429-2009-HW6key - SB3 HW6 Key_v2(corrected 2.2(b 3 7(e...

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SB3 HW6 Key_v2 (corrected 2.2(b), 3, 7(e), bonus) by An-Chi Wei Oct 28, 2009 1. Q2.1 1/ 2 1 2 2 2 1 2 2 1 2 1 2 2 1 2 2 1 2 1/ 2 1/ 2 , (0) ( ) (0) ( ) 1 ( ) ( ) ( ) ( ) ( ) ln 2 ( ) 2 t t t t dY Y Y dt sY s Y Y s s Y s s s s Y t e e e Y t e t α α α α β β α α β α β β α α α β β β β β β α α α α α β β β β β β α α α α α = = = = + + + = + = + = + = + = ± ± ± 0 50 100 150 200 250 300 350 400 450 500 0 10 20 30 40 50 60 time(min) Y(t) beta1=1/hr, beta2=1/min, alpha=1/hr
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m m 2. Q2.2 (a)The dynamic equation for the concentration of mRNA of gene Y: - β : mRNA production rate, α : mRNA degradation rate The dynamic equation for the concentration of protein due to produ m m m m dY Y dt β α = ction of p copies per mRNA and degradation rate - m dY pY Y dt α α = m ( ) (i)With fast RNA dynamics ( >> ), the quasi steady state assumption (mRNA reaches steady state with respect to slower process) can be used. Thus, 0 - - effective m m m m m m m b dY Y dt dY pY Y p Y Y dt α α β α β α α β α α = = = = = m protein production rate (ii) with slow RNA dynamics( α << α ) ( ) ( ), ( ) ( ) ( ) ( ) ( ) ( ), ( ) ( )( ) 1 ( ) ( ) ( ) ( ) a m m m m m m m m m m m m m t t m m m m m p sY s Y s Y s s s s p Y s p sY s pY s Y s Y s s s s s e e Y t p α α β β α β β α α β α α α α β αα α α α α α α = = = + = = = + + + = + + ± ± ± ± ± ± ± ± 1/ 2 m 1/ 2 1/ 2 s α << α , 1 ( ) ( ) (1 ) ( ) (1 ) 2 ln 2 m m m t t m m m m m t m m m m m p e Y t p e p p Y t e t α α α β β αα α α αα β β αα αα α + = = = =
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