Multiscale modelling of coupled Ca2+ channels using coloured stochastic Petri nets

Abstract

Stochastic modelling of coupled Ca2+ channels is a challenge, especially when the coupling of the channels, as determined by their spatial arrangement relative to each other, has to be considered at multiple spatial scales. In this study, the authors address this problem using coloured stochastic Petri nets (SPNc) as high-level description to generate continuous-time Markov chains. The authors develop several models with increasing complexity. They first apply SPNc to model single clusters of coupled Ca2+ channels arranged in a regular or irregular lattice, where they describe how to represent the geometrical arrangement of Ca2+ channels relative to each other using colours. They then apply this modelling idea to construct more complex models by modelling spatially arranged clusters of channels. The authors' models can be easily reproduced and adapted to different scenarios.

Fig. 1

Two‐state model of Ca²⁺ ‐activated Ca²⁺ channels

Fig. 2

Two coupled Ca²⁺ channels assuming two states for each channel. ‘O’ denotes ‘open’ and ‘C’ ‘closed’

Fig. 3

Stochastic Petri net models a For the CTMC of a two‐state model of one Ca²⁺ channel in Fig. 1 b For the CTMC of the coupled two‐state models of two Ca²⁺ channels in Fig. 2. A place connected by a modifier arc (dotted line) is used for the rate function of the transition connected by this arc. The mappings of parameters are as follows: k _p = k ⁺, c _i = c_∞ , m = η, k _m = k⁻ , c _12 = c ₁₂, c _21 = c ₂₁

Fig. 4

Two equivalent coloured Petri net models that are built based on Fig. 3b with the mean field assumption a The all () function on the arc (Open, Associate) represents the effect of all open channels on one channel. #(Open) in the rate function of Associate returns the number of open channels b A new place NumOpen is used to explicitly obtain the number of open channels, which is then used in the rate function of Associate. The declarations: constant N = 2; colorset Dot = Dot with dot; CS = int with 1–N; variable x : CS. all () on place Closed is an initial marking specification function, which means that all the colours in a colour set are used to set the initial marking with the same coefficient (here it is 1)

Fig. 5

Arrangements of coupled Ca²⁺ channels in a 3 × 3 grid (M = 3, N = 3) a CS = CS _Rec, a channel is allocated to each cell of the grid (marked by coordinates) b CS = CS _Cir4, channels are only positioned in the middle circular region, that is, five cells with coordinates c CS = CS _Ir5, channels are positioned in an irregular region, that is, five cells with coordinates

Fig. 6

${\mathcal{S}\mathcal{P}\mathcal{N}}^{\mathcal{C}}$ models for two coupled Ca²⁺ channels arranged in a 2D grid With M = 1, N = 2 and CS = CS _Rec, these models correspond to the ones in Fig. 4. See Table 1 for all declarations. By adjusting the values of M and N and defining a subset of CS _Rec for a regular or irregular arrangement, we can model Ca²⁺ channels of any number and any arrangement

Fig. 7

${\mathcal{S}\mathcal{P}\mathcal{N}}^{\mathcal{C}}$ model of a 2D grid of weakly coupled clusters, each containing a 2D grid of strongly coupled Ca²⁺ channels See Table 2 for all declarations

Fig. 8

Simulation plots of one simulation run for the model of an array of clusters in Fig. 7 Declarations are shown in Table 2. Parameters: η = 2, c_∞ = 0.05, k ⁺ = 1.5, k⁻ = 0.5, c * = 0.1 a Ignoring (f_s = 0) the effect of neighbouring clusters b Considering (f_s = 0.0078) the effect of neighbouring clusters