Team:HBUT-China/Model

Overview

Achievements

1. In our modeling work, we established the system dynamics equations according to reaction details.

2. We found some model parameters in literature; for others we derived the optimization model for unknown parameters based on the quasi-steady-state-approximation (QSSA), which can be calculated by the MEIGO toolbox in MATLAB.

3. In addition, we established the logistical regression equation for cytotoxicity of Ni.

4. For the incomplete transcription repression, we have used the Hill equation to describe.

5. Finally, we gave the experimentation group some advice about the design of their experiments and provided data analysis of the entire process.

Modeling

ASSUMPTIONS

We made four assumptions in order to simplify some aspects of the model. Many similar assumptions have been made in the literature.

1)The degradation of and other substances are linear.

2)Each cell has the same state.

3)The reaction in the process satisfies the first-order kinetic reaction.

Chemical Species

Chemical species	Symbol	Description
*N_crB*	N	Concentration of repressor protein
*RFP*	R	Concentration of red fluorescent protein
*promoter_RFP-N_crB*	P-N	Concentration of promoter-NcrB complex
*N_crB-Ni*	P-Ni	Concentration of NcrB-Ni complex
Ni	Ni	Intracellular concentration of Ni
*Ni_gx*	Ni_gx	Extracellular concentration of Ni
promoter_RFP promoter_NcrB plasmid	P	Concentration of plasmid. The promoters are all occupy the same plasmid.
*mRNA_RFP*	RR	Concentration of RFP’s mRNA
*mRNA_NcrB*	RN	Concentration of NcrB’s mRNA

1. Transcription Repression

In the culture media without Ni, E.coli has the following transcriptional repression process:

Figure 1

Reactions

Differential Equations

Applying mass action kinetic laws, we obtain the following set of differential equations.

2. Release Repression

In the culture medium containing Ni, the repression is released and the following reaction process is produced.

Figure 2

Reactions

Differential Equations

Applying mass action kinetic laws, we obtain the following set of differential equations (2.1) ~ (2.7).

3. Quasi-steady-state approximation (QSSA)

In the following derivation, we add the cup mark to parameters that could not be found in literature, for example . Parameters found in literature are listed in the parameters table.

Applying the quasi-steady-state approximation to the equation (2.10), the following derivation can be found.

It can be changed to the equation (3.1).

Also, applying the quasi-steady-state approximation to equation (2.8), the following derivation can be found.

It can be change to equation (3.3).

Substituting equation (3.3) into equation (3.1) gives:

Substituting equation (3.3) into equation (2.9)

,

gives:

then, by the indefinite integral gives:

Substituting equation (3.3) into equation (2.5)

gives:

Applying the quasi-steady-state approximation to the equation (3.7), give the following equation (3.8).

It can be changed to equation (3.9).

The following is the derivation of (symbol: ).

Substituting the equation (3.2) into equation (2.6) ,

gives:

Substituting equation (3.4) into equation (3.10), gives:

Substituting equation (3.9) into equation (3.11), gives:

However, considering the complexity of substituting the quasi-steady-state solver operator again, we simply use the approximates to , then (3.13) is given.

Substituting equation (3.6) into equation (3.13), and simplifying, gives:

where,

4. Incomplete Repression

The incomplete repression of promoters is a major issue in our fluorescence system. This issue was particularly observed in following diagram.

Figure 3

We use the Hill equation to model this process.

The common formulation of the Hill equation is as follows.

Consequently, we modeled the incomplete repression by using the ratio of occupied promoter concentration to total promoter concentration.

In view of the fact that is difficult to measure and and have the same trend as , we used the concentration of nickel ions to approximate the equation (4.2).

So, the fluorescence intensity can be obtained by the following ideal formula (does not consider cytotoxicity of nickel ions - this is addressed in the next section).

The parameters for the production functions are:

– basal expression level of promoter

– maximal expression level of promoter

– half maximal effective concentration of

– Hill coefficient for induction.

we can use regression to obtain the Hill coefficient for induction .

Added Variable (4.2), The differential equation (2.6) was updated by:

so, we get:

where,

. 5. The Cytotoxicity of Nickel Ions

We found that a high concentration of nickel ions will reduce the expression of fluorescence, because nickel ions have cytotoxicity.

The July 20 experimental data is as follows:

Figure 4

Using the Logistic equation to fit the data (the incubation time was 6 hours).

We get the following results:

Coefficients (with 95% confidence bounds):

a = 2.615 (2.501, 2.73)

b = 0.9272 (0.7015, 1.153)

r = 1.836 (1.47, 2.201)

Goodness of fit:

SSE: 0.01289

R-square: 0.9985

Adjusted R-square: 0.9977

RMSE: 0.05677

We find that the adjusted R-square is close to 1, indicating that the regression equation for fitting data is very appropriate. The figure of regression is as follows:

Figure 5

Based on the previous data and graphic, the toxicity function can be established as:

Equation (2.7) is then modified to:

6. Calculating Unknown Parameters

Solving for these parameters can be regarded as the optimization problem, meanings the sum of the errors of the numerical solution of the ordinary differential equation and the original experimental data should be smallest.

The optimization model is as follows:

where RFP_i,j is experimental data in time i and at index j of nickel ions concentration.

Using the MEIGO Toolbox (in MATLAB 2015a) to solve optimization model (6.1), we can obtain the unknown parameters.

In the toolbox, we can choose different algorithms. These algorithms have differing stability and efficiency.

We realized that the global algorithm has better convergence, but it’s run time is too long. In this situation, the local algorithm also meets the requirements.

Parameters and References

Our exhaustive list of parameters is summarized in the table below.

Parameter	Value	Description	Source/Rationale
α_RN	0.0014nM·min^-1	mRNA production from NcrB DNA	Found this rate from [4]
β_N	0.0093nM·min^-1	NcrB protein synthesis rate from NcrB mRNA	Found this rate from [5]
κ_-P-N	0.1 to 1.0 nM	Rate of dissociation of promoter-NcrB to form complex	Range defined relative to experience, not founded in literatures.
κ_+P-N	0.1 to 1.0 nM	Rate of dissociation of promoter-NcrB to form complex	Range defined relative to experience, not founded in literatures.
γ_p	0.007min^-1	Production of promoter	Found this rate from [3]
κ_-Ni	0.0045nM·min^-1	Rate of Ni ions releasing	Found this rate from [2]
κ_+Ni	0.1 to 1.0 nM	Rate of Ni enteringl	Range defined relative to experience, not founded in literatures.
κ_-N-Ni	0.1 to 1.0 nM	Rate of dissociation of NcrB-Ni to form complex	Range defined relative to experience, not founded in literatures.
κ_+N-Ni	0.1 to 1.0 nM	Rate of dissociation of NcrB-Ni to form complex	Range defined relative to experience, not founded in literatures.
Κ	8 nM	Formation rate of NcrB-Ni	Found this rate for NcrB and promoter dissociation simultaneously with NcrB and Ni formation from [1]
α_RR	0.0014 nM·min^-1	mRNA production from RFP DNA	Found this rate from [4]
β_RFP	0.0093 nM·min^-1	RFP production from RFP mRNA	Found this rate from [5]