Our ultimate goal is to establish the sequence design guideline for templates and helpers used in Three-Way-Junction SDA (TWJ-SDA). In this section, we focus on the Signal-to-Noise Ratio (S/N Ratio) of TWJ-SDA and build an ordinary differential equation (ODE) Model to predict templates and helpers that will result in a higher S/N ratio. Since the first ODE Model required improvement due to discrepancies with experimental results, we decided to create a new ODE Model. This new Model is closer to the experimental results by taking into account the influence of ab initio synthesis. By combining this new ODE Model with the template and helper conditions for sequence-specific amplification as determined in the TWJ Specificity section, we developed software to assist in the design of optimal templates and helpers for TWJ-SDA. This software will lower the hurdle to use our amplification and detection systems for various miRNAs, and will greatly expand the applicability of our project, POIROT.
Figure 1.
Based on the results of miRNA research at Dry Lab (For more information, see Model_miRNA_Selection), we decided to use TWJ-SDA, which has high sequence specificity, for our amplification system. For TWJ-SDA, we need to design the sequence of a helper and a template. Previous studies have investigated optimal helpers and templates for individual microRNAs (miRNAs) 1. However, no general guidelines for the design of TWJ-SDA helpers and templates have been investigated. Our ultimate goal is to create guidelines for optimal helpers and templates design.
TWJ Specificity section (Model_TWJ_Specificity) helped to identify the causes of TWJ-SDA sequence specificity. This allowed us to know the conditions for sequence specificity. In this section, we focus on the Signal-to-Noise Ratio (S/N ratio) of TWJ-SDA and investigate the conditions for helpers and templates to increase the S/N ratio. The S/N ratio of TWJ-SDA can even be less than 1 if the helper and template are not well designed. Therefore, Seeking conditions for a high S/N ratio is crucial to achieving the desired results.
In designing the TWJ-SDA helper and template, it has to be decided how many bases the target miRNA and template are complementary and how many bases the helper and template are complementary. We tried to identify these two conditions for a high S/N ratio by using an ODE Model. An ODE Model of TWJ-SDA was also used to study TWJ specificity, but in this section it is the S/N ratio that is of particular interest. Therefore, we decided to create a different ODE Model from the ODE Model used in TWJ Specificity section.
The amplification reaction of TWJ-SDA start from the three pathways in Figure 2.
Figure 2. The three pathways of TWJ to amplification. When the target is present, amplification begins at Path A, Path B, and Path C. When target is absent, amplification begins only at Path C.
We simulated TWJ-SDA by creating the ODE Model that represents the reactions in Figure 2.
The equation of the ODE Model is Equation1.
ODE for TWJ-SDA
\[ \begin{equation} \begin{aligned} \dfrac{d[X]}{dt} &= -k_1 [X] [T] + \dfrac{k_1}{a_1} [XT] - k_2 [X] [H] + \dfrac{k_2}{a_2} [XH] \notag\\ \dfrac{d[T]}{dt} &= -k_1 [X] [T] + \dfrac{k_1}{a_1} [XT] - k_3 [XH] [T] + \dfrac{k_3}{a_3} [XHT] - k_5 [H] [T] + \dfrac{k_5}{a_5} [HT] \notag\\ \dfrac{d[H]}{dt} &= -k_2 [X] [H] + \dfrac{k_2}{a_2} [XH] - k_4 [XT] [H] + \dfrac{k_4}{a_4} [XHT] - k_5 [H] [T] + \dfrac{k_5}{a_5} [HT] \notag\\ \dfrac{d[XT]}{dt} &= k_1 [X] [T] - \dfrac{k_1}{a_1} [XT] - k_4 [XT] [H] + \dfrac{k_4}{a_4} [XHT] \notag\\ \dfrac{d[XH]}{dt} &= k_2 [X] [H] - \dfrac{k_2}{a_2} [XH] - k_3 [XH] [T] + \dfrac{k_3}{a_3} [XHT] \notag\\ \dfrac{d[XHT]}{dt} &= k_3 [XH] [T] - \dfrac{k_3}{a_3} [XHT] + k_4 [XT] [H] - \dfrac{k_4}{a_4} [XHT] - \dfrac{k_{\text{cat}} P_0}{32 m c} [XHT] \notag\\ \dfrac{d[HT]}{dt} &= k_5 [H] [T] - \dfrac{k_5}{a_5} [HT] - \dfrac{k_{\text{cat}} P_0}{32 m c} [HT] \notag\\ \dfrac{d[W]}{dt} &= \dfrac{k_{\text{cat}} P_0}{32 m c} [XHT] - k_n [W] [N] + \dfrac{k_{\text{cat}} P_0}{23 m c} [Z] \notag\\ \dfrac{d[Z]}{dt} &= -\dfrac{k_{\text{cat}} P_0}{23 m c} [Z] + k_n [W] [N] \notag\\ \dfrac{d[P1]}{dt} &= \dfrac{k_{\text{cat}} P_0}{23 m c} [Z] \notag\\ \end{aligned} \end{equation} \] where, \[ \begin{equation} \begin{aligned} c &= 1 + \dfrac{[XHT] + [Z] + [W] + [HT]}{m}\notag\\ \end{aligned} \end{equation} \]
