Amplification Comparison


Abstract


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Our goal is to develop a home detection device for glaucoma. To this end, we explored the use of strand displacement amplification (SDA) methods that do not require complex temperature changes. In Dry lab, we constructed an ordinary differential equation (ODE) model to select appropriate amplification reactions. First, based on the experimental data, we fitted the ODE model parameters that were unknown and experimentally challenging to determine in order to replicate the experimental results in Wet lab. The ODE model revealed that three-way junction (TWJ-SDA) alone was insufficient to activate Cas due to low amplification efficiency. Based on previous researches, then, we proposed the alternative amplification systems, including TWJ-SDA > Exponential amplification reaction (EXPAR), TWJ-SDA > 2step-SDA and TWJ-SDA > 3step-SDA. About 2step-SDA and 3step-SDA were introduced precisely at Optimise Amplification. The amplification efficiency, the positive-to-negative signal ratio (P/N ratio), and robustness to enzyme activity and time were compared by using the ODE model. As a result, we proposed using TWJ-SDA > 3step-SDA as the best amplification system and suggested the optimal initial template concentration.

Fitting the parameters


Purpose

In constructing the ODE model, it is necessary to define the parameters used within the model. However, in this amplification system, the reaction rate constants for nucleic acid binding and nicking, and the Michaelis constant were unknown and difficult to determine experimentally. The goal of this section is to appropriately estimate these parameters based on wet lab experimental results and construct an ODE model that replicates a portion of these results.

Methods

In constructing the ODE model, the parameters like these were generally determined through fitting. We developed several ODE models throughout the project. As several ODE modellings of EXPAR had been constructed, the parameters on EXPAR were fitted first and the fitted parameters were used in the ODE models of other amplification systems.

In EXPAR, the process begins with the target binding to the template, followed by a polymerase extension reaction that produces double strands (Figure 1.). Next, nickase recognizes the nicking recognition sequence within the double strands, leading to nicking. The polymerase then initiates a new extension reaction from the nicking site, displacing the downstream single-strand DNA (ssDNA) encountered during synthesis and producing the full double-strands DNA (dsDNA). Since the displaced ssDNA has the same sequence as the target, it binds to the template again, initiating further extension reactions and achieving overall exponential amplification.

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Figure 1. The reaction mechanism of EXPAR.

The ODE model for EXPAR in this project was constructed with the references to multiple prior studies that made ODE models of EXPAR (Figure 2.) 1, 2 and the reaction mechanism of ultrasensitive DNA amplification reactions 3, which is similar to that of EXPAR. As X could bind to the 5' end of T, ODE models were built in these studies based on different scenarios, such as X binding to both the 3' and 5' ends of T or only to the 3' end. These ODE models were constructed for each case by Dry lab. Still, since no significant difference was observed in the amplification behaviour of the reaction products, we opted to construct the simpler ODE model, assuming X binds only to the 3' end of T to reduce the number of parameters.

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Figure 2. The model of EXPAR.
X: target. T: template. XT: the complex of X and T. W: the complete dsDNA. Z: the nicked dsDNA.

This ODE model is formulated as shown in Equcation 1. The way to determine the reaction rates in this ODE model follows prior research 3, with the assumption that binding and dissociation reactions follow the law of mass action and enzyme reactions follow the Michaelis-Menten equation. For more assumptions in the construct of ODE model, see Assumption.
The experiments of EXPAR were conducted at 50 ℃, with an initial T concentration of 4.40e-2 µM and an initial X concentration ranging from 0 M (NC) to 1 pM-1 µM SYBR Green Ⅰ was used to detect fluorescence, which indicates the presence of dsDNA. Bst 2.0 polymerase, optimal at 65 ℃, was employed, and Nt.BstNBI nickase, optimal at 55 ℃, was used. Their initial concentrations were 1.50e-2 U/µL and 9.38e-2 U/µL, respectively. The parameters used in the ODE model are listed below in Table 1. Notably, \(k_{cat}P_{0}\) is expressed as such because \(k_{cat}\) and \(P_{0}\) appear only as a product and not individually. Given that 1 unit of Bst 2.0 polymerase is defined as the amount of the enzyme required to incorporate 25 nmol of dNTP into acid-insoluble material in 30 minutes at 65 ℃ 4, U is translated to mol as \(1U=25 nmol/1800 s/k_{cat}\), eliminating the need to measure the exact value of \(k_{cat}\). Since the fitting experiments were conducted at 50℃, where polymerase activity is reduced to 30-50% 5, we assumed 40% activity for the calculations. For the initial nickase concentration, we referred to prior studies 3 modelling the reaction in which the same enzyme was used and 0.2 U/µL corresponded to 2.6e-2 µM. Nt.BstNBI Nickase retains 100% activity at 50℃. Additionally 6, the experimental data suggested its activity remains constant during the reaction, though a prior research considered the heat inactivation of nickase. The association constants were calculated using NUPACK. For more detailed experimental methods, see Experiments_EXPAR.

\[ \begin{align} \dfrac{d[X]}{dt}&=-k_{1}[X][T]+\dfrac{k_{1}}{a}[XT]+\dfrac{k_{cat}}{17}\dfrac{P_{0}}{mc}[Y]\notag\\ \dfrac{d[T]}{dt}&=-k_{1}[X][T]+\dfrac{k_{1}}{a}[XT]\notag\\ \dfrac{d[XT]}{dt}&=k_{1}[X][T]-\dfrac{k_{1}}{a}[XT]-\dfrac{k_{cat}}{27}\dfrac{P_{0}}{mc}[XT]\notag\\ \dfrac{d[W]}{dt}&=\dfrac{k_{cat}}{27}\dfrac{P_{0}}{mc}[XT]-k_{2}[N][W]+\dfrac{k_{cat}}{17}\dfrac{P_{0}}{mc}[Y]\notag\\ \dfrac{d[Y]}{dt}&=k_{2}[N][W]-\dfrac{k_{cat}}{17}\dfrac{P_{0}}{mc}[Y]\notag\\ \end{align} \]

Equation 1. The equations of EXPAR.

