By building a mathematical model, we demonstrate the principles of the project and provide guidance for optimizing product design and its application.
Gray value-density model The relationship between the gray value and the added substance concentration
We have established a grayscale model to describe our results in antigen fixation, antibody immobilization and detection, and to digitize the results into accurate numbers. This model can guide us to more accurately determine the relationship between the reaction results and our effects in three phases. Especially in the detection, the concentration of the analyte can be embodied and digitized, thus giving us a better and specific detection limit.
For the gray value calculation of a resulting image, we first grayscale it. In order to calculate the gray value of the color picture, we have grayed out with MATLAB. At the same time, in order to give a clear grayscale picture, we performed image equalization and gave a balanced picture[Attachment 1]. After obtaining the grayscale image, we use the formula (1) to calculate the grayscale of the image. In order to obtain accurate data more conveniently, we use the programming language to design a small software [Attachment 2], which can calculate the average grayscale of the image. There is an error in the data for the no grayscale picture. According to our experimental results, it is about 5% when the gray value is not too low. Compared with the equalized grayscale image, the error is about 10%.
$$ g_w = \sum_{i=0}^{255} i \times T_i \div T \tag{1} $$
For the experimental results, we can understand whether it is fixed antigen, immobilized antibody or detection. In the initial stage, the gray value is positively correlated with the concentration of the additive; and as the concentration of the additive increases, the gray value is not proportional to the increase. There are two models that satisfy these properties, namely equation (2) and formula (3):
Michaelis-Menten model: $$ y = \frac{\alpha x}{\beta + x} \tag{2} $$
Exponential growth model: $$ y = \alpha (1- exp(-\beta x)) \tag{3} $$
We substituting the above two formulas into the literature we examined[2], where model (2) is in good agreement. At the same time model (2) if the enzyme is compared to our slide, the substrate is analogous to our addition. The resulting enzymatic reaction rate-substrate concentration and gray value-addition concentration have a good fit. So we decided to use this model. And in the other literature[1] of the model, the results obtained are also consistent. The gray value and concentration relationship in this model are shown in Fig. 1(a), This applies to both fixed antigens, immobilized antibodies and assays. The different added substances are shown in Fig. 2(b).
(a) Addition of added substances |
(b) Relationship between different additions and gray values (joined separately) |
|
(c) Fixed antigen grayscale model curve |
|
Fig.1 Gray value and additive concentration relationship |
In the step of immobilizing the antigen, we control the amount of antigen added, the fixed time, and the brightness and the observation magnification at the time of sampling, and use a large amount of experimental data to average and reduce the error. Experimental Results After data processing, the Figures 1(c), Equations \(\mathrm{Y_1}\), can be obtained. The resulting image was used to predict other data and found good agreement. Explain that the gray value-concentration model we built is in line with our experiments. Thus, we further established the model on antibody immobilization and obtained the results of equations \(\mathrm{Z_1}\). From the model and our experimental data, it can be stated that the gray value and the additive concentration are positively correlated when the concentration is not large. From the model and our experimental data, it can be stated that the gray value and the additive concentration are positively correlated when the concentration is not very large and more than 0.4 μg/ml.
$$ y = 6.4904 x \quad \mathrm{(Y_1)} $$
$$ y = 9.3506 x \quad \mathrm{(Z_1)} $$
There is a bias when the antigen is not fixed. Therefore, we perform a statistical average of the gray values of the unfixed antigen. The \(\mathrm{Y_1}\) equation is the result of correcting the zero point deviation and the post-processing of the comparison relation \(\mathrm{Ψ_q}\). \(\mathrm{Z_1}\) can be obtained by treating the modified antibody model in the same manner.
[1]Zhang Jiao, Su Xiuxia, Yang Dong, Luan Chonglin,Label-free liquid crystal biosensor for cecropin B detection,Talanta,Volume 186,2018,Pages 60-64,ISSN 0039-9140, https://doi.org/10.1016/j.talanta.2018.04.004.
[2]Li Xian, Li Guang, Yang Meng, Chen Long-Cong, Xiong Xing-Liang,Gold nanoparticle based signal enhancement liquid crystal biosensors for tyrosine assays,Sensors and Actuators B: Chemical,Volume 215,2015,Pages 152-158,ISSN 0925-4005, https://doi.org/10.1016/j.snb.2015.03.054.
