In this part, we have two main aims to achieve:
1. Estimating the expression quantity of ferritin and preparing for temperature calculation in the next part.
2. Analyzing the noise in ferritin expression and identifying its sability.
(If you want to learn more details about the ferritin or Magnetic Receiver, you can click here)
Deterministic Model
With the purpose of controlling the expression, we used PQE-80L, which has two Lac I binding sites, to construct our genetic circuit.
The schematic figure is shown below:
And the schematic diagram can be simplified as follow:
Using Michaelis-Menten kineticsmass-action laws, we constructed the following set of differential equations:
Because the vector we used was PQE-80L, which has two Lac I binding sites, we created a parameter “α” to describe the probability that RNA polymerase can bind to the promoter of interest.
Parameter Choice
Modeling Induction Effect with α
We created the parameter “α” with thermodynamic model. We made the assumption that the gene expression level is proportional to the equilibrium probability that RNA polymerase is bound to the promoter of interest.
This statistical mechanics provide a framework for computing these probabilities.Once these states have been enumerated, we weight each of them according to the Boltzmann Law: if "e" is the energy of a state, its statistical weight is exp(-ε/kBT). Finally, to compute the probability of promoter occupancy, we constructed the ratio of the sum of the weights for the favorable outcome (i.e. promoter occupied) to the sum over all of the weights[1].
In our modeling, there are three states for the lac I binding sites: none of them is occupied, either of them is occupied and both of them are occupied. And only in the third state can the gene be transcribed.
So the “α” can be described as follow:
In order to simplify the representation, we divided the numerator and denominator by exp(-ε1/kBT), then we can get a simple form:
In the formula, Δε stands for:
and its another form is
and the “a”in the formula means:
So we can get a simple from of “α”:
Results
With “α” calculated, we can simulate the induction effect of IPTG as shown blow.
From the result shown above, we can drew the conclusion that the leak from the gene expression was terrible and this result got great feedback from our wet lab experiments.
Sensitivity Analysis
In this part we used local sensitivity analysis to study the impact of small fluctuations on the model outputs and observe which parameter takes the greatest importance.
Local sensitivity analysis involves computing the relative change of the steady state with respect to a change in the parameter. So we used Matlab and a finite-difference approximation of the derivative, sensitivities were calculated for 5% changes in the parameters. The sensitivity below was defined as the ratio of the output with changed parameters and the output with unchanged parameters. So the larger the ratio was, the more sensitive the parameter was.
The sensitivity coefficient was defined as the ratio of the value with changed parameters and the value with unchanged parameters:
And the sensitivity of some important parameters were shown as follow:
From the table above we could find that the degradation rates showed great influence on the result of the deterministic model.
Stochastic Model and Noise Analysis
As a magnetic receiver, we need a relative stable steady-state for the coming heating process. So the noise from the ferritin’s expression should be really small.
Consequently, we constructed the stochastic model with the condition that the promoter was fully induced by IPTG. We used the Gillespie SSA to simulate the reaction and calculate its noise through the stationary distribution of ferritin molecules[23].
According to the stationary distribution of ferritin molecules, the noise we calculated was 0.0058, which was very small. It means that ferritin’s expression level would be stable for a long time. So in the heating process, we could get a relative stable change in the whole-cell temperature distribution.
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