Duality for dyadic intervals

In an earlier paper, Romanowska, Ślusarski and Smith described a duality between the category of (real) polytopes (finitely generated real convex sets considered as barycentric algebras) and a certain category of intersections of hypercubes, considered as barycentric algebras with additional constant operations. This paper is a first step in finding a duality for dyadic polytopes, analogues of real convex polytopes, but defined over the ring [Formula: see text] of dyadic rational numbers instead of the ring of reals. A dyadic [Formula: see text]-dimensional polytope is the intersection with the dyadic space [Formula: see text] of an [Formula: see text]-dimensional real polytope whose vertices lie in the dyadic space. The one-dimensional analogues are dyadic intervals. Algebraically, dyadic polytopes carry the structure of a commutative, entropic and idempotent groupoid under the operation of arithmetic mean. Such dyadic polytopes do not preserve all properties of real polytopes. In particular, there are infinitely many (pairwise non-isomorphic) dyadic intervals. We first show that finitely generated subgroupoids of the groupoid [Formula: see text] are all isomorphic to dyadic intervals. Then, we describe a duality for the class of dyadic intervals. The duality is given by an infinite dualizing (schizophrenic) object, the dyadic unit interval. The dual spaces are certain subgroupoids of the square of the dyadic unit interval with additional constant operations. A second paper deals with a duality for dyadic triangles.

Calcium Sparks in Cardiac Cells in Silico

We simulate elementary calcium release events (sparks) in a single calcium release unit in ventricular myocyte. Previously developed and tested electron-conformational model of the stochastic dynamics of RyR-channels is integrated to the calcium dynamics model in the cardiac cell. This approach allows to observe RyRs opening/closing in details on the macromolecular level during the calcium dynamics course. We simulate calcium diffusion in the dyadic space and “domino-like” RyR’s activation during the so-called “calcium induced-calcium release process”. Ca2+ sparks initiation, spread and termination are investigated in the computer experiments. Sparks’ initiation and termination rate dependence on the Ca2+ diffusion velocity is observed. We show that sarcoplasmic reticulum lumen local depletion and RyR’s stochastic attrition could be the reasons of Ca2+ spark termination.

Effect of Ca2+Efflux Pathway Distribution and Exogenous Ca2+Buffers on Intracellular Ca2+Dynamics in the Rat Ventricular Myocyte: A Simulation Study

We have used a previously published computer model of the rat cardiac ventricular myocyte to investigate the effect of changing the distribution of Ca2+efflux pathways (SERCA, Na+/Ca2+exchange, and sarcolemmal Ca2+ATPase) between the dyad and bulk cytoplasm and the effect of adding exogenous Ca2+buffers (BAPTA or EGTA), which are used experimentally to differentially buffer Ca2+in the dyad and bulk cytoplasm, on cellular Ca2+cycling. Increasing the dyadic fraction of a particular Ca2+efflux pathway increases the amount of Ca2+removed by that pathway, with corresponding changes in Ca2+efflux from the bulk cytoplasm. The magnitude of these effects varies with the proportion of the total Ca2+removed from the cytoplasm by that pathway. Differences in the response to EGTA and BAPTA, including changes in Ca2+-dependent inactivation of the L-type Ca2+current, resulted from the buffers acting as slow and fast “shuttles,” respectively, removing Ca2+from the dyadic space. The data suggest that complex changes in dyadic Ca2+and cellular Ca2+cycling occur as a result of changes in the location of Ca2+removal pathways or the presence of exogenous Ca2+buffers, although changing the distribution of Ca2+efflux pathways has relatively small effects on the systolic Ca2+transient.

Activation of calcium release assessed by calcium release-induced inactivation of calcium current in rat cardiac myocytes

In mammalian cardiac myocytes, calcium released into the dyadic space rapidly inactivates calcium current ( ICa). We used this Ca2+ release-dependent inactivation (RDI) of ICa as a local probe of sarcoplasmic reticulum Ca2+ release activation. In whole cell patch-clamped rat ventricular myocytes, Ca2+ entry induced by short prepulses from —50 mV to positive voltages caused suppression of peak ICa during a test pulse. The negative correlation between peak ICa suppression and ICa inactivation during the test pulse indicated that RDI evoked by the prepulse affected only calcium channels in those dyads in which calcium release was activated. Ca2+ ions injected during the prepulse and during the subsequent tail current suppressed peak ICa in the test pulse to a different extent. Quantitative analysis indicated that equal Ca2+ charge was 3.5 times less effective in inducing release when entering during the prepulse than when entering during the tail. Tail Ca2+ charge injected by the first voltage-dependent calcium channel (DHPR) openings was three times less effective than that injected by DHPR reopenings. These findings suggest that calcium release activation can be profoundly influenced by the recent history of L-type Ca2+ channel activity due to potentiation of ryanodine receptors (RyRs) by previous calcium influx. This conclusion was confirmed at the level of single RyRs in planar lipid bilayers: using flash photolysis of the calcium cage NP-EGTA to generate two sequential calcium stimuli, we showed that RyR activation in response to the second stimulus was four times higher than that in response to the first stimulus.