Ion channel model reduction using manifold boundaries

Abstract

Mathematical models of voltage-gated ion channels are used in basic research, industrial and clinical settings. These models range in complexity, but typically contain numerous variables representing the proportion of channels in a given state, and parameters describing the voltage-dependent rates of transition between states. An open problem is selecting the appropriate degree of complexity and structure for an ion channel model given data availability. Here, we simplify a model of the cardiac human Ether-à-go-go related gene (hERG) potassium ion channel, which carries cardiac I_Kr, using the manifold boundary approximation method (MBAM). The MBAM approximates high-dimensional model-output manifolds by reduced models describing their boundaries, resulting in models with fewer parameters (and often variables). We produced a series of models of reducing complexity starting from an established five-state hERG model with 15 parameters. Models with up to three fewer states and eight fewer parameters were shown to retain much of the predictive capability of the full model and were validated using experimental hERG1a data collected in HEK293 cells at 37°C. The method provides a way to simplify complex models of ion channels that improves parameter identifiability and will aid in future model development.

Keywords: electrophysiology; identifiability; ion channel; mathematical model; model reduction.

Figure 1.

(a) Eigenvector components of the initial (sloppiest) and final parameter direction at the end of the geodesic path for an MBAM iteration (the fourth MBAM iteration in the results section for revision r3 → r4). (b) Eigenvalue spectra of g at the start and end of the geodesic path. (c) A plot of the geodesic path (black line) in a slice of log parameter space from the starting point denoted by a black circle. The plot is coloured according to evaluations of the cost function given in equation (2.1), such that darker shades of blue represent worse agreement with the full model output.

Figure 2.

(a) Evolution of the Markov chain hERG model by Wang et al. [14] through subsequent iterations of the MBAM. Each structure shows a dynamic model (starting with the full model, r0) and the parameter changes which took place to get to the next reduced model, as in table 1, guided by a grey arrow showing the direction of model reduction (from r0 to r8). As in table 1, parameters highlighted blue →0 and red →∞ in the next reduction. The O′ and O″ states for the Wang-r5 and Wang-r8 models relate to the actual open probability through the relations O = O′/(1 + ϕ₃exp ( − p₄V)) and O = O″/(1 + ϕ₇exp (p₈V)), respectively. The dotted red line around the O″ state in the final reduction denotes that the maximal channel conductance →∞. (b) The eigenvalue spectra of g for selected models (denoted by coloured stars) under the shortened ‘staircase’ protocol described in §2.2.

Figure 3.

A practical assessment of model identifiability. Plotted are the inferred parameter values in (a) the full Wang model [14] and (b) a reduced model (Wang-r6) for 50 repeats of fitting from different initial guesses to experimental ‘staircase’ calibration protocol hERG channel currents at 37°C [16]. The 30 best parameter sets are shown in each case, from lowest (green) to highest (blue) likelihood. Note that in (b) the 30 inferred parameter sets are overlapping. (c) Two parameter sets of the full Wang model which show large divergence for many parameters but are both consistent with the experimental data, giving highly similar model outputs in response to the same input voltage protocol, as shown in (d).

Figure 4.

(a) A comparison of the full Wang model and Wang-r6/Wang-r8 reduced model fits to experimental data under the ‘staircase’ calibration protocol. (b) Prediction of the full Wang model and Wang-r6/Wang-r8 reduced models under a complex action potential waveform validation protocol. (c) Predictions of the full Wang model and Wang-r6/Wang-r8 reduced models under shortened versions of traditional activation and inactivation protocols. (d) Comparisons of summary data between the full Wang model and Wang-r6/Wang-r8 reduced models and experiment corresponding to the data shown in (c). All experimental data were recorded in HEK293 cells at 37°C [16] (see §2.3 for details).