Fitting records at a single concentration with constraints
Constraints. Scheme 1 (Fig. 1) has 14 rate constants, but one of them (k_{+1a}) was always determined by microscopic reversibility so there are 13 free parameters to be fitted. This can be reduced to 10 free parameters if it is assumed that the binding to site a is the same whether or not site b is occupied, and vice versa. This assumption implies that the two different binding sites behave independently of each other while the channel is shut. This is plausible, given the distance between the sites, but it is not inevitable. Nevertheless this assumption of independence has been made in earlier studies. It implies imposition of the following three constraints:
 (9)
These, together with the microscopic reversibility constraint, assure also that:
 (10)
When a single lowconcentration record is fitted in bursts (see Methods), there is no information available about how frequently the channel is activated, so whether or not the above constraints are applied, it is necessary to supply more information in order to get a fit. This was done in two ways. Either (a) one of the rate constants (k_{+2a}) was fixed at an arbitrary value such as 10^{8}m^{−1} s^{−1} (the effects of error in this value are investigated below), or (b) an EC_{50} value was specified, and used to calculate one of the rate constants (see Methods). In either case the number of free parameters is reduced to nine.
Initial guesses based on the two binding sites being similar. As with any iterative fitting method, initial guesses for the free parameters have to be supplied. It is always important to check that the same estimates are obtained with different initial guesses. It is quite possible, if the fit is very insensitive to the value of one of the rate constants, for convergence to be obtained with the initial guess being hardly changed. This does not mean that it was a good guess, but merely that the data contain next to no information about that particular rate constant; it is easy to get spurious corroboration of one's prejudices. And in a complex problem like this it is quite possible that the likelihood surface will have more than one maximum; a bad guess may lead you to the wrong maximum. This problem can be illustrated by what happens when attempts are made to start the fit of scheme 1 with guesses that make the two binding sites almost the same, when in fact they are different. In general it seems like quite a good idea to start from a ‘neutral’ guess like this, but in practice it can give problems. (Note, too, that all the calculations assume that eigenvalues are distinct, so it is inadvisable to start with guesses that are identical.)
Figure 2 shows the distributions of 1000 estimates obtained from fitting a single record at a low ACh concentration, 30 nm, with the constraints in eqns (9) and (10), and with k_{+1a}=k_{+2a} fixed at 1 × 10^{8}m^{−1} s^{−1} (half its true value in this case). The resolution imposed before fitting was 25 μs, as in most experiments. The fitting was done in bursts of openings that corresponded to individual activations of the channel, defined by t_{crit}= 3.5 ms, and the likelihood calculation for each burst was started and ended with CHS vectors (see Methods). In this case the rate constants, rather than their logarithms, were the free parameters. The true rate constants (those used for the simulation) are shown in Table 1, and the initial guesses for the fitting are shown as ‘guess 1’ in column 3 of Table 1.
Table 1. Rate constants used for simulation, and as initial guesses for fits   True 1  Guess 1  Guess 2  True 2  Guess 3  Guess 4 


α_{2}  s^{−1}  2000  1500  1500  2000  1500  1500 
β_{2}  s^{−1}  52000  50000  50000  50000  50000  50000 
α_{1a}  s^{−1}  6000  13000  2000  4000  2000  10000 
β_{1a}  s^{−1}  50  50  20  80  20  50 
α_{1b}  s^{−1}  50000  15000  80000  40000  80000  20000 
β_{1b}  s^{−1}  150  10  300  10  30  30 
k_{−2a}  s^{−1}  1500  6000  1000  12000  1000  20000 
k_{+2a}  M^{−1} s^{−1}  2.0 × l0^{8}  1.0 × 10^{8}  1.0 × 10^{8}  0.5 × 10^{8}  1.0 × 10^{8}  1.0 × 10^{8} 
k_{−2b}  s^{−1}  10000  5000  20000  2000  20000  1000 
k_{+2b}  M^{−1} s^{−1}  4.0 × l0^{8}  1.0 × 10^{8}  1.0 × 10^{8}  5.0 × 10^{8}  4.0 × 10^{8}  2.0 × 10^{8} 
k_{−1a}  s^{−1}  1500  6000  1000  400  1000  1000 
