Mean field strategies induce unrealistic non-linearities in calcium puffs

Abstract

Mean field models are often useful approximations to biological systems, but sometimes, they can yield misleading results. In this work, we compare mean field approaches with stochastic models of intracellular calcium release. In particular, we concentrate on calcium signals generated by the concerted opening of several clustered channels (calcium puffs). To this end we simulate calcium puffs numerically and then try to reproduce features of the resulting calcium distribution using mean field models were all the channels open and close simultaneously. We show that an unrealistic non-linear relationship between the current and the number of open channels is needed to reproduce the simulated puffs. Furthermore, a single channel current which is five times smaller than the one of the stochastic simulations is also needed. Our study sheds light on the importance of the stochastic kinetics of the calcium release channel activity to estimate the release fluxes.

Keywords: 02.30.Jr; 82.33.-z; 87.16.A-; 87.16.Vy; 87.17.-d; calcium signals; mean field model; puffs; stochastic model.

Figure 1

Puff current and fluorescence time course for the stochastic and mean field models. (A,B) Stochastic puff model: simulations of the stochastic puff model for a cluster of 15 IP₃Rs (A) and 33 IP₃Rs (B). The spatial distribution of channels is shown in the upper square plots (500 nm × 500 nm). The lower plots show the time evolution of the total current released and the resulting fluorescence signal. Red curves correspond to simulations in which the channels follow the CD model and blue lines, CI model. (C,D) Mean field puff models: simulations of the mean field puff models. In (C) the simulations of the linear (green) and the non-linear (red) mean field models are such that the Ca²⁺ current is ∼0.25 pA. The number of channels are 3 in the linear mean field and 17 in the non-linear mean field model. In (D) the release current is ∼0.5 pA and the number of channels is 6 (linear model) and 33 (non-linear model). The resulting fluorescence time course is shown in the bottom plot and the distribution of channels over a 500-nm × 500-nm region is shown in the upper plot.

Figure 2

Non-linear relationship between the amplitude and the release current. Puff amplitude as function of the maximum released current for experimental puffs (adapted from Bruno et al., 2010) and puffs simulated with the stochastic, linear and non-linear models. In the simulations of the stochastic puff model, IP₃Rs follow the CD kinetic mode.

Figure 3

Non-linearities as a function of the number of available channels. (A) Puff amplitude as a function of the number of available channels, N_p for puffs simulated with the stochastic model and the CD IP₃R kinetic model. (B) Average Ca²⁺ current (Equation 5) released during a puff simulated using the stochastic model as a function of the number of available channels. The black line corresponds to the best fit to a non-linear function of the form I = I₀₁N_p for N_p ≤ 22 and $I=I_{02}\sqrt{N_p}$ for N_p > 22 which gives I₀₁ = 0.019 pA and I₀₂ = 0.105 pA. I₀₁ should not identified as the actual single IP₃R current but as an effective value that follows from the assumption that all channels open and close simultaneously.

Figure 4

The semi-analytical puff model supports the results of the stochastic puff model. (A) Average current as a function of N_p for the semi-analytical model and for simulated puffs obtained using the stochastic puff model with the CD IP₃R kinetic model. (B) Effective single channel current as a function of N_p for puffs simulated with the stochastic model using the CD IP₃R kinetic model. In (A,B), black circles and squares correspond to the semi-analytical model using the CD and the CI kinetic model, respectively. Small red squares and error bars represent mean and SD for puffs simulated with the stochastic model.