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A complex voltage-gated sodium channel. (A) An example of the text-based method to enter a model, as explained in Tutorial 8, is shown. (B) Ensemble response to a typical IV protocol consisting of 20-ms steps to voltages ranging from −60 to +60 mV (in 10-mV increments), from a holding potential of −90 mV (the corresponding settings file NaChannelIV.set can be found on https://github.com/mikpusch/MarkovEditor.git). (C) A single channel simulated for a 20-ms pulse to 0 mV (holding potential −90 mV). (D) Ensemble gating currents. (E) Stochastic gating current response to a voltage step to 0 mV (holding potential −90 mV) simulated for 50 channels.
Strong collision model simulation of q(t) for the biphasic potential with unequal well frequencies. Parameters were Q = 7, VB = 2, ωA = 0.5, and ωB = 0.25 (dashed trace in Fig 2). The simulation used a time step of 0.01 and consisted of 234 steps with a collision rate of α = 2.5; q(t) was stored every 210 steps. (a) The q(t) at every 1,000 stored sample point during the simulation. (b) Autocorrelation function of q(t) (black) fit with a monoexponential (reddish–purple, dashed line) and biexponential (blue, dotted line) decay function; 10 replicate simulations were averaged to produce the final autocorrelation function. The inset shows the fast initial decay of the autocorrelation function, which is well described by the biexponential fit. The fitted parameters are amplitudes a1 = 6.0 and a2 = 77.1, and decay times τ1 = 30.7 and τ2 = 10,440. All parameters are dimensionless, as described in the text. The simulated dynamics in the potential are shown in Supplemental Movie S2.
The 2-state random-coil model for the biphasic potential with unequal well frequencies. (a) Sample of ω(t) at times during the simulation with parameters chosen, so <ω(t)2>1/2 = 9.60 × 10−6. (b) Autocorrelation of ω(t) (black) fit with monoexponential (reddish–purple, dashed line) and biexponential (blue, dotted line) functions. The inset shows the fast initial decay of the autocorrelation function, which is well described by the biexponential fit. The fitted parameters are amplitudes a1 = 4.52 × 10−11 and a2 = 4.49 × 10−11 and decay times τ1 = 32.0 and τ2 = 10,420; <δω(t)2>1/2τ2 = 0.1 and an estimated value of RBWR = 4.69 × 10−7. (c) Sample of Re[s+(t)] at times during the simulation. (d) Real part of the autocorrelation of s+(t) (black) fit with a monoexponential (reddish–purple, dashed line) function with initial amplitude fixed at 1.0 and decay time constant of 2.47 × 106, yielding Rsc = 4.05 × 10−7 in good agreement with RBWR. All parameters are dimensionless, as described in the text.
The 2-state random-coil model for the biphasic potential with equal well frequencies. (a) Sample of ω(t) at times during the simulation with parameters chosen, so <ω(t)2>1/2 = 1.96 × 10−5. (b) Autocorrelation of ω(t) (black) fit with monoexponential (reddish–purple, dashed line) and biexponential (blue, dotted line) functions. The inset shows the fast initial decay of the autocorrelation function, which is well described by the biexponential fit. The fitted parameters are amplitudes a1 = 1.67 × 10−10 and a2 = 2.15 × 10−10 and decay times τ1 = 8.5 and τ2 = 5,110; <δω(t)2>1/2τ2 = 0.1 and an estimated value of RBWR = 1.10 ×10−6. (c) Sample of Re[s+(t)] times during the simulation. (d) Real part of the autocorrelation of s+(t) (black) fit with a monoexponential (reddish–purple, dashed line) function with initial amplitude fixed at 1.0 and decay time constant of 9.60 × 105, yielding Rsc = 1.04 × 10−6 in good agreement with RBWR. All parameters are dimensionless, as described in the text.
