A comparison of three models of the inositol trisphosphate receptor

The inositol (1,4,5)-trisphosphate receptor (IPR) plays a crucial role in calcium dynamics in a wide range of cell types, and is often a central feature in quantitative models of calcium oscillations and waves. We compare three mathematical models of the IPR, fitting each of them to the same data set to determine ranges for the parameter values. Each of the fits indicates that fast activation of the receptor, followed by slow inactivation, is an important feature of the model, and also that the speed of inositol trisphosphate (IP₃) binding cannot necessarily be assumed to be faster than Ca²⁺ activation. In addition, the model which assumed saturating binding rates of Ca²⁺ to the IPR demonstrated the best fit. However, lack of convergence in the fitting procedure indicates that responses to step increases of [Ca²⁺] and [IP₃] provide insufficient data to determine the parameters unambiguously in any of the models.     

Markov chain Monte Carlo fitting of single-channel data from inositol trisphosphate receptors

In many cell types, the inositol trisphosphate receptor (IPR) is one of the important components that control intracellular calcium dynamics, and an understanding of this receptor (which is also a calcium channel) is necessary for an understanding of calcium oscillations and waves. Recent advances in experimental techniques now allow for the measurement of single-channel activity of the IPR in conditions similar to its native environment, and these data can be used to determine the rate constants in Markov models of the IPR. We illustrate a parameter estimation method based on Markov chain Monte Carlo, which can be used to fit directly to single-channel data, and determining, as an intrinsic part of the fit, the times at which the IPR is opening and closing. We show, using simulated data, the most complex Markov model that can be unambiguously determined from steady-state data and show that non-steady-state data is required to determine more complex models.     

Calcium Oscillations and Waves Generated by Multiple Release Mechanisms in Pancreatic Acinar Cells

We explore the dynamic behavior of a model of calcium oscillations and wave propagation in the basal region of pancreatic acinar cells [Sneyd, J., et al., Biophys. J. 85: 1392–1405, 2003]. Since it is known that two principal calcium release pathways are involved, inositol trisphosphate receptors (IPR) and ryanodine receptors (RyR), we study how the model behavior depends on the density of each receptor type. Calcium oscillations can be mediated either by IPR or RyR. Continuous increases in either RyR or IPR density can lead to the appearance and disappearance of oscillations multiple times, and the two receptor types interact via their common effect on cytoplasmic calcium concentration and the subsequent effect on the total amount of calcium inside the cell. Increases in agonist concentration can stimulate oscillations via the RyR by increasing calcium influx. Using a two time-scale approach, we explain these complex behaviors by treating the total amount of cellular calcium as a slow parameter. Oscillations are controlled by the shape of the slow manifold and where it intersects the nullcline of the slow variable. When calcium diffusion is included, the existence of traveling waves in the model equation is strongly dependent on the interplay between the total amount of calcium in the cell and membrane transport, a feature that can be experimentally tested. Our results help us understand the behavior of a model that includes both receptors in comparison to the properties of each receptor type in isolation.     

Dynamics of Calcium Fluxes in Nonexcitable Cells: Mathematical Modeling

Mathematical models simulating the dynamics of calcium redistribution (elicited by experimental interference with the pathways of calcium fluxes) in cellular compartments have been developed, based on a minimal scheme of the pathways of calcium fluxes in nonexcitable cells suspended in calcium-free medium. The models are consistent with available experimental data. All parameters are quantitatively related to the intrinsic properties of calcium adenosine triphosphatases (ATPases) and cellular membranes; there is no interdependence between the parameters. The models can be used as the basis for quantitative analysis and interpretation of experimental data. The activities of plasma membrane and sarcoendoplasmic reticulum calcium ATPases (PMCA and SERCAs) are governed by different mechanisms. PMCA is likely to undergo transitions from inactive to active to “dormant” (not identical to the initial) and back to inactive states, the mean duration of the cycle lasting for minutes or longer. The sequence of the transitions is initiated, presumably, by an increase in cytosolic calcium concentration. The transition of PMCA from inactive to active (at least at low rates of increase in cytosolic calcium concentration) is likely to be slower than that from active to dormant. SERCA, presumably, transits from inactive to active state in response to increases in calcium leakage from calcium stores. Whereas PMCA extrudes excess calcium (a definite quantity of it) in a short pulse, SERCA retakes calcium back into the stores permanently at a high rate. The models presented here may be the best means for the moment to quantitatively relate the dynamics of calcium fluxes in nonexcitable cells with known or putative properties of the mechanisms underlying activation of calcium ATPases.     

