finalex_f06_soln

finalex_f06_soln - 10.34 Final Exam Fall 2006 Solution...

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10.34 Final Exam - Fall 2006 Solution Overall: Average = 62.5 St. Dev. = 11.25 1) (20 points total) Average = 14.2 St. Dev. = 3.8 You are using the Gillespie Algorithm in a Kinetic Monte Carlo simulation of a biochemical process forming a protein. This process occurs at only one active site in a cell, and it initiates quickly, so there is almost always a growing protein attached to the site. This process runs correctly most of the time, but in about one case in a million the protein folds the wrong way. Ideally, the simulation could precisely determine what fraction of the cells will contain N={0, 1, 2, …} copies of the misfolded protein after two days. The simulation includes two processes: addition of an amino acid to the growing protein (time constant approximately 1 millisecond) and misfolding of the growing protein (time constant approximately 10 5 seconds). In the simulation, the protein detaches from the active site immediately when the final amino acid is added, and a new protein immediately starts growing at the site. The rates of both processes vary a little bit depending on the size of the growing chain; these details are included in your simulation. The protein consists of 100 amino acids. a) (10 points) Approximately how many random numbers will the computer have to generate in order to simulate what happens in a single cell over the course of two days? In the Gillespie algorithm, one needs to generate 2 random numbers for each reaction event; one determines the time until the event, while the other determines the reaction that will take place at the given time. So we need to estimate the tau for the system in order to estimate the number of time steps that will be taken to reach 2 days (172800 seconds). 1 1 1 τ = + 5 = 0.001 sec event 0.001 s 1 × 10 s So, in order to reach our goal of 2 days of simulation time, we need to divide the total time by the time per event (on average) to get to total number of events that will take place (on average). This is: N events = t Total [ 0,1 ]) note : ln ( rand [ 0,1 ]) = 1 ln ( rand [ 0,1 ] ) = 0.693 −⋅ ln ( rand N events == t Total = 172800 sec = 1.728 × 10 8 events sec 0.001 event Since we said that each event requires the generation of 2 random numbers, the answer is: N rand # = 3.456 × 10 8 1 Cite as: William Green, Jr., course materials for 10.34 Numerical Methods Applied to Chemical Engineering, Fall 2006. MIT OpenCourseWare (http://ocw.mit.edu), Massachusetts Institute of Technology. Downloaded on [DD Month YYYY].

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b) (10 points) Approximately how many cells will you need to simulate to compute the ensemble average N unfolded ( t = 2 days ) to a standard error of less than ± 1.0% ? If the CPU time for the simulation is dominated by the cost of generating random numbers, and generating each random number takes 1 microsecond of CPU time, approximately how many CPU hours will be required for to compute this ensemble average to this level of accuracy using Gillespie’s Algorithm?
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finalex_f06_soln - 10.34 Final Exam Fall 2006 Solution...

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