A Model of CatSper Channel Mediated Calcium Dynamics in Mammalian Spermatozoa

CatSpers are calcium (Ca²⁺) channels that are located along the principal piece of mammalian sperm flagella and are directly linked to sperm motility and hyperactivation. It has been observed that Ca²⁺ entry through CatSper channels triggers a tail to head Ca²⁺ propagation in mouse sperm, as well as a sustained increase of Ca²⁺ in the head. Here, we develop a mathematical model to investigate this propagation and sustained increase in the head. A 1-d reaction-diffusion model tracking intracellular Ca²⁺ with flux terms for the CatSper channels, a leak flux, and plasma membrane Ca²⁺ clearance mechanism is studied. Results of this simple model exhibit tail to head Ca²⁺ propagation, but no sustained increase in the head. Therefore, in this model, a simple plasma membrane pump-leak system with diffusion in the cytosol cannot account for these experimentally observed results. It has been proposed that Ca²⁺ influx from the CatSper channels induce additional Ca²⁺ release from an internal store. We test this hypothesis by examining the possible role of Ca²⁺ release from the redundant nuclear envelope (RNE), an inositol 1,4,5-trisphosphate (IP₃) gated Ca²⁺ store in the neck. The simple model is extended to include an equation for IP₃ synthesis, degradation, and diffusion, as well as flux terms for Ca²⁺ in the RNE. When IP₃ and the RNE are accounted for, the results of the model exhibit a tail to head Ca²⁺ propagation as well as a sustained increase of Ca²⁺ in the head.     

Reduced Model Captures Mg-RNA Interaction Free Energy of Riboswitches

The stability of RNA tertiary structures depends heavily on Mg²⁺. The Mg²⁺-RNA interaction free energy that stabilizes an RNA structure can be computed experimentally through fluorescence-based assays that measure Γ₂₊, the number of excess Mg²⁺ associated with an RNA molecule. Previous explicit-solvent simulations predict that the majority of excess Mg²⁺ ions interact closely and strongly with the RNA, unlike monovalent ions such as K⁺, suggesting that an explicit treatment of Mg²⁺ is important for capturing RNA dynamics. Here we present a reduced model that accurately reproduces the thermodynamics of Mg²⁺-RNA interactions. This model is able to characterize long-timescale RNA dynamics coupled to Mg²⁺ through the explicit representation of Mg²⁺ ions. KCl is described by Debye-Hückel screening and a Manning condensation parameter, which represents condensed K⁺ and models its competition with condensed Mg²⁺. The model contains one fitted parameter, the number of condensed K⁺ ions in the absence of Mg²⁺. Values of Γ₂₊ computed from molecular dynamics simulations using the model show excellent agreement with both experimental data on the adenine riboswitch and previous explicit-solvent simulations of the SAM-I riboswitch. This agreement confirms the thermodynamic accuracy of the model via the direct relation of Γ₂₊ to the Mg²⁺-RNA interaction free energy, and provides further support for the predictions from explicit-solvent calculations. This reduced model will be useful for future studies of the interplay between Mg²⁺ and RNA dynamics.     

Reduced Model Captures Mg2+-RNA Interaction Free Energy of Riboswitches

The stability of RNA tertiary structures depends heavily on Mg²⁺. The Mg²⁺-RNA interaction free energy that stabilizes an RNA structure can be computed experimentally through fluorescence-based assays that measure Γ₂₊, the number of excess Mg²⁺ associated with an RNA molecule. Previous explicit-solvent simulations predict that the majority of excess Mg²⁺ ions interact closely and strongly with the RNA, unlike monovalent ions such as K⁺, suggesting that an explicit treatment of Mg²⁺ is important for capturing RNA dynamics. Here we present a reduced model that accurately reproduces the thermodynamics of Mg²⁺-RNA interactions. This model is able to characterize long-timescale RNA dynamics coupled to Mg²⁺ through the explicit representation of Mg²⁺ ions. KCl is described by Debye-Hückel screening and a Manning condensation parameter, which represents condensed K⁺ and models its competition with condensed Mg²⁺. The model contains one fitted parameter, the number of condensed K⁺ ions in the absence of Mg²⁺. Values of Γ₂₊ computed from molecular dynamics simulations using the model show excellent agreement with both experimental data on the adenine riboswitch and previous explicit-solvent simulations of the SAM-I riboswitch. This agreement confirms the thermodynamic accuracy of the model via the direct relation of Γ₂₊ to the Mg²⁺-RNA interaction free energy, and provides further support for the predictions from explicit-solvent calculations. This reduced model will be useful for future studies of the interplay between Mg²⁺ and RNA dynamics.     

