Multi-scale models of local control of calcium induced calcium release.

Calcium-induced-calcium-release in cardiac myocytes is the release of Ca(2+) from the sarcoplasmic reticulum (SR) triggered by Ca(2+) entering the cell through L-type Ca(2+) channels. The Ca(2+) is released through ryanodine receptors which 'sense' local [Ca(2+)] in the small region (the diadic space) positioned between the t-tubules and the SR. The length-scale of a single diad is of the order of 10nm and the diffusion time-scale is of order of 1 micros with each cell containing approximately 10,000 diadic spaces which act independently. However, typically one is interested in Ca(2+) currents at the whole cell level and higher. This is a multi-scale problem and cannot be solved by direct computation. In this paper we develop a general framework for deriving approximate solutions of these models.

A simplified local control model of calcium-induced calcium release in cardiac ventricular myocytes.

Calcium (Ca2+)-induced Ca2+ release (CICR) in cardiac myocytes exhibits high gain and is graded. These properties result from local control of Ca2+ release. Existing local control models of Ca2+ release in which interactions between L-Type Ca2+ channels (LCCs) and ryanodine-sensitive Ca2+ release channels (RyRs) are simulated stochastically are able to reconstruct these properties, but only at high computational cost. Here we present a general analytical approach for deriving simplified models of local control of CICR, consisting of low-dimensional systems of coupled ordinary differential equations, from these more complex local control models in which LCC-RyR interactions are simulated stochastically. The resulting model, referred to as the coupled LCC-RyR gating model, successfully reproduces a range of experimental data, including L-Type Ca2+ current in response to voltage-clamp stimuli, inactivation of LCC current with and without Ca2+ release from the sarcoplasmic reticulum, voltage-dependence of excitation-contraction coupling gain, graded release, and the force-frequency relationship. The model does so with low computational cost.

Genomics, proteomics and bioinformatics of human heart failure.

Unraveling the molecular complexities of human heart failure, particularly end-stage failure, can be achieved by combining multiple investigative approaches. There are several parts to the problem. Each patient is the product of a complex set of genetic variations, different degrees of influence of diets and lifestyles, and usually heart transplantation patients are treated with multiple drugs. The genomic status of the myocardium of any one transplant patient can be analysed using gene arrays (cDNA- or oligonucleotide-based) each with its own strengths and weaknesses. The proteins expressed by these failing hearts (myocardial proteomics) were first investigated over a decade ago using two-dimensional polyacrylamide gel electrophoresis (2DGE) which promised to resolve several thousand proteins in a single sample of failing heart. However, while 2DGE is very successful for the abundant and moderately expressed proteins, it struggles to identify proteins expressed at low levels. Highly focused first dimension separations combined with recent advances in mass spectrometry now provide new hope for solving this difficulty. Protein arrays are a more recent form of proteomics that hold great promise but, like the above methods, they have their own drawbacks. Our approach to solving the problems inherent in the genomics and proteomics of heart failure is to provide experts in each analytical method with a sample from the same human failing heart. This requires a sufficiently large number of samples from a sufficiently large pool of heart transplant patients as well as a large pool of non-diseased, non-failing human hearts. We have collected more than 200 hearts from patients undergoing heart transplantations and a further 50 non-failing hearts. By combining our expertise we expect to reduce and possibly eliminate the inherent difficulties of each analytical approach. Finally, we recognise the need for bioinformatics to make sense of the large quantities of data that will flow from our laboratories. Thus, we plan to provide meaningful molecular descriptions of a number of different conditions that result in terminal heart failure.

Molecular interactions between two long-QT syndrome gene products, HERG and KCNE2, rationalized by in vitro and in silico analysis.

The cardiac delayed rectifier potassium current mediates repolarization of the action potential and underlies the QT interval of the ECG. Mutations in either of the two molecular components of the rapid delayed rectifier (I(K,r)), HERG and KCNE2, have been linked to heritable or acquired long-QT syndrome. Mechanisms whereby mutations of KCNE2 produce fatal cardiac arrhythmias characteristic of long-QT syndrome remain unclear. In this study, we characterize functional interactions between HERG and KCNE2 with a view to defining underlying mechanisms for action potential prolongation and long-QT syndrome. Whereas coexpression of hKCNE2 with HERG alters both kinetics and density of ionic current, incorporation of these effects into a quantitative model of the action potential predicts that only changes in current density significantly affect repolarization. Thus, the primary functional consequence of hKCNE2 on action potential morphology is through modulation of I(K,r) density, as predicted by the model. Mutations associated with long-QT syndrome that result only in modest changes of gating kinetics may be epiphenomena or may modulate action potential repolarization via interaction with alternative pore-forming potassium channel alpha subunits.

