Encoded in CellML by Catherine Lloyd

Bioengineering Institute, University of Auckland

Model Status

This model has been curated by Penny Noble using Flavio Fenton's Java code as a reference (See http://thevirtualheart.org/ for Java applet rendering of model - Java code is available from Dr Fenton.) An artificial stimulus component has been added this model to allow it to reproduce the action potential simulation shown in Figure 4 of the publication. The model is known to run and integrate in the PCEnv and COR CellML environments. A PCEnv session file is also associated with this model.

ValidateCellML detects unit inconsistency within this model.

Model Structure

In contrast to the earlier Purkinje fibre ionic current models of D. Noble (1962) and R.E. McAllister, D. Noble and R.W. Tsien (1975), the G.W. Beeler and H. Reuter 1977 model was developed to describe the mammalian ventricular action potential. Not all the ionic currents of the Purkinje fibre model are present in ventricular tissue; therefore, this model is simpler than the MNT model. The total ionic flux is divided into only four discrete, individual ionic currents (see the figure below). The main additional feature of the Beeler-Reuter ionic current model is a representation of the intracellular calcium ion concentration.

The complete original paper reference is cited below:

Reconstruction of the action potential of ventricular myocardial fibres, Beeler, G.W. and Reuter, H. 1977, Journal of Physiology, 268, 177-210. PubMed ID: 874889

A schematic diagram describing the current flows across the cell membrane that are captured in the BR model.

The network defined in the CellML description of the Beeler-Reuter model. A key describing the significance of the shapes of the components and the colours of the connections between them is in the notation guide. For simplicity, not all the variables are shown.

The membrane physically contains the currents as indicated by the blue arrows in the figure above. The currents act independently and are not connected to each other. Several of the channels encapsulate and contain further components which represent activation and inactivation gates. The addition of an encapsulation relationship informs modellers and processing software that the gates are important parts of the current model. It also prevents any other components that aren't also encapsulated by the parent component from connecting to its gates, effectively hiding them from the rest of the model.

The breakdown of the model into components and the definition of encapsulation and containment relationships between them is somewhat arbitrary. When considering how a model should be broken into components, modellers are encouraged to consider which parts of a model might be re-used and how the physiological elements of the system being modelled are naturally bounded. Containment relationships should be used to provide simple rendering information for processing software (ideally, this will correspond to the layout of the physical system), and encapsulation should be used to group sets of components into sub-models.

The original paper reference is cited below:

Computational Mechanics of the Heart - From Tissue Structure to Ventricular Function, M. P. Nash, P. J. Hunter, 2000. Journal of Elasticity, Volume 61, Numbers 1-3, 113-141   DOI: 10.1023/A:1011084330767

A FieldML beta format demonstration of the model is available in the FieldML directory of this workspace.

To launch the model, please select 'Zinc Viewer' under navigation on the right.

To launch the model, please select 'Zinc Viewer' under navigation on the right.

With default setting the model produce isometric contraction (parameter isotonic=0). To switch from isomentric mode to isotonic mode isotonic parameter should be set 1. Then set parameter F_afterload to the value less then the maximum force calculated at isometric mode. Make sure that the model is run in isometric mode first and see the value of the maximum of isometric force. If F_afterload is not less than this value, model performs isometric mode instead of isotonic.

The version of the model used in this study actually deals only with one viscous element VS1, i.e. we assumed here the coefficient of viscosity of the element VS2 to be constantly equal to 0.

Model Structure

ABSTRACT: A mathematical model of the cardiomyocyte electromechanical function is used to study contribution of mechanical factors to rhythm disturbances in the case of the cardiomyocyte calcium overload. Particular attention is paid to the overload caused by diminished activity of the sodium-potassium pump. It is shown in the framework of the model, where mechano-calcium feedback is accounted for that myocardium mechanics may significantly enhance arrhythmogenicity of the calcium overload. Specifically, a role of cross-bridge attachment/detachment processes, a role of mechanical conditions of myocardium contractions (length, load), and a role of myocardium viscosity in the case of simulated calcium overload have been revealed. Underlying mechanisms are analyzed. Several approaches are designed in the model and compared to each other for recovery of the valid myocardium electrical and mechanical performance in the case of the partially suppressed sodium-potassium pump.

