Markov chain models of coupled calcium channels: Kronecker representations and iterative solution methods

Understanding Calmodulin Variants Affecting Calcium-Dependent Inactivation of L-Type Calcium Channels through Whole-Cell Simulation of the Cardiac Ventricular Myocyte

Mutations in the calcium-sensing protein calmodulin (CaM) have been linked to two cardiac arrhythmia diseases, Long QT Syndrome 14 (LQT14) and Catecholaminergic Polymorphic Ventricular Tachycardia Type 4 (CPVT4), with varying degrees of severity. Functional characterization of the CaM mutants most strongly associated with LQT14 show a clear disruption of the calcium-dependent inactivation (CDI) of the L-Type calcium channel (LCC). CPVT4 mutants on the other hand are associated with changes in their affinity to the ryanodine receptor. In clinical studies, some variants have been associated with both CPVT4 and LQT15. This study uses simulations in a model for excitation-contraction coupling in the rat ventricular myocytes to understand how LQT14 variant might give the functional phenotype similar to CPVT4. Changing the CaM-dependent transition rate by a factor of 0.75 corresponding to the D96V variant and by a factor of 0.90 corresponding to the F142L or N98S variants, in a physiologically based stochastic model of the LCC prolonger, the action potential duration changed by a small amount in a cardiac myocyte but did not disrupt CICR at 1, 2, and 4 Hz. Under beta-adrenergic simulation abnormal excitation-contraction coupling was observed above 2 Hz pacing for the mutant CaM. The same conditions applied under beta-adrenergic stimulation led to the rapid onset of arrhythmia in the mutant CaM simulations. Simulations with the LQT14 mutations under the conditions of rapid pacing with beta-adrenergic stimulation drives the cardiac myocyte toward an arrhythmic state known as Ca2+ overload. These simulations provide a mechanistic link to a disease state for LQT14-associated mutations in CaM to yield a CPVT4 phenotype. The results show that small changes to the CaM-regulated inactivation of LCC promote arrhythmia and underscore the significance of CDI in proper heart function.

Critical Requirements for the Initiation of a Cardiac Arrhythmia in Rat Ventricle: How Many Myocytes?

Cardiovascular disease is the leading cause of death worldwide due in a large part to arrhythmia. In order to understand how calcium dynamics play a role in arrhythmogenesis, normal and dysfunctional Ca²⁺ signaling in a subcellular, cellular, and tissued level is examined using cardiac ventricular myocytes at a high temporal and spatial resolution using multiscale computational modeling. Ca²⁺ sparks underlie normal excitation-contraction coupling. However, under pathological conditions, Ca²⁺ sparks can combine to form Ca²⁺ waves. These propagating elevations of (Ca²⁺)_i can activate an inward Na⁺-Ca²⁺ exchanger current (I_NCX) that contributes to early after-depolarization (EADs) and delayed after-depolarizations (DADs). However, how cellular currents lead to full depolarization of the myocardium and how they initiate extra systoles is still not fully understood. This study explores how many myocytes must be entrained to initiate arrhythmogenic depolarizations in biophysically detailed computational models. The model presented here suggests that only a small number of myocytes must activate in order to trigger an arrhythmogenic propagating action potential. These conditions were examined in 1-D, 2-D, and 3-D considering heart geometry. The depolarization of only a few hundred ventricular myocytes is required to trigger an ectopic depolarization. The number decreases under disease conditions such as heart failure. Furthermore, in geometrically restricted parts of the heart such as the thin muscle strands found in the trabeculae and papillary muscle, the number of cells needed to trigger a propagating depolarization falls even further to less than ten myocytes.

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.

Calcium waves in a grid of clustered channels with synchronous IP3 binding and unbinding

Calcium signals in cells occur at multiple spatial scales and variable temporal duration. However, a physical explanation for transitions between long-lasting global oscillations and localized short-term elevations (puffs) of cytoplasmic Ca²⁺ is still lacking. Here we introduce a phenomenological, coarse-grained model for the calcium variable, which is represented by ordinary differential equations. Due to its small number of parameters, and its simplicity, this model allows us to numerically study the interplay of multi-scale calcium concentrations with stochastic ion channel gating dynamics even in larger systems. We apply this model to a single cluster of inositol trisphosphate (IP ₃) receptor channels and find further evidence for the results presented in earlier work: a single cluster may be capable of producing different calcium release types, where long-lasting events are accompanied by unbinding of IP ₃ from the receptor (Rückl et al., PLoS Comput. Biol. 11, e1003965 (2015)). Finally, we show the practicability of the model in a grid of 64 clusters which is computationally intractable with previous high-resolution models. Here long-lasting events can lead to synchronized oscillations and waves, while short events stay localized. The frequency of calcium releases as well as their coherence can thereby be regulated by the amplitude of IP ₃ stimulation. Finally the model allows for a new explanation of oscillating [IP ₃], which is not based on metabolic production and degradation of IP ₃.

