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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(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 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]