Team:LCG-UNAM-Mexico/Modeling

The objective of our modelling is to accurately describe and predict the behavior of the system and its response given an inducing signal. Also, we aim to better know and understand the  system through the identification of critical parameters and species, and thus be able to obtain the desired dynamics.
Our system is composed of 13 species and 11 coupled biochemical reactions that completely describe it. This can be represented through a set of ordinary differential equations (ODEs). The simulations were done using Simbiology, a package from Matlab.

FIG 1: Our system is conformed by two regulation mechanisms. The first mechanism is the one controlled by us through AHL. LuxR and AiiA compete to bind AHL when it enters the cell. AiiA efficiently degrades AHL, while LuxR and AHL form a dimer. This dimer serves as an activator of cI*, which represses RcnA. The second of these mechanisms is the natural regulation of RcnA in response to the intracellular nickel concentration. When there is no nickel inside the cell, RcnR represses RcnA. However, when nickel enters the cell, it forms a dimer with RcnR and changes its conformation so it no longer represses RcnA. RcnA is then free to start pumping Ni out of the cell. We are keeping this because it is damaging to the bacteria to have the pump always on, and otherwise it would need a constant supply of AHL.

Metabolites and enzymes relevant to the model

Reactions

You can click on the next image to see a table of our reactions with their kinetics.

* The equations are numbered like this because those we had initially defined evolved into this final list throughout the summer. We didn't want to change all references made to these equations so we just adjusted the numbering.

Ordinary Differential Equations

We are taking into account the following set of ODEs, based on the biochemical reactions above. This set accurately and completely describes our model. Please click on the image to see a higher resolution.

Assumptions of the model

The complete model uses 18 kinetic parameters and 11 biochemical reactions. We got 13 of these parameters researching the literature, and of the other 5 we estimated 2. The remaining 3 we adjusted to the observed results. Reaction kinetics were gotten from the literature, and if no evidence was found then we assumed it to be Law of Mass Action.

1. Degradation of AHL by AiiA

AiiA + AHL → AiiA Kinetics: Michaelis-Menten1,2 Parameters: k1cat = 27.97 s-1       K1m = 3.723 mM = 224.20427E5 molecules Flux:

2. Complex formation and dissociation between AHL and LuxR

AHL + LuxR ↔ AHL:LuxR Kinetics: Mass Action3 Parameters: k2 = 10 -5 molecules-1 s-1     k-2 = 3.33 x 10 -3 s-1   Flux: Notes: The complex formation is slow and its dissociation is fast, so with few AHL and LuxR the complex concentration is negligible.

2.1. Dimer formation and dissociation between AHL:LuxR complexes

2 AHL:LuxR ↔ (AHL:LuxR):(AHL:LuxR) Kinetics: Mass Action3 Parameters: k2.1 = 10 -5 molecules-1 s-1     k-2.1 = 10 -2 s-1   Flux:

3.1. CI synthesis induced by AHL and LuxR complexes dimer

ρcI + (AHL:LuxR):(AHL:LuxR) → ρcI + (AHL:LuxR):(AHL:LuxR) + CI Kinetics: Mass Action3 Parameters: k3on = 10 -2 molecules-1 s-1       Flux:

3.2. Constitutive CI synthesis

ρcI → ρcI + CI Kinetics: Mass Action3 Parameters: k3off = 4 x 10 -2 s-1       Flux: Notes: To give more stability to the off state in the model, the rate constant in the presence of the inducer is lower than the constitutive rate constant, regardless the implication of a greater threshold to achieve the on state3.

4. Natural degradation of CI

CI → Ø Kinetics: Mass Action3 Parameters: k4 = 0.002888 s-1       Flux: Notes: The half life of CI with LAA tail is 4 minutes8. Andersen JB et al.9 conclude that LAA tail and LVA tail modified the half life of GFP in a similar extent. Given this value, the rate constant was calculated..

4.1. Dimer formation and dissociation between CI molecules

2 CI ↔ CI:CI Kinetics: Mass Action3 Parameters: k4.1 = 0.00001 molecules-1 s-1 k-4.1 = 0.01 s-1 Flux: Notes: Kenneth et al. estimated the change in free Gibbs energy in this reaction (with wildtype CI) as -11.1 kcal/mol,10 which leads to an equilibrium constant of 8.32186E16 molecules-1. This implies that the forward rate constant should be at least sixteen orders of magnitude greater than the reverse rate constant, which means a constant repression of RcnA even with the constitutive CI synthesis. A parameter scan was run to determine the range of values that gives the desired behavior and the rate constants were chosen arbitrarily within this range. These values are comparable to others typical biochemical parameters. It has been shown that kinetic parameters can be modified by changing amino acid sequences (for example, CI half life is reduced by adding a LVA tail in the C-terminal), it’s proposed that it’s possible to engineer the protein to reach an acceptable dissociation constant.

