Biology Reference
In-Depth Information
If the cell is held vertically and contains a light, high-viscosity fluid, a heavy, low-viscosity
fluid can be layered on top with no need for injection ( Figure 26.2 b). In either case, the result
is the formation of branches by the low-viscosity fluid as it penetrates the high-viscosity one.
The model was described by the pioneering Swiss paleobotanist Jean-Jacques Scheuchzer
exactly three centuries ago, in an attempt to explain the existence of geological dendrites, 3
although the physical mechanism involved was not understood at the time, being analysed
in detail in the late 1950s by Saffman and Taylor. 4
The phenomenon can be described mathematically using differential equations, but can
also be understood intuitively. The low-viscosity fluid advances because it has internal pres-
sure (from the syringe, for example). Low-viscosity fluids distribute internal pressure quickly
and efficiently, so that pressure within the interface will be approximately equal whatever the
shape of the boundary. High-viscosity fluids move comparatively slowly, so ahead of an
advancing boundary they show a pressure gradient, squeezed between the advance of the
boundary and the reluctance of more distant high-viscosity fluid to get out of the way: where
the boundary is circular, this gradient will be symmetrical ( Figure 26.3 a). If the boundary
develops a small bulge through random fluctuations, the high-pressure zone is taken
forwards and there is a local steepening of the pressure gradient in the high-viscosity fluid
ahead of it ( Figure 26.3 b). The steeper the pressure gradient, the faster the high-viscosity fluid
will move out of the way and the faster the boundary will advance ( Figure 26.3 c). A small
bulge therefore advances as a finger. The flat sides of the finger have only a very shallow pres-
sure gradient leading away from them, so the finger is not inclined to broaden. Any bulges
that might develop on the side (again from random fluctuations) will themselves become
fingers, and a branching system is created.
Creation of more interface between the liquids is energetically expensive (surface tension).
This acts to limit the thinness of the tips of the fingers (very fine fingers would have a large
surface area:volume ratio). The shapes of the fingers are therefore a compromise between the
effects of pressure in the low-viscosity fluid (from the syringe), which tends to favour
numerous long, fine spikes, and surface tension that tends to favour fewer broader ones.
This can be explored practically by using the same liquids in identical Hele-Shaw cells,
and injecting the low-viscosity one with less or more enthusiasm. )
How likely is it that viscous fingering offers an explanation (rather than an analogy) for
branching morphogenesis of epithelial tubules? The following facts can be martialled to
support the idea. First, cell lines developed from branching epithelia (for example, mammary
cells and mIMCD3 urinary-collecting duct cells) show no evidence of forming branches in
simple two-dimensional culture or in liquid suspension, even when treated with ramogens,
but they will produce branching tubules when cultured in three-dimensional collagen
gels. 5,6,7 Given that they can be hydrolysed slowly by secreted enzymes, these essential
gels approximate to being very high-viscosity liquids. Second, there is the so-far unexplained
requirement for tension in some organs (at least in culture) if branching is to take place effi-
ciently. It was observed many years ago that rudiments of organs such as the kidney develop
much better if cultured at a medium air-interface so that surface tension presses them
outwards. Recent experiments have shown that specific reduction of surface tension (using
) Safety point for practical classes d glass and even Perspex Hele-Shaw plates can explode if students become
competitive about the idea of 'high pressure' (as they invariably do): issue eye protection.
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