Designed Protein Degradation Method Based on

Trim21 And Nanobody -- Basic Design

# Basic Design

## Introduction

Compared with general model whose circuit directly express the protein and functions, the construction of model is complicated. Our trim21-antibody system needs to bind to target protein after expressing trim21 and antibody to complete the task. So we made the following design for the model:As is shown in the flow chart, we split the whole process into three parts: the plasmid expresses GFP antibody and trim21, antibody binds to antigen, the ubiquitination by Trim21 to the final degradation, which are connected by bridges (concentration of antibody and complex).Then the final ODE equations are constructed.

*Fig.1 Three parts of the whole process.*

## Assumption and Justification

About the model：

1、Due to the stable environment of cell culture and the sustainable supply of energy required by cells by culture media, cells with high degree of confluence will pass through. It is believed that the cell proliferation rate is basically stable and the intracellular chemical reaction rate constant is basically unchanged.

2、Because intracellular protein turnover was basically stable under the involvement of intracellular e3 enzyme, and the endogenous trim21 protein level in the cell lines studied was low, we ignored the effect of the endogenous trim21 enzyme on the degradation of target protein in our study object, and believed that protein degradation was basically completed by the exogenous trim21 expression.

3、The reaction rate can be described by the law of mass action.

About the data:

1. The data we obtain from wet-lab experiment are reliable.

2. All the results are trustworthy in the process of statistical processing and data calculation.

## Equation

*Fig.2 A generalization of the equations involved in the model.*

## Data Analysis

However,the models now constructed are soulless.
Since the kinetic models require a lot of parameter values to describe genetic process,some of which are not mentioned in literature or related to a large amount of implicit assumptions involved in the process of substitute in such a parameter into an ODE model,we performed parameter simulation based on our datasets to ensure as much as possible that our insights from the model analysis will be impact future teams.

The least squares method is a mathematical optimization technique,which finds the best function match for the data by minimizing the sum of the squares of the errors(
).
In 1829, Gauss proved that no unbiased estimator would be superior to the ordinary least squares estimator. Moreover, in the case where the error conforms to the normal distribution, the least squares is equivalent to the maximum likelihood, which minimizes the empirical risk .In the parameter estimation, the least squares method and the partial derivative are used to obtain the parameters by solving for points with a gradient of 0. This method can expediently obtain the unknown data while ensuring the accuracy, which is explicit to be expressed by formula.

Simbiology offers the parameter fitting method based on the least squares method, which can create interfaces that connect both parametric equations and data. We used it to process our data for parameter fitting and got the following results.Implementation code uploaded here.

*Fig.3 All parameters fitted through the least square method.*

*Fig.4 Comparison of simulated and measured values of the model.*

Further more, the models we built are so complicated that we believe the results of the least squares method also need to be checked due to the limitations of the algorithm and the amount of data. To validate and further ensure the validity of the model, we next use another method--the neural network in machine learning, to analyze the model and find a direct link between input and output.

Neural networks have the ability to learn and construct models of nonlinear complex relationships after learning from the initialization inputs and their relationships.It can also infer the unknown relationships between the data, allowing the model to generalize and predict unknown data. This method simplifies the complexity of the model,rather than specifically considering each parameter and each step of the reaction. Generally, it does not have strict requirements like the least squares method for the accuracy of the equation, which could obtain the mapping relation between the plasmid dosage and protein degradation rate more directly.
Back Propagation (BP) Neutral Network is a kind of Multilayer Feed Forward Neutral Networks (MIFFN), which can be effectively used in big data analysis and prediction.
Figure shows how BP Neutral Network works:

*Fig.5 .A schematic diagram of a neural network*

Establish a typical BPNN model :

There are 8 specific steps to set up a typical BPNN model:

Step1: Initialize the weight and threshold of the net.

Step2: Calculate the input and output of the first layer.

Step3: Calculate the input of the middle layer.

Step4: Calculate the output of the middle layer

Step5: Calculate the input and output of the third layer.

Step6: Introduce a power function E.

Step7: The adjust of weight and threshold between adjacent layers.

Step8: Restore the result.

We use 4 indexes, including time,the transfected plasmid concentration of gfp and the transfected plasmid concentration of Trim21-nanobody to train the neutral network. The training parameters and the corresponding test results are shown as below.

*Fig.6 Our BPNN model for big data analysis and prediction. Value 1,2,3,4 are represented as different neutral network with different parameters. We explicitly account for the relation between input and output.*

*Fig.7 Different parameter Settings correspond to different test results.*

Then，we can use them to test the model obtained through the least squares method.Implementation code uploaded here.