7.4.P10
Prerequisites:
Many biological systems are welldescribed by the laws of statistical physics. A simple yet often powerful approach is to think of a system as having only two states. For example, an ion channel may be open or closed. In this problem, consider a simple model of membrane channels for ions: The system is described by a Boltzmann distribution with only two states, with energies ε_{1 }(open) and ε_{2} (closed). Assume the “open” state is the state of higher energy, so that ε_{1} > ε_{2}.
1. If the probability of finding an ion channel open is p_{open }and the probability of finding the ion channel closed is p_{closed }, which of the expressions below best represents the relative probability of open to closed, R = p_{open} /p_{closed }?
2. Since channels are either open or closed, it must be the case that p_{open} + p_{closed} = 1. You can therefore express the absolute probability of finding the channel open as p_{open} / (p_{open} + p_{closed}) and work out an explicit expression for it. Which of the expressions below best represents the absolute probability of finding the channel open?
3. The two curves in the figure to the right show the probabilities of finding an ion channel either open or closed as a function of ε/k_{B}T = (ε_{1}  ε_{2})/k_{B}T. Think about the way accessible energy states, probability, and temperature work together, examine the figure, and identify the true statements in this list: 

 The top curve is the probability of finding the ion channel open.
 The top curve is the probability of finding the ion channel closed.
 The curves show that the probability of finding the ion channel open is greater at low temperature.
 The curves show that the probability of finding the ion channel open is greater at high temperature.
 If (ε_{1} ε_{2}) is much less than k_{B}T, the relative probability R is approximately 1.
4.1 At approximately what value of ε/k_{B}T are the ion channels three times more likely to be closed than open?
4.2 If a membrane could change its value of ε, what change would it make to open more ion channels?
Joe Redish & Bill Dorland 2/7/18
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