Microdomain calcium fluctuations as a colored noise process

Analog Signaling With the "Digital" Molecular Switch CaMKII

Molecular switches, such as the protein kinase CaMKII, play a fundamental role in cell signaling by decoding inputs into either high or low states of activity; because the high activation state can be turned on and persist after the input ceases, these switches have earned a reputation as "digital." Although this on/off, binary perspective has been valuable for understanding long timescale synaptic plasticity, accumulating experimental evidence suggests that the CaMKII switch can also control plasticity on short timescales. To investigate this idea further, a non-autonomous, nonlinear ordinary differential equation, representative of a general bistable molecular switch, is analyzed. The results suggest that switch activity in regions surrounding either the high- or low-stable states of activation could act as a reliable analog signal, whose short timescale fluctuations relative to equilibrium track instantaneous input frequency. The model makes intriguing predictions and is validated against previous work demonstrating its suitability as a minimal representation of switch dynamics; in combination with existing experimental evidence, the theory suggests a multiplexed encoding of instantaneous frequency information over short timescales, with integration of total activity over longer timescales.

Partial Autoinformation to Characterize Symbolic Sequences

An information-theoretic approach to numerically determine the Markov order of discrete stochastic processes defined over a finite state space is introduced. To measure statistical dependencies between different time points of symbolic time series, two information-theoretic measures are proposed. The first measure is time-lagged mutual information between the random variables X_n and X_n+k, representing the values of the process at time points n and n + k, respectively. The measure will be termed autoinformation, in analogy to the autocorrelation function for metric time series, but using Shannon entropy rather than linear correlation. This measure is complemented by the conditional mutual information between X_n and X_n+k, removing the influence of the intermediate values X_n+k−1, …, X_n+1. The second measure is termed partial autoinformation, in analogy to the partial autocorrelation function (PACF) in metric time series analysis. Mathematical relations with known quantities such as the entropy rate and active information storage are established. Both measures are applied to a number of examples, ranging from theoretical Markov and non-Markov processes with known stochastic properties, to models from statistical physics, and finally, to a discrete transform of an EEG data set. The combination of autoinformation and partial autoinformation yields important insights into the temporal structure of the data in all test cases. For first- and higher-order Markov processes, partial autoinformation correctly identifies the order parameter, but also suggests extended, non-Markovian effects in the examples that lack the Markov property. For three hidden Markov models (HMMs), the underlying Markov order is found. The combination of both quantities may be used as an early step in the analysis of experimental, non-metric time series and can be employed to discover higher-order Markov dependencies, non-Markovianity and periodicities in symbolic time series.

Fluorescence correlation spectroscopy experiments to quantify free diffusion coefficients in reaction-diffusion systems: The case of Ca^{2+} and its dyes

Many cell signaling pathways involve the diffusion of messengers that bind and unbind to and from intracellular components. Quantifying their net transport rate under different conditions then requires having separate estimates of their free diffusion coefficient and binding or unbinding rates. In this paper, we show how performing sets of fluorescence correlation spectroscopy (FCS) experiments under different conditions, it is possible to quantify free diffusion coefficients and on and off rates of reaction-diffusion systems. We develop the theory and present a practical implementation for the case of the universal second messenger, calcium (Ca^{2+}) and single-wavelength dyes that increase their fluorescence upon Ca^{2+} binding. We validate the approach with experiments performed in aqueous solutions containing Ca^{2+} and Fluo4 dextran (both in its high and low affinity versions). Performing FCS experiments with tetramethylrhodamine-dextran in Xenopus laevis oocytes, we infer the corresponding free diffusion coefficients in the cytosol of these cells. Our approach can be extended to other physiologically relevant reaction-diffusion systems to quantify biophysical parameters that determine the dynamics of various variables of interest.

Calcium Ion Fluctuations Alter Channel Gating in a Stochastic Luminal Calcium Release Site Model

Stochasticity and small system size effects in complex biochemical reaction networks can greatly alter transient and steady-state system properties. A common approach to modeling reaction networks, which accounts for system size, is the chemical master equation that governs the dynamics of the joint probability distribution for molecular copy number. However, calculation of the stationary distribution is often prohibitive, due to the large state-space associated with most biochemical reaction networks. Here, we analyze a network representing a luminal calcium release site model and investigate to what extent small system size effects and calcium fluctuations, driven by ion channel gating, influx and diffusion, alter steady-state ion channel properties including open probability. For a physiological ion channel gating model and number of channels, the state-space may be between approximately 10⁶-10⁸ elements, and a novel modified block power method is used to solve the associated dominant eigenvector problem required to calculate the stationary distribution. We demonstrate that both small local cytosolic domain volume and a small number of ion channels drive calcium fluctuations that result in deviation from the corresponding model that neglects small system size effects.

FCS experiments to quantify Ca2+ diffusion and its interaction with buffers

Ca²⁺ signals are ubiquitous. One of the key factors for their versatility is the variety of spatio-temporal distributions that the cytosolic Ca²⁺ can display. In most cell types Ca²⁺ signals not only depend on Ca²⁺ entry from the extracellular medium but also on Ca²⁺ release from internal stores, a process which is in turn regulated by cytosolic Ca²⁺ itself. The rate at which Ca²⁺ is transported, the fraction that is trapped by intracellular buffers, and with what kinetics are thus key features that affect the time and spatial range of action of Ca²⁺ signals. The quantification of Ca²⁺ diffusion in intact cells is quite challenging because the transport rates that can be inferred using optical techniques are intricately related to the interaction of Ca²⁺ with the dye that is used for its observation and with the cellular buffers. In this paper, we introduce an approach that uses Fluorescence Correlation Spectroscopy (FCS) experiments performed at different conditions that in principle allows the quantification of Ca²⁺ diffusion and of its reaction rates with unobservable (non-fluorescent) Ca²⁺ buffers. To this end, we develop the necessary theory to interpret the experimental results and then apply it to FCS experiments performed in a set of solutions containing Ca²⁺, a single wavelength Ca²⁺ dye, and a non-fluorescent Ca²⁺ buffer. We show that a judicious choice of the experimental conditions and an adequate interpretation of the fitting parameters can be combined to extract information on the free diffusion coefficient of Ca²⁺ and of some of the properties of the unobservable buffer. We think that this approach can be applied to other situations, particularly to experiments performed in intact cells.

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.

Calcium, TRPC channels, and regulation of the actin cytoskeleton in podocytes: towards a future of targeted therapies

With more than 6,000 new pediatric patients with treatment-resistant nephrotic syndrome in the US each year alone, the unmet need for novel, podocyte-specific therapies is substantial. Recently, the established therapeutic benefit of angiotensin-converting enzyme (ACE) inhibitors and angiotensin receptor blockers (ARB) was used as a starting point to gain insight into the pathomechanism of primary podocytopathies. A calcium (Ca(2+))-mediated pathway has been identified that connects the angiotensin type 1 receptor (AT1R) to podocyte cytoskeletal dynamics, essential for a functioning glomerular filtration barrier. This discovery provided an important missing piece in our understanding of the pathomechanism of filter barrier damage, revealing Ca(2+) signaling as critical for podocyte health and disease. The identification of the two Ca(2+) permeant channels TRPC5 and TRPC6 as mediators of this pathway not only bolstered the importance of podocyte cytoskeleton dynamics but also revealed promising drug targets for treatment-resistant nephrotic syndrome. This review will focus on this novel signaling pathway in primary podocytopathies and its implications for next-generation therapies for glomerular disease.

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.