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Hi Dave, Paul & others,
The many complicated formulations of how diffusion occurs are modelled
mainly on the formally similar process of heat conduction along rods
and through slabs etc, where industrial application needs have led to
more widespread analysis (eg. classic book by Carslaw & Jaeger). The
rate of supply will indeed vary according to whether the oxygen is
removed instantaneously at the far end, or allowed to build up, in the
long term though not initially. In terms of the evolutionary pressure
to branch further, I don't think this argument is valid. The reason is
that the rate of movement of diffusant along a channel or tube in the
direction of its length is independent of the width of the channel
until you get down to molecular dimensions (wall effects). If you
imagine that all the branched-off channels have the same diameter as
the original one, that would be similar to instantaneously widening
the pathway (increases cross-sectional area), which would have a
serious effect (the first part of the pathway would be limiting).
However, this is not how tracheae branch and sub-branch again, but
rather like the branches of a tree, with the deeper branches getting
finer and finer until they enter the muscles; I'm not sure if the
actual aggregate cross-section remains exactly constant, but it
probably approaches that. Most of the tracheal system has hydrophobic
walls which is partly why they don't fill with water when an insect is
immersed, but the tips of the final branches, tracheoles, are
hydrophylic and contain fluid where gas exchange occurs. When the
muscles are active, I think it was Wigglesworth who showed that this
fluid withdraws deeper into the muscle so the interface moves closer
to site of oxygen demand.
I cut a few corners in the earlier e-mail: some insects in some parts
of their bodies actively pump the tracheal system locally by
contracting nearby muscles, so this presumably aids movement of gases
along, beyond the constraints of simple diffusion. The most obvious
example though adapted for a different use would be that of the
Madagascar hissing cockroach, which when alarmed expels air rapidly
from the enlarged tracheae connected to one pair of spiracles. And in
addition to the series of segmental hearts, blood is also circulated
up long appendages like the antennae by little pumping stations
('accessory hearts') at their bases.
I assumed that the old story about sauropods in swamps was still
correct, but Paul sets this on its head. I agree that the likely
reason why the large carboniferous dragonflies could function is that
their bodies while very long were not all that wide, from memory of a
diorama somewhere illustrating this. The tracheae basically run across
the body segments from the sides, but in addition there are
longitudinal tubes that connect up the tracheae between segments in
most insects as well.
Interesting, in light of the earlier recommendation of Vincent
Dethier's beautifully written books (the comment on flight in
particular though I'd forgotten that he treated that), it has emerged
more recently that small insects don't fly by the same mechanism as
larger birds. From Michael Dickinson's work, air viscosity is really
important at small size, and small insects generate lift quite
differently by having air vortices roll off the wings. This raises
the interesting question of whether those large ancient dragonflies
would have used more conventional lift mechanisms as they were as
large or larger than many modern birds. I haven't followed this
recently but assume that people will now be looking at hummingbirds.
Dave, there's a lot of info about heart rate in different insect
species (tables in a book by J. C. Jones for instance), and I guess
that circulation rate goes up as size does, but am not sure if this is
well documented without further digging. You may know already that
fancy insect flight muscle doesn't work well when cold, which is
another likely reason for sunning, to increase its temperature. It is
known that this happens during warm-up when some moths vibrate their
wings for several minutes -- the temperature in flight muscle goes up
by several degrees, allowing them finally to take off.
Steve, Halifax
Quoting David & Alison Webster <dwebster@glinx.com>:
> ----- Original Message ----- From: "Steve Shaw" <srshaw@DAL.CA>
> To: <naturens@chebucto.ns.ca>
> Sent: Sunday, October 03, 2010 1:38 PM
> Subject: Re: [NatureNS] Wasp question (long)
>
>
>> Hi Paul, Derek, Andy, Dave and all,
> So if you
>> double the width of the insect, you double the length of the tube
>> pathway (*2), but the rate of movement of the gases O2 and CO2 will
>> drop to one quarter of what it had been before, at the tissue end
>> (1/(2 squared) = 1/4). This is believed to be one of the main factors
>> that limits the ultimate size of insects, such that at large size they
>> simply cannot supply O2 to the tissues fast enough by passive
>> diffusion. The most energetically expensive tissue known is insect
>> flight muscle, to give an idea of why this might be important.
>
> Hi Steve & All, Oct 7, 2010
> It is correct that if delivery of a gas by diffusion to the end
> of a tube (of uniform diameter) is J then delivery to the end of
> same diameter tube will be J/4, if tube length is doubled.
>
> And-- the above applies (I think) whether removal of gas from the
> sink end of the tube is at a very rapid rate relative to the rate of
> diffusion or at a slow rate; if removal is rapid then the
> concentration gradient will be steep and if slow then shallow.
>
> But if this dimensional effect were relevant, why over all these
> years, would not evolutionary pressures have developed sufficient
> branching in larger insects to compensate for this effect ?
>
> Unless I have missed something, the answer is quite simple. Most
> insects have evolved with flight somewhere in the background and if
> you are going to fly then small is beautiful. For example, fairly
> large Mayflies can almost float in thermals generated by a canoe or
> wharf at calm twilight. And smaller insects can ride the tiny
> thermals generated by shrubs or other localized heat sources.
>
> Also, unless transfer of O2 from the trachea to tissue fluids,
> followed by flow away from the trachea, is very efficient, and it is
> difficult to see how it could be*, then the really steep
> concentration gradient would be at the gas/liquid interface.
>
> *Based on O2 solubility in water at 20oC; 20% O2 in air would be in
> equilibrium with about 0.6% O2 by volume in water.
>
> Insects, especially rapid flie