For our engineering, to transform Clostridium tyrobutyricum so that it can survive under aerobic conditions, we need to predict the strength of the promoter to assess whether the performance of the promoter meets our experimental requirements.There is no doubt that one of the basic foundations of predicting promoter strength is to perform accurate, quantitative estimates and simulations. Generally, the way to do this includes accurate measurements and simulations using mathematical methods.

For this, we refer to the relevant content of Proceedings of the 26th Annual International Conference of the IEEE EMBSSan Francisco, CA, USA •September 1-5,2004^[1]. We selected the continuous deterministic model and built the oxygen inhibited promoter model based on it. Through this model, we can analyze the performance of promoters in advance to guide the experiment. These theoretical models will be improved iteratively once the experimental results are achieved. In addition to the oxygen inhibition promoter model, we also refer to the Logistic equation and build a model for the growth of engineering bacteria on this basis, which helps us to control the growth of engineering bacteria macroscopically^[2].

(1) Assist the wet lab team in verifying the feasibility and correctness of the experiments through mechanism-based modeling.

(2) Obtain experimental data from the wet lab and improve the model we developed.

After we modified the VGB promoter, cells can survive in the aerobic environment. We use ordinary differential equations to simulate the growth of cells under different oxygen concentrations. First of all, we use Hill equation to describe the process that oxygen inhibits the strength of VGB promoter and thus affects the expression of perR protein. Secondly, we used the Logistic equation to simulate the cell growth curve, and we also considered the inhibition of perR protein concentration on cell growth under aerobic conditions.

(1) The initial concentration of perR was assumed to be 0.

(2) The initial value of OD₆₀₀ was assumed to be 0.156.

(3) The oxygen was seen as the repressor to repress synthesis of perR protein.

(4) The repression of oxygen on gene expression was assumed to follow Hill equation characteristics.

Before fitting the parameters of the model, we use the established model to preliminarily simulate the OD₆₀₀ values under different oxygen concentrations, and obtain the general trend to guide the preliminary experiment.We simulated the growth of engineering bacteria with different oxygen concentrations through our model. This provides a reference for our experiment to select the appropriate oxygen concentration to cultivate engineering bacteria.

After phased results of the experiment were achieved, we optimized the model by fitting the parameters of the model with the experimental data.

After building the preliminary model, we get the model parameters by fitting the experimental data to verify the correctness of the model. We use the gradient descent method to iteratively obtain the model parameters so that the model can better fit the experimental data. All the initial value of iteration of model parameters was assumed to be 1. Our iterative fitting process is realized by writing Python programs. In order to better observe the iterative process, we choose a relatively small learning rate. For the loss function of the model, we choose the mean square error to evaluate the model.

In fact, in the process of parameter fitting using gradient descent method, we found that the selection of iteration step size, also known as learning rate, has a great impact on the final result. At first, we chose a step size of 0.01 during iteration. We found that even if we iterated more times, the final result kept jumping up and down in the attachment of the optimal solution, and could never fall to the optimal solution. When the step size is 0.01, with the increase of iteration times, the iteration result as follows:

According to Fig 5, we can see the decreasing speed of the mean square error of models with different iteration steps. In general, if the iteration step size is selected relatively large, the result of gradient decline may fluctuate around the optimal value. If the iteration step size is selected relatively small, the convergence speed of the model will slow down. Therefore, the best strategy is to use a dynamic iteration step size. If the current iteration result is far from the optimal solution, use a larger iteration step size. If the current iteration result is close to the optimal solution, use a smaller iteration step size. However, it is not easy to find a suitable strategy to adjust the iteration step size for different problems.

Although the result of parameter fitting looks good, the model after parameter determination cannot fit all experimental data. On the one hand, due to the limitation of experimental time and other factors, we lack a large number of experimental data on the value of OD₆₀₀ at different oxygen concentrations. On the other hand, because some parameters in the model are difficult to be determined by experiment or parameter fitting, the accuracy of the model is low. When we change the oxygen concentration, the discrimination of the output results of our model is relatively small. Therefore, we fine tune the model and optimize the model by simplifying the equations to better simulate the growth curve under different oxygen. We ignored the whole process of oxygen inhibiting the strength of VGB promoter, thereby reducing the expression of perR and promoting cell growth, and directly established a differential equation between oxygen and cell growth. We simplified the whole process into that oxygen has a positive effect on cell growth referring to Hill equation.

(1) The initial value of OD₆₀₀ was assumed to be 0.156.

(2) The oxygen was seen as the repressor to repress synthesis of perR protein. The repression of oxygen on gene expression was assumed to follow Hill equation characteristics.

(3) The positive effect of oxygen concentration on cell growth was assumed to follow Hill equation characteristics.

After the optimization of our model, we reduce some parameters that are difficult to determine, thus reducing the error of the model. At the same time, according to the simulation results of the model, we can specifically see the impact of oxygen concentration on cell growth.