Characterization of Mg Inhibition of Mitochondrial Ca Uptake by a Mechanistic Model of Mitochondrial Ca Uniporter

Ca²⁺ is an important regulatory ion and alteration of mitochondrial Ca²⁺ homeostasis can lead to cellular dysfunction and apoptosis. Ca²⁺ is transported into respiring mitochondria via the Ca²⁺ uniporter, which is known to be inhibited by Mg²⁺. This uniporter-mediated mitochondrial Ca²⁺ transport is also shown to be influenced by inorganic phosphate (Pi). Despite a large number of experimental studies, the kinetic mechanisms associated with the Mg²⁺ inhibition and Pi regulation of the uniporter function are not well established. To gain a quantitative understanding of the effects of Mg²⁺ and Pi on the uniporter function, we developed here a mathematical model based on known kinetic properties of the uniporter and presumed Mg²⁺ inhibition and Pi regulation mechanisms. The model is extended from our previous model of the uniporter that is based on a multistate catalytic binding and interconversion mechanism and Eyring's free energy barrier theory for interconversion. The model satisfactorily describes a wide variety of experimental data sets on the kinetics of mitochondrial Ca²⁺ uptake. The model also appropriately depicts the inhibitory effect of Mg²⁺ on the uniporter function, in which Ca²⁺ uptake is hyperbolic in the absence of Mg²⁺ and sigmoid in the presence of Mg²⁺. The model suggests a mixed-type inhibition mechanism for Mg²⁺ inhibition of the uniporter function. This model is critical for building mechanistic models of mitochondrial bioenergetics and Ca²⁺ handling to understand the mechanisms by which Ca²⁺ mediates signaling pathways and modulates energy metabolism.     

Caffeine-induced Ca2+ oscillations in type I horizontal cell of carp retina: A mathematical model

Oscillations in intracellular free Ca²⁺ concentration ([Ca²⁺]_i) have been observed in a variety of cell types. In the present study, we constructed a mathematical model to simulate the caffeine-induced [Ca²⁺]_i oscillations based on experimental data obtained from isolated type I horizontal cell of carp retina. The results of model analysis confirm the notion that the caffeine-induced [Ca²⁺]_i oscillations involve a number of cytoplasmic and endoplasmic Ca²⁺ processes that interact with each other. Using this model, we evaluated the importance of store-operated channel (SOC) in caffeine-induced [Ca²⁺]_i oscillations. The model suggests that store-operated Ca²⁺ entry (SOCE) is elicited upon depletion of the endoplasmic reticulum (ER). When the SOC conductance is set to 0, caffeine-induced [Ca²⁺]_i oscillations are abolished, which agrees with the experimental observation that [Ca²⁺]_i oscillations were abolished when SOC was blocked pharmacologically, verifying that SOC is necessary for sustained [Ca²⁺]_i oscillations.     