Role of the calcium-independent transient outward current I(to1) in shaping action potential morphology and duration.

The Kv4.3-encoded current (I:(Kv4.3)) has been identified as the major component of the voltage-dependent Ca(2+)-independent transient outward current (I:(to1)) in human and canine ventricular cells. Experimental evidence supports a correlation between I:(to1) density and prominence of the phase 1 notch; however, the role of I:(to1) in modulating action potential duration (APD) remains unclear. To help resolve this role, Markov state models of the human and canine Kv4.3- and Kv1.4-encoded currents at 35 degrees C are developed on the basis of experimental measurements. A model of canine I:(to1) is formulated as the combination of these Kv4.3 and Kv1.4 currents and is incorporated into an existing canine ventricular myocyte model. Simulations demonstrate strong coupling between L-type Ca(2+) current and I:(Kv4.3) and predict a bimodal relationship between I:(Kv4.3) density and APD whereby perturbations in I:(Kv4.3) density may produce either prolongation or shortening of APD, depending on baseline I:(to1) current level.

Direct histological validation of diffusion tensor MRI in formaldehyde-fixed myocardium.

Diffusion tensor MRI is emerging as a rapid, nondestructive method to map myocardial fiber organization. It accurately measures myofiber orientation in hearts bathed in or perfused with cardioplegic solution. This study shows it also accurately maps the fibrous architecture of formalin-fixed hearts. Fiber orientations obtained by MRI and histology at the same locations in an excised portion of rabbit ventricle differed on average by 3.7 degrees (SD = 6.4 degrees, N = 70), a closer correspondence than achieved with previous preparations. The longer acquisition times afforded by fixed-heart imaging provides better accuracy, and should enable high-resolution reconstruction of the entire ventricular architecture. Magn Reson Med 44:157-161, 2000.

Modeling short-term interval-force relations in cardiac muscle.

This study employs two modeling approaches to investigate short-term interval-force relations. The first approach is to develop a low-order, discrete-time model of excitation-contraction coupling to determine which parameter combinations produce the degree of postextrasystolic potentiation seen experimentally. Potentiation is found to increase 1) for low recirculation fraction, 2) for high releasable fraction, i.e., the maximum fraction of Ca(2+) released from the sarcoplasmic reticulum (SR) given full restitution, and 3) for strong negative feedback of the SR release on sarcolemmal Ca(2+) influx. The second modeling approach is to develop a more detailed single ventricular cell model that simulates action potentials, Ca(2+)-handling mechanisms, and isometric force generation by the myofilaments. A slow transition from the adapted state of the ryanodine receptor produces a gradual recovery of the SR release and restitution behavior. For potentiation, a small extrasystolic release leaves more Ca(2+) in the SR but also increases the SR loading by two mechanisms: 1) less Ca(2+)-induced inactivation of L-type channels and 2) reduction of action potential height by residual activation of the time-dependent delayed rectifier K(+) current, which increases Ca(2+) influx. The cooperativity of the myofilaments amplifies the relatively small changes in the Ca(2+) transient amplitude to produce larger changes in isometric force. These findings suggest that short-term interval-force relations result mainly from the interplay of the ryanodine receptor adaptation and the SR Ca(2+) loading, with additional contributions from membrane currents and myofilament activation.

Reconstruction of cardiac ventricular geometry and fiber orientation using magnetic resonance imaging.

An imaging method for the rapid reconstruction of fiber orientation throughout the cardiac ventricles is described. In this method, gradient-recalled acquisition in the steady-state (GRASS) imaging is used to measure ventricular geometry in formaldehyde-fixed hearts at high spatial resolution. Diffusion-tensor magnetic resonance imaging (DTMRI) is then used to estimate fiber orientation as the principle eigenvector of the diffusion tensor measured at each image voxel in these same hearts. DTMRI-based estimates of fiber orientation in formaldehyde-fixed tissue are shown to agree closely with those measured using histological techniques, and evidence is presented suggesting that diffusion tensor tertiary eigenvectors may specify the orientation of ventricular laminar sheets. Using a semiautomated software tool called HEARTWORKS, a set of smooth contours approximating the epicardial and endocardial boundaries in each GRASS short-axis section are estimated. These contours are then interconnected to form a volumetric model of the cardiac ventricles. DTMRI-based estimates of fiber orientation are interpolated into these volumetric models, yielding reconstructions of cardiac ventricular fiber orientation based on at least an order of magnitude more sampling points than can be obtained using manual reconstruction methods.

