(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
)Δ
t
) that in the interval between
t
and
t
+Δ
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 closedopen transition. Start by sketching an ion
channel current trace and show what is meant by the waiting time from one
closedopen 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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 Winter '09
 Bacteria, Ki Database

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