TWJ specificity


Abstract


We created a model in the Dry Lab to reproduce the amplification results when mutations were introduced into the target in NJ Amplification and TWJ Amplification. By analyzing the model, we found that TWJ Amplification has superior sequence specificity compared to NJ Amplification because, in TWJ Amplification, the binding process involves the formation of a TWJ complex through multiple steps, resulting in a lower binding constant. This finding suggests that adjusting the thermodynamic stability of the TWJ complex can improve specificity, allowing us to propose sequence designs with superior specificity from the Dry lab.

Understand why TWJ Amplification has sequence specificity


Purpose

In Non-Junction (NJ) Amplification, the amplification reaction proceeds even with miRNAs that differ by several bases, whereas in TWJ Amplification, it has been experimentally reported in prior studies that amplification does not occur even with miRNAs that differ by only a single base from the target miRNA, demonstrating high sequence specificity 1. We aimed to enhance the sequence specificity in Three-Way Junction (TWJ) Amplification by carefully designing the template and Helper DNA. To achieve this, it is crucial to accurately understand the factors contributing to the high sequence specificity in TWJ Amplification. Although prior reports suggest that this specificity arises from thermodynamic factors within the TWJ structure, the understanding is not yet developed enough to inform precise sequence design. To address this, we attempted to construct a new model to better understand why TWJ Amplification exhibits higher sequence specificity than NJ Amplification. Just as developing a treatment requires an accurate identification of the disease's cause, designing templates or primers with higher specificity requires a precise understanding of the underlying factors that give TWJ Amplification its superior specificity.

Wet Results

In our Wet Lab, we demonstrated that TWJ Amplification shows higher specificity than NJ Amplification by comparing cases where the target is hsa-let-7b (let-7b) with cases where mutations are introduced in the sequence, similarly to previous studies. Our next step is to build a model that can replicate these experimental results.

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Figure 1. Comparison of sequence specificity between NJ and TWJ by our Wet Lab

Methods

Models for Complex Formation and Amplification

We created models (Model 0 and Model 1) for NJ Amplification and TWJ Amplification, where the target binds with the template and helper to form double and TWJ complexes, respectively. To make it simple, in Model 1, we assumed that the target, helper and template bind simultaneously to form the TWJ complexes.

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Figure 2. Model 0: NJ Complex Formation.

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Figure 3.

Model 1: TWJ Complex Formation

We use Model 0 and 1 for the simulation of complex formation, and Model P for the strand elongation by polymerase after the complex formation. In other words, we simulate NJ Amplification and TWJ Amplification by replacing T→YT3 in Figure 4. with Model 0 and 1, respectively. Here, we assume that W and Z react with SYBR GREEN and that fluorescence value corresponds to X+Z. Although Figure 4. represents the case for NJ Amplification, note that the corresponding parts for W and Z will change in the case of TWJ Amplification.

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Figure 4. Model P

The Model above can be written using Mass Action as the following system of Ordinary Differential Equations (ODEs).

Equation 1. ODE for NJ Amplification (Model 0 & Model P)

\[ \begin{equation} \begin{aligned} \frac{d[X]}{dt} &= -k_1 [X] [T] + \frac{k_1}{a_1} [XT] \notag\\ \frac{d[T]}{dt} &= -k_1 [X] [T] + \frac{k_1}{a_1} [XT] \notag\\ \frac{d[XT]}{dt} &= k_1 [X] [T] - \frac{k_1}{a_1} [XT] - \frac{k_{\text{cat}} P_0}{32 m_1 c} [XT] \notag\\ \frac{d[W]}{dt} &= \frac{k_{\text{cat}} P_0}{32 m_1 c} [XT] - k_2 [W] [N] + \frac{k_{\text{cat}} P_0}{23 m_2 c} [Z] \notag\\ \frac{d[Z]}{dt} &= -\frac{k_{\text{cat}} P_0}{23 m_2 c} [Z] + k_2 [W] [N] \notag\\ \frac{d[N]}{dt} &= -b [N] \notag\\ \end{aligned} \tag{1} \end{equation} \]