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Table 1. The parameters of EXPAR

The mean fluorescence of the three experiments for each target concentration, excluding the background, was as follows (Figure 3.).

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The fluorescence was predicted by the simulation as follows (Figure 4.). Since the fluorescence is measured by using SYBR Green Ⅰ, which basically detects double-stranded DNA, it was assumed that the total concentration of W and Z was proportional to the fluorescence intensity, as well as the prior study 1 that also performed fitting of an ODE model of EXPAR. When the concentration graph reaches a plateau, the total concentration of W and Z was approximately equal to the initial template concentration of 4.4e-2 μM as there was almost no T or XT in the simulation. Therefore, this value 4.4e-2 was assumed to correspond to the plateau fluorescence value of 3.00e5 in Figure 3. Both the fluorescence graph from the wet experiment and the concentration graph from the dry simulation reach a plateau. After reaching the plateau, however, the concentration remains stable in the simulation, while the experimental fluorescence continues to increase. According to advice from Professor Komiya in Human Practices, this might be due to SYBR Green Ⅰ emitting slight fluorescence in response to increasing amounts of X, which is much more abundant than dsDNA.

The parameters \(k_{1},k_{2},m\) was fitted using a genetic algorithm with a population size of 100 and 50 generations, and the used data was located between the dotted lines \((0, 2.80e+5)\) in Figure 4. In the simulation, the initial fluorescence was set to 0, but in the wet experiment, when the initial X concentration was high, the amplification began immediately, and before the solution was placed in the measurement device, the graph of the fluorescence started to rise. Hence, in Dry lab simulation, it was assumed that the reaction started, and that 1 minute later the fluorescence measurement began. The amplification of the negative control (NC) and of the high concentrations like 100 nM and 1 μM, where the fluorescences reached at 2.80e5 within the first minute was excluded from the data used in the fitting. The amplification of NC is thought to occur because of ab initio synthesis, which refers free dNTPs bind to the template without primers, triggering the extension reaction 7.

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A genetic algorithm is a method that explores solutions by representing data (a set of proposed solutions) as "individuals" with genes and repeatedly performing operations such as selection, crossover, and mutation, selecting individuals with high fitness. In this project, we used a genetic algorithm to fit the reaction rate constants of nucleic acid binding and nicking, and the Michaelis constant.
First, in each generation, 100 individuals were determined with genes representing the three constants, and the fitness was defined by how closely the simulated results matched the experimental data based on the mean squared error (MSE). Individuals with smaller MSEs were considered fitter. The algorithm was designed so that, for each generation, individuals were defined to undergo crossover with probability of 50 %, mutation with that of 20 %, and copied with that of 30 %. For crossover, in which two individuals need to be selected, mutation and copying, in which one individual needs to be selected, a selection process known as "tournament selection" was used. The tournament selection involves randomly picking several individuals and selecting the best one among them. In this fitting process, two individuals were randomly chosen, and the individual the fitness of which is the higher was selected. The tournament selection was occured twice for crossover, and once for mutation and copying. By limiting the number of participants in each tournament, the randomness of this algorithm increases, promoting the diversity in the selected individuals.

For crossover, the "Blend Crossover" method was applied. Blend Crossover generates the new genes of the child individual by mixing the genes of two parent individuals. The gene of the child individual is randomly selected from an extended range of the parent individuals' genes, controlled by a parameter called \(\alpha\). Let's assume the two parent individuals \(P_{1}\),\(P_{2}\) have genes \(x_{1}\),\(x_{2}\), respectively. When performing Blend Crossover, the gene c of the child individual is randomly chosen from the following range.

\begin{align} c \in \left[ x_\text{min} - \alpha \cdot (x_\text{max} - x_\text{min}), x_\text{max} + \alpha \cdot x_\text{max} - x_\text{min} \right]\notag\\ \end{align}

In these equation, \(x_{\text{min}}, x_{\text{max}}\) were defined as follows.

\begin{align} x_{\text{min}} &= \min(x_1, x_2)\notag\\ x_{\text{max}} &= \max(x_1, x_2)\notag\\ \end{align}

\(\alpha\) is the parameter controlling the degree of expansion in the range for the child individual's genes and typically takes values in the range \(0 \leq \alpha \leq 1\). In this way, Blend Crossover allows the child individual's genes to explore beyond the range of the parent genes, expanding the search space and reducing the risk of getting stuck in a local optimum.

Next, in the mutation process for this fitting, each gene of an individual had a 20 % probability of being modified by adding a random value drawn from a Gaussian distribution. The Gaussian distribution has a mean of 0 and a standard deviation of 0.01. The child individual's gene is calculated as follows

\begin{align} x_{\text{new}} = x + N(0, 0.01)\notag\\ \end{align}

The small standard deviation allows for fine-tuning of the genes, and by gradually changing the genes, the search range is expanded, preventing the algorithm from settling into a local optimum. Mutation is essential for exploring solutions that cannot be found by crossover alone.

Additionally, copying occurs with a 30 % probability, and these three processes were repeated until the number of the child generation reaches 100. After that, the child generation became the current generation, and the same process was repeated for 50 generations. Finally, the individual with the highest fitness was selected and its genes were used as the fitted constants.

Result

The parameters obtained through fitting were listed in Table. 1, and the fluorescence of the simulation results compared with that of the experimental results was shown below (Figure 5).

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Conclusion

The purpose of this section was to determine the unknown parameters in the ODE model of the amplification system in this project, specifically the reaction rate constants for nucleic acid binding, the nicking reaction, and the Michaelis constant, which are tough to obtain experimentally. By employing a genetic algorithm, we were able to fit the experimental results of EXPAR with the simulation results from the ODE model to evaluate these unknown constants. The determined values are summarised in Table 1.

The amplification rate of TWJ-SDA


Purpose

In the previous section discussing specificity, the high specificity of TWJ-SDA (Figure 6.) was demonstrated. For more precise information, see Model. However, the amplification efficiency of TWJ-SDA is limited by the nature of the reaction, where the target binds to the template and proceeds through strand displacement, resulting in a maximum yield of amplification products determined by the target concentration. The purpose of this section was to evaluate whether TWJ-SDA alone had sufficient effeciency as the amplification system in this project by using simulations based on the ODE model of TWJ-SDA. While the presence of amplification can be examined by observing the fluorescence, the actual concentration of the amplification products cannot be measured, making this modelling essential.

