Stochastic binding of Ca²⁺ ions in the dyadic cleft; continuous vs Random walk description of diffusion

¹ Simula Research Laboratory
² Simula Reearch Laboratory

^* To whom correspondence should be addressed. E-mail: hake{at}simula.no.

Submitted on April 23, 2007
Revised on August 6, 2007
Accepted on 18 December 2007

	Abstract

Ca²⁺ signalling in the dyadic cleft in ventricular myocytes is fundamentally discrete and stochastic. We study the stochastic binding of single Ca ions to receptors in the cleft using two different models of diffusion: a stochastic and discrete Random Walk (RW) model, and a deterministic continuous model. We investigate whether the latter model, together with a stochastic receptor model, can reproduce binding events registered in fully stochastic RW simulations. By evaluating the continuous model goodness-of-fit, for a large range of parameters, we present evidence that it can. Further, we show that the large fluctuations in binding rate observed at the level of single time steps are integrated and smoothed at the larger time scale of binding events, which explains the continuous model goodness-of-fit. With these results we demonstrate that the stochasticity and discreteness of the Ca²⁺ signalling in the dyadic cleft, determined by single binding events, can be described using a deterministic model of Ca²⁺ diffusion together with a stochastic model of the binding events, for a specific range of physiological relevant parameters. Time-consuming RW simulations can thus be avoided. We also present a new analytical model of bi-molecular binding probabilities, which we use in the RW simulations and the statistical analysis.

Key Words: Monte Carlo, discrete, mathematical modelling, partial differential equation, ryanodine receptors, signalling micro domain