Parameter inference for stochastic reaction models of ion channel gating from whole-cell voltage-clamp data

Figures

• 

  Figure 1.. (a): a cell in solution with its membrane voltage Vm clamped to a voltage protocol Vcmd that may change over time (b, top). As Vm varies, the channels change their conformation xx, e.g. Closed (C) – Open (O) (b, middle), thus increasing or decreasing the flow of ions through them. The ionic current yy=Iion (b, bottom) is measured with a voltage-clamp amplifier acting as a membrane current ammeter. The stochastic model of ion channel gating is written in a state-space formulation (d), where the dynamics are described by stochastic reaction networks, as shown in plot panel (c) for the simplistic case of a C–O model. The unknown parameters are estimated by optimizing the log-likelihood (d), and the resulting first two moments of xx and yy are returned, as displayed in plot panel (e) for a C–O model. The second moment of xx and yy is shown as 95% confidence interval (upper and lower lines) around the mean (middle line).

• 

  Figure 2.. (a): Graphical representation of the four-state ion channel gating model from Beattie et al. [12] with the configurations: open conducting O′, inactive I, closed C, and inactive-closed IC. (b): The five-state gating model used in this study and detailed in §2f. It extends the four-state model by splitting the open state O′into open conducting O and open non-conducting (flickering) F states. In both models, arrows represent transitions between ion channel configurations whose rates are voltage-dependent, except for the flickering parameters γ and δ of the five-state model.

• 

  Figure 3.. (a): The voltage-clamp protocol used in this study and detailed in §2a. It consists of the command voltage (y-axis) over time (x-axis). (b): Synthetic stochastic trajectories (thin lines) over time, for each ion channel configuration (colours), simulated from the five-state model by using the simulation scheme from §2d. The first two moments of the state vector xx are obtained from the differential moment equations (2.18), where the variance is displayed as a 95% confidence interval (shaded area) around the mean (line). The model parameters were set consistent with §3a, with a total number of channels equal to η=100. (c): Synthetic ionic current traces (grey) corresponding to the ion channel configurations of plot panel (b), and simulated from the measurement model of equation (2.11), with a measurement noise variance set to σ2=0 nA2. The first two moments of the ionic current are obtained from equation (2.23), where the variance is displayed as a 95% confidence interval (blue-shaded area) around the mean (blue line).

• 

  Figure 4.. (a): A synthetic current trace (grey) generated under η=1000 and σ2=10−5 nA2, along with the first two moments of the model fit, defined by equation (2.23) and obtained with our proposed inference method from §2e, where the variance is displayed as a 95% confidence interval (blue-shaded area) around the mean (blue line). (b): A zoom of plot panel (a) on a portion of the sinusoidal part of the voltage-clamp protocol. (c): Scatter plot of the estimated against true parameters ψψ, on a log–log scale, under different combinations of total number of channels η (colours) and variance σ2 of the measurement error (symbols). Each replicate correspond to a particular synthetic trace. The diagonal grey line identifies the true values.

• 

  Figure 5.. Profile negative log-likelihood (PL, black line) and Fisher Information Matrix-based parabolas (FIM, grey lines) evaluated for the logarithm of each parameter (plot panels). The ML estimates are indicated by blue star symbols. All profile negative log-likelihoods are displayed on the same scale for the y-axis, so as to compare their level of identifiability. The red horizontal lines identify the threshold −l(ψψ^)+1/2χ1−α,12, representing a 95% confidence interval.

• 

  Figure 6.. (a): Scatter plot of estimated (y-axis) against true (x-axis) parameters, on a log-log scale, under each level of sampling frequency ω (colours), with a total number of channels η=1000 and measurement error variance σ2=10−5nA2. The corresponding relative error between the estimated and true parameters are displayed in panel (b), where ‖⋅‖2 is the Euclidean norm. (c): One synthetic current trace obtained from the full trace with a sampling frequency of ω=5% (grey), and the corresponding model fit, obtained with our inference method, in terms of the first two moments defined by equation (2.23), where the variance is displayed as a 95% confidence interval (blue-shaded area) around the mean (blue line). Model fits under different values of sampling frequencies are displayed in electronic supplementary material, figure S2. (d): A zoom of plot panel (c) on a portion of the sinusoidal part of the voltage-clamp protocol.