k_{+1a}  M^{−1} s^{−1}  2.0 × 10^{8}  1.0 × 10^{8}  1.0 × 10^{8}  0.2 × 10^{8}  1.0 × 10^{8}  0.5 × 10^{7} 
k_{−1b}  s^{−1}  10000  5000  20000  100  20000  1000 
k_{+1b}  m^{−1} s^{−1}  4.0 × 10^{8}  1.0 × 10^{8}  1.0 × 10^{8}  3.0 × 10^{8}  4.0 × 10^{8}  2.0 × 10^{8} 
E_{2}  —  26  —  —  25  —  — 
E_{1a}  —  0.0083  —  —  0.02  —  — 
E_{1b}  —  0.003  —  —  0.00025  —  — 
K_{2a}  μM  7.5  —  —  240  —  — 
K_{2b}  μM  25  —  —  4  —  — 
K_{1a}  μM  7.5  —  —  20  —  — 
K_{lb}  μM  25  —  —  0.333  —  — 
On each histogram of the 1000 estimates, the true value is marked with an arrow. The distribution of the estimates of α_{2}, the shutting rate for diliganded channels, in Fig. 2A has two peaks. One, shown enlarged in the inset, is close to the true value of α_{2}= 2000 s^{−1}. This peak contains 73 % of all estimates and these have a mean of 2045 ± 174 s^{−1} (coefficient of variation 8.5 %), so these estimates have a slight positive bias but are quite good. The other 27 % of estimates of α_{2} are much bigger, nowhere near the true value. A similar picture is seen with the estimates of β_{2} shown in Fig. 2B. Again 73 % of estimates (the same 73 %) are near the right value, β_{2}= 52 000 s^{−1}, and the other 27 % are much too big. The main peak has a mean of 52 736 ± 3692 s^{−1}, the coefficient of variation being 7.0 %, slightly lower than for α_{2}.
Figure 2C shows that there is essentially no difference between the ‘goodness of fit’, as measured by the maximum value of the loglikelihood attained in the ‘experiments’ that gave good estimates, and those that gave estimates that were much too fast. All of the fits fall clearly into either the ‘right solution’ or into the ‘fast solution’ peaks, apart from 10 or so (1 %) that are smeared in between the two main peaks. This behaviour resembles the very simplest version of the missed event problem, which is known to have two solutions (see Discussion).
When the estimate of α_{2} is plotted against the value of β_{2} from the same fit, in Fig. 2D, it is clear that the two values are very strongly correlated – the fits that give good estimates of α_{2} also give good estimates of β_{2}, and vice versa. This phenomenon will be discussed below.
Initial guesses based on the two binding sites being different. When similar experiments are simulated, but with initial guesses for the fit that start from the supposition that the two binding sites are not similar, these better guesses very rarely lead to the incorrect ‘fast solution’. The guesses used for each of the 1000 fits are shown in column 4 of Table 1 (‘guess 2’). The results are shown in Figs 3–5.
In this case none of the 1000 fits converged on the incorrect ‘fast solution’. The mean of 1000 estimates of α_{2} was 2016.5 ± 146.4 s^{−1}, compared with a true value of 2000 s^{−1} (Fig. 3A). The coefficient of variation (CV) of the estimates is 7.3 % and there is a very slight positive bias of +0.82 % (calculated as a fraction of the true value). For β_{2} the mean was 52 285 ± 3248 s^{−1}, compared with a true value of 52 000 s^{−1} (Fig. 3B). The CV was 6.2 %, and bias +0.55 %. Again the estimates of α_{2} and β_{2} show a positive correlation (Fig. 3E), though over the narrower range of values found here it is much more nearly linear than seen in Fig. 2D. The ratio of these two quantities, E_{2}=β_{2}/α_{2}, represents the efficacy for diliganded openings (Colquhoun, 1998). Because of the strong positive correlation between α_{2} and β_{2}, this ratio is better defined than either rate constant separately. The 1000 estimates of E_{2} shown in Fig. 3C have mean of 25.96 ± 32 (true value 26), so their CV is 2.9 % with an insignificant bias of −0.16 %. The total dissociation rate of agonist from diliganded receptors, k_{−2a}+k_{−2b}, was also welldefined. The distribution of 1000 estimates shown in Fig. 3D has a mean of 11 463 ± 573 s^{−1}, compared with a true value of 11 500 s^{−1}. The CV was 5.0 %, and bias −0.32 %. This is somewhat more precise that the two separate values, k_{−2a}=k_{−1a} (CV = 12 %, bias = 1.8 %) and k_{−2b}=k_{−1b} (CV = 6.0 %, bias = 0.1 %) (see Fig. 4F, H). In this example the negative correlation between these two values was modest (r=−0.274) so their sum is more precise than their separate values to a correspondingly modest extent.
