RyR clusters in a rat ventricular cardiomyocyte during 21 the first 30 ms of the calcium transient. The effects of RyR cluster density and mitochondria acting 22 as diffusion barriers were examined with the model. Confocal fluorescence microscopy images 23 were simulated and analysed using CaCLEAN. Detection recall and precision were found to vary 24 between 0.69 and 0.82 in densely and sparsely-packed cases, respectively. The maximum relevant 25 z-depth for accurate cluster detection also varied between 280 and 610 nm in densely and 26 sparsely-packed cases. 27 28 Manuscript submitted to bioRxiv Tian et al. (2017) recently proposed an adaption of the CLEAN method from radio astronomy 38 (Högbom, 1974) to detect Ca2+ release sites from fluorescence confocal imaging of live cardiomy-39 ocytes. Using this algorithm (CaCLEAN) the authors demonstrated the possibility of detecting the 40 time-dependent functional response of RyR clusters in live cell preparations using widely available 41 experimental methods. Further, if proven in this context, similar methods may extend to other 42 cellular processes that can be approximated as an impulse response of a collection of point sources 43 in a fluorescence microscopy signal. 44 Several questions remain however regarding the suitability and performance of CaCLEAN. In 45 their original article, Tian et al. (2017) state that based on analysis of Ca2+ transients with varying 46 distance and convolved noise, CaCLEAN is able to resolve local transients with centres 1 m apart. 47 370 Astrophysics Supplement Series. 1974 Jun; 15:417.