Fig 1.
Schematic illustration of the stochastic shielding approximation, for a graph representing 24 transitions (directed edges) interconnecting eight states (vertices).
One of the states (black disk) is distinguishable from the rest (white disks). For example, the black disk could represent a conducting ion channel state, while the white disks could represent non-conducting states. Left: Numerical simulation of the full process is computationally expensive: each blue trace superimposed on an edge represents independently generated stochastic forcing, but not all edges make significant contributions to fluctuations in the state of interest. Right: Rather than simulate the full process, the stochastic shielding approximation reduces the number of independent noise sources (blue edges) used to drive the stochastic process on the graph, while preserving the dynamical behavior of a particular projection of the random process.
Fig 2.
(A) Colquhoun & Hawkes’ five-state model for the nicotinic acetylcholine (ACh) receptor [30]. White disks: closed (non-conducting) states (Mj = 0). Gray disks: open (conducting) states (Mj = 1). Nodes 1-5 (large black numbers) are defined in Table 1. Transitions 1-10 (small blue numbers) are defined in Table 2. The opening of the channel requires the binding of acetylcholine. Transitions 2, 6, and 10 are driven by ACh concentration. Transitions 3, 4, 7, 8 are directly observable through a conductance change. (B) Timescale separation (ratio of non-zero eigenvalues of the graph Laplacian) as a function of ACh concentration. (C) Edge importance Rk for k ∈ {1, …, 10} for each edge in the graph as a function of [ACh], see Eq 42. (D) Sample trace of the model exhibiting burstiness of the channel for low agonist concentration, here [ACh] = 0.5μM. (E) Zoomed in version of the burst in (D) labeled by the red arrow.
Table 1.
Colquhoun & Hawkes’ five-state model for the nicotinic acetylcholine receptor [30], cf. Fig 2A.
Definition of the states and the measurement vector M (normalized conductance). Mi = 1 means that state i is open (conducting/observable) and Mi = 0 means that state i is closed (non-conducting/non-observable).
Table 2.
Colquhoun & Hawkes’ five-state model for the nicotinic acetylcholine receptor [30], cf. Fig 2A.
Definition of the edges and the transition rates αij. The acetylcholine concentration is c ≥ 0. Bold font denotes edges with [ACh]-dependent transition rates.
Fig 3.
An illustration of the general 3-state chain with per capita transition rates αk for k = 1, 2, 3, 4.
State 3 (black) is the observable state (or open/conducting state) of the system, and all other states are not observable (or closed/non-conducting states). By convention, we identify α1 ≡ α12, α2 ≡ α21, α3 ≡ α23, and α4 ≡ α32.
Fig 4.
Stochastic shielding and the power spectrum when all transition rates equal unity.
Panel A shows that the majority of the power comes from the observable edges (red dashed line), as expected from the edge importance measure and the stochastic shielding method. Black line is the total power spectral density (S) for the observed process X, red dashed line is the PSD (S3,4) for the approximate process X3,4 with noise from observable edges preserved, blue dashed line is PSD (S1,2) for the approximation X1,2 with noise from hidden edges preserved. Panel B shows trajectories (Gaussian version of the model) of the full observed process with all noise sources included (black trace), the approximate process with noise preserved on the observable edges (red), and the approximate process with noise preserved on the hidden edges (blue). The red trace closely follows the black trace, whereas the blue trace only tracks mean behavior and misses most fluctuations; X3,4 is the best approximation of X, in agreement with the stochastic shielding method.
Table 3.
Detailed description of each case for the 3-state model.
The first seven cases correspond to the chain that has the third state observable (see Fig 3). The last five cases have the middle state as the observable state. Transition rates αk for k = 1, 2, 3, 4 are given by columns 3-6. The final column shows the characterization of each case into one of six different types determined by their edge importance graph as a function of parameter α (see right most column in Figs 5 and 6).
Fig 5.
All possible cases for the 3-state chain with state 3 conducting.
Left column: 3-state diagram with accelerated/decelerated edges labeled 1 or α where α ∈ [10−4, 104]. Middle column: logarithm of the ratio of the two non-zero eigenvalues (λ2/λ3) versus α. This shows the “timescale separation” (present when λ2/λ3 ≪ 1). Right column: relative edge importance Rk versus α.
Fig 6.
All possible cases for the 3-state chain with middle state 2 conducting.
In this case there are no hidden transitions, and hence no stochastic shielding effect. Left column: 3-state diagram with accelerated/decelerated edges labeled as 1 or α where α ∈ [10−4, 104]. Middle column: logarithm of ratio of the two non-zero eigenvalues (λ2/λ3) versus α. This shows the timescale separation (present when λ2/λ3 ≪ 1). Right column: relative edge importance Rk versus α.
Fig 7.
Factors contributing to edge importance reversal.
The relative importance due to the hidden edges, η = R12/(R12 + R23), was calculated for an ensemble of 3-state chains (100,000 samples, see text for details). Relative edge importance is inverted when η > 0.5. Left column shows η plotted versus stationary occupancy probability of node 3 (π3, panel A), node 2 (π2, C), and the ratio of nodes 2 to 3 (π2/π3, E). The corresponding plot for π1 appears similar to that for π3 (not shown). Edge importance can be inverted for any values of π1 and π3, but requires π2 ≲ 1/6. Right column shows η plotted versus timescale separation (ν = λ3/λ2, B), relative fraction of flux generated by the hidden edges (ΔJ = (J12 − J23)/(J12 + J23), D), and ratio of relaxation times for isolated 2-state systems corresponding to the hidden versus observable transitions (τ12/τ23, F). Edge importance reversal requires timescale separation (|λ3| ≳ 15|λ2| or ν ≳ 15), larger mean flux along the observable edges than the hidden edges (J23 > J12), and faster relaxation along the visible edges than along the hidden edges (τ12 > τ23). None of these conditions alone are sufficient. However, panel G shows η versus the two factors F1 and F2 in the exact expression for η, black ‘+’ line is F1 ⋅ F2 (see Eq 22).
Fig 8.
Stochastic shielding and the power spectrum in the 3-state chain for the case where edge importance is reversed (α = 10 shown here).
Panel A shows that the majority of the power comes from the hidden edges (blue) for low frequencies, but the red and blue curves cross at ω ≈ 3, so for high frequencies the majority of the power comes from the observable edges (red). This switch in dominant spectral contributions is reflected in the corresponding Gaussian model trajectories shown in Panel B with the blue trace closely following the black trace, and the red trace deviating from the black trace. This shows that X1,2 best approximates the full process X in this case (whereas X3,4 is the best approximation in the uniform transition rate case shown in Fig 4).
Fig 9.
Stochastic shielding and the power spectrum in the acetylcholine receptor model.
For a high concentration ([ACh] = 100 μM shown here), edge importance is not reversed. Panel A shows that the majority of the power comes from the visible edges at all frequencies (S3,4, red trace), in agreement with the usual edge importance ranking. The corresponding Gaussian model trajectories shown in Panel B illustrate that X3,4 is the best approximation to X, whereas the other approximation only captures average behavior.
Fig 10.
Stochastic shielding and the power spectrum in the acetylcholine receptor model.
For a low concentration ([ACh] = 0.5 μM shown here), edge importance is reversed. Panel A shows that the majority of the power comes from the hidden edges at low frequencies (S5,6, blue trace) and from the visible edges at high frequencies (S3,4, red trace), just as we saw in the 3-state model under the case of edge importance reversal. The corresponding Gaussian model trajectories shown in Panel B illustrate that X5,6 (blue) is the best approximation to X, although it misses some the fluctuations captured by X3,4 (red).