Equation Chapter 1 Section 1Modelling

Our experiment creates a biological device for measuring Ni2+ concentration. Therefore, the main purpose of our modeling is to find the functional relationship between the ambient Ni2+ concentration (Nienvior.) and the luminescent intensity (LI) expressed by Lux. Firstly, we established a mathematical model to describe the dynamic process of the whole experiment, compared the predicted results with the experimental data, and adjusted the model with the experimental data to obtain the functional relationship between Nienvior. and LI.

Our model is divided into three parts. Firstly, NikABCDE is an ABC superfamily transport system, whose function is to transport Ni2+. Secondly, NcrB protein is the core regulatory system, which can bind with Ni2+ and then release the inhibition of promoter PncrA. Thirdly, as the inhibition of PncrA is released, the fluorescent protein gene of LuxCDABE connected downstream of PncrA can express and luminescence. Models are established for these three parts, and the effect of Nienvior. concentration on luminescent intensity is obtained.

Symbol Description

NikABCDE Transports Ni2+

NikABCDE is an ABC superfamily transport system, and can transport Ni2+ for E.coli. Among them, two transmembrane proteins, NikB and NikC, constitute the transmembrane core of the transport system. NikA is the periplasmic binding protein (PBP), which can transfer captured nickel ions to the NikBC core, and NikD and NikE are two cytoplasmic proteins (De Pina, et al., 1995). Navarro, et al., 1993).

NikABCDE system for transporting Ni2 + process shows below

ke is the binding rate of nikABCDE and Ni2+ , and the value of ke  is 0.1uM(Englert, 2010).

1. Assumption

        Ni2 + diffusion equilibrium assumption

when Ni2 + are transported into the cell, internal balance of Ni2+ concentration will be influenced, which can lead to uneven distribution of Ni2+concentration instantaneously. With a certain amount of time for the free diffusion, the Ni2+ concentration in the cell can be steady. So we assume that Ni2+ transported into the cell will have an uniform distribution instantaneously. And this assumption is completely true in the case of lower transport rate of the transporter protein.

2. Model

Reference diffusion balance model, establish ODE of Ni2 + diffusion model. From the law of mass action, the ODE between Ni and NiE is obtained

(1.1)

(1.2)

Regulation of PncrA Mediated by NcrB

When the bacteria were in a low-concentration Ni2+ environment, the repressor protein NcrB binds to promoter PncrA, making LuxCDABE downstream of PncrA unable to express. When bacteria are in a high concentration of Ni2+ environment, the intracellular Ni2+ concentration accumulates, NcrB binds to Ni2+, changes the conformational mode of NcrB, which falls off PncrA. The PncrA promoter is transcribed, and the LuxCDABE expression downstream of PncrA promoter emitted fluorescence. In this model, a kinetic model is established, and the functional relationship between Ni2+ concentration and Luminescent Intensity is deduced.

1. Assumption

        Assumption of NcrB

For NcrB, we make the assumption that there are three forms in the cells within the NCRB, respectively is free NCRB, NCRB - PncrA compounds, as well as the NCRB Ni2+ complexes. And in the process of inhibition of Ni2+ contact NcrB protein, which can be treated as competition between competition between Ni2 + and PncrA . In the process of competition, free NcrB can be regarded as an intermediate transition state. Specifically,  NcrB is detached from the promoter and becomes a free NcrB, which then binds to the Ni2+ to form a NcrB-Ni2+ complex, so the concentration of NcrB in the intermediate state can be seen as a constant.

        Transcription Assumption

1) The total transcription rate is mainly determined by the amount of substrate – NcrB dissociates with PncrA and the intensity of PncrA.

2) The promoter PncrA intensity (in polymerase per second) is a constant that represents the maximum expression rate of transcription.

3) The total transcription rate is the function of the maximum transcription rate and the fraction of the signal molecule bound to its binding site.

4) For constitutive expression of genes - NcrB, we can assume that it is the maximum transcript.

        Translation Assumption

1) The main factor influencing the translation rate is the rate of ribosomes across the mRNA, which can be approximated by constant rates.

2) The translation rate is limited by the amount of available mRNA transcripts that bind with ribosome.

3) The translation rate is a function of the ribosome velocity and the mRNA transcript concentration.

2. Model

Using the above parameters, we can construct a series of ODE representing our biosensor network.

