Robert Hinch

A Mathematical Analysis of the Generation and Termination of Calcium Sparks

Calcium sparks are local regenerative releases of Ca(2+) from a cluster of ryanodine receptors on the sarcoplasmic reticulum. During excitation-contraction coupling in cardiac cells, Ca(2+) sparks are triggered by Ca(2+) entering the cell via the T-tubules (Ca(2+)-induced Ca(2+) release). However under conditions of calcium overload, Ca(2+) sparks can be triggered spontaneously. The exact process by which Ca(2+) sparks terminate is still an open question, although both deterministic and stochastic processes are likely to be important. In this article, asymptotic methods are used to analyze a single Ca(2+) spark model, which includes both deterministic and stochastic biophysical mechanisms. The analysis calculates both spark frequencies and spark duration distributions, and shows under what circumstances stochastic transitions are important. Additionally, a model of the coupling of the release channels via the FK-binding protein is analyzed.

An analytical study of the physiology and pathology of the propagation of cardiac action potentials

Many cardiac diseases are caused by the abnormal propagation of electrical waves. Previous experimental and modelling work is reviewed, then a detailed study of the mathematics of cardiac propagation is presented. Pathologies are examined in the context of the models by varying parameters in the models to mimic different pathological states. Ionic models of cells are simplified to form analytically tractable models of the propagation of electrical cardiac waves. The roles that sodium channel activation and inactivation play in determining the conduction velocity are studied in detail, and the roles of resting potential currents in conduction block are calculated. The effect of curvature on the conduction velocity is examined, and the conditions in which curvature leads to conduction block and fibrillation are discussed. Hyperkalaemia (important during ischaemia) is modelled, and the model correctly describes the bi-phasic relation between propagation velocity and extracellular potassium.

Mechanisms and Models of Cardiac Excitation-Contraction Coupling

Intracellular calcium (Ca2+) concentration plays an important regulatory role in a number of cellular processes. Cellular influx of Ca2+ activates intracellular signaling pathways that in turn regulate gene expression. Studies have identified over 300 genes and 30 transcription factors which are regulated by intracellular Ca2+ [1,2]. Fluctuation of intracellular Ca2+ levels is also known to regulate intracellular metabolism by activation of mitochondrial matrix dehydrogenases. The subsequent effects on the tri-carboxylic acid cycle increase the supply of reducing equivalents (NADH, FADH2), stimulating increased flux of electrons through the respiratory chain [3]. Most importantly, Ca2+ is a key signaling molecule in excitation-contraction (EC) coupling, the process by which electrical activation of the cell is coupled to mechanical contraction and force generation.

Functional significance of Na+/Ca2+ exchangers co-localization with ryanodine receptors.

Abstract:  Co‐localization of Na+/Ca2+ exchangers (NCX) with ryanodine receptors (RyRs) is debated. We incorporate local NCX current in a biophysically detailed model of L‐type Ca2+ channels (LCCs) and RyRs and study the effect of NCX on the regulation of Ca2+‐induced Ca2+ release and the shape of the action potential. In canine ventricular cells, under pathological conditions, e.g., impaired LCCs, local NCXs become an enhancer of sarcoplasmic reticulum release. Under such conditions incorporation of local NCXs is critical to accurately capture mechanisms of excitation–contraction coupling.

MODELING OF WOLFF PARKINSON WHITE SYNDROME

Wolff–Parkinson–White syndrome is a disease where an arrhythmia is caused by the ventricles being electrically excited by an additional accessory pathway that links the atria to the ventricles. The spread of the activation wave from this pathway to the ventricles is modeled using a simplified model of Hodgkin–Huxley sodium channel kinetics, in a two ion-channel model. The model is investigated both analytically (using an asymptotic analysis) and numerically, and both methods are shown to give the same result. It is found that for a given width of the accessory pathway, there is a critical sodium channel density needed for the activation wave to spread from the pathway to the tissue. This result provides an explanation for the success of class-I anti-arrhythmic drugs in treating Wolff–Parkinson–White syndrome.