Caffeine-induced Ca2+ oscillations in type I horizontal cell of carp retina: A mathematical model

Oscillations in intracellular free Ca²⁺ concentration ([Ca²⁺]_i) have been observed in a variety of cell types. In the present study, we constructed a mathematical model to simulate the caffeine-induced [Ca²⁺]_i oscillations based on experimental data obtained from isolated type I horizontal cell of carp retina. The results of model analysis confirm the notion that the caffeine-induced [Ca²⁺]_i oscillations involve a number of cytoplasmic and endoplasmic Ca²⁺ processes that interact with each other. Using this model, we evaluated the importance of store-operated channel (SOC) in caffeine-induced [Ca²⁺]_i oscillations. The model suggests that store-operated Ca²⁺ entry (SOCE) is elicited upon depletion of the endoplasmic reticulum (ER). When the SOC conductance is set to 0, caffeine-induced [Ca²⁺]_i oscillations are abolished, which agrees with the experimental observation that [Ca²⁺]_i oscillations were abolished when SOC was blocked pharmacologically, verifying that SOC is necessary for sustained [Ca²⁺]_i oscillations.     

Mathematical modelling of human P2X-mediated plasma membrane electrophysiology and calcium dynamics in microglia

Regulation of cytosolic calcium (Ca²⁺) dynamics is fundamental to microglial function. Temporal and spatial Ca²⁺ fluxes are induced from a complicated signal transduction pathway linked to brain ionic homeostasis. In this paper, we develop a novel biophysical model of Ca²⁺ and sodium (Na⁺) dynamics in human microglia and evaluate the contribution of purinergic receptors (P2XRs) to both intracellular Ca²⁺ and Na⁺ levels in response to agonist/ATP binding. This is the first comprehensive model that integrates P2XRs to predict intricate Ca²⁺ and Na⁺ transient responses in microglia. Specifically, a novel compact biophysical model is proposed for the capture of whole-cell patch-clamp currents associated with P2X₄ and P2X₇ receptors, which is composed of only four state variables. The entire model shows that intricate intracellular ion dynamics arise from the coupled interaction between P2X₄ and P2X₇ receptors, the Na⁺/Ca²⁺ exchanger (NCX), Ca²⁺ extrusion by the plasma membrane Ca²⁺ ATPase (PMCA), and Ca²⁺ and Na⁺ leak channels. Both P2XRs are modelled as two separate adenosine triphosphate (ATP) gated Ca²⁺ and Na⁺ conductance channels, where the stoichiometry is the removal of one Ca²⁺ for the hydrolysis of one ATP molecule. Two unique sets of model parameters were determined using an evolutionary algorithm to optimise fitting to experimental data for each of the receptors. This allows the proposed model to capture both human P2X₇ and P2X₄ data (hP2X₇ and hP2X₄). The model architecture enables a high degree of simplicity, accuracy and predictability of Ca²⁺ and Na⁺ dynamics thus providing quantitative insights into different behaviours of intracellular Na⁺ and Ca²⁺ which will guide future experimental research. Understanding the interactions between these receptors and other membrane-bound transporters provides a step forward in resolving the qualitative link between purinergic receptors and microglial physiology and their contribution to brain pathology.     

Stochastic reaction-diffusion modeling of calcium dynamics in 3D dendritic spines of Purkinje cells

Calcium (Ca²⁺) is a second messenger assumed to control changes in synaptic strength in the form of both long-term depression and long-term potentiation at Purkinje cell dendritic spine synapses via inositol trisphosphate (IP₃)-induced Ca²⁺ release. These Ca²⁺ transients happen in response to stimuli from parallel fibers (PFs) from granule cells and climbing fibers (CFs) from the inferior olivary nucleus. These events occur at low numbers of free Ca²⁺, requiring stochastic single-particle methods when modeling them. We use the stochastic particle simulation program MCell to simulate Ca²⁺ transients within a three-dimensional Purkinje cell dendritic spine. The model spine includes the endoplasmic reticulum, several Ca²⁺ transporters, and endogenous buffer molecules. Our simulations successfully reproduce properties of Ca²⁺ transients in different dynamical situations. We test two different models of the IP₃ receptor (IP₃R). The model with nonlinear concentration response of binding of activating Ca²⁺ reproduces experimental results better than the model with linear response because of the filtering of noise. Our results also suggest that Ca²⁺-dependent inhibition of the IP₃R needs to be slow to reproduce experimental results. Simulations suggest the experimentally observed optimal timing window of CF stimuli arises from the relative timing of CF influx of Ca²⁺ and IP₃ production sensitizing IP₃R for Ca²⁺-induced Ca²⁺ release. We also model ataxia, a loss of fine motor control assumed to be the result of malfunctioning information transmission at the granule to Purkinje cell synapse, resulting in a decrease or loss of Ca²⁺ transients. Finally, we propose possible ways of recovering Ca²⁺ transients under ataxia.     

