A thermodynamic model of the cardiac sarcoplasmic/endoplasmic Ca(2+) (SERCA) pump

The mechanisms of exercise improving cardiovascular function by stimulating Piezo1 and TRP ion channels: a systemic review

Mechanosensitive ion channels are widely distributed in the heart, lung, bladder and other tissues, and plays an important role in exercise-induced cardiovascular function promotion. By reviewing the PubMed databases, the results were summarized using the terms "Exercise/Sport", "Piezo1", "Transient receptor potential (TRP)" and "Cardiovascular" as the keywords, 124-related papers screened were sorted and reviewed. The results showed that: (1) Piezo1 and TRP channels play an important role in regulating blood pressure and the development of cardiovascular diseases such as atherosclerosis, myocardial infarction, and cardiac fibrosis; (2) Exercise promotes cardiac health, inhibits the development of pathological heart to heart failure, regulating the changes in the characterization of Piezo1 and TRP channels; (3) Piezo1 activates downstream signaling pathways with very broad pathways, such as AKT/eNOS, NF-κB, p38MAPK and HIPPO-YAP signaling pathways. Piezo1 and Irisin regulate nuclear localization of YAP and are hypothesized to act synergistically to regulate tissue mechanical properties of the cardiovascular system and (4) The cardioprotective effects of exercise through the TRP family are mostly accomplished through Ca2+ and involve many signaling pathways. TRP channels exert their important cardioprotective effects by reducing the TRPC3-Nox2 complex and mediating Irisin-induced Ca2+ influx through TRPV4. It is proposed that exercise stimulates the mechanosensitive cation channel Piezo1 and TRP channels, which exerts cardioprotective effects. The activation of Piezo1 and TRP channels and their downstream targets to exert cardioprotective function by exercise may provide a theoretical basis for the prevention of cardiovascular diseases and the rehabilitation of clinical patients.

In silico study of the mechanisms of hypoxia and contractile dysfunction during ischemia and reperfusion of hiPSC cardiomyocytes

Interconnected mechanisms of ischemia and reperfusion (IR) has increased the interest in IR in vitro experiments using human induced pluripotent stem cell-derived cardiomyocytes (hiPSC-CMs). We developed a whole-cell computational model of hiPSC-CMs including the electromechanics, a metabolite-sensitive sarcoplasmic reticulum Ca2+-ATPase (SERCA) and an oxygen dynamics formulation to investigate IR mechanisms. Moreover, we simulated the effect and action mechanism of levosimendan, which recently showed promising anti-arrhythmic effects in hiPSC-CMs in hypoxia. The model was validated using hiPSC-CM and in vitro animal data. The role of SERCA in causing relaxation dysfunction in IR was anticipated to be comparable to its function in sepsis-induced heart failure. Drug simulations showed that levosimendan counteracts the relaxation dysfunction by utilizing a particular Ca2+-sensitizing mechanism involving Ca2+-bound troponin C and Ca2+ flux to the myofilament, rather than inhibiting SERCA phosphorylation. The model demonstrates extensive characterization and promise for drug development, making it suitable for evaluating IR therapy strategies based on the changing levels of cardiac metabolites, oxygen and molecular pathways.

Ionic Mechanisms of Propagated Repolarization in a One-Dimensional Strand of Human Ventricular Myocyte Model

Although repolarization has been suggested to propagate in cardiac tissue both theoretically and experimentally, it has been challenging to estimate how and to what extent the propagation of repolarization contributes to relaxation because repolarization only occurs in the course of membrane excitation in normal hearts. We established a mathematical model of a 1D strand of 600 myocytes stabilized at an equilibrium potential near the plateau potential level by introducing a sustained component of the late sodium current (INaL). By applying a hyperpolarizing stimulus to a small part of the strand, we succeeded in inducing repolarization which propagated along the strand at a velocity of 1~2 cm/s. The ionic mechanisms responsible for repolarization at the myocyte level, i.e., the deactivation of both the INaL and the L-type calcium current (ICaL), and the activation of the rapid component of delayed rectifier potassium current (IKr) and the inward rectifier potassium channel (IK1), were found to be important for the propagation of repolarization in the myocyte strand. Using an analogy with progressive activation of the sodium current (INa) in the propagation of excitation, regenerative activation of the predominant magnitude of IK1 makes the myocytes at the wave front start repolarization in succession through the electrical coupling via gap junction channels.

BondGraphs.jl: composable energy-based modelling in systems biology

SUMMARY

BondGraphs.jl is a Julia implementation of bond graphs. Bond graphs provide a modelling framework that describes energy flow through a physical system and by construction enforce thermodynamic constraints. The framework is widely used in engineering and has recently been shown to be a powerful approach for modelling biology. Models are mutable, hierarchical, multiscale, and multiphysics, and BondGraphs.jl is compatible with the Julia modelling ecosystem.

AVAILABILITY AND IMPLEMENTATION

BondGraphs.jl is freely available under the MIT license. Source code and documentation can be found at https://github.com/jedforrest/BondGraphs.jl.

Multiscale computational modeling of the effects of 2'-deoxy-ATP on cardiac muscle calcium handling

2'-Deoxy-ATP (dATP), a naturally occurring near analog of ATP, is a well-documented myosin activator that has been shown to increase contractile force, improve pump function, and enhance lusitropy in the heart. Calcium transients in cardiomyocytes with elevated levels of dATP show faster calcium decay compared with cardiomyocytes with basal levels of dATP, but the mechanisms behind this are unknown. Here, we design and utilize a multiscale computational modeling framework to test the hypothesis that dATP acts on the sarcoendoplasmic reticulum calcium-ATPase (SERCA) pump to accelerate calcium re-uptake into the sarcoplasmic reticulum during cardiac relaxation. Gaussian accelerated molecular dynamics simulations of human cardiac SERCA2A in the E1 apo, ATP-bound and dATP-bound states showed that dATP forms more stable contacts in the nucleotide binding pocket of SERCA and leads to increased closure of cytosolic domains. These structural changes ultimately lead to changes in calcium binding, which we assessed using Brownian dynamics simulations. We found that dATP increases calcium association rate constants to SERCA and that dATP binds to apo SERCA more rapidly than ATP. Using a compartmental ordinary differential equation model of human cardiomyocyte excitation-contraction coupling, we found that these increased association rate constants contributed to the accelerated rates of calcium transient decay observed experimentally. This study provides clear mechanistic evidence of enhancements in cardiac SERCA2A pump function due to interactions with dATP.

Local Control Model of a Human Ventricular Myocyte: An Exploration of Frequency-Dependent Changes and Calcium Sparks

Calcium (Ca2+) sparks are the elementary events of excitation-contraction coupling, yet they are not explicitly represented in human ventricular myocyte models. A stochastic ventricular cardiomyocyte human model that adapts to intracellular Ca2+ ([Ca2+]i) dynamics, spark regulation, and frequency-dependent changes in the form of locally controlled Ca2+ release was developed. The 20,000 CRUs in this model are composed of 9 individual LCCs and 49 RyRs that function as couplons. The simulated action potential duration at 1 Hz steady-state pacing is ~0.280 s similar to human ventricular cell recordings. Rate-dependence experiments reveal that APD shortening mechanisms are largely contributed by the L-type calcium channel inactivation, RyR open fraction, and [Ca2+]myo concentrations. The dynamic slow-rapid-slow pacing protocol shows that RyR open probability during high pacing frequency (2.5 Hz) switches to an adapted "nonconducting" form of Ca2+-dependent transition state. The predicted force was also observed to be increased in high pacing, but the SR Ca2+ fractional release was lower due to the smaller difference between diastolic and systolic [Ca2+]SR. Restitution analysis through the S1S2 protocol and increased LCC Ca2+-dependent activation rate show that the duration of LCC opening helps modulate its effects on the APD restitution at different diastolic intervals. Ultimately, a longer duration of calcium sparks was observed in relation to the SR Ca2+ loading at high pacing rates. Overall, this study demonstrates the spontaneous Ca2+ release events and ion channel responses throughout various stimuli.

