Martin Falcke

Reading the patterns in living cells —the physics of ca2+ signaling

Ca2+ is one of the most important messengers. It transmits signals inside living cells and takes part in intercellular coordination. The dynamics of the Ca2+ concentration shows a transition from elemental, stochastic events to global events like waves and oscillations. This transition renders it an ideal tool for studying basic concepts of pattern formation, especially since access to the most important experimental parameters is given. Ca2+ dynamics in living cells has been a major topic of biophysical modelling in the last 15 years. Modelling has reached the level of predictive power. The theoretical analysis of waves provided new insight into the mechanisms of Ca2+ signaling and led to new concepts of analysis of wave equations with concentration dependent diffusion and novel wave bifurcations. Modelling of oscillations provided understanding especially of complex oscillations and allowed to extract information about the underlying cellular parameters and mechanisms. The investigation of the stochastic aspects of intracellular Ca2+ dynamics demonstrated the fundamental role of fluctuations arising from the control of the release channel by Ca2+ and IP3. This review presents an overview of current theoretical research on Ca2+ dynamics in living cells driven by the inositol 1,4,5-trisphosphate receptor channel.

On the Role of Stochastic Channel Behavior in Intracellular Ca2+ Dynamics

I present a stochastic model for intracellular Ca(2+) oscillations. The model starts from stochastic binding and dissociation of Ca(2+) to binding sites on a single subunit of the IP(3)-receptor channel but is capable of simulating large numbers of clusters for many oscillation periods too. I find oscillations with variable periods ranging from 17 s to 120 s and a standard deviation well in the experimentally observed range. Long period oscillations can be perceived as nucleation phenomenon and can be observed for a large variety of single channel dynamics. Their period depends on the geometric characteristics of the cluster array. Short periods are in the range of the time scale of channel dynamics. Both long and short period oscillations occur for parameters with a nonoscillatory deterministic regime.

Clustering of InsP3 receptors by InsP3 retunes their regulation by InsP3 and Ca2

The versatility of Ca2+ signals derives from their spatio-temporal organization. For Ca2+ signals initiated by inositol-1,4,5-trisphosphate (InsP3), this requires local interactions between InsP3 receptors (InsP3Rs) mediated by their rapid stimulation and slower inhibition by cytosolic Ca2+. This allows hierarchical recruitment of Ca2+ release events as the InsP3 concentration increases. Single InsP3Rs respond first, then clustered InsP3Rs open together giving a local ‘Ca2+ puff’, and as puffs become more frequent they ignite regenerative Ca2+ waves. Using nuclear patch-clamp recording, here we demonstrate that InsP3Rs are initially randomly distributed with an estimated separation of ∼1 μm. Low concentrations of InsP3 cause InsP3Rs to aggregate rapidly and reversibly into small clusters of about four closely associated InsP3Rs. At resting cytosolic [Ca2+], clustered InsP3Rs open independently, but with lower open probability, shorter open time, and less InsP3 sensitivity than lone InsP3Rs. Increasing cytosolic [Ca2+] reverses the inhibition caused by clustering, InsP3R gating becomes coupled, and the duration of multiple openings is prolonged. Clustering both exposes InsP3Rs to local Ca2+ rises and increases the effects of Ca2+. Dynamic regulation of clustering by InsP3 retunes InsP3R sensitivity to InsP3 and Ca2+, facilitating hierarchical recruitment of the elementary events that underlie all InsP3-evoked Ca2+ signals.

Release Currents of IP3 Receptor Channel Clusters and Concentration Profiles

We simulate currents and concentration profiles generated by Ca(2+) release from the endoplasmic reticulum (ER) to the cytosol through IP(3) receptor channel clusters. Clusters are described as conducting pores in the lumenal membrane with a diameter from 6 nm to 36 nm. The endoplasmic reticulum is modeled as a disc with a radius of 1-12 microm and an inner height of 28 nm. We adapt the dependence of the currents on the trans Ca(2+) concentration (intralumenal) measured in lipid bilayer experiments to the cellular geometry. Simulated currents are compared with signal mass measurements in Xenopus oocytes. We find that release currents depend linearly on the concentration of free Ca(2+) in the lumen. The release current is approximately proportional to the square root of the number of open channels in a cluster. Cytosolic concentrations at the location of the cluster range from 25 microM to 170 microM. Concentration increase due to puffs in a distance of a few micrometers from the puff site is found to be in the nanomolar range. Release currents decay biexponentially with timescales of <1 s and a few seconds. Concentration profiles decay with timescales of 0.125-0.250 s upon termination of release.

Buffers and Oscillations in Intracellular Ca2+ Dynamics

I model the behavior of intracellular Ca(2+) release with high buffer concentrations. The model uses a spatially discrete array of channel clusters. The channel subunit dynamics is a stochastic representation of the DeYoung-Keizer model. The calculations show that the concentration profile of fast buffer around an open channel is more localized than that of slow buffers. Slow buffers allow for release of larger amounts of Ca(2+) from the endoplasmic reticulum and hence bind more Ca(2+) than fast buffers with the same dissociation constant and concentration. I find oscillation-like behavior for high slow buffer concentration and low Ca(2+) content of the endoplasmic reticulum. High concentration of slow buffer leads to oscillation-like behavior by repetitive wave nucleation for high Ca(2+) content of the endoplasmic reticulum. Localization of Ca(2+) release by slow buffer, as used in experiments, can be reproduced by the modeling approach.

Calcium signals driven by single channel noise.