Model for rhizodeposition and CO2 efflux from planted soil and its validation by 14C pulse labelling of ryegrass

A model for rhizodeposition and root respiration was developed and parameterised based on ¹⁴C pulse labelling of Lolium perenne. The plants were grown in a two-compartment chamber on a loamy Haplic Luvisol under controlled laboratory conditions. The dynamics of ¹⁴CO₂ efflux from the soil and ¹⁴C content in shoots, roots, micro-organisms, dissolved organic carbon (DOC) and soil were measured during the first 11 days after labelling. Modelled parameters were estimated by fitting on measured ¹⁴C dynamics in the different pools. The model and the measured ¹⁴C dynamics in all pools corresponded well (r ²=0.977). The model describes well ¹⁴CO₂ efflux from the soil and ¹⁴C dynamics in shoots, roots and soil, but predicts unsatisfactorily the ¹⁴C content in micro-organisms and DOC. The model also allows for division of the total ¹⁴CO₂ efflux from the soil in ¹⁴CO₂ derived from root respiration and ¹⁴CO₂ derived from rhizomicrobial respiration by use of exudates and root residues. Root respiration and rhizomicrobial respiration amounted for 7.6% and 6.0% of total assimilated C, respectively, which accounts for 56% and 44% of root-derived ¹⁴CO₂ efflux from the soil planted with 43-day-old Lolium perenne, respectively. The sensitivity analysis has shown that root respiration rate affected the curve of ¹⁴CO₂ efflux from the soil mainly during the first day after labelling. The changes in the exudation rate influenced the ¹⁴CO₂ efflux later than first 24 h after labelling.     

Quantitative aspects of drug-receptor interactions. I. Ca2+ and cholinergic receptor activation in smooth muscle: a basic model for drug-receptor interactions

An attempt has been made to devise a general model of drug-receptor interactions as it relates to the initiation of mechanical responses. A key feature of this model is the regulatory role played by membrane-bound Ca²⁺ (Ca_mem²⁺).The effects on the mechanical responsiveness of guinea pig ileal longitudinal muscle of four muscarinic agonists derived from and including the highly active cis-2-methyl-4-dimethylaminomethyl-1, 3-dioxolane methiodide have been studied. The concentration-response (isotonic contraction) curves of these four agonists at normal Ca_ext²⁺-levels show evidence of cooperativity (n_H > 1) and this was found to increase dramatically with decreasing [Ca_ext²⁺]. A three step model has been proposed, based on that previously advanced by Hurwitz &; Suria (1971), in which activation of the acetylcholine receptor initiates a Ca²⁺ translocation mechanism supplying the contractile machinery with Ca²⁺. Arguments are advanced to suggest that two sources of Ca²⁺ are thus utilized: membrane-bound (Ca_mem²⁺) and free extracellular (Ca_ext²⁺), the former being responsible for the initial phasic contraction and the latter for the slower phase of contractile development.Analysis of the theoretical model shows that the cooperativity of the concentration-response relationships derives not from the initial agonist-receptor interaction but from the subsequently initiated Ca²⁺ translocation step so that [Ca_int²⁺] ∝ [Ca_ext²⁺]ⁿ. The limiting value of n is found to be 6 and to be the same for agonists and partial agonists. According to this model intrinsic activity is determined by the linkage between the agonist-receptor complex and the Ca²⁺ translocation process.The general findings of this work are discussed in terms of an equilibrium between Ca²⁺-associated and Ca²⁺-dissociated membrane states. The similarities to other Ca²⁺ dependent processes are emphasized.     

