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| September |
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MODELING:
Lac promoter synthesis rate:
The effect of stochasticity on the Lac Operon: An evolutionary perspective from van Hoek F et al (2007)
The article's objective, as you can infer from the title, is to evaluate the effect of stochasticity in the evolution of a Promoter. In order to do so they built a comprehensive model including every parameter involved in transcription and translation. They measure some parameters but they depend mostly on literature to define them.
They do both a deterministic and a stochastic analysis. To generate a stochastic model they added one parameter, the average burst size of protein translation (protein translation occurs in bursts, after an mRNA is synthesized, several proteins can be translated from the same mRNA). This was possible because when an mRNA molecule is translated it can not be degraded. Therefore after each translation it can either be translated again (p) or be degraded (1-p). This suggests that protein production occurs in bursts with a burst size geometrically distributed. Afterwards, they compared the noise levels in their model with experimental noise measurements and found correlation.
To model transcription they used a two-dimensional Hill-function dependent on the cAMP and allolactose concentration. (repress the glucose and lactose operon via cAMP and activates the operon via allolactose).
They use 11 biochemical parameters, including three of special importance for us:
a, Transcription rate when the RNA Polymerase is bound to the DNA, but CRP and Laci are not. Initial value: 1.1 × 10-7 mM/min
b, The transcription rate when both RNA Polymerase and CRP are bound, but Laci is not bound to the DNA. Initial value: 2.2 × 10-5 mM/min
c, Leakiness, the transcription rate when RNA Polymerase is not bound to the DNA. Initial value 5.5 × 10-10 mM/min
They modeled binomially protein degradation, assuming that when cells divide, their proteins are randomly divided between the cells. However in a population of non-Dividing cells this "dilution" can not be taken in account.
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MODELING:
Exploring Sensibility Analysis:
Normalizing sensitivity
- None: dx(t)/dt
- Medium: 1/x(t)*dx(t)/dt
- Complete: k/x(t)*dx(t)/dt (Allows you to compare dimensionless.)
How is it measured? Are the k values moved in a range? Or is it a property of the system?
Wilkinson (1978).
The parameters are systematically perturbed from their given values…
… change from the given value… (although it is recommended to define them in each system).
Ingalls & Sauro (2002)
- Before the analysis, it is recommended that you detect the 'preserved structures' (linear units, eg moieties).
How to reduce the system
- The response coefficient, defined above, provides a measure of the difference between this ‘‘perturbed trajectory’’ and the ‘‘nominal’’ (unperturbed) trajectory at each time t: As time tends to infinity, each trajectory will converge to its steady state, and so the response coefficient will converge to the steady-state response of MCA.
- At steady state, these coefficients reduce to their standard MCA counterparts—flux responses.
NOTE: The sensitivity analysis is sensitive to the initial concentrations of metabolites.
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To-do List:
Electrodes and measurement method:
There are, broadly speaking, four options to choose from:
1) The faculty of physiology at UNAM has a sensor for variations of voltages of orders that could be useful (Question: If it is not specific for nickel, is there a way to filter the noise?).
Among the benefits versus the other possibilities: sensitivity appears to be very good and we know this because similar experiments have been done previously. This in turn gives us the assurance that the sensor has already been tested in other biological systems in line with the results expected. Plus, a member of our team already knows how to use this system. On the other hand, the fact that the Insitute is part of the UNAM has the advantage of working with people from the same team, not counting the enormous advantage of being physically close.
2) At the University of Guadalajara, there is a device that measures the medium resistivity in the orders of 10^-9 Molar. We have not checked with sufficient detail the operation of this system, but at least the sensitivity offered is very promising. Furthermore, we believe that the metal used in a phase of measurement reacts specifically with nickel also producing a easily measurable and identifiable optical effect.
Among the advantages this option provides are: that it is sensitive and that the software used to process and record each measurement is very comprehensive and drops the noise reliably. The main disadvantage, is that the apparatus is in a laboratory of the UdG, which means that we would have to carry biological material.
3) Someone offered to buy the specific sensor for nickel and lend it to us during the measuring stage. We need to contact him and describe the project and what we need at the time of sensing. His only requirement is that he appears as a collaborator in the experimental publication resulting from this research.
4) Finally, most certainly not least, Trejo is still building up the sensor as we had planned initially. He has progressed well and in about two weeks it will be ready.
- We decided to wait for Option 4 and, as a backup, the support of Dr. Pena (1), but this does not rule out the option 3, which will be investigated for further details such as shipping time and specificity of the device.
- We need to define the requirements for the bioparts and make the oligos. The oligos are being designed and probably by Friday they will be sent.
- Design of experiments to estimate parameters (probable date September 10-12).
