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

Description of chemical systems by means of response functions

In this paper we introduce a formalism that allows to describe the response of a part of a biochemical system in terms of renewal equations. In particular, we examine under which conditions the interactions between the different parts of a chemical system, described by means of linear ODEs, can be represented in terms of renewal equations. We show also how to apply the formalism developed in this paper to some particular types of linear and non-linear ODEs, modelling some biochemical systems of interest in biology (for instance, some time-dependent versions of the classical Hopfield model of kinetic proofreading). We also analyse some of the properties of the renewal equations that we are interested in, as the long-time behaviour of their solution. Furthermore, we prove that the kernels characterising the renewal equations derived by biochemical system with reactions that satisfy the detail balance condition belong to the class of completely monotone functions.

How noise can generate calcium spike-type oscillations in deterministic equilibrium modes

Stochastic excitability of spiking oscillatory regimes in the calcium kinetics is studied on the basis of the Li-Rinzel conceptual model. The probabilistic mechanisms of the noise-induced generation of large-amplitude oscillations in parametric zones, where the original deterministic model has only stable equilibria, are investigated numerically and analytically. A parametric statistical description of interspike intervals is curried out and the phenomenon of coherence resonance is discussed. For the analytical study of the stochastic excitement, the confidence domain method using a stochastic sensitivity technique is applied. In this analysis, a key role of mutual arrangement of the confidence ellipses and separatrices detaching the sub- and supercritical regions is demonstrated. It is shown that in the Li-Rinzel model such separatrices are the stable manifolds of the saddle equilibria and the transient semiattractors.

A Tale of two receptors

Calcium (Ca²⁺) oscillations in hepatocytes have a wide dynamic range. In particular, recent experimental evidence shows that agonist stimulation of the P2Y family of receptors leads to qualitatively diverse Ca²⁺ oscillations. We present a new model of Ca²⁺ oscillations in hepatocytes based on these experiments to investigate the mechanisms controlling P2Y-activated Ca²⁺ oscillations. The model accounts for Ca²⁺ regulation of the IP₃ receptor (IP₃R), the positive feedback from Ca²⁺ on phospholipase C (PLC) and the P2Y receptor phosphorylation by protein kinase C (PKC). Furthermore, PKC is shown to control multiple cellular substrates. Utilising the model, we suggest the activity and intensity of PLC and PKC necessary to explain the qualitatively diverse Ca²⁺ oscillations in response to P2Y receptor activation.

Endoplasmic Reticulum Calcium Pumps and Tumor Cell Differentiation

Endoplasmic reticulum (ER) calcium homeostasis plays an essential role in cellular calcium signaling, intra-ER protein chaperoning and maturation, as well as in the interaction of the ER with other organelles. Calcium is accumulated in the ER by sarco/endoplasmic reticulum calcium ATPases (SERCA enzymes) that generate by active, ATP-dependent transport, a several thousand-fold calcium ion concentration gradient between the cytosol (low nanomolar) and the ER lumen (high micromolar). SERCA enzymes are coded by three genes that by alternative splicing give rise to several isoforms, which can display isoform-specific calcium transport characteristics. SERCA expression levels and isoenzyme composition vary according to cell type, and this constitutes a mechanism whereby ER calcium homeostasis is adapted to the signaling and metabolic needs of the cell, depending on its phenotype, its state of activation and differentiation. As reviewed here, in several normal epithelial cell types including bronchial, mammary, gastric, colonic and choroid plexus epithelium, as well as in mature cells of hematopoietic origin such as pumps are simultaneously expressed, whereas in corresponding tumors and leukemias SERCA3 expression is selectively down-regulated. SERCA3 expression is restored during the pharmacologically induced differentiation of various cancer and leukemia cell types. SERCA3 is a useful marker for the study of cell differentiation, and the loss of SERCA3 expression constitutes a previously unrecognized example of the remodeling of calcium homeostasis in tumors.

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.

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.

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.

