Obtaining transition rates from single-channel data without initial parameter seeding

Figures

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  Figure 1.. Information loss from kinetic scheme to processed data.a) Kinetic scheme showing the connection framework of a five-state ion-channel with two open (grey and labeled 1 and 2) and three closed conformations (colored and labeled 3, 4, and 5), and transition rates are given in s-1. b) The upper panel illustrates how the channel in A moves between the five states as time progresses. The middle panel demonstrates loss of information as only transitions between open and closed states are observed. The bottom panel shows the observed data after the loss of brief events to limited time-resolution.

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  Figure 2.. Numerical root finding. The challenge of numerically locating multiple roots is a balance between search resolution and computation time.a) Seven-state kinetic scheme with three open (grey) and four closed conformations (black). The identity of the 14 transition rates is noted as color-coded small letters from a to n with constrained parameters in black. b) The left-hand table gives example parameters for input in the model shown in A. With these parameters; the four roots are located as demonstrated in the right-hand plots. The lower plot is an expansion of the upper plot at the region indicated by the arrow.

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  Figure 3.. Nelder-Mead optimization with an initial guess. a) Five-state kinetic scheme with two open (grey) and three closed conformations (black) and ten transition rates given as color-coded small letters from a to j with constrained parameters in black. MR denotes that parameter c is set to maintain microscopic reversibility. b) The table shows how NM with a basis in an initial guess estimate parameter values that approach the true values. c) NM improves the parameters given as an initial guess over 510 iterations; of these, those that change the fitness value are included in the plot. The eight parameters are color-coded as in a and b, and the right-hand insert shows the true values. The solid grey line follows the right-hand Y-axis and illustrates how the fitness value improves.

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  Figure 4.. Rate optimization with GA and PS before NMa) Scheme specifying how three closed (black) and two open (gray) states are connected with rates labeled as small letters with color-codes that apply to all figure panels. The box specifies constrained parameters, MR is short for microscopic reversibility. The right-hand table gives rate values for individual parameters. The table shows initial rate estimates by GA and PS, and the final estimates from NM using the best GA or PS estimate as input. b) Progress of GA fitting with the best set of parameters from a population of 200 is plotted over 92 iterations. The remaining iterations are from NM optimization based on the best GA solution. From a 300 iteration NM optimization, only 35 iterations where the fitness value changes are included. The eight parameters are color-coded as in a and b, and the right-hand insert shows the true values. The solid grey line follows the right-hand Y-axis and illustrates how the fitness value improves. c) Each panel illustrates how the function value changes with the given parameter when the remaining rates are fixed. The upper values in each panel are based on the final GA estimate while the lower values…

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  Figure 5.. Rate optimization with PS, NM and expanding limitsa) Scheme specifying how three closed (black) and two open (gray) states are connected with rates labeled as small letters with color-codes that apply to all figure panels. The box specifies constrained parameters; MR is short for microscopic reversibility. b) Rate values for individual parameters. The table shows the true rates that simulated data is based on and rates after an initial PS search with limits 10 to 2000 s-1. Furthermore, two consecutive rounds of NM optimization improve the best ranking PS estimate, one with limits 1 to 20000 s-1 (NM#1) and one with limits 0.1 to 200000 s-1 (NM#2). c) Rate estimation begins with 19 rounds of PS optimization; each iteration represents the best estimate of 100 particles. Iteration 20 to 45 covers the first round of NM optimization; the remaining follow the final round of NM. Only iterations where the fitness value change are included in the plot. The right-hand box shows the true values while the right Y-axis and solid grey line track the progressing fitness value. d) Plots for each parameter showing the results of 20 optimization trials.

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  Figure 6.. Seven-state model optimization with PS, NM and expanding limitsa) Scheme specifying how four closed (black) and three open (gray) states are connected with rates labeled as small letters with color-codes that apply to all figure panels. The box specifies constrained parameters. b) Rate values for individual parameters. The table shows the true rates that data simulation is based on. Third column show rates after an initial PS search with limits 10 to 2000 s-1. Furthermore, two consecutive rounds of NM optimization improve the best ranking PS estimate, one with limits 1 to 20000 s-1 (NM#1) and one with limits 0.1 to 200000 s-1 (NM#2). c) Rate estimation begins with 11 rounds of PS optimization; each iteration represents the best estimate of 100 particles. Iteration 11 to 33 covers the first round of NM optimization; the remaining follow the final round of NM. Only iterations where the fitness value change are included in the plot. The right-hand box shows the true values while the right Y-axis and solid grey line track the progressing fitness value. d) Plots for each parameter showing the results of 20 optimization trials.