Equation 2. ODE for TWJ Amplification(Model 1 & Model P)

\[ \begin{equation} \begin{aligned} \frac{d[X]}{dt} &= -k_1 [X] [P] [T] + \frac{k_1}{a_1} [XPT] \notag\\ \frac{d[P]}{dt} &= -k_1 [X] [P] [T] + \frac{k_1}{a_1} [XPT] \notag\\ \frac{d[T]}{dt} &= -k_1 [X] [P] [T] + \frac{k_1}{a_1} [XPT] \notag\\ \frac{d[XPT]}{dt} &= k_1 [X] [P] [T] - \frac{k_1}{a_1} [XPT] - \frac{k_{\text{cat}} P_0}{32 m_1 c} [XPT] \notag\\ \frac{d[W]}{dt} &= \frac{k_{\text{cat}} P_0}{32 m_1 c} [XPT] - k_2 [W] [N] + \frac{k_{\text{cat}} P_0}{23 m_2 c} [Z] \notag\\ \frac{d[Z]}{dt} &= -\frac{k_{\text{cat}} P_0}{23 m_2 c} [Z] + k_2 [W] [N] \notag\\ \frac{d[N]}{dt} &= -b [N] \notag\\ \end{aligned} \tag{2} \end{equation} \]

Nucleic acid Sequences

The binding constants in Model 0 and 1 depend on the nucleotide sequences. By using the sequences actually used in the Wet Lab, more accurate binding constants can be determined.

\[ \begin{array}{|c|c|c|} \hline \textbf{Type} & \textbf{Name} & \textbf{Sequence(5' \, \text{to} \, 3')} \\ \hline \text{target (p.m.)} & \text{let-7b} & \text{UGAGGUAGUAGGUUGUGUGGUU} \\ \hline \text{target (m.m.)} & \text{mismatch-5} & \text{UGAGAUAGUAGGUUGUGUGGUU} \\ \hline \text{primer (helper)} & & \text{AACCACACAACCCCAAA} \\ \hline \text{template (NJ)} & & \text{AAGTGTGTGTGTCCTCGCTGAG}\\ &&\text{GAACCACACAACCTACTACCTCATTT} \\ \hline \text{template (TWJ)} & & \text{AAGTGTGTGTGTCCTCGCTGA}\\ &&\text{GGTTGTTTTGGAATACTACCTCATTT} \\ \hline \end{array} \]

Table 1. target and primer Sequence

First, we compare the template that is fully complementary to the target (let-7b) with a sequence that has a single base mutation at the 5th position from the 5' end of let-7b (mismatch-5). NJ Amplification cannot distinguish between let-7b and mismatch-5 while TWJ Amplification can differentiate between the two (Figure 14.). According to Figure 14., with NJ Amplification, let-7b and mismatch-5 are amplified similarly. However, with TWJ Amplification, let-7b is amplified more strongly than mismatch-5.

Parameters

We assume that parameters other than binding constant a are independent of the sequence. (Dry lab_Model_Amplification_comparison) From onwards, we will continue to use this for the constants of Model P. The binding constant a is determined by using the Nearest Neighbor (NN) model, by deriving the free energy of the binding from Equation 3. and then applying Equation 4. The NN model is a theory used to work out the stability of DNA duplexes. Besides the binding of each base pair in the sequence, we take \(\Delta H\) and \(\Delta S\), which change depending on the adjacent base pairs, into consideration, to compute \(\Delta G\) of the binding of the entire duplex. Since, the experimental temperature is 37 ℃, \(\Delta G\) of the entire sequence can be calculated as follows.

\begin{equation} \begin{aligned} \Delta G = \Delta H - T\Delta S~(T = 310.15 K) \end{aligned} \tag{3} \end{equation}

\(\Delta G\) is from Table2 2, and Table3 3.

The conversion of \(\Delta G\) due to the Buffer concentration was based on 4.

The binding constant a can be written using \(\Delta G\) as follows.

\begin{equation} \begin{aligned} a = \exp\left(\frac{-\Delta G}{RT}\right) \end{aligned} \tag{4} \end{equation}

Based on Mathews, D. H., & Turner, D. H. (2002), when forming TWJ complexes, we need to add \(\Delta G^\circ_{37\text{MBL}} = 3.64\ (kcal/mol)\) to the total free energy5. Furthermore, according to Hooyberghs, J., Van Hummelen, P., & Carlon, E. (2009), the difference in the free energy between let-7b and mismatch-5 is \(\Delta\Delta G =\Delta G_{\text{m.m.}} - \Delta G_{\text{p.m.}}= 2.7\ (kcal/mol)\) 6.