A mathematical model of parathyroid hormone response to acute changes in plasma ionized calcium concentration in humans

A complex bio-mechanism, commonly referred to as calcium homeostasis, regulates plasma ionized calcium (Ca²⁺) concentration in the human body within a narrow range which is crucial for maintaining normal physiology and metabolism. Taking a step towards creating a complete mathematical model of calcium homeostasis, we focus on the short-term dynamics of calcium homeostasis and consider the response of the parathyroid glands to acute changes in plasma Ca²⁺ concentration. We review available models, discuss their limitations, then present a two-pool, linear, time-varying model to describe the dynamics of this calcium homeostasis subsystem, the Ca-PTH axis. We propose that plasma PTH concentration and plasma Ca²⁺ concentration bear an asymmetric reverse sigmoid relation. The parameters of our model are successfully estimated based on clinical data corresponding to three healthy subjects that have undergone induced hypocalcemic clamp tests. In the first validation of this kind, with parameters estimated separately for each subject we test the model’s ability to predict the same subject’s induced hypercalcemic clamp test responses. Our results demonstrate that a two-pool, linear, time-varying model with an asymmetric reverse sigmoid relation characterizes the short-term dynamics of the Ca-PTH axis.     

Models of cardiac excitation-contraction coupling in ventricular myocytes

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

Isothermal titration calorimetric procedure to determine protein-metal ion binding parameters in the presence of excess metal ion or chelator

Determination of binding parameters for metal ion binding to proteins usually requires preceding steps to remove protein-bound metal ions. Removal of bound metal ions from protein is often associated with decreased stability and inactivation. We present two simple isothermal titration calorimetric procedures that eliminate separate metal ion removal steps and directly monitor the exchange of metal ions between buffer, protein, and chelator. The concept is to add either excess chelator or metal ion to the protein under investigation and subsequently titrate with metal ion or chelator, respectively. It is thereby possible in the same experimental trial to obtain both chelator-metal ion and protein-metal ion binding parameters due to the different thermodynamic "fingerprints" of chelator and protein. The binding models and regression routines necessary to analyze the corresponding binding isotherms have been constructed. Verifications of the models have been done by titrations of mixtures of calcium chelators (BAPTA, HEDTA, and EGTA) and calcium ions and they were both able to account satisfactorily for the observed binding isotherms. Therefore, it was possible to determine stoichiometric and thermodynamic binding parameters. In addition, the concept has been tested on a recombinant alpha-amylase from Bacillus halmapalus where it proved to be a consistent procedure to obtain calcium binding parameters.     

NMDA receptors are required in memory formation in Drosophila mushroom body

Dynamics of changes in cytosolic calcium concentration resulting from facilitation of calcium leakage from the stores and (or) blocking the pathways of its reuptake back into the stores or extrusion out of the cell (or both) have been investigated experimentally. It has been found that: (a) no mechanisms other than the membrane leakage, PMCA or SERCA, are involved in the discharge of calcium stores and calcium extrusion or reuptake; (b) the discharge of calcium stores in the absence of both its extrusion and reuptake back into the stores depends only on membrane leakage, the asymptotic calcium concentration in cytosol depending only on the initial content of the stores and being independent of the leakage; (c) the dynamics of the activity of both PMCA and SERCA depend on the initial rate of calcium influx, the dynamics differing from each other at high initial rates of calcium influx; (d) whereas there is no observable background activity of PMCA, background activity of SERCA is observed.     