Exact stochastic simulation of a calcium microdomain reveals the impact of Ca²⁺ fluctuations on IP₃R gating

In this study, we numerically analyzed the nonlinear Ca(2+)-dependent gating dynamics of a single, nonconducting inositol 1,4,5-trisphosphate receptor (IP₃R) channel, using an exact and fully stochastic simulation algorithm that includes channel gating, Ca(2+) buffering, and Ca(2+) diffusion. The IP₃R is a ubiquitous intracellular Ca(2+) release channel that plays an important role in the formation of complex spatiotemporal Ca(2+) signals such as waves and oscillations. Dynamic subfemtoliter Ca(2+) microdomains reveal low copy numbers of Ca(2+) ions, buffer molecules, and IP₃Rs, and stochastic fluctuations arising from molecular interactions and diffusion do not average out. In contrast to models treating calcium dynamics deterministically, the stochastic approach accounts for this molecular noise. We varied Ca(2+) diffusion coefficients and buffer reaction rates to tune the autocorrelation properties of Ca(2+) noise and found a distinct relation between the autocorrelation time τac, the mean channel open and close times, and the resulting IP₃R open probability PO. We observed an increased PO for shorter noise autocorrelation times, caused by increasing channel open times and decreasing close times. In a pure diffusion model the effects become apparent at elevated calcium concentrations, e.g., at [Ca(2+)] = 25 μM, τac = 0.082 ms, the IP₃R open probability increased by ≈20% and mean open times increased by ≈4 ms, compared to a zero noise model. We identified the inactivating Ca(2+) binding site of IP₃R subunits as the primarily noise-susceptible element of the De Young and Keizer model. Short Ca(2+) noise autocorrelation times decrease the probability of Ca(2+) association and consequently increase IPvR activity. These results suggest a functional role of local calcium noise properties on calcium-regulated target molecules such as the ubiquitous IP₃R. This finding may stimulate novel experimental approaches analyzing the role of calcium noise properties on microdomain behavior.

Reliable encoding of stimulus intensities within random sequences of intracellular Ca2+ spikes

Ca(2+) is a ubiquitous intracellular messenger that regulates diverse cellular activities. Extracellular stimuli often evoke sequences of intracellular Ca(2+) spikes, and spike frequency may encode stimulus intensity. However, the timing of spikes within a cell is random because each interspike interval has a large stochastic component. In human embryonic kidney (HEK) 293 cells and rat primary hepatocytes, we found that the average interspike interval also varied between individual cells. To evaluate how individual cells reliably encoded stimuli when Ca(2+) spikes exhibited such unpredictability, we combined Ca(2+) imaging of single cells with mathematical analyses of the Ca(2+) spikes evoked by receptors that stimulate formation of inositol 1,4,5-trisphosphate (IP3). This analysis revealed that signal-to-noise ratios were improved by slow recovery from feedback inhibition of Ca(2+) spiking operating at the whole-cell level and that they were robust against perturbations of the signaling pathway. Despite variability in the frequency of Ca(2+) spikes between cells, steps in stimulus intensity caused the stochastic period of the interspike interval to change by the same factor in all cells. These fold changes reliably encoded changes in stimulus intensity, and they resulted in an exponential dependence of average interspike interval on stimulation strength. We conclude that Ca(2+) spikes enable reliable signaling in a cell population despite randomness and cell-to-cell variability, because global feedback reduces noise, and changes in stimulus intensity are represented by fold changes in the stochastic period of the interspike interval.

Mesoscopic behavior from microscopic Markov dynamics and its application to calcium release channels

A major challenge in biology is to understand how molecular processes determine phenotypic features. We address this fundamental problem in a class of model systems by developing a general mathematical framework that allows the calculation of mesoscopic properties from the knowledge of microscopic Markovian transition probabilities. We show how exact analytic formulae for the first and second moments of resident time distributions in mesostates can be derived from microscopic resident times and transition probabilities even for systems with a large number of microstates. We apply our formalism to models of the inositol trisphosphate receptor, which plays a key role in generating calcium signals triggering a wide variety of cellular responses. We demonstrate how experimentally accessible quantities, such as opening and closing times and the coefficient of variation of inter-spike intervals, and other, more elaborated, quantities can be analytically calculated from the underlying microscopic Markovian dynamics. A virtue of our approach is that we do not need to follow the detailed time evolution of the whole system, as we derive the relevant properties of its steady state without having to take into account the often extremely complicated transient features. We emphasize that our formulae fully agree with results obtained by stochastic simulations and approaches based on a full determination of the microscopic system's time evolution. We also illustrate how experiments can be devised to discriminate between alternative molecular models of the inositol trisphosphate receptor. The developed approach is applicable to any system described by a Markov process and, owing to the analytic nature of the resulting formulae, provides an easy way to characterize also rare events that are of particular importance to understand the intermittency properties of complex dynamic systems.

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.

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.