Using the parameters above and determining the binding constants of Model 0 and 1 (\(a\) and \(a'\) respectively), we can obtain Table 2.

\[ \begin{array}{|c|c|c|} \hline \textbf{} & \textbf{let-7b} & \textbf{mismatch-5} \\ \hline \textbf{Model 0} & a=5.0 \times 10^{11} & a=3.8 \times 10^{9} \\ \hline \textbf{Model 1} & a'=7.97 \times 10^{5} & a'=6.11 \times 10^{3} \\ \hline \end{array} \]

Table 2. Binding constant for Model 0, Model 1

Results

We simulated the amplification of W+Z using Model 0 for the complex formation of NJ Amplification and Model 1 for the complex formation of TWJ Amplification. (Figure 5.)

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Figure 5. The simulation result of the amplification of W+Z using Model 0 and 1

Note that, with NJ Amplification, the graph for let-7b and mismatch-5 completely overlap, because there is almost no difference between the two. From Figure 5. we can see that Model 0 is sufficient to show the lack of specificity in NJ Amplification. Model 1 shows that TWJ Amplification cannot distinguish between let-7b and mismatch-5, and this contradicts the eXPerimental results shown in Figure 1. This suggests that Model 1 is insufficient to show the characteristics of TWJ Amplification. Therefore, we decided to develop a new model.

New Methods

We could not reproduce the fact that TWJ Amplification can distinguish between let-7b and mismatch-5 in Model 1. In this section, we devise a new model that can differentiate the two.

New Complex Formation Models

The previous Model 1 was a model where 3 nucleic acids bind at once. Based on Dr. Sakuraba's advice from Human Practices, we created Model 2, where a TWJ complex is formed through a two-step binding process. Note that, since the binding is multi-step, there are multiple pathways to form a TWJ complex. Here, we ignored the reaction in which the helper and the template bind, in this model. This is because it can be predicted that when the helper and template bind, the polymerase elongation reaction proceeds even without the target, and we assume that this is the reason why the graph of fluorescence signal rises for NC. We ignored the pathway where the helper and template bind, because there is no need to take the rise of the graph for NC into account when we want to compare the specificity. Here, we call the pathways \(X→XT→XPT, X→XP→XPT\) as \(1→4, 2→3\), respectively. (Hereafter, we will refer to \(X→XT, XT→XPT, X→XP, XP→XPT\) as reactions \(1, 2, 3, 4\), respectively.)

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Figure 6. Model2: TWJ New Complex Formation

As before, we will replace Model 2 by Y→YT3 in Model P. In this case, ODE can be written as Equation 5.

\begin{equation} \begin{aligned} \frac{d[X]}{dt} &= -k_1 [X] [T] + \frac{k_1}{a_1} [XT] - k_2 [X] [P] + \frac{k_2}{a_2} [XP] \notag\\ \frac{d[T]}{dt} &= -k_1 [X] [T] + \frac{k_1}{a_1} [XT] - k_3 [XP] [T] + \frac{k_3}{a_3} [XPT] \notag\\ \frac{d[P]}{dt} &= -k_2 [X] [P] + \frac{k_2}{a_2} [XP] - k_4 [XT] [P] + \frac{k_4}{a_4} [XPT] \notag\\ \frac{d[XT]}{dt} &= k_1 [X] [T] - \frac{k_1}{a_1} [XT] - k_4 [XT] [P] + \frac{k_4}{a_4} [XPT] \notag\\ \frac{d[XP]}{dt} &= k_2 [X] [P] - \frac{k_2}{a_2} [XP] - k_3 [XP] [T] + \frac{k_3}{a_3} [XPT] \notag\\ \frac{d[XPT]}{dt} &= k_3 [XP] [T] - \frac{k_3}{a_3} [XPT] + k_4 [XT] [P] - \frac{k_4}{a_4} [XPT] - \frac{k_{\text{cat}} P_0}{32 m_1 c} [XPT] \notag\\ \frac{d[W]}{dt} &= \frac{k_{\text{cat}} P_0}{32 m_1 c} [XPT] - k_5 [W] [N] + \frac{k_{\text{cat}} P_0}{23 m_2 c} [Z] \notag\\ \frac{d[Z]}{dt} &= -\frac{k_{\text{cat}} P_0}{23 m_2 c} [Z] + k_5 [W] [N] \notag\\ \frac{d[N]}{dt} &= -b [N] \notag\\ \end{aligned} \tag{5} \end{equation}