The parameters for singly liganded receptors are generally less precisely estimated than those for diliganded receptors, especially when unconstrained (see below), but quite reasonable estimates can be found if the constraints in eqns (9) and (10) are true, as in the present case. Figure 4 shows the distributions of the estimates of the other free parameters for the same simulations as those shown in Fig. 3. These are the singly liganded opening and shutting rates, α_{1a}, β_{1a}, α_{1b} and β_{1b}, and the binding rate constants, k_{−1a}, k_{−1b} and k_{+1b}. In these fits k_{+1a}=k_{+2a} was fixed arbitrarily at 1 × 10^{8}m^{−1}s^{−1}, half its true value. It can be seen that the estimates of all of these parameters are tolerably good, apart from β_{1b}, which is, on average about half of its true value. This happens because k_{+1a}=k_{+2a} was fixed at half of its true value; if we fix k_{+2a} at its true value, 2 × 10^{8}m^{−1}s^{−1}, then good estimates of β_{1b} are found too. It is natural to ask, why it is primarily the estimate of β_{1b} that is affected by an error in the fixed value of k_{+1a}=k_{+2a}? There is a good intuitive reason for this happening. Inspection of the expressions for the equilibrium occupancies for scheme 1 (Fig. 1) shows that the relative frequencies of the two sorts of singlyliganded openings is given by:
 (11)
Furthermore, the frequency of openings with both sites occupied, relative to the frequency with only the a site occupied is given by:
 (12)
where c is the agonist concentration, and the corresponding relative frequency when only the b site is occupied is given by:
 (13)
The fit is sensitive to the values of these ratios of opening frequencies (in this particular case the open states are not connected to each other, so they are simply the ratios of the areas of the three components of the open time distribution). All three ratios will be unaffected by a decrease in the value of k_{+1a}=k_{+2a}, if, at the same time, β_{1b} is reduced by the same factor. Attempting to compensate for a reduction in k_{+1a}=k_{+2a} in other ways does not work. For example a concomitant increase in β_{1a} in eqn (11) can keep f_{1} unchanged, but will result in changes in eqns (12) and (13). It is only by decreasing β_{1b} that the predicted relative frequencies of the three sorts of openings will be unchanged.
Figure 5A–D shows the distributions of the equilibrium constants, calculated for each of the 1000 fits from the rate constants shown in Fig. 3 and Fig. 4. Figure 5A and B shows the two ‘efficacies’ for singly liganded openings, E_{1a} (=β_{1a}/β_{1a}) for when only the a site is occupied, and E_{1b} (=β_{1b}/α_{1b}) for when only the b site is occupied. The estimates are tolerable apart from the bias caused by specification of an incorrect value for k_{+1a}=k_{+2a}. The equilibrium constants for binding to a and b sites, K_{a} and K_{b}, are shown in Fig. 5 C and D (the notation K_{a} can be used because the constraints imply that K_{1a}=K_{2a} and similarly for the b site). Apart from the bias caused by specification of an incorrect value for k_{+1a}=k_{+2a}, the estimates are not too bad (CV = 11.6 % for K_{a} but larger (CV = 17.5 %) for K_{b}. The plot in Fig. 5E shows that there is quite a strong negative correlation (r=−0.74) between the estimates of k_{−1a}=k_{−2a} and of β_{1b}. Figure 5F shows a stronger positive correlation (r=+0.92) between the estimates of k_{+1b}=k_{+2b} and of β_{1a}. Correlations of this magnitude are a sign of ambiguity in the separate values of the parameters concerned.
The quality of the fit obtained in a single simulated experiment.Figures 3–5 showed the distributions of 1000 estimates of rate constants. In practice, experiments are analysed one at a time, and after the estimates of the rate constants have been obtained, the extent to which they describe the observations is checked. Figure 6 shows examples of these checks in the case of a single experiment that was simulated under exactly the same conditions as were used to generate Figs 3–5. More details of these plots are given in Methods (see Checking the quality of the fit).
