Ion Channel Modeling beyond State of the Art: A Comparison with a System Theory-Based Model of the Shaker-Related Voltage-Gated Potassium Channel Kv1.1.

The mathematical modeling of ion channel kinetics is an important tool for studying the electrophysiological mechanisms of the nerves, heart, or cancer, from a single cell to an organ. Common approaches use either a Hodgkin-Huxley (HH) or a hidden Markov model (HMM) description, depending on the level of detail of the functionality and structural changes of the underlying channel gating, and taking into account the computational effort for model simulations. Here, we introduce for the first time a novel system theory-based approach for ion channel modeling based on the concept of transfer function characterization, without a priori knowledge of the biological system, using patch clamp measurements. Using the shaker-related voltage-gated potassium channel Kv1.1 (KCNA1) as an example, we compare the established approaches, HH and HMM, with the system theory-based concept in terms of model accuracy, computational effort, the degree of electrophysiological interpretability, and methodological limitations. This highly data-driven modeling concept offers a new opportunity for the phenomenological kinetic modeling of ion channels, exhibiting exceptional accuracy and computational efficiency compared to the conventional methods. The method has a high potential to further improve the quality and computational performance of complex cell and organ model simulations, and could provide a valuable new tool in the field of next-generation in silico electrophysiology.

A hybrid model of intercellular tension and cell-matrix mechanical interactions in a multicellular geometry.

Epithelial cells form continuous sheets of cells that exist in tensional homeostasis. Homeostasis is maintained through cell-to-cell junctions that distribute tension and balance forces between cells and their underlying matrix. Disruption of tensional homeostasis can lead to epithelial-mesenchymal transition (EMT), a transdifferentiation process in which epithelial cells adopt a mesenchymal phenotype, losing cell-cell adhesion and enhancing cellular motility. This process is critical during embryogenesis and wound healing, but is also dysregulated in many disease states. To further understand the role of intercellular tension in spatial patterning of epithelial cell monolayers, we developed a multicellular computational model of cell-cell and cell-substrate forces. This work builds on a hybrid cellular Potts model (CPM)-finite element model to evaluate cell-matrix mechanical feedback of an adherent multicellular cluster. Cellular movement is governed by thermodynamic constraints from cell volume, cell-cell and cell-matrix contacts, and durotaxis, which arises from cell-generated traction forces on a finite element substrate. Junction forces at cell-cell contacts balance these traction forces, thereby producing a mechanically stable epithelial monolayer. Simulations were compared to in vitro experiments using fluorescence-based junction force sensors in clusters of cells undergoing EMT. Results indicate that the multicellular CPM model can reproduce many aspects of EMT, including epithelial monolayer formation dynamics, changes in cell geometry, and spatial patterning of cell-cell forces in an epithelial tissue.

Revealing the Concealed Nature of Long-QT Type 3 Syndrome.

BACKGROUND: Gain-of-function mutations in the voltage-gated sodium channel (Nav1.5) are associated with the long-QT-3 (LQT3) syndrome. Nav1.5 is densely expressed at the intercalated disk, and narrow intercellular separation can modulate cell-to-cell coupling via extracellular electric fields and depletion of local sodium ion nanodomains. Models predict that significantly decreasing intercellular cleft widths slows conduction because of reduced sodium current driving force, termed "self-attenuation." We tested the novel hypothesis that self-attenuation can "mask" the LQT3 phenotype by reducing the driving force and late sodium current that produces early afterdepolarizations (EADs). METHODS AND RESULTS: Acute interstitial edema was used to increase intercellular cleft width in isolated guinea pig heart experiments. In a drug-induced LQT3 model, acute interstitial edema exacerbated action potential duration prolongation and produced EADs, in particular, at slow pacing rates. In a computational cardiac tissue model incorporating extracellular electric field coupling, intercellular cleft sodium nanodomains, and LQT3-associated mutant channels, myocytes produced EADs for wide intercellular clefts, whereas for narrow clefts, EADs were suppressed. For both wide and narrow clefts, mutant channels were incompletely inactivated. However, for narrow clefts, late sodium current was reduced via self-attenuation, a protective negative feedback mechanism, masking EADs. CONCLUSIONS: We demonstrated a novel mechanism leading to the concealing and revealing of EADs in LQT3 models. Simulations predict that this mechanism may operate independent of the specific mutation, suggesting that future therapies could target intercellular cleft separation as a compliment or alternative to sodium channels.

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.

Population Density and Moment-based Approaches to Modeling Domain Calcium-mediated Inactivation of L-type Calcium Channels.