Role of myocardial viscoelasticity in disturbances of electrical and mechanical activity in calcium overloaded cardiomyocytes: mathematical modeling

Cardiomyocyte Ca²⁺ overload is closely linked to cardiac arrhythmias. We have earlier shown in a mathematical model that myocardium mechanical activity may contribute to rhythm disturbances induced by Ca²⁺ overload in cardiomyocytes with reduced Na⁺-K⁺ pump work (Sulman et al., 2008). The same model is used here to address possible contribution of the passive mechanical properties of cardiac muscle (i.e. myocardial viscous and elastic properties) to the arrhythmogenesis. In a series of contractions at regular pacing rate of 75 beats/min a model with higher viscosity demonstrated essentially earlier appearance of extrasystoles due to a faster cardiomyocyte Ca²⁺ loading up to a level triggering spontaneous Ca²⁺ releases from the sarcoplasmic reticulum. The model predicts that myocardial elasticity also may affect arrhythmogenesis in cardiomyocytes overloaded with Ca²⁺. Contribution of the mechanical properties of the myocardial tissue to the arrhythmia has been analyzed for wide ranges of both viscosity and elasticity coefficients. The results suggest that myocardial viscoelastic properties may be a factor affecting Ca²⁺ handling in cardiomyocytes and contributing to cardiac mechano-electric feedback in arrhythmogenesis.     

Confocal laser scanning microscopy reveals voltage-gated calcium signals within hippocampal dendritic spines

The induction of long-term potentiation (LTP) is generally assumed to be triggered by Ca²⁺ entry into dendritic spines via NMDA receptor-gated channels. A previous computational model proposed that spines serve several functions in this process. First, they compartmentalize and amplify increases in [Ca²⁺]_i. Second, they augment the nonlinear relationship between synaptic strength and the probability or magnitude of LTP induction. Third, they isolate the metabolic machinery responsible for LTP induction from increases in [Ca²⁺]_i produced by voltage-gated Ca²⁺ channels in the dendritic shaft. Here we examine this last prediction of the model using methods that combine confocal microscopy with simultaneous neurophysiological recordings in hippocampal brain slices. Either of two Ca²⁺-sensitive dyes were injected into CA1 pyramidal neurons. Direct depolarization of the neurons via the somatic electrode produced clear increases in Ca²⁺ signals within the dendritic spines, a result that was not predicted by the previous spine model. Our new spine model suggests that some of this signal could theoretically result from Ca²⁺-bound dye diffusing from the dendritic shaft into the spine. Dye diffusion alone cannot, however, explain the numerous cases in which the Ca²⁺ signal in the spine was considerably larger than that in the adjacent dendritic shaft. The latter observations raise the possiblity of voltage-gated Ca²⁺ entry directly into the spine or else perhaps via Ca²⁺-dependent Ca²⁺release. The new spine model accommodates these observations as well as several other recent experimental results. 1994 John Wiley & Sons, Inc.     

Inhibition of human red cell Nak-ATPase by magnesium and potassium.

The inhibition of NaK-ATPase (EC 3.6.1.3) from human red cells by Mg²⁺ is markedly dependent on the relative concentrations of Na⁺ and K⁺. Inhibition increases with increasing K⁺ and decreases with increasing Na⁺. The inhibition appears to be a combined effect of Mg²⁺ and K⁺ at sites distinct from the sites at which these cations activate the enzyme. The kinetics of activation of the enzyme by Na⁺ with inhibitory levels of Mg²⁺ and K⁺ are biphasic, indicating both low and high affinity Na⁺ sites. At noninhibitory levels of Mg²⁺ and K⁺ only high affinity Na⁺ sites are seen. The results are inconsistent with any model in which Mg²⁺ and K⁺ compete with Na⁺ at a single site. A kinetic model is proposed to explain the mechanism of inhibition by Mg²⁺ and K⁺.     