Usually, the occurrence of random cell behavior is appointed to small copy numbers of molecules involved in the stochastic process. Recently, we demonstrated for a variety of cell types that intracellular Ca2+ oscillations are sequences of random spikes despite the involvement of many molecules in spike generation. This randomness arises from the stochastic state transitions of individual Ca2+ release channels and does not average out due to the existence of steep concentration gradients. The system is hierarchical due to the structural levels channel - channel cluster - cell and a corresponding strength of coupling. Concentration gradients introduce microdomains which couple channels of a cluster strongly. But they couple clusters only weakly; too weak to establish deterministic behavior on cell level. Here, we present a multi-scale modelling concept for stochastic hierarchical systems. It simulates active molecules individually as Markov chains and their coupling by deterministic diffusion. Thus, we are able to follow the consequences of random single molecule state changes up to the signal on cell level. To demonstrate the potential of the method, we simulate a variety of experiments. Comparisons of simulated and experimental data of spontaneous oscillations in astrocytes emphasize the role of spatial concentration gradients in Ca2+ signalling. Analysis of extensive simulations indicates that frequency encoding described by the relation between average and standard deviation of interspike intervals is surprisingly robust. This robustness is a property of the random spiking mechanism and not a result of control.

Chemical turbulence and standing waves in a surface reaction model: The influence of global coupling and wave instabilities

Among heterogeneously catalyzed chemical reactions, the CO oxidation on the Pt(110) surface under vacuum conditions offers probably the greatest wealth of spontaneous formation of spatial patterns. Spirals, fronts, and solitary pulses were detected at low surface temperatures (T<500 K), in line with the standard phenomenology of bistable, excitable, and oscillatory reaction-diffusion systems. At high temperatures (T greater, similar 540 K), more surprising features like chemical turbulence and standing waves appeared in the experiments. Herein, we study a realistic reaction-diffusion model of this system, with respect to the latter phenomena. In particular, we deal both with the influence of global coupling through the gas phase on the oscillatory reaction and the possibility of wave instabilities under excitable conditions. Gas-phase coupling is shown to either synchronize the oscillations or to yield turbulence and standing structures. The latter findings are closely related to clustering in networks of coupled oscillators and indicate a dominance of the global gas-phase coupling over local coupling via surface diffusion. In the excitable regime wave instabilities in one and two dimensions have been discovered. In one dimension, pulses become unstable due to a vanishing of the refractory zone. In two dimensions, turbulence can also emerge due to spiral breakup, which results from a violation of the dispersion relation.

Impact of Mitochondrial Ca2+ Cycling on Pattern Formation and Stability

Energization of mitochondria significantly alters the pattern of Ca2+ wave activity mediated by activation of the inositol (1,4,5) trisphosphate (IP3) receptor (IP3R) in Xenopus oocytes. The number of pulsatile foci is reduced and spiral Ca2+ waves are no longer observed. Rather, target patterns of Ca2+ release predominate, and when fragmented, fail to form spirals. Ca2+ wave velocity, amplitude, decay time, and periodicity are also increased. We have simulated these experimental findings by supplementing an existing mathematical model with a differential equation for mitochondrial Ca2+ uptake and release. Our calculations show that mitochondrial Ca2+ efflux plays a critical role in pattern formation by prolonging the recovery time of IP3Rs from a refractory state. We also show that under conditions of high energization of mitochondria, the Ca2+ dynamics can become bistable with a second stable stationary state of high resting Ca2+ concentration.

Discrete stochastic modeling of calcium channel dynamics.

We propose a discrete stochastic model for calcium dynamics in living cells. A set of probabilities for the opening/closing of calcium channels is assumed to depend on the calcium concentration. We study this model in one dimension, analytically in the limit of a large number of channels per site N, and numerically for small N. As the number of channels per site is increased, the transition from a nonpropagating region of activity to a propagating one changes from one described by directed percolation to that of deterministic depinning in a spatially discrete system. Also, for a small number of channels a propagating calcium wave can leave behind a novel fluctuation-driven state.

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

Mathematical analysis of Ca2+ signals in single cells reveals how cells can encode stimulus intensity in the frequency of Ca2+ spikes. Apparently Random Signals Encode Information Reliably Repetitive Ca2+ spikes occur in many cells in response to stimuli that activate an intracellular signaling cascade that involves Ca2+ released from internal stores. These repetitive spikes are believed to represent the intensity of the stimulus, such that increasing the stimulus increases the frequency of the spikes. But the time between spikes (interspike interval) is random within a cell, and cells in a population exhibit variable spiking frequencies. Thurley et al. performed single-cell Ca2+ imaging of primary liver cells and human embryonic kidney (HEK) 293 cells to examine the properties of Ca2+ spikes in response to extracellular ligands under various conditions. Mathematical analysis revealed that, although the interspike interval had a random element, there was a consistent fold change in this interval across populations of cells responding to different amounts of the ligands. Thus, a common change to a random element enables the cells to properly interpret signal intensity from the frequency of repetitive Ca2+ spikes. Ca2+ is a ubiquitous intracellular messenger that regulates diverse cellular activities. Extracellular stimuli often evoke sequences of intracellular Ca2+ 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 Ca2+ spikes exhibited such unpredictability, we combined Ca2+ imaging of single cells with mathematical analyses of the Ca2+ 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 Ca2+ spiking operating at the whole-cell level and that they were robust against perturbations of the signaling pathway. Despite variability in the frequency of Ca2+ 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 Ca2+ 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.