The relationship between residual antigen concentration and time
In order to optimize the antigen fixation time, the relationship between the remaining antigen concentration and the fixed time was made.
From this, the reaction formula \(\mathrm{α}\) can be obtained and the pattern diagram ω1 is given, where \(A\) is the antigen and \(M\) is the concentration of the reaction site on the slide. Since the addition liquid fixing and treatment methods are unchanged, \(M_0\) does not change.
$$A+M\overset{K_a}{\underset{K_d}{\rightleftharpoons}}A-M \tag{ α }$$
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Fig.ω1 Fixed antigen pattern |
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, the following relationships can be listed:
$$-\frac{d[A]}{dt}=K_a(1-\theta)[M_0][A]-K_d\theta[M_0]$$
If \(θ\) is determined, that is, the proportion of the number of fixed positions of the slide, it is usually the maximum concentration under the premise of disturbing the liquid crystal. Experimental data gives that \(θ\) is generally optimal between 0.05 and 0.1. Let \(b=\frac{K_a}{K_d}\), the relationship can be got:
$$[A]=C_1exp[-K_a(1-\theta)[M_0]t]+\frac{\theta}{b(1-\theta)}$$
In this way, the relationship between the residual concentration of the added antigen and the reaction time can be obtained (4):
$$[A]=C_1exp[mt]+n\tag{4}$$
Therefore, based on this relationship, the relationship between the residual antigen concentration and time can be predicted, as shown in Fig. 2(a) and (b).
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(a) One Concentration |
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(b) The relationships of multiple concentration |
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Fig.2 Relationship between residual antigen concentration and time when immobilized antigen |
Combining the formula (2) in the content of the gray scale model, the relationship between the fixed time and the gray value of the relationship (5) can be obtained. As shown in Fig. 3.
$$\mathrm{Grayscale \ value}=\frac{\sigma_1exp[mt]+\Psi_1}{\beta_0+C_1exp[mt]}\tag{5}$$
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Fig.3 Relationship between antigen fixation time and result gray value |
It can be concluded that the reaction can be stopped in certain stage 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 to optimize the concentration ratio for them.
After fixing the antigen. For immobilized antibodies, there is a reaction \(β\) and the pattern diagram ω2:
$$A'+R_T\overset{K_a}{\underset{K_d}{\rightleftharpoons}}R_TA' \tag{ β }$$
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Fig.ω2 Fixed antibody pattern |
If \(θ_2\) is defined, it represents the effect that is desired to be achieved when immobilizing the antibody, and typically the antibody concentration can destroy the grayscale percentage of the liquid crystal to a defined full brightness. At this time, there is another relationship (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 the experimental data, it can be given that \(\theta_2\) meets the requirements when it reaches 0.5~0.7. Under the preconditions, relation (8) and Fig. 4 can be derived:
$$\frac{[R_{T0}]}{A_0}=K\theta_2+b\tag{8}$$
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Fig.4 The relationship between the optimal ratio of antibody antigen concentration and θ2 |
There are then two scenarios. Scheme 1: immobilized antigen, test substance and antibody mixed detection. Scheme 2: After the antigen is fixed and the excess antibody is added, the test substance is added for competitive reaction. These two options need to be compared.
At present, the gray scale model and the dynamic model are discussed in the first scheme.
When determining the residual antigen concentration and time relationship when immobilizing an antigen and the residual antibody concentration and time relationship when the antibody is attached, the model can be used to give an optimal concentration ratio of the immobilized antigen and the antibody by combining given \(θ\) and \(θ_2\).
At the same time, it can be concluded that the proportional relationship is closely related to the expected \(θ_2\) and has no obvious relationship with the expected \(θ\).
The relationship between the Grayscale value and the reaction time of the liquid to be tested
When inspecting the finished product, it is desirable to obtain the shortest test time to determine the credibility of the test results. Therefore, the relationship between the test results and time is completed. The reaction \(\mathrm{γ}\) and the pattern diagram ω3 are as follows:
$$R_TA\overset{K_{d1}}{\underset{K_{a1}}{\rightleftharpoons}}A+R_T$$
$$P+R_T\overset{K_{a2}}{\underset{K_{d2}}{\rightleftharpoons}}R_TP \tag{ γ }$$
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Fig.ω3 Detection mode diagram |
For detection, it is assumed that the immobilized antigen is not easily detached outside the reaction, and the amount of free antibody during the reaction is very small and remains substantially unchanged. Using the steady-state and kinetic knowledge of the antibody, 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:
$$[R_TA]=Cexp[xt]+y\tag{10}$$
$$\mathrm{Grayscale \ value}=\frac{\sigma exp[xt]+ \Psi }{\beta+Cexp[xt]}\tag{11}$$
According to this relationship, a detection curve expressed by a fixed antigen-antibody concentration can be obtained. The relationship between the high-sensitivity detection curve and the low-sensitivity detection curve (slide detection sensitivity and positive correlation of antigen-antibody pairs) can also be obtained. Fig. 5(a)(b)(c) can be listed:
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(a) Relationship between fixed antigen-antibody concentration and detection time |
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(b) Relationship between different fixed antigen-antibody concentration and detection time |
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(c) Gray value and detection time relationship diagram |
Fig.5 Predictive graph during detection |
From the analysis results, the detection time can be optimized to achieve the most effective time. At the same time, for a fixed antigen-antibody pair, this 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.
