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

AHL: LuxR

Reaction 3

Conflict: k3 (ON) <k3 (OFF)?
Reference: Goryachev et al. (2006)

The references they use where they obtained parameters were not specific for this parameters (?) In fact, one mentions the rate of RNA polymerase in HUMAN!
-> Check whether the article mentions how they got the parameter, or search through the references.

They do explain why in the model, the k3 (ON) is in principle "very small":
<<common to all models considered here, is that the stability of the state "off" defined by the constitutive Transcription levels of I and R comes at a price of high value for the critical self Extracellular concentration.>>


And it seems that this explains a bit the criteria that determined the parameters used, although it does not appear in references such as:
<<For each layout we attempted to identify a set of parameters that optimize the functional fitness of the network. The search in the parameter space is constrained by requesting that the kinetic parameters must remain in the biologically realistic range and the resulting network should demonstrate the behavior compatible with our present understanding of the phenomenon quorum sensing.>>

Natural degradation of cI *

Reaction 4

Real life time of cI?

The half-life of modified cI is 4 minutes, according to Elowitz & Leibler (2000). They analyze the tail LAA, and JB Andersen et al (1998) conclude that the queues LAA and LVA confer about the same time of life to GFP.

Reaction rate?

Once we get the half-life time of the protein, how do we calculate the rate of reaction and the flow?

The half-life of a reaction (t1 / 2) is the time it takes for half of the reagents to become products. In a first order reaction, t1 / 2 is a constant and can be calculated from the rate constant, as follows:

t1 / 2 =-ln (0.5) / k = 0.693 / k

This reciprocal relationship between the half life time and the rate constant is very useful to make an estimate of the timea given reaction will take place in. Thus, for k = 0.01 s-1, the half life time would be about 70 s. For k = 10 s-1, the half life time would be about 0.07 s or 70 milliseconds. The average life time of the reactions of the first order is also independent of the initial concentration. If the first half of the molecules react in aprox 20 s, half of the remaining molecules will also take 20 s to react, and so on. The fact that the lifetime average in an unimolecular reaction is a constant means that, at any time of the reaction, a constant fraction of reactive molecules have enough energy to overcome the kinetic barrier and become molecules of product. This makes sense because the energy of a set of molecules is distributed randomly according to a Boltzmann distribution.

RT Sauer (1999); http://mit.ocw.universia.net/7.51/f01/pdf/fa01-lec02.pdf.

NOTE: A first order reaction is the type A → B.

Checking parameters

Reaction 6

The value that had been found before (reference 7 from the model document) k6 (Pl) = 0.20mM / h is really the flow of the reaction (which is why the units are mM / h). That is, how many mRNA molecules are being produced per unit of time. The system in measuring this parameter is a derivative of pBR322 plasmid, pTrc99A. It has the same origin of replication and the number of copies is estimated at 15-20 (doi: 10.1016/S0264-410X (02) 00292-X) but under certain conditions where replication is limited (?) it appears to be between 3-5 copies.

From here we can calculate the value of k+ of the reaction in our system if we consider that each promoter acts independently and we multiply by the ratio between the number of copies of our plasmid and theirs.

Tasks

1. Parameters.
- cI Dimerization.
- Nickel extrusion . (Estimate,?).
- RcnA degradation.
- Nickel Internalization (Experimentally,?).
- Initial concentrations (aiiA, LuxR? (Define arbitrary medium), ? (by number of copies of the plasmid; change to molar), Niext (experimental)).

2. Reviewing tools in SimBiology (sensitivity analysis, parameter estimation, moiety conservation).

3. Stationary states of the system (if there is multistationarity).

4. Jacobian.

5. Analysis of the stechiometric matrix (also analyze the null space and its transposed).

6. Electrochemical theory (the difference between potential and the concentration of nickel).

7. Electrodes.
->We need to check how the measurement device we will use is going.
-> Take into account the possibility of asking for support to Dr. Peña.