The above model is based on the law of mass action. The polymerase and nickase reactions follow the Michaelis-Menten equation. The parameter c found in the denominator of the Michaelis-Menten term is acknowledging the competition between DNA species XHT, Z, W, HT for polymerase 2. The amplification reaction from Path C is strictly different from that from Path A and Path B. We assumed that once the amplification reaction starts, the helper doesn't leave the template and the polymerase reaction proceeds in the same way as in XPT in HT, where there is no target miRNA. Therefore, in the equation, we assumed that the reaction proceeds from HT to W.
The binding rate between nucleic acids was set to \(k_1 = k_2 = k_3 = k_4 = k_5\) based on the study of Rauzan et al. 3. The binding speed was obtained by genetic algorithm fitting as described in the Amplification Comparison section (for more information, see Model_Amplification_Comparison).
The parameter values used are as follows.
We have simulated 12 different combinations of association constants \(a_1, a_2, a_3, a_4, a_5\). These association constants were calculated by Nearest-Neighbor Model 7, 8. These 12 combinations of association constants can be seen at Model_Sequence design.
The above ODE Model simulation was performed using python. The code used for simulation is available at the following link.
Please visit our GitLab.
We used the results from Wet Lab to validate The ODE Model. As the target miRNA, hsa-miR-30d-5p was used. Amplification of TWJ-SDA was monitored by fluorescence from SYBR Green II. Experiments were performed under 4 different complementary strand length for helper and template (5 bp, 6 bp, 7 bp, 8 bp) and 3 complementary strand lengths for miRNA and template (10 bp, 11 bp, 12 bp), totaling 12 different conditions. For each condition, three experiments were performed with target miRNA 0nM and 10nM. The S/N ratio was calculated as Signal/Noise, where Signal and Noise were calculated by subtracting the value at 0 min as background from the amount of fluorescence at 60 min when miRNA was 10 nM and 0 nM, respectively.
The fluorescence in the experiment is mainly due to ssDNA, and in the simulation, it is assumed to correspond to the amplification product (P1) of TWJ-SDA. In the simulation, the concentration of P1 after 60 min when the target miRNA is 10 nM is Signal and the concentration of P1 after 60 min when the target miRNA is 0 nM is Noise. Signal and Noise units are in \(\mu M\). S/N ratio was calculated as Signal / Noise.
Figure 3. The heatmap of Signal, Noise, and S/N ratio when the target miRNA is hsa-miR-30d-5p. a56 is association constant when the complementary length between helper and template is 6 bp. a16 is association constant when the complementary length between target miRNA and template is 6 bp.
Figure 3 shows the Signal, Noise, and S/N ratio results when \(a_5\) and \(a_1\) are varied. A larger \(a_5\) means a longer length of the complementary strand of helper and template. On the other hand, a larger \(a_1\) means a longer length of the complementary strand of target miRNA and template. A 10-fold increase in \(a_5\) and \(a_1\), respectively, corresponds to an increase of about 1 bp in the length of the complementary strand. The results of this simulation indicate that the S/N ratio increases as the length of the helper and template complementary strand decreases.
