# Homework Help: Enzyme Kinetics Problem

1. Oct 6, 2008

### physicsstudent14

1. The problem statement, all variables and given/known data

Using the scheme below, define the fractional saturation of individual IPTG binding sites, YPI, and the fractional saturation of the dimer with operator DNA, YP2O in terms of the concentrations of intermediate dimer species. This Adair-like mass action scheme can be modeled as a 2-state monod-wyman-changeux mechanism in which IPTG = I, Operator DNA = O, repressor dimer = P2

Derive expressions for YPI and YO as a function of , [O], and the K's in the above scheme. Rearrange the expression for YPI so the apparent Adair constants for IPTG binding are a function of [O], Ko, K1o, K2o. Rearrange the expression for Yo so the apparent association equilibrium constant for O binding is a function of , K1, K2, Ko, K1o, K2o

2. Relevant equations

3. The attempt at a solution

YPI = ([P2I] + 2[P2I2] + [P2OI] + 2[P2OI2])/2([P2] + [P2O] + [P2I] + [P2OI] + [P2I2] + [P2OI2])

K1 = [P2I]/([P2])
K2 = [P2I2]/([P2I])
Ko = [P2O]/([P2][O])
K1o = [P2OI]/([P2I][O])
K2o = [P2OI2]/[([P2I2][O])

Definition of relative probabilities
[P2] = 1
[P2I] = K1
[P2I2] = K1K22
[P2O] = Ko[O]
[P2OI] =K1oK1[O]
[P2OI2] = K2oK1K2[O]2

Yo = (Ko[O] +K1oK1[O] +2K2oK1K2[O])/(2(1 + Ko[O] +K1 +K1oK1[O] + K1K22 + K2oK1K2[O]2))

Yo = (Ko[O] + K1(K1o[O] + 2K2oK2[O]))/(2(1 + Ko[O] + K1(1 + K1o[O] + K2 +K2oK2[O])))

YPI = (K1 + 2K1K2[SUP]2[/SUP] + K[SUB]1o[/SUB]K[SUB]1[/SUB][I][O] + 2K[SUB]2o[/SUB]K[SUB]1[/SUB]K[SUB]2[/SUB][O][I][SUP]2[/SUP])/(2(1 + K[SUB]o[/SUB][O] + K[SUB]1[/SUB][I]+ K[SUB]1o[/SUB]K[SUB]1[/SUB][I][O] + K[SUB]1[/SUB]K[SUB]2[/SUB][I][SUP]2[/SUP] + K[SUB]2o[/SUB]K[SUB]1[/SUB]K[SUB]2[/SUB][O][I][SUP]2[/SUP]))

Y[SUB]PI[/SUB] = (K[SUB]1[/SUB][I](1 + K[SUB]1o[/SUB][O] + 2K[SUB]2[/SUB][I] + 2K[SUB]2o[/SUB]K[SUB]2[/SUB][O][I]))/(2(1 + K[SUB]o[/SUB][O] + K[SUB]1[/SUB][I](1 + K[SUB]1o[/SUB][O] + K[SUB]2[/SUB][I] + K[SUB]2o[/SUB]K[SUB]2[/SUB][O][I])))

I'm mainly interested in knowing if my solution makes sense. I am also having trouble with rearranging the expression so that it is a function of the the values indicated in the question. My rearrangements look so disgusting and more convoluted. Thank you very much. I truly appreciate all your help.[/I][/I][/I][/I][/I][/I][/I][/I][/I][/I][/I][/I]