The opening and closing behaviour of ion channels

From a cell physiological point of view, the mechanism of the opening and closing of ion channels in nerve cells is very important. However, it is not so easy to perform experiments at molecular level, so these mechanisms must be inferred indirectly from measurements of total cell performance. It is possible to measure the electrical flux between two clamps attached to a dendrite of a single cell. It varies somewhat in time, and the first question is: can we make use of these variations in time to disentangle the opening and closing of individual channels? With aid of a model for the behaviour of these channels, it is possible indeed. The argument runs as follows. Assume that the electrical flux is proportional to the number of ion channels that are open. Assume further that the probability that a randomly chosen closed channel will open in an incremental time interval equals . Similarly, the probability that a randomly chosen open channel will close in that interval is assumed to be . For the moment we will assume that the probability rates and $\mu$ are constant and that the probability that a channel opens as well as closes in is negigibly small. When denotes the number of channels between the clamps, the number of channels that are open and $t$ the time, the probability that there are open channels at time $t$ is given by

To simplify the notation, we assume that . If we let approach to zero, we arrive at

This equation describes the opening and closing behaviour of ion channels, in which we consider , and $\mu$ as parameters to be estimated from a continuous registration of the electrical flux, so the number of open channels. With a very small probability, all channels are open at a given time, so when we wait long enough, the maximum flux over a very long time bears information about the number of channels . However, it is very hard to keep conditions constant over such long a period. Therefore we will not make use of this possibility. What we will do is use the mean, the variance and the autocovariance function to lead us to the three parameters. The mean and variance can be deduced from the equilibrium distribution for , i.e. the case where . Using (16.2) we obtain

This we recognize as the binomial distribution with mean and variance . The straightforward way to obtain the covariance function is first to solve from (16.2) from the initial condition . Let us call this solution . When we denote the equilibrium distribution by , the (equilibrium) autocovariance function is given by . The problem now is that it is very laborious to obtain . In this case it is helpful to note that we do not need it explicitely, we only need it as a weighted sum over all and . We therefore try to convert the set of differential equations of all the 's into one for the autocovariance function. We note that , with . From (16.2), we obtain

After some manipulation, we obtain the simple expression

(16.5)

which leads to . So the autocovariance function decreases exponentially with rate . It is instructive to relate this model formulation with a very simple deterministic one, where we assume that is large enough to allow a continuous approximation. When the opening of channels is proportional to the number of closed channels and the closing of channels is proportional to the number of open channels, we have . The solution is , and we recognize as the relaxation time. Back to the stochastic model now. The simplest way to proceed is to uncover from . We can do this e.g. by plotting against time and fit a straight line. Although this procedure must be classified as "quick and dirty", it is not at all obvious how to formulate a "clean" procedure. Next we multiply by and obtain $\mu$. Subsequently we uncover and . The ability to disentangle the opening from the closing rate can be very valuable in the experimental research to which (environmental) factors influence both mechanisms. The constraint that and $\mu$ are constant can be relaxed to the constraint that the rate at which they vary is small with respect to .
Filename: ex016; Date: 1990/09/02; Author: Bas Kooijman