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Unformatted text preview: 1 Stochastic Systems Richard Bertram Department of Mathematics and Programs in Neuroscience and Molecular Biophysics Florida State University Tallahassee, Florida 32306 2 The HodgkinHuxleytype models of nerve activity are based on ionic currents that are produced as the result of ion ux through channels. We have thus far assumed that there are enough channels so that a deterministic approach to the frac tion of open channels is appropriate. However, this assump tion is not always valid. For example, patch clamp electrical recordings measure the electrical activity in a small patch of membrane, where there are only a few channels. Figure 1: Cellattached patch clamp illustration. (en.wikipedia.org/wiki/Patch clamp) 3 Figure 2: Current measurement using cellattached patch clamp. (www.biophysj.org/cgi/content/full/82/6/3056) 4 As another example, in some cells there are only a few (100 or less) channels of a particular type. In this case, the vari ance in the number of open channels can be quite large com pared with the mean, so once again the deterministic approach (which is a description of the mean) is questionable. As a nal example, in some cases the noise in the system can be a key component for signal ampli cation and transmission. Leaving out the noise fundamentally changes the signal transduction process. Single channel as a 2state Markov process A simple ion channel has a single closed and single open state: C O k _ k + Figure 3: State diagram for a single ion channel The state of the system is given by the random variable 5 s ∈ C, O . De ne P c ( t ) = Prob[ s = C, t ] (1) P o ( t ) = Prob[ s = O, t ] (2) The parameter k + is the C → O rate (units of ms 1 ). If the channel is closed at time t , the probability that it will open by time t + ¡ t is Prob[ s = O, t + ¡ t  s = C, t ] = k + ¡ t . (3) This is a conditional probability . Must multiply by the probability that the channel is in state C at time t . Finally, Prob[ C → O ] = Prob[ s = O, t + ¡ t  s = C, t ] × P c ( t ) = k + ¡ tP c ( t ) . (4) C O A closed channel must either open or stay closed, so Prob[ s = C, t + ¡ t  s = C, t ] = 1 k + ¡ t (5) 6 and therefore, Prob[ C → C ] = Prob[ s = C, t + ¡ t  s = C, t ] × P c ( t ) = (1 k + ¡ t ) P c ( t ) . (6) C O Similarly, Prob[ O → C ] = k ¡ tP o ( t ) (7) C O and Prob[ O → O ] = (1 k ¡ t ) P o ( t ) . (8) C O What is the probability that the channel is closed at time t + ¡ t ? 7 P c ( t + ¡ t ) = Prob[ C → C ] + Prob[ O → C ] = (1 k + ¡ t ) P c ( t ) + k ¡ tP o ( t ) (9) C O Similarly, P o ( t + ¡ t ) = Prob[ O → O ] + Prob[ C → O ] = (1 k ¡ t ) P o ( t ) + k + ¡ tP c ( t ) (10) C O These transition probabilities can be written in matrix/vector form as ~ P ( t + ¡ t ) = T ~ P ( t ) (11) where ~ P = P c P o (12) 8 and where T is the transition probability matrix : T = Prob[ s = C, t + ¡ t  s = C, t ] Prob[ s = C, t + ¡ t  s = O, t ] Prob[ s = O, t + ¡ t  s = C, t ] Prob[...
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 Fall '07
 Zhang
 Calculus, Trigraph, TA, Kc, proility tht, t time

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