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Assignment 3 Solutions - AMATH/BIOL 382 Problem Set 3...

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AMATH/BIOL 382 Problem Set 3 Solutions 1. (i) Running the file # Assignment 3 Problem 2i s1’=Vmax1-Vmax2*S1/(Km2+s1) s2’=Vmax2*S1/(Km2+s1)-Vmax3*s2/(Km3+s2) # parameters par Vmax1=1,Vmax2=2,Vmax3=4,Km2=4,Km3=3 done yields steady state ( s 1 , s 2 ) = (4 , 1). (ii) Perturbing V max 2 to V max 2 = 2 . 01 results in a new steady state of ( s 1 , s 2 ) = (3 . 96 , 1). This corresponds to a sensitivity of zero in s 2 and of 3 . 96 - 4 2 . 01 - 2 = - 4 Checking at V max 2 = 1 . 99 gives a similar result 4 . 04 - 4 1 . 99 - 2 = - 4 The absolute sensitivity of s 1 with respect to V max 2 is thus 4. The relative senstivity is 2 4 ( - 4) = - 2. The relative sensitivity in s 2 is zero. (iii) With this new nominal parameter set, we find steady state ( s 1 , s 2 ) = ( 1 3 , 1). Making a slight change to V max 2 = 1 . 01 results in a new steady state of ( s 1 , s 2 ) = (0 . 329 , 1). This corresponds to a sensitivity of zero in s 2 and of 0 . 329 - 0 . 333 1 . 01 - 1 = - 0 . 43 The absolute sensitivity of s 1 with respect to V max is thus -0.43. The relative senstivity is 1 1 / 3 ( - 0 . 43) = - 1 . 3. The relative sensitivity in s 2 is zero.
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