The Role of Ca2+ Sparks in Force Frequency Relationships in Guinea Pig Ventricular Myocytes

Calcium sparks are the elementary Ca2+ release events in excitation-contraction coupling that underlie the Ca2+ transient. The frequency-dependent contractile force generated by cardiac myocytes depends upon the characteristics of the Ca2+ transients. A stochastic computational local control model of a guinea pig ventricular cardiomyocyte was developed, to gain insight into mechanisms of force-frequency relationship (FFR). This required the creation of a new three-state RyR2 model that reproduced the adaptive behavior of RyR2, in which the RyR2 channels transition into a different state when exposed to prolonged elevated subspace [Ca2+]. The model simulations agree with previous experimental and modeling studies on interval-force relations. Unlike previous common pool models, this local control model displayed stable action potential trains at 7 Hz. The duration and the amplitude of the [Ca2+]myo transients increase in pacing rates consistent with the experiments. The [Ca2+]myo transient reaches its peak value at 4 Hz and decreases afterward, consistent with experimental force-frequency curves. The model predicts, in agreement with previous modeling studies of Jafri and co-workers, diastolic sarcoplasmic reticulum, [Ca2+]sr, and RyR2 adaptation increase with the increased stimulation frequency, producing rising, rather than falling, amplitude of the myoplasmic [Ca2+] transients. However, the local control model also suggests that the reduction of the L-type Ca2+ current, with an increase in pacing frequency due to Ca2+-dependent inactivation, also plays a role in the negative slope of the FFR. In the simulations, the peak Ca2+ transient in the FFR correlated with the highest numbers of SR Ca2+ sparks: the larger average amplitudes of those sparks, and the longer duration of the Ca2+ sparks.

The Na+/K+ pump dominates control of glycolysis in hippocampal dentate granule cells

Cellular ATP that is consumed to perform energetically expensive tasks must be replenished by new ATP through the activation of metabolism. Neuronal stimulation, an energetically demanding process, transiently activates aerobic glycolysis, but the precise mechanism underlying this glycolysis activation has not been determined. We previously showed that neuronal glycolysis is correlated with Ca2+ influx, but is not activated by feedforward Ca2+ signaling (Díaz-García et al., 2021a). Since ATP-powered Na+ and Ca2+ pumping activities are increased following stimulation to restore ion gradients and are estimated to consume most neuronal ATP, we aimed to determine if they are coupled to neuronal glycolysis activation. By using two-photon imaging of fluorescent biosensors and dyes in dentate granule cell somas of acute mouse hippocampal slices, we observed that production of cytoplasmic NADH, a byproduct of glycolysis, is strongly coupled to changes in intracellular Na+, while intracellular Ca2+ could only increase NADH production if both forward Na+/Ca2+ exchange and Na+/K+ pump activity were intact. Additionally, antidromic stimulation-induced intracellular [Na+] increases were reduced >50% by blocking Ca2+ entry. These results indicate that neuronal glycolysis activation is predominantly a response to an increase in activity of the Na+/K+ pump, which is strongly potentiated by Na+ influx through the Na+/Ca2+ exchanger during extrusion of Ca2+ following stimulation.

SBML to bond graphs: From conversion to composition

The Systems Biology Markup Language (SBML) is a popular software-independent XML-based format for describing models of biological phenomena. The BioModels Database is the largest online repository of SBML models. Several tools and platforms are available to support the reuse and composition of SBML models. However, these tools do not explicitly assess whether models are physically plausible or thermodynamically consistent. This often leads to ill-posed models that are physically impossible, impeding the development of realistic complex models in biology. Here, we present a framework that can automatically convert SBML models into bond graphs, which imposes energy conservation laws on these models. The new bond graph models are easily mergeable, resulting in physically plausible coupled models. We illustrate this by automatically converting and coupling a model of pyruvate distribution to a model of the pentose phosphate pathway.

Network thermodynamics of biological systems: A bond graph approach

Edmund Crampin (1973-2021) was at the forefront of Systems Biology research and his work will influence the field for years to come. This paper brings together and summarises the seminal work of his group in applying energy-based bond graph methods to biological systems. In particular, this paper: (a) motivates the need to consider energy in modelling biology; (b) introduces bond graphs as a methodology for achieving this; (c) describes extensions to modelling electrochemical transduction; (d) outlines how bond graph models can be constructed in a modular manner and (e) describes stoichiometric approaches to deriving fundamental properties of reaction networks. These concepts are illustrated using a new bond graph model of photosynthesis in chloroplasts.

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.

Nanoscale organization of ryanodine receptor distribution and phosphorylation pattern determines the dynamics of calcium sparks

Super-resolution imaging techniques have provided a better understanding of the relationship between the nanoscale organization and function of ryanodine receptors (RyRs) in cardiomyocytes. Recent data have indicated that this relationship is disrupted in heart failure (HF), as RyRs are dispersed into smaller and more numerous clusters. However, RyRs are also hyperphosphorylated in this condition, and this is reported to occur preferentially within the cluster centre. Thus, the combined impact of RyR relocalization and sensitization on Ca²⁺ spark generation in failing cardiomyocytes is likely complex and these observations suggest that both the nanoscale organization of RyRs and the pattern of phosphorylated RyRs within clusters could be critical determinants of Ca²⁺ spark dynamics. To test this hypothesis, we used computational modeling to quantify the relationships between RyR cluster geometry, phosphorylation patterns, and sarcoplasmic reticulum (SR) Ca²⁺ release. We found that RyR cluster disruption results in a decrease in spark fidelity and longer sparks with a lower amplitude. Phosphorylation of some RyRs within the cluster can play a compensatory role, recovering healthy spark dynamics. Interestingly, our model predicts that such compensation is critically dependent on the phosphorylation pattern, as phosphorylation localized within the cluster center resulted in longer Ca²⁺ sparks and higher spark fidelity compared to a uniformly distributed phosphorylation pattern. Our results strongly suggest that both the phosphorylation pattern and nanoscale RyR reorganization are critical determinants of Ca²⁺ dynamics in HF.

Ryanodine receptors (RyRs) are ion channels located on the membrane of the sarcoplasmic reticulum that are responsible for an increase in cytosolic Ca²⁺ during cell excitation. Here, we investigate how the geometry of RyR clusters combined with spatial phosphorylation patterns impacts on Ca²⁺ spark generation and kinetics. The findings from our study show that phosphorylation pattern and both RyR cluster shape and dispersion have implications on Ca²⁺ spark activity and provide insights into altered Ca²⁺ dynamics during HF.

A mathematical model to quantify RYR Ca2+ leak and associated heat production in resting human skeletal muscle fibers

Cycling of Ca2+ between the sarcoplasmic reticulum (SR) and myoplasm is an important component of skeletal muscle resting metabolism. As part of this cycle, Ca2+ leaks from the SR into the myoplasm and is pumped back into the SR using ATP, which leads to the consumption of O2 and generation of heat. Ca2+ may leak through release channels or ryanodine receptors (RYRs). RYR Ca2+ leak can be monitored in a skinned fiber preparation in which leaked Ca2+ is pumped into the t-system and measured with a fluorescent dye. However, accurate quantification faces a number of hurdles. To overcome them, we developed a mathematical model of Ca2+ movement in these preparations. The model incorporated Ca2+ pumps that move Ca2+ from the myoplasm to the SR and from the junctional space (JS) to the t-system, Ca2+ buffering by EGTA in the JS and myoplasm and by buffers in the SR, and Ca2+ leaks from the SR into the JS and myoplasm and from the t-system into the myoplasm. The model accurately simulated Ca2+ uptake into the t-system, the relationship between myoplasmic [Ca2+] and steady-state t-system [Ca2+], and the effect of blocking RYR Ca2+ leak on t-system Ca2+ uptake. The magnitude of the leak through the RYRs would contribute ∼5% of the resting heat production of human muscle. In normal resting fibers, RYR Ca2+ leak makes a small contribution to resting metabolism. RYR-focused pathologies have the potential to increase RYR Ca2+ leak and the RYR leak component of resting metabolism.

Functional characterization of Legionella pneumophila Cu+ transport ATPase. The activation by Cu+ and ATP

Cu⁺-ATPases are integral membrane proteins belonging to the IB subfamily of the P-type ATPases that couple Cu⁺ transport to the hydrolysis of ATP. As some structural and functional particularities arise for Cu⁺-ATPases, several authors suggest that some of the reaction steps of the Albers-Post model postulated for other P-ATPases may be different. In this work we describe a functional characterization of Legionella pneumophila Cu⁺-ATPase (LpCopA), the first P_IB-ATPase whose structure was determined by X-ray crystallography. Cu⁺-ATPase activity of the enzyme presents a maximum at ∼37 °C and pH 6.6-6.8. Phospholipids enhance LpCopA Cu⁺-ATPase activity in a non-essential mode where optimal activity is achieved at an asolectin molar fraction of 0.15 and an amphiphile-protein ratio of ~30,000. As described for other P-ATPases, Mg²⁺ acts as an essential activator. Furthermore, Cu⁺-ATPase activity dependence on [Cu⁺] and [ATP] can both be described by a sum of two hyperbolic functions. Based on that, and the [Cu⁺] and [ATP] dependencies of the best fitting parameters of the hyperbolae pointed above, we propose a minimal reaction scheme for the catalytic mechanism that shares the basic reaction steps of the Albers-Post model for P-type ATPases. The reaction scheme postulated contemplates two different binding affinities for a single ATP (apparent affinities of 0.66 and 550 μM at [Cu⁺] → ∞) and binding of at least 2 Cu⁺ with different affinities as well (apparent affinities of 1.4 and 102.5 μM at [ATP] → ∞).

