Fig 1.
Reaction scheme and IP3R model.
The biochemical processes included in the model are illustrated in (A). Cytosolic calcium can exit the cytosol to the extracellular space or the endoplasmic reticulum (ER) at a (total) rate α, lumping together the effects of ER and plasma membrane pumps. Likewise, Ca2+ can enter the cytosol from the extracellular space or from the ER via IP3R-independent flow, with (total) rate γ, emulating calcium channels from the plasma membrane. When an IP3R channel opens, calcium enters the cytosol through the channel at rate μ. Phospholipase Cδ (PLCδ), once activated by calcium binding, produces IP3 at rate δ. Like Ca2+, IP3 can bind the IP3R channel and is removed with rate β. (B) Our model of the kinetics of the IP3R channel is an 8-state Markov model adapted from [46, 56]. Each IP3R channel monomer is associated with 3 binding sites, two calcium binding sites and one IP3 binding site. Occupancy states are designated by a triplet {i, j, k} where i stands for the occupation of the first Ca binding site (i = 1 if bound, 0 else), j for that of the IP3 binding site and k for the second Ca site. The first calcium binding site has higher affinity than the second. The open state is state {110}, where the first Ca and the IP3 sites are bound but not the second Ca site. (C) Spatial parameters for the particle-based model. The molecules are positioned within uniformly distributed clusters, with η IP3R in each cluster. Hence η = 1 corresponds to uniformly distributed IP3R (no clustering), while the degree of clustering increases with η (for constant total IP3R number). To account for potential co-localization between IP3R-dependent and IP3R-independent calcium sources, the influx of IP3R-independent calcium (at rate γ) occurs within distance Rγ of an IP3R. Thus, low values of Rγ emulate co-localization between IP3R-dependent and IP3R-independent Ca2+ influx sources.
Table 1.
Parameter values and initial conditions of the 2D model.
a.u: arbitrary unit. In 2d, by definition, a MC time unit is 100 Δt and one MC space unit is set by the interaction radius of IP3R, i.e. space unit. δ, β, μ, γ, b1, b2 and b3 are first order constants, in (MC time unit)−1. Diffusion coefficients DCa and DIP3 are expressed in (MC space unit)2.(MC time unit)−1 whereas α, a1, a2, a3 are expressed in (MC space unit)2.(MC time unit)−1.
Fig 2.
(A) Spontaneous transients are observed in simulations of the particle-based and the Gillespie’s SSA model but not in the Mean Field model. (B) The three models display the same basal calcium level when μ, the calcium influx rate through open IP3R channels, increases. The higher variability in the stochastic models reflects the integer value of basal calcium (either 49 or 50, depending on simulations). (C) Quantification of calcium transients in the stochastic models (calcium peak frequency and mean peak amplitude). No significant difference between the two models was observed. (D) Excitability of the Mean-Field model: increasing quantities of exogenous IP3 molecules were injected at time t = 20Δt, after model equilibration. The amplitude of the resulting calcium response (D1) was quantified depending on the amount of IP3 injected and the value of the binding rate constant to the first calcium IP3R site, a1 (D2). Parameter values for the particle-based model: DCa = DIP3 = ∞ (perfect mixing) and η = 1, Rγ = 200, i.e. no IP3R channels clustering, and no co-localization of IP3R with IP3R-independent Ca2+ sources. For SSA and particle-based models, the figure shows the average ± standard deviation over 20 simulations.
Fig 3.
The particle-based model produces different calcium activity regimes depending on parameter values.
Color-coded map of variation of the peak frequency, expressed as the number of calcium peaks per MC time step (A) and as the maximal number of IP3R channel open per peak (B). The color scale is given for each map. The black area corresponds to the stationary regime. Note that the x and y-axis scales in (A) and (B) are not regularly spaced. The symbols ★, ◼ and ● locate parameter pairs that are illustrative of the three dynamical regimes shown in (C): stationary (★, μ = 5, a1 = 0.5), blips (◼, μ = 25, a1 = 2) and puffs (●, μ = 75, a1 = 7). Red crosses show the locations of peaks from automatic detection. DCa = DIP3 = ∞, η = 1, Rγ = 200.
Fig 4.
Ca2+ diffusion modulates the temporal characteristics of the signals upon co-localization.
