Single Ion Channels

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Single Ion Channels
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Overview

• Biology
• Modeling
• Paper

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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

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Ion Channels
Source Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 404
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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

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Ion Channels

• Gating examples

Source Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 407
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Transmitter-Gated Channel in Postsynaptic Cell
Source Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 418
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Voltage-Gated Na Channel in Nerve Axon
Source Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 413
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Voltage-Gated Na Channel in Nerve Axon (contd)
Source Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 407
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Stress-Activated Ion Channel in Ear
Source Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 408
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How Ion Channels Are Observed
Source Alberts et al., Essential Cell Biology,
Second Edition, 2004, p. 406
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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

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Modeling Ion Channels

• Behaviors C and H tried to model
• Duration of state (Probability Distribution
  Function)
• Open, Shut, Blocked
• Transition probabilities
• Open to Shut

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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

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Duration of State (contd)

• Cumulative distribution of open-channel
  lifetimes
• F(t) Prob(open lifetime ? t) 1 exp(-?t)
• Forms an exponentially increasing curve to Prob
  1
• PDF of open-channel lifetime
• 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, ?

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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

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Bursts

• Geometric Distribution
• P(r) (?12 ?21) r-1 ?13
• ? 13 (1- ? 12)
• Example
• Two openings the open channel first blocks ? 12,
  then reopens ? 21, and finally shuts.
• Product of these three probabilities (? 12 ? 21)
  ? 13

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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.

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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)

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Three-State Model Diagram and Q Matrix
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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

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Five-State Model Diagram and Q Matrix
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How its used

• Subset matrices
• Q
• P

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Five-State Q Matrix, Partitioned Into Open and
Shut State Sets
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Example Shut time distribution for three-state
model

• Standard method
• f(t) (?/?kBxB)?exp(-?t)(kBxB/?kBxB)k-Bex
  p(-k-Bt)
• Two shut states intercommunicate through open
  state
• ? and kB 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)

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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