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Figure 6. The reaction mechanism of TWJ-SDA.

Method

We constructed the ODE model for TWJ-SDA based on the reaction mechanism of TWJ-SDA (Figure 7.). In this model, the binding of the target to the template and that of primer to the template were not considered, as they did not contribute to the overall amplification efficiency and the robustness of the amplification system, which is a key criterion for evaluating the amplification system. By excluding these complexes, it was also aimed to reduce the number of parameters in this ODE model.

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Figure 7. The model of TWJ-SDA.
C: the complex of target and primer. T: template. CT: the complex of C and T. W: the complete dsDNA. Z: the nicked dsDNA. X: the product.

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Table 2. The concentrations of TWJ-SDA complexes.

Based on the above model diagram (Figure 7.), the following ODE was formulated (Equation 2.). Considering practical device implementations, the simulation was conducted setting the temperature, which was required to determine enzyme activity, to 37 ℃. Since the amplification efficiency did not change significantly when there was an ample amount of template, excess initial template concentration of 1 M and the initial target concentration of 100 fM were assumed in the simulation. The enzyme concentrations were optimised based on Wet experiments. For Polymerase, we used Bst. Large Fragment, which has an optimal temperature of 65 ℃, with an initial concentration of 6.25e-2 U/µL, and for Nickase, we used Nb.BbvCI, which has an optimal temperature of 37 ℃, with an initial concentration of 1.20e-1 U/µL. As for the sequences, the target was designated as a biomarker for glaucoma, with the template and primer being complementary to it.

The constants used in the ODE were referenced from fitting, NEB, and NUPACK (Tab. 3). The definition of 1 unit of the Polymerase, Bst Large Fragment, is the amount of enzyme required to incorporate 10 nmol of dNTP into acid-insoluble material in 30 minutes at 65 ℃ 9, expressed as \(1 U=10 nmol/1800 s/k_{cat}\). The Polymerase activity at 37 ℃ is calculated to be 12.5%, as it ranges from 10 % to 15 % 10. For the initial concentration of Nickase, we performed unit conversion from U to mol in the same manner as done in EXPAR, assuming the Nickase, Nt. BbvCI, operated at 100 % activity at the optimal temperature of 37 ℃ 6.

\begin{align} \frac{d[C]}{dt} &= -k_{1}[C][T] + \frac{k_{1}}{a}[CT]\notag \\ \frac{d[T]}{dt} &= -k_{1}[C][T] + \frac{k_{1}}{a}[CT]\notag \\ \frac{d[CT]}{dt} &= k_{1}[C][T] - \frac{k_{1}}{a}[CT] - \frac{k_{cat}}{26}\frac{P_{0}}{mc}[CT]\notag \\ \frac{d[W]}{dt} &= \frac{k_{cat}}{26}\frac{P_{0}}{mc}[CT] - k_{2}[N][W] + \frac{k_{cat}}{18}\frac{P_{0}}{mc}[Z]\notag \\ \frac{d[Z]}{dt} &= k_{2}[N][W] - \frac{k_{cat}}{18}\frac{P_{0}}{mc}[Z]\notag \\ \frac{d[X]}{dt} &= \frac{k_{cat}}{18}\frac{P_{0}}{mc}[Z]\notag\\ \end{align}

Equation 2. The equations of TWJ-SDA.

table3

Table 3. The parameters of TWJ-SDA

Results

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When considering the reaction time upon device implementation, it was estimated to be around 40 minutes. However, the simulation results (Figure 8.) indicated that after 40 minutes, the concentration of the amplification product reached approximately only 3 pM. This concentration is far from the required 2.5 nM concentration of the PAM sequence needed for Cas activation, which was shown by the Wet result, making it clear that achieving this efficiency with TWJ-SDA alone is virtually impossible. For more information about the result of Cas activation, see Results.

Conclusion

From the above, it is evident that in our project to develop a glaucoma detection device, the amplification efficiency of TWJ-Amp is insufficient. The ODE model suggests that it is necessary to combine TWJ-Amp with a more efficient amplification reaction to achieve the desired results.

Optimise Amplification


Purpose

The TWJ-SDA model indicated that it was insufficient for the amplification of PAM-containing dsDNA to the approximately required 2.5 nM for Cas recognition from the result of experiment. For more information about the result of Cas activation, see Results.
Therefore, it was necessary to use the amplified product of TWJ-SDA as a target and conduct other reactions with high amplification efficiency. Based on a review of prior studies, linking EXPAR and Multi step-SDA 11 were proposed, suggesting TWJ-SDA > EXPAR, TWJ-SDA > 2step-SDA and TWJ-SDA > 3step-SDA as the overall amplification system (Figure 9., 10., 11.). However, experimentally conducting these three amplification systems was challenging due to cost constraints (money, effort, and time). Consequently, the objective of this section was to propose the optimal amplification system and initial concentrations using simulations based on the ODE model.

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Figure 9. The reaction mechanism ofTWJ-SDA > EXPAR.

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Figure 10. The reaction mechanism of TWJ-SDA > 2step-SDA.

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Figure 11. The reaction mechanism of TWJ-SDA > 3step-SDA.

