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TABLE II

Results of the Least Squares Nonlinear Fitting of the Data under Schemes 2–7

Scheme
2 (EC50 free)
2
3
4
5
6
7
Df 13 15 15 15 
SS (pA290, 512 960, 362 83,500 47,205 324,371 51,071 53,187 
Macroscopic parameters        
EC50 (μM) 44.9 Constrained to 9.8 μM      
pOmax 0.0145 0.0032 0.109 0.158 0.985 0.54 0.92 
pOmin 7 × 10−6 0.004 0.0004 
Kinetic rates        
kon (s−1μM−110.2 ± 11% 7.90 ± 12% 18.0 ± 81% 15.98 ± 120% 5.85 ± 11% 9.28 ± 17% 12.0 ± 24% 
kon2 (s−1μM−1  16.8 ± 64% 16.3 ± 85%    
kon3 (s−1μM−1  8.42 ± 52% 11.6 ± 120%    
koff (s−11,869 ± 39% 121 ± 24% 0.017 ± 250% 0.019 ± 560% 190 ± 10% 1,871 ± 62% 6,731 ± 65% 
koff2 (s−1  175 ± 56% 380 ± 120%    
koff3 (s−1  4,541 ± 30% 6,822 ± 61%    
γ (s−127.9 ± 9% 42.1 ± 23%  43.54 ± 98%  1.07 ± 105% 242 ± 75% 
δ (s−13,491 ± 17% 632 ± 9%  3,718 ± 17%  3,875 ± 33% 9,035 ± 11% 
α (s−11,829 ± 9% 110,104 ± ∞% 29.0 ± 13% 1,088 ± 33% 10.2 ± 53% 776 ± 120% 110 ± 66% 
α2 (s−1  1,189 ±410% 0.246 ±123%    
β (s−127.3 ± 14% 382 ± ∞% 24.08 ± ∞% 540 ± 99% 769 ± 7% 914 ± 110% 1,380 ± 9% 
β2 (s−1  3,198 ± 18% 0.033 ± 160%    
ε1 (s−1  1,936 ± 7% 79.0 ± 84%    
ε2 (s−1  8.48 ± 822% 3.20 ± 491%    
ζ1 (s−1  0.298 ± 680% 31.16 ± 123%    
ζ2 (s−1  5.9*107 ± 490% 4.53 ± 113%    
Allosteric factors        
Akoff     49.6 ± 17% 1.00 ± 34% 0.309 ± 316% 
Akon     4.51 ± 31% 1.77 ± 31% 1.00 ± 182% 
Aα     48.9 ± 19% 2.30 ± 103% 1.00 ± 106% 
Aβ     4.58 ± 41% 0.77 ± 121% 0.309 ± 226% 
Bkoff      65.1 ± 72% 4,295 ± 270% 
Bkon      1.15 ± 110% 1.00 ± 520% 
Bδ      33.3 ± 115% 4,300 ± 260% 
Bγ      2.25 ± 33% 1.00 ± 300% 
Cα      0.19 ± 230% 10.1 ± 240% 
Cβ      635 ± ∞% 62.1 ± 380% 
Cδ      1.00 ± ∞% 5.31 ± 440% 
Cγ      123 ± 300% 118 ± 250% 
Scheme
2 (EC50 free)
2
3
4
5
6
7
Df 13 15 15 15 
SS (pA290, 512 960, 362 83,500 47,205 324,371 51,071 53,187 
Macroscopic parameters        
EC50 (μM) 44.9 Constrained to 9.8 μM      
pOmax 0.0145 0.0032 0.109 0.158 0.985 0.54 0.92 
pOmin 7 × 10−6 0.004 0.0004 
Kinetic rates        
kon (s−1μM−110.2 ± 11% 7.90 ± 12% 18.0 ± 81% 15.98 ± 120% 5.85 ± 11% 9.28 ± 17% 12.0 ± 24% 
kon2 (s−1μM−1  16.8 ± 64% 16.3 ± 85%    
kon3 (s−1μM−1  8.42 ± 52% 11.6 ± 120%    
koff (s−11,869 ± 39% 121 ± 24% 0.017 ± 250% 0.019 ± 560% 190 ± 10% 1,871 ± 62% 6,731 ± 65% 
koff2 (s−1  175 ± 56% 380 ± 120%    
koff3 (s−1  4,541 ± 30% 6,822 ± 61%    
γ (s−127.9 ± 9% 42.1 ± 23%  43.54 ± 98%  1.07 ± 105% 242 ± 75% 
δ (s−13,491 ± 17% 632 ± 9%  3,718 ± 17%  3,875 ± 33% 9,035 ± 11% 
α (s−11,829 ± 9% 110,104 ± ∞% 29.0 ± 13% 1,088 ± 33% 10.2 ± 53% 776 ± 120% 110 ± 66% 
α2 (s−1  1,189 ±410% 0.246 ±123%    
β (s−127.3 ± 14% 382 ± ∞% 24.08 ± ∞% 540 ± 99% 769 ± 7% 914 ± 110% 1,380 ± 9% 
β2 (s−1  3,198 ± 18% 0.033 ± 160%    
ε1 (s−1  1,936 ± 7% 79.0 ± 84%    
ε2 (s−1  8.48 ± 822% 3.20 ± 491%    
ζ1 (s−1  0.298 ± 680% 31.16 ± 123%    
ζ2 (s−1  5.9*107 ± 490% 4.53 ± 113%    
Allosteric factors        
Akoff     49.6 ± 17% 1.00 ± 34% 0.309 ± 316% 
Akon     4.51 ± 31% 1.77 ± 31% 1.00 ± 182% 
Aα     48.9 ± 19% 2.30 ± 103% 1.00 ± 106% 
Aβ     4.58 ± 41% 0.77 ± 121% 0.309 ± 226% 
Bkoff      65.1 ± 72% 4,295 ± 270% 
Bkon      1.15 ± 110% 1.00 ± 520% 
Bδ      33.3 ± 115% 4,300 ± 260% 
Bγ      2.25 ± 33% 1.00 ± 300% 
Cα      0.19 ± 230% 10.1 ± 240% 
Cβ      635 ± ∞% 62.1 ± 380% 
Cδ      1.00 ± ∞% 5.31 ± 440% 
Cγ      123 ± 300% 118 ± 250% 