Properties of two human atrial cell models in tissue: restitution, memory, propagation, and reentry

To date, two detailed ionic models of human atrial cell electrophysiology have been developed, the Nygren et al. model (NM) and the Courtemanche et al. model (CM). Although both models draw from similar experimental data, they have vastly different properties. This paper provides the first systematic analysis and comparison of the dynamics of these models in spatially extended systems including one-dimensional cables and rings, two-dimensional sheets, and a realistic three-dimensional human atrial geometry. We observe that, as in single cells, the CM adapts to rate changes primarily by changes in action potential duration (APD) and morphology, while for the NM rate changes affect resting membrane potential (RMP) more than APD. The models also exhibit different memory properties as assessed through S1-S2 APD and conduction velocity (CV) restitution curves with different S1 cycle lengths. Reentrant wave dynamics also differ, with the NM exhibiting stable, non-breaking spirals and the CM exhibiting frequent transient wave breaks. The realistic atrial geometry modifies dynamics in some cases through drift, transient pinning, and breakup. Previously proposed modifications to represent atrial fibrillation-remodeled electrophysiology produce altered dynamics, including reduced rate adaptation and memory for both models and conversion to stable reentry for the CM. Furthermore, proposed variations to the NM to reproduce action potentials more closely resembling those of the CM do not substantially alter the underlying dynamics of the model, so that tissue simulations using these modifications still behave more like the unmodified NM. Finally, interchanging the transmembrane current formulations of the two models suggests that currents contribute more strongly to RMP and CV, intracellular calcium dynamics primarily determine reentrant wave dynamics, and both are important in APD restitution and memory in these models. This finding implies that the formulation of intracellular calcium processes is as important to producing realistic models as transmembrane currents.     

Clarification and application of an ion parametric resonance model for magnetic field interactions with biological systems

Theoretical models proposed to date have been unable to clearly predict biological results from exposure to low-intensity electric and magnetic fields (EMF). Recently a predictive ionic resonance model was proposed by Lednev, based on an earlier atomic spectroscopy theory described by Podgoretskii and Podgoretskii and Khrustalev. The ion parametric resonance (IPR) model developed in this paper corrects mathematical errors in the earlier Lednev model and extends that model to give explicit predictions of biological responses to parallel AC and DC magnetic fields caused by field-induced changes in combinations of ions within the biological system. Distinct response forms predicted by the IPR model depend explicitly on the experimentally controlled variables: magnetic flux densities of the AC and DC magnetic fields (B_ac and B_dc, respectively); AC frequency (f_ac); and, implicitly, charge to mass ratio of target ions. After clarifying the IPR model and extending it to combinations of different resonant ions, this paper proposes a basic set of experiments to test the IPR model directly which do not rely on the choice of a particular specimen or endpoint. While the fundamental bases of the model are supported by a variety of other studies, the IPR model is necessarily heuristic when applied to biological systems, because it is based on the premise that the magnitude and form of magnetic field interactions with unhydrated resonant ions in critical biological structures alter ion-associated biological activities that may in turn be correlated with observable effects in living systems. © 1994 Wiley-Liss, Inc.     

Multilevel complexity of calcium signaling: Modeling angiogenesis

Intracellular calcium signaling is a universal,evolutionary conserved and versatile regulator of cell biochemistry.The complexity of calcium signaling and related cell machinery can be investigated by the use of experimental strategies,as well as by computational approaches.Vascular endothelium is a fascinating model to study the specific properties and roles of calcium signals at multiple biological levels.During the past 20 years,live cell imaging,patch clamp and other techniques have allowed us to detect and interfere with calcium signaling in endothelial cells(ECs),providing a huge amount of information on the regulation of vascularization(angiogenesis) in normal and tumoral tissues.These data range from the spatiotemporal dynamics of calcium within different cell microcompartments to those in entire multicellular and organized EC networks.Beside experimental strategies,in silico endothelial models,specifically designed for simulating calcium signaling,are contributing to our knowledge of vascular physiol-ogy and pathology.They help to investigate and predict the quantitative features of proangiogenic events moving through subcellular,cellular and supracellular levels.This review focuses on some recent developments of computational approaches for proangiogenic endothelial calcium signaling.In particular,we discuss the creation of hybrid simulation environments,which combine and integrate discrete Cellular Potts Models.They are able to capture the phenomenological mechanisms of cell morphological reorganization,migration,and intercellular adhesion,with single-cell spatiotemporal models,based on reaction-diffusion equations that describe the agonist-induced intracellular calcium events.     