Cardiac Alternans Occurs through the Synergy of Voltage- and Calcium-Dependent Mechanisms

Cardiac alternans is characterized by alternating weak and strong beats of the heart. This signaling at the cellular level may appear as alternating long and short action potentials (APs) that occur in synchrony with alternating large and small calcium transients, respectively. Previous studies have suggested that alternans manifests itself through either a voltage dependent mechanism based upon action potential restitution or as a calcium dependent mechanism based on refractoriness of calcium release. We use a novel model of cardiac excitation-contraction (EC) coupling in the rat ventricular myocyte that includes 20,000 calcium release units (CRU) each with 49 ryanodine receptors (RyR2s) and 7 L-type calcium channels that are all stochastically gated. The model suggests that at the cellular level in the case of alternans produced by rapid pacing, the mechanism requires a synergy of voltage- and calcium-dependent mechanisms. The rapid pacing reduces AP duration and magnitude reducing the number of L-type calcium channels activating individual CRUs during each AP and thus increases the population of CRUs that can be recruited stochastically. Elevated myoplasmic and sarcoplasmic reticulum (SR) calcium, [Ca2+]myo and [Ca2+]SR respectively, increases ryanodine receptor open probability (Po) according to our model used in this simulation and this increased the probability of activating additional CRUs. A CRU that opens in one beat is less likely to open the subsequent beat due to refractoriness caused by incomplete refilling of the junctional sarcoplasmic reticulum (jSR). Furthermore, the model includes estimates of changes in Na+ fluxes and [Na+]i and thus provides insight into how changes in electrical activity, [Na+]i and sodium-calcium exchanger activity can modulate alternans. The model thus tracks critical elements that can account for rate-dependent changes in [Na+]i and [Ca2+]myo and how they contribute to the generation of Ca2+ signaling alternans in the heart.

Investigating two kinds of cellular alternans and corresponding TWA induced by impaired calcium cycling in myocardial ischemia

BACKGROUND

The utility of T wave alternans (TWA) in identifying arrhythmia risk has been demonstrated. During myocardial ischemia (MI), TWA could be induced by cellular alternans. However, the relationship between cellular alternans patterns and TWA patterns in MI has not been investigated thoroughly.

METHODS

We set MI conditions to simulate alternans. Either prolonging Ca²⁺ release or increasing spark-induced sparks (secondary sparks) can give rise to different patterns of APD alternans and TWA. In addition, different ischemic zones and reduced conduction velocity are also considered in one dimensional simulation.

RESULTS

Delay of Ca²⁺ release can produce discordant Ca²⁺-driven alternans in single cell simulation. Increasing secondary sparks leads to concordant alternans. Correspondingly, morphology and magnitude of TWA vary in two different cellular alternans. Epi ischemia results in alternans concentrating in the first half of T wave. Endo and transmural ischemia lead to fluctuations in the second half of T wave. In addition, slowing conduction velocity has no effect on TWA magnitude.

CONCLUSION

Specific ionic channel dysfunction and ischemic zones affect TWA patterns.

Calibration of ionic and cellular cardiac electrophysiology models

Cardiac electrophysiology models are among the most mature and well-studied mathematical models of biological systems. This maturity is bringing new challenges as models are being used increasingly to make quantitative rather than qualitative predictions. As such, calibrating the parameters within ion current and action potential (AP) models to experimental data sets is a crucial step in constructing a predictive model. This review highlights some of the fundamental concepts in cardiac model calibration and is intended to be readily understood by computational and mathematical modelers working in other fields of biology. We discuss the classic and latest approaches to calibration in the electrophysiology field, at both the ion channel and cellular AP scales. We end with a discussion of the many challenges that work to date has raised and the need for reproducible descriptions of the calibration process to enable models to be recalibrated to new data sets and built upon for new studies. This article is categorized under: Analytical and Computational Methods > Computational Methods Physiology > Mammalian Physiology in Health and Disease Models of Systems Properties and Processes > Cellular Models.

Cardiomyocyte calcium handling in health and disease: Insights from in vitro and in silico studies

Calcium (Ca²⁺) plays a central role in cardiomyocyte excitation-contraction coupling. To ensure an optimal electrical impulse propagation and cardiac contraction, Ca²⁺ levels are regulated by a variety of Ca²⁺-handling proteins. In turn, Ca²⁺ modulates numerous electrophysiological processes. Accordingly, Ca²⁺-handling abnormalities can promote cardiac arrhythmias via various mechanisms, including the promotion of afterdepolarizations, ion-channel modulation and structural remodeling. In the last 30 years, significant improvements have been made in the computational modeling of cardiomyocyte Ca²⁺ handling under physiological and pathological conditions. However, numerous questions involving the Ca²⁺-dependent regulation of different macromolecular complexes, cross-talk between Ca²⁺-dependent regulatory pathways operating over a wide range of time scales, and bidirectional interactions between electrophysiology and mechanics remain to be addressed by in vitro and in silico studies. A better understanding of disease-specific Ca²⁺-dependent proarrhythmic mechanisms may facilitate the development of improved therapeutic strategies. In this review, we describe the fundamental mechanisms of cardiomyocyte Ca²⁺ handling in health and disease, and provide an overview of currently available computational models for cardiomyocyte Ca²⁺ handling. Finally, we discuss important uncertainties and open questions about cardiomyocyte Ca²⁺ handling and highlight how synergy between in vitro and in silico studies may help to answer several of these issues.

Exploring Impaired SERCA Pump-Caused Alternation Occurrence in Ischemia

Impaired sarcoplasmic reticulum (SR) calcium transport ATPase (SERCA) gives rise to Ca²⁺ alternans and changes of the Ca2+release amount. These changes in Ca²⁺ release amount can reveal the mechanism underlying how the interaction between Ca²⁺ release and Ca²⁺ uptake induces Ca²⁺ alternans. This study of alternans by calculating the values of Ca²⁺ release properties with impaired SERCA has not been explored before. Here, we induced Ca²⁺ alternans by using an impaired SERCA pump under ischemic conditions. The results showed that the recruitment and refractoriness of the Ca²⁺ release increased as Ca²⁺ alternans occurred. This indicates triggering Ca waves. As the propagation of Ca waves is linked to the occurrence of Ca²⁺ alternans, the “threshold” for Ca waves reflects the key factor in Ca²⁺ alternans development, and it is still controversial nowadays. We proposed the ratio between the diastolic network SR (NSR) Ca content (Ca_nsr) and the cytoplasmic Ca content (Ca_i) (Ca_nsr/Ca_i) as the “threshold” of Ca waves and Ca²⁺ alternans. Diastolic Ca_nsr, Ca_i, and their ratio were recorded at the onset of Ca²⁺ alternans. Compared with certain Ca_nsr and Ca_i, the “threshold” of the ratio can better explain the comprehensive effects of the Ca²⁺ release and the Ca²⁺ uptake on Ca²⁺ alternans onset. In addition, these ratios are related with the function of SERCA pumps, which vary with different ischemic conditions. Thus, values of these ratios could be used to differentiate Ca²⁺ alternans from different ischemic cases. This agrees with some experimental results. Therefore, the certain value of diastolic Ca_nsr/Ca_i can be the better “threshold” for Ca waves and Ca²⁺ alternans.

Mechanisms Underlying Spontaneous Action Potential Generation Induced by Catecholamine in Pulmonary Vein Cardiomyocytes: A Simulation Study

Cardiomyocytes and myocardial sleeves dissociated from pulmonary veins (PVs) potentially generate ectopic automaticity in response to noradrenaline (NA), and thereby trigger atrial fibrillation. We developed a mathematical model of rat PV cardiomyocytes (PVC) based on experimental data that incorporates the microscopic framework of the local control theory of Ca²⁺ release from the sarcoplasmic reticulum (SR), which can generate rhythmic Ca²⁺ release (limit cycle revealed by the bifurcation analysis) when total Ca²⁺ within the cell increased. Ca²⁺ overload in SR increased resting Ca²⁺ efflux through the type II inositol 1,4,5-trisphosphate (IP₃) receptors (InsP₃R) as well as ryanodine receptors (RyRs), which finally triggered massive Ca²⁺ release through activation of RyRs via local Ca²⁺ accumulation in the vicinity of RyRs. The new PVC model exhibited a resting potential of −68 mV. Under NA effects, repetitive Ca²⁺ release from SR triggered spontaneous action potentials (APs) by evoking transient depolarizations (TDs) through Na⁺/Ca²⁺ exchanger (AP_TDs). Marked and variable latencies initiating AP_TDs could be explained by the time courses of the α1- and β1-adrenergic influence on the regulation of intracellular Ca²⁺ content and random occurrences of spontaneous TD activating the first AP_TD. Positive and negative feedback relations were clarified under AP_TD generation.

