Introduction Our Assumptions The Model The Amplification The Binding Equilibrium An aside on the efficiency of RET Modelling Fluorescent Activity Discussion Parameters:

Introduction

With the hope of optimising the BRET RCA reaction mixture composition, we modelled the kinetics involved with the assay. This would allow us to tweak protein, probe and luciferin concentrations, in order to either boost the signal itself, or reduce the relative noise from the nano luciferase. We were able to generate a model for the BRET system, and would have used further optimization to test assay composition, had we not failed to purify the BRET proteins, preventing us from even conducting the assay. We were however still able to determine the optimal wavelength and time to sample the reaction mixture.

Our Assumptions

The active RCA probe is a limiting reagent (luciferin and ZFs are all in excess) and the initial rate of reaction is what we are modelling.
We assume that, despite the unbound RCA product being produced continuously, the bound and unbound forms are immediately at equilibrium, due to the very low dissociation constants.
The steady state approximation was used to characterise the enzyme kinetics of NanoLuc (NLuc)
We assume a maximum rate - all RCA probes are bound to miR399f, as are the phi 29 polymerases, at time = 0.

The Model

Almost all data used was found in literature, and sources can be found at the bottom of this page - the only exceptions are distances between functional groups, determined via alphafold, for which the files are attached.
RCA (Rolling circle amplification) involves the continuous amplification of a circular single stranded DNA probe, by phi29 polymerase. This results in the continuous synthesis of the complement to the circular probe.

The Amplification

We begin modelling with the rate of RCA production in nanomoles per second: K = 200, and is the maximum number of RCA product motifs (per probe), given the number of oligos. The polymerase produces 2280 nucleotides per minute, the RCA probe is 69 nucleotides long.

$$\begin{align} \frac{\mathrm{d} \mathit{RCA\ Product}}{\mathrm{d}t} = [\mathit{RCA\ Probe}] \times (K - [\mathit{RCA\ Product}]) \times 2280 \div (60 \times 69 \times K) \end{align}$$

The Binding Equilibrium

The very low dissociation constant strongly favours the bound form of the zinc finger protein conjugates in the following systems of equilibrium

$$\begin{align} \frac{[\mathit{Free\ RCA\ product}][\mathit{NLuc}]}{[\mathit{RCA\ product\ with\ NLuc}]}=2nM \end{align}$$

Assuming they are in equilibrium and all given concentrations are in nM →[Free RCA product][NLuc]=2[RCA product with NLuc]

Likewise

$$\begin{align} [\mathit{RCA\ product\ with\ mNeon}][\mathit{NLuc}] &=2[\mathit{Fully\ bound\ RCA\ product}]\\ [\mathit{Free\ RCA\ product}][\mathit{mNeon}] &=11[\mathit{RCA\ product\ with\ mNeon}]\\ [\mathit{RCA\ product\ with\ NLuc}][\mathit{mNeon}] &=11[\mathit{Fully\ bound\ RCA\ product}]\\ \end{align}$$

Overall, this gives a system of simultaneous equations as below, modelling binding and dissociation of the zinc finger proteins:

$$\begin{align} \frac{\mathrm{d} \mathit{RCA\ Product}}{\mathrm{d}t}&= [\mathit{RCA\ probe}] ([\mathit{RCA\ product}]-K) \times 2280 \div (60 \times 69 \times K) \\ [\mathit{Free RCA product}][\mathit{NLuc}]&=2[\mathit{RCA product with NLuc}] \\ [\mathit{RCA product with mNeon}][\mathit{NLuc}]&=2[\mathit{Fully bound RCA product}] \\ [\mathit{Free RCA product}][\mathit{mNeon}]&=11[\mathit{RCA product with mNeon}] \\ [\mathit{RCA product with NLuc}][\mathit{mNeon}]&=11[\mathit{Fully bound RCA product}] \\ \end{align}$$

Both the amplification and equilibrium combined give rise to the initial kinetic model of the concentrations of reagents in the BRET assay mixture.

An aside on the efficiency of RET

Let Eff be the efficiency of Nluc for excitation of mNeon green:
This is derived using Forster efficiencies, an equation given to determine the efficiency of RET, taking into account integral overlap of donor emission and acceptor excitation spectra, as well as the change in wavelength.

$$\begin{align} \mathit{Eff} = \mathit{Forster\ Efficiency} \times \mathit{quantum\ yield_\mathit{mNeon}} \end{align}$$

Modelling Fluorescent Activity

Here we use classic Michaelis-Menten kinetics to determine the rate of NLuc activity, where furizamine is the substrate, $K_{cat} = 6.6 \mathrm{s}^{-1}$ and $K_m = 360\mathrm{nM}$. The optimal detection wavelength and time for a given detection bandwidth was then determined, as was the signal to noise ratio for the detected wavelength.

