2_Simplification%20of%20kinetic%20analysis

2_Simplification%20of%20kinetic%20analysis - Four...

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1 Four Approaches to Simplify Kinetic Analysis In Mathematical Modeling Reactions near equilibrium => [A] = [A] - [A] eq << [A] eq Steady state 0 t Steady state [A] 0 t ] A [ Equilibrium A state of balance between opposing forces or actions Equilibrium of a chemical reaction Forward rate = Reverse rate Dimensional analysis => Ignore small dimensionless terms equilibrium equilibrium steady state steady state A B C [A], [B], and [C] can be at steady state but the reactions cannot be at equilibirum equilibirum since they are not reversible. A B C It is possible to have [A], [B], and [C] be at steady state and the reactions be at equilibirum . Example 1 Example 2 ] [ ] [ ] ][ [ ] [ MN k MN k N M k dt MN d r f M + N P k MN k f k r At the steady state of [MN], K k K k k k MN r r f / k 1 [M][N] ] [ If the reversible reaction is at equilibrium , K k k [M][N] ] MN [ r f or K A
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2 Quasi- Seemingly, close to, or can be treated as Quasi-Steady State and Quasi-Equilibrium In kinetic analysis, Assuming d[A]/dt 0 in the analysis Quasi-steady state of [A] Assuming R f R r in the analysis Quasi-equilibrium of reversible reaction X Quasi Quasi -equilibrium equilibrium Quasi Quasi -steady state steady state Quasi -equilibrium equilibrium Equilibrium Example: Consider the following reactions: AB C k 1 k 2 k -2 ] A [ k dt ] A [ d 1 ] C [ k ] B [ k ] A [ k dt ] B [ d 2 2 1 ] C [ k ] B [ k dt ] C [ d 2 2 If k 1 << k 2 and k -2 , k 2 [B] k -2 [C] (i.e., quasi-equilibrium) Thus, ] [ ] [ 1 A k dt B d 0 ] [ dt C d But (since k 1 is small, k 1 [A] can be on the same order of magnitude as k 2 [B] - k -2 [C].) When k 2 [B] = k -2 [C] (i.e., equilibrium), ] [ ] [ 1 A k dt B d Similarly, Quasi -steady steady steady steady (see Matlab HW1) Another Quasi-Equilibrium Analysis (To be Shown in Enzymatic Reaction Lecture)
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3 If all connected reactions R j are very slow compared to R f and R r of the reversible reaction => Quasi-equilibrium A Sufficient But Not Necessary Condition for Quasi-equilibrium j i [products] ] [reactants R 1 R 2 R j R 3 R 4 R 5 because reversible reactions always reach equilibrium as t  A Sufficient But Not Necessary Condition for Quasi-Steady State If [A] << [ ] of other molecules over the entire period of time, [A] is at quasi-steady state. For example, [ES] in enzymatic reactions.
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This note was uploaded on 01/17/2010 for the course BME 100 taught by Professor Yuan during the Fall '07 term at Duke.

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2_Simplification%20of%20kinetic%20analysis - Four...

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