• 

  Figure 7.. (a): Scatter plot of the estimated against true number of channels η, on a log-log scale, obtained by our proposed method (black) and the standard approach (blue). The 45° diagonal red line indicates the true values. (b): Histogram of the estimated measurement noise variance σ2, on a log scale, provided by either our proposed method (red) and the standard approach (blue). Results displayed in both left and right panels are obtained from the analyses of the synthetic ionic current traces generated under a measurement noise variance σ2 set to 10−6 (top), 10−5 (middle) and 10−4 nA2 (bottom).

• 

  Figure 8.. (a): Graphical representation (left) and reaction rates (right) of the eight-state ion channel gating model used in this study. It extends the five-state model of figure 2b by splitting the closed configuration into three separate states C, C′, C″, and the inactive-closed configuration into two separate states IC and IC′. (b): Scatter plot of estimated (y-axis) against true (x-axis) parameters, on a log–log scale, obtained by applying our inference method of §2e to calibrate the eight-state ion channel gating model of plot panel (a) to fit the N=100 synthetic traces, independently. (c): One synthetic stochastic current trace (grey) simulated with the Euler–Maruyama electronic supplementary algorithm 1 from the eight-state ion channel gating network of plot panel (a) with the dynamic parameters set to the values reported in table 2 and a total number of ion channels set to η=10000, together with the corresponding model fit, in terms of the first two moments defined by equation (2.23), where the variance is displayed as a 95% confidence interval (blue-shaded area) around the mean (blue line).

• 

  Figure 9.. (a): A synthetic current trace (grey line) simulated from the true generative data process of the five-state model under the simulation setting of §3a using the dynamic parameters of table 1, under a total number of ion channels η=1000 and a measurement error variance σ2=10−5 nA2, together with the corresponding model fit, in terms of the first two moments defined by equation (2.23), obtained with either our proposed inference method based on the DMEs (blue) from §2e, or its Kalman filter counterpart from electronic supplementary material, section K (red). In both cases, the variance is displayed as a 95% confidence interval (shaded area) around the mean (line). (b): A zoom of plot panel (a) on a portion of the sinusoidal part of the voltage-clamp protocol. (c): Scatter plot of the estimated (y-axis) against true (x-axis) parameters, on a log–log scale, obtained with the DME-based (blue) and the EKF-based (red) approaches, independently from the N=100 simulated stochastic traces. (d): Box plots of the relative error between the estimated and true parameters, across the N=100 simulated stochastic traces, obtained with the DME-based and the EKF-based approaches, where ‖⋅‖2 is the…

• 

  Figure 10.. (a): Log–log scatter plot of the estimated against true parameters, obtained after fitting the five-state model (empty dots) and the four-state model (stars) using the proposed inference method. The estimates for the total number of ion channels are highlighted in red. (b): Synthetic ionic current trace (black), and the first two moments of the model fit, defined by equation (2.23), obtained with the four-state (red) and the five-state (blue) formulations, where the variance is displayed as a 95% confidence interval (blue- and red-shaded areas) around the mean (blue and red lines).

• 

  Figure 11.. (a): Ionic current trace recorded from cell 5 (black line) and the first two moments, defined by equation (2.23), obtained by fitting the five-state model, where the variance is displayed as a 95% confidence interval (blue-shaded area) around the mean (blue line). Data of the remaining cells and the corresponding model fits are displayed in figure 12. (b): Estimated logarithmic parameters for each cell (colours), whose values are reported in table 3. (c): Data (grey stars) of single-channel conductance (y-axis) observed at three levels of extracellular potassium concentration (x-axis), consistent with [49], and the corresponding fit of the Michaelis–Menten model defined by equation (2.28), including a zoom around the observed extracellular potassium concentration (bottom-right corner). The grey full dots indicate the estimated lower and upper bounds of the single-channel conductance, obtained with equation (2.29) from §2g.

• 

  Figure 12.. Each plot panel shows the ionic current trace recorded from each cell, and the first two moments, defined by equation (2.23), obtained by fitting the five-state model, where the variance is displayed as a 95% confidence interval (blue-shaded area) around the mean (blue line). Parameter estimates for each cell are displayed in figure 11b and reported in table 3. Data of cell 5 and the corresponding model fit are displayed in figure 11a.