Methods

We constructed the ODE models of the three amplification systems (Figure 12., 13., 14.; Equation. 3., 4., 5.; Table 4., 5., 6.). In the previous ODE models for EXPAR and TWJ-SDA, the binding of amplification products to the 5' end of the template was not considered. However, since when combining multiple amplification reactions, this binding can inhibit subsequent amplification reactions, the ODE models that account for this factor were constructed. Since there was no existing ODE model for Multi step-SDA in prior studies, the ODE models were constructed based on the reaction mechanisms (Figure 9, 10, 11) as those of EXPAR and TWJ-SDA. Assuming practical device implementations, the simulation was conducted setting the temperature, which is required to determine enzyme activity, to 37 ℃, and using enzyme concentrations optimised through actual Wet experiments. For Polymerase, Bst. Large Fragment, which has an optimal temperature of 65 ℃, was used with an initial concentration of 8.00e-2 U/µL, and for Nickase, Nb.BbvCI, which has an optimal temperature of 37 ℃, was used with an initial concentration of 2.00e-1 U/µL. As for sequences, they could be freely changed for the amplification products, including TWJ-SDA, and were discussed in other sections. Thus, they were omitted from these ODE models discussion as they were irrelevant to the overall amplification efficiency and the robustness of the amplification system. The constants used in the ODE were determined in the same way as for TWJ-SDA. For binding constants, we used the same constants if the base pairs near the binding sites remain unchanged (Table 4., 5., 6.).

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Figure 12. The model of TWJ-SDA > EXPAR

\begin{align} \frac{d[C]}{dt} &= -k_{1}[C][T_{1}] + \frac{k_{1}}{a_{1}}[CT_{1}] - k_{1}[C][X_{2}T_{1}] + \frac{k_{1}}{a_{1}}[X_{2}CT_{1}]\notag \\ \frac{d[T_{1}]}{dt} &= -k_{1}[C][T_{1}] + \frac{k_{1}}{a_{1}}[CT_{1}] - k_{1}[X_{2}][T_{1}] + \frac{k_{1}}{a_{2}}[X_{2}T_{1}]\notag \\ \frac{d[CT_{1}]}{dt} &= k_{1}[C][T_{1}] - \frac{k_{1}}{a_{1}}[CT_{1}] - k_{1}[X_{2}][CT_{1}] + \frac{k_{1}}{a_{2}}[X_{2}CT_{1}] - \frac{k_{cat}}{28}\frac{P_{0}}{mc}[CT_{1}]\notag \\ \frac{d[W_{1}]}{dt} &= \frac{k_{cat}}{28}\frac{P_{0}}{mc}[CT_{1}] - k_{2}[N][W_{1}] + \frac{k_{cat}}{22}\frac{P_{0}}{mc}[Z_{1}]\notag \\ \frac{d[Z_{1}]}{dt} &= k_{2}[N][W_{1}] - \frac{k_{cat}}{22}\frac{P_{0}}{mc}[Z_{1}] + \frac{k_{cat}}{6}\frac{P_{0}}{mc}[X_{2}CT_{1}]\notag \\ \frac{d[X_{2}CT_{1}]}{dt} &= k_{1}[CT_{1}][Z_{2}] - \frac{k_{1}}{a_{2}}[X_{2}CT_{1}] - \frac{k_{cat}}{6}\frac{P_{0}}{mc}[X_{2}CT_{1}] + k_{1}[C][X_{2}T_{1}] - \frac{k_{1}}{a_{1}}[X_{2}CT_{1}]\notag \\ \frac{d[X_{2}T_{1}]}{dt} &= k_{1}[X_{2}][T_{1}] - \frac{k_{1}}{a_{2}}[X_{2}T_{1}] - k_{1}[C][X_{2}T_{1}] + \frac{k_{1}}{a_{1}}[X_{2}CT_{1}]\notag \\ \frac{d[X_{2}]}{dt} &= -k_{1}[X_{2}][CT_{1}] + \frac{k_{1}}{a_{2}}[X_{2}CT_{1}] - k_{1}[X_{2}][T_{1}] + \frac{k_{1}}{a_{2}}[X_{2}T_{1}]\notag \\ &\quad + \frac{k_{cat}}{22}\frac{P_{0}}{mc}[Z_{1}] + \frac{k_{cat}}{22}\frac{P_{0}}{mc}[Z_{2}] - k_{1}[X_{2}][T_{3}] + \frac{k_{1}}{a_{2}}[X_{2}T_{3}]\notag \\ \frac{d[T_{2}]}{dt} &= k_{1}[X_{2}][T_{2}] - \frac{k_{1}}{a_{2}}[X_{2}T_{2}^{3}] + k_{1}[X_{2}][T_{2}] - \frac{k_{1}}{a_{2}}[X_{2}T_{2}^{5}]\notag \\ \frac{d[X_{2}T_{2}^{5}]}{dt} &= k_{1}[X_{2}][T_{2}] - \frac{k_{1}}{a_{2}}[X_{2}T_{2}^{5}] - k_{1}[X_{2}][X_{2}T_{2}^{5}] + \frac{k_{1}}{a_{2}}[X_{2}T_{2}] - \frac{k_{cat}}{22}\frac{P_{0}}{mc}[X_{2}T_{2}^{5}]\notag \\ \frac{d[X_{2}T_{2}^{3}]}{dt} &= k_{1}[X_{2}][T_{2}] - \frac{k_{1}}{a_{2}}[X_{2}T_{2}^{3}] - k_{1}[X_{2}][X_{2}T_{2}^{3}] + \frac{k_{1}}{a_{2}}[X_{2}T_{2}]\notag \\ \frac{d[X_{2}T_{2}]}{dt} &= k_{1}[X_{2}][X_{2}T_{2}^{5}] - \frac{k_{1}}{a_{2}}[X_{2}T_{2}] + k_{1}[X_{2}][X_{2}T_{2}^{3}] - \frac{k_{1}}{a_{2}}[X_{2}T_{2}] - \frac{k_{cat}}{4}\frac{P_{0}}{mc}[X_{2}T_{2}]\notag \\ \frac{d[Z_{2}]}{dt} &= k_{2}[N][W_{2}] - \frac{k_{cat}}{22}\frac{P_{0}}{mc}[Z_{2}] + \frac{k_{cat}}{4}\frac{P_{0}}{mc}[X_{2}T_{2}]\notag \\ \frac{d[W_{2}]}{dt} &= \frac{k_{cat}}{26}\frac{P_{0}}{mc}[X_{2}T_{2}^{5}] - k_{2}[N][W_{2}] + \frac{k_{cat}}{22}\frac{P_{0}}{mc}[Z_{2}]\notag \\ \frac{d[T_{3}]}{dt} &= -k_{1}[X_{2}][T_{3}] + \frac{k_{1}}{a_{2}}[X_{2}T_{3}]\notag \\ \frac{d[X_{2}T_{3}]}{dt} &= k_{1}[X_{2}][T_{3}] - \frac{k_{1}}{a_{2}}[X_{2}T_{3}] - \frac{k_{cat}}{26}\frac{P_{0}}{mc}[X_{2}T_{3}]\notag \\ \frac{d[W_{4}]}{dt} &= \frac{k_{cat}}{26}\frac{P_{0}}{mc}[X_{2}T_{3}] \end{align}

Equation 3. The equations of TWJ-SDA > EXPAR.