Reactions for activity of the luciferase, and the interchange between active and inactive forms of the BRET system

$$\begin{align} \frac{\mathrm{d} \mathit{Furizamine}}{\mathrm{d}t} &= - \frac{6.6([\mathit{Free NLuc}]+[\mathit{RCA product with NLuc}]+[\mathit{Fully bound inactive RCA product}])}{360 + [\mathit{Furizamine}]} \\ \frac{\mathrm{d} \mathit{Fully\ bound\ inactive\ RCA\ product}}{\mathrm{d}t}&=\frac{1}{3.05 \times 10^9}[\mathit{Fully\ bound\ active\ RCA\ product}]- \frac{Eff\times6.6[\mathit{Fully\ bound\ inactive\ RCA\ product}]}{360 + [\mathit{Furizamine}]} + \frac{\mathrm{d} \mathit{Fully\ bound\ RCA\ product}}{\mathrm{d}t} \\ \frac{\mathrm{d} \mathit{Fully\ bound\ active\ RCA\ product}}{\mathrm{d}t} &= \frac{Eff \times 6.6[\mathit{Fully\ bound\ inactive\ RCA\ product}]}{360 + [\mathit{Furizamine}]} -\frac{1}{3.05 \times 10^9}[\mathit{Fully\ bound\ active\ RCA\ product}] \\ \end{align}$$

$$\begin{align} \mathit{mNeon\ fluorescence} &= \frac{Eff \times 6.6 \times [\mathit{Fully\ bound\ inactive\ RCA\ product}]}{360 + [\mathit{Furizamine}]} \times N_A \\ \mathit{NLuc\ fluorescence} &= \frac{\mathit{quantum\ yield}_\mathit{NLuc} \times 6.6([\mathit{Free\ NLuc}]+[\mathit{RCA\ Product\ with\ NLuc}]+(Eff-1)) \times [\mathit{Fully\ bound\ inactive\ RCA\ product}]}{360 + [\mathit{Furizamine}]} \times N_A \end{align}$$

The best mneon:nluc ratio is 79.17305934920131 with median wavelength 524.0 nm, with a range of wavelengths spanning 20 nms, at time 61.5 seconds.
The fraction of max mneon intensity is 0.9310502309840681. The Green/Blue ratio is 11.782125153370348.

These graphs demonstrate the effectiveness of BRET as a detection method, with significant signal to noise differences at the optimal sampling wavelength. Using the results from this model, we have determined that for RCA BRET assays with miR399f, it is best to sample at 61.5 seconds and 524nm. Unfortunately we had difficulties expressing the Nano Luciferase, however if we were to pursue RCA BRET further, next steps would involve using the model to optimise the reaction mixture concentrations.

Discussion

Integrating BRET and RCA kinetics is both novel and useful, allowing for optimisation of the signal amplification of the assay. Combining the chemistry of enzyme kinetics and dissociation equilibria, with the physics of RET poses a unique model with potential in other BRET or FRET systems. If we had successfully purified BRET proteins we would have used this model further, in order to optimise protein concentrations to minimise light noise from the luciferase. However, the actual kinetics are fairly simple, with somewhat unrealistic assumptions. This is especially the case for the assumption of the instantaneous equilibrium of bound and unbound RCA products. However this could hopefully be improved upon by other people modelling similar systems.

We hope that our model poses value to Other iGEM teams: past teams have modelled aspects of BRET (Eindhoven 2019 and Shanghai Tech 2023), yet the kinetics and rates associated with the assay remain uncharacterised. This shows that BRET is not an uncommon tool, yet there is no current kinetic model foundation for the system. Hopefully, despite being designed for RCA, other teams will be able to adjust the reactions and parameters to fit other BRET systems. Hopefully this would allow other teams to optimise assay timings and wavelengths in their iGEM projects. - or even go further and optimise other aspects of the assay as we had hoped to. ALl code has been commented and attached below.

Parameters:

Parameter	Value	Source
Rate of Phi 29	2280 nt/min	https://www.neb.com/en-gb/faqs/2011/01/20/at-what-rate-does-phi29-dna-polymerase-add-nucleotides-to-a-primed-single-stranded-template
Zif268 Kd (NLuc Conjugate)	2nM	https://www.sciencedirect.com/science/article/pii/S0021925820890851
AZP4 Kd (MNeon Green Conjugate)	11nM	https://pubs.acs.org/doi/10.1021/bi020095c
NLuc Kinetic Properties	Km = 360nM, 6.6Kcat s-1	https://www.nature.com/articles/ncomms13718#Sec28
Forster Distance for MNeon and Nluc	51 Angstroms	https://pmc.ncbi.nlm.nih.gov/articles/PMC7550199/#S1
RET distance	20nt (from spacer length) ~ 68 Angstroms 28.86 Angstroms	Green - NLuc Orange - MNeon Pink + Purple - ds DNA, the RCA amplicon and DNA oligo from Alphafold (See Below) https://drive.google.com/file/d/1ts5GcUWW20R-C0xXBOVUOrSfiIN-bNue/view?usp=drive_link
Nluc Quantum Yield	0.28	https://www.nature.com/articles/ncomms13718#Sec28
MNeon Quantum Yield	0.80	https://www.fpbase.org/protein/mneongreen/
NLuc Emission Spectrum		https://www.fpbase.org/protein/nanoluc/
MNeon Absorbance and Emission		https://www.fpbase.org/protein/mneongreen/