[From Bruce Abbott (941115.1830 EST)]
Rick Marken (941115.1320) --
Bruce Abbott (941115.1400 EST) --
Kahn: Explain it to them.
Was Kahn a reviewer of my Science paper?
(b) NutSave is NOT the reinforcer here, as I took pains to explain. To
repeat: Certain consequences of tumbling are NOT random. Tumbling
while moving up the nutrient gradient usually makes things worse;
tumbling while moving down the nutrient gradient usually makes things
better. Better = reinforcement, worse = punishment.
This is exactly what the Science reviewer said; fascinating. If this is the
way your program works then you have managed to overcome the laws of
probability.
[Enterprise attempts to raise its deflector shields, but these are useless in
the Motaran Nebula. Kahn fires a photon torpedo at the fleeing Enterprise...]
O.K., let's try geometry:
1. Best-case scenario (e. coli moving directly toward nutrient source):
Nutrient
>
dNut | / dNut
before Tumble | / after Tumble to e.g., 45 degree angle
···
*
dNut is the rate at which Nutrient levels are changing around e. coli as
it moves forward.
dNut is increasing at maximum possible rate before tumble.
dNut after tumble MUST be less than or equal to dNut before tumble.
Same logic holds for line directly away from nutrient source, in which
case dNut after tumble MUST be greater than or equal to dNut before
Tumble.
2. General-case scenario (e. coli moving toward source at an angle).
Nutrient
\ /
dNut before \ p / dNut after tumble
tumble \ /
* p marks region where dNut >= after tumble
dNut is increasing before tumble.
Probability, p, that dNut after tumble > dNut before tumble = angle/360
(here about 80 degrees), so p = 80/360 = 0.222.
For any angle < 90 degrees with respect to Nutrient source, any tumble
within + or - that angle with respect to nutrient source will produce an
increase in dNut. The probability that pNut will decrease is 1-p, which
in this case is 0.7777. The same logic holds for angles away from the
nutrient source.
3. Worst-case scenario (e. coli moving oblique to source).
Nutrient
dNut before tumble p dNut after tumble
-------- * --------
Any angle up to + or - angle before tumble will produce an increase in
dNut. Angle = to + or - angle before tumble will produce no change.
Angle greater than + or - angle before tumble will produce a decrease in
dNut. p = 180/360 = 0.50, 1-p = 0.50
Thus the only time p = 0.50 is when e. coli is traveling on a tangent to
a circle around the nutrient source.
Apparently the laws of probability, the ones that make p = 0.50 in all these
scenarios according to your calculations, are DIFFERENT in California than
they are here in Indiana....?
In fact, tumbling while moving up the nutrient gradient DOES NOT "usually
makes things worse" and tumbling while moving down the nutrient gradient
DOES NOT usually makes things better. I just ran a little simulation here on
my paper weight -- er.. Macintosh -- and I find that the probability of
things getting worse after a tumble when moving up the gradient is .503835;
the probability of things getting better after a tumble when moving up the
gradient is .495761; the probability of things getting better after a tumble
when moving down the gradient is .495046; the probability of things
getting worse after a tumble when moving down the gradient is .505351. The
calculated probabilities for each of these events is 0.5. My estimates are
based on only 5000 trials (don't want to burn up the ol' paper weight).
Must be some strange new way of defining "better" and "worse." Why don't you
give my way a try? Use:
Nutrient Rate of Change Before a Tumble - Nutrient Rate of Change After a
Tumble. Positive is better, negative is worse. Notice that Nutrient Rate of
Change is what is being compared, not Nutrient Level.
No Kahn job.
Cheers,
Bruce