Figure 4. The heatmap of experimental and simulated S/N ratio. The red dots are the conditions under which the experiment was conducted. The first number in the red dot label indicates the length of the complementary strand between target and template (Length of TT), and the second number indicates the length of the complementary strand between helper and template (Length of HT). The white areas in the heat map on the right are areas where the S/N ratio is greater than 1000.
Here we compare the simulation results with the results from Wet Lab. The left figure shows the experimental results, and the Wet Lab results suggest that the S/N ratio does not increase by a factor of 10 or more as in simulation.Furthermore, in the simulation, the smaller \(a_5\) is, the higher the S/N ratio is, but this doesn't seem to be the case in the experimental results. In all cases where length of TT was 10 bp, 11 bp, and 12 bp, the S/N ratio was the highest when the length of HT was 6 bp.
Comparing the simulation results and experimental results, the S/N ratio in simulation is more than 10, but in the experimental results, the S/N ratio is not that large. In addition, the helper and template conditions that give the maximum S/N ratio are different between the simulation predictions and the experimental results. The reason why the S/N ratio of the simulation is more than 10 is that the path C reaction is almost non-existent when the length of HT is short, and the Noise is very small. The simulation doesn't predict the experimental results at all. We thought that this model needs to be improved.
What aspects of reality did our Model not represent well? From the comparison of simulation and experimental results, we thought that the difference from reality might be that the Noise is very small relative to the Signal when the length of HT is short. As a result of the literature review, we came up with a remedy. Zyrina's research shows that when there is a polymerase and some DNA, the polymerase works to produce a complementary strand even if there is no primer present 9. We hypothesized that this phenomenon called ab initio synthesis might also be occurring in TWJ Amplification. If ab initio synthesis is occurring, we can say that the amplification product (P1) is amplified by factors other than Path A, B, and C in both Signal and Noise cases. If we assume that ab initio synthesis is taking place, it will no longer be the case that the Noise is very small compared to Signal when the length of HT is short.
Considering the effect of ab inito synthesis, the ODE of the improved model is as in Equation 2. We have assumed that P1 is increasing at a constant rate by ab initio synthesis. Therefore, the change from the previous model is that \(a_b\) is added to the right-hand side of \(\dfrac{d[P1]}{dt}\). the value of \(a_b\) was determined so that the S/N ratio scale is consistent with the simulation and experimental results.
New ODE for TWJ-SDA
\[ \begin{equation} \begin{aligned} \dfrac{d[X]}{dt} &= -k_1 [X] [T] + \dfrac{k_1}{a_1} [XT] - k_2 [X] [H] + \dfrac{k_2}{a_2} [XH] \notag\\ \dfrac{d[T]}{dt} &= -k_1 [X] [T] + \dfrac{k_1}{a_1} [XT] - k_3 [XH] [T] + \dfrac{k_3}{a_3} [XHT] - k_5 [H] [T] + \dfrac{k_5}{a_5} [HT] \notag\\ \dfrac{d[H]}{dt} &= -k_2 [X] [H] + \dfrac{k_2}{a_2} [XH] - k_4 [XT] [H] + \dfrac{k_4}{a_4} [XHT] - k_5 [H] [T] + \dfrac{k_5}{a_5} [HT] \notag\\ \dfrac{d[XT]}{dt} &= k_1 [X] [T] - \dfrac{k_1}{a_1} [XT] - k_4 [XT] [H] + \dfrac{k_4}{a_4} [XHT] \notag\\ \dfrac{d[XH]}{dt} &= k_2 [X] [H] - \dfrac{k_2}{a_2} [XH] - k_3 [XH] [T] + \dfrac{k_3}{a_3} [XHT] \notag\\ \dfrac{d[XHT]}{dt} &= k_3 [XH] [T] - \dfrac{k_3}{a_3} [XHT] + k_4 [XT] [H] - \dfrac{k_4}{a_4} [XHT] - \dfrac{k_{\text{cat}} P_0}{32 m c} [XHT] \notag\\ \dfrac{d[HT]}{dt} &= k_5 [H] [T] - \dfrac{k_5}{a_5} [HT] - \dfrac{k_{\text{cat}} P_0}{32 m c} [HT] \notag\\ \dfrac{d[W]}{dt} &= \dfrac{k_{\text{cat}} P_0}{32 m c} [XHT] - k_n [W] [N] + \dfrac{k_{\text{cat}} P_0}{23 m c} [Z] \notag\\ \dfrac{d[Z]}{dt} &= -\dfrac{k_{\text{cat}} P_0}{23 m c} [Z] + k_n [W] [N] \notag\\ \dfrac{d[P1]}{dt} &= \dfrac{k_{\text{cat}} P_0}{23 m c} [Z] + a_b\notag\\ \end{aligned} \end{equation} \] where, \[ \begin{equation*} \begin{aligned} c &= 1 + \dfrac{[XHT] + [Z] + [W] + [HT]}{m}\notag\\ \end{aligned} \end{equation*} \]
The parameters used are as follows.
