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

MODELING:
We read chapter 2 of Predictive Microbiology by T.A. McMeekin et al., 1993. It taught us the standard stages of a modeling process, the purpose of mathematical models.
The suggested stages of the modeling process are:
- Planning
- Data collection and analysis
- Mathematical description
- Validation and maintenance (Including Function Fitting, Model Validation and Model comparisons)

Remarkably this chapter also mentioned the changes of the conductivity of the medium depending on the cellular metabolism, therefore pointing out the importance of having a cellular population which isn’t growing; this to prevent the noise and minimize the conductance change due to bacterial metabolism.

MODELING ( System to be analyzed):
System proposed by Iwig et al. (2006):

Fig. 6 from Iwig Jeffrey S., Rowe Jessica L. and Chivers Peter T., 2006. Nickel homeostasis in Escherichia coli – the rcnR-rcnA efflux pathway and its linkage to NikR function. Mol Microbiol. 62(1), 252–262.

Our proposed system (modified from Iwig et al. (2006)):

RcnR, LuxR and aiiA synthesis rates will be assumed as constitutive and already on stationary state. Nickel entry to the cell will only be mediated via Unk (Unknown) (NikABCDE is only active during anaerobic phase and our experiment will be held in aerobic conditions) and is also constitutive as long as extracellular nickel isn’t rate-limiting.

We also defined the variables and equations for our first approach and built our stoichiometric matrix. This can be found in the modeling section.

MODELING:
We defined our preliminary model’s assumptions and our equations with their respective stoichiometric matrix. The current assumptions and equations can be found in the Modeling section.

NOTES: 
- (d/dt)x=Ʃ(entering flows) - Ʃ(exiting flows)
- Law of mass action: The flow is equal to the reaction constant times the product of the mass of the reactants.

TO-DO LIST:
-Start doing simulations with Gepasi, Copasi or Matlab.
-Investigate the underlying dynamic of each equation (Law of Mass Action, Hill, Michaelis-Menten, etc.)
-Read the articles on our webpage, especially Adam Arkin’s and the one regarding lambda phage.

MODELING:

MISSING REACTION:
We forgot to take into account one reaction, cI* degradation (cI* -» 0). Now it has been added to the model.

ASSUMPTIONS:
There were some changes in our assumptions.

CHECK:
- It is suggested to check more in-depth if nickel natural usage in the cell can really be considered irrelevant for our experiment (assumption 5).
- Daniela identified the metabolic pathways which involve nickel in E. coli. Here is the link.

MODEL:
The first simulation in Matlab was generated. It worked! Everyone should test it and correct or suggest anything necessary.

TO-DO LIST:
Defining parameters is our topmost priority (reaction rates, kinetic constants and concentrations).

Today we discussed the following with Miguel:

BIOPARTS:
There was information in regards to the transformation of the bioparts. Chiba’s team explained to us that their biopart’s DNA was not in the registry however it was mistakenly set as available. However they assured us that they have the functional bioparts, so we will have to request some DNA directly to the team.

ELECTRODES:
Miguel thinks that measuring a conductivity change in the medium will be sufficient. The temporary plan is to create a document gathering everything we know on the subject and that everyone helps investigating as much as possible.

PARAMETERS FOR THE MODEL:
This is further explained below, in modeling session, but basically what Miguel told us is that we can’t count with experimental data for the moment. He suggested that we should search for average half-lives and protein kinetics in general and use those parameters as a first approach.

CONTROLS:
Now that the experimental work has inevitably been delayed, that time can be invested in defining controls, both positive and negative for the experiment.

MODELING:
Defining parameters:
Obtaining the kinetic parameters experimentally is impossible right now; we first need the working bioparts. Right now we could get an average half-life of other proteins, which share characteristics with our desired protein (for example, average half-life of membrane proteins to define RcnA’s half-life) and trust that they won’t differ much. Once the experimental work resumes we could compare our parameters with those obtained experimentally.

The concentrations that will be experimentally determined (such as aiiA) will be calculated using the average synthesis rate of the given promoter, and the proteins degradation rate.
In the case of nickel and AHL, whose concentrations are arbitrary, we’ll use the model itself to estimate the range that will be experimentally tested; it is also necessary to evaluate if this concentrations are biologically feasible.

Notes on the model:

1. Rules are apparently unnecessary, at least in our case, given that the declared properties naturally occur in our model; for example, in the case of total nickel, it is unnecessary to specify that it will be constant, given that for every lost molecule of internal nickel one molecule of external nickel is gained, and vice versa. Therefore the sum will be constant. The rules will therefore only be used to illustrate this phenomenon.
2. Null is a special variable of the program to define the degradation of a protein, the model was corrected to use this variable.

TO-DO LIST:
Parameters: We are going to focus on this.