Electrophysiological modeling of cardiac ventricular function: from cell to organ.

Three topics of importance to modeling the integrative function of the heart are reviewed. The first is modeling of the ventricular myocyte. Emphasis is placed on excitation-contraction coupling and intracellular Ca2+ handling, and the interpretation of experimental data regarding interval-force relationships. Second, data on use of diffusion tensor magnetic resonance (DTMR) imaging for measuring the anatomical structure of the cardiac ventricles are presented. A method for the semi-automated reconstruction of the ventricles using a combination of gradient recalled acquisition in the steady state (GRASS) and DTMR images is described. Third, we describe how these anatomically and biophysically based models of the cardiac ventricles can be implemented on parallel computers.

Modeling gain and gradedness of Ca2+ release in the functional unit of the cardiac diadic space.

A model of the functional release unit (FRU) in rat cardiac muscle consisting of one dihydropyridine receptor (DHPR) and eight ryanodine receptor (RyR) channels, and the volume surrounding them, is formulated. It is assumed that no spatial [Ca2+] gradients exist in this volume, and that each FRU acts independently. The model is amenable to systematic parameter studies in which FRU dynamics are simulated at the channel level using Monte Carlo methods with Ca2+ concentrations simulated by numerical integration of a coupled system of differential equations. Using stochastic methods, Ca(2+)-induced Ca2+ release (CICR) shows both high gain and graded Ca2+ release that is robust when parameters are varied. For a single DHPR opening, the resulting RyR Ca2+ release flux is insensitive to the DHPR open duration, and is determined principally by local sarcoplasmic reticulum (SR) Ca2+ load, consistent with experimental data on Ca2+ sparks. In addition, single RyR openings are effective in triggering Ca2+ release from adjacent RyRs only when open duration is long and SR Ca2+ load is high. This indicates relatively low coupling between RyRs, and suggests a mechanism that limits the regenerative spread of RyR openings. The results also suggest that adaptation plays an important modulatory role in shaping Ca2+ release duration and magnitude, but is not solely responsible for terminating Ca2+ release. Results obtained with the stochastic model suggest that high gain and gradedness can occur by the recruitment of independent FRUs without requiring spatial [Ca2+] gradients within a functional unit or cross-coupling between adjacent functional units.

Optimal detection of flash intensity differences using rod photocurrent observations.

The rod photocurrent contains two noise components that may limit the detectability of flash intensity increments. The limits imposed by the low- and high-frequency noise components were assessed by computing the performance of an optimal detector of increments in flash intensity. The limits imposed by these noise components depend on the interval of observation of the photocurrent signal. When the entire photocurrent signal, lasting 3 or more seconds, is observed, the low-frequency component of the photocurrent noise (attributed to the quantal noise of the incoming light, as well as random isomerizations of enzymes within the phototransduction cascade) is the most significant limitation on detectability. When only the first 380 ms or less is observed, the high-frequency component of the noise (due to the thermal isomerizations of the cGMP-gated channel) presents a significant limit on the detectability of flashes.

Comparison of putative cooperative mechanisms in cardiac muscle: length dependence and dynamic responses.

Length-dependent steady-state and dynamic responses of five models of isometric force generation in cardiac myofilaments were compared with similar experimental data from the literature. The models were constructed by assuming different subsets of three putative cooperative mechanisms. Cooperative mechanism 1 holds that cross-bridge binding increases the affinity of troponin for Ca2+. In the models, cooperative mechanism 1 can produce steep force-Ca2+ (F-Ca) relations, but apparent cooperativity is highest at midlevel Ca2+ concentrations. During twitches, cooperative mechanism 1 has the effect of increasing latency to peak as the magnitude of force increases, an effect not seen experimentally. Cooperative mechanism 2 holds that the binding of a cross bridge increases the rate of formation of neighboring cross bridges and that multiple cross bridges can maintain activation of the thin filament in the absence of Ca2+. Only cooperative mechanism 2 can produce sarcomere length (SL)-dependent prolongation of twitches, but this mechanism has little effect on steady-state F-Ca relations. Cooperativity mechanism 3 is designed to simulate end-to-end interactions between adjacent troponin and tropomyosin. This mechanism can produce steep F-Ca relations with appropriate SL-dependent changes in Ca2+ sensitivity. With the assumption that tropomyosin shifting is faster than cross-bridge cycling, cooperative mechanism 3 produces twitches where latency to peak is independent of the magnitude of force, as seen experimentally.

Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart failure, II: model studies.

Ca2+ transients measured in failing human ventricular myocytes exhibit reduced amplitude, slowed relaxation, and blunted frequency dependence. In the companion article (O'Rourke B, Kass DA, Tomaselli GF, Kääb S, Tunin R, Marbán E. Mechanisms of altered excitation-contraction coupling in canine tachycardia-induced heart, I: experimental studies. Circ Res. 1999;84:562-570), O'Rourke et al show that Ca2+ transients recorded in myocytes isolated from canine hearts subjected to the tachycardia pacing protocol exhibit similar responses. Analyses of protein levels in these failing hearts reveal that both SR Ca2+ ATPase and phospholamban are decreased on average by 28% and that Na+/Ca2+ exchanger (NCX) protein is increased on average by 104%. In this article, we present a model of the canine midmyocardial ventricular action potential and Ca2+ transient. The model is used to estimate the degree of functional upregulation and downregulation of NCX and SR Ca2+ ATPase in heart failure using data obtained from 2 different experimental protocols. Model estimates of average SR Ca2+ ATPase functional downregulation obtained using these experimental protocols are 49% and 62%. Model estimates of average NCX functional upregulation range are 38% and 75%. Simulation of voltage-clamp Ca2+ transients indicates that such changes are sufficient to account for the reduced amplitude, altered shape, and slowed relaxation of Ca2+ transients in the failing canine heart. Model analyses also suggest that altered expression of Ca2+ handling proteins plays a significant role in prolongation of action potential duration in failing canine myocytes.

Cardiac sodium channel Markov model with temperature dependence and recovery from inactivation.

A Markov model of the cardiac sodium channel is presented. The model is similar to the CA1 hippocampal neuron sodium channel model developed by Kuo and Bean (1994. Neuron. 12:819-829) with the following modifications: 1) an additional open state is added; 2) open-inactivated transitions are made voltage-dependent; and 3) channel rate constants are exponential functions of enthalpy, entropy, and voltage and have explicit temperature dependence. Model parameters are determined using a simulated annealing algorithm to minimize the error between model responses and various experimental data sets. The model reproduces a wide range of experimental data including ionic currents, gating currents, tail currents, steady-state inactivation, recovery from inactivation, and open time distributions over a temperature range of 10 degrees C to 25 degrees C. The model also predicts measures of single channel activity such as first latency, probability of a null sweep, and probability of reopening.

Modeling the cellular basis of altered excitation-contraction coupling in heart failure.

Ca transients measured in failing human ventricular myocytes exhibit reduced amplitude and slowed relaxation [Beuckelmann, D.J., Nabauer, M., Erdmann, E., 1992. Intracellular calcium handling in isolated ventricular myocytes from patients with terminal heart failure. Circulation 85, 1046-1055; Gwathmey, J.K., Copelas, L., MacKinnon, R., Schoen, F.J., Feldman, M.D., Grossman, W., Morgan, J.P., 1987. Abnormal intracellular calcium handling in myocardium from patients with end-stage heart failure. Circ. Res. 61, 70-76; Kaab, S., Nuss, H. B., Chiamvimonvat, N., O'Rourke, B., Pak, P.H., Kass, D.A., Marban, E., Tomaselli, G.F., 1996. Ionic mechanism of action potential prolongation in ventricular myocytes from dogs with pacing-induced heart failure. Circ. Res. 78(2); Li, H.G., Jones, D.L., Yee, R., Klein, G.J., 1992. Electrophysiologic substrate associated with pacing-induced hert failure in dogs: potential value of programmed stimulation in predicting sudden death. J. Am. Coll. Cardiol. 19(2), 444-449; Vermeulen, J.T., McGuire, M.A., Opthof, T., Colonel, R., Bakker, J.M.T.d., Klopping, C., Janse, M.J., 1994. Triggered activity and automaticity in ventricular trabeculae of failing human and rabbit hearts. Cardiovasc. Res. 28, 1547-1554.] and blunted frequency dependence [Davies, C.H., Davia, K., Bennett, J.G., Pepper, J.R., Poole-Wilson, P.A., Harding, S.E., 1995. Reduced contraction and altered frequency response of isolated ventricular myocytes from patients with heart failure. Circulation, 92, 2540-2549; Hasenfuss, G., Reinecke, H., Studer, R., Meyer, M., Pieske, B., Holtz, J., Holubarsch, C., Posival, H., Just, H., Drexler, H., 1994. Relation between myocardial function and expression of sarcoplasmic reticulum Ca-ATPase in failing and nonfailing human myocardium. Circ. Res. 75, 434-442; Hasenfuss, G., Reinecke, H., Studer, R., Pieske, B., Meyer, M., Drexler, H., Just, H., 1996. Calcium cycling proteins and force-frequency relationships in heart failure. Basic Res. Cardiol. 91, 17-22; Monte, F.D., O'Gara, P., Poole-Wilson, P.A., Yacoub, M., Harding, S.E., 1995. Cell geometry and contractile abnormalities of myocytes from failing human left ventricle.

