[From Bill Powers (970809.1515 MDT)]

Just a (relatively) quick note while resting up for the Saturday night

banquet and the end of the meeting (which has been fine, as usual).

Richard Kennaway, 970806.1840 BST --

My absence came at an inopportune time with respect to untangling the

E. Coli discussion, especially my first abortive attempts to write the

equations for an E. coli model. I understand now why the reference signal

in my original model had no effect, and Rick Marken has come up with an

alternative to my model (which is actually the same as Bruce Abbott's PCT

model) that does have a reference signal with an effect, albeit a rather

unusual one.

The problem with my original model: the criterion for a tumble was that the

present error be greater than the previous error (squared, or absolute

value; both work). This means that only the first derivative of error was

considered, which is, of course, independent of the actual level of error

and therefore independent of the reference signal setting. This is why

changing the reference signal had no effect.

Rick proposed a much simpler model, in which the error operates the delay

mechanism directly. The delay mechanism (in both models) accumulates the

error signal into a dummy variable, and when that variable reaches a fixed

limit a tumble occurs and the variable is reset to zero. If L is the limit,

the time to the next tumble is just the integral of (L/e)*dt. Now, since we

are using the error signal directly, the reference setting does make a

difference.

We set up a simple model in which the gradient is uniform in the x

direction, increasing to the right. The rate of change of nutrient

concentration (the hypothesized controlled variable) is called n, and it is

compared against a reference concentration rate n', so that

e = n' - n.

This exploration is still in an early stage so we don't even have a gain

factor in the equation.

The time to the next tumble is L/e as described above. L can probably

absorb the missing gain constant.

When a tumble occurs, the new angle of travel (theta) relative to the

x-axis (which is also the direction of the gradient) produces a value of n

equal to

n = nmax*cos(theta),

where nmax is the product of the gradient and the speed of swimming.

Thus the time to the next tumble becomes

tt = L/(n' - nmax*cos(theta))

where theta is the current angle of travel.

Some precautions have to be taken to prevent tt from becoming actually

infinite (dividing by zero) or negative (impossible). We're still

exploring, so we haven't seen all the effects yet.

The average value of n, nbar, can be computed as

nbar = sum(n*tt)/sum(tt)

where the sum is taken over a long enough time that all angles of travel

are equally represented. The ratio nbar/nmax (with speed and gradient

properly included) would give the rate of progress up the gradient as a

fraction of the maximum possible rate. For present purposes we're only

interested in the achieved rate of change of concentration.

If we plot tt radially in polar coordinates as a function of theta, we can

see the bias on the distribution of directions. It is greatest when the

reference signal is set to nmax (or a smidgen higher). This produces a long

narrow ellipse pointed toward the positive x axis. At this setting of the

reference signal, the actual rate of change of concentration is equal to

the reference signal.

Raising the reference signal n' above this optimum value means that the

ellipse becomes less eccentric, and the achieved value of n falls below n'.

With n' less than the