Notice that the fit looks excellent despite the 2fold error in the (fixed) value of k_{+1a}=k_{+2a}, and the consequent error in β_{1b}. Figure 6A–C shows the data as histograms, for (A) all open times, (B) all shut times and (C) open times that are adjacent to short (up to 100 μs) shut times. On each of these histograms, the solid line that is superimposed on (not fitted to) the data is the appropriate HJC distribution calculated from scheme 1 using the values of the rate constants that were obtained for the fit and the imposed time resolution of 25 μs. The fitting was done as described for Figs 3–5. The HJC distributions (solid blue lines in Fig. 6A–C), were, as always, calculated from the exact expressions up to 3t_{res} (i.e. up to 75 μs in this case), and thereafter from the asymptotic form. The green line in Fig. 6B shows the asymptotic form plotted right down to t_{res}= 25 μs. It is seen to become completely indistinguishable from the exact value for intervals above about 40 μs, thus justifying the claim that the calculations are essentially exact. For the apparent open times in Fig. 6, the exact and asymptotic were hardly distinguishable right down to 25 μs (see Hawkes et al. 1992, for more details).
In Fig. 6A and B, the red dashed line shows the estimate of the ideal distribution (no missed events) calculated from the fitted rate constants (see Methods for details). It is clear from Fig. 6A that the apparent open times are greatly extended by the failure to detect many brief shuttings.
The conditional distribution in Fig. 6C shows that short openings very rarely occur adjacent to short shuttings (the dashed line shows the HJC distribution of all open times longer than 25 μs: see Methods).
Figure 6D shows a conditional mean open time plot. The diamond symbols show the data. Each represents the mean apparent open time for openings that are adjacent to shut times within a specified range. Seven shut time ranges were specified (see legend) and the means of the open times (blue diamonds) are plotted with their standard deviations (bars). The HJC predictions (calculated from the fitted rate constants and a resolution of 25 μs, as in Colquhoun et al. 1996) are shown, for the same ranges, as red circles. The dashed red line shows the theoretical continuous relationship between mean open time and apparent shut times, but this cannot be used directly as a test of fit, because shut time ranges must be used that are wide enough to encompass a sufficient number of observations.
Figure 6E and F shows the observed and the predicted dependency plot, respectively, for the same ‘experiment’ (see Methods). The dependency plot calculated from the fitted rate constants by the HJC method (Fig. 6F) shows that the shortest apparent shut times are much more likely to occur adjacent to long apparent openings than next to short openings, and that long apparent shut times are predicted to be rather more common adjacent to short shut times. The ‘observations’ (Fig. 6E) are qualitatively similar, but exact comparison is difficult with 3D plots, and a large number of observations is needed to get a smooth 3D plot.
The quality of internal estimates of variance and correlation. In the last fit of the set of 1000 shown in Figs 3–5, the Hessian matrix was calculated as described in Methods. The approximate standard deviations for the parameter estimates, and the correlations between pairs of estimates, were compared with the values measured directly from the 1000 fits. The values are shown in Tables 2 and 3.