Equation 5. ODE for TWJ Amplification(Model 2 & Model P)

Parameters

Given that the differences in the reaction rate constants are small compared to the differences in binding constants due to the nucleic acid sequence variance, we assume \(k1=k2=k3=k4=11.4\) 7. The absolute values are based on the k1 obtained from the Fitting below. (Dry lab_Model_Amplification_comparison) The values of \(a_1, a_2, a_3, a_4\) are derived from the NN model and are presented in Table 3.

\[ \begin{array}{|c|c|c|} \hline \textbf{Model 2} & \textbf{let-7b} & \textbf{mismatch-5} \\ \hline a_1 & 3.06 & 2.35 \times 10^{-2} \\ \hline a_2 & 7.65 \times 10^{2} & 7.65 \times 10^{2} \\ \hline a_3 & 4.05 \times 10^{1} & 3.10 \times 10^{-1} \\ \hline a_4 & 3.81 \times 10^{3} & 3.81 \times 10^{3} \\ \hline \end{array} \]

Table 3. Binding constant for Model 2

As before, we will use mismatch-5 for the miRNA with a single base mutation. It is important to note that since the mutation is occurring at the fifth base from the 5' end of the target, only the binding constants \(a_1, a_3\) will show changes.

Results

The simulation result of the amplification of W+Z using Model 2 is shown below. (Figure 7.)

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Figure 7. The simulation result of the amplification of W+Z using Model 2

Figure 7. shows that by changing from Model 1 to Model 2, TWJ Amplification can distinguish between let-7b and mismatch-5, which matches the eXPerimental results obtained from the Wet Lab. What is crucial here is understanding why the change from Model 1 to Model 2 allows us to replicate the specificity of TWJ Amplification. By comprehending this, we can identify the underlying reasons for the specificity of TWJ Amplification.

Analysis

We developed two distinct models, Model 1 and Model 2, as representations of the TWJ complex in TWJ Amplification. However, based on the discussions thus far, we can conclude that Model 2 is the superior model. In this section, we will eXPlore the fundamental differences between Model 1 and Model 2.

Breaking Down Model2

As mentioned above, there are two differences between Model 1 and 2: the number of reaction pathways leading to complex formation and whether the complex is formed in a single step or two-step process. It is challenging to clarify which of these factors are significant in the differences in Model 1 and 2. By breaking down Model 2 to consolidate the two reaction pathways into one, we will isolate the difference between Model 1 and Model 2. This approach will allow us to identify the fundamental cause of the differences in specificity between the two models.

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Figure 8. Model 2 (1→4): Reaction pathway 1→4

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Figure 9. Model 2 (2→3): Reaction pathway 2→3

We simulated the amplification of W+Z using Model 2 (1→4) and Model 2 (2→3) as before.

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Figure 10. The simulation result of the amplification of W+Z using the broken down Model 2

From Figure 10., we can see that both Model 2 (1→4) and Model 2 (2→3) can distinguish between let-7b and mismatch-5. This indicates that the higher specificity of Model 2 compared to Model 1 is more significantly attributed to the two-step binding process rather than the existence of two reaction pathways.

Analysis on the Broken Down Model 2

In the previous section, we confirmed that both Model 2 (1→4) and Model 2 (2→3) can distinguish between let-7b and mismatch-5 when the reaction pathways are consolidated into one. In this section, we will demonstrate that the high specificity of TWJ Amplification is important because the reaction is multi-step, which keeps the binding constants low by manipulating the binding constants in both Model 2 (1→4) and Model 2 (2→3).

Model 2 (1→4)

Let \(W + Z (\text{p.m.})\) denote the value of \(W + Z\) when the target sequence is perfectly complementary to the template, and let \(W + Z (\text{m.m.})\) denote the value of \(W + Z\) when the target sequence contains a mutation. As in the previous methods, we assume that the mutation occurs at the fifth nucleotide from the 5' end of the target. Using Model 2 (1→4), we varied \(a_1\) and \(a_4\) as \(a_1 = 10^{n_1}\) and \(a_4 = 10^{n_2}\text{, where} -2 \leq n_1, n_2 < 8\), and examined \(\frac{F'}{F} = \frac{W + Z (\text{m.m.})}{W + Z (\text{p.m.})}\) at \(t = 3000\) (Figure 11.).

