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Unformatted text preview: Computational Methods in Biology (Spring 2011) Assignment 3 (due in class on March 14) This assignment focuses on stochastic models and simulation methods, using ion channel gating as the model system. At the end of this sheet there is a brief descrip- tion of how random (actually pseudo-random) numbers are generated on a computer. 1. [20 points] Consider the gating of a single two-state ion channel, with C → O transition probability k + and O → C probability k- (units of both are ms- 1 ). Write a program for a Monte Carlo simulation of the gating of this channel for a duration of 500 msec, and time step ¡ t = 0 . 5 msec. Let S = 0 when the channel is in a closed state, and S = 1 when it is in an open state. Plot S vs. time. Also, compute the running sample mean of S and plot it along with S . [Sample mean = ∑ n i S ( i ) n , where n is the current iteration number, t = n ¡ t , and S ( i ) is S at iteration i .] Do this for three combinations of the transition probabilities ( k + ,k- ): (0.05,0.05), (0.3,0.3), (0.05,0.3). Comment on how the channel dynamics and sample means di er for these di erent combinations. Use your simulation to calculate the sample mean open and closed dwell times. Plot these vs. time. Compare the values of the sample mean dwell times at time t = 500 ms to the expected equilibrium mean dwell times from probability theory. 2. [20 points] Next consider an ensemble of 5 identical and independent ion channels. This could represent, for example, a patch of neural membrane containing 5 channels whose activity is monitored by a patch clamp electrode. (a) Simulate the stochastic gating of the ensemble using ve Markov variables, one for the state of each channel. Suppose that the transition probabilities for each channel are ( k + ,k- ) = (0 . 05 , . 05). Turn in a plot with three panels. The rst two should show the states of two of the ve channels versus time. The third panel should have two superimposed curves. One curve should show the number of open channels versus time. The other curve should show the sample mean of the number of open channels. According to probability theory, what is the expected mean number of open channels? Is this consistent with your sample mean from the Monte Carlo simulation?channels?...
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- Fall '07
- Calculus, Trigraph, smple men, men open nd, nd stohsti models, deterministi nd stohsti