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 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

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

Voltage-Gated Na+ Channel in Nerve Axon (cont’d)

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Stress-Activated Ion Channel in Ear

Source: Alberts et al., Essential Cell Biology, Second Edition, 2004, p. 408

How Ion Channels Are Observed

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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

Modeling Ion Channels

Behaviors C and H tried to modelDuration 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: 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,

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

Two openings the open channel first blocks 12, then reopens 21, and finally shuts.

Product of these three probabilities ( 12 21) 13

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.

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)

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]