Representative simulations of the particle-based model showing both calcium trace and number of open IP3R for co-localized calcium sources (Rγ = 0) in the case of slow calcium diffusion (A) or perfect-mixing of calcium (B). The red crosses show peak locations from automatic detection. The impact of calcium diffusion coefficient DCa on peak frequency (C) and the amount of puff (D) are shown for different values of the co-localization parameter Rγ: from Rγ = 0 (IP3R are not clustered but co-localized with other calcium sources) to Rγ = 100 (IP3R are neither clustered nor co-localized). The puff ratio quantifies the fraction of peaks that are puffs. (E) and (F) respectively present the probabilities that IP3R closure results from binding of a Ca2+ to the inactivating site (probability to switch to state {111}, P110−>111) or unbinding of an IP3 (probability to switch to state {100}, P110−>100) depending on DCa and on Rγ. Probability of closure due to Ca2+ unbinding from activating site, P110−>010 can be deduced from 1 = P110−>010 + P110−>100+P110−>111. Data are presented as mean ± standard deviation over 20 simulations. Lines are guide for the eyes. Note that the x-axis scale in (C), (D), (E) and (F) is not regularly spaced. Other parameters: η = 1 (no clustering), a1 = 1 a.u, μ = 50 a.u.
Fig 5.
IP3R clustering modulates calcium signals when co-localized.
Representative simulations of the particle-based model with the corresponding IP3R distribution over space, the calcium trace and number of open IP3R for weakly co-localized calcium sources (Rγ = 10) in the case of uniform distribution of the IP3R (A) or strongly clustered IP3R (B) are illustrated. The red crosses show peak locations from automatic detection. The impact of IP3R cluster size η on calcium peak frequency (C) and on the amount of puffs (D) are shown for different values of the cluster size: from η = 1 (IP3R are not clustered) to η = 50 (strong clustering). Data are presented as mean ± standard deviation over 20 simulations. Lines are guide for the eyes. Other parameters: DCa = 0.1 a.u, a1 = 1 a.u, μ = 50 a.u.
Fig 6.
3d model simulations in fine astrocyte processes successfully reproduce calcium microdomains signals.
(A) Experimental monitoring of the spontaneous local Ca2+ signals in astrocytic sponge-like processes. Panel A1 shows a ‘summed projection’ of a confocal time lapse image stack of a GCaMP6s-expressing astrocyte. Panel A2 illustrates magnification of the boxed region of panel A1. Panel A3 displays spontaneous calcium traces from the regions of interest shown in (A2). (B) The 3d geometry used for the 3D model is a cylinder of length Lastro = 1 μm and radius Rastro = 0.1 μm, with ER as a thinner cylinder inside. The interior volume is roughly 0.03 fL. (C) Representative simulations of calcium dynamics within the above cylinder with the “No-GCaMP”, “GCaMP” and “GC+Buf” simulations. The raw signal corresponds to cytosolic free calcium concentration for the “No-GCaMP” model and to calcium-bound GCaMP concentration for “GCaMP” and “GC+Buf” models. For all simulation types, parameter values were partly taken from the literature and partly adjusted for fitting calcium traces shown in A (reported in Table 2). (D) Quantitative comparisons of the spontaneous calcium signals measured experimentally (black bars) or simulated with the “No-GCaMP”, “GCaMP” or “GC+Buf” models (white bars). The compared quantities are peaks amplitude in terms of ΔF/F ratio (D1), their frequency (measured in min−1 for each μm2 area, D2) and duration (expressed as full width at half maximum, FWHM) D3. Significance is assigned by * for p ≤ 0.05, ** for p ≤ 0.01, *** for p ≤ 0.001.
Fig 7.
The kinetics and concentration of GECIs strongly influence calcium dynamics.
(A) Quantitative comparisons of the spontaneous calcium signals measured with “GCaMP6s” or “GCaMP6f” as fluorescent reporters. The compared quantities are peak amplitude in terms of ΔF/F ratio (A1), frequency (measured in min−1 for each μm2 area, A2) and duration (expressed as full width at half maximum, FWHM) (A3). (B) The impact of the concentration of GCaMP6s in the system on basal concentration of GCaMP-Ca (B1), on the GCaMP-Ca peak amplitude (B2), frequency (B3) and duration (B4) are shown for different values of [GCaMP6s]. Significance is assigned by * for p ≤ 0.05, ** for p ≤ 0.01, *** for p ≤ 0.001. Data are presented as mean ± standard deviation over 20 simulations. Lines are guide for the eyes.
Table 2.
Parameter values and initial conditions of the 3d model.
The parameter values for the 3d model listed here correspond to the “GCaMP” model. The parameter values for the “No-GCaMP” 3d model are the same except that GCaMP6s concentration equals 0 nM. In the “GCaMP6f” model variant, Gcampf = 1.05 × 107 M−1.s−1 and Gcampb = 3.93s−1. Parameter values in the 3d model have been adjusted to optimize the match with experimental data as described in the Methods section. Note that the values for calcium and IP3 binding or unbinding to IP3R, i.e. the ai’s and bj’s parameters below, are smaller in our model than in the literature, probably because our model is not cooperative. For GCaMP6s and GCaMP6f, we used the diffusion coefficient of calmodulin. The initial number of Ca2+ ions was adjusted so that the measured basal GCaMP6s-Ca concentration was around 300nM [86, 124].