sb3-hw04-2016-key - JHU 580.429 SB3 HW4 The Laplace...

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JHU 580.429 SB3 HW4 The Laplace transform operator is L , defined as L [ f ( t )] = ˜ f ( s ) = R 0 dte - st f ( t ) . The convolution operator is ? , defined as f ? g ( t ) = R t 0 dt 0 f ( t - t 0 ) g ( t 0 ) . The real part of a complex variable z = x + iy is denoted ( z ) = x . 1. The MAPK signaling cascade usually has three levels denoted k ∈ { 1 , 2 , 3 } . The activation state of level k at time t is denoted x k ( t ) . When activation is weak, a linear model may be appropriate: ˙ x 1 ( t ) = β ( t ) - α 1 x 1 ( t ) ˙ x 2 ( t ) = b 2 x 1 ( t ) - α 2 x 2 ( t ) ˙ x 3 ( t ) = b 3 x 2 ( t ) - α 3 x 3 ( t ) . The input is under external control, β ( t ) = β 0 sin ( ω t ) . At time 0, x k ( 0 ) = ˙ x k ( 0 ) = 0 for k 1 ... 3. Provide all results in terms of model parameters { β 0 , ω , b , α 1 , α 2 , α 3 } , as well as s or t as appropriate. For simplicity assume that each α k is different. (a) What is ˜ β ( s ) ? ˜ β ( s ) = Z 0 dt e - st β 0 sin ( ω t ) (1) = β 0 Z 0 dt e - ( s - i ω ) t (2) = β 0 - 1 s - i ω e - ( s - i ω ) t 0 (3) = β 0 ω s 2 + ω 2 (4) (b) What is ˜ x 1 ( s ) ? ˜ x 1 ( s ) = ˜ β ( s ) s + α 1 (5) = β 0 ω ( s + α 1 )( s 2 + ω 2 ) (6) (c) What is ˜ x 2 ( s ) ?
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