eRCA & RCA Models

RCA Overview

In 2022, Lambert iGEM created a mathematical model for rolling circle amplification (RCA) (see Fig. 1). RCA can be summarized by the steps below (see RCA):

• Hybridization: The target microRNA (miRNA) hybridizes to a linear DNA padlock probe, bringing the ends closer together (Graugnard et al., 2010).
• Ligation: SplintR ligase circularizes the padlock probe (Jin et al., 2016; Lohman et al., 2013).
• Amplification: The miRNA acts as a primer for phi29 DNA polymerase (Esteban et al., 1993) to amplify the padlock probe sequence, thus synthesizing the rolling circle product (RCP) made of interspaced repeats of the miRNA sequence and the reporting mechanism sequence.
• Quantification: Linear probes are short DNA sequences tagged with either a fluorophore or quencher molecule. When the linear probes bind to the RCP, they are in close proximity, allowing the quencher to absorb the light emitted by the fluorophore. This results in a quantifiable decrease in fluorescence.

Figure 1. Diagram of the reactions of the RCA pathway in MATLAB Simbiology software

Last year, Lambert iGEM modeled the RCA to guide and validate our wet-lab results. To accomplish this task, we relied on the Mass Action Laws and Michaelis-Menten Kinetics to simulate each reaction in the pathway.

This year, as part of our ongoing commitment to enhancing the RCA model, we conducted a scientific literature review to source estimated parameters such as degradation values for rolling circle product (RCP) and the nicking endonuclease Nb.BbvCI, which is utilized in exponential RCA (Shen et al., 2019). We sourced a number of ‘k’ constants associated with the array of reversible reactions in our system. We then used equilibrium equations to derive the specific value of these reverse reactions. This process has notably improved results in our model simulations resulting in higher accuracy.

Development

To assist in interpreting and validating RCA reactions, Lambert iGEM incorporated a deterministic ordinary differential equations (ODE) model to simulate the reaction and correlate fluorescence caused by the RCP-fluorophore-quencher (RFQ) to the initial miRNA concentration.

Signaling Pathway

The team used MATLAB’s SimBiology software to diagram each reaction using the Law of Mass Action and Michaelis-Menten equations. The Mass Action equations are used to diagram the collision of reactants in a solution, which are needed for most reactants in our reaction pathways, while Michaelis-Menten equations are needed to properly diagram catalytic enzyme kinetics (Park, 2022; Zi, 2012). The pathway begins with the introduction of miRNA into the system and ends with the RFQ (see Fig. 2).

Figure 2. Diagram of the reactions of the Updated RCA pathway in MATLAB Simbiology software

Biochemical Reactions

The model contains a total of 3 biochemical reactions (see Fig. 2).

Initial Values of Species

In order to simulate our reactions, our team needed to input values to initialize each component in our model. The initial value of each species was obtained from wet lab protocols:

Parameters

Rate constants and degradation rates are required to input into our Mass Action Laws and Michaelis-Menten equations. These values were derived from literature or estimated if the values could not be found. The value of each parameter is shown below:

Ordinary Differential Equations

As mentioned previously, Lambert iGEM incorporated the Mass Action Kinetic Laws and Michaelis-Menten Equations to represent the reactions for the RCA model.

Results

Lambert iGEM’s 2022 RCA protocol requires samples to be incubated at a very specific temperature for eight hours. However, phi29-XT DNA polymerase, an optimized enzyme with improved thermostability and sensitivity, shortens this to a two-hour amplification time (New England Biolabs). Therefore, we simulated the updated RCA model containing reverse reaction values and updated parameters, such as phi29-XT. The comparison of the activity of both the original RCA (see Fig. 3) and revised RCA (see Fig. 4) models clearly demonstrates the larger quantity of DNA phi29-XT produces .

Figure 3. Relationship between miRNA concentrations ranging from 0 to 25 picomoles and RFU based on simultations completed with the original RCA Model

Figure 4. Relationship between miRNA concentrations ranging from 0 to 25 picomoles and RFU based on simultations completed with the updated RCA Model

eRCA Overview

This year, we also investigated exponential rolling circle amplification (eRCA), an adaptation of RCA that produces exponentially greater fluorescence per unit of miRNA, therefore improving the accuracy and sensitivity of the reaction. The eRCA process includes the addition of the nicking endonuclease Nb.BbvCI to cleave the continuously produced RCP at specifically placed nicking sites. This allows the miRNA sequence to be cut out of the RCP, bind to another padlock probe, and begin the amplification process again.

Development

In order to validate our wetlab experimentation, Lambert iGEM incorporated a deterministic ordinary differential equations (ODE) model to simulate the eRCA reaction (see Fig.4). Our model converted several reactions in the eRCA system into ODEs and created a loop that updates miRNA values to ensure the reaction continues until the initial amount of padlock probes are depleted. Lambert iGEM also reached out to Dr. Mark Styczynski from the Georgia Institute of Technology to aid in the identification of parameters and the steps for creating a model that functions exponentially rather than linearly.

Signaling Pathway

The team used MATLAB’s SimBiology software to diagram each reaction using the Law of Mass Action and Michaelis-Menten equations. The Mass Action equations are used to diagram the collision of reactants in a solution, which is needed for most reactants in our reaction pathways, while Michaelis-Menten equations are needed to properly diagram catalytic enzyme kinetics (Park, 2022; Zi, 2012). The pathway begins with the introduction of miRNA into the system and ends with the RFQ (see Fig. 5).

Figure 5. Diagram of the reactions of the eRCA pathway in MATLAB Simbiology software

Biochemical Reactions

The model contains a total of 3 biochemical reactions (see Fig. 5).