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Figure 11. The change in the specificity when varied the values of $a_1, a_4$ in Model 2 (1→4)

When let-7b and mismatch-5 were used as targets, the values of \(a_1 and a_4\) were as listed in Table 3, and under these conditions, Model 2 (1→4) exhibited specificity. However, Figure 11. shows that Model 2 (1→4) loses specificity as the values of \(a_1 and a_4\) increase. It is important to note that specificity is more sensitive to changes in \(a_1 than a_4\). The high specificity observed in Model 2 (1→4) suggests that maintaining a small binding constant, especially \(a_1\), by making the reaction multistep is crucial.

While decreasing \(a_1\) makes the reverse reaction more likely, it is not immediately obvious why the ratio \(\frac{W + Z (\text{m.m.})}{W + Z (\text{p.m.})}\) decreases. Therefore, we will next investigate the reason for this.

The forward reaction, reverse reaction, and net flux for \(X+T→XT\) can be expressed as follows. From here, we will use analogous notation to represent the reaction flux.

Equation 6. Flux, \(X\_XT\_f, X\_XT\_b, X\_XT \)

\begin{equation} \begin{aligned} X\_XT\_f &= k_1 [X] [T] \notag\\ X\_XT\_b &= \frac{k_1}{a_1} [XT] \notag\\ X\_XT&=k_1 \left( [X] [T] - \frac{1}{a_1} [XT] \right) \notag\\ \end{aligned} \tag{6} \end{equation}

From Figure 11., we can see that decreasing \(a_1\) will reduce \(\frac{X\_XT(\text{m.m.})}{X\_XT(\text{p.m.})}\), resulting in the increase in the specificity. Note that when \(a_1=1e+7\), the two curves overlap almost completely.

\begin{equation} \begin{aligned} \frac{X\_XT(\text{m.m.})}{X\_XT(\text{p.m.})} = \frac{X\_XT\_f(\text{m.m.}) - X\_XT\_b(\text{m.m.})}{X\_XT\_f(\text{p.m.}) - X\_XT\_b(\text{p.m.})} \notag\ \end{aligned} \tag{7} \end{equation}
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Figure 12. The change in the \(X\_XT\) of p.m. and m.m. when varied the values of \(a_1\) in Model 2 (1→4)

Regardless of the value of \(a_1, X_XT_f\) can be considered almost unchanged. On the other hand, as \(a_1\) decreases, \(X\_XT_b\) increases. Therefore, the value of Equation 7. increases, and specificity improves. This can be understood with simple arithmetic. For example, assume that \(X\_XT\_f=1\) regardless of the magnitude of \(a_1\) (since Eq. (7) focuses on the ratio, the absolute values are not of great importance). If the values of \(X\_XT\_b\) are \(0.001, 0.01, 0.05, 0.5\) when \(a_1=1e+7,\ 7.6e+4,\ 1e+3,\ 7.6\), respectively, the values of Equation 7. become \(0.99\) and \(0.52\). This demonstrates that increasing \(X\_XT\_b\) by decreasing \(a_1\) is crucial for enhancing specificity.

Model 2 (2→3)

In the previous section, we varied \(a_1 and a_4\) in Model 2 (1→4). In this section, we investigated Model 2 (2→3) by varying \(a_2\_ \text{and}\_ a_3 \_\text{as}\_ a_2 = 10^{n_1} \_ \text{and}\_ a_3 = 10^{n_2}\), where \(-2 \leq n_1, n_2 \leq 7\), and examined \(\frac{F'}{F} = \frac{W + Z (\text{m.m.})}{W + Z (\text{p.m.})}\) at \(t=3000\) (Figure 13.).

