single ion channels. overview biology modeling paper
TRANSCRIPT
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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 importantEssential 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 (cont’d)
Sou
rce:
Alb
erts
et a
l., E
ssen
tial
Cel
l Bio
logy
, Sec
ond
Edi
tion
, 200
4, 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
Sou
rce:
Alb
erts
et a
l., E
ssen
tial
Cel
l Bio
logy
, Sec
ond
Edi
tion
, 200
4, 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 modelDuration 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 (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,
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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(AB) 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 openProbability 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 it’s 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) = (/+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)
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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]