Assessing the role of internal phosphorus recycling on eutrophication in four lakes in China and Malaysia

Internal phosphorus recycling (IPR) is an important nutrient source driving algal growth and eutrophication in lakes. The complexity of eutrophication behaviours caused by high IPR complicates lake management and undermines restoration efforts. Hence, knowledge about the possible types of bifurcation behaviours caused by high IPR is essential for effective and sustainable lake eutrophication management. For this purpose, numerical bifurcation analysis is performed on an algae‑phosphorus model to investigate how IPR drives complex and rich eutrophication behaviours in two tropical and two subtropical lakes. The two tropical lakes are Tasik Harapan and Sunway Lagoon in Malaysia, while the two subtropical lakes are Lake Fuxian and Lake Taihu in China. For each specified level of IPR, co-dimension one bifurcation analysis is performed by means of XPPAUT. Co-dimension two bifurcation analysis is then carried out by means of MatCont. At low IPR, Lake Fuxian exhibits reversible behaviour, accompanied by higher external phosphorus loading (EPL) thresholds. Lake Fuxian is also more conducive to stable equilibrium and its lake dynamics are easily predictable. At moderate IPR, Sunway Lagoon is likely to exhibit stable equilibrium, accompanied by possible shifting between two stable steady states (hysteresis behaviour) and oscillations. With higher IPR, Lake Taihu and Tasik Harapan are prone to irreversibility, accompanied by lower EPL thresholds. Because of increased complexity in lake dynamics in Lake Taihu and Tasik Harapan, small changes in EPL or in algal mortality rates could trigger various transitions in lake dynamics. Overall, high IPR can trigger unexpected sharp increases in algal concentration and can reduce the resilience of an oligotrophic lake. For shallow lakes, high IPR would cause unexpected sharp increases in algal concentrations, undermine resilience of lakes, complicate lake management, and delay lake recovery process.     

On calcium-buffer dynamics within the excess buffer regime

In intracellular calcium signaling, calcium buffers has been recognized for their role in reshaping and localizing the calcium concentration profile in the vicinity of the channel, as well as reducing the effective diffusion of free calcium. In the presence of an excess of endogenous or exogenous buffers, linearization of the reaction-diffusion system describing the calcium-buffer dynamics has been instrumental in understanding the extent of the microdomain formation and in quantifying the apparent diffusion of the free calcium. In these linearized models, the conclusions are usually drawn from the steady-state solutions upon the opening of the channel. In this work, using the joint Laplace-Fourier method, we give an explicit integral transient solution, as well as, the long-time asymptotic behavior of the linearized calcium-buffer dynamics. The results confirm and emphasize the long stated intuitions on the diffusive character of the calcium-buffer dynamics. Numerical validations of our analytical results will be discussed.     

Genetically encoded calcium indicators for studying long-term calcium dynamics during apoptosis