Initial Values of Species

In order to simulate our reactions, our team needed to input values to initialize each component in our model. The initial value of each species was obtained from wet lab protocols:

Parameters

Rate constants and degradation rates are required to input into our Mass Action Laws and Michaelis-Menten equations. These values were derived from literature or estimated if the values could not be found. The value of each parameter is shown below:

Ordinary Differential Equations

As mentioned previously, Lambert iGEM incorporated the Mass Action Kinetic Laws and Michaelis-Menten Equations to represent the reactions for the eRCA model.

Results

Our eRCA model provides an accurate mathematical representation of the reaction. According to the model, eRCA produces an exponentially larger decrease in fluorescence than RCA. The trend of decreasing fluorescence is much clearer in the eRCA model than in the RCA model. This makes it more reliable to derive accurate miRNA levels from the RFU values obtained from the eRCA model (See Fig. 6).

Figure 6. Relationship between miRNA concentrations ranging from 0 to 0.49 picomoles and RFU based on simultations completed with the eRCA Model

The model simulation has produced a graph that clearly illustrates an exponential relationship between RFU and miRNA concentration (see Fig. 6). This outcome serves to corroborate the notion that eRCA, which is based on the premise of an exponential increase in RCP concentration, can significantly enhance the accuracy of wetlab experiments (see RCA: eRCA). The observed exponential relationship further reinforces the concept that as RCP concentration increases exponentially, the margin of error in miRNA concentration detection decreases. Notably, the simulation results for miRNA concentrations ranging from 0 to 0.49 picomolar above provide compelling evidence of this decrease in error (see Fig. 6). In this range, the points on the graph are distinctly separated, leading to a remarkable reduction in measurement error indicating the effect of the model on wetlab accuracy.

Assumptions

Our RCA and eRCA ODE models assume all rate constants remain constant due to a lack of external environmental factors. Thus, this model replicates only the miRNA-padlock probe binding process to simulate RCP production and RQF. The model also assumes the total quantity of padlock probes, SplintR ligase, phi29 DNA polymerase, fluorophores, and quenchers is known initially and remains constant over time. Finally, DNA polymerase activity is expected to remain the same regardless of the primer, miRNA, or dNTP concentration.

Capillary Tube RCA Models

Capillary RCA Overview:

Lambert iGEM designed capillary rolling circle amplification (cpRCA), a frugal, capillary tube-based diagnostic for coronary artery disease (CAD). cpRCA works as an extension of exponential rolling circle amplification (eRCA) or rolling circle amplification (RCA) to serve as a time-effective diagnostic tool.

To corroborate the results from cpRCA, we created an updated RCA model that utilized the SYBRsafe-based reporting mechanism. The model used the biochemical equations from the RCA model while modifying specific parameters to fit the capillary tube setting, enabling us to map the initial miRNA concentration to the final amount of SYBRsafe fluorescence (See Fig. 7).

Figure 7. Diagram of the reactions of the cpRCA pathway in MATLAB Simbiology software

Results

The capillary model offers a precise simulation of the cpRCA pathway. When initialized with a 0.00166 picomoles of miRNA—a value determined through our wet lab experiments—the model accurately predicted a count of approximately 47 molecules for the R_Sybr complex, which represents the RCP-SYBR™ Safe interaction (See Fig. 8). This outcome substantiates the validity of our cpRCA protocols, as the simulated count of 47 molecules closely aligns with the expected value of around 50. Not only will this count likely reduce overlap that can negatively affect the miRNA count, but has been validated with our wetlab results, having a cpRCA reaction work at 50 molecules of miRNA (see RCA: outputs).

Figure 8. Simulation results of RCP-SYBR™ Safe complexes over 4 hours

DNA and miRNA Diffusion

To validate the functionality of cpRCA, we created a model to examine the diffusion of microRNA (miRNA) and rolling circle product (RCP) within the capillary tube which is crucial for the imaging process and formation of fluorescent dots indicating miRNA concentration. This model enabled us to simulate and analyze the diffusion and concentration of miRNA and RCP under different experimental conditions, providing insights into the correlation between diffusion patterns and fluorescent dot formation. Additionally, the model utilized various parameters and optimized the accuracy of the simulation by studying tube dimensions, reaction times, and the impact of miRNA/RCP concentrations on diffusion and fluorescent signals.

Development and Results

In order to simulate the diffusion of the miRNA and RCP within the capillary tube we utilized a random walk model in conjunction with the Stokes-Einstein law of diffusion.

As illustrated in Figure 9, these computational simulations provide insights into the dynamic behavior of miRNA and RCP within the capillary tube. The results confirm that significant diffusion of RCP is unlikely, and there is minimal overlap between the RCP and miRNA species. This validates the imaging step in the cpRCA pathway that relies on fluorescent dots for accurate detection and quantification of miRNA.

Figure 9. 20 minute simulation of miRNA and DNA positions inside a capillary tube

Limitations

The model uses the input parameters to provide a simulation of the diffusion tailored towards cpRCA. However, the model has many limitations and may not be an accurate representation of the diffusion of the species. The Stokes-Einstein equation’s assumptions pose limitations when applied to DNA and miRNA diffusion in a capillary tube. Specifically, DNA and miRNA are not spherical and can interact with other particles, violating the equation’s premises. Additionally, the capillary’s confined geometry can lead to wall interactions affecting diffusion, and biological fluids like blood are often non-Newtonian, altering viscosity dynamically. Consequently, while our results provide initial insights, they should be considered approximations that require refinements to account for these biological complexities.

References