Identification of structures for ion channel kinetic models

Fig 1. Reduction in model search space by enumerating unique topologies as a function of the states. A) Blue states are non-open while the open state (root) is colored green. All 36 permutations of three state rooted topologies are depicted. Permutations outlined in black represent the original six possible rooted topologies of three states. Orange shading represents the three isomorphic permutations of the six possible rooted topologies while the three unshaded topologies correspond to the unique rooted topologies as in B. The yellow shaded topologies are the remaining 30 permutations. A reduction from 36 rooted graph permutations to three unique topologies is depicted in C. C) Results of a similar graphical enumeration analysis for rooted topologies with 4+ states. D) Biophysically inspired restriction of the maximum degree in a graph to 4. Applying this restriction to the unique topologies results in a reduction in a model search space as enumerated in the table. After further restricting the maximum cycle length in a graph to size 4 after the degree restrictions, final graph counts are displayed in E as function of the number of states. E) Enumeration summary table of rooted graph…

Fig 2. Identification of possible structures for the Ito,f dataset. A) Optimization iterations completed as a function of free rate constants. Weighting of dots represent the frequency of models completing a specified number of iterations. Most models complete the generous maximum iteration limit (40,000,000) with 4 and 5 free rate parameters while few models complete less iterations due to early stopping from the overfitting criterion. Models with 6 and 7 free rate parameters may run for longer (maximum 80,000,000 iterations displayed) while a greater fraction of models are terminated early due to overfitting criterion. Data points include the multiple starting conditions for each model to reduce the dependence of the minimum solution on initial conditions B) Distribution of normalized costs for each model with multiple starts after completing optimization iterations as depicted in A. Especially bad Sobol starting conditions are asterisked. The absolute minimum costs are outlined in the dashed grey box. The point of diminishing returns is at 6 and 7 free rate constants. C) Topologies producing acceptable fits and D) unacceptable fits for Ito,f. Acceptable fits include models with four…

Fig 3. Identification of possible model structures for the INa HEK dataset A) Representative acceptable model topologies of the more complex HEK INa dataset. Topologies have at least 8 free rate constants. B) Representative unacceptable model topologies. Topologies have fewer than 8 free rate constants or are sparsely connected. C) Representative acceptable and unacceptable model fits to steady state activation and inactivation D) Representative fits to fast recovery from inactivation E) Representative fits to recovery from use dependent block which includes timescales of fast, intermediate, and slow recovery from inactivation. F) Representative model fits to the time constant of 50% inactivation of the peak sodium current G) Simulated action potentials with representative acceptable and unacceptable INa modeled currents. Unacceptable models have varying degrees of late INa current) H) Corresponding representative acceptable and unacceptable INa currents during the action potentials in G with a magnified inset highlighting the late INa current.

Fig 4. Summary of Markov model performance for all topologies studied. A) and B) Performance summary of all Ito,f topologies studied. Unacceptable and acceptable labels are as previously defined. Tentative yellow topologies produced acceptable voltage protocol fits but did not perform as other acceptable models in the action potential validation. In A) more states help topologies find tentative and acceptable solutions. B) Topologies with lower root degrees (open state connections) tend to create acceptable models C and D) Performance summary of all INa topologies studied. Unacceptable and acceptable labels are as previously defined. Tentative yellow topologies produced acceptable voltage protocol fits but did not perform like other acceptable models in the action potential validation (different degrees of repolarization failure). C) Topologies with more than the minimum number of edges tend to yield more tentative or acceptable Markov models. D) As edges increase, topologies most benefit from higher root degrees (more open state connections) to serve as successful Markov models. Panels C and D do not show unacceptable models for clarity.

Fig 5. Validation of acceptable model solutions in cellular and tissue simulations. Model solutions show diverse behavior in cellular and tissue simulations despite good fidelity fits to the training voltage protocols. A) Representative shallow and steep slopes of acceptable Ito,f models in a dynamic restitution protocol. Slopes are calculated from DI intervals between 60 and 80 ms. Most acceptable model solutions have shallower slopes between 0.2–0.4. There are 12 acceptable model solutions that have slopes higher than 0.4 indicating larger Ito,f currents (see S10 Fig). B) Conduction velocity variability results from variability in the precise INa gating. Despite low-cost protocol fits (S11 Fig), the activation of the INa current varies between 4 and 8 ms with conduction velocities of 67 (blue) and 56 (orange) cm/s, respectively for two representative acceptable models. Dashed lines represent the corresponding action potentials upstrokes. Plotted below are all acceptable INa model conduction velocities in a 100-cell ten Tusscher cable. Reported conduction velocities are from Cell 50.