[Attachment 1] Name:(Picture grayscale processing code) https://github.com/zkingri/gray
[Attachment 2] Except for all documents in Annex 1. https://github.com/zkingri/gray
Kinetic model The relationship between residual antigen concentration and time
In order to optimize the antigen fixation time, we did the relationship between the remaining antigen concentration and time when the antigen was immobilized.
In our step of fixing the antigen concentration, we can get the reaction formula \(\mathrm{\alpha}\) and give the pattern diagram ω1, where \(A\) is the antigen and \(M\) is the reaction site concentration on the slide. Since the addition liquid fixation and treatment methods are unchanged, \(M_0\) does not change.
$$A+M\overset{K_a}{\underset{K_d}{\rightleftharpoons}}A-M \tag{ α }$$
If \(\theta=\frac{[A-M]}{M_0}\), that is, the position occupied by the immobilized antigen accounts for the percentage of all positions of the entire slide. Then by the dynamics relationship, we can list the relationship:
$$-\frac{d[A]}{dt}=K_a(1-\theta)[M_0][A]-K_d\theta[M_0]$$
If we determine a \(\theta\), that is, we want the percentage of the fixed position on the slide, generally the maximum concentration under the premise of disturbing the liquid crystal. It is given by our experimental data that \(\theta\) is generally optimal between 0.05 and 0.1. Let \(b=\frac{K_a}{K_d}\), we can get the relationship:
$$[A]=C_1exp[-K_a(1-\theta)[M_0]t]+\frac{\theta}{b(1-\theta)}$$
In this way we can get the relationship between the residual concentration of the added antigen and the reaction time (4):
$$[A]=C_1exp[mt]+n\tag{4}$$
Thus, from the relationship we can predict the relationship between the residual antigen concentration and time, as shown in Fig. 2(a) and (b).
(a) One Concentration |
The relationships of multiple concentration |
|
Fig.2 Relationship between residual antigen concentration and time when immobilized antigen |
Combined with formula (2) inside the grayscale model content, we can obtain the following relationship (5), the relationship between antigen fixed time and gray value. As shown in Fig. 3.
$$Grayscale value=\frac{\sigma_1exp[mt]+\Psi_1}{\beta_0+C_1exp[mt]}\tag{5}$$
Fig.3 Relationship between antigen fixation time and result gray value |
It can be concluded that the reaction can be stopped by 90% or 80% of the reaction, because it is uneconomical to continue the reaction. At the same time, for the initial antigen concentration change, the initial large reaction starts to be fast, while the initial value is small, the reaction rate starts to be small; at the same time, the optimum reaction time for the different concentrations of the added antigen is about the same, and has little relationship with the concentration. Finally, there is a clear relationship between the fixed time of the antigen and the gray value.
Adding antigen concentration and adding antibody concentration ratio relationship
In order to the final result was as expected by making the solubility of the configured antigen-antibody immobilization solution. It is necessary for us to optimize the concentration ratio for them.
After fixing the antigen. For immobilized antibodies, there is a reaction \(β\) and the pattern diagram \(ω_2\):
$$A'+R_T\overset{K_a}{\underset{K_d}{\rightleftharpoons}}R_TA' \tag{ β }$$
 |
Fig.ω2 Fixed antibody pattern |
If we define a \(θ_2\), which represents the effect that we would like to achieve when immobilizing an antibody, it is generally the antibody concentration that can disrupt the percentage of gray that the liquid crystal reaches our defined full brightness. At this time, there is also a relation (6):
$$[R_T]=C_2exp[Qt]+P\tag{6}$$
When the reaction is sufficient, it is assumed to be \(t_0\). For the relationship between the reaction \(\beta\):
$$K_a(1-\theta_2)[R_T]=K_d\theta_2$$
$$\theta_2=\frac{[R_TA']}{[R_{T0}]}$$
Combining relation (4), we can get relation (7):
$$\frac{[R_{T0}]}{A_0}=\alpha+\frac{\beta}{C_1+n}\tag{7}$$
Among them, \(\alpha\) and \(\beta\) have a large correlation with \(\theta_2\), but have little correlation with \(\theta\). From our experimental data, it can be given that \(\theta_2\) meets our requirements when it reaches 0.5~0.7. Under the preconditions, relation (8) and Fig. 4 can be derived:
$$\frac{[R_{T0}]}{A_0}=K\theta_2+b\tag{8}$$
Fig.4 The relationship between the optimal ratio of antibody antigen concentration and θ2 |
When we determine the residual antigen concentration and time relationship when fixing the antigen and the residual antibody concentration and time relationship when immobilizing the antibody, we can use this model to give the optimal concentration ratio for immobilized antigen and antibody by combining the given \(\theta\) and \(\theta_2\).