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Table 4. The parameters of TWJ-SDA > EXPAR.

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Figure 13. The model of TWJ-SDA > 2step-SDA.

\begin{align} \frac{d[C]}{dt} &= - k_{1} [T_{1}] [C] + \frac{k_{1}}{a_{1}} [CT_{1}] - k_{1} [C] [X_{2}T_{1}] + \frac{k_{1}}{a_{4}} [X_{2}CT_{1}] \notag\\ \frac{d[T_{1}]}{dt} &= - k_{1} [T_{1}] [C] + \frac{k_{1}}{a_{1}} [CT_{1}] - k_{1}[T_{1}][X_{2}] + \frac{k_{1}}{a_{3}} [X_{2}T_{1}] \notag\\ \frac{d[CT_{1}]}{dt} &= k_{1} [C] [T_{1}] - \frac{k_{1}}{a_{1}} [CT_{1}] - k_{1} [CT_{1}] [X_{2}] + \frac{k_{1}}{a_{2}} [X_{2}CT_{1}] - \frac{k_{cat}}{28} \frac{P_{0}}{mc} [CT_{1}] \notag\\ \frac{d[Z_{1}]}{dt} &= k_{2} [N] [W_{1}] - \frac{k_{cat}}{22} \frac{P_{0}}{mc} [Z_{1}] + \frac{k_{cat}}{6} \frac{P_{0}}{mc} [X_{2}CT_{1}] \notag\\ \frac{d[W_{1}]}{dt} &= \frac{k_{cat}}{28} \frac{P_{0}}{mc} [CT_{1}] - k_{2} [N] [W_{1}] + \frac{k_{cat}}{22} \frac{P_{0}}{mc} [Z_{1}] \notag\\ \frac{d[X_{2}CT_{1}]}{dt} &= k_{1} [CT_{1}] [X_{2}] - \frac{k_{1}}{a_{2}} [X_{2}CT_{1}] - \frac{k_{cat}}{6} \frac{P_{0}}{mc} [X_{2}CT_{1}] + k_{1} [C] [X_{2}T_{1}] - \frac{k_{1}}{a_{4}} [X_{2}CT_{1}] \notag\\ \frac{d[X_{2}T_{1}]}{dt} &= - \frac{k_{1}}{a_{3}} [X_{2}T_{1}] + k_{1} [T_{1}] [X_{2}] - k_{1} [C] [X_{2}T_{1}] + \frac{k_{1}}{a_{4}} [X_{2}CT_{1}] \notag\\ \frac{d[X_{2}]}{dt} &= \frac{k_{cat}}{22} \frac{P_{0}}{mc} [Z_{1}] - k_{1} [T_{1}] [X_{2}] + \frac{k_{1}}{a_{3}} [X_{2}T_{1}] - k_{1} [CT_{1}] [X_{2}] + \frac{k_{1}}{a_{2}} [X_{2}CT_{1}] + \frac{k_{1}}{a_{5}} [X_{2}T_{2}] \notag\\ & - k_{1} [X_{2}] [T_{2}] - k_{1} [X_{2}] [X_{3}T_{2}] + \frac{k_{1}}{a_{8}} [X_{3}X_{2}T_{2}] \notag\\ \frac{d[T_{2}]}{dt} &= - k_{1} [T_{2}] [X_{2}] + \frac{k_{1}}{a_{5}} [X_{2}T_{2}] - k_{1} [T_{2}] [X_{3}] + \frac{k_{1}}{a_{7}} [X_{3}T_{2}] \notag\\ \frac{d[X_{2}T_{2}]}{dt} &= k_{1} [X_{2}] [T_{2}] - \frac{k_{1}}{a_{5}} [X_{2}T_{2}] - \frac{k_{cat}}{26} \frac{P_{0}}{mc} [X_{2}T_{2}] - k_{1} [X_{2}T_{2}] [X_{3}] + \frac{k_{1}}{a_{6}} [X_{3}X_{2}T_{2}] \notag\\ \frac{d[Z_{2}]}{dt} &= k_{2} [N] [W_{2}] - \frac{k_{cat}}{24} \frac{P_{0}}{mc} [Z_{2}] + \frac{k_{cat}}{2} \frac{P_{0}}{mc} [X_{3}X_{2}T_{2}] \notag\\ \frac{d[W_{2}]}{dt} &= \frac{k_{cat}}{26} \frac{P_{0}}{mc} [X_{2}T_{2}] - k_{2} [N] [W_{2}] + \frac{k_{cat}}{24} \frac{P_{0}}{mc} [Z_{2}] \notag\\ \frac{d[X_{3}X_{2}T_{2}]}{dt} &= k_{1} [X_{2}T_{2}] [X_{3}] - \frac{k_{1}}{a_{6}} [X_{3}X_{2}T_{2}] - \frac{k_{cat}}{2} \frac{P_{0}}{mc} [X_{3}X_{2}T_{2}] + k_{1} [X_{2}] [X_{3}T_{2}] - \frac{k_{1}}{a_{8}} [X_{3}X_{2}T_{2}] \notag\\ \frac{d[X_{3}T_{2}]}{dt} &= k_{1} [T_{2}] [X_{3}] - \frac{k_{1}}{a_{7}} [X_{3}T_{2}] - k_{1} [X_{2}] [X_{3}T_{2}] + \frac{k_{1}}{a_{8}} [X_{3}X_{2}T_{2}] \notag\\ \frac{d[X_{3}]}{dt} &= \frac{k_{cat}}{24} \frac{P_{0}}{mc} [Z_{2}] - k_{1} [T_{2}] [X_{3}] + \frac{k_{1}}{a_{7}} [X_{3}T_{2}] - k_{1} [X_{2}T_{2}] [X_{3}] + \frac{k_{1}}{a_{6}} [X_{3}X_{2}T_{2}] + \frac{k_{1}}{a_{9}} [X_{3}T_{3}] \notag\\ & - k_{1} [X_{3}] [T_{3}] - k_{1} [X_{3}] [X_{4}T_{3}] + \frac{k_{1}}{a_{12}} [X_{4}X_{3}T_{3}] \notag\\ \frac{d[T_{3}]}{dt} &= - k_{1} [T_{3}] [X_{3}] + \frac{k_{1}}{a_{9}} [X_{3}T_{3}] - k_{1} [T_{3}] [X_{4}] + \frac{k_{1}}{a_{11}} [X_{4}T_{3}] \notag\\ \frac{d[X_{3}T_{3}]}{dt} &= k_{1} [X_{3}] [T_{3}] - \frac{k_{1}}{a_{9}} [X_{3}T_{3}] - \frac{k_{cat}}{26} \frac{P_{0}}{mc} [X_{3}T_{3}] - k_{1} [X_{3}T_{3}] [X_{4}] + \frac{k_{1}}{a_{10}} [X_{4}X_{3}T_{3}] \notag\\ \frac{d[Z_{3}]}{dt} &= k_{2} [N] [W_{3}] - \frac{k_{cat}}{24} \frac{P_{0}}{mc} [Z_{3}] + \frac{k_{cat}}{2} \frac{P_{0}}{mc} [X_{4}X_{3}T_{3}] \notag\\ \frac{d[W_{3}]}{dt} &= \frac{k_{cat}}{26} \frac{P_{0}}{mc} [X_{3}T_{3}] - k_{2} [N] [W_{3}] + \frac{k_{cat}}{24} \frac{P_{0}}{mc} [Z_{3}] \notag\\ \frac{d[X_{4}X_{3}T_{3}]}{dt} &= k_{1} [X_{3}T_{3}] [X_{4}] - \frac{k_{1}}{a_{10}} [X_{4}X_{3}T_{3}] - \frac{k_{cat}}{2} \frac{P_{0}}{mc} [X_{4}X_{3}T_{3}] + k_{1} [X_{3}] [X_{4}T_{3}] - \frac{k_{1}}{a_{12}} [X_{4}X_{3}T_{3}] \notag\\ \frac{d[X_{4}T_{3}]}{dt} &= k_{1} [T_{3}] [X_{4}] - \frac{k_{1}}{a_{11}} [X_{4}T_{3}] - k_{1} [X_{3}] [X_{4}T_{3}] + \frac{k_{1}}{a_{12}} [X_{4}X_{3}T_{3}] \notag\\ \frac{d[X_{4}]}{dt} &= \frac{k_{cat}}{24} \frac{P_{0}}{mc} [Z_{3}] - k_{1} [T_{3}] [X_{4}] + \frac{k_{1}}{a_{11}} [X_{4}T_{3}] - k_{1} [X_{3}T_{3}] [X_{4}] + \frac{k_{1}}{a_{10}} [X_{4}X_{3}T_{3}] \notag \end{align}

