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Assignment 1 Solutions

# Assignment 1 Solutions - AMATH/BIOL 382 Problem Set 1...

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AMATH/BIOL 382 Problem Set 1 Solutions 1. One can begin by observing that equation two factors as 0 = 2 a ss (2 - b ss ) . The case a ss = 0 can be ruled out, since this cannot satisfy the first equation. Thus b ss = 2. Substituting into the first equation gives 0 = 3 - 4 a ss - 4 a ss so a ss = 3 / 8. Substituting into the other equations gives c ss = 2 · 2 · (3 / 8) 2 . 5 = 3 5 d ss = 2 · 2 · (3 / 8) = 3 2 . as indicated. 2. Closed Network a) The system is described by d dt a ( t ) = - k 1 a ( t ) b ( t ) d dt b ( t ) = - k 1 a ( t ) b ( t ) + k 2 d ( t ) d dt c ( t ) = k 1 a ( t ) b ( t ) - k 3 c ( t ) d dt d ( t ) = k 1 a ( t ) b ( t ) - k 2 d ( t ) d dt e ( t ) = k 3 c ( t ) d dt f ( t ) = k 3 c ( t ) b) We note that the following combinations of species each satisfy a conservation: b ( t ) + d ( t ) = T 1 a ( t ) + c ( t ) + e ( t ) = T 2 e ( t ) - f ( t ) = T 3 , for constants T 1 , T 2 , T 3 whose values are set by the initial conditions. These can be verified by considering flow in the network or by considering combinations of rates of change. (This set of three conservations is not unique, e.g. a ( t ) + c ( t ) + f ( t ) is also constant.) 1

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Setting d ( t ) = T 1 - b ( t ) e ( t ) = T 2 - a ( t ) - c ( t ) f ( t ) = - T 3 + e ( t ) and substituting into the DEs for the remaining three variables, we have d dt a ( t ) = - k 1 a ( t ) b ( t ) d dt b ( t ) = - k 1 a ( t ) b ( t ) + k 2 ( T 1 - b ( t )) d dt c ( t ) = k 1 a ( t ) b ( t ) - k 3 c ( t ) . c) Steady state values of the concentrations must satisfy 0 = - k 1 a ss b ss 0 = - k 1 a ss b ss + k 2 ( T 1 - b ss ) 0 = k 1 a ss b ss - k 3 c ss .
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Assignment 1 Solutions - AMATH/BIOL 382 Problem Set 1...

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