Thermodynamics of Cation Binding to the Sarcoendoplasmic Reticulum Calcium ATPase Pump and Impacts on Enzyme Function

Sarcoendoplasmic reticulum Ca²⁺-ATPase (SERCA) is a transmembrane pump that plays an important role in transporting calcium into the sarcoplasmic reticulum (SR). While calcium (Ca²⁺) binds SERCA with micromolar affinity, magnesium (Mg²⁺) and potassium (K⁺) also compete with Ca²⁺ binding. However, the molecular bases for these competing ions' influence on the SERCA function and the selectivity of the pump for Ca²⁺ are not well-established. We therefore used in silico methods to resolve molecular determinants of cation binding in the canonical site I and II Ca²⁺ binding sites via (1) triplicate molecular dynamics (MD) simulations of Mg²⁺, Ca²⁺, and K⁺-bound SERCA, (2) mean spherical approximation (MSA) theory to score the affinity and selectivity of cation binding to the MD-resolved structures, and (3) state models of SERCA turnover informed from MSA-derived affinity data. Our key findings are that (a) coordination at sites I and II is optimized for Ca²⁺ and to a lesser extent for Mg²⁺ and K⁺, as determined by MD-derived cation-amino acid oxygen and bound water configurations, (b) the impaired coordination and high desolvation cost for Mg²⁺ precludes favorable Mg²⁺ binding relative to Ca²⁺, while K⁺ has limited capacity to bind site I, and (c) Mg²⁺ most likely acts as inhibitor and K⁺ as intermediate in SERCA's reaction cycle, based on a best-fit state model of SERCA turnover. These findings provide a quantitative basis for SERCA function that leverages molecular-scale thermodynamic data and rationalizes enzyme activity across broad ranges of K⁺, Ca²⁺, and Mg²⁺ concentrations.

Insights on the impact of mitochondrial organisation on bioenergetics in high-resolution computational models of cardiac cell architecture

Recent electron microscopy data have revealed that cardiac mitochondria are not arranged in crystalline columns but are organised with several mitochondria aggregated into columns of varying sizes spanning the cell cross-section. This raises the question—how does the mitochondrial arrangement affect the metabolite distributions within cardiomyocytes and what is its impact on force dynamics? Here, we address this question by employing finite element modeling of cardiac bioenergetics on computational meshes derived from electron microscope images. Our results indicate that heterogeneous mitochondrial distributions can lead to significant spatial variation across the cell in concentrations of inorganic phosphate, creatine (Cr) and creatine phosphate (PCr). However, our model predicts that sufficient activity of the creatine kinase (CK) system, coupled with rapid diffusion of Cr and PCr, maintains near uniform ATP and ADP ratios across the cell cross sections. This homogenous distribution of ATP and ADP should also evenly distribute force production and twitch duration with contraction. These results suggest that the PCr shuttle and associated enzymatic reactions act to maintain uniform force dynamics in the cell despite the heterogeneous mitochondrial organization. However, our model also predicts that under hypoxia activity of mitochondrial CK enzymes and diffusion of high-energy phosphate compounds may be insufficient to sustain uniform ATP/ADP distribution and hence force generation.

Mammalian cardiomyocytes contain a high volume of mitochondria, which maintains the continuous and bulk supply of ATP to sustain normal heart function. Previously, cardiac mitochondria were understood to be distributed in a regular, crystalline pattern, which facilitated a steady supply of ATP at different workloads. Using electron microscopy images of cell cross sections, we recently found that they are not regularly distributed inside cardiomyocytes. We created new spatially accurate computational models of cardiac cell bioenergetics and tested whether this heterogeneous distribution of mitochondria causes non-uniform energy supply and contractile force production in the cardiomyocyte. We found that ATP and ADP concentrations remain uniform throughout the cell because of the activity of creatine kinase (CK) enzymes that convert ATP produced in the mitochondria into creatine phosphate. Creatine phosphate rapidly diffuses to the myofibril region where it can be converted back to ATP for the contraction cycle in a timely manner. This mechanism is called the phosphocreatine shuttle (PCr shuttle). The PCr shuttle ensures that different areas of the cell produce the same amount of force regardless of the mitochondrial distribution. However, our model also shows that when the cellular oxygen supply is limited—as can be the case in conditions such as heart failure—the PCr shuttle cannot maintain uniform ATP and ADP concentrations across the cell. This causes a non-uniform acto-myosin force distribution and non-uniform twitch duration across the cell cross section. Our study suggests that mechanisms other than the PCr shuttle may be necessary to maintain uniform supply of ATP in a hypoxic environment.

Ryanodine receptor dispersion disrupts Ca2+ release in failing cardiac myocytes

Reduced cardiac contractility during heart failure (HF) is linked to impaired Ca²⁺ release from Ryanodine Receptors (RyRs). We investigated whether this deficit can be traced to nanoscale RyR reorganization. Using super-resolution imaging, we observed dispersion of RyR clusters in cardiomyocytes from post-infarction HF rats, resulting in more numerous, smaller clusters. Functional groupings of RyR clusters which produce Ca²⁺ sparks (Ca²⁺ release units, CRUs) also became less solid. An increased fraction of small CRUs in HF was linked to augmented ‘silent’ Ca²⁺ leak, not visible as sparks. Larger multi-cluster CRUs common in HF also exhibited low fidelity spark generation. When successfully triggered, sparks in failing cells displayed slow kinetics as Ca²⁺ spread across dispersed CRUs. During the action potential, these slow sparks protracted and desynchronized the overall Ca²⁺ transient. Thus, nanoscale RyR reorganization during HF augments Ca²⁺ leak and slows Ca²⁺ release kinetics, leading to weakened contraction in this disease.

The muscle cells of the heart coordinate how they contract and relax in order to produce the heartbeat. During heart failure, these cells become less able to contract. As a result the heart becomes inefficient, pumping less blood around the body.

For the cardiac muscle cells to contract, the levels of calcium ions in the cells needs to rapidly increase. In failing hearts, these increases in calcium ion levels are smaller, slower and less well coordinated. It was not known what causes these changes, making it difficult to treat heart failure.

Calcium ions are released in cardiac muscle cells through protein channels called ryanodine receptors. These receptors form clusters that allow them to synchronize when they open and close. Could the reorganization of ryanodine receptors account for the problems seen in failing hearts? To investigate, Kolstad et al. examined rat hearts using a technique called super-resolution microscopy. This showed that the clusters of ryanodine receptors break apart during heart failure to form smaller clusters. Further experiments showed that calcium ions ‘leak’ from these smaller clusters, reducing the amount of calcium that can be released into cardiac muscle cells during each heartbeat. Released calcium also spreads between the dispersed clusters, resulting in a slower rise of the calcium levels in the cells. Both changes contribute to weakened contractions of cells in failing hearts.

Therefore, heart failure can be traced back to very small rearrangements of the ryanodine receptors. This understanding will help researchers as they investigate new ways to treat heart failure.

A thermodynamic framework for modelling membrane transporters

Membrane transporters contribute to the regulation of the internal environment of cells by translocating substrates across cell membranes. Like all physical systems, the behaviour of membrane transporters is constrained by the laws of thermodynamics. However, many mathematical models of transporters, especially those incorporated into whole-cell models, are not thermodynamically consistent, leading to unrealistic behaviour. In this paper we use a physics-based modelling framework, in which the transfer of energy is explicitly accounted for, to develop thermodynamically consistent models of transporters. We then apply this methodology to model two specific transporters: the cardiac sarcoplasmic/endoplasmic Ca²⁺ ATPase (SERCA) and the cardiac Na⁺/K⁺ ATPase.