MODEL (Searching for parameters):

To be defined:
Half-lives, Promoter efficiency, Dissociation rates and Initial concentrations

Missing Information:
- Average half-life of membrane proteins
- Average half-life of proteins with an LVA tail
- Average half-life of transcriptional factors
- Relative strength of the promoters and their efficiency
- Efficiency or reaction rate of the channels
- Efficiency of the repressors (cI, RcnR)
- Efficiency of the inducers (AHL:LuxR)
- Rate of complex formation ( AHL:LuxR)

Investigate enzymes average half-life:
Article: Arribas E., Muñoz-Lopez A., Garcia-Meseguer M.J., Lopez-Najera A., Avalos L., Garcia-Molina F., Garcia-Moreno M., Varon R., 2008. Mean Lifetime and First-Passage Time of the Enzyme Species Involved in an Enzyme Reaction. Application to Unstable Enzyme Systems. Bull. Math. Biol. 70: 1425–1449

HALF-LIVES:
- aiiA: The Imperial College iGEM site states that aiiA half-life is of 24 hours. The value of analogue proteins is determined in other articles.
- cI: cI has a half-life 40 minutes (Arkin, Ross, McAdams 1998). In another study, done in E. coli, cI is reported to have a half-life of 10 hours (Parsell, Silber & Sauer 1990), the amino acid sequence reported in that study is the same as the cI from the biopart we are using. In another study (Andersen et al 1998), they determined the effect of the LVA tail on a GFP. They designed constructions with different LVA tails attached to a mutant GFP with a half-life greater than one day. The addition of the LVA tail decreased the half-life to 40 minutes, which would equal to ~1/36 of the half-life of the wild type.
- LuxR: On Manefield et al. (2002), although a half-life isn’t directly measured there are some experiments in E. coli which suggest that LuxR half-life is greater than 45 minutes and lower than 90 minutes (figure 2ª). On the Imperial College’s iGEM webpage they defined it as 60 minutes.
- RcnA: Some important facts about RcnA that could help us calculate its half-life:
It has a length of 274 residues with 6 transmembrane domains, a mass of 32 kDa and a zone very rich in histidines (17 histidines, 3 aspartatates and 3 glutamates in a region of 26 amino acids). Source: Rodrigue, Effantin & Mandrand-Berthelot (2005).  On page 1076 of the Lehninger there is a table, modified from Bachmair, Finley & Varshavsky (1986), which relates proteins half-life with its amino terminal residue.

PROMOTER EFFICIENCY:
- Pm (promoter repressed by cI): In Reinitz y Vaisnys (1990), they reported that the Gibbs free energy of the binding of E. coli’s polymerase to the promoter is -12.5kcal mol-1. They also reminded us that an association constant k, can formally be translated to Gibbs free energy through the following equation: k = exp(ΔG/RT), where R is the universal gas constant, T is the temperature in Kelvin degrees (310°K, equals to 37°C) and ΔG is the difference in the free energy of Gibbs between the bound and unbound states. They also report that the synthesis rate for cro under control of the Pm promoter is 4.7 x 10-9 mol/min (We still have to discuss if this could be linked with RcnA).
- p(AHL:LuxR): The maximum transcription rate of a sequence under control of the AHL:LuxR activator is reported as 1.4 x 10-2 ms-1, by Goryachev, Toh & Lee (2006).

DISSOCIATION RATE:
- cI: It seems that the dimerization of cI is important for its activation, therefore we are considering including it in the equations. According to Babic & Little (2007) cI slowly dimerizes. In Reinitz y Vaisnys (1990) dimeric cI dissociation rate is defined as 2.0 x 10-8 M.

Others:
Michaelis-Menten: Describes the kinetics of many enzymatic reactions. This model is only valid when the substrate concentration is much greater than that of the enzyme, and when the enzyme-substrate complex maintains a constant concentration.
* Implies law of mass action

For our equations first approach we gathered all their information in a document which included the name, kinetics, references and notes for each equation.

MODELING:

Michaelis-Menten Kinetics
We read Lehninger Principles of Biochemistry (Many Enzymes Catalyze Reactions with Two or More Substrates, pp.207) to better understand Michaelis-Menten Kinetics. Mariana summarized and condensed the information in a document for everyone else to read.

MODELING:
We checked the dynamics of our equations further developing the beforementioned document
Reminder:
According to Michaelis-Menten: V0=[kcat*Et*S]/[km+S] where Vmax=kcat*Et (as long as the enzyme is saturated).

TO-DO LIST:
- Find missing parameters and start evaluating the model
- Add the dimerization of cI* and of the complex AHL:LuxR to the model
- Add the regulation by RcnR to define the initial intracellular nickel concentration
- Include the constitutive synthesis of cI* in the model.

MODELING:
Analyzed Hill’s Coefficient.

TO-DO LIST:
- Ka(ON)<k4(OFF)? Check reference
- Is k7 the right parameter? Which are the units of this parameter?
- Hills Cooperativity
- Missing parameters
- Initial Concentrations
- Include Dimerization