Table 2. Approximate standard deviations obtained from the Hessian matrix in a single fit, compared with the values calculated directly from 1000 fits  Estimate  SD from one run  SD from 1000 fits 
α_{2}  2006.8  174  146 
β_{2}  53158.5  3891  3248 
α_{1a}  5825.8  197  193 
β_{1a}  39.8  6.6  8.4 
α_{1b}  46364.1  4031  4839 
β_{1b}  99.1  14.4  11.9 
k_{−1a}  1231.8  148  171 
k_{−1b}  9917.7  632  596 
k_{+1b}  3.26 × 10^{8}  0.53 × 10^{8}  0.64 × 10^{8} 
Table 3. Approximate correlations obtained from the Hessian matrix in a single fit, compared with the values calculated directly from 1000 fits  α_{2}  β_{2}  α_{1a}  β_{1a}  α_{1b}  β_{1b}  k_{−1a}  k_{−1b}  k_{+1b} 


α_{2}          
β_{2}  0.937         
 0.916         
α_{1a}  −0.043  −0.068        
 −0.006  −0.049        
β_{1a}  0.013  0.040  −0.033       
 0.042  0.075  0.011       
α_{1b}  −0.055  −0.095  0.489  −0.018      
 0.054  0.002  0.447  0.000      
β_{1b}  −0.107  −0.062  −0.025  −0.606  0.281     
 −0.087  −0.079  −0.023  −0.596  0.399     
k_{−1a}  −0.007  0.011  0.018  0.763  0.047  −0.791    
 0.028  0.055  0.025  0.799  0.029  −0.736    
k_{−1b}  −0.542  −0.525  0.193  −0.192  0.064  0.366  −0.219   
 −0.456  −0.451  0.188  −0.261  0.075  0.363  −0.275   
k_{+1b}  −0.106  −0.097  −0.040  0.915  −0.106  −0.588  0.669  0.028  
 −0.070  −0.064  0.027  0.916  −0.079  −0.585  0.700  −0.035  
There is good general agreement between the errors and correlations that are predicted in this particular ‘experiment’ and the values actually found by repetition of the experiment 1000 times. The calculation of errors via the Hessian matrix thus produces, at least in this case, a good prediction of what the real errors and correlations will be. Of course, in real life it is not so easy to repeat an experiment under exactly the same conditions. When experiments are repeated at different times, and with different batches of cells, we (Gibb et al. 1990; Hatton et al. 2003) and others (e.g. Milone et al. 1997; Bouzat et al. 2000) have often found quantitative differences between repeated experiments that are beyond what would be expected from experimental error.
Use of an EC_{50} value as a constraint. The fixing of a rate constant at an arbitrary value (as in Figs 3–5) is obviously an unsatisfactory solution to the problem of the patch containing an unknown number of channels. In real life we do not know the true value of a rate constant, and there are two ways to circumvent this problem. One is to fit simultaneously results at several different concentrations (see below). Another is to use an independently determined EC_{50} value to constrain the missing rate constant, (see Methods). The EC_{50} for the true rates in Table 1 is 3.3 μm. Rather than fixing k_{+1a}=k_{+2a} at an arbitrary value, its value is calculated at each iteration from the specified EC_{50} plus the values of the other rate constants.
When 1000 fits were done, like those shown in Figs 3–5, but with k_{+2a} calculated from the (correct) EC_{50} (3.3 μm), reasonable estimates were obtained for all nine free rate constants, including β_{1b}, for which the mean of all 1000 estimates was 158.4 ± 43.6 s^{−1} (true value 150 s^{−1}). The results for all the rate constants, with the specified EC_{50} being the correct value, are shown in Fig. 7 and Fig. 8.
As always, the rate constants for the diliganded receptor are better defined than those that refer to the two separate sites, but even the worst estimates are tolerable. This applies to the binding equilibrium constants for the two binding sites too, which are quite scattered. The distribution of K_{a} (Fig. 8E) has a CV = 21.8 % and bias =−1.4 %, and the distribution of K_{b} (Fig. 8F) has a CV = 17.3 % and bias =−1.3 %. However these two quantities show a strong (though not linear) negative correlation (Fig. 8G). Therefore it is not surprising that their product, K_{a}K_{b}, is rather more precisely determined, as shown in Fig. 8H, which has a CV = 6.35 % and bias −1.3 %. It is this product that occurs in those terms that refer to diliganded receptors in the expressions for equilibrium state occupancies.
The success of this procedure depends, of course, on having an accurate value for the EC_{50}, undistorted by desensitisation (unless desensitisation is part of the mechanism to be fitted). In general it will be best if the EC_{50} can be determined from a onechannel Popen curve determined under conditions similar to those used for the HJCFIT data. To test the effects of using an incorrect EC_{50}, the simulations were repeated but using an EC_{50} that was half, or double, the correct value.
When an EC_{50} of 6.6 μm (twice its correct value) was used, most of the parameters were still estimated quite well. The exceptions were β_{1b}, and k_{+1a}=k_{+2a}, both of which were too small, by factors of 4.0 and 4.3 respectively, as shown in Fig. 9.