A 3D diffusional-compartmental model of the calcium dynamics in cytosol, sarcoplasmic reticulum and mitochondria of murine skeletal muscle fibers

Variations of free calcium concentration ([Ca2+]) are powerful intracellular signals, controlling contraction as well as metabolism in muscle cells. To fully understand the role of calcium redistribution upon excitation and contraction in skeletal muscle cells, the local [Ca2+] in different compartments needs to be taken into consideration. Fluorescent probes allow the determination of [Ca2+] in the cytosol where myofibrils are embedded, the lumen of the sarcoplasmic reticulum (SR) and the mitochondrial matrix. Previously, models have been developed describing intracellular calcium handling in skeletal and cardiac muscle cells. However, a comprehensive model describing the kinetics of the changes in free calcium concentration in these three compartments is lacking. We designed a new 3D compartmental model of the half sarcomere with radial symmetry, which accounts for diffusion of Ca2+ into the three compartments and simulates its dynamics at rest and at various rates of stimulation in mice skeletal muscle fibers. This model satisfactorily reproduces both the amplitude and time course of the variations of [Ca2+] in the three compartments in mouse fast fibers. As an illustration of the applicability of the model, we investigated the effects of Calsequestrin (CSQ) ablation. CSQ is the main Ca2+ buffer in the SR, localized in close proximity of its calcium release sites and near to the mitochondria. CSQ knock-out mice muscles still preserve a near-normal contractile behavior, but it is unclear whether this is caused by additional SR calcium buffering or a significant contribution of calcium entry from extracellular space, via stored-operated calcium entry (SOCE). The model enabled quantitative assessment of these two scenarios by comparison to measurements of local calcium in the cytosol, the SR and the mitochondria. In conclusion, the model represents a useful tool to investigate the impact of protein ablation and of pharmacological interventions on intracellular calcium dynamics in mice skeletal muscle.

Bond graph modelling of the cardiac action potential: implications for drift and non-unique steady states

Mathematical models of cardiac action potentials have become increasingly important in the study of heart disease and pharmacology, but concerns linger over their robustness during long periods of simulation, in particular due to issues such as model drift and non-unique steady states. Previous studies have linked these to violation of conservation laws, but only explored those issues with respect to charge conservation in specific models. Here, we propose a general and systematic method of identifying conservation laws hidden in models of cardiac electrophysiology by using bond graphs, and develop a bond graph model of the cardiac action potential to study long-term behaviour. Bond graphs provide an explicit energy-based framework for modelling physical systems, which makes them well suited for examining conservation within electrophysiological models. We find that the charge conservation laws derived in previous studies are examples of the more general concept of a 'conserved moiety'. Conserved moieties explain model drift and non-unique steady states, generalizing the results from previous studies. The bond graph approach provides a rigorous method to check for drift and non-unique steady states in a wide range of cardiac action potential models, and can be extended to examine behaviours of other excitable systems.

Bond graph modelling of chemoelectrical energy transduction

Energy‐based bond graph modelling of biomolecular systems is extended to include chemoelectrical transduction thus enabling integrated thermodynamically compliant modelling of chemoelectrical systems in general and excitable membranes in particular. Our general approach is illustrated by recreating a well‐known model of an excitable membrane. This model is used to investigate the energy consumed during a membrane action potential thus contributing to the current debate on the trade‐off between the speed of an action potential event and energy consumption. The influx of Na+ is often taken as a proxy for energy consumption; in contrast, this study presents an energy‐based model of action potentials. As the energy‐based approach avoids the assumptions underlying the proxy approach it can be directly used to compute energy consumption in both healthy and diseased neurons. These results are illustrated by comparing the energy consumption of healthy and degenerative retinal ganglion cells using both simulated and in vitro data.

β-alanine supplementation improves in-vivo fresh and fatigued skeletal muscle relaxation speed

Purpose

In fresh muscle, supplementation with the rate-limiting precursor of carnosine, β-alanine (BA), results in a decline in muscle half-relaxation time (HRT) potentially via alterations to calcium (Ca²⁺) handling. Accumulation of hydrogen cation (H⁺) has been shown to impact Ca²⁺ signalling during muscular contraction, carnosine has the potential to serve as a cytoplasmic regulator of Ca²⁺ and H⁺ coupling, since it binds to both ions. The present study examined the effect of BA supplementation on intrinsic in-vivo isometric knee extensor force production and muscle contractility in both fresh and fatigued human skeletal muscle assessed during voluntary and electrically evoked (nerve and superficial muscle stimulation) contractions.

Methods

Twenty-three males completed two experimental sessions, pre- and post- 28 day supplementation with 6.4 g.day⁻¹ of BA (n = 12) or placebo (PLA; n = 11). Isometric force was recorded during a series of voluntary and electrically evoked knee extensor contractions.

Results

BA supplementation had no effect on voluntary or electrically evoked isometric force production, or twitch electromechanical delay and time-to-peak tension. There was a significant decline in muscle HRT in fresh and fatigued muscle conditions during both resting (3 ± 13%; 19 ± 26%) and potentiated (1 ± 15%; 2 ± 20%) twitch contractions.

Conclusions

The mechanism for reduced HRT in fresh and fatigued skeletal muscle following BA supplementation is unclear. Due to the importance of muscle relaxation on total energy consumption, especially during short, repeated contractions, BA supplementation may prove to be beneficial in minimising contractile slowing induced by fatigue.

Trial registration

The trial is registered with Clinicaltrials.gov, ID number NCT02819505.

Electrophysiological properties of computational human ventricular cell action potential models under acute ischemic conditions

Acute myocardial ischemia is one of the main causes of sudden cardiac death. The mechanisms have been investigated primarily in experimental and computational studies using different animal species, but human studies remain scarce. In this study, we assess the ability of four human ventricular action potential models (ten Tusscher and Panfilov, 2006; Grandi et al., 2010; Carro et al., 2011; O'Hara et al., 2011) to simulate key electrophysiological consequences of acute myocardial ischemia in single cell and tissue simulations. We specifically focus on evaluating the effect of extracellular potassium concentration and activation of the ATP-sensitive inward-rectifying potassium current on action potential duration, post-repolarization refractoriness, and conduction velocity, as the most critical factors in determining reentry vulnerability during ischemia. Our results show that the Grandi and O'Hara models required modifications to reproduce expected ischemic changes, specifically modifying the intracellular potassium concentration in the Grandi model and the sodium current in the O'Hara model. With these modifications, the four human ventricular cell AP models analyzed in this study reproduce the electrophysiological alterations in repolarization, refractoriness, and conduction velocity caused by acute myocardial ischemia. However, quantitative differences are observed between the models and overall, the ten Tusscher and modified O'Hara models show closest agreement to experimental data.

In Silico Investigation into Cellular Mechanisms of Cardiac Alternans in Myocardial Ischemia

Myocardial ischemia is associated with pathophysiological conditions such as hyperkalemia, acidosis, and hypoxia. These physiological disorders may lead to changes on the functions of ionic channels, which in turn form the basis for cardiac alternans. In this paper, we investigated the roles of hyperkalemia and calcium handling components played in the genesis of alternans in ischemia at the cellular level by using computational simulations. The results show that hyperkalemic reduced cell excitability and delayed recovery from inactivation of depolarization currents. The inactivation time constant τ_f of L-type calcium current (I_CaL) increased obviously in hyperkalemia. One cycle length was not enough for I_CaL to recover completely. Alternans developed as a result of I_CaL responding to stimulation every other beat. Sarcoplasmic reticulum calcium-ATPase (SERCA2a) function decreased in ischemia. This change resulted in intracellular Ca (Ca_i) alternans of small magnitude. A strong Na⁺-Ca²⁺ exchange current (I_NCX) increased the magnitude of Ca_i alternans, leading to APD alternans through excitation-contraction coupling. Some alternated repolarization currents contributed to this repolarization alternans.

Changes in mitochondrial morphology and organization can enhance energy supply from mitochondrial oxidative phosphorylation in diabetic cardiomyopathy

Diabetic cardiomyopathy is accompanied by metabolic and ultrastructural alterations, but the impact of the structural changes on metabolism itself is yet to be determined. Morphometric analysis of mitochondrial shape and spatial organization within transverse sections of cardiomyocytes from control and streptozotocin-induced type I diabetic Sprague-Dawley rats revealed that mitochondria are 20% smaller in size while their spatial density increases by 53% in diabetic cells relative to control myocytes. Diabetic cells formed larger clusters of mitochondria (60% more mitochondria per cluster) and the effective surface-to-volume ratio of these clusters increased by 22.5%. Using a biophysical computational model we found that this increase can have a moderate compensatory effect by increasing the availability of ATP in the cytosol when ATP synthesis within the mitochondrial matrix is compromised.