6. RcnA production

ρ → ρ + RcnA Kinetics: Cooperative inhibition (Hill kinetics)4,5,6,7 Parameters: n5 =1.9 Flux: Notes: ΔGCI:CI-OR1=-11.6 kcal/mol ΔGCI:CI-OR2=-10.1 kcal/mol ΔGCI:CI-OR1-OR2=-23.8 kcal/mol ν6(Pl)=20mM/h=3346.111 molecules/s with 20 promoter copies (ρ0)7. The promoter in our construction is Pr, which is similar to Pl, the one used to estimated the parameter7.

7. Nickel efflux by RcnA

RcnA + Niint → RcnA + Niext Kinetics: Mass Action3 Parameters: k7 = ? Flux: Notes: Experimentally measured

8. Natural degradation of RcnA

RcnA → Ø Kinetics: Mass Action3 Parameters: k8 = 1.666E-4 s-1 Flux: Notes: This kinetic parameter wasn’t found in our bibliographic search and personal communication with Peter T. Chivers (Washington University School of Medicine) confirmed that this parameter is unknown. The value used is the degradation rate of LacY, the lactose permease of E. coli, which is also a transmembran protein.11

9. Nickel import by unknown channel

Unk + Niext → Unk + Niint Kinetics: Mass Action3 Parameters: k9 = ? Flux: Notes: Experimentally measured

NOTE: The average volume of an E. coli cell is 10^-15 liters.

Defining the initial state of the system

The initial concentrations of the constitutive proteins (AiiA, LuxR, CI -constitutive synthesis- and CI:CI -due to constitutive synthesis-) were estimated based on the efficiency rate of their promoters, number of promoters per cell, degradation rate of their mRNAs, translation efficiency and degradation rate of the proteins. Initial concentrations of AHL:LuxR complex, the dimer of complexes, CI and CI:CI due to complex activation were set to 0, given these are all due to the action of AHL. Number of copies of both cI and rcnA promoters are 10 based on plasmid copy number. RcnA and Unk were estimated experimentally and set consistent to the observed rate. Concentration of AHL and nickel is determined by us to obtain the desired results.

AHL: It’s an arbitrary and adjustable value. Different outcomes can be observed manipulating this initial value.
Nickel (total): It’s an arbitrary and adjustable value. Different outcomes can be observed manipulating this initial value.
Unk: Both the Unk concentration and its rate constant are unknown. They are arbitrarily defined in such a way that the flux of the reaction 9 is consistent with experimental measurements.
     [Unk]= (SLOPE) molecules
Ni_int: The initial concentration of Nickel inside the cell is estimated based on experimental measurements in absence of AHL.
     [Ni_int]= (SLOPE) molecules
ρ and ρCI: Their concentration is defined by the copy number of the plasmids that contain them.
    [ρ]= 10 molecules
    [ρcI]= 10 molecules
CI and CI:CI: Given the constitutive synthesis and degradation rate of CI, as well as its dimerization constant, CI and CI:CI concentrations are estimated in absence of AHL.
    [CI]= 138 molecules
    [CI:CI]= 19 molecules
RcnA: Given the synthesis and degradation rate of RcnA, as well as the constitutive concentration of CI:CI, RcnA concentration is estimated in absence of AHL.
    [RcnA]= 33150 molecules
AiiA: The constant concentration of AiiA is calculated taking into account the following parameters retrieved from literature:
         - pLac average transcription rate^1,2: 0.003 s^-1
         - mRNA average degradation rate^1,3: 0.00766 s^-1
         - Average translation rate^1,3: 0.31333 s^-1
         - AiiA degration rate: 0.00012 s^-1
The half life of RcnA with LVA tail is approximately 2 minutes¹⁴; Andersen JB et al. found that this tail reduces the half life of GFP forty-eight times.9 Therefore the half life of wildtype AiiA can be estimated to 96 minutes.
    [AiiA]= 10000 molecules
LuxR: The constant concentration of LuxR is calculated taking into account the following parameters retrieved from literature:
        - pTet average transcription rate^12,15: 0.003 s^-1
        - mRNA average degradation rate¹³: 0.00766 s^-1
        - LuxR translation rate¹⁶: 0.556 s^-1
        - LuxR degration rate¹⁶: 9.627E-5 s^-1
    [LuxR]= 22000 molecules