Figure 5. The heatmap of simulated Signal and Noise.
Figure 5 shows the Signal and Noise results of the new model. The difference from the previous model is that the Noise is more than \(0.3 \ \mu M\) even when the length of HT is short. This is due to the ab initio synthesis added in the new model.
Figure 6. The heatmap of simulated S/N ratio. The red dots are the conditions under which the experiment was conducted. The first number in the red dot label indicates the length of the complementary strand between target and template (Length of TT), and the second number indicates the length of the complementary strand between helper and template (Length of HT).
Figure 6 shows the S/N ratio results of the new model. The new model with ab initio synthesis influence showed that S/N ratio was less than 10, which is consistent with the experimental results. In the old model, the shorter the length of HT, the higher S/N ratio, but in the new model, the shorter the length of HT is not necessarily better. It can be said that the new model is able to generally reproduce the trend of experimental results.
We will specifically examine whether this new model accurately predicts helper and template with large S/N Ratio.
Figure 7. Comparison of predictions of S/N ratio in simulation and experimental results of S/N ratio. The horizontal axis shows the rank order of large and small S/N ratio in the simulation and the vertical axis shows the rank order of large and small S/N ratio in the experiment. The coefficient of determination is 0.55.
Figure 7 shows the consistency between the order of S/N ratio in the simulation and the order of S/N ratio in the experiment. If the simulation perfectly predicts the experimental results, all plots lie on the line \(y=x\). From Figure 7, we can say that the new ODE Model predicts the experimental results to some extent.
We will integrate the discussion in this section with the constraints about template and helper that were identified in TWJ Specificity (Model_TWJ_Specificity). TWJ Specificity section found that the association constant \(a_3\) must be less than \(10^3\) to have sequence specificity. To predict the optimal TWJ-SDA template and helper, we need to integrate the prediction of the new model made in this section with the constraints obtained in TWJ Specificity section.
Integrated sequence design guidelines can predict templates and helpers that allow TWJ-SDA to have high S/N ratio and sequence specificity. This allows us to omit many experiments looking for optimal conditions. Our project POIROT detects glaucoma using miRNA biomarker. The POIROT system can be applied to detect for various other diseases by changing the biomarker miRNAs. One of the hardest hurdles in its application is the design of TWJ-SDA template and helper. Therefore, we have created a software that predicts the optimal template and helper for each miRNA according to the integrative sequence design guidelines.
Our TWJ-SDA sequence design software works as follows:
Click here for more information on how to use the software
We designed the sequence with the goal of a high S/N ratio in TWJ-SDA. The ODE Model was improved to be consistent with the experimental results, and the improved model was able to predict the S/N ratio of the experimental results to some extent. By integrating this ODE Model with the constraints for sequence specificity obtained in TWJ Specificity section, we create software to design the optimal TWJ-SDA template and helper. This software will lower the hurdles to using our amplification and detection systems for various miRNAs, and will make the applicability of POIROT take a leap.
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