## 3 Results and Discussion  
### FRET Theory  
In every FRET method, a donor fluorophore (D) initially absorbs the energy due to the excitation of incident light and transfer the excitation energy to a nearby chromophore, the acceptor (A). Energy transfer manifests itself through a decrease or quenching of the donor fluorescence and a reduction of excited state lifetime accompanied also by an increase in acceptor fluorescence intensity [19].  
There are four criteria that must be satisfied in order for FRET to occur. These are:  
* The fluorescence emission spectrum of the donor molecule must overlap the absorption acceptor molecule.
* Donor and acceptor molecules must be in close proximity to one another (typically 1-10 nm).
* The transition dipole orientations of the donor and acceptor must be approximately parallel to each other.
* The fluorescence lifetime of the donor molecule must be of sufficient duration to allow the FRET to occur.  
The non-radiative energy transfer (\(k_{T}\)) from donor to acceptor is given by:  
\[k_{T}(r)\ =\ \frac{1}{\tau_{D}}(\frac{R_{0}}{r})^{6} \tag{1}\]  
where (\(r\)) is the distance between the donor and acceptor and \(R_{0}\) is the Forster distance, the distance at which the resonance energy transfer is 50% efficient.  
The FRET efficiency (E) can be estimated from the Equation (1) using the FRET distance (\(R_{0}\), the distance (\(r\)) between the donor and acceptor, and the ratio of the acceptor/donor (\(n\)):  
\[E\ =\ \frac{nR_{0}^{6}}{r^{6}+n^{6}} \tag{2}\]  
The FRET efficiency (E) decreases exponentially by the sixth power of the distance. As the distance between the donor and the acceptor is longer than \(2R_{0}\), the FRET efficiency reaches zero. So, the washing step is not necessary because unhybridized reporter probes are too far from the QDs to affect the FRET signal. The FRET distance (\(R_{0}\)) can be estimated from Equation (2) with the quantum yield (QY) of QDs and the spectral overlap between the emission wavelength of the donor and the absorption wavelength of the acceptor as I as in Equation (3),  
\[R_{0}^{6}\ =\ \left[\frac{900(\ln 10)k^{2}\varnothing_{D}}{128\pi^{5}Nn^{4}} \right]\int_{0}^{\infty}F_{D}(\lambda)\varepsilon_{A}(\lambda)\lambda^{4}d\lambda \tag{3}\]  
where \(F_{D}\) is the fluorescence intensity of the donor without presence acceptor; \(\varepsilon_{A}\) is the extinction coefficient of the acceptor (in M\({}^{-1}\) cm\({}^{-1}\)); \(\lambda\) is the wavelength (in nm); \(\Phi_{D}\) is the fluorescence QDs of the donor in the absence of acceptor; \(n\) is the refractive index of the medium; \(k^{2}\) is the orientation factor between donor (D) and acceptor (A) (2/3 for randomly oriented dipoles); and \(N\) is the Avogadro's number. The integral part of Equation (3) is known as the spectral overlap integral \(J\) (\(\lambda\)) and is given by:  
\[J(\lambda)\ =\ \int_{0}^{\infty}F_{D}(\lambda)\varepsilon_{A}(\lambda)\lambda^{4}d\lambda \tag{4}\]  
The energy transfer efficiency can be expressed as:  
\[E\ =\ \frac{\tau_{D}k_{T}}{1+\tau_{D}k_{T}}\ =\ 1-\frac{-F_{DA}}{F_{D}}\ =\ 1- \frac{DA}{D} \tag{5}\]  
where \(F_{DA}\), and \(\tau_{\rm DA}\) are the relative fluorescence intensity and fluorescence lifetime of the donor in the presence of the acceptor, respectively, and \(\tau_{\rm D}\) is fluorescence lifetime of the donor without the presence of the acceptor.