Relationship between plant species richness and biomass in a coastal Maine Quercus-Pinus forest

Abstract. Several researchers have hypothesized that plant species richness has a unimodal relationship with biomass, while others have argued for a linear relationship. Data from various types of herbaceous communities show some support for the unimodal hypothesis, but this has not been tested extensively for forests and questions remain concerning its generality. We used linear and quadratic regression models to examine the relationship between overstory biomass and richness in a coastal Maine Quercus-Pinus forest across and within cover types using data from two plot sizes (2500-m² quadrats and 625-m² sub-quadrats). Understory data from 1-m² plots were also analysed. Richness was quadratically related to biomass at both plot sizes for all cover types combined, but the amount of variation explained by the models was very low (R²s < 0.09). Richness and biomass were not significantly related at either plot size for the mixed mesic cover type, the most common type in the forest. The best fit (R²= 0.43) was obtained with a quadratic model for the conifer cover type at the sub-quadrat level, with the quadratic model for the 1-m² data having the second highest R² (0.24). Across all six data sets, the quadratic model was the only one with a significant fit in two cases, and had considerably higher R²s (1.3–1.9 ×) in two others. The remaining two data sets could not be fit with a significant model of either type. For this forest, these results suggest little support for a linear relationship between plant species richness and biomass and variable, often weak, support for a unimodal relationship. Density, a potentially confounding variable in this type of analysis, was only weakly correlated with richness and was not found to alter the relationship between biomass and richness.     

Carbon Dioxide Variation in a Hardwood Forest Stream: An Integrative Measure of Whole Catchment Soil Respiration

The concentration of CO₂ in stream water is a product of not only instream metabolism but also upland, riparian, and groundwater processes and as such can provide an integrative measure of whole catchment soil respiration. Using a 5-year dataset of pH, alkalinity, Ca²⁺, and Mg²⁺ in surface water of the West Fork of Walker Branch in eastern Tennessee in conjunction with a hydrological flowpath chemistry model, we investigated how CO₂ concentrations and respiration rates in stream, bedrock, and soil environments vary seasonally and interannually. Dissolved inorganic carbon concentration was highest in summer and autumn (P < 0.05) although the proportion as free CO₂ (pCO₂) did not vary seasonally (P > 0.05). Over the 5 years, pCO₂ was always supersaturated with respect to the atmosphere ranging from 374 to 3626 ppmv (1.0- to 10.1-fold greater than atmospheric equilibrium), and CO₂ evasion from the stream to the atmosphere ranged from 146 to 353 mmol m⁻² d⁻¹. Whereas pCO₂ in surface water exhibited little intra-annual or interannual variation, distinct seasonal patterns in soil and bedrock pCO₂ were revealed by the catchment CO₂ model. Seasonally, soil pCO₂ increased from a winter low of 8167 ppmv to a summer high of 27,068 ppmv. Driven by the seasonal variation in gas levels, evasion of CO₂ from soils to the atmosphere ranged from 83 mmol m⁻² d⁻¹ in winter to 287 mmol m⁻² d⁻¹ in summer. The seasonal variation in soil CO₂ tracked soil temperature (r ²= 0.46, P < 0.001) and model-derived estimates of CO₂ evasion rate from soils agreed with previously reported fluxes measured using chambers (Pearson correlation coefficient = 0.62, P < 0.05) supporting the model assumptions. Although rates of CO₂ evasion were similar between the stream and soils, the overall rate of evasion from the channel was only 0.4% of the 70,752 mol/d that evaded from soils due to the vastly different areas of the two subsystems. Our model provides a means to assess whole catchment CO₂ dynamics from easily collected and measured stream-water samples and an approach to study catchment scale variation in soil ecosystem respiration. Received 24 July 1997; accepted 14 November 1997.     

