1. Single Ion Channels

2. Overview
Biology
Modeling
Paper

3. 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

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

5. 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

6. Ion Channels Gating examples
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 407

7. Transmitter-Gated Channel in Postsynaptic Cell
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 418

8. Voltage-Gated Na+ Channel in Nerve Axon
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 413

9. Voltage-Gated Na+ Channel in Nerve Axon (cont’d)
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 407

10. Stress-Activated Ion Channel in Ear
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 408

11. How Ion Channels Are Observed
Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 406
Patch Clamp Recording
Fine glass tube forms seal against a patch of membrane and detaches it
Microelectrode measures current flowing through surrounded channels
Vary concentrations of ions on either side to test which ions go through
Voltage can be “clamped” at certain value to measure effect of membrane potential on openings and closings

12. 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

13. Modeling Ion Channels Behaviors C and H tried to model
Duration of state (Probability Distribution Function)
Open, Shut, Blocked
Transition probabilities
Open to Shut

14. 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

15. Duration of State (cont’d)
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, 

16. 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

17. Bursts Geometric Distribution P(r) = (12 21) ^r-1 13 Example
 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

18. 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.

19. 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)

20. Three-State Model Diagram and Q Matrix

21. 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

22. Five-State Model Diagram and Q Matrix

23. How it’s used
Subset matrices Q P

24. Five-State Q Matrix, Partitioned Into Open and Shut State Sets

25. 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)

26. 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]