We present a population density and moment-based description of the stochastic dynamics of domain [Formula: see text]-mediated inactivation of L-type [Formula: see text] channels. Our approach accounts for the effect of heterogeneity of local [Formula: see text] signals on whole cell [Formula: see text] currents; however, in contrast with prior work, e.g., Sherman et al. (Biophys J 58(4):985-995, 1990), we do not assume that [Formula: see text] domain formation and collapse are fast compared to channel gating. We demonstrate the population density and moment-based modeling approaches using a 12-state Markov chain model of an L-type [Formula: see text] channel introduced by Greenstein and Winslow (Biophys J 83(6):2918-2945, 2002). Simulated whole cell voltage clamp responses yield an inactivation function for the whole cell [Formula: see text] current that agrees with the traditional approach when domain dynamics are fast. We analyze the voltage-dependence of [Formula: see text] inactivation that may occur via slow heterogeneous domain [[Formula: see text]]. Next, we find that when channel permeability is held constant, [Formula: see text]-mediated inactivation of L-type channels increases as the domain time constant increases, because a slow domain collapse rate leads to increased mean domain [[Formula: see text]] near open channels; conversely, when the maximum domain [[Formula: see text]] is held constant, inactivation decreases as the domain time constant increases. Comparison of simulation results using population densities and moment equations confirms the computational efficiency of the moment-based approach, and enables the validation of two distinct methods of truncating and closing the open system of moment equations. In general, a slow domain time constant requires higher order moment truncation for agreement between moment-based and population density simulations.

Ca(2+)-activation kinetics modulate successive puff/spark amplitude, duration and inter-event-interval correlations in a Langevin model of stochastic Ca(2+) release.

Through theoretical analysis of the statistics of stochastic calcium (Ca(2+)) release (i.e., the amplitude, duration and inter-event interval of simulated Ca(2+) puffs and sparks), we show that a Langevin description of the collective gating of Ca(2+) channels may be a good approximation to the corresponding Markov chain model when the number of Ca(2+) channels per Ca(2+) release unit (CaRU) is in the physiological range. The Langevin description of stochastic Ca(2+) release facilitates our investigation of correlations between successive puff/spark amplitudes, durations and inter-spark intervals, and how such puff/spark statistics depend on the number of channels per release site and the kinetics of Ca(2+)-mediated inactivation of open channels. When Ca(2+) inactivation/de-inactivation rates are intermediate-i.e., the termination of Ca(2+) puff/sparks is caused by an increase in the number of inactivated channels-the correlation between successive puff/spark amplitudes is negative, while the correlations between puff/spark amplitudes and the duration of the preceding or subsequent inter-spark interval are positive. These correlations are significantly reduced or change signs when inactivation/de-inactivation rates are extreme (slow or fast) and puff/sparks terminate via stochastic attrition.

Calcium homeostasis in a local/global whole cell model of permeabilized ventricular myocytes with a Langevin description of stochastic calcium release.

Population density approaches to modeling local control of Ca(2+)-induced Ca(2+) release in cardiac myocytes can be used to construct minimal whole cell models that accurately represent heterogeneous local Ca(2+) signals. Unfortunately, the computational complexity of such "local/global" whole cell models scales with the number of Ca(2+) release unit (CaRU) states, which is a rapidly increasing function of the number of ryanodine receptors (RyRs) per CaRU. Here we present an alternative approach based on a Langevin description of the collective gating of RyRs coupled by local Ca(2+) concentration ([Ca(2+)]). The computational efficiency of this approach no longer depends on the number of RyRs per CaRU. When the RyR model is minimal, Langevin equations may be replaced by a single Fokker-Planck equation, yielding an extremely compact and efficient local/global whole cell model that reproduces and helps interpret recent experiments that investigate Ca(2+) homeostasis in permeabilized ventricular myocytes. Our calculations show that elevated myoplasmic [Ca(2+)] promotes elevated network sarcoplasmic reticulum (SR) [Ca(2+)] via SR Ca(2+)-ATPase-mediated Ca(2+) uptake. However, elevated myoplasmic [Ca(2+)] may also activate RyRs and promote stochastic SR Ca(2+) release, which can in turn decrease SR [Ca(2+)]. Increasing myoplasmic [Ca(2+)] results in an exponential increase in spark-mediated release and a linear increase in nonspark-mediated release, consistent with recent experiments. The model exhibits two steady-state release fluxes for the same network SR [Ca(2+)] depending on whether myoplasmic [Ca(2+)] is low or high. In the later case, spontaneous release decreases SR [Ca(2+)] in a manner that maintains robust Ca(2+) sparks.

Discrete-state stochastic models of calcium-regulated calcium influx and subspace dynamics are not well-approximated by ODEs that neglect concentration fluctuations.

Cardiac myocyte calcium signaling is often modeled using deterministic ordinary differential equations (ODEs) and mass-action kinetics. However, spatially restricted "domains" associated with calcium influx are small enough (e.g., 10(-17) liters) that local signaling may involve 1-100 calcium ions. Is it appropriate to model the dynamics of subspace calcium using deterministic ODEs or, alternatively, do we require stochastic descriptions that account for the fundamentally discrete nature of these local calcium signals? To address this question, we constructed a minimal Markov model of a calcium-regulated calcium channel and associated subspace. We compared the expected value of fluctuating subspace calcium concentration (a result that accounts for the small subspace volume) with the corresponding deterministic model (an approximation that assumes large system size). When subspace calcium did not regulate calcium influx, the deterministic and stochastic descriptions agreed. However, when calcium binding altered channel activity in the model, the continuous deterministic description often deviated significantly from the discrete stochastic model, unless the subspace volume is unrealistically large and/or the kinetics of the calcium binding are sufficiently fast. This principle was also demonstrated using a physiologically realistic model of calmodulin regulation of L-type calcium channels introduced by Yue and coworkers.