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Figure 13. The change in the specificity when varied \(a_2, a_3\) in Model 2 (2→3)

When let-7b and mismatch-5 were used as targets, the values of \(a_2\) and \(a_3\) were as listed in Table 3, and under these conditions, Model 2 (2→3) exhibited specificity. However, Figure 13. shows that Model 2 (2→3) loses specificity as the values of \(a_2\) and \(a_3\) increase. It is important to note that specificity is more sensitive to changes in \(a_3\) than \(a_2\). The high specificity observed in Model 2 (2→3) suggests that maintaining a small binding constant, especially \(a_3\), by making the reaction multistep is crucial.

By making \(a_3\) small, \(XP\_XPT\_b\) increases, and due to the same reason as the previous section,
\begin{equation} \begin{aligned} \frac{XP\_XPT(\text{m.m.})}{XP\_XPT(\text{p.m.})} = \frac{XP\_XPT\_f(\text{m.m.}) - XP\_XPT\_b(\text{m.m.})}{XP\_XPT\_f(\text{p.m.}) - XP\_XPT\_b(\text{p.m.})} \notag\ \end{aligned} \tag{8} \end{equation}

will become smaller. What is significant in the specificity in Model 2 (2→3) is that \(XP\_XPT\_b\) increases as \(a_3\) decreases.

Conclusions

By breaking down Model 2 and analyzing Model 2 (1→4) and Model 2 (2→3), each of which has a single reaction pathway, we were able to identify the essential reason for the high specificity of TWJ Amplification. TWJ Amplification keeps the binding constant low by making the complex formation a multistep process. Among the binding constants, it is important that the one that becomes smaller when there is a single-nucleotide mismatch in the target is kept low—\(a_1\) in Model 2 (1→4) and \(a_3\) in Model 2 (2→3). Furthermore, we found that the threshold for specificity is \(a_1, a_3 < 1e+3\). When \(a_1\) and \(a_3\) are below \(1e+3\), specificity is maintained regardless of the values of \(a_2\) and \(a_4\).

The above conclusions are based on the case where the mutation is located at the fifth nucleotide from the 5' end (mismatch-5), but when the mutation is located at the 15th nucleotide from the 5' end (e.g., mismatch-15), it is \(a_2\) and \(a_4\) that are important for specificity, not \(a_1\) and \(a_3\). Specifically, due to the symmetry in Model 2, \(a_2\) corresponds to \(a_1\), and \(a_4\) corresponds to \(a_3\). The threshold values can also be determined through a similar discussion as for mismatch-5.

The Difference in the Specificity due to the Mutation Position

Up to this point, we have only considered sequences with mutations located at the 5th base position from the 5' end of the target (mismatch-5). As shown in Figure 14., NJ Amplification cannot distinguish between let-7b and mismatch-5, whereas TWJ Amplification can. However, even with TWJ Amplification, there are cases where single-base mutations cannot be distinguished from let-7b, depending on the position of the mutation. These are when the mutation occurs at the 1st, 9th, or 11th base position from the 5' end (mismatch-1, mismatch-9, mismatch-11). This occurs when the mutation is located near the strand's end or near the junction structure of the TWJ complex.

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Figure 14. The comparison of the specificity in NJ Amplification and TWJ Amplification with single base mutation, obtained from the Wet Lab

To consider why the specificity varies depending on the position of the mutation, we will now consider sequences with mutations located at the 1st, 9th, and 11th base positions from the 5' end (mismatch-1, mismatch-9, mismatch-11). Using the \(\Delta\Delta G\) values from Table 4, we will simulate mismatch-1, mismatch-9, and mismatch-11 and compare the results with those of mismatch-5 and let-7b.

\[ \begin{array}{|c|c|} \hline & \Delta\Delta G \, (\text{kcal/mol}) \\ \hline \textbf{mismatch-1} & 1.6 \\ \hline \textbf{mismatch-9} & 1.9 \\ \hline \textbf{mismatch-11} & 2.3 \\ \hline \end{array} \]
Table 4 Binding constant for Model2
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Figure 15. The comparison of the specificity in mismatch-1, mismatch-5, mismatch-9, and mismatch-11

Although the difference in specificity is not as pronounced as in Figure 14., we can confirm through simulation that mismatch-1, mismatch-9, and mismatch-11 exhibit lower specificity compared to mismatch-5. This is because mutations located in the middle of the double strand result in larger \(\Delta\Delta G\) values. When \(\Delta\Delta G\) is larger, the ratio \(\dfrac{a(\text{m.m.})}{a(\text{p.m.})}\) becomes smaller, and the specificity increases. The fact that we cannot observe as much difference in specificity as seen in Figure 14. suggests that the model used in this analysis is insufficient.

