Goal:

Our aim is to investigate and predict the yield of anthocyanins after adding different amounts of β-estradiol, and thus, to find out the optimum amount of β-estradiol to increase the yield of anthocyanins.

Equations:

The binding of a ligand molecule to its receptor is usually considered to be a one-step reversible reaction, and thus:

From the law of mass action, it follows that the rate of complex formation is:

The steady state means that the above reversible reactions are in dynamic equilibrium, and the rate of production and consumption of the receptor-ligand complex is equal. Therefore:

From there we obtain the steady-state dissociation coefficient for the reaction :

We define the proportion of bound receptors (complexes), F, as the number of all ligand-bound receptors as a proportion of the total number of receptors, we can have:

Setup and Assumptions:

1. Setup description: The added inducer β-estradiol is the ligand, and the corresponding receptor protein initiates downstream transcriptional expression of the downstream gene after binding with inducer, and what we end up measuring is the expression of the gene ---Anthocyanin content.

2. Assumptions

The receptor specifically initiates transcription of downstream genes only after binding to the ligand, and will not initiate transcription at all otherwise

The expression of downstream genes is proportional to the number of receptor bindings and can only be triggered upon the existence of bound receptors, which means that when the number of receptor bindings is zero, the expression of downstream genes expression of the downstream gene is zero

The total number of recptor proteins are all at the same level before we are adding inducer to it. Thus, we can evaluate the number of total bindings by the proportion of total bindings

d)The addition of the inducer influences the overall gene expression of the cell. Here we assume that it is a simple linear relationship as no evidence found indicating otherways.

Data collection:

Carrot root tips of hairy roots were selected and inoculated in M medium with Cefotaxime (200 mg-L-1) and incubated at 25 at 90 rpm in a shaker shaken from light and the medium was changed weekly. On day 30, after the total volume of medium had been increased to 50 mL, estradiol was added to reach final concentrations of 2, 4, 6, 8 and 10 μM, with three biological replicates for each concentration. The culture was continued for 120 h in shaking protected from light. Afterwards, the roots were collected, ground in liquid nitrogen, lyophilised and the anthocyanin content was determined.

Modeling process (ligand binding, with linear stress):

By assuming that the concentration of the added inducer is , the binding proportion of the receptors is , and the expression of the downstream gene is (content of anthocyanins), we can obtain the following mathematical model:

In this equation, the -mx term is the effect of the inducer on the overall gene expression of the cell. A is the ratio of the expression of the downstream gene to the fraction of receptors bound With previously described relationship between the fraction of bindings to the concentration of ligand, we can have here that

where Kd is the dissociation coefficient of binding. Combining these two equations, we can easily form the model mapping our experimental variable input to the output

Through the python code, we performed regression and have the following expression:

Here, y represents the anthocyanin content (relative unities of anthocyanin /g dry weight) and x represents the concentration of β-estradiol（µM).

Conclusion:

With our modeling, we are able to find out the amount of β-estradiol in order to produce desirable yield of anthocyanin. It is both theoretical and practical, and we tested the model using the results of our experiments. From the fitted model, we can convincingly conclude that a higher dosage of inducer will only further increase the production of anthocyanin in a small amount if not at all. And if to put this into real world scenario, such model will also help us figure out the best production-to-investment ratio within plausible range

Python Code:

      def model(x, A, k, m, epsilon=1e-15):
        return (x / (k + x)) * A - m * x

      popt, pcov = curve_fit(model, X, Y, p0=[1, 1, 1], bounds=(0, np.inf))
      A, k, m = popt
      X_plot = np.linspace(X.min(), X.max(), 1000)
      Y_plot = model(X_plot, *popt)

      fig = plt.figure(figsize=(8, 6))
      plt.scatter(X, Y, c="tab:blue", label="experimental data")
      plt.plot(
        X_plot,
        Y_plot,
        c="tab:orange",
        label=r"model fit($R^2=" f"{R2(df['AC'], model(df['BE'], *popt)):.3f}$)",
      )
      x_m = np.sqrt(A * k / m) - k
      plt.axhline(
        model(x_m, *popt), linestyle="--", c="tab:orange", label=f"limit={model(x_m, *popt):.3f}"
      )
      plt.xlabel(r"Concentration of $\beta$-estradiol in $\mu$M")
      plt.ylabel(r"Relative units of anthocyanin per gram of dry weight")
      plt.legend()
      plt.show()

      print()
      display(
        Math(
          r"\hat{y}="
          f"{popt[0]:.3f}"
          r"\cdot \frac{x}{"
          f"{popt[1]:.3f}"
          r" + x}-"
          f"{popt[2]:.3f}x"
          )
      )
    

Reference:

Zuo J;Niu QW;Chua. “Technical Advance: An Estrogen Receptor-Based Transactivator XVE Mediates Highly Inducible Gene Expression in Transgenic Plants.” The Plant Journal : for Cell and Molecular Biology, U.S. National Library of Medicine, 24 Feb. 2000, pubmed.ncbi.nlm.nih.gov/11069700/.

Tsang, Wing, and Assa Lifshitz. “Single-Pulse Shock Tube.” Handbook of Shock Waves, Academic Press, 2 Sept. 2007, www.sciencedirect.com/science/article/pii/B9780120864300500403.