Ionic selectivity in L-type calcium channels by electrostatics and hard-core repulsion

A physical model of selective “ion binding” in the L-type calcium channel is constructed, and consequences of the model are compared with experimental data. This reduced model treats only ions and the carboxylate oxygens of the EEEE locus explicitly and restricts interactions to hard-core repulsion and ion–ion and ion–dielectric electrostatic forces. The structural atoms provide a flexible environment for passing cations, thus resulting in a self-organized induced-fit model of the selectivity filter. Experimental conditions involving binary mixtures of alkali and/or alkaline earth metal ions are computed using equilibrium Monte Carlo simulations in the grand canonical ensemble. The model pore rejects alkali metal ions in the presence of biological concentrations of Ca²⁺ and predicts the blockade of alkali metal ion currents by micromolar Ca²⁺. Conductance patterns observed in varied mixtures containing Na⁺ and Li⁺, or Ba²⁺ and Ca²⁺, are predicted. Ca²⁺ is substantially more potent in blocking Na⁺ current than Ba²⁺. In apparent contrast to experiments using buffered Ca²⁺ solutions, the predicted potency of Ca²⁺ in blocking alkali metal ion currents depends on the species and concentration of the alkali metal ion, as is expected if these ions compete with Ca²⁺ for the pore. These experiments depend on the problematic estimation of Ca²⁺ activity in solutions buffered for Ca²⁺ and pH in a varying background of bulk salt. Simulations of Ca²⁺ distribution with the model pore bathed in solutions containing a varied amount of Li⁺ reveal a “barrier and well” pattern. The entry/exit barrier for Ca²⁺ is strongly modulated by the Li⁺ concentration of the bath, suggesting a physical explanation for observed kinetic phenomena. Our simulations show that the selectivity of L-type calcium channels can arise from an interplay of electrostatic and hard-core repulsion forces among ions and a few crucial channel atoms. The reduced system selects for the cation that delivers the largest charge in the smallest ion volume.     

Modeling the calcium sequestration system in isolated guinea pig cardiac mitochondria

Under high Ca²⁺ load conditions, Ca²⁺ concentrations in the extra-mitochondrial and mitochondrial compartments do not display reciprocal dynamics. This is due to a paradoxical increase in the mitochondrial Ca²⁺ buffering power as the Ca²⁺ load increases. Here we develop and characterize a mechanism of the mitochondrial Ca²⁺ sequestration system using an experimental data set from isolated guinea pig cardiac mitochondria. The proposed mechanism elucidates this phenomenon and others in a mathematical framework and is integrated into a previously corroborated model of oxidative phosphorylation including the Na⁺/Ca²⁺ cycle. The integrated model reproduces the Ca²⁺ dynamics observed in both compartments of the isolated mitochondria respiring on pyruvate after a bolus of CaCl₂ followed by ruthenium red and a bolus of NaCl. The model reveals why changes in mitochondrial Ca²⁺ concentration of Ca²⁺ loaded mitochondria appear significantly mitigated relative to the corresponding extra-mitochondrial Ca²⁺ concentration changes after Ca²⁺ efflux is initiated. The integrated model was corroborated by simulating the set-point phenomenon. The computational results support the conclusion that the Ca²⁺ sequestration system is composed of at least two classes of Ca²⁺ buffers. The first class represents prototypical Ca²⁺ buffering, and the second class encompasses the complex binding events associated with the formation of amorphous calcium phosphate. With the Ca²⁺ sequestration system in mitochondria more precisely defined, computer simulations can aid in the development of innovative therapeutics aimed at addressing the myriad of complications that arise due to mitochondrial Ca²⁺ overload.     

Some implications of a constant fiber stress hypothesis in the diastolic left ventricle

A thick-wall incompressible, elastic sphere was used as a model for the diastolic rat left ventricle. A model for myocardial nonhomogeneity was derived assuming that fiber (circumferential) stress was independent of position in the ventricular wall. The theoretical implications of the resulting constitutive relations together with the spherical model were analyzed in the context of large deformation elasticity theory. It was found that muscle stiffness at a given level of uniaxial stress increased monotonically from the endocardium to the epicardium. In addition, fiber stress was found to be essentially a linear function of transmural pressure above a pressure of 6 g/cm². It was also shown theoretically that neglecting the nonhomogeneity of the myocardium resulted in a state of stress which differed significantly from that predicted by the nonhomogeneous model. For example, at a transmural pressure of 14 g/cm², fiber stress in the nonhomogenous model was equal to 17 g/cm² while fiber stress in the homogeneous model varied between 100 g/cm² at the endocardial surface and 2 g/cm² at the epicardial surface. The change in muscle stiffness with position which characterized the nonhomogeneous model also tended to linearize the highly curvilinear radial stress distribution predicted by the homogeneous model at a given transmural pressure.     