Equation 4. The equations of TWJ-SDA > 2step-SDA

table5

Table 5. The parameters of TWJ-SDA > 2step-SDA

hogehoge

Figure 14. The model of TWJ-SDA > 3step-SDA.

\begin{align} \frac{d[C]}{dt} &= - k_{1} [T_{1}] [C] + \frac{k_{1}}{a_{1}} [CT_{1}] - k_{1} [C] [X_{2}T_{1}] + \frac{k_{1}}{a_{1}} [X_{2}CT_{1}]\notag \\ \frac{d[T_{1}]}{dt} &= - k_{1} [T_{1}] [C] + \frac{k_{1}}{a_{1}} [CT_{1}] - k_{1} [T_{1}] [X_{2}] + \frac{k_{1}}{a_{2}} [X_{2}T_{1}]\notag \\ \frac{d[CT_{1}]}{dt} &= k_{1} [C] [T_{1}] - \frac{k_{1}}{a_{1}} [CT_{1}] - k_{1} [CT_{1}] [X_{2}] + \frac{k_{1}}{a_{2}} [X_{2}CT_{1}] - \frac{k_{cat}P}{28 mc} [CT_{1}]\notag \\ \frac{d[X_{1}]}{dt} &= k_{2N} [W_{1}] - \frac{k_{cat}P}{22 mc} [X_{1}] + \frac{k_{cat}P}{6 mc} [X_{2}CT_{1}]\notag \\ \frac{d[W_{1}]}{dt} &= \frac{k_{cat}P}{28 mc} [CT_{1}] - k_{2N} [W_{1}] + \frac{k_{cat}P}{22 mc} [X_{1}]\notag \\ \frac{d[X_{2}CT_{1}]}{dt} &= k_{1} [CT_{1}] [X_{2}] - \frac{k_{1}}{a_{2}} [X_{2}CT_{1}] - \frac{k_{cat}P}{6 mc} [X_{2}CT_{1}] + k_{1} [C] [X_{2}T_{1}] - \frac{k_{1}}{a_{1}} [X_{2}CT_{1}]\notag \\ \frac{d[X_{2}T_{1}]}{dt} &= - \frac{k_{1}}{a_{2}} [X_{2}T_{1}] + k_{1} [T_{1}] [X_{2}] - k_{1} [C] [X_{2}T_{1}] + \frac{k_{1}}{a_{1}} [X_{2}CT_{1}]\notag \\ \frac{d[X_{2}]}{dt} &= \frac{k_{cat}P}{22 mc} [X_{1}] - k_{1} [T_{1}] [X_{2}] + \frac{k_{1}}{a_{2}} [X_{2}T_{1}] - k_{1} [CT_{1}] [X_{2}] + \frac{k_{1}}{a_{2}} [X_{2}CT_{1}]\notag \\ &\quad + \frac{k_{1}}{a_{2}} [X_{2}T_{2}] - k_{1} [X_{2}] [T_{2}] - k_{1} [X_{2}] [X_{3}T_{2}] + \frac{k_{1}}{a_{2}} [X_{3}X_{2}T_{2}]\notag \\ \frac{d[T_{2}]}{dt} &= - k_{1} [T_{2}] [X_{2}] + \frac{k_{1}}{a_{2}} [X_{2}T_{2}] - k_{1} [T_{2}] [X_{3}] + \frac{k_{1}}{a_{6}} [X_{3}T_{2}]\notag \\ \frac{d[X_{2}T_{2}]}{dt} &= k_{1} [X_{2}] [T_{2}] - \frac{k_{1}}{a_{2}} [X_{2}T_{2}] - \frac{k_{cat}P}{26 mc} [X_{2}T_{2}] - k_{1} [X_{2}T_{2}] [X_{3}] + \frac{k_{1}}{a_{3}} [X_{3}X_{2}T_{2}]\notag \\ \frac{d[X_{3}]}{dt} &= k_{2N} [W_{2}] - \frac{k_{cat}P}{24 mc} [X_{3}] + \frac{k_{cat}P}{2 mc} [X_{3}X_{2}T_{2}]\notag \\ \frac{d[W_{2}]}{dt} &= \frac{k_{cat}P}{26 mc} [X_{2}T_{2}] - k_{2N} [W_{2}] + \frac{k_{cat}P}{24 mc} [X_{3}]\notag \\ \frac{d[X_{3}X_{2}T_{2}]}{dt} &= k_{1} [X_{2}T_{2}] [X_{3}] - \frac{k_{1}}{a_{3}} [X_{3}X_{2}T_{2}] - \frac{k_{cat}P}{2 mc} [X_{3}X_{2}T_{2}] + k_{1} [X_{2}] [X_{3}T_{2}] - \frac{k_{1}}{a_{2}} [X_{3}X_{2}T_{2}]\notag \\ \frac{d[X_{3}T_{2}]}{dt} &= k_{1} [T_{2}] [X_{3}] - \frac{k_{1}}{a_{3}} [X_{3}T_{2}] - k_{1} [X_{2}] [X_{3}T_{2}] + \frac{k_{1}}{a_{2}} [X_{3}X_{2}T_{2}]\notag \\ \frac{d[X_{3}]}{dt} &= \frac{k_{cat}P}{24 mc} [X_{3}] - k_{1} [T_{2}] [X_{3}] + \frac{k_{1}}{a_{3}} [X_{3}T_{2}] - k_{1} [X_{2}T_{2}] [X_{3}] + \frac{k_{1}}{a_{3}} [X_{3}X_{2}T_{2}]\notag \\ &\quad + \frac{k_{1}}{a_{3}} [X_{3}T_{3}] - k_{1} [X_{3}] [T_{3}] - k_{1} [X_{3}] [X_{4}T_{3}] + \frac{k_{1}}{a_{3}} [X_{4}X_{3}T_{3}]\notag \\ \frac{d[T_{3}]}{dt} &= - k_{1} [T_{3}] [X_{3}] + \frac{k_{1}}{a_{3}} [X_{3}T_{3}] - k_{1} [T_{3}] [X_{4}] + \frac{k_{1}}{a_{4}} [X_{4}T_{3}]\notag \\ \frac{d[X_{3}T_{3}]}{dt} &= k_{1} [X_{3}] [T_{3}] - \frac{k_{1}}{a_{3}} [X_{3}T_{3}] - \frac{k_{cat}P}{26 mc} [X_{3}T_{3}] - k_{1} [X_{3}T_{3}] [X_{4}] + \frac{k_{1}}{a_{4}} [X_{4}X_{3}T_{3}]\notag \\ \frac{d[X_{4}]}{dt} &= k_{2N} [W_{3}] - \frac{k_{cat}P}{24 mc} [X_{4}] + \frac{k_{cat}P}{2 mc} [X_{4}X_{3}T_{3}]\notag \\ \frac{d[W_{3}]}{dt} &= \frac{k_{cat}P}{26 mc} [X_{3}T_{3}] - k_{2N} [W_{3}] + \frac{k_{cat}P}{24 mc} [X_{4}]\notag \\ \frac{d[X_{4}X_{3}T_{3}]}{dt} &= k_{1} [X_{3}T_{3}] [X_{4}] - \frac{k_{1}}{a_{4}} [X_{4}X_{3}T_{3}] - \frac{k_{cat}P}{2 mc} [X_{4}X_{3}T_{3}] + k_{1} [X_{3}] [X_{4}T_{3}] - \frac{k_{1}}{a_{3}} [X_{4}X_{3}T_{3}]\notag \\ \frac{d[X_{4}T_{3}]}{dt} &= k_{1} [X_{4}] [T_{3}] - \frac{k_{1}}{a_{4}} [X_{4}T_{3}] - k_{1} [X_{3}] [X_{4}T_{3}] + \frac{k_{1}}{a_{3}} [X_{4}X_{3}T_{3}]\notag \\ \end{align}

Equation 6. The equations of TWJ-SDA > 3step-SDA.

hogehoge

Table 6. The parameters of TWJ-SDA > 3step-SDA

The amplificataion optimization was based on these ODE models. Firstly, for each amplification system, a lot of simulations were conducted using constructed ODE models with several initial template concentrations. The initial template concentrations (the initial \(T_{n})\) concentration described as \((T_{n0})\) were proposed by Wet, assuming amplification would occur in the experiment, with \(T_{10}\) set to 50 pM, 500 pM, and 5 nM. The ratio of the initial template concentrations in consecutive reactions varied from 1 to 5 in increments of 0.5. Those simulations in which the initial concentration exceeded 1µM were excluded because it was predicted that amplification reactions would be hindered by crowding in the reaction environment. The reason why we used the concentration ratio as the parameter was that the initial template concentration of one reaction was expected to be greater than that of the previous reaction, and this could be satisfied by setting the concentration ratio to 1 or more. Then, the two criteria were established that must be achieved at the very least. The first was the amplification efficiency. More specifically, the concentration of dsDNA containing the PAM sequence after 40 minutes, which is considered as the amplification time when used as a device, must be at least 2.5 nM, the concentration required for Cas activation. For more information about the result of Cas activation, see Results.

