Penetration Depth in 3D Immunostaining (Part II)

Permeability, diffusibility, and penetration

Permeability usually refers to how accessible or reachable a deeper region is compared to a shallower region. It is usually related to the number of available paths for one to get from the tissue surface to its core, independent of the size of the travelling molecule. It usually scales with porosity. 

Diffusibility refers to how far a given molecule can move in a given environment without a driving force. It is governed by its size, charge, solvent viscosity, and temperature as per the Stokes-Einstein equation. The displacement due to diffusion is always less than the diffusion distance. For example, the displacement will be more restricted in a less permeable tissue matrix, such as in a polymer-crosslinked or embedded matrix, meaning that more distances must be travelled for a net displacement or penetration depth.

Penetration of staining refers to the maximal depth of specific staining achievable, and is usually correlated with how far the antibody can go into the tissue from the surface. Note that penetration depth should be quantified from the nearest surface, which should be quantified in 3D, and in most cases is not equal to a single 2D z-slice view and with quantification relative to the sliced tissue perimeter, which can lead to substantial overestimation of penetration depth.

What governs immunostaining penetration? 

Counterintuitively, the penetration depth of antiodies is not limited by its diffusivity but its reaction with fixed tissue antigens. This is because as antibodies enter the tissue, they get consumed by the fixed antigens, depleting the concentration gradients and hence limiting their penetration. Hence, penetration depth and uniformity decrease with

• a denser distribution of the tissue antigen,
• a larger the surface area of the tissue, 
• less antibody you add at one time,
• higher antibody affinity, and
• the use of polycloncal antibody versus monoclonal antibody. Because one antigen can trap multiple antibodies, the antibodies get depleted faster as they move across the tissue.

It has been shown that the diffusivity of whole IgG antibodies (~150 kDa in size) is around 4 times slower in a 4% acrylamide-embedded formalin-fixed tissue matrix than in water. As the dense hydrogel mesh can be expected to slow the diffusion displacement of macromolecules, a simply formalin-fixed (non-embedded) tissue will give faster IgG diffusivity. Quantitatively, a given distance needs <4x the time for 150kDa IgG to travel in a tissue than in water. However, an increase in penetration depth does not scale with an increase in incubation time during staining, suggesting that reaction is the critical component in limiting antibody penetration. A rigorous treatment on how different factors at play can be described by using the reaction-diffusion-advection model (https://www.sciencedirect.com/science/article/pii/S2667237523000772).