Ca Alternans in a Cardiac Myocyte Model that Uses Moment Equations to Represent Heterogeneous Junctional SR Ca

Multiscale whole-cell models that accurately represent local control of Ca²⁺-induced Ca²⁺ release in cardiac myocytes can reproduce high-gain Ca²⁺ release that is graded with changes in membrane potential. Using a recently introduced formalism that represents heterogeneous local Ca²⁺ using moment equations, we present a model of cardiac myocyte Ca²⁺ cycling that exhibits alternating sarcoplasmic reticulum (SR) Ca²⁺ release when periodically stimulated by depolarizing voltage pulses. The model predicts that the distribution of junctional SR [Ca²⁺] across a large population of Ca²⁺ release units is distinct on alternating cycles. Load-release and release-uptake functions computed from this model give insight into how Ca²⁺ fluxes and stimulation frequency combine to determine the presence or absence of Ca²⁺ alternans. Our results show that the conditions for the onset of Ca²⁺ alternans cannot be explained solely by the steepness of the load-release function, but that changes in the release-uptake process also play an important role. We analyze the effect of the junctional SR refilling time constant on Ca²⁺ alternans and conclude that physiologically realistic models of defective Ca²⁺ cycling must represent the dynamics of heterogeneous junctional SR [Ca²⁺] without assuming rapid equilibration of junctional and network SR [Ca²⁺].     

Mechanisms by which cytoplasmic calcium wave propagation and alternans are generated in cardiac atrial myocytes lacking T-tubules-insights from a simulation study

This study investigated the mechanisms underlying the propagation of cytoplasmic calcium waves and the genesis of systolic Ca²⁺ alternans in cardiac myocytes lacking transverse tubules (t-tubules). These correspond to atrial cells of either small mammals or large mammals that have lost their t-tubules due to disease-induced structural remodeling (e.g., atrial fibrillation). A mathematical model was developed for a cluster of ryanodine receptors distributed on the cross section of a cell that was divided into 13 elements with a spatial resolution of 2 μm. Due to the absence of t-tubules, L-type Ca²⁺ channels were only located in the peripheral elements close to the cell-membrane surface and produced Ca²⁺ signals that propagated toward central elements by triggering successive Ca²⁺-induced Ca²⁺ release (CICR) via Ca²⁺ diffusion between adjacent elements. Under control conditions, the Ca²⁺ signals did not fully propagate to the central region of the cell. However, with modulation of several factors responsible for Ca²⁺ handling, such as the L-type Ca²⁺ channels (Ca²⁺ influx), SERCA pumps (sarcoplasmic reticulum (SR) Ca²⁺ uptake), and ryanodine receptors (SR Ca²⁺ release), Ca²⁺ wave propagation to the center of the cell could occur. These simulation results are consistent with previous experimental data from atrial cells of small mammals. The model further reveals that spatially functional heterogeneity in Ca²⁺ diffusion within the cell produced a steep relationship between the SR Ca²⁺ content and the cytoplasmic Ca²⁺ concentration. This played an important role in the genesis of Ca²⁺ alternans that were more obvious in central than in peripheral elements. Possible association between the occurrence of Ca²⁺ alternans and the model parameters of Ca²⁺ handling was comprehensively explored in a wide range of one- and two-parameter spaces. In addition, the model revealed a spontaneous second Ca²⁺ release in response to a single voltage stimulus pulse with SR Ca²⁺ overloading and augmented Ca²⁺ influx. This study provides what to our knowledge are new insights into the genesis of Ca²⁺ alternans and spontaneous second Ca²⁺ release in cardiac myocytes that lack t-tubules.