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DSolve: Gene production concentration

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  • Subject: [mg131708] DSolve: Gene production concentration
  • From: jes <jespervng at gmail.com>
  • Date: Sun, 22 Sep 2013 02:46:41 -0400 (EDT)
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Hi i have never used Mathematica, i was wondering if it would be an nice tool to use for a course i am taking. Could someone shown me this would be solved and plotted using Mathematica ? 

A gene Y with simple regulation is produced at a constant rate Beta1. The production rate suddenly shifts to a different rate Beta2.

Calculate and plot the gene product concentration Y(t)

Solution:

Let us mark the time when the shift occurs as t = 0. Before the shift Y reaches steady state at a level Y(t=0) = Tst = Beta1/a after the shift.

dY/dt = Beta2 - a*Y 
The solution of such an equation is generally

Y = C1 + C2 * e^(-at) 
Where the constants C1 and C2 need to be determined so that Y(t=0) = Beta1/a and Y after a long time reaches its new steady state Beta2/a. This yields the following sum of an exponential and a constant:

Y(t) = Beta2/a + (Beta1/a - Beta2/a) e^(-at) 
Take the derivative with respect to time, dY/dt, and verify that the equation is fulfilled.
And plot the gene product concentration Y(t)



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