CPMG relaxation dispersion for 2-state (filled circles, solid line) and random-coil models (open circles, dashed line). Relaxation rate constants shown as circles were obtained as described in Figure 11. Simulated points were fit with Eq. 21 augmented by a constant offset parameter. Optimized values of τex = 10,330 and 10,200 for the telegraph and random-coil models, respectively, in agreement with the results shown in Figures 9 and 10. The optimized offset was 0 for the telegraph model and 1.36 × 10−9 for the random-coil model. The limiting relaxation rate constant for the random-coil model agrees well with the value of 1.44 × 10−9 obtained as the product of the amplitude and decay time for the fast component of the autocorrelation function shown in Figure 10b, confirming that the apparent plateau represents the contribution from dynamics processes faster than the CPMG pulsing rates. Data have been normalized by the relaxation rate constant in the absence of pulsing, RBWR, for display.
The 2-state telegraph model for the biphasic potential with unequal well frequencies. (a) Sample of ω(t) at times during the simulation with parameters chosen, so <δω(t)2>1/2 = 9.60 × 10−6. (b) Autocorrelation of ω(t) (black) fit with a monoexponential function (reddish–purple, dashed line), with amplitude <δω(t)2> = 9.17 × 10−11 and decay time τc = 10,440; <δω(t)2>1/2τc = 0.1 and an estimated value of RBWR = <δω(t)2>τc = 9.58 × 10−7. The inset shows only the monoexponential decay. (c) Sample of Re[s+(t)] at times during the simulation. (d) Real part of the autocorrelation of s+(t) (black) fit with a monoexponential (reddish–purple, dashed line) function with initial amplitude fixed at 1.0 and decay time constant of 1.13 × 106, yielding Rsc = 8.83 × 10−7 in good agreement with RBWR. All parameters are dimensionless, as described in the text.
CPMG relaxation dispersion for the strong collision model, the biphasic potential with unequal well frequencies, and the telegraph signal mapping of resonance frequencies. Parameters were Q = 7, VB = 2, ωA = 0.5, and ωB = 0.25 (dashed trace in Fig 3). The simulations used a time step of 0.01, and a collision rate of α = 2.5; q(t) was stored every 210 steps. Values of τcp were 1.05 × 104 and 2.62 × 103. (a) Values of s+(t) for 640 trajectories of length 229 steps were averaged to obtain < s+(t) >. (b) Autocorrelation functions of s+(t) were calculated, as in Figure 8, for 20 individual trajectories of 234 steps and averaged. In each figure, black lines are simulated results. Fits with single exponential functions are shown for absence of CPMG block (blue, dotted line), CPMG block with τcp = 1.05 × 104 (green, dash–dotted line), and CPMG block with τcp = 2.62 × 103 (reddish–purple, dashed line). The decay time constants in the absence of an applied CPMG pulse train agree well between the NMR signal and its autocorrelation function. The decay rates in the absence of the CPMG sequence are (a) 9.49 × 10−7 and (b) 9.16 × 10−7 (in good agreement with the results shown in Fig 8), the decay rates are (a) 2.07 × 10−7 and (b) 2.29 × 10−7 for τcp = 1.05 × 104, and the decay rates are (a) 1.97 × 10−8 and (b) 2.26 × 10−8 for τcp = 2.62 × 103.
NMR spectra for 2-site chemical exchange. Parameters were p1 = 0.8, p2 = 0.2, ω1 = −200 s−1, ω2 = 800 s−1, and Δω = 1,000 s−1, yielding the average resonance frequency p1ω1 + p2ω2 = 0 for convenience. The values of kex are 250 s−1 (dash–dotted line), 1,000 s−1 (dashed line), 2,500 s−1 (solid line), and 4,000 s−1 (dotted line), corresponding to slow, intermediate, fast, and very fast exchange on the chemical shift timescale, respectively. The inset shows a vertical expansion of the region from 250 to 1,000 s−1. The transverse relaxation rate constants obtained for the major peak in each spectrum by fitting a Lorentzian line shape function over the region −500 to 300 s−1 were 49.7, 113.8, 61.6, and 39.5 s−1, respectively. Spectra were calculated from the Bloch–McConnell equations (1).