Panel I: Scheme of the ionic currents accounted for in the cardiomyocyte model. The currents modeled ( i_x) contribute to the development of action potential and/or calcium transient. Here i_CaL-Ca2+ influx via L-type Ca2+ channels; i_rel-Ca2+ release from terminal cisterns (TC) triggered by Ca2+ entered from dyadic space (DS); Ca_Trop-concentration of Ca2+ bound by specific troponin C to activate the contractile element (CE) from the rheological scheme (see Panel II); B_1, B_2-concentration of Ca2+ bound by a fast and a slow buffer; i_up -Ca2+ uptake by the sarcoplasmic reticulum pump into longitu-dinal reticulum (LR); i_tr -Ca2+ diffusion between LR and TC; CaS-concentration of Ca2+ complexes with calsequestrin; i_NaCa -sodium–calcium exchange current; i_bCa -background Ca2+ current. Panel II: rheological scheme of a single cardiomyocyte/homogeneous myocardium sample including contractile element CE, three passive elastic elements: parallel element PE , series element SE, extra-series element XSE and two viscous elements VS1 and VS2. The version of the model used in this study actually deals only with one viscous element VS1, i.e. we assumed here the coefficient of viscosity of the element VS2 to be constantly equal to 0.

The complete original paper reference is cited below:

Mathematical modeling of mechanically modulated rhythm disturbances in homogeneous and heterogeneous myocardium with attenuated activity of na+ -k+ pump Bulletin of Mathmatical Biology, Volume 70, Number 3, 910-949. PubMed ID: 18259823

The current CellML model implementation runs in OpenCell. The results have been validated against the original Matlab code.

Model Structure

This model presents a phenomenological ODE model of baroreflex open-loop control of heart rate. The signal flow of the model is diagrammed in Figure 1. An aortic blood pressure signal (the driving signal for the model) is transduced by afferent baroreceptor nerve fibers in the wall of the aortic arch into a neural (firing rate) signal. This transduction process is governed by mechanical strain in the wall of the aorta. This neural signal is then relayed and further processed by the central and peripheral nervous systems into parallel sympathetic and parasympathetic tone signals. Sympathetic and parasympathetic tones drive release of norepinephrine and acetylcholine, respectively, into the neuromuscular junction of the sinoatrial node of the heart, thereby modulating the pacemaker activity at the sinoatrial node. The sinoatrial node is the ultimate effector of heart rate, which is the model output.

Bugenhagen SM, Cowley AW Jr, Beard DA. Identifying physiological origins of baroreflex dysfunction in salt-sensitive hypertension in the Dahl SS rat. Physiological Genomics.42:23-41, 2010

This CellML model will not run in OpenCell as integration failed at t=0 and h=2.19671e-13, the corrector convergence test failed repeatedly or with |h| = hmin Need UNITS and INITIAL VALUES for concentrations

Model Structure

Mathematical models of the cell cycle can contribute to an understanding of its basic mechanisms. Modern simulation tools make the analysis of key components and their interactions very effective. This paper focuses on the role of small modules and feedbacks in the gene-protein network governing the G1/S transition in mammalian cells. Mutations in this network may lead to uncontrolled cell proliferation. Bifurcation analysis helps to identify the key components of this extremely complex interaction network. RESULTS: We identify various positive and negative feedback loops in the network controlling the G1/S transition. It is shown that the positive feedback regulation of E2F1 and a double activator-inhibitor module can lead to bistability. Extensions of the core module preserve the essential features such as bistability. The complete model exhibits a transcritical bifurcation in addition to bistability. We relate these bifurcations to the cell cycle checkpoint and the G1/S phase transition point. Thus, core modules can explain major features of the complex G1/S network and have a robust decision taking function.