There and back again: Iterating between population-based modeling and experiments reveals surprising regulation of calcium transients in rat cardiac myocytes

While many ion channels and transporters involved in cardiac cellular physiology have been identified and described, the relative importance of each in determining emergent cellular behaviors remains unclear. Here we address this issue with a novel approach that combines population-based mathematical modeling with experimental tests to systematically quantify the relative contributions of different ion channels and transporters to the amplitude of the cellular Ca(2+) transient. Sensitivity analysis of a mathematical model of the rat ventricular cardiomyocyte quantified the response of cell behaviors to changes in the level of each ion channel and transporter, and experimental tests of these predictions were performed to validate or invalidate the predictions. The model analysis found that partial inhibition of the transient outward current in rat ventricular epicardial myocytes was predicted to have a greater impact on Ca(2+) transient amplitude than either: (1) inhibition of the same current in endocardial myocytes, or (2) comparable inhibition of the sarco/endoplasmic reticulum Ca(2+) ATPase (SERCA). Experimental tests confirmed the model predictions qualitatively but showed some quantitative disagreement. This guided us to recalibrate the model by adjusting the relative importance of several Ca(2+) fluxes, thereby improving the consistency with experimental data and producing a more predictive model. Analysis of human cardiomyocyte models suggests that the relative importance of outward currents to Ca(2+) transporters is generalizable to human atrial cardiomyocytes, but not ventricular cardiomyocytes. Overall, our novel approach of combining population-based mathematical modeling with experimental tests has yielded new insight into the relative importance of different determinants of cell behavior.

Superresolution modeling of calcium release in the heart

Stable calcium-induced calcium release (CICR) is critical for maintaining normal cellular contraction during cardiac excitation-contraction coupling. The fundamental element of CICR in the heart is the calcium (Ca²⁺) spark, which arises from a cluster of ryanodine receptors (RyR). Opening of these RyR clusters is triggered to produce a local, regenerative release of Ca²⁺ from the sarcoplasmic reticulum (SR). The Ca²⁺ leak out of the SR is an important process for cellular Ca²⁺ management, and it is critically influenced by spark fidelity, i.e., the probability that a spontaneous RyR opening triggers a Ca²⁺ spark. Here, we present a detailed, three-dimensional model of a cardiac Ca²⁺ release unit that incorporates diffusion, intracellular buffering systems, and stochastically gated ion channels. The model exhibits realistic Ca²⁺ sparks and robust Ca²⁺ spark termination across a wide range of geometries and conditions. Furthermore, the model captures the details of Ca²⁺ spark and nonspark-based SR Ca²⁺ leak, and it produces normal excitation-contraction coupling gain. We show that SR luminal Ca²⁺-dependent regulation of the RyR is not critical for spark termination, but it can explain the exponential rise in the SR Ca²⁺ leak-load relationship demonstrated in previous experimental work. Perturbations to subspace dimensions, which have been observed in experimental models of disease, strongly alter Ca²⁺ spark dynamics. In addition, we find that the structure of RyR clusters also influences Ca²⁺ release properties due to variations in inter-RyR coupling via local subspace Ca²⁺ concentration ([Ca²⁺]_ss). These results are illustrated for RyR clusters based on super-resolution stimulated emission depletion microscopy. Finally, we present a believed-novel approach by which the spark fidelity of a RyR cluster can be predicted from structural information of the cluster using the maximum eigenvalue of its adjacency matrix. These results provide critical insights into CICR dynamics in heart, under normal and pathological conditions.

A human ventricular myocyte model with a refined representation of excitation-contraction coupling

Cardiac Ca(2+)-induced Ca(2+) release (CICR) occurs by a regenerative activation of ryanodine receptors (RyRs) within each Ca(2+)-releasing unit, triggered by the activation of L-type Ca(2+) channels (LCCs). CICR is then terminated, most probably by depletion of Ca(2+) in the junctional sarcoplasmic reticulum (SR). Hinch et al. previously developed a tightly coupled LCC-RyR mathematical model, known as the Hinch model, that enables simulations to deal with a variety of functional states of whole-cell populations of a Ca(2+)-releasing unit using a personal computer. In this study, we developed a membrane excitation-contraction model of the human ventricular myocyte, which we call the human ventricular cell (HuVEC) model. This model is a hybrid of the most recent HuVEC models and the Hinch model. We modified the Hinch model to reproduce the regenerative activation and termination of CICR. In particular, we removed the inactivated RyR state and separated the single step of RyR activation by LCCs into triggering and regenerative steps. More importantly, we included the experimental measurement of a transient rise in Ca(2+) concentrations ([Ca(2+)], 10-15 μM) during CICR in the vicinity of Ca(2+)-releasing sites, and thereby calculated the effects of the local Ca(2+) gradient on CICR as well as membrane excitation. This HuVEC model successfully reconstructed both membrane excitation and key properties of CICR. The time course of CICR evoked by an action potential was accounted for by autonomous changes in an instantaneous equilibrium open probability of couplons. This autonomous time course was driven by a core feedback loop including the pivotal local [Ca(2+)], influenced by a time-dependent decay in the SR Ca(2+) content during CICR.

Membrane potential and Ca2+ concentration dependence on pressure and vasoactive agents in arterial smooth muscle: A model

A mathematical model incorporating junctional and stretch-activated microdomains and 37 protein components describes the myogenic response in arterial smooth muscle cells.

Arterial smooth muscle (SM) cells respond autonomously to changes in intravascular pressure, adjusting tension to maintain vessel diameter. The values of membrane potential (V_m) and sarcoplasmic Ca²⁺ concentration (Ca_in) within minutes of a change in pressure are the results of two opposing pathways, both of which use Ca²⁺ as a signal. This works because the two Ca²⁺-signaling pathways are confined to distinct microdomains in which the Ca²⁺ concentrations needed to activate key channels are transiently higher than Ca_in. A mathematical model of an isolated arterial SM cell is presented that incorporates the two types of microdomains. The first type consists of junctions between cisternae of the peripheral sarcoplasmic reticulum (SR), containing ryanodine receptors (RyRs), and the sarcolemma, containing voltage- and Ca²⁺-activated K⁺ (BK) channels. These junctional microdomains promote hyperpolarization, reduced Ca_in, and relaxation. The second type is postulated to form around stretch-activated nonspecific cation channels and neighboring Ca²⁺-activated Cl⁻ channels, and promotes the opposite (depolarization, increased Ca_in, and contraction). The model includes three additional compartments: the sarcoplasm, the central SR lumen, and the peripheral SR lumen. It incorporates 37 protein components. In addition to pressure, the model accommodates inputs of α- and β-adrenergic agonists, ATP, 11,12-epoxyeicosatrienoic acid, and nitric oxide (NO). The parameters of the equations were adjusted to obtain a close fit to reported V_m and Ca_in as functions of pressure, which have been determined in cerebral arteries. The simulations were insensitive to ±10% changes in most of the parameters. The model also simulated the effects of inhibiting RyR, BK, or voltage-activated Ca²⁺ channels on V_m and Ca_in. Deletion of BK β1 subunits is known to increase arterial–SM tension. In the model, deletion of β1 raised Ca_in at all pressures, and these increases were reversed by NO.

Interplay Between Intracellular Ca(2+) Oscillations and Ca(2+)-stimulated Mitochondrial Metabolism

Oscillations of cytosolic Ca(2+) concentration are a widespread mode of signalling. Oscillatory spikes rely on repetitive exchanges of Ca(2+) between the endoplasmic reticulum (ER) and the cytosol, due to the regulation of inositol 1,4,5-trisphosphate receptors. Mitochondria also sequester and release Ca(2+), thus affecting Ca(2+) signalling. Mitochondrial Ca(2+) activates key enzymes involved in ATP synthesis. We propose a new integrative model for Ca(2+) signalling and mitochondrial metabolism in electrically non-excitable cells. The model accounts for (1) the phase relationship of the Ca(2+) changes in the cytosol, the ER and mitochondria, (2) the dynamics of mitochondrial metabolites in response to cytosolic Ca(2+) changes, and (3) the impacts of cytosol/mitochondria Ca(2+) exchanges and of mitochondrial metabolism on Ca(2+) oscillations. Simulations predict that as expected, oscillations are slowed down by decreasing the rate of Ca(2+) efflux from mitochondria, but also by decreasing the rate of Ca(2+) influx through the mitochondrial Ca(2+) uniporter (MCU). These predictions were experimentally validated by inhibiting MCU expression. Despite the highly non-linear character of Ca(2+) dynamics and mitochondrial metabolism, bioenergetics were found to be robust with respect to changes in frequency and amplitude of Ca(2+) oscillations.

Calcium Sparks in the Heart: Dynamics and Regulation

Calcium (Ca²⁺) plays a central role in the contraction of the heart. It is the bi-directional link between electrical excitation of the heart and contraction. Electrical excitation initiates Ca²⁺influx across the sarcolemma and T-tubular membrane that triggered calcium release from the sarcoplasmic reticulum. Ca²⁺sparks are the elementary events of calcium release from the sarcoplasmic reticulum. Therefore, understanding the dynamics of Ca²⁺sparks is essential for understanding the function of the heart. To this end, numerous experimental and computational studies have focused on this topic, exploring the mechanisms of calcium spark initiation, termination, and regulation and what role these play in normal and patho-physiology. The proper understanding of Ca²⁺ spark regulation and dynamics serves as the foundation for our insights into a multitude of pathological conditions may develop that can be the result of structural and/or functional changes at the cellular or subcellular level. Computational modeling of Ca²⁺ spark dynamics has proven to be a useful tool to understand Ca²⁺ spark dynamics. This review addresses our current understanding of Ca²⁺ sparks and how synchronized SR Ca²⁺ release, in which Ca²⁺ sparks is a major pathway, is linked to the different cardiac diseases, especially arrhythmias.

