Reaction of MP with RNA molecules available at the reaction site

The reaction between a nucleic acid sequence and their target RNA site is limited either by the diffusion to the site or the chemical reaction of both molecules. When the reaction is the limiting step, the time taken for it to occur is longer than the diffusional time (treaction >> tdiffusion). When the opposite happens, the diffusion is the limiting step, which means that the 2 molecules immediately react at their first encounter. The pathway that the MPs undergo in this step is basically translated in the velocity of the two processes. The DNA-MP hybridization reaction kinetics, where a single-stranded DNA MP species (A) binds to a target RNA (B) to form a duplex (AB), can be expressed as48:
\({A+B\ }_{\overset{\leftarrow}{k2\ \ \ }}\text{\ AB}\)
And the reaction rate of AB formation is given by
\(\frac{d[AB]}{\text{dt}}=k_{1}\left[A\right]\left[B\right]-k_{2}[AB]\)(Eq. 4)
where [A], and [B] are the concentrations of the reactants A and B, [AB] is the concentration of the product AB, k1is the association reaction constant of second order kinetics (M-1s-1), and k2 is the dissociation reaction constant of first order (s-1). The total concentration of the two reactants is given by:
A0 = [A] + [AB] (Eq. 5)
and
B0 = [B] + [AB] (Eq. 6)
Many hybridization reactions are carried out with excess of MPs to maximize the fluorescence signal and the reaction speed. As demonstrated before, the MPs are in excess compared to the available RNA molecules for hybridization, such that A0>> B0 and reaction kinetics becomes a pseudo first order. The reaction rate equation for pseudo-first order kinetics can be expressed as
\(\frac{d[AB]}{\text{dt}}=k_{1}A_{0}(B_{0}-[AB])-k_{2}[AB]\)(Eq. 7)
At the initial conditions, [AB] = 0 and integrating the above first order equation it is possible to obtain:
\(\left[\text{AB}\right]=B_{0}\frac{A_{0}}{K+A_{0}}(1-e^{-\left(K+A_{0}\right)k_{1}t})\)(Eq. 8)
where K denotes the equilibrium dissociation constant defined as k2/k148. The characteristic time scale of the reaction was calculated using Eq. 8, knowing the association and dissociation constant values available in the literature and assuming a complete hybridization, i.e. all the available RNA molecules are hybridized with the MPs. In this setting, [B] = 0, and [AB] = [B0]. At t=0, [A] = 2.0x10-7 M and [B] = 1.03x10-6 M (4x106 ribosomes) was considered for HeLa cells, whereas [B] = 3.54x10-5 M (3x104ribosomes) was considered for E. coli cells49. For simplification. it was assumed that there was a homogeneous distribution of probes and RNA in all cells.
Regarding the kinetic parameters, no values are available for the specific MP sizes chosen for this study, to the best of the authors’ knowledge. Instead, a k1 = 2.9x106M-1s-1 and a k2= 0.3 s-1 for a 16 bp DNA probe51 was used, which are values for HeLa cells. Using these values and Eq. 8, it was possible to obtain the characteristic time of the reaction, which is approximately of 0.02 s for HeLa cells, and 9.8x10-4 s for E. coli cells.