Team:LCG-UNAM-Mexico/Notebook/2008-October

Hill Kinetics

REFERENCE: Segel; Enzyme kinetics: Behaviour and Analysis of rapid equilibrium and Steady-state Enzyme systems.

Multiple Inhibition Analysis

v = kp[ES]          ->         v/[Et]=kp[ES]/([E]+[IE]+[EI]+[IEI]+[ES])

Ks=[E][S]/[ES]   ->         [ES]=[E][S]/Ks  
Ki=[I][E]/[IE]     ->         [IE]=[I][E]/Ki    
Ki=[EI][I]/[IEI]   ->         [IEI]=[EI][I]/Ki=[I]2[E]/Ki2

v/[Et]=([E][S]/Ks)kp/([E]+([I][E]/Ki)+([E][I]/Ki)+([I]2[E]/Ki2)+([E][S]/Ks)) 
v/[Et]=([S]/Ks)kp/(1+2([I]/Ki)+([I]2/Ki2)+([S]/Ks))                       
v=([S]/Ks)kp[Et]/(1+2([I]/Ki)+([I]2/Ki2)+([S]/Ks))=([S]/Ks)Vmax/(1+2([I]/Ki)+([I]2/Ki2)+([S]/Ks))

…with cooperativity    
v=([S]/Ks)Vmax/(1+2([I]/Ki)+([I]2/aKi2)+([S]/Ks)) 
* a factor

It can be written in Hill's terms (if the cooperativity is strong). 

System: cI repression

Inhibitor:        cI:cI      (I)       
“Enzyme”:       ρ          (ρ)       
Substrate:          -          
Product:        RcnA     (P)

Union sites:         OR2 & OR1         
I in OR1             ->         ρI        
I in OR2             ->         Iρ

K5-1=[I][ρ]/[ρI] 
K5-2=[I][ρ]/[Iρ] 
a & b cooperativity factor

* K5=[ρ][I]2/[IρI]

K5=K5-1·K5-2·a

ΔGº=ΔGº1+ ΔGº2+ ΔGº12

1/K5=exp(-ΔGº/RT)=exp(ΔGº1/RT)+ exp(ΔGº2/RT)+ exp(ΔGº12/RT)

ρI         ->         ΔGº1=-11.7 kcal/mol    
Iρ         ->         ΔGº2=-10.1 kcal/mol    
Coop.   ->         ΔGº12=-2 kcal/mol        
ΔGº =-23.8 kcal/mol

v=k6·ρ  
v/[ρt]=kp[ρ]/([ρ]+[Iρ]+[ρI]+[IρI])          
v/[ρt]=[ρ]kp/([ρ]+([ρ][I]/K5-1)+([I][ρ]/K5-2)+([I]2[ρ]/K52))
v/[ρt]= kp/(1+([I]/K5-1)+([I]/K5-2)+([I]2/K5))        
v= kp[ρt]/(1+([I]/K5-1)+([I]/K5-2)+([I]2/K5))

NOTE: We are ommiting the fact that cI:cI will be "kidnapped" by the promotor. This doesn't seem important since we only have 10 molecules of the promoter per cell, comparing with 150 basal molecules (without the AHL signal) of the dimer.

Estimating the amount of AiiA per cell

AiiA is under the control of the lac promoter. The transcription and mRNA degradation rates help us estimate the amount of mRNA present on the cell.

“The half-life of protein A is assumed to be around 10 minutes which is similar to what is used in Elowitz’s repressilator model [1]. Furthermore, we assume that a more aggressive degradation tail can enable half-times on the order of two minutes for protein B.”

Modeling the Lux/AiiA Relaxation Oscillator

Christopher Batten

In the paper AiiA is called protein B. Therefore the degradation rate for AiiA with an aggressive degradation tail is 0.0058 s-1. This would give us a lower limit.

“Transcription initiation rate, km

Malan et al. (1984) measured the transcription initiation rate at P1 and report the following value: km ≈ 0.18min-1

mRNA degradation rate, jM

Kennell and Riezman (1977), measured a lacZ mRNA half-life of 1.5 min: ξM = 0.46min-1

lacZ mRNA translation initiation rate, кB

From Kennell and Riezman (1977), translation starts every 3.2 s at the lacZ mRNA. This leads to the following translation initiation rate: кB ≈ 18.8min-1”

Santillán M. and Mackey M. C. (2004). Influence of Catabolite Repression and Inducer Exclusion on the Bistable Behavior of the lac Operon. Biophys J 86:1282–1292

We modified both transcription initiation and translation rates by multiplying both rates by 4. This due to the fact that LuxR is a four times smaller than LacZ:

LacZ has a length of 1024 aa

LuxR has a length of 250 aa

Simulating with simbiology, AiiA reaches stationary state at almost 3500 molecules per cell.

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