Regulation of cardiac cellular bioenergetics: mechanisms and consequences

The regulation of cardiac cellular bioenergetics is critical for maintaining normal cell function, yet the nature of this regulation is not fully understood. Different mechanisms have been proposed to explain how mitochondrial ATP production is regulated to match changing cellular energy demand while metabolite concentrations are maintained. We have developed an integrated mathematical model of cardiac cellular bioenergetics, electrophysiology, and mechanics to test whether stimulation of the dehydrogenase flux by Ca²⁺ or Pi, or stimulation of complex III by Pi can increase the rate of mitochondrial ATP production above that determined by substrate availability (ADP and Pi). Using the model, we show that, under physiological conditions the rate of mitochondrial ATP production can match varying demand through substrate availability alone; that ATP production rate is not limited by the supply of reducing equivalents in the form of NADH, as a result of Ca²⁺ or Pi activation of the dehydrogenases; and that ATP production rate is sensitive to feedback activation of complex III by Pi. We then investigate the mechanistic implications on cytosolic ion homeostasis and force production by simulating the concentrations of cytosolic Ca²⁺, Na⁺ and K⁺, and activity of the key ATPases, SERCA pump, Na⁺/K⁺ pump and actin-myosin ATPase, in response to increasing cellular energy demand. We find that feedback regulation of mitochondrial complex III by Pi improves the coupling between energy demand and mitochondrial ATP production and stabilizes cytosolic ADP and Pi concentrations. This subsequently leads to stabilized cytosolic ionic concentrations and consequentially reduced energetic cost from cellular ATPases.

Altered FoF1 ATP synthase and susceptibility to mitochondrial permeability transition pore during ischaemia and reperfusion in aging cardiomyocytes

Aging is a major determinant of the incidence and severity of ischaemic heart disease. Preclinical information suggests the existence of intrinsic cellular alterations that contribute to ischaemic susceptibility in senescent myocardium, by mechanisms not well established. We investigated the role of altered mitochondrial function in the adverse effect of aging. Isolated perfused hearts from old mice (> 20 months) displayed increased ischaemia-reperfusion injury as compared to hearts from adult mice (6 months) despite delayed onset of ischaemic rigor contracture. In cardiomyocytes from aging hearts there was a more rapid decline of mitochondrial membrane potential (Δψm) as compared to young ones, but ischaemic rigor shortening was also delayed. Transient recovery of Δψm observed during ischaemia, secondary to the reversal of mitochondrial FoF1 ATP synthase to ATPase mode, was markedly reduced in aging cardiomyocytes. Proteomic analysis demonstrated increased oxidation of different subunits of ATP synthase. Altered bionergetics in aging cells was associated with reduced mitochondrial calcium uptake and more severe cytosolic calcium overload during ischaemia-reperfusion. Despite attenuated ROS burst and mitochondrial calcium overload, mitochondrial permeability transition pore (mPTP) opening and cell death was increased in reperfused aged cells. In vitro studies demonstrated a significantly reduced calcium retention capacity in interfibrillar mitochondria from aging hearts. Our results identify altered FoF1 ATP synthase and increased sensitivity of mitochondria to undergo mPTP opening as important determinants of the reduced tolerance to ischaemia-reperfusion in aging hearts. Because ATP synthase has been proposed to conform mPTP, it is tempting to hypothesise that oxidation of ATP synthase underlie both phenomena.

Energy-based analysis of biochemical cycles using bond graphs

Thermodynamic aspects of chemical reactions have a long history in the physical chemistry literature. In particular, biochemical cycles require a source of energy to function. However, although fundamental, the role of chemical potential and Gibb's free energy in the analysis of biochemical systems is often overlooked leading to models which are physically impossible. The bond graph approach was developed for modelling engineering systems, where energy generation, storage and transmission are fundamental. The method focuses on how power flows between components and how energy is stored, transmitted or dissipated within components. Based on the early ideas of network thermodynamics, we have applied this approach to biochemical systems to generate models which automatically obey the laws of thermodynamics. We illustrate the method with examples of biochemical cycles. We have found that thermodynamically compliant models of simple biochemical cycles can easily be developed using this approach. In particular, both stoichiometric information and simulation models can be developed directly from the bond graph. Furthermore, model reduction and approximation while retaining structural and thermodynamic properties is facilitated. Because the bond graph approach is also modular and scaleable, we believe that it provides a secure foundation for building thermodynamically compliant models of large biochemical networks.

EAD and DAD mechanisms analyzed by developing a new human ventricular cell model

It has long been suggested that the Ca(2+)-mechanisms are largely involved in generating the early afterdepolarization (EAD) as well as the delayed afterdepolarization (DAD). This view was examined in a quantitative manner by applying the lead potential analysis to a new human ventricular cell model. In this ventricular cell model, the tight coupled LCC-RyR model (CaRU) based on local control theory (Hinch et al. 2004) and ion channel models mostly based on human electrophysiological data were included to reproduce realistic Ca(2+) dynamics as well as the membrane excitation. Simultaneously, the Ca(2+) accumulation near the Ca(2+) releasing site was incorporated as observed in real cardiac myocytes. The maximum rate of ventricular repolarization (-1.02 mV/ms) is due to IK1 (-0.55 mV/ms) and the rest is provided nearly equally by INCX (-0.20 mV/ms), INaL (-0.16 mV/ms) and INaT (-0.13 mV/ms). These INaL and INaT components are due to closure of the voltage gate, which remains partially open during the plateau potential. DADs could be evoked by applying high-frequency stimulations supplemented by a partial Na(+)/K(+) pump inhibition, or by a microinjection of Ca(2+). EADs was evoked by retarding the inactivation of INaL. The lead potential (VL) analysis revealed that IK1 and IKr played the primary role to reverse the AP repolarization to depolarizing limb of EAD. ICaL and INCX amplified EAD, while the remaining currents partially antagonized dVL/dt. The maximum rate of rise of EAD was attributable to the rapid activation of both ICaL (45.5%) and INCX (54.5%).

Translating intracellular calcium signaling into models

The rich experimental data on intracellular calcium has put theoreticians in an ideal position to derive models of intracellular calcium signaling. Over the last 25 years, a large number of modeling frameworks have been suggested. Here, I will review some of the milestones of intracellular calcium modeling with a special emphasis on calcium-induced calcium release (CICR) through inositol-1,4,5-trisphosphate and ryanodine receptors. I will highlight key features of CICR and how they are represented in models as well as the challenges that theoreticians face when translating our current understanding of calcium signals into equations. The selected examples demonstrate that a successful model provides mechanistic insights into the molecular machinery of the Ca²⁺ signaling toolbox and determines the contribution of local Ca²⁺ release to global Ca²⁺ patterns, which at the moment cannot be resolved experimentally.

Mitochondrial calcium uptake

Calcium (Ca(2+)) uptake into the mitochondrial matrix is critically important to cellular function. As a regulator of matrix Ca(2+) levels, this flux influences energy production and can initiate cell death. If large, this flux could potentially alter intracellular Ca(2+) ([Ca(2+)]i) signals. Despite years of study, fundamental disagreements on the extent and speed of mitochondrial Ca(2+) uptake still exist. Here, we review and quantitatively analyze mitochondrial Ca(2+) uptake fluxes from different tissues and interpret the results with respect to the recently proposed mitochondrial Ca(2+) uniporter (MCU) candidate. This quantitative analysis yields four clear results: (i) under physiological conditions, Ca(2+) influx into the mitochondria via the MCU is small relative to other cytosolic Ca(2+) extrusion pathways; (ii) single MCU conductance is ∼6-7 pS (105 mM [Ca(2+)]), and MCU flux appears to be modulated by [Ca(2+)]i, suggesting Ca(2+) regulation of MCU open probability (P(O)); (iii) in the heart, two features are clear: the number of MCU channels per mitochondrion can be calculated, and MCU probability is low under normal conditions; and (iv) in skeletal muscle and liver cells, uptake per mitochondrion varies in magnitude but total uptake per cell still appears to be modest. Based on our analysis of available quantitative data, we conclude that although Ca(2+) critically regulates mitochondrial function, the mitochondria do not act as a significant dynamic buffer of cytosolic Ca(2+) under physiological conditions. Nevertheless, with prolonged (superphysiological) elevations of [Ca(2+)]i, mitochondrial Ca(2+) uptake can increase 10- to 1,000-fold and begin to shape [Ca(2+)]i dynamics.

