C lets now imagine that time is discretized into a

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(c) Let’s now imagine that time is discretized into a series of steps of length Δ t . At the initial instant, imagine that the channel is closed. What is the probability that in the next time interval Δ t that it will switch to open? What is the probability that in that same time interval Δ t that it will stay closed? Using those insights, now write an expression for the probability ( p closed ( t t ) that in the interval between t and t t the channel will switch from closed to open, having stayed closed the entire time until then. Make a corresponding derivation for p on ( t ). What can you say about the waiting time distributions? What are the time constants for the open and closed waiting time distributions? Using these distributions, compute the average time that the channel stays in each of the states. Make sure to compute this as an average and explain what integrals you write down and why. Given the nature of ion channel current traces, explain how you could go about determining these distributions and finding these average times. Hint: Remember that the exponential is characterized by the interesting property lim N →∞ (1 - x N ) N = e - x , (2) and use the fact that N = t/ Δ t . (d) Now we follow up on the results of part (c) to ask a more sophisticated question by considering the distribution of waiting times from one closed- open transition to the next closed-open transition. Start by sketching an ion channel current trace and show what is meant by the waiting time from one closed-open transition to the next. In particular, make a cogent argument that this waiting time distribution is given by p successive ( t ) = Z t 0 p open ( t - τ ) p closed ( τ ) dτ. (3) Then, using the results of part (c) for p open ( t ) and p closed ( t ), obtain an analytic form for this distribution and plot it and explain its features and significance.
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