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

Hill's cooperativity
5th Reaction
Reminder:

A + B <--> AB           
Ka=Keq=[AB]/[A][B]=1/Kd     
θ=[AB]/([AB]+[A])=[B]/([B]+Kd)

MWC Model (Cooperativity)           
A + nB <--> ABn       
Ka=Keq=[ABn]/[A][B]n=1/Kd  
θ=[B]n/([B]n+Kd)       
log(θ/(1- θ))=nlog(B)-log(kd)                        …Hill's equation

Suppression mediated by cI:
ρ + nCI <--> ρ:CIn     (k+, k-) 
Keq=Ka=[ρ:CIn]/[ρ][CI]n      
Si ρ0=[ρ]+[ρ:CIn]      
… ρ0=[ρ]+Keq[ρ][CI]n
=> ρ= (ρ0/Keq)/((1/keq)+[CI]n)

Flow= k+[ρ][CI]n = K+((ρ0/Keq)/((1/Keq)+[CI]n))[CI]n

Flow= k+([ρ0]/Keq) [CI]n / ((1/Keq)+[CI]n)

=> Vm= k+([ρ0]/Keq)   &   Kp=1/Keq=Kd

So:
Keq = exp( -ΔG / R T )          
k+ = (KB/h) T exp( -ΔG / R T ) = (KB/h) T Keq

Hill's Cooperativity

5th Reaction, conflict ...

If we consider that:

Keq = exp (-ΔG / R T)

k + = (KB / h) T exp (-ΔG / R T) = (KB / h) T Keq

and given that the flow is (k + / Keq) [ρ0] [CI] n / ((1/Keq) + [CI] n), the value of the maximum speed of the flow loses its meaning.

The speed limit is being determined by (k + / Keq) [ρ0], but k + / Keq = (KB / h) * T, and we know that [ρ0] is arbitrary, i.e., Vmax is no longer based on the reaction as such, which does not make sense.

For example: Take the same reaction that we are considering, the maximum speed of the flow of the reaction would be the same with the promoter that has the operators of CI, that if you used one with a random sequence, so, whether we repeated the experiment, with the same temperature and the same concentration of DNA and an equal number of copies of the sequence, the maximum speed reached by the flow would be the same for the real promoter as for for any sequence, without taking any consideration with their affinity for their substrates... That does not makes sense!

The proposed explanation is that the equation used to determine k + does not fit our model, we should explore other possibilities.

Hill's Cooperativity:
5th Reaction, resolving the conflict...

The error in the previous approach is that we were considering ΔG to be the same for both equations (for Keq & k+).

The explanation of why these two values are different is very clear when we look at the graph below. Recalling what the two constants represent:

We know that the balance depends solely on the difference between Gibbs free energy of the substrate and the product (ΔG 'th), The one with less energy will be favored in the balance, while the rate of reaction depends on the activation energy needed for the conversion (ΔG ‡). A reaction reaches equilibrium faster or slower depending on the rate of reaction (depending on how big is ΔG ‡), but the balance of it as such does not change.

Thus:
        Keq = exp (- ΔG 'º / R T)
        k + = (KB / h) T exp (- ΔG ‡ / RT) ≠ (KB / h) T Keq

GROUP MEETING
Experimental work

Objectives:
- Build the bioparts.
- Transform the bacteria with the construction that we have.
- Design the experiments to test our construction.
- Build the system.
- Collaborate with the modeling group.

To do:
- Extract DNA of the strain to get RcnA.
- Get the bioparts catalog.
- We need to have a large number of plasmids that we can use, amplifying the bioparts.
- Transformation of the bacteria with bioparts.

Currently:
- There are plasmids.
- There are parts already amplified and in a plasmid.

Problems:
- There was no DNA that we needed in the catalog.
- The oligos were delayed 2 week and a half.
- Issues to extract the plasmid from the colonies.
- Make a PCR ligation with the three parts and amplify with the ends (it did not work).
- With the enzyme used: Increased frequency of spontaneous mutation of all the enzymes that exist.
An error every thousand base pairs.
- There is a problem with tetracycline. You get false positives.

Can be done:
- A part with RcnA and can be linked to the plasmid.
- In the others we have to link and restrict, and re-link and restrict once more and re-connect the last time in the final plasmid.
- HindIII can be used with the big biopart to verify the sequence.

Electrodes:
- Are they specific for Nickel?

Our response to the IPN team:

Hello,
We apologize for the late reply, but we had to discuss carefully our answer.
First of all, we think you are confused about what our project really is. We want to make bacteria to modify the extracellular nickel concentration in response to an external signal (AHL in this case), and of course, be able to predict to what extent the concentration of the input signal will affect the amount of nickel in the medium. To achieve this, it is true we have to synchronize our cell population at the beginning. This is easy to do and doesn't represent any technical problems.
We are very conscious of the facts you tell us, first: we know the half-life of the lactones is relatively long (24 hrs as you say). That's why we are including AiiA under a constitutive promoter in our model, which degrades AHL very efficiently. This will ensure AHL does not saturate the medium. Second, we know AiiA does not diffuse freely through the cell membrane. However, we don't need that to happen, as each cell will degrade its own AHL (yes, we are assuming that all AHL will enter a cell within a window of time).
In other words, we do not need to synchronize the bacterial population more than in the first step. We are considering that some cells may respond earlier than others. However, we are assuming that, as we are not changing the physical nor chemical conditions, the proportion of cells responding "earlier" will remain constant, thus allowing us to draw some conclusions of the behaviour of the population as a whole. We hope you see why the synchrony is no longer important for our project.

To summarize what we plan to do, AHL will enter the cell and form a dimer with LuxR (which is under a constitutive promoter, so AHL is the only limiting step). This will start the transcription of cI*, which will repress the expression of RcnA. RcnA is the nickel efflux pump, and thus we are aiming to predict the amount of AHL necessary to get the desired extracellular nickel concentration.
We are doing small moves. At first, we only want to make one successful assay. We hope that in the near future we will be able to use the response time of the system to generate a succession of desired nickel concentrations, thus generating a song.
We hope this letter answers your questions,

LCG-UNAM-Mexico Team
Cuernavaca, Morelos

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