MODELING

The goal of our experiment is to develop an alternative method to quickly visualize the response of E. coli to exogenous autoinducer by expressing the already present fusion proteins of GFP to the autoinducer receptor. As hypothesized, this new system for inducing protein production is expected to be quicker. The expression of protein relies on various factors, especially, on the level of the monomers of GFP attached to the receptor already present in the cell and the degradation rate of these monomers which is associated with the degradation of the TraR receptor. In the following discussion, we represent an analytical approach for the modeling of the protein production based on the factors stated above.

1)      Time taken for Diffusion of Auto-Inducer across cell membrane (T1~ seconds)

For calculation purposes, we assume that the auto-inducer diffusion across a cell membrane is identical to passive diffusion of small molecules through a membrane. The diffusion of a molecule across a cell membrane is dependent upon the parameters of the surface area and the volume of the cell as well as the permeability of the cell wall to the auto-inducer. The primary assumption for the diffusion of auto-inducer across a cell membrane in this case would be the presence of excess amounts of auto-inducer outside the cell than inside of the cell at all the times.

The diffusion rate of AI can be described by Fick’s Second Law as indicated by the following equation:

Where,

D1 = Diffusion Constant for auto-inducer = 7*10^-6 cm2/s

C = Concentration of AI

Solving the above equation with a fixed boundary condition described below:

C{x=0, t} = Cs = 1µm (This is the concentration of AI outside the cell which will be a constant known amount fixed by us)

C{x, t = 0} = Co = 0 (This is the initial concentration of AI inside the cell at a distance of thickness of the cell membrane)

2)      Time taken for binding of AI to the TraR receptor (T2)

This can be solved as ligand-receptor binding kinetics. The equation for excess Ligand present (or AI in our case) is

Where,

c = number of the complexes of AI+TraR receptor/cell = (Assuming we need 5000 GFP molecules, c = 10000 molecules)

kf= forward rate constant or binding constant=1.4*10^-4 m-1 s-1; kr = reverse rate constant or dissociation constant = 10^-2 s-1

R = number of unoccupied receptors/cell = 23000 – c (23,000 is the maximum steady state TraR-GFP fusion molecules)

L = number of AI/cell = 300

Solving, minimum T2= 130 seconds

3)          Time taken for transcription + translation in our GFP reporter system+

Time taken for the accumulation of nGFP and cGFP molecules in cell in our system (T3=0)

Time taken for above steps for traditional GFP system is

The GFP will be fluorescent after the maturation (post-translational modification + accumulation) and expression of dimer GFP. The equation will be based on the maturation, the cell growth rate and degradation of the dimer GFP and is detailed below:

Index for variables:

                m: maturation rate (accounts for post-translation modification and accumulation)

D: Degradation of GFP dimer.

                u: Growth Rate of bacteria

Equation :

Literature Values:

·          Maturation can be described as a first-order reaction and the constant m can be calculated based on the time constant of GFP maturation whose range has been determined from 30 minutes to 2hours.  So, m= {0.5, 2} hr-1. (Leveau et.al, 2005)

·         The GFP dimer is hypothesized to be a very stable molecule and its degradation can be determined based on its half-life of 2.8hr. Thus, D= (ln2/2.8) hr-1 (Halter et.al, 2007)

·         Growth rate of bacteria ranges from 0.5-1.5 hr-1

·         It has been determined that the concentration required for detection of cytoplasmic GFP above typical cellular auto-fluorescence is 0.1-0.2µM which roughly translates to a range of 5000-10000 GFP molecules per cell. (Patterson et al., 1997)

Integrating the above equation and putting in the boundary condition of 5000 GFP molecules for the cell to fluoresces,

Similar calculations are performed for the boundary condition of 10000 GFP molecules.

Ø  When the values of the parameters are inputted in the above equation, the time required for accumulation of enough GFP molecules to be detected using standard microscopy produces a range of 0.5-2 hr.

Ø  The studies conducted in our parent lab suggest that the fluorescence reaches maximum at time range of 1-3 hours. Thus, our modeling is consistent with experimental results.

