Background

Our team's trying to synthesize delphinidin with seven enzymes. The whole process can be divided into two separate parts: converting coumarin into naringenin, and then converting naringenin into delphinidin. In this series of reactions, naringenin plays a very important role in both parts, so it is very important for us to monitor the generation of naringenin. The efficiency of traditional HPLC method is particularly low and cannot monitored reactions in a timely fashion. We decide to use a naringenin biosensor instead.

We have found a naringenin sensor once used in a previous iGEM competition. We also referred to the original paper of the sensor for coding sequence and design principles, and we updated the gene circuit for our pETDuet vector.

In order to study the sensor response to the presence of naringenin, we planned to vary naringenin concentrations and bacterial culture conditions exhausting all combinations, which eventually would add up to more than 60 experimental groups. Apparently it would have taken too long to get the results. and we decided to conduct the experiment with modeling.

Based on our experimental results, we found that the concentration of naringenin in the generation process ranged from 10uM to 1000uM. We hope that by adjusting, the concentration range that makes the biosensor sensitive can also stay in this range. In order to give better guidance to the experimental team and suggestion possible directions for improvement, we have established a mathematical model to theoretically calculate the value of the sensitive concentration range of the naringenin sensor. Then we can tweak the naringenin concentration to activate response more gently and easily to be observed in experiments.

Hypothesis

For the sake of convenience, when a quantity is not one of our experimental variables, we set it to a constant, and the values are referred from the literature.
(1) We assume that cell numbers are at a plateau, ignoring growth, division and death when they are abundant and in good condition
(2) We assume adequate nutrition so that there won't be any fluctuations due to nutritional deficiencies
(3) Naringenin can enter and exit cells freely, without considering the rate for transporting through the barrier such as the membrane.
(4) Naringenin added to the cell does not degrade.

Equations

Figure: Detection mechanism for FdeR regulator, referenced from work [1].

Our sensor detects naringenin in three steps, and we've written out these three steps separately. First, promoter PfedR initiates the expression of the FdeR protein at the rate of α1. The FdeR protein is degraded at the rate of δ1. Over time, t, the concentration of FdeR protein was accumulated in the cells to [P]. In summary, the change rate of fdeR protein concentration can be expressed as follows.

Next, the naringenin concentration is [Nar], and n naringenin molecules bind to the m FdeR protein to form the receptor-naringenin-binding body concentration [PNar]. Therefore, the binding rate of naringenin and fedR protein can be expressed as the following eq.

Third, based on Hill equations

At the inducer concentration of X (X= [PNar]), PfdeA expresses GFP at rate α, and GFP is leaky expressed at rate ε, and is degraded at the rate of δ A. The m is the Hill constant. X__M is the X value when the trend of the model changes.

Solution

Because we want to use the sensor to detect naringenin in a relatively stable condition, the sensor's rate of protein production can be assumed zero. According to this equation.

We change the formula for zero protein detection rate, by adding naringenin on the right-hand side

And here’s the second formula for the rate at which naringenin binds to the protein. We call the combination of naringenin and protein “Q” for short

Then we plug this into the formula with the binding rate of naringenin and protein

Lastly, we substitute this simplified formula into the variable X of the Hill equation

Substitute Trial Numbers for the Parameters

We plug all the constants into the equation, in this part for instance we let α1 to be 1.

And finally, we get these formulas for plotting. Afterwards we changed the value of α 1
Whenα1=1

Whenα1=0.1

Whenα1=0.01

Whenα1=0.001

Mapping

The black, red, cyan and yellow lines represent α1 at 0.01, 0.001 and 0.0001, and 0.5E-4, respectively. The plotted results show that moving along the positive X axis the production of green fluorescence first rises quickly and then gradually stabilizes and asymptotes to maximum, which is one of our successes.

And in order to make the reaction slower, we tried to change the variables including α, α1, δ and it turned out that when we change α1, i.e., when we slowed down the rate of fedR protein production, we could slow down the rate of reaction. Different plots for values of α1 showed that α1 at 0.001 would best meet our expectations for a convenient experimental observation.

Future Plan

In the future, we will take into account the variables that we have assumed as constant at the moment. We will measure the transportation of naringenin through the cell membrane to accurately calculate the optimal naringenin concentration and many other possible variables we described in the hypothesis part. We will fix problems on our naringenin bio-sensor, and find the relationship between the naringenin concentration and GFP fluorescent intensity. We will fit the experimental data to our model and finetune our sensor For it to work within our target concentration range of naringenin-from10uM to 1000uM.

References

[1] S. Siedler, S. G. Stahlhut, S. Malla, J. Maury, A. R. Neves “Novel biosensors based on flavonoid-responsive transcriptional regulators introduced into Escherichia coli”, Metabolic Engineering 21 (2014) 2–8