           (1.3)

              (1.4)

                (1.5)

              (1.6)

is mRNA production rate from NcrB DNA, which values 0.0014nM·min-1(Junhua, 2012). is translation rate of NcrB mRNA, which values 0.0093nM·min-1 (Pengfei, 2013).

3. Model Simplification

To simplify the calculation, ignore the parts that have little impact, and simplify the process as follows

k2 is the reaction rate constant for the binding of Ni2+ and NcrB, the value of k2 is 8nM (Xinming Yang. 2013).

It can be seen from the assumption that R is the intermediate constant value during the change process, and R0 is the total amount of intracellular NcrB, which is a function of copy number B of plasmid

(1.7)

Intracellular PncrA has two states, that is, binding to NcrB and free

(1.8)

The binding process of Ni2+ and R can be regarded as steady state equilibrium,

(1.9)

(1.10)

Luminescence Process of LuxCDABE

When Ni2+ is combined with NcrB, it can make promoter PncrA transcription, and then LuxCDABE expression and luminescence in the downstream. Luminescent intensity is related to the amount of LuxCDABE expression, and the amount of LuxCDABE expression is related to the amount of promoter PncrA transcription.

1. Assumption

        All of the detected promoter PncrA were produced by the unbinding of NcrB

        The luminescent intensity is positively correlated with the promoter concentration

2. Model

The luminescent intensity can be modelled as below:

(1.11)

Relationship between LI and t

The purpose of modeling is to obtain the relationship between Luminescent Intensity LI and Ni2+ concentration in vitro, solve (1.1)~(1.10), and substitute (1.11) to get the analytical solution of LI.

(1.12)

while

           (1.13)

to simplify the formula, we discuss it further

LI can't go up indefinitely, or Ni2+ can't go up indefinitely, so r1, r2 <0，γ>0. When the ,

(1.14)

It can be seen that when the external Ni2+ in vitro is a constant value, the final LI will tend to be a constant value

 Relationship between Ni2+ in vitro and LImax

From the derivation, the relationship between lg(Nienvior.) in vitro and LImax is linear. However, according to the experimental data, LImax shows a downward trend after reaching a certain concentration, and tends to 0 as the concentration increases. Therefore, we suspect that this is related to the cytotoxicity of Ni2+ and related experiments have been conducted. Through the experimental results, the modified model of LI and lg(Nienvior.) is established, and the original model was modified with this model. Compared with the experimental results, the model is in good agreement.

1. The linear correlation between LI and lg(Nienvior.)

When the external Ni2+ in vitro is a constant value, the final LI will tend to be a constant value. On this basis, we discuss how Nienvior is positively correlated with LImax.

When t→∞

(1.15)

while

(1.16)

according to mode，the relation between LImax and Nienvior. is  positive correlation.

We analyze the experimental data and find, within a certain range of Nienvior , concentration LImax and lg(Nienvior) are linear correlation.

2. Model Verification By Experiment

We make a line graph of experimental data as figure 4

We find that LImax and lg(Nienvior) tend to be a linear function with a positive correlation when lg(Nienvior) is in the range of (-∞，-3]. However, after the value of lg(Nienvior) is greater than -3, the value of LImax decreases significantly. Considering the influence of high concentration of nickel ions on cytotoxicity, the cytotoxicity of nickel ions is analyzed.

3. The Cytotoxicity of Nickel Ions

Using the Logistic equation to fit the data

                      

We can get the following results:

Coefficients (with 95% confidence bounds):

       a = 1.086  (1.027, 1.146)

       b = 0.2424  (-0.4756, 0.9605)

       c =7.07  (1.872, 12.27)

Goodness of fit:

  SSE: 0.005441

  R-square: 0.9971

  Adjusted R-square: 0.9957

  RMSE: 0.03688

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

Figure 5 The Cytotoxicity of Nickel Ions

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

(1.17)

4. Modifying the Model

According to the above part, the model is modified as follows:

Considering cytotoxicity, the model is modified. LImax is modified to be . With the Matlab, we get the revised relationship between LI and lg(Nienvior.)

Compare the revised model with the data.

According to the modified model, the relationship between LImax and lg(Nienvior) is predicted, as shown in figure 6. In the figure, the red dots represent the measurement data of the real experiment. It can be seen that our model has a good agreement with the experiment.

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