[From Bill Powers (970413.2221 MST)]
Bruce Abbott (970413.1805 EST)--
As the rate of nutrient increase goes from negative through zero to highly
positive, the intertumble time decreases proportionately. However, in "e.
coli-type reorganization," as I understand it, reorganization (the
equivalent of tumbling) ceases when there is no error in intrinsic
variables. (Reorganization is the output of a system or systems
monitoring long-term error in these variables.)
Many forms of the model are possible.
In the typical real E. coli, the model that would fit the best would have a
positive reference signal for the time rate of change of concentration of an
attractant. The error signal is converted to a delay in the next tumble
through a particular constant of proportionality: delay = mean delay +
k*error. With a small gain constant, the difference between maximum and
minimum delay would be small; the higher the gain, the greater the
difference would be. To fit the model to the actual behavior, both the
reference level and the gain constant would have to be adjusted, as well as
the mean value.
The real E. coli behavior covers a range that can be modeled by assuming
that different bacteria have different reference signals. Some bacteria are
reported to "always tumble," and others to "never tumble." There is a
distribution between these limits. However, in perfusion experiments,
Koshland showed that an "always-tumbling mutant" can be made to cease
tumbling by a rapid enough increase of concentration of the attractant in
the perfusing stream, while a "never-tumbling mutant" could be made to
tumble by a rapid enough _decrease_. The swimming speed of the bacterium is
not great enough, for such "mutants", to bring the rate of change of
concentration up to, or down to, the reference level, but in the perfusion
experiments much faster rates of change could be generated (the bacteria are
"tethered" in a gel).
The reason I put "mutants" in quotes is that this oddball behavior might be
explained by the existence of a higher level control system that varies the
reference signal for the tumbling system. If some bacteria are replete, the
reference level for the rate of change of concentration might be zero, or
even negative. Similarly, for very hungry bacteria, the reference level
might be set so high that swimming can never increase the rate of change
enough to slow the rate of tumbling. Of course the latter bacteria are
doomed, because if they always tumble they will not progress toward the
source of the nutrient except by a low-probability chance.
I don't know if this second level of control exists; one would have to
observe bacteria as they approach high concentrations of the nutrient, and
see if they cease to tumble (and of course, to swim -- both would have to
stop). Koshland reported no observations that would answer this question.
An incidental observation reported by Koshland. When the rate of change
greatly exceeds the normal level, the flagellae actually reverse their
direction of spin and the bacterium moves backward, at a lower speed. This
suggests that "tumbling" might simply be the result of the individual
flagellae reversing direction at different levels of the error signal near
zero, so there would be a range in which some are spinning forward and some
backward. Since there are six or seven flagellae, there can be various
combinations of forward and backward spin, leading to random-seeming changes
in orientation. It's only when all the flagellae are turning in the same
direction that they form a single spiral and generate straight-line
movement. One of the interesting aspects of the topology of flagellae is
that if they are all spinning in unison, they form a single spiral even
though the "motors" are located off-axis relative to each other. Hard to
visualise why they don't get tangled up! Of course if they're not all
spinning the same way, they can't form a single spiral. Some are pushing,
some are pulling, and a tumble results.
Best,
Bill P.