Application to Sequence Design


It has been determined that Model 2 is more suitable than Model 1 as a model for TWJ Amplification. Additionally, for Model 2 (1→4) and Model 2 (2→3) to exhibit specificity, it is necessary for \(a_1\) and \(a_3\) to be small. In actual TWJ Amplification, does reducing \(a_1\) and \(a_3\) similarly increase specificity? In previous discussions, \(a_1\) and \(a_4\), as well as \(a_2\) and \(a_3\), were varied simultaneously in the broken down models of Model 2. In this section, we will consider how to design sequences that enhance specificity by varying \(a_1\) and \(a_3\) simultaneously in the full Model 2.

To begin, in order to evaluate the specificity of mismatch-5, we will assume \(\Delta\Delta G = 2.7\) kcal/mol and vary \(a_1\) and \(a_3\) as \(a_1 = 10^{n_1}\) and \(a_3 = 10^{n_2}\), where \(-2 \leq n_1, n_2 < 8\). We will then examine \(\frac{F'}{F} = \frac{W + Z (\text{m.m.})}{W + Z (\text{p.m.})}\) at \(t=3000\).

\begin{equation} \begin{aligned} \Delta G_1 + \Delta G_2 &= \text{const.} \notag\\ \Delta G_1 + \Delta G_4 &= \Delta G_2 + \Delta G_3 = \text{const.} \notag\\ \end{aligned} \tag{9} \end{equation}

If the target length is constant and \(\Delta G\) is equal in the pathways \(1→4, 2→3\), it could be approximated to Eq. (9). Hence, the relationship among \(a_1, a_2, a_3, a_4\) can be written as Equation 10.

\begin{equation} \begin{aligned} a_2 &\propto \frac{1}{a_1} \notag\\ a_4 &= \frac{a_2 \cdot a_3}{a_1} \notag\\ a_1 &< a_3 \notag\\ \end{aligned} \tag{10} \end{equation}
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Figure 16_1. The comparison of the specificity when \(a_1, a_3\) are varied in Model 2 (mismatch-5)

It can be predicted from Figure 16. that when \(a_1 < a_3 < 1e+3\), there will be sufficient specificity. In fact, when target, template, and helper are of the sequence in the Table 1, \(a_1=3.06, a_3=4.05e+1\) can be obtained, and it can be shown that the specificity is sufficient from Figure 16.

Let the length of the binding region between the target and the template be \(tt\), and the length of the binding region between the helper and the template be \(ht\). We will investigate at what length of \(ht\) the specificity decreases, with \(tt\) being fixed. Here, the target is let-7b, and the template is the sequence of the template (TWJ) from Table 1. By varying \(ht\) to complement the template, we modify \(a_3\).

\[ \begin{array}{|c|c|c|} \hline \textbf{ht} & \Delta G \, (\text{kcal/mol}) & a_3 \\ \hline 3 & -8.76 & 1.50 \\ \hline 4 & -9.78 & 7.86 \\ \hline 5 & -10.8 & 4.05 \times 10^{1} \\ \hline 6 & -11.5 & 1.39 \times 10^{2} \\ \hline 7 & -12.0 & 2.96 \times 10^{2} \\ \hline 8 & -12.4 & 5.13 \times 10^{2} \\ \hline 9 & -13.1 & 1.75 \times 10^{3} \\ \hline 10 & -14.3 & 1.17 \times 10^{4} \\ \hline 11 & -15.8 & 1.35 \times 10^{5} \\ \hline \end{array} \]

Table 5. Ht, \(\Delta{G}, a_3\)

From Table 5, when the target and template sequences are set to those in Table 1, we can predict that in order to maintain sufficient specificity, \(ht > 8\) is incompatible.

