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).