At the same time, it can be concluded that the proportional relationship is closely related to our expected \(\theta_2\), and there is no clear relationship with our expected \(\theta\).
The relationship between the Grayscale value and the reaction time of the liquid to be tested
When our finished product is inspected, we want to obtain the minimum detection time and determine the credibility of the test results. Therefore, the relationship between the test result and time is made. The reaction \(\gamma\) and the pattern diagram ω3 are as follows:
$$R_TA\overset{K_{d1}}{\underset{K_{a1}}{\rightleftharpoons}}A+R_T$$
$$P+R_T\overset{K_{a2}}{\underset{K_{d2}}{\rightleftharpoons}}R_TP \tag{ γ }$$
|
|
Figure. ω3 Detection mode diagram |
For the detection, we assume that the immobilized antigen is not easily detached outside of the reaction, and the amount of free antibody during the reaction is very small and remains substantially unchanged. Using this free antibody steady-state and kinetic knowledge, we derive the relationship (9):
\begin{equation}\begin{split} [R_TA]=&Cexp[-(K_{d2}+K_{a2}[R_T])t]\\&+\frac{K_{d2}(K_{d2}[R_TA_0]+[R_T])-K_{a2}([P_0]-[R_TA_0]+[R_T])[R_T]}{K_{d2}+K_{a2}[R_T]} \end{split}\tag{9}\end{equation}
equal meaning between \([R_TA_0]\) and \([R_TA]_0\).
By simplification of relation (9) and combining relation (2) inside the grayscale model content, relations (10) and (11) can be obtained:
$$[R_TA]=Cexp[xt]+y\tag{10}$$
$$Grayscale value=\frac{\sigma exp[xt]+Psi}{\beta+Cexp[xt]}\tag{11}$$
According to the relationship, We can obtain a detection curve expressed by the fixed antigen-antibody concentration. The relationship between high and low sensitivity detection curves can also be obtained (slide detection sensitivity and antigen-antibody positive correlation).And we can list Fig. 5 (a)(b)(c):
(a) Relationship between fixed antigen-antibody concentration and detection time |
(b) Relationship between different fixed antigen-antibody concentration and detection time |
(c) Gray value and detection time relationship diagram |
Fig.5 Predictive graph during detection |
From the analysis results, we can optimize the detection time in order to achieve the most effective time. At the same time, for a fixed antigen-antibody pair, the value is proportional to the initial reaction rate; the optimal time is approximately the same. Finally, there is a clear relationship between the detection time and the gray value relationship.
LIQUID CRYSTAL MOLECULES PERTURBATIONS MODEL
The field of synthetic biology relies heavily on mathematical models that help simulate and predict the process and consequence of biological systems. We developed two models, the radial domino model and the blooming model to describe the potential perturbation mechanism and interaction modes between protein particles and liquid crystal molecules. The protein density threshold can be calculated by this model.
Purpose
An important part of our liquid crystal experiment is to determine an optimal antigen concentration, so we creatively proposed two models to calculate the β2-M density microscopically. The trouble is that we do not know the range of antigen molecules that can perturb liquid crystal molecules. Therefore, we consulted the relevant parameters, proposed two models, calculated the range of the single antigen molecule to perturb the liquid crystal, and finally determined the threshold of the antigen concentration, which was consistent with our experimental results.
Parameters
 |
|
 |
|
|
Fig.6 The chemical structure (A) and the simulated structure (B) of
5CB |
|
|
|
|
Fig.7 The simulated structure of Nanoantibody |
Fig.8 The simulated structure of β2-MG |
Hypothesis
|
|
Fig.9 Two possible mechanisms for β2-M to disturb liquid crystal |
The process of protein perturbation of liquid crystal molecules is particularly complicated, so we can only guess the disturbance mechanism on our own. Inspired by dominoes and the blossoming of flowers, we proposed that there might be two mechanisms of disruption. We called them RADIAL DOMINO MODEL and BLOOMING MODEL
|
|
Fig.10 RADIAL DOMINO MODEL |
Simplify the β2-M and 5CB as ball and stick. The distance between each two sticks is equal. Due to the interference of the ball, the first stick next to the ball will be inclined at an angle. The first stick will affect the next stick until the last stick is aligned vertically.
|
|
Fig.11 BLOOMING MODEL |
This model is actually an extension of the previous model. Simplify the 5CB into one side of the polygons. As the molecular shells increases, the angle between the two sides of the polygon will tend to be 90°.