The second is the ratio of P/N ratio. Here, the simulation with a target concentration of 100 fM was considered Positive, and one with 1 fM was considered Negative. The criterion was set so that the ratio of the concentration of dsDNA containing the PAM sequence after 40 minutes at 100 fM to that at 1 fM would be at least 50, which is predicted to be the minimum requirement for quantification in lateral flow assay (LFA).

Using the simulations of several initial template concentrations from each amplification system that met these two criteria, the amplification systems were compared. Two aspects of robustness were used as the comparison criteria. The first was the robustness of PAM-containing dsDNA. By comparing the ratio of the dsDNA concentration containing the PAM sequence at 35 minutes to that at 40 minutes, the robustness of the PAM sequence to time was compared across systems, and the robustness to polymerase was compared by analysing the ratio of the dsDNA concentration containing the PAM sequence after 40 minutes to that when the \(k_{cat}P\), was reduced to 80 %, as well as the robustness of the PAM sequence to nickase when \(k_{2}\) was reduced to 80 %.

The second was the robustness of P/N ratio. By comparing P/N ratio at 35 minutes to that at 40 minutes, the robustness of the P/N ratio sequence to time was compared across systems, and the robustness to polymerase was compared by analysing P/N ratio after 40 minutes to that when the polymerase activity, represented by \(k_{cat}P\), was reduced to 80 %, as well as the robustness of the P/N ratio to nickase when \(k_{2}\) was reduced to 80 %.

Finally, in the selected optimal amplification system, considering the inhibition of amplification due to crowding in the reaction environment, which could not be accounted for in these ODE models, the optimal initial template concentration was chosen to minimise the total initial template concentration.

Results

First, the histogram of the concentrations of dsDNA including the PAM sequence at 40 minutes in TWJ-SDA > EXPAR at multiple initial template concentrations was as follows (Figure 15.). The values located to the right of the red vertical line indicating 2.5 nM met the criteria for amplification efficiency.

hogehoge

Regarding the initial template conditions of simulations that met the criteria for amplification efficiency, the histogram of P/N ratio was as follows (Figure 16.). The values located to the right of the red vertical line indicating 50 met the criteria for the P/N ratio.

hogehoge

In TWJ-SDA > EXPAR, there were 22 patterns of initial template concentration sets that met the criteria for both amplification efficiency and P/N ratio. Next, the histogram of the concentrations of dsDNA including the PAM sequence in TWJ-SDA > 2step-SDA at various initial template concentrations was shown below (Figure 17.). Since there were no values located to the right of the red vertical line indicating 2.5 nM, none met the criteria for amplification efficiency, suggesting that TWJ-SDA > 2step-SDA was unsuitable as an amplification system in this project.

hogehoge

The histogram of the concentrations of dsDNA including the PAM sequence in TWJ-SDA > 3step-SDA at various initial template concentrations was shown below (Figure 18.). The values located to the right of the red vertical line indicating 2.5 nM meet the criteria for amplification efficiency.

hogehoge

Regarding the initial template conditions of simulations that met the criteria for amplification efficiency, the histogram of P/N ratio was as follows (Figure 19.). The values located to the right of the red vertical line indicating 50 met the criteria for the P/N ratio.

hogehoge

In the TWJ-SDA > 3step-SDA system, there were 995 patterns of initial template concentration sets that met the criteria for amplification efficiency and P/N ratio. Based on these results, the robustness of TWJ-SDA > EXPAR and TWJ-SDA > 3step-SDA were compared in terms of time and enzyme. The histograms of the six robustness values at initial template concentrations meeting the criteria for amplification efficiency and P/N ratio in each amplification system are shown below (Figure 20. - Figure 31.).

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The closer these robustness values are to 1, the more robust the system is considered to be. In all six values, it was believed that TWJ-SDA > 3step-SDA demonstrated greater robustness. Therefore, TWJ-SDA > 3step-SDA was proposed as the optimal amplification system for this project.

Next, we determined the initial template concentrations in TWJ-SDA > 3step-SDA that satisfied the criteria for amplification efficiency and P/N ratio while minimising the sum of the initial template concentrations. The optimal initial template concentrations are as follows: T10 = 5 nM, T20 = 15 nM, T30 = 45 nM, T40 = 135 nM, and T50 = 473 nM.

Conclusion

The purpose of this section was to estimate the optimal amplification system and initial template concentrations through simulations using the ODE model. Based on the perspectives of amplification efficiency, P/N ratio, and robustness to time and enzymes, it was proposed to perform amplification under the conditions of T10 = 5 nM, T20 = 15 nM, T30 = 45 nM, T40 = 135 nM, and T50 = 473 nM using TWJ-SDA > 3step-SDA.

Assumption


In constructing the ODE model for amplification reactions, the following assumptions were made:

  • The binding and dissociation of nucleic acids, as well as the nicking by nickase, follow the law of mass action.
  • The extension reaction by polymerase follows the Michaelis-Menten equation.
  • The enzyme activity remains constant within the same reaction.
  • The binding constants follow the equilibrium concentration ratios of reactants and products predicted by NUPACK.
  • The reaction rate constants of nucleic acids, the turnover number of polymerase, and the Michaelis constant are equal across EXPAR, SDA, and TWJ.
  • In the case of a single amplification reaction, the impact of the amplification product attaching to the 3' end of the template can be ignored.
  • In EXPAR, the fluorescence generated by SYBR-Green I primarily derives from the complete double strand W and Z, and additionally from excess X.
  • In EXPAR, one minute has elapsed from the actual start of the reaction to the fluorescence measurement.
  • The enzyme activity decreases as the temperature deviates from the optimal range.
  • The inhibition of reactions due to the congestion of the reaction system is ignored.

References


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