There are two good reasons for trying to identify kinetic mechanisms for receptors. Firstly, it is only by doing so that one can study sensibly the effect of structure changes in agonists (for example, does the change in structure alter the ability to bind, or the ability of the agonist to activate the receptor once bound?). Secondly, it is only by doing so that the effect of mutations in a receptor can be studied rationally (for example, does the mutated residue form part of the agonist binding site?). These questions have been reviewed by Colquhoun (1998).

In order to answer the questions of interest, two things must be done. First a qualitative reaction scheme must be postulated, and then values for the rate constants in the scheme must be found. In many ways the first step is the harder, because unless the reaction scheme is a sufficiently good description of actual physical structural reality, it cannot be expected that physically meaningful conclusions can be drawn from it.

The only sort of receptor for which it has so far been possible to achieve these aims are the agonist-activated ion channels, and then only by observation of single ion channels. In earlier studies (e.g. Colquhoun & Sakmann, 1981, 1985), rate constants in the mechanism could not be estimated directly. Rather, individual distributions (shut times, open times, burst lengths etc.) were fitted separately with arbitrary mixtures of exponential distributions (e.g. Colquhoun & Sigworth, 1995), and correlations between these quantities were measured separately. It was not possible to take into account all of the information in the record simultaneously, so information from individual distributions had to be cobbled together in a rather arbitrary way to infer values for the rate constants in the mechanism. It was also not possible to make proper allowance for the inability to resolve brief events in a single channel record. Since that time, better methods of analysis have been devised, the most appealing of which is to maximise the likelihood of the entire sequence of open and shut times. The appeal of this method stems from the facts that (a) it provides estimates of the rate constants in a specified mechanism directly from the observations, (b) it is based on measurements of open and shut times (an ‘idealisation’ of the observed record), so the user has a chance to check the data going into the calculation, (c) the calculation can be carried out without having to choose arbitrarily which particular distributions to plot and (d) it takes into account correctly the fact that in most real records subsequent intervals are not independent of one another (it is common, for example, to find that long open times are followed on average by short shut times), and uses all of the information in the observed record in a single calculation (Fredkin *et al.* 1985). Since, in the usual general treatment of ion channels, the rate constants for the connections between each pair of states are tabulated in the ** Q** matrix, it may be said that the method provides an estimate of the

**matrix. The method was first proposed by Horn & Lange (1983), but at that time it was not possible to allow for the fact that brief events cannot be seen in experimental records. The implementation of the method by Ball & Sansom (1989) had the same problem. Since many brief events are missed in most experimental records, the method was not useable in practice until this problem had been solved. Ball & Sansom (1988**

*Q**a*,

*b*) gave the exact solution for the missed events problem in the form of its Laplace transform, and various approximate solutions have been proposed too, the best of which appears to be that of Crouzy & Sigworth (1990) (see Hawkes

*et al.*1990; Colquhoun & Hawkes, 1995

*b*). However there is no longer any need for approximations because the exact solution to the problem has been found by Hawkes

*et al.*(1990, 1992).

Two computer programs are available for doing direct maximum likelihood fitting of rate constants, MIL (Qin *et al.* 1996, 1997) and HJCFIT (Colquhoun *et al.* 1996). The MIL program is available at http://www.qub.buffalo.edu/index.html and HJCFIT from http://www.ucl.ac.uk/Pharmacology/dc.html. The former uses (a corrected form of) the approximate missed event method of Roux & Sauve (1985); the latter uses the exact solution.

It is the responsibility of anyone who proposes an estimation method to describe the properties of the estimators, and in this paper we describe some of the properties of estimates of rate constants found with HJCFIT. This provides the background for the method, and the necessary justification for the use of HJCFIT to analyse experimental results on nicotinic receptor channels in the accompanying paper (Hatton *et al.* 2003).