Diffusion of the MPs in the bulk liquid

To calculate the characteristic time in the bulk solution, i.e. the time necessary for a MP to reach a cellular interface, the diffusion coefficient in the bulk (D0) first has to be determined. Subsequently the characteristic time of this phase must be calculated, by applying the mean-square displacement of a molecule in the medium (MSD).
The diffusion coefficient can simply be determined by considering the DNA mobility in water14 according to Eq. 1:
\(D_{0}=4.9x10^{-6}\left[\text{bp}\ \text{size}\right]^{-0.72}\)(Eq. 1)
where D0 represents the diffusion and bp size represents the number of nucleotides present in the DNA. This equation is valid for DNA strands ranging from 21 to 6000 bp but was developed for double stranded DNA in water. Despite the lack of studies regarding the relationship between single stranded DNA and its size, it has been demonstrated that a rhodamine-labelled 30 bp single-stranded DNA has a diffusion coefficient of 6.2x10-11m2/s in a water solution15. This value is in the same order of magnitude as a of 9.34x10-11 and 3.44x10-11m2/s calculated with Eq. 1, for a 10 bp and 40 bp MP, respectively.
It is also important to bear in mind that the hybridization solution is more viscous than water, due to the presence of dextran sulphate16, which may affect diffusion by decreasing the motion of the MP. As such, the diffusion coefficients for 10bp and 40 bp DNA-MPs were also calculated using the viscosity values of two 10% (wt/vol) dextran-hybridization solutions: one containing a low molecular weight dextran of 10 kDa (with a viscosity of 2 mPa.s at 30 ºC)17 and another containing 500 kDa dextran (with a viscosity of 57.2 mPa.s, at 25 ºC)16 (supplementary material). The dextran of 500 kDa is used in particular cases to enhance the probe hybridization rate by artificially increasing the MP’s concentration due to the reduction of the available physical space16,18,19. Since Eq. 1 does not take into consideration physicochemical parameters such as the viscosity, the diffusion coefficient for a 500 kDa hybridization solution was calculated using the Stokes-Einstein (SE) equation adapted for non-spherical DNA-MP molecules (Supplementary material). This value was used to calculate the characteristic time of the MP’s diffusion in a highly viscous hybridization solution and to modulate the MP’s diffusion in an extreme scenario. In the case of the hybridization solution containing dextran sulphate of 10 kDa, Eq. 1 provides similar values compared to the SE equation. The molecular weight of the dextran used in the hybridization solution can be a limiting factor for diffusion in FISH. Particularly for gram-positive bacteria, low-viscosity solutions have shown to provide better results in terms of fluorescent signal16.
To determine the MSD and calculate the characteristic time, the following three-dimensional equation was applied:
< r(n)2 > = 6D0t (Eq. 2)
Where r is the average displacement of a molecule over time, D0 is the diffusion coefficient, as calculated above, and t is the time interval of the simulation20. For simplicity, Eq. 1 was used to calculate the characteristic time of diffusion of the MPs in a hybridization solution containing 10kDa dextran.
To calculate the MSD, it was assumed that the MPs and the cells are homogeneously distributed in the bulk solution. That way, the total volume of the bulk was divided by the number of cells present in the solution, assuming that each cell occupies a defined geometric volume (Fig. 2).
In the case of non-spherical cells, the distance between each cell, equally distributed in the medium, is dependent on its size and orientation. As such, in here, the different layers of the cells were individually characterized, using representative organisms – E. coli for gram-negative bacteria, B. subtilis for gram-positive bacteria and HeLa for animal cells. Rod-shaped cells can be considered as spherocylinders with radius r and total length L, while spherical cells are modelled as spheres with radius r (as further demonstrated in Table 1). Bacterial cells often present a rod-shape and, in here, Hela cells were considered spherical. Since the cell-size of the models selected for gram-positive and –negative bacteria is similar, an average L and r were used instead (Table 1).
In the bulk solution, the size of the different cells influences the diffusional time. Animal cells are approximately 20 times larger than bacterial cells, and thus the probability of a MP to encounter a cell in the same bulk volume is higher, resulting in a faster characteristic time. In addition, it is important to notice that this means that for less concentrated samples the characteristic time associated with the diffusion in the bulk medium will also increase (Fig. 3).
By combining the diffusion coefficients calculated using Eq. 1 with Eq. 2, the characteristic diffusion times of MPs of different sizes and in different cells were determined. In the bulk solution, a 10 bp MP takes around 5.55 s to encounter a bacterial cell and 1.84 s to encounter an animal cell, while a 40 bp MP takes 15 s and 5 s, respectively. Moreover, the diffusion coefficients calculated for the 10 and 40 bp MPs in the bulk solution (using Eq. 1) were respectively 9.3x10-11 m2/s and 3.4x10-11 m2/s, which are similar to the ones determined by Robertson et al. 21 for similar-length linear DNA molecules in buffer solutions (specifically, 9.7x10-11 m2/s for an 11 bp-DNA molecule and 4.37x10-11 m2/s for a 45 bp-DNA molecule).