Unifying principles of calcium wave propagation - Insights from a three-dimensional model for atrial myocytes

Atrial myocytes in a number of species lack transverse tubules. As a consequence the intracellular calcium signals occurring during each heartbeat exhibit complex spatio-temporal dynamics. These calcium patterns arise from saltatory calcium waves that propagate via successive rounds of diffusion and calcium-induced calcium release. The many parameters that impinge on calcium-induced calcium release and calcium signal propagation make it difficult to know a priori whether calcium waves will successfully travel, or be extinguished. In this study, we describe in detail a mathematical model of calcium signalling that allows the effect of such parameters to be independently assessed. A key aspect of the model is to follow the triggering and evolution of calcium signals within a realistic three-dimensional cellular volume of an atrial myocyte, but with low computational costs. This is achieved by solving the linear transport equation for calcium analytically between calcium release events and by expressing the onset of calcium liberation as a threshold process. The model makes non-intuitive predictions about calcium signal propagation. For example, our modelling illustrates that the boundary of a cell produces a wave-guiding effect that enables calcium ions to propagate further and for longer, and can subtly alter the pattern of calcium wave movement. The high spatial resolution of the modelling framework allows the study of any arrangement of calcium release sites. We demonstrate that even small variations in randomly positioned release sites cause highly heterogeneous cellular responses. This article is part of a Special Issue entitled: 13th European Symposium on Calcium.

Application of stochastic automata networks for creation of continuous time Markov chain models of voltage gating of gap junction channels

The primary goal of this work was to study advantages of numerical methods used for the creation of continuous time Markov chain models (CTMC) of voltage gating of gap junction (GJ) channels composed of connexin protein. This task was accomplished by describing gating of GJs using the formalism of the stochastic automata networks (SANs), which allowed for very efficient building and storing of infinitesimal generator of the CTMC that allowed to produce matrices of the models containing a distinct block structure. All of that allowed us to develop efficient numerical methods for a steady-state solution of CTMC models. This allowed us to accelerate CPU time, which is necessary to solve CTMC models, ∼20 times.

Multiscale modelling of coupled Ca2+ channels using coloured stochastic Petri nets

Stochastic modelling of coupled Ca2+ channels is a challenge, especially when the coupling of the channels, as determined by their spatial arrangement relative to each other, has to be considered at multiple spatial scales. In this study, the authors address this problem using coloured stochastic Petri nets (SPNc) as high-level description to generate continuous-time Markov chains. The authors develop several models with increasing complexity. They first apply SPNc to model single clusters of coupled Ca2+ channels arranged in a regular or irregular lattice, where they describe how to represent the geometrical arrangement of Ca2+ channels relative to each other using colours. They then apply this modelling idea to construct more complex models by modelling spatially arranged clusters of channels. The authors' models can be easily reproduced and adapted to different scenarios.

Reduction of calcium release site models via moment fitting of phase-type distributions

Models of calcium (Ca(2 +)) release sites derived from continuous-time Markov chain (CTMC) models of intracellular Ca(2 +) channels exhibit collective gating reminiscent of the experimentally observed phenomenon of Ca(2 +) puffs and sparks. In order to overcome the state-space explosion that occurs in compositionally defined Ca(2 +) release site models, we have implemented an automated procedure for model reduction that replaces aggregated states of the full release site model with much simpler CTMCs that have similar within-group phase-type sojourn times and inter-group transitions. Error analysis based on comparison of full and reduced models validates the method when applied to release site models composed of 20 three-state channels that are both activated and inactivated by Ca(2 +). Although inspired by existing techniques for fitting moments of phase-type distributions, the automated reduction method for compositional Ca(2 +) release site models is unique in several respects and novel in this biophysical context.

