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Single Ion Channels
Overview
Biology
 Modeling
 Paper

Ion Channels

What they are
 Protein
molecules spanning lipid bilayer membrane of
a cell, which permit the flow of ions through the
membrane
 Subunits form channel in center
 Distinguished from simple pores in a cell membrane
by their ion selectivity and their changing states, or
conformation
 Open and close at random due to thermal energy;
gating increases the probability of being in a certain
state
Ion Channels
Source: Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 404
Ion Channels

Why they are important
 Essential
bodily functions such as
transmission of nerve impulses and hearing
depend on them
 Membrane potential created by ion channels
is basis of all electrical activity in cells
 Transmit ions at much faster rate (1000 x)
than carrier proteins, for example
Ion Channels

Gating examples
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 407
Transmitter-Gated Channel in
Postsynaptic Cell
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 418
Voltage-Gated Na+ Channel in
Nerve Axon
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 413
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 407
Voltage-Gated Na+ Channel in
Nerve Axon (cont’d)
Stress-Activated Ion Channel in
Ear
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 408
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 406
How Ion Channels Are Observed
Modeling


Mathematical models mimic behavior in the real
world by representing a description of a system,
theory, or phenomenon that accounts for its
known or inferred properties and may be used
for further study of its characteristics. Scientists
rely on models to study systems that cannot
easily be observed through experimentation or
to attempt to determine the mechanism behind
some behavior.
Advantages
Modeling Ion Channels

Behaviors C and H tried to model
 Duration
of state (Probability Distribution
Function)

Open, Shut, Blocked
 Transition

probabilities
Open to Shut
Duration of State of Random Time
Intervals





Length of time in a particular state (open, shut, blocked)
PDF based on Markovian assumption that the last
probability depends on the state active at time t, not on
what has happened earlier
Open channel must stretch its conformation to overcome
energy barrier in order flip to shut conformation
Each stretch is like binomial trial with a certain probability
of success for each trial
Stretching is on a picosecond time scale, so P is small
and N is large, and binomial distribution approaches
Poisson distribution
Duration of State (cont’d)

Cumulative distribution of open-channel lifetimes:



PDF of open-channel lifetime:




F(t) = Prob(open lifetime  t) = 1 – exp(-t)
Forms an exponentially increasing curve to Prob = 1
f(t) =  exp(-t)
Forms an exponentially decaying curve
Exponential distribution as central to stochastic processes as
normal (bell-curve) distribution is to classical statistics
Mean = 1/(sum of transition rates that lead away from
the state); in this case, 
Transition Probabilities
where the transition leads when it
eventually does occur
 Two transition types of interest


the number of oscillations within a burst
 the probability that a certain path of
transitions will occur
Bursts
Geometric Distribution
 P(r) = (12 21) ^r-1 13



13 = (1-  12)
Example
openings the open channel first blocks 
12, then reopens  21, and finally shuts.
 Product of these three probabilities ( 12 
21)  13
 Two
Pathways
Markov events are independent
 from conditional probability, P(AB) is
P(A) * P(B) if A and B are independent.
 Easily calculated by using the one-step
transition probability matrix which contains
probability of transitioning from one state
to another in a single step.

2 State Model
Duration of state = 1/
 Transition Probabilities

 Open
to shut to open
 Probability of open to shut * Probability of shut
to open * Probability of open to shut
(Conditional Check this)
Three-State Model Diagram and Q
Matrix
Computation of the Models
Equation approach – as the system
increases in states the possible routes
also increases which complicates the
probability equations (openings per burst)
Matrix approach – single computer program
to numerically evaluate the predicted
behavior given only the transition rates
between states
Five-State Model Diagram and Q
Matrix
How it’s used
Subset matrices
Q
P

Five-State Q Matrix, Partitioned
Into Open and Shut State Sets
Example: Shut time distribution
for three-state model

Standard method

f(t) = (/+k+BxB)’exp(-’t)+(k+BxB/+k+BxB)k-Bexp(-k-Bt)
 Two shut states intercommunicate through open state
  and k+B: transitions from open state
 ’ and k-B: transitions to open state

Q-Matrix method
f(t) = S exp(QFFt)(-QFF)uF
 S is a 1 x kF row vector with probabilities of starting a shut time in each
of the kF shut states
 QFF is a kF x kF matrix with the shut states from the Q matrix
 uF is a kF x 1 column vector whose elements are all 1 (sums over the F
states)

Conclusion




Matrix notation makes it possible to write a
general program for analyzing behavior of
complex mechanisms
Matrix is constrained by the number of states
which can be observed
The nature of random systems means that they
must be modeled using stochastic mechanisms
The microscopic size of ion channels
necessitates generalizing to a system by
observing [a subset]