The journey: from the bulk solution to the target RNA site

As stated above, the transport of chemical species within physical systems is ubiquitous on different engineering disciplines, such as chemical and biological engineering. Taking as an example the oxygen transport in bioreactors, it is possible to make an analogy with the transport of the MP from the bulk solution to the reaction site within the cells (Fig. 1).
In bioreactors, the oxygen is transferred from the gas bubbles through different interfaces, until it reaches the cells for uptake (Fig. 1A)9,10. This way, the first interfaces involve the transfer of oxygen from the gas bubbles to the liquid phase (gas-liquid interface), and the diffusion of oxygen through the bulk liquid (liquid-cell interface). The subsequent interfaces involve the cellular uptake of oxygen from the bulk to and within the cytoplasm to the reaction site. In this model, the variation of the dissolved oxygen’s concentration in time within the bioreactor depends on the oxygen transfer rate (OTR) and on the oxygen uptake rate (OUR).
Analogously to oxygen transfer, for the MP to reach its target, it first has to diffuse from the bulk solution, and cross the cellular membrane/envelope and the cytoplasm until it reaches the RNA site for hybridization. The diffusion process is based on a concentration gradient over time (Fig. 1B). It is important to notice that, in contrast to oxygen in aerobic bioreactors, the MPs are not continuously added to the system. Instead, the quantity of the MP applied is significantly higher than the number of cells/RNA available for the reaction, in order to ensure a high concentration gradient throughout the entire process. In fact, in a standard in vitro FISH procedure there are around 200 MPs per ribosome. It is important to notice that the steps regarding oxygen transfer from the interior of the bubble to the gas-liquid interface and then to the bulk (Fig. 1A, step 1-3) are not applicable in a probe diffusion scenario, as the MPs are already in suspension.
Additionally, in the oxygen transport models, a liquid film around the cells/solid particle is considered (Fig. 1A step 5 and Fig. 1B step 2). The film theory is the simplest premise for interfacial mass transfer, and assumes the existence of a stagnant film, also called unstirred layer, near every interface. It represents the resistance to mass transfer that might occur near an interface, and is almost always hypothetical since the motion of the fluid occurs even in a solid interface12. Due to the lack of available data for the MP’s concentrations in the interface, an overall process, where the cellular membrane/envelope is the first barrier for the MP’s diffusion from the bulk solution to the cytoplasm of the cells, can be considered instead (Fig. 1B, step 2).
The characteristic time is defined as an estimation of the order of magnitude of the time required for a process to reach equilibrium. In chemical engineering, it describes the dynamics of a system, for instance the residence time in well-stirred tanks13. In here, the characteristic time will give an estimation of the limiting step for diffusion and provide further insights on the circumstances under which FISH needs to be approached as a diffusional problem or a reaction kinetics problem.
In the following sections, the characteristic times associated with the transport through the bulk liquid, the diffusion across the cell membrane/envelope and in the cytoplasm, and, finally, the reaction of the MPs with their target RNA will be estimated for 10 and 40 base pairs (bp) DNA-MPs. The diffusion of the MPs through the cell membrane/envelope is further characterized taking into account the cellular differences between E. coli , Bacillus subtilis(B. subtilis ) and HeLa cells, as respective models of gram-negative bacteria, gram-positive bacteria and animal cells.