Model studies of the role of mechano-sensitive currents in the generation of cardiac arrhythmias.

Mechano-electrical feedback is studied by incorporating linear, instantaneously activating mechano-sensitive conductances into single cardiac cell models, as well as one- and two-dimensional cardiac network models. The models qualitatively reproduce effects of maintained mechanical stretch on experimentally measured action potential characteristics such as amplitude, maximum diastolic potential, peak upstroke velocity, and conduction velocity. Models are also used to simulate stretch-induced depolarizations, action potentials, and arrhythmias produced by pulsatile volume changes in left ventricle of dog. The mechano-sensitive conductance threshold for a stretch-induced action potential is closely related to the magnitude of the time-independent K+ current, IK1, which offsets inward mechano-sensitive current. Activation of mechano-sensitive conductances in small, spatially localized region of cells can evoke graded depolarizations, propagating ectopic beats, and if timed appropriately, spiral reentrant waves. Mechano-sensitive conductance changes required to evoke these responses are well within the physiologically plausible range. Results therefore indicate that many mechano-electrical feedback effects can be modeled using linear, instantaneously activating mechano-sensitive conductances. As an example of how stretch can occur in real human hearts, magnetic resonance images with saturation tagging are used to reconstruct the three-dimensional left ventricular wall motion. In patients with infarcts or recent ischemic events, "paradoxical deformation" is observed in that regions of myocardium are stretched rather than contracted during systole. In contrast, normal hearts contract uniformly with no stretch during systole. Paradoxical deformations in ischemic hearts may therefore present one possible substrate for the mechanically induced arrhythmias modeled above.

Cardiac Ca2+ dynamics: the roles of ryanodine receptor adaptation and sarcoplasmic reticulum load.

We construct a detailed mathematical model for Ca2+ regulation in the ventricular myocyte that includes novel descriptions of subcellular mechanisms based on recent experimental findings: 1) the Keizer-Levine model for the ryanodine receptor (RyR), which displays adaptation at elevated Ca2+; 2) a model for the L-type Ca2+ channel that inactivates by mode switching; and 3) a restricted subspace into which the RyRs and L-type Ca2+ channels empty and interact via Ca2+. We add membrane currents from the Luo-Rudy Phase II ventricular cell model to our description of Ca2+ handling to formulate a new model for ventricular action potentials and Ca2+ regulation. The model can simulate Ca2+ transients during an action potential similar to those seen experimentally. The subspace [Ca2+] rises more rapidly and reaches a higher level (10-30 microM) than the bulk myoplasmic Ca2+ (peak [Ca2+]i approximately 1 microM). Termination of sarcoplasmic reticulum (SR) Ca2+ release is predominately due to emptying of the SR, but is influenced by RyR adaptation. Because force generation is roughly proportional to peak myoplasmic Ca2+, we use [Ca2+]i in the model to explore the effects of pacing rate on force generation. The model reproduces transitions seen in force generation due to changes in pacing that cannot be simulated by previous models. Simulation of such complex phenomena requires an interplay of both RyR adaptation and the degree of SR Ca2+ loading. This model, therefore, shows improved behavior over existing models that lack detailed descriptions of subcellular Ca2+ regulatory mechanisms.