429-2009-HW5key

429-2009-HW5key - Systems Bioengineering III Homework 5 Key Rahul Karnik 1 We are given the following ODEs ˙ x 1 t = k 1 S t k f 1 x 2 t α 1 x t

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Unformatted text preview: Systems Bioengineering III: Homework 5 Key Rahul Karnik October 12, 2009 1. We are given the following ODEs: ˙ x 1 ( t ) = k 1 S ( t ) + k f 1 x 2 ( t )- α 1 x ( t ) ˙ x 2 ( t ) = k 2 x 1 ( t )- α 2 x 2 ( t ) Diagram of pathway From the ODEs, we can see that: • There are 2 genes in the pathway. • Gene x 1 is stimulated by the incoming signal S ( t ) and is also activated by gene x 2 (positive feedback). • Gene x 2 is stimulated by gene x 1 . Using the above observations, we can draw the diagram below: 1 The arrows in the diagram all represent activation. The boxes are genes. Laplace domain closed-form expressions Let us first write the ˜ x 1 ( s ) in terms of ˜ S ( s ) and ˜ x 2 ( s ). ˙ x 1 ( t ) = k 1 S ( t ) + k f 1 x 2 ( t )- α 1 x ( t ) s ˜ x 1 ( s ) + x 1 (0) = k 1 ˜ S ( s ) + k f 1 ˜ x 2 ( s )- α 1 ˜ x 1 ( s ) ˜ x 1 ( s ) = k 1 s + α 1 ˜ S ( s ) + k f 1 s + α 1 x 2 ( s ) [ ∵ x 1 (0) = 0] (1) Next, let us write ˜ x 2 ( s ) in terms of ˜ x 1 ( s ) so we can eliminate it from the equation above. ˙ x 2 ( t ) = k 2 x 1 ( t )- α 2 x 2 ( t ) s ˜ x 2 ( s ) + x 2 (0) = k 2 ˜ x 1 ( s )- α 2 ˜ x 2 ( s ) ˜ x 2 ( s ) = k 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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429-2009-HW5key - Systems Bioengineering III Homework 5 Key Rahul Karnik 1 We are given the following ODEs ˙ x 1 t = k 1 S t k f 1 x 2 t α 1 x t

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