On the thermodynamic efficiency of Ca²⁺-ATPase molecular machines

Experimental studies have shown that the activity of the reconstituted molecular pump Ca(2+)-ATPase strongly depends on the thickness of the supporting bilayer. It is thus expected that the bilayer structure will have an impact on the thermodynamic efficiency of this nanomachine. Here, we introduce a nonequilibrium-thermodynamics theoretical approach to estimate the thermodynamic efficiency of the Ca(2+)-ATPase from analysis of available experimental data about ATP hydrolysis and Ca(2+) transport. We find that the entropy production, i.e., the heat released to the surroundings under working conditions, is approximately constant for bilayers containing phospholipids with hydrocarbon chains of 18-22 carbon atoms. Our estimates for the heat released during the pump operation agree with results obtained from separate calorimetric experiments on the Ca(2+)-ATPase derived from sarcoplasmic reticulum. We show that the thermodynamic efficiency of the reconstituted Ca(2+)-ATPase reaches a maximum for bilayer thicknesses corresponding to maximum activity. Surprisingly, the estimated thermodynamic efficiency is very low, ∼12%. We discuss the significance of this result as representative of the efficiency of other nanomachines, and we address the influence of the experimental set-up on such a low efficiency. Overall, our approach provides a general route to estimate thermodynamic efficiencies and heat dissipation in experimental studies of nanomachines.

The relative influences of phosphometabolites and pH on action potential morphology during myocardial reperfusion: a simulation study

Myocardial ischemia-reperfusion (IR) injury represents a constellation of pathological processes that occur when ischemic myocardium experiences a restoration of perfusion. Reentrant arrhythmias, which represent a particularly lethal manifestation of IR injury, can result when ischemic tissue exhibits decreased excitability and/or changes of action potential duration (APD), conditions that precipitate unidirectional conduction block. Many of the cellular components that are involved with IR injury are modulated by pH and/or phosphometabolites such as ATP and phosphocreatine (PCr), all of which can be manipulated in vivo and potentially in the clinical setting. Using a mathematical model of the cardiomyocyte that we previously developed to study ischemia and reperfusion, we performed a series of simulations with the aim of determining whether pH- or phosphometabolite-related processes play a more significant role in generating changes in excitability and action potential morphology that are associated with the development of reentry. In our simulations, persistent shortening of APD, action potential amplitude (APA), and depolarization of the resting membrane potential were more severe when ATP and PCr availability were suppressed during reperfusion than when extracellular pH recovery was inhibited. Reduced phosphometabolite availability and pH recovery affected multiple ion channels and exchangers. Some of these effects were the result of direct modulation by phosphometabolites and/or acidosis, while others resulted from elevated sodium and calcium loads during reperfusion. In addition, increasing ATP and PCr availability during reperfusion was more beneficial in terms of increasing APD and APA than was increasing the amount of pH recovery. Together, these results suggest that therapies directed at increasing ATP and/or PCr availability during reperfusion may be more beneficial than perturbing pH recovery with regard to mitigating action potential changes that increase the likelihood of reentrant arrhythmias.

Nucleotide activation of the Ca-ATPase

We have used fluorescence spectroscopy, molecular modeling, and limited proteolysis to examine structural dynamics of the sarcoplasmic reticulum Ca-ATPase (SERCA). The Ca-ATPase in sarcoplasmic reticulum vesicles from fast twitch muscle (SERCA1a isoform) was selectively labeled with fluorescein isothiocyanate (FITC), a probe that specifically reacts with Lys-515 in the nucleotide-binding site. Conformation-specific proteolysis demonstrated that FITC labeling does not induce closure of the cytoplasmic headpiece, thereby assigning FITC-SERCA as a nucleotide-free enzyme. We used enzyme reverse mode to synthesize FITC monophosphate (FMP) on SERCA, producing a phosphorylated pseudosubstrate tethered to the nucleotide-binding site of a Ca(2+)-free enzyme (E2 state to prevent FMP hydrolysis). Conformation-specific proteolysis demonstrated that FMP formation induces SERCA headpiece closure similar to ATP binding, presumably due to the high energy phosphoryl group on the fluorescent probe (ATP·E2 analog). Subnanosecond-resolved detection of fluorescence lifetime, anisotropy, and quenching was used to characterize FMP-SERCA (ATP·E2 state) versus FITC-SERCA in Ca(2+)-free, Ca(2+)-bound, and actively cycling phosphoenzyme states (E2, E1, and EP). Time-resolved spectroscopy revealed that FMP-SERCA exhibits increased probe dynamics but decreased probe accessibility compared with FITC-SERCA, indicating that ATP exhibits enhanced dynamics within a closed cytoplasmic headpiece. Molecular modeling was used to calculate the solvent-accessible surface area of FITC and FMP bound to SERCA crystal structures, revealing a positive correlation of solvent-accessible surface area with quenching but not anisotropy. Thus, headpiece closure is coupled to substrate binding but not active site dynamics. We propose that dynamics in the nucleotide-binding site of SERCA is important for Ca(2+) binding (distal allostery) and phosphoenzyme formation (direct activation).

Computational approaches to understand cardiac electrophysiology and arrhythmias

Cardiac rhythms arise from electrical activity generated by precisely timed opening and closing of ion channels in individual cardiac myocytes. These impulses spread throughout the cardiac muscle to manifest as electrical waves in the whole heart. Regularity of electrical waves is critically important since they signal the heart muscle to contract, driving the primary function of the heart to act as a pump and deliver blood to the brain and vital organs. When electrical activity goes awry during a cardiac arrhythmia, the pump does not function, the brain does not receive oxygenated blood, and death ensues. For more than 50 years, mathematically based models of cardiac electrical activity have been used to improve understanding of basic mechanisms of normal and abnormal cardiac electrical function. Computer-based modeling approaches to understand cardiac activity are uniquely helpful because they allow for distillation of complex emergent behaviors into the key contributing components underlying them. Here we review the latest advances and novel concepts in the field as they relate to understanding the complex interplay between electrical, mechanical, structural, and genetic mechanisms during arrhythmia development at the level of ion channels, cells, and tissues. We also discuss the latest computational approaches to guiding arrhythmia therapy.

Modelling cardiac calcium sparks in a three-dimensional reconstruction of a calcium release unit

Triggered release of Ca2+ from an individual sarcoplasmic reticulum (SR) Ca(2+) release unit (CRU) is the fundamental event of cardiac excitation–contraction coupling, and spontaneous release events (sparks) are the major contributor to diastolic Ca(2+) leak in cardiomyocytes. Previous model studies have predicted that the duration and magnitude of the spark is determined by the local CRU geometry, as well as the localization and density of Ca(2+) handling proteins. We have created a detailed computational model of a CRU, and developed novel tools to generate the computational geometry from electron tomographic images. Ca(2+) diffusion was modelled within the SR and the cytosol to examine the effects of localization and density of the Na(+)/Ca(2+) exchanger, sarco/endoplasmic reticulum Ca(2+)-ATPase 2 (SERCA), and calsequestrin on spark dynamics. We reconcile previous model predictions of approximately 90% local Ca(2+) depletion in junctional SR, with experimental reports of about 40%. This analysis supports the hypothesis that dye kinetics and optical averaging effects can have a significant impact on measures of spark dynamics. Our model also predicts that distributing calsequestrin within non-junctional Z-disc SR compartments, in addition to the junctional compartment, prolongs spark release time as reported by Fluo5. By pumping Ca(2+) back into the SR during a release, SERCA is able to prolong a Ca(2+) spark, and this may contribute to SERCA-dependent changes in Ca(2+) wave speed. Finally, we show that including the Na(+)/Ca(2+) exchanger inside the dyadic cleft does not alter local [Ca(2+)] during a spark.

Kinetic and mesoscopic non-equilibrium description of the Ca(2+) pump: a comparison

We analyse the operation of the Ca(2+)-ATPase ion pump using a kinetic cycle diagram. Using the methodology of Hill, we obtain the cycle fluxes, entropy production and efficiency of the pump. We compare these results with a mesoscopic non-equilibrium description of the pump and show that the kinetic and mesoscopic pictures are in accordance with each other. This gives further support to the mesoscopic theory, which is less restricted and also can include the heat flux as a variable. We also show how motors can be characterised in terms of unidirectional backward fluxes. We proceed to show how the mesoscopic approach can be used to identify fast and slow steps of the model in terms of activation energies, and how this can be used to simplify the kinetic diagram.