When an EC_{50} of 1.65 μm (half its correct value) was used, the errors were worse. The estimates of the ‘diliganded parameters’, α_{2}, β_{2} and total dissociation rate, k_{−2a}+k_{−2b}, were still very good, as were the estimates of β_{1a} and α_{1b} (data not shown). The distributions of the estimates of the other parameters were all centred on means that were more or less incorrect. The largest errors were again in β_{1b}, and k_{+1a}=k_{+2a}, both of which were too big on average, by factors of 3.4 and 4.4 respectively. The means for the other rate constants were too big on average by factors that varied from 0.95 for k_{−2b}=k_{−1b}, to 1.35 for β_{1a}. Some of the results are shown in Fig. 10.
Correlations between parameter estimates. The correlation between estimates of two different parameters is a purely statistical phenomenon. It has already been illustrated in Figs 2D, 3E, 4E, 5E, 5F and 8G. If the estimates are precise enough the correlations vanish. It is quite distinct from the correlation between, for example, adjacent open and shut times (see Fig. 6C–F) which is a physical property of the mechanism, and gives interesting information about it (e.g. Fredkin et al. 1985; Colquhoun & Hawkes, 1987). The statistical correlation between parameter estimates resembles the negative correlation seen between repeated estimates of the slope and intercept of a straight line, or the positive correlation seen between the EC_{50} and maximum when fitting a Langmuir binding curve. It is merely a nuisance that limits the speed and accuracy of the fitting process. The correlation can be seen, in the form of a correlation coefficient, from the calculation of the covariance matrix (see Methods), as illustrated in Table 3.
Figure 11 shows in graphical form the correlations between all possible pairs of parameters, for the set of simulated fits shown in Fig. 7 and Fig. 8. They are arranged as in the correlation matrix shown in Table 3.
The effect of the strong correlation between the estimates of α_{2} and β_{2} on the fitting process is illustrated in Fig. 12, for an experiment on wild type human receptor (30 nm ACh, see Hatton et al. (2003). In this case the correlation coefficient between estimates of α_{2} and β_{2} was r= 0.915, a typical value. The likelihood surface is in 10dimensional space, and so cannot be represented. Fig. 12A shows a 3D ‘cross section’ of the actual likelihood surface that was constructed by calculating the likelihood for various values of α_{2} and β_{2}, with the seven other free parameters fixed at their maximum likelihood values.
The correlation appears as a diagonal ridge (coloured pink). Along this ridge, the values of α_{2} and β_{2} change roughly in parallel (so the efficacy, E_{2}=β_{2}/α_{2}, does not change much), and the likelihood increases only slowly towards its maximum (marked red). Figure 12B shows a contour representation of the same surface near its maximum. Dashed lines show the coordinates of the maximum point, the maximum likelihood estimates being α_{2}= 1524 s^{−1} and β_{2}= 50 830 s^{−1}. The contours are shown also for log(likelihood) values of L=L_{max}− 0.5 and L=L_{max}− 2.0. The tangents to these contours provide 0.5 and 2.0unit likelihood intervals for the estimates of α_{2} and β_{2} (these correspond roughly to one and two standard deviations, but being asymmetrical they provide better error estimates: see Colquhoun & Sigworth, 1995).
The effect of this correlation on the fitting process is illustrated in Fig. 13.
The vertical axis gives the likelihood that corresponds to the values of α_{2} and β_{2} that are reached at various stages during the fitting process. The initial guess is marked at the bottom of the graph, and the likelihood increases during the course of the fit. At first the increase is rapid but there is a long final crawl along a diagonal ridge near the maximum. This involves many changes of direction and slows the fitting process considerably, not least with the simplex method employed in HJCFIT. In this case the rate constants, not their logarithms, were used as the free parameters. However fitting the logarithms of the rates (see Methods) speeds up the fit and speed is not a problem in practice.
The effects of fitting as though the binding sites were independent when they are not. It is quite possible to obtain good fits to lowconcentration data even if it is assumed incorrectly that the binding sites are independent. The rate constants in Table 1 (labelled ‘true 2’) were used to simulate 1000 experiments. These rates represent sites that interact (see Hatton et al. 2003). The microscopic equilibrium constant for binding to the a site when the b site is vacant, K_{1a}=k_{−1a}/k_{+1a}= 20 μm, but for binding to the a site when the b site is occupied K_{2a}=k_{−2a}/k_{+2a}= 240 μm, so binding at the a site has a lower affinity if the b site is occupied; there is negative cooperativity in the binding of agonist to the shut channel (see Jackson, 1989 and Hatton et al. 2003). Likewise for binding to the b site K_{1b}=k_{−1b}/k_{+1b}= 0.33 μm, but when the a site is occupied K_{2b}=k_{−2b}/k_{+2b}= 4 μm. Again there is negative cooperativity in the binding of agonist to the shut channel.
