Fig 1.
Schematic of the Wild-Type Na+ Channel.
The wild-type Na+ channel formulation contains 8 states: 3 closed states (C3, C2, C1), 1 open state (O), a fast- and slow-inactivated state (IF and IS, respectively), and two close-inactivated states (IC3, IC2). Included are cartoon representations of the gating structure closely associated with the kinetic state of the channel. Note the movement of the S4 voltage sensors (red ovals around the channel pore) as the channel traverses the closed states (C3 → C2 → C1 → O). Note also the fast inactivation gate (green ball), the III-IV linker, occluding the channel pore on movement from O → IF and C → IC, IF [17–19].
Table 1.
Parameters to be optimized in the drug free Na+ channel.
Fig 2.
Schematic of the optimization procedure.
The first step in the algorithm gathers the experimental data; experiments are then simulated. The algorithm compares the difference, or error, between experiment and what the model predicts (see inset: red arrows indicate error, the simulated experiment is denoted by the solid line, the experiment data are the dots). The total error is then summed. The routine perturbs the parameter set, and iterates again. This routine finishes when the model agrees sufficiently well (defined as the change in cost function between iteration n and n+1) with the experiment. In other words, when the error between what the model predicts and the actual experiment falls below a predefined value.
Table 2.
Example Code and Key Commands.
Fig 3.
Wild-type Na+ channel kinetics–pre- and postoptimization.
In each panel, the points are experiment, and the lines are simulation. Blue traces indicate preoptimization using initial guesses as described in the text; red traces indicate the optimized parameters. The protocols are as follows: steady state availability (or inactivation) (Panel A), steady state activation (Panel B), recovery from inactivation at -100 mV (Panel C), recovery from use-dependent block (Panel D), and time to 50% decay of Na+ current (Tau50%) (Panel E). The model was further constrained by mean open time at -30 mV. Data are from [3,9,10]. Voltage protocols are shown as insets.
Fig 4.
Ratio of optimized parameter to initial value.
As can be seen, many parameters needed minimal optimization to accurately fit the wealth of data from multiple protocols. Some parameters, however, varied markedly from their initial value (e.g. b12). See Table 1 and S1 Supplementary Information for the initial, and optimized parameter values.
Fig 5.
Robustness Analysis of Optimized Parameters.
An optimization routine was set up such that a “true” global minimum was defined as the optimized parameters. The previously found optimized values were perturbed by a random number with a 5% (n = 3 runs), 10% (n = 3 runs), or 25% (n = 3 runs) deviation. The deviated parameters were used as the initial guesses, and the optimization algorithm was restarted. The graph shows the averaged parameter values from each set of runs, normalized to the optimized starting value. A value of 1 with no error bar would mean that the strategy found the exact optimized value. The error bars show ±1 standard deviation, normalized to the average value found from the runs. As can be seen, the algorithm performed best with smaller deviations from the optimum value, and many parameters were sufficiently constrained, even with large (25%) perturbations.
Fig 6.
In panel (A,) a computed Kd curve is generated from Kd = Kd0*e(-d*V*F/(R*T)) with Kd0 = 11.2 (Kd at 0 mV). Kd-100mV = 175.8 μM. See Table 3. Panel B and C are the results from optimization of neutral rate constants for flecainide using a neutral flecainide derivative (NUFL). Panel (B) is use-dependent block at 10 Hz for 10 μM and 100 μM flecainide. Blue bars are experiment [10], and red bars are the result of the simulation. Panel C is recovery from UDB with 100 μM NUFL. Panels (D–H) are the results of the optimization for charged flecainide under a variety of protocols: (D)–steady state availability, (E)–tonic block (1-pulse block), (F)–use-dependent block (UDB), (G)–recovery from UDB, and (H)–frequency dependent use-dependent block. Protocols are shown as insets.
Table 3.
Model State Specific Affinities of Drug to the Sodium Channel Flecainide.
Fig 7.
Analysis of initial conditions in charged flecainide model.
Shown are the results of 3 additional optimization routines for charged flecainide, all starting from different initial conditions. The black trace is from a sequential design strategy of 400 initial iterations (100 each) of SSA, Block, RUDB, and FDUDB. The optimization was then continued to convergence. Red inputs indicated an empirically derived (“hand tuned”) set of initial guesses (see Table 4). Blue traces indicate an initial vector of all 1’s–indicating that the drug bound rate constant = 1* the drug free rate constant. See text for further details.
Table 4.
Summary data of charged flecainide parameters starting from different initial conditions.