Next, we will research how to improve the sequence to enhance the specificity for mismatch-1. Assuming \(\Delta\Delta G (\text{kcal/mol}) = 1.6\), we vary \(a_1\) and \(a_3\) as \(a_1 = 10^{n_1}\) and \(a_3 = 10^{n_2}\), where \(-2 \leq n_1, n_2 < 8\), and investigate \(\frac{F'}{F} = \frac{W + Z (\text{m.m.})}{W + Z (\text{p.m.})}\) at \(t=3000\).

hogehoge

Figure 16_2. The comparison of the specificity when \(a_1\), \(a_3\) are varied in Model 2 (mismatch-1)

Note that for mismatch-1, the range of \(a_3\) that yields specificity is slightly narrower compared to mismatch-5. Considering that in Figure 15., sufficient specificity was not obtained in the simulation when \(ht=5\), we can predict that to maintain sufficient specificity, \(ht \leq 4\) is required.

When examining ways to enhance the specificity of mismatch-15 or mismatch-17, the parameters to adjust are \(a_2\) and \(a_4\). As in the above discussion, adjusting the length of the binding regions between the helper and the target, as well as between the helper and the template, can maintain or improve specificity.

Future


We were able to replicate the fact obtained in the wet lab that the specificity for mismatch-5 is high and the specificity for mismatch-1 is low in TWJ Amplification using the model created in the Dry lab. Although we predicted the conditions to improve the specificity for mismatch-1 using this model, we did not have the time to confirm it in the Wet Lab. If we could verify the Dry lab predictions in the Wet Lab, it would further enhance the reliability of the model and allow it to be used for predicting other phenomena. Additionally, while we demonstrated that the specificity for mismatch-1, 9, and 11 is higher than that for mismatch-5, we were unable to reproduce the significant differences observed in the Wet Lab. Moving forward, it will be necessary to further investigate the reasons behind this.

Appendix


Based on the previous discussions, it can be anticipated that the high sequence specificity of TWJ Amplification is achieved by making the complex binding a multi-step process, particularly by keeping \(a_3\) small, which increases \(XP\_XPT\_b\) and ensures specificity. In this section, we will investigate the strong involvement of \(XP\_XPT\_b\) in the reaction specificity by manipulating \(k_2\) and \(k_3\) in Model 2 (2→3). We will vary \(k_2 = 10^{n_1}\) and \(k_3 = 10^{n_2}\) within the range of \(-4 \leq k_2, k_3 \leq 5\) and examine \(\frac{F'}{F} = \frac{W + Z (\text{m.m.})}{W + Z (\text{p.m.})}\) at \(t=3000\).

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Figure 17. The comparison of the specificity when \(k_2, k_3\) are varied in Model2(2→3)

We can see from Figure 17. that the specificity is robust with respect to \(k_2\). We will analyze why the specificity improves when changing \(k_2\): \(1e+1\) (const.), \(k_3\): \(1e-3\)→\(1e-1\).

\begin{equation} \begin{aligned} XP\_XPT &= k_3 [XP] [T] - \frac{k_3}{a_3} [XPT] \notag\\ XPT\_W &= \frac{k_{\text{cat}} P_0}{32 m_1 c} [XPT] \notag\\ \end{aligned} \tag{11} \end{equation}

From Equation 11., we can immediately understand that \(XP\_XPT\) increases as \(k_3\) increases, which results in the increase in the amount of \(XPT\).

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Figure 18. Examining \(XP\_XPT\), \(XPT\_W\) in Model 2 (2→3)

We can see from Figure 18. that increasing \(k_3\) decreases \(\frac{XP\_XPT(\text{mismatch-5})}{XP\_XPT(\text{let-7b})}\), which means that the specificity improved. Then why does \(\frac{XP\_XPT(\text{mismatch-5})}{XP\_XPT(\text{let-7b})}\) decrease when \(XP\_XPT\) increases?

\begin{equation} \begin{aligned} XP\_XPT\_f = k_3 [XP] [T] \notag\\ XP\_XPT\_b = \frac{k_3}{a_3} [XPT] \notag\\ \end{aligned} \tag{12} \end{equation}

From Equation 12., increasing \(k_3\) by a factor of \(1e+2\) results in \(XP\_XPT\_f\) also increasing by a factor of approximately \(1e+2\). However, since \(k_3\) significantly enhances \(XPT\), \(XP\_XPT\_b\) becomes more than \(1e+2\) times larger. Therefore, as discussed previously, \(\frac{XP\_XPT(\text{mismatch-5})}{XP\_XPT(\text{let-7b})}\) decreases. This again highlights the importance of \(XP\_XPT\_b\).

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