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Unformatted text preview: Notes on a genetic network Patrick De Leenheer * February 18, 2009 If necessary, first review section 3.4.1 Phase-plane analysis: Linear systems. These notes describe a simple genetic network consisting of two genes. The product of gene 1 has concentration x , and the product of gene 2 has concentration y . We assume that the product of each gene inhibits the transcription of the other gene. In practice, this happens when the gene product of one gene binds to the site of the DNA corresponding to the other gene, thereby inhibiting transcription. The model is as follows: ˙ x =- γ 1 x + x x + αy + 1 = x- γ 1 + 1 x + αy + 1 (1) ˙ y =- γ 2 y + y y + βx + 1 = y- γ 2 + 1 y + βx + 1 (2) where γ 1 ,γ 2 ∈ (0 , 1) represent the decay rates of the proteins, and the state vector ( x,y ) has non-negative components. The production rate of protein 1 is given by the function f ( x,y ) = x x + αy + 1 , where α > 0. This function is increasing in x ( ∂f/∂x > 0) and non-increasing in y ( ∂f/∂y ≤ 0). Thus, the gene product of gene 1 promotes transcription and translation of gene 1, whereas the gene product of gene 2 inhibits this process. Similarly, the production rate of protein 2 is given by the (non-negative and bounded) function g ( x,y ) = y y + βx + 1 , where β > 0, which has similar properties as f , namely it is increasing in y and non-increasing in x ....
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This note was uploaded on 09/12/2011 for the course MAP 4305 taught by Professor Deleenheer during the Summer '06 term at University of Florida.
- Summer '06