Team:NYU Abu Dhabi/Model
Mathematical Modelling
As part of out project, we sought to understand the molecular reactions in more depth and thereby improve our results. We set out to model RPA, that has only been modeled once before. We found that the model in the given paper is insufficient and does not yield the generally correct responses. Accordingly, we built our model, but found that there is a lack of rate constants available for many of the reactions of binding happening during Recombinase Polymerase Amplification. We believe, however, that as our model helped us understand our project better, so it will provide researchers with a new way to look at modelling RPA and hopefully produce more reliable models.
Firstly, we identified the components of RPA, as supplied by both the foremost supplier, TwistDX, and other research papers. We have then researched and agreed upon a sequence of events that occurs in RPA based on the one used by the majority of researchers in the field. We must point out, however, that there is disagreement about the particularities of the individual binding reactions as well as to what complexities they should be modeled since many of the proteins involved are multimeric with their own respective association reactions. We felt that a general model could be agreed upon and be accepted and that is what we provide here. Realizing based on the equations that possible limiting reagents could be the SSB protein, and that decreasing its binding affinity could help rapidify the amplification.
Furthermore, what was of essential importance to our project was that the commonly identified constant stirring conditions necessary for the RPA model to hold place can be supported by our microfluidics. At the microfluidic level, the reagents have been lyophilized and put into close proximity to each other in the microfluidic channel. With the small amount of reagents used, this approach justifies the approximations made in the model, and we can say that the well-stirred state’s results are provided by the microfluidic system.
Before we continue with the mathematical modelling, some helpful synonyms and symbolics.
The following species were used throughout our modelling:
Species Name |
Symbol Used |
|---|---|
Recombinase |
\(R\) |
Primer |
\(Pr\) |
DNA |
\(D\) |
DNA Polymerase |
\(P\) |
Single Strand Binding Protein (SSB) |
\(S\) |
A specific binding of recombinase to a primer to form a closed complex |
\(R:Pr\) |
A specific binding of recombinase and primer to the DNA to form a closed complex |
\(R:Pr:D\) |
A specific binding of recombinase, primer, SSB protein to the DNA to form a closed complex |
\(R:Pr:D:S\) |
A specific binding of recombinase, primer, SSB protein, DNA Polymerase to the DNA to form a closed complex |
\(R:Pr:D:S:P\) |
A specific binding of primer, SSB protein, DNA Polymerase to the DNA to form a closed complex |
\(Pr:D:S:P\) |
The following rate constants were deduced from the reactions below the table. An important note we must raise is that most of the rate constants here identified have not been measured for empirical values. It is for this reason that our model could not have been used to provide simulations which is common practice. We think, though, that our identification of the errors in the only RPA model provided, as well as our proposal of a different view into the RPA world, from which we drew important conclusions as well as microfluidic deductions is more than enough to state the contribution of this model to our project.
The following parameters were used:
Parameter |
Variable Name |
|---|---|
Forward rate constant for the binding of Recombinase and Primer |
\(k1f\) |
Reverse rate constant for the binding of Recombinase and Primer |
\(k1r\) |
Forward rate constant for the binding of Recombinase, Primer to DNA |
\(k2f\) |
Reverse rate constant for the binding of Recombinase, Primer to DNA |
\(k2r\) |
Forward rate constant for the binding of Recombinase, Primer, DNA to SSB protein |
\(k3f\) |
Reverse rate constant for the binding of Recombinase, Primer, DNA to SSB protein |
\(k3r\) |
Forward rate constant for the binding of Recombinase, Primer, DNA, SSB protein to DNA Polymerase |
\(k4f\) |
Reverse rate constant for the binding of Recombinase, Primer, DNA, SSB protein to DNA Polymerase |
\(k4r\) |
Forward rate constant for the unbinding of Recombinase from the Recombinase, Primer, DNA, SSB protein, DNA Polymerase complex |
\(k5f\) |
Reverse rate constant for the unbinding of Recombinase from the Recombinase, Primer, DNA, SSB protein, DNA Polymerase complex |
\(k5r\) |
Rate constant for the replication of DNA with the complex consists of Primer, DNA, SSB Protein, DNA Polymerase |
\(kf\) |
$$ 1. R + Pr \leftrightharpoons R:Pr $$ $$ 2. R:Pr + D \leftrightharpoons R:Pr:D $$ $$ 3. R:Pr:D + S \leftrightharpoons R:Pr:D:S $$ $$ 4. R:Pr:D:S + P \leftrightharpoons R:Pr:D:S:P $$ $$ 5. R:Pr:D:S:P \leftrightharpoons R + Pr:D:S:P $$ $$ 6. Pr:D:S:P \to 2D + S + P + Pr + D $$
Based on the reaction model above, the following equations could be deduced:
