429-2009-Q8key - and S ( t ) a signal that we can control....

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580.429 SBE3 2009 Quiz 8 Name: Suppose that mRNA M produces protein P . The ODE dynamics for M and P are ! M ( t ) = ! M " ( t ) # $ M M ( t ) ! P ( t ) = kM ( t ) # P P ( t ) At time 0, M = P = 0, and ( t ) is a unit step at t = 0. 1. (2 pts) After a long time, the mRNA concentration reaches a steady-state value. What is the value? M at steady state = M / M 2. (2 pts) What model parameters do we look at to know whether a separation of mRNA and protein timescales is a good approximation? What condition involving these parameters do we require? Parameters: M , P Condition: M ! P 3. (2 pts) When separation of timescales is valid with fast mRNA response, we can write ! P ( t ) ! P # ( t ) $ % P P ( t ) where P is an effective production rate. Provide a formula for this effective rate in terms of model parameters. P = k M / M 4. (2 pts) Consider the gene circuit given by the ODEs ! X ( t ) = S ( t ) " X ( t ) ! Y ( t ) = !$ ( X > K ) " Y ( t ) ! Z ( t ) = ( X > K ) ( Y > K ) " Z ( t ) with / > K
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Unformatted text preview: and S ( t ) a signal that we can control. At time 0, X = Y = Z = 0, and S ( t ) = 1 for t > 0. At what time do X , Y , and Z each reach their half-maximal values? Define K = " (1 / )ln[1 " K / ( / )] as the time to get from 0 to K , and 1/2 = (1 / )ln 2 as the time to get from 0 to half maximum. X : 1/2 Y : K + 1/2 Z : 2 K + 1/2 5. (2 pts) Same question as 4, but now X , Y , and Z start at their maximal values at time 0, and S ( t ) = 0 for t > 0. At what time do X , Y , and Z decay to their half-maximal values? Re-define K = (1 / )ln[( / ) / K ] and keep 1/2 = (1 / )ln 2 . Here Z decays with Y due to AND logic. X : 1/2 Y : K + 1/2 Z : K + 1/2...
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This note was uploaded on 03/30/2010 for the course SBE 580.429 taught by Professor Joelbader during the Fall '09 term at Johns Hopkins.

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