4)      Time Required for the dimerization rate of GFP: (T4)

 Since in the previous calculation, we assumed that the number of cGFP and nGFP molecules are the same, the dimerization reaction for the formation of GFP protein from the monomers can be represented as a second order reaction. Here, we are considering that the monomers are molecules which will randomly collide to result in the formation of dimers. Thus, the second order integrated equation for the dimerization rate will be given as follows.

Index for variables:

z: number of dimers formed

k: rate constant for the reaction

i: number of monomers to start with

Equation :

The integrated second order reaction is written as (in case of dimerization) by:

So, the time can be represented as

So, based on equation 3, time required for the dimerization rate of GFP can be calculated by inputting the boundary values for the number of monomers and dimers at steady state in the cell.

·         Derived values of the parameters

 i = 10000-23500  molecules/cell;

z = 5000 molecules/cell;

k = 10^-5 m-1s-1 (m is molecules per cell) (from literature)

Ø  The minimum value of time required for dimerization is calculated to be

T4 is 3 seconds and the average range is 3 – 200 seconds depending on the stability of TraR.

Since the cells consist of nGFP and cGFP DNA, there will be an accumulation of monomer GFP proteins within the cell. Thus, as regards to the time taken for the accumulation of the monomers after the addition of AI is zero. This result leads us to the conclusion that the nGFP and cGFP molecules are already present at a steady state level within the cells. The steady state level of the two monomers will be determined by the promoter activity, and the degradation of monomers associated with the degradation of the TraR receptor. The following analysis represents a mathematical representation of the above statements as well as assumptions.

·         Principle Assumption:

To simplify the calculations as well as the dimerization reaction, we are assuming that similar number of nGFP and cGFP molecules are present in the cell and that they have reached steady state.

Index for variables:

P: Promoter activity;

t1/2: half-life of traR;

i: number of immature nGFP or cGFP molecules (assume they are similar in number)

Thus, the rate equation to describe the change in concentration of immature nGFP or cGFP molecules is determined as:

Analysis of whether increasing TraR stability  would improve the response time

Equation :

Degradation Rate = ln2/ t1/2

(P *t1/2)/ln2

Thus,

Literature Values:

·         The promoter activity of GFP ranges from 0.4 to 0.04 (molecules/cell)/second. (Goryachev et.al, 2005)

·         The half life of TraR can be varied from 3 minutes to 680 minutes based on the methods described in Costa et al., 2012.

So, the value of i ranges from 734-2350 molecules in terms of lowest promoter activity while when the promoter activity is optimum (0.4) which we assume the case to be in our cells, the value of i ranges from 100-23500 molecules.

Total time taken for protein expression in traditional GFP system= T1+T2+T3+T4 ~ 60-180 mins

Total time taken for protein expression in our novel GFP system = 2-6 minutes

Conclusion:

The important thing to note here is that the removal of transcription, translation and accumulation time from the system.  Thus, this showcases a faster method of protein production.

ØThus, our model represents that our novel GFP reporter system reduces the time required for GFP expression  by a factor of 30, supporting our hypothesis.

Resources:

1.          Goryachev et al., A. "Systems analysis of a quorum sensing network: Design constraints imposed by the functional requirements, network topology and kinetic constants." BioSystems. 83.2006 (2005): 178-187. Print.

2.          Patterson et al., H. "Use of the Green Fluorescent Protein and Its Mutants in Quantitative Fluorescence Microscopy." Biophysical Journal. 73. (1997): 2782-90. Print.

3.          Costa, E, and S Winans. "The quorum-sensing protein TraR of Agrobacterium tumefaciens is susceptible to intrinsic and TraM-mediated proteolytic instabilitymmi_8037 807..815." Molecular Microbiology. 84.5 (2012): 807-15. Print.

4.          Leveau, J, and S Lindow. "Predictive and Interpretive Simulation of Green Fluorescent Protein Expression in Reporter Bacteria." Journal of Bacteriology. 183.23 (2001): 6752-62. Print.