2.6 Data analysis
Curve fitting was done in Microsoft Excel, using the Solver Add-in. Steady-state inactivation (SSI ) curves were fitted using the Boltzmann function: I = Imax /{1 + exp[VpV1/2 / -k ]}, where Vp is the pre-pulse potential, V1/2 is the voltage where the curve reached its midpoint, and k is the slope factor.
Recovery from inactivation (RFI ) experiments were fitted by mono- or bi-exponential function: I =A1 *[1-exp(-tip /τ1 )]n+A2 *[1-exp(-tip /τ2 )], where τ1 and τ2 are the fast and slow time constants, A1 , andA2 are their contributions to the amplitude (A2 = 0, for monoexponential fitting), andtip is the duration of the interpulse interval. We observed that recovery (not only in control, but in the presence of riluzole, and with covalently bound azido-riluzole as well) was steeper than a simple exponential function, therefore we introduced the exponentn . When its value was unconstrained, best fits produced n= 2.35 ± 0.65 in control (range: 1.22 to 3.15) and n = 3.19 ± 0.51 in the presence of 100 µM riluzole (range: 1.95 to 5.54). However, the value of the exponent affected the value of the time constant, as we have discussed before(Lukacs et al., 2018). For this reason, we chosen = 1.5, the lowest exponent that could acceptably fit our data, and used it for fitting throughout the experiments.
State-dependent onset (SDO ) data were fitted with either single or double exponential functions: I =A1 *exp(-tp /τ1 ) +A2 *exp(-tp /τ2 ), where A1 and A2 are the relative amplitudes of the two components, and tpis the duration of the pulse.
Averaging of SSI , RFI , and SDO curves was not done by calculating the mean of measured points because this would introduce an error in the slope of curves. Instead, parameters from individually fitted experimental curves were averaged (arithmetic mean for amplitudes, membrane potential values, and slope factors; geometric mean for time constants), and the curve was constructed using the averaged parameters.
Apparent affinity (IC50 ) values were calculated from either the extent of inhibition or from the onset and offset time constants. Apparent affinity is a useful way to represent the potency of sodium channel inhibitors under specific circumstances. This potency can change radically even on the sub-millisecond time scale, as seen for example in the 3PT protocol, depending on the temporal pattern of membrane potential, due to the interrelated dynamics of binding and gating. Apparent affinity values can be calculated from a single inhibition value, using the simplified Hill equation: when a one-to-one binding is assumed, the Hill equation is reduced to Inh =cc /(cc + IC50 ), where Inh is the inhibited fraction and cc is the drug concentration. The calculation is most accurate at ~50% inhibition, but becomes increasingly inaccurate as inhibition approaches either 0 or 100%.
Apparent affinity can also be calculated from onset and offset time constants (obtained by single exponential fitting of amplitudes in the3PT protocol), if we suppose a single-step binding reaction:IC50 = cc *τon /(τoff -τon ). In the case of riluzole, as we will discuss below, the inhibited fraction and the bound fraction were not equivalent, therefore we expected that the IC50values calculated by the two ways will differ (see Fig. S1 in supporting information).