106.

[Cover] [Abbreviated Contents] [Contents] [Index]

Page 200
2.5.5—
Ion Channel Kinetics:
Random or Deterministic?
An ion channel protein in the cell membrane switches between states that are open or closed to the flow of ions. We have already seen that the open and closed times have fractal properties. The fractal analysis describes the statistical properties of the open and closed times. It does not tell us whether the switches between the open and closed states are generated by a random or by a deterministic process.
It has been known for 40 years that a random process can generate the statistical properties of the open and closed times found in the experimental data. We showed that a deterministic process could also generate the same statistical properties. For example, the following random and deterministic processes each generate the number of times that a channel is open or closed for a duration of time t that is proportional to a single exponential of the form exp(-kt). More complex models can be constructed from pieces consisting of these models that will have numbers of open and closed times proportional to the sum of single exponential terms.
1—
Random:
Markov Model
There is a certain probability at each time interval that the channel will continue in its present state and a certain probability that it will switch states. Although the probabilities are known, the time at which the switch occurs is set by chance.
2—
Deterministic:
Iterated Map Model
Let x(n) be the value of the current through the channel at a time n. The value of the current x(n+ 1) at the next point in time n+ 1 is a function of the current x(n) at time n. This function can be shown on a plot of x(n+l) versus x(n). In this plot, the y-axis is the value of the current at the next point in time and the x-axis is the value of the current at the present time. One example of such a function is shown by the heavy lines.
To compute the values of the current with time, start with the first value x(l) of the current on the x-axis, move up vertically to the heavy line, and then move left to read off the next value x(2) on the y-axis. Then start with this new value x(2) on the x-axis and repeat the procedure. This type of model is called an iterative map.

 
[Cover] [Abbreviated Contents] [Contents] [Index]


Fractals and Chaos Simplified for the Life Sciences
Fractals and Chaos Simplified for the Life Sciences
ISBN: 0195120248
EAN: 2147483647
Year: 2005
Pages: 261

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