The original paper reference is cited below:

'Bifurcation analysis of the regulatory modules of the mammalian G1/S transition', Maciej Swat, Alexander Kel,Hanspeter Herzel 2004 Bioinformatics, 10, 1506-1511. PubMed ID: 15231543

Image illustrating model 'Bifurcation analysis of the regulatory modules of the mammalian G1/S transition'

This CellML model will run in OpenCell but does not reproduce published results. This is the 20% Oxygen version, two more versions to follow.

Model Structure

Our small-scale bioreactor with a gas-permeable membrane has previously been shown to allow direct oxygenation of primary hepatocytes in collagenous sandwich cultures. In this work the applicability of this system for studying the response of hepatocytes to different ambient oxygen concentrations above the normoxic situation and the effect of the antioxidant vitamin C (ASC) on hepatocyte functionality in different oxygen cultures were evaluated. Determination of the parameters for functional levels (albumin and urea synthesis, biotransformation) combined with the model calculation of these parameters using a product inhibition model based on the concentration of hydrogen peroxide (H2 O2) indicated a relevance of oxygen levels for the maintenance of hepatic-specific functions. A model for the dynamics of oxidative stress was proposed to predict the time course of the qualitative concentrations of H2 O2 and the superoxide radical (O2 • − ) in different oxygen cultures. The 20% and 30% O2 cultures exhibited similar differentiated hepatic functions that were unequivocally lost in the 40% O2 cultures apparently due to oxidative damage. Metabolic functions in the 20% and 30% O2 cultures could be improved by ASC. These findings suggest that our bioreactor model not only allows a prediction of the liver response to variations in oxygen environments, but also serves as an in vitro tool to screen new compounds for antioxidant capacity.

The original paper reference is cited below:

'Improvement of metabolic performance of primary hepatocytes in hyperoxic cultures by vitamin C in a novel small-scale bioreactor, Stephanie Schmitmeier, Angelika Langsch, Wolfgang Schmidt-Heck, Inka Jasmund, Augustinus Bader 2007 Journal of Membrane Science, 298, 30-40.

Image illustrating model 'Improvement of metabolic performance of primary hepatocytes in hyperoxic cultures by vitamin C in a novel small-scale bioreactor '

This CellML model should work in OpenCell and COR to reproduce published results. The a model is based on equations 2 (wihout mitochondria) and the b model is based on equatios 2 and 3 (with mitochondria). Six other versions are as follows: 2a and 2b = equations 3 and 4, 3a and 3b = equations 5 and 6, 4a and 4b = equations 7 and 8. Does not reproduce published results but runs in OpenCell.

Model Structure

In a mathematical model for simple calcium oscillations [Biophys. Chem. 71 (1998) 125], it has been shown that mitochondria play an important role in the maintenance of constant amplitudes of cytosolic Ca[2+] oscillations. Simple plausible rate laws for [Ca+] fluxes across the inner mitochondrial membrane have been used in this model. Here we show that it is possible to use the same rate laws as a plug-in element in other existing mathematical models and obtain the same effect on amplitude regulation. This result appears to be universal, independent of the type of model and the type of Ca oscillations. We demonstrate this on two models for spiking Ca oscillations [J. Biol. Chem. 266 (1991) 11068; Cell Calcium 14 (1993) 311] and on two recent models for bursting Ca oscillations; one of them being a receptor-operated model [Biophys. J. 79 (2000) 1188] and the other one being a store-operated model [BioSystems 57 (2000) 75].

The original paper reference is cited below:

'Mitochondria regulate the amplitude of simple and complex calcium oscillations', Vladimir Grubelnik, Ann Zahle Larsen, Ursula Kummer, Lars Folke Olsen, Marko Marhl, 2001 Biophysical Chemistry, 94, 59-74. PubMed ID: 11744191

Schematic Diagram of Grubelnik model