These values were used to simulate the experiments, but during the fit, the (inappropriate) constraints in eqns (9) and (10) were applied. The initial guesses shown in Table 1 (‘guess 3’) also obeyed these constraints. A single low (30 nm) concentration was used and k_{+1a} (assumed, incorrectly, to be the same as k_{+2a} was constrained to give the specified EC_{50} (9.697 μm, its correct value). The results were fitted in bursts (t_{crit}= 3.5 ms), with CHS vectors (see Methods). Although good fits could be obtained to the distributions of apparent open and shut times, many of the parameter estimates were quite wrong, as shown in Fig. 14 and Fig. 15.
Figure 14A and B shows that the estimates of rates constants in a single fit (actually the last of the 1000 fits) predict well the distributions of apparent open time, and apparent shut time. Figure 14C shows that the conditional open time distribution, for openings that are adjacent to the shortest shut times (25–100 μs), is also predicted well. However the fact that something is wrong is shown, in this case, by the dependence of mean open time on adjacent shut time (Fig. 14D). Although the prediction of the fit is quite good for the shortest shut times (as shown also in Fig. 14C), and for the longest shut times, the prediction is quite bad for shut times between about 0.3 and 30 ms. This is also visible in the conditional apparent open time distribution shown in Fig. 14E. This shows the distribution of apparent open times that are adjacent to shut times in the range 0.5–10 ms, and the predicted fit is bad.
Examples are shown in Fig. 15 of the distributions of rate constants obtained in 1000 fits that were done under the same conditions as the single fit shown in Fig. 14. Despite the grossly incorrect assumptions (and the somewhat subtle indication of imperfect fit shown in Fig. 14D and E), the estimates of the ‘diliganded’ rate constants, α_{2} and β_{2} are nevertheless quite good (Fig. 15A and B). The estimates of the total dissociation rate from diliganded receptors, k_{−2a}+k_{−2b}, was estimated reasonably well too (Fig. 15C), though with some bias (true value, 14 000 s^{−1}, mean of 1000 estimates 14 900 s^{−1} with a CV of 6.3 % and bias +6.4 %). However, as might be expected, the rate constants that refer to the two separate sites are not wellestimated, being anything from poor to execrable. The estimates for β_{1a} and α_{1b} were poor (bias +61 % and +12 % respectively), but the estimates of β_{1a} (shown in Fig. 15D) and β_{1b} were worse (bias −63 %, CV 30 % for β_{1a}; bias +135 %, CV 12.1 % for β_{1b}), and the estimates of the association and dissociation rates were inevitably very poor. For example the estimates of k_{−1a} (true value 400 s^{−1}) and of k_{−2a} (true value 2000 s^{−1}) were constrained by the fit to be the same, and had a mean slightly below either true value, 367 s^{−1}; this distribution is shown in Fig. 15E and F (on two different scales, to allow display of the arrow that indicates the true values of k_{−1a} (Fig. 15E) and of k_{−2a} (Fig. 15F).
Nonindependent binding sites
Up to now, the two binding sites have always been assumed to be different from each other, but independent of one another. If it is allowed that the binding of the agonist to one site can affect the binding to another site, so the constraints in eqns (9) and (10) can no longer be applied, there are 13 free rate constants to be estimated, rather than 10. In Fig. 14 and Fig. 15, the effects were investigated of fitting as though the two sites were independent when they are not. We now describe attempts to fit all 13 rate constants in the case where the sites are not independent.