The EC50 was forced to be 9.8 μM for all fits except for the version of Scheme 2 indicated as EC50 free. The data set is the same illustrated in Fig. 11. Df indicates the number of free parameters to be optimized by the nonlinear algorithm. SS indicates the sum of squared residuals. Except for Scheme 2 there was very little increase in the SS as a result of fixing the EC50. The macroscopic parameters pOmax (maximum open probability for a saturating pulse of ATP) and pOmin (the open probability with no ATP present) were calculated based on the microscopic rates from each fit shown below. The EC50 was also calculated for Model 2. Standard errors of the estimated kinetic rates and allosteric constants were calculated on a logarithmic scale and presented as percentages. They were obtained using a bootstrap approach (Kraushaar and Jonas, 2000).

The kinetic rates of the fits are defined as follows. For Schemes 2, 5, 6, and 7 there was a single binding rate (kon) and unbinding rate (koff), while for Schemes 3 and 4 there were three independent rates designated as kon, kon2, and kon3 for binding and koff, koff2, and koff3 for unbinding. When flipping was present, the flipping rate was designated as δ and the unflipping rate as γ. β is the opening rate and α the closing rate from the fully liganded (and if relevant, fully flipped) state. For Schemes 3 and 4, additional values are provided for the opening to (β2) and closing from (α2) a second open state. For models 3 and 4 in which there are two open states, ζ is the rate of closing to, and ε the rate of opening from, a final closed state from which only opening is possible. The subscript 1 or 2 indicates the identity of the open state involved in the transition.

Allosteric factors of Schemes 5–7 are designated as follows. Binding–gating coupling factors start with an A. Akon indicates the factor of increase in the binding rate produced by each ligand. Akoff indicates the factor of decrease in the unbinding rate per ligand. Aα indicates the factor of decrease in the closing rate per ligand and Aβ the factor of increase in the opening rate per ligand. Binding–flipping coupling factors start with a B. Bkon is factor of increase in the binding rate to the binding domain produced by flipping of the whole channel (Scheme 6) or by the flipping of the same domain referred by the binding rate. Bkoff is the factor of decrease in the unbinding rate produced by the flipping of the whole channel or of the relevant domain. Bγ and Bδ in Scheme 6 are the factor of increase and decrease of the flipping and unflipping rate produced by each ligand. In Scheme 7 they are the factor of increase and decrease of the flipping and unflipping rates produced at the ligand domain by the presence of a bound ligand. Flipping–gating coupling factors start with a C. Cδ and Cγ indicate the factor of increase in flipping rate and decrease in unflipping rate produced by the opening of the channels, whereas Cβ and Cα indicate the factor of increase in opening and of decrease in closing produced by the flipping either of the whole channel (Scheme 6) or by each flipping domain (Scheme 7).

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