429-2009-Q7key - Y grows without bound. In a biological...

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580.429 2009 Quiz 7 Name: TA/Section: Consider a system described by the ODE ! X ( t ) = ! " # X ( t ) with X = 0 at t = 0 . 1. (2 pts) What is the long-time steady-state value of X , termed X ! ? X ! = " / 2. (2 pts) At time t 1/2 , X ( t 1/2 ) = X ! / 2 . What is t 1/2 ? t 1/2 = ln 2 / 3. (2 pts) Now consider a system with positive feedback, ! Y ( t ) = 0 + 1 Y / K " Y , with 1 / K < . In fact, take 1 / K = / 2 . What value of 0 ensures that Y ! = X ! ? / = 0 / ( # 0.5 ) = 2 0 / 0 = / 2 4. (2 pts) What is t 1/2 for Y (assuming Y = 0 at t = 0 )? t 1/2 = ln 2 / ( / 2) = 2 ln 2 / 5. (2 pts) What happens qualitatively if 1 / K > in the ODE for Y ? What would happen in a real biological system? The ODE says that
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Unformatted text preview: Y grows without bound. In a biological system, saturation would eventually limit the product of Y to reach a steady-state plateau. Remember that the linear term Y / K came from an approximation for a Hill function. If we kept the Hill function itself, we would get saturation and the system would resemble strong feedback with the possibility of bistability ( Y supporting its own production through positive auto-regulation)....
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