Using Jensen's inequality to explain the role of regular calcium oscillations in protein activation

Oscillations of cytosolic Ca(2 +) are very important for cellular signalling in excitable and non-excitable cells. The information of various extracellular stimuli is encoded into oscillating patterns of Ca(2 +) that subsequently lead to the activation of different Ca(2 +)-sensitive target proteins in the cell. The question remains, however, why this information is transmitted by means of an oscillating rather than a constant signal. Here we show that, in fact, Ca(2 +) oscillations can achieve a better activation of target proteins than a comparable constant signal with the same amount of Ca(2 +) used. For this we use Jensen's inequality that describes the relation between the function value of the average of a set of argument values and the average of the function values of the arguments from that set. We analyse the role of the cooperativity of the binding of Ca(2 +) and of zero-order ultrasensitivity, which are two properties that are often observed in experiments on the activation of Ca(2 +)-sensitive target proteins. Our results apply to arbitrary oscillation shapes and a very general decoding model, thus generalizing the observations of several previous studies. We compare our results with data from experimental studies investigating the activation of nuclear factor of activated T cells (NFAT) and Ras by oscillatory and constant signals. Although we are restricted to specific approximations due to the lack of detailed kinetic data, we find good qualitative agreement with our theoretical predictions.

Models of cardiac excitation-contraction coupling in ventricular myocytes

Mathematical and computational modeling of cardiac excitation-contraction coupling has produced considerable insights into how the heart muscle contracts. With the increase in biophysical and physiological data available, the modeling has become more sophisticated with investigations spanning in scale from molecular components to whole cells. These modeling efforts have provided insight into cardiac excitation-contraction coupling that advanced and complemented experimental studies. One goal is to extend these detailed cellular models to model the whole heart. While this has been done with mechanical and electrophysiological models, the complexity and fast time course of calcium dynamics have made inclusion of detailed calcium dynamics in whole heart models impractical. Novel methods such as the probability density approach and moment closure technique which increase computational efficiency might make this tractable.

Entropically modified spiking ability and periodicity in clustered channels

It has been shown that, in models of excitable clusters of voltage-gated sodium channels, the spontaneous rate of action potential and signal encoding ability exhibit multiple peaks at different cluster sizes due to an entropy effect in small system. In this paper, we show that similar results can be found in excitable cluster of ligand-gated calcium channels. Furthermore, we demonstrate that the periodicity of spontaneous Ca2+ spikes elicited by the cluster reveals multiple maxima at small discrete cluster sizes.

Puff-wave transition in an inhomogeneous model for calcium signals

In many cell types, calcium ion channels on the endoplasmic reticulum membrane occur in a clustered distribution. The channels generate either localized puffs, each comprising channels of only one cluster, or global calcium waves. In this work we model the calcium system as a two-dimensional lattice of active elements distributed regularly in an otherwise passive space. We address an important feature of the puff-wave transition, which is the difference in lifetime of puffs at a few hundred milliseconds and long-lived global waves with periods of several seconds. We show that such a lifetime difference between puffs and waves can be understood with strongly reduced ordinary differential equations modified by a time-scale factor that takes into account the coupling strength of active and passive regions determined by the Ca2+ diffusion coefficient. Furthermore, we show that the point model can also describe very well the dependence of Ca2+ oscillation characteristics on the cluster-cluster distance in the case of large diffusivity.

Reduction of calcium release site models via fast/slow analysis and iterative aggregation/disaggregation

Mathematical models of calcium release sites derived from Markov chain models of intracellular calcium channels exhibit collective gating reminiscent of the experimentally observed phenomenon of calcium puffs and sparks. Such models often take the form of stochastic automata networks in which the transition probabilities of each channel depend on the local calcium concentration and thus the state of the other channels. In order to overcome the state-space explosion that occurs in such compositionally defined calcium release site models, we have implemented several automated procedures for model reduction using fast/slow analysis. After categorizing rate constants in the single channel model as either fast or slow, groups of states in the expanded release site model that are connected by fast transitions are lumped, and transition rates between reduced states are chosen consistent with the conditional probability distribution among states within each group. For small problems these conditional probability distributions can be numerically calculated from the full model without approximation. For large problems the conditional probability distributions can be approximated without the construction of the full model by assuming rapid mixing of states connected by fast transitions. Alternatively, iterative aggregation/disaggregation may be employed to obtain reduced calcium release site models in a memory-efficient fashion. Benchmarking of several different iterative aggregation/disaggregation-based fast/slow reduction schemes establishes the effectiveness of automated calcium release site reduction utilizing the Koury-McAllister-Stewart method.