In every case that has been investigated so far it has proved impossible to get good estimates of all 13 rate constants under conditions that can be realised in practice (this is true, at least, for the values of the rate constants used here). In particular, they cannot be obtained under conditions where the number of channels that are present in a lowconcentration record is unknown. At high concentrations it is possible to obtain long stretches of record that are known to contain only one channel, but high concentration records alone do not contain enough information about singly liganded states to allow estimation of all 13 rate constants. The same is true if high concentration records are fitted simultaneously with low concentration records, the latter being fitted in bursts because of the unknown number of channels in the patch. Fixing one of the rate constants, or determining one of them from a known EC_{50}, does not help either, and improving the resolution from 25 μs to 10 μs does not solve the problem. In most of these simulations, the problem lay in separating the values of k_{+1a} and k_{+1b}, the estimates of all the other rate constants being good or at least acceptable. The estimates of these two rate constants were very smeared, with a tendency to approach unreasonably large values, no doubt as a result of the strong positive correlation seen between them (their ratio was better determined). Values for k_{−1b} were poor too. Despite the nearuseless estimates of at least two of the rate constants, good predictions of the data (such as those shown in Fig. 6 and Fig. 14) could be obtained, as might be expected from the large number of free parameters.
Examples of poorly estimated parameters are shown in Fig. 17. These are from an attempt to fit all 13 parameters with a large amount of highresolution data, three low concentrations (10 nm, 30 nm and 100 nm) fitted simultaneously, with a resolution of 10 μs. However all three records were fitted in bursts (t_{crit}= 3.5 ms), with CHS vectors, to avoid any assumption about the number of channels in the patch. All the parameters were reasonably estimated apart from the three shown: k_{+1a} (Fig. 17A) and k_{+1b} (Fig. 17C) are very smeared, and k_{−1b} (Fig. 17B) is poor. This is a reflection of the strong positive correlation between k_{+1a} and k_{+1b}, (Fig. 17D); the correlation coefficient was +0.61 in this plot (which is curtailed by an upper limit of 10^{10}m^{−1} s^{−1} placed on any association rate constant during the fit. The correlation coefficient between estimates of the equilibrium constants for the first bindings, K_{1a} and K_{1b}, was very strong indeed (+0.99995 in this case).
The only way in which it has proved possible, so far, to get good estimates of all 13 rate constants, is by simultaneous fit of records at two concentrations, with the assumption that only one channel is present at the low concentration(s) as well as at the high concentration(s). In other words the low concentration records are not fitted in bursts, but the likelihood is calculated from the entire sequence, including all shut times. Good estimates of all 13 free rate constants could be obtained by simultaneous fit of two low concentrations (10 nm and 100 nm), or by simultaneous fit of a low concentration (30 nm) and a high concentration (10 μm), as long as it was supposed that the low concentration record(s) originated from one channel. In this case the entire shut time distribution is predicted by the fit, not only shut times up to t_{crit}. Fits predicted from one such simulated experiment are shown in Fig. 18A–D.
The distributions of both apparent open times (Fig. 18A and C) and of apparent shut time (Fig. 18B and D) are predicted well by a single set of rate constants at both concentrations. The shut time distributions (Fig. 18B and D) include all shut times (above t_{res}), and the longer time between activations at the lower concentration is obvious. The estimates of all 13 rate constants found in 1000 such fits were good, but the only ones shown are the distributions of k_{+1a}, k_{−2b} and k_{+1b} (Fig. 18E–G). The estimates of these are now quite good, whereas the estimates of these rates shown in Fig. 17A–C (with more and better data, but fitted in bursts) were bad. The estimates of k_{+1a} and k_{+1b} are now essentially independent (Fig. 18H), rather than strongly correlated (Fig. 17D). The set of 1000 fits exemplified in Fig. 18 gave CVs of about 5 % or less for α_{2}, β_{2}, β_{1a}, α_{1b}, k_{−2a}, (k_{−2a}+k_{−2b}), and for the corresponding equilibrium constant E_{2} (all with bias less than 0.4 %); CVs of about 5–10 % were found for β_{1b}, k_{+2a}, k_{−1a}, k_{+1a}, k_{+1b}, and for the corresponding equilibrium constants E_{1b}, K_{2a}, K_{1b} (all with bias less than 1 %). The least precise estimates were for β_{1a}, (CV = 11.5 %, bias =+0.84 %), k_{−1b} (CV = 11.4 %, bias =+0.96%), k_{−2b} (CV = 18.7 %, bias =−1.2 %), k_{+2b} (CV = 17.6 %, bias =−0.7 %), and for E_{1a} (CV = 12.8 %, bias =+1.3 %), K_{2b} (CV = 13.1 %, bias =+0.1 %), K_{1a} (CV = 12.6 %, bias =+1.2 %).