Calculation
1. Obtain the spacing of 5CB molecules
The forces between two molecules are covalent bonds, ionic bonds, metal bonds, hydrogen bonds and Van der Waals force. Because no other forces are present, the main force is Van der Waals force. The point at which the force becomes repulsive rather than attractive as two molecules near one another is called the van der Waals contact distance. Van der Waals force includes electrostatic force, dispersion force and induction force.
|
|
Fig.12 van der Waals potential energy |
We can use a simplified formula: the Lennard-Jones potential[1].
$$E=\frac{A}{r^{12}}-\frac{B}{r^6}$$
where \(A=4\epsilon\sigma^{12}\), \(B=4\epsilon\sigma^6\).
However, there is no sigma Lennard-Jones parameters (Angstrom) epsilon Lennard-Jones parameters (kcal/mol) about 5CB, so we calculate the van der Waals contact distance(\(r_0\)) by statistical mechanics[2].
In statistical mechanics, the radial distribution function \(g(r)\) in a system of particles (atoms, molecules, colloids, etc.), describes how density varies as a function of distance from a reference particle.
|
|
Fig.13 The sketch map of radial distribution function |
The radial distribution function:
$$\rho g(r)4\pi r^2=dN$$
where \(\rho\) is the density of 5CB, \(N\) is the total number of 5CB molecules. Integrate the above formula:
$$\int_0^\infty \rho g(r)4\pi r^2 \mathrm{d}r=\int_0^N\mathrm{d}N=N$$
$$g(r)=\frac{\mathrm{d}N}{\rho 4\pi r^2}$$
Molecular dynamics calculation:
$$g(r)=\frac{1}{\rho 4\pi r^2}\frac{\sum_{i=1}^r\sum_{j=1}^N \Delta N(r\to r+\delta r)}{N\times T}$$
Where \(N\) is the number of molecules and \(T\) is the calculated time (steps), \(\Delta N\) is the number of molecules between \(r\) and \(r+\delta r\).
The radial distribution function can be computed via computer simulation methods. Because the peak of the radial distribution function appears at 0.5nm, so the spacing of 5CB molecules is around 0.5nm.
2. Calculate the range of β2-M that can perturb 5CB
We assume 5CB molecule as a ellipse (\(a\)=1nm, \(b\)=0.1nm), we got the tilt angles of three molecules by Matlab.
|
|
Fig.14 The tilt angles are 43°, 64°, 90° respectively. |
|
|
\(\therefore\)Maximum radius of β2-M that can perturb 5CB (Fig. 10):
\(R_1=[2\times\sin (64^\circ )+1]\mathrm{nm}=\)2.8nm
3. Verification
In the liquid crystal experiment, the droplets dropped on the slide have a radius of about 2 mm.
$$\therefore S=\pi R^2=4\pi\times 10^{12}\mathrm{nm^2}$$
The molecular weight of β2-M is 11800, the volume of the droplet is 10μL.
The area disturbed by each protein molecule is:
$$r=2.8\mathrm{nm}$$
$$S_0=\pi r^2=7.84\pi\mathrm{nm^2}$$
When the picture is just full colorful, the number of molecules contained in the droplet is:
$$N=\frac{S}{S_0}=5.1\times 10^{11}$$
$$m=\frac{11800\times 5.1\times 10^{11}}{6.02\times 10^{23}}=10^{-8}\mathrm{g}$$
\(\therefore\)Maximum concentration is:
\(c=\frac{m}{V}=\)1000ng/mL
That is to say, when the concentration reaches 1000ng/mL, the image starts to be completely bright. This is in good agreement with our experimental results. The following are the results of the liquid crystal experiment.
|
|
|
A. 500ng/ml(β2-M) |
B. 600ng/ml(β2-M) |
|
|
|
C. 700ng/ml(β2-M) |
D. 1000ng/ml(β2-M) |
Photographs of different antigen concentrations(photographed through a polarizing microscope) |
|
[1] Jones J E. On the determination of molecular fields. From the equation of state of a gas[J]. Proc. R. Soc. Lond. A, 1924, 106(738): 463-477.
[2] Haynes W M. CRC handbook of chemistry and physics[M]. CRC press, 2014.