Intracellular calcium release is essential for regulating almost all cellular functions. Specific spatio-temporal patterns of cytosolic calcium elevations are critical determinants of cell fate in response to pro-apoptotic cellular stressors. As the apoptotic program can take hours or days, measurement of long-term calcium dynamics are essential for understanding the mechanistic role of calcium in apoptotic cell death. Due to the technical limitations of using calcium-sensitive dyes to measure cytosolic calcium little is known about long-term calcium dynamics in living cells after treatment with apoptosis-inducing drugs. Genetically encoded calcium indicators could potentially overcome some of the limitations of calcium-sensitive dyes. Here, we compared the performance of the genetically encoded calcium indicators GCaMP6s and GCaMP6f with the ratiometric dye Fura-2. GCaMP6s performed as well or better than Fura-2 in detecting agonist-induced calcium transients. We then examined the utility of GCaMP6s for continuously measuring apoptotic calcium release over the course of ten hours after treatment with staurosporine. We found that GCaMP6s was suitable for measuring apoptotic calcium release over long time courses and revealed significant heterogeneity in calcium release dynamics in individual cells challenged with staurosporine. Our results suggest GCaMP6s is an excellent indicator for monitoring long-term changes cytosolic calcium during apoptosis.     

Disease‐structured N‐mixture models: A practical guide to model disease dynamics using count data

Obtaining inferences on disease dynamics (e.g., host population size, pathogen prevalence, transmission rate, host survival probability) typically requires marking and tracking individuals over time. While multistate mark–recapture models can produce high‐quality inference, these techniques are difficult to employ at large spatial and long temporal scales or in small remnant host populations decimated by virulent pathogens, where low recapture rates may preclude the use of mark–recapture techniques. Recently developed N‐mixture models offer a statistical framework for estimating wildlife disease dynamics from count data. N‐mixture models are a type of state‐space model in which observation error is attributed to failing to detect some individuals when they are present (i.e., false negatives). The analysis approach uses repeated surveys of sites over a period of population closure to estimate detection probability. We review the challenges of modeling disease dynamics and describe how N‐mixture models can be used to estimate common metrics, including pathogen prevalence, transmission, and recovery rates while accounting for imperfect host and pathogen detection. We also offer a perspective on future research directions at the intersection of quantitative and disease ecology, including the estimation of false positives in pathogen presence, spatially explicit disease‐structured N‐mixture models, and the integration of other data types with count data to inform disease dynamics. Managers rely on accurate and precise estimates of disease dynamics to develop strategies to mitigate pathogen impacts on host populations. At a time when pathogens pose one of the greatest threats to biodiversity, statistical methods that lead to robust inferences on host populations are critically needed for rapid, rather than incremental, assessments of the impacts of emerging infectious diseases.     

异质种群动态模型:破碎化景观动态模拟的新途径

景观破碎化导致物种以异质种群方式存活,使得基于异质种群动态模拟破碎化景观动态成为可能。异质种群动态模型的发展为景观动态模拟奠定了良好基础。根据空间处理方式的不同,异质种群模型可分为三大类,可不同程度地用于描述破碎化景观动态。(1)空间不确定异质种群模型,假定所有局域种群间均等互联,模型中不包含空间信息,仅能用于景观斑块动态描述;(2)空间确定异质种群模型,假设局域种群在二维空间上以规则格子形式排列,是一种准现实的空间处理方式,可用于景观动态的简单描述;(3)空间现实异质种群模型,包含了破碎化景观中局域种群的几何特征,可直接用于真实景观动态的模拟研究。空间现实的和基于个体的异质种群模型不但是未来异质种群模型发展的主流,也将成为未来破碎化景观动态研究的重要工具。为了更加准确完整地描述破碎化景观动态,不但应该综合运用已有的各种异质种群模型方法,更要引进新模型来刎画多物种、多变量、高维度、复杂连接的破碎化景观格局与过程。     

