The measurement of membrane current was discussed in the preceding lecture on the voltage clamp technique. The membrane current represents the movement of ions across the membrane through special ion specific channels. In the fifty years since the initial voltage clamp experiments, there is abundant experimental evidence to show that the channels act in a binary fashion, i.e., they are either open or closed. When open, a channel will allow the passage of ions. The macroscopic conductance of the membrane to the ion in question depends on the number of open channels. Graded variation in the macroscopic conductance is due to graded change in the number of open channels.
If the rate constant of channels opening (from the closed state) is &alpha and the rate constant of channels closing from the open state is &beta then the channel dynamics can be represented as follows:
The number of closed channels that become open at this time is: &alpha NC(t), and the number of open channels becoming closed at this time is: &beta No(t). If we consider a small interval of time &delta t, then the change in the number of open channels can be written as:
&delta No(t) = [&alpha NC(t) - &beta No(t)] &delta t
This can be rewritten as a differential equation:
The opening and closing of individual channels behaves like a random process. Therefore, even under steady state conditions, channels are individually opening and closing, and the mean ionic current is seen as the macroscopic current. If the current through an individual channel is recorded (as can be done with a patch clamp recording), it appears as a random binary process as shown in Figure 1 with the current taking one of two possible values (either the channel is open or it is closed – no intermediate conducting state):
Figure 1
When more channels are in the area of observation we will see several current levels resulting from many channels being open simultaneously as show in Figure 3.
Figure 2
The rate equation for an individual channel can be written in terms of the probabilities as follows:
This has a solution of the form:
The current in a single channel is very difficult to observe, but macroscopic current fluctuations due to a small population of ion channels can be more easily seen. The above expression holds even if the number of channels is increased, as only the constant A will change in value.
Since po(t) is a random process, the power spectrum of the above expression can be determined as the squared magnitude of the Fourier transform (the stylised F, “ℑ”, is used to inidcate the Fourier transform). The squared Fourier magnitude of po(t) is:
If this Fourier magnitude spectrum is plotted on a log-log scale, the resulting function is shown in Figure 3.
Figure 3
The function described in the above equation and shown in Figure 3 is called a Lorentzian. The corner frequency can be determined directly from the equations as:
The low frequency spectral power can be shown to be:
where I is the mean current measured and N is the number of channels available in the area of membrane under observation.
And further, the variance of the fluctuations can be shown to be:
Using fc, S(0), I, &sigma from the fluctuations spectrum, and estimated value of N we can calculate, the rate constants a and b using the above set of three equations.
The Fourier spectrum of simulated channel fluctuations with a fitted Lorentzian curve is shown below in Figure 4:
Figure 4
Note on calculating the spectrum of the current fluctuations
In order to calculate the spectrum of the fluctuations, a block of data is multiplied by a suitable window (say, Hamming window) and then transformed using an FFT routine. The square of the Fourier transform can then be calculated.
Note on simulation of the channels
The channel fluctuations can be modelled as a Poisson random process. Such a random process mimics the experimental data quite well as shown in the accompanying simulation.
Simulation of Fluctutuation Analysis
Suresh Devasahayam
July 2004