They are still working on the buildings, but there will be a first meeting on Tuesday, Sept. 23, at 4:00 pm.
Wiki:
- Update the notebook.
- Update the section of the model.
- We need to solve the problem of space.
- Correct the image format.
Model:
- Simulation.
- Pending data.
- Analysis (... stochastic processes?).
- We are working on it... We've had some problems with the simulation, and we are doing sensitivity analysis and parameter's scanning.
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MODELING:
Converting units:
Reaction 1.
3.723mM = ? Molecules
M = mole/liter
The volume of a bacterium is 10^-15L
3.723mM = 37.23x10-18 mol at 10-15 liters
37.23 x10-18 mol = 224.20427x105 molecules
* 1 mol = 6.02214x1023 molecules
Reaction 6.
The flow in 20 plasmids is 20mM/h
Therefore, in 10 plasmids it would be 10mM/h
Flow = 10 mM/h = 10mM/3600s = 0.00278mM/s
0.00278mM = 0.0278x10-18mol in 10-15 liters
0.0278x10-18mol = 1.67415x10^5 molecules.
The flow in the cell is 1.67415x10^4 molecules / s with 10 copies (plasmids).
ν = k * [promoter]
1.67415x10^4 molecules / s = k * 10 molecules
-> k = 1.67415x10^3 molecules / s
Reaction 5.
ΔG ° =- 23.81 kcal / mol
Keq = exp (-ΔG º / RT)
*Units supposedly do not affect this formula's usage
Correction of the synthesis reaction of cI:
Units of the k3ON, estimated in reference 3 are 1/molecules*seconds, which means that the reaction is of second order.
In that same article, they suggest that the sole presence of the dimer ensures the production of cI with k3ON rate (that is, that bonding is efficient). Since the estimated values do not consider the intermediate step of the promoter's union and the complex, we should not consider it.
3.1 ρcI + (AHL: LuxR): (AHL: LuxR) -> CI + ρcI + (AHL: LuxR): (AHL: LuxR)
k3ON
Unknown parameters:
| cI Dimerization |
k4.1 & k-4.1 |
| Suppression by CI |
V5max or k5 |
| Nickel Extrusion |
k7 |
| RcnA Degradation |
k8 |
| Nickel Internalization |
k9
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MODELING:
Dimerization of cI
k4.1 & k-4.1?
2 cI <-> cI: cI
¿Quasi-equilibrium?
The initial concentration of cI varies over time, but the proportion is preserved.
...avoid kinetic analysis of the fast reactions, ie to take into consideration only their equilibrium constants instead of considering their rates (Kholodenko et al. 1998).
Keq = [cI:cI]/[cI][cI] -> [cI:cI] = Keq [cI] 2
d(cI:cI 2cI)/dt = v+ - v-
d(Keq[cI]^2+2cI)/dt = v+ - v-
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MODELING:
Simulation: The previous crisis was overcome... Things seem to be working, however, there are still some parameters missing, but it has been decided that they will be estimated by adjusting the model and experimentally when we have the opportunity.
Data: There are a few missing parameters and we know that most of them are not available, so we have given up the search. What has yet to be defined in terms of data are concentrations of AiiA and LuxR, considering that they are constant in the model.
We will also design experiments to obtain the missing parameters through the experimental measurements (viable).
Analysis: - Stoichiometric matrix: There was a meeting today in the morning with Osbaldo to review its analysis, we will share the information as soon as possible.
- Sensitivity analysis: Although we don't yet understand the particular units in which SimBiology returns the results, the first graphics that display the basic parameters of the model are ready and they are the ones involved in the degradation of AHL and its dimerization with LuxR (the beginning of the cascade) and the ones regarding the entry and exit of Nickel. We must do this analysis later, as we have seen that this is sensitive to the initial concentrations of metabolites, which are not yet fully defined.
- Stationary States and Jacobian: They were stopped briefly because we need the parameters for further analysis.
WET LAB:
Requirements: Urgent! We need to send the oligos required to be synthesized; We are working on it, and it is our priority.
Electrodes: The device is not ready.
Design of experiments: The first meeting will be today.
Funds & Jambouree: We are still waiting for some sponsors to reply.
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MODELING:
Correcting reaction 5
The Keq is dimensionless, but for formality in the reaction, it is sometimes necessary to add units.
In the case of the reaction 5, the only information we have is the ΔGº of the reaction, so the Keq is calculated as exp (-ΔG º / RT), which returns a value without units.
As the definition of Keq is given in concentration, we interpret this value in terms of molarity. We know that once defined, they are completely interconvertibles: Molar <-> moles <-> molecules and since we are working on molecules for our model, we make the relevant adjustments.
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