Calcium transport mechanisms in muskrat and rat hearts

Mammalian hearts experience calcium overload during extreme and prolonged hypoxia and the calcium overload may lead to enzyme activation and cell death. Several calcium transport systems were examined in muskrat hearts and compared to those found in rat hearts to determine if there is a species difference that might be related to the muskrats' superior ability to survive hypoxia. Radiolabeled nitredendipine binding was determined in rat and muskrat hearts to estimate the density of voltage gated calcium channels in surface membranes. There were no species differences. Calcium release channel density in the sarcoplasmic reticulum was estimated by the determination of radiolabeled ryanodine binding in muskrat and rat heart SR membranes. No differences were revealed between species. The SR uptake of calcium was measured in SR membranes from the hearts of the two species. No differences were found in the B(max) values, however, the muskrat SR membranes did have a slightly lower K(m) value. There were large species differences in Na(+)/Ca(2+) exchange in SL membranes with the muskrat heart having approximately 3.5 times the transport capacity of rat SL membranes. During hypoxic conditions in which there is extensive ATP depletion leading to [Na(+)](i) accumulation and discharge of cellular membrane potential, the Na(+)/Ca(2+) exchanger may operate in the reverse mode and import calcium into the cell and accelerate hypoxic damage. Prior to reaching this state a robust Na(+)/Ca(2+) exchange would facilitate the maintenance of normal diastolic calcium levels and calcium cycling. Muskrats hearts are hypoxia tolerant by virtue of their ability to reduce metabolic demand and generate ATP anaerobically thus, maintaining a favorable ATP balance. Therefore, the relative overexpression of Na(+)/Ca(2+) exchangers in muskrat hearts may be beneficial in the preservation of contractile function and calcium homeostasis in this freshwater diving mammal.     

Dynamics of human atrial cell models: restitution, memory, and intracellular calcium dynamics in single cells

Mathematical models of cardiac cells have become important tools for investigating the electrophysiological properties and behavior of the heart. As the number of published models increases, it becomes more difficult to choose a model appropriate for the conditions to be studied, especially when multiple models describing the species and region of the heart of interest are available. In this paper, we will review and compare two detailed ionic models of human atrial myocytes, the Nygren et al. model (NM) and the Courtemanche et al. model (CM). Although both models include the same transmembrane currents and are largely based on the same experimental data from human atrial cells, the two models exhibit vastly different properties, especially in their dynamical behavior, including restitution and memory effects. The CM produces pronounced rate adaptation of action potential duration (APD) with limited memory effects, while the NM exhibits strong rate dependence of resting membrane potential (RMP), limited APD restitution, and stronger memory, as well as delayed afterdepolarizations and auto-oscillatory behavior upon cessation of rapid pacing. Channel conductance modifications based on experimentally measured changes during atrial fibrillation modify rate adaptation and memory in both models, but do not change the primary rate-dependent properties of APD and RMP for the CM and NM, respectively. Two sets of proposed changes to the NM that yield a spike-and-dome action potential morphology qualitatively similar to the CM at slow pacing rates similarly do not change the underlying dynamics of the model. Moreover, interchanging the formulations of all transmembrane currents between the two models while leaving calcium handling and ionic concentrations intact indicates that the currents strongly influence memory and the rate adaptation of RMP, while intracellular calcium dynamics primarily determine APD rate adaptation. Our results suggest that differences in intracellular calcium handling between the two human atrial myocyte models are responsible for marked dynamical differences and may prevent reconciliation between the models by straightforward channel conductance modifications.     

Fluctuations in the pollen tube tip-focused calcium gradient are not reflected in nuclear calcium level: a comparative analysis using recombinant yellow cameleon calcium reporter

For the first time in pollen tubes, both cytoplasmic and nuclear calcium have been imaged to allow comparative analysis of calcium dynamics in these two compartments with high spatial and temporal dynamics. An improved cameleon (YC2.1) calcium reporter was expressed cytoplasmically in both Lilium longiflorum and Nicotiana tabacum pollen tubes and the periodically fluctuating tip-focused calcium gradient typical of normal growth was recorded by ratio image analysis. The nucleoplasmin targeting sequence was then used to localise expressed YC2.1 to the vegetative nucleus of N. tabacum pollen tubes to permit imaging of nuclear location, shape and calcium dynamics. Nuclear-targeted YC2.1 (NupYC2.1) showed an absence of any obvious regular fluctuations in nuclear calcium levels during tube extension in vitro with typical growth rate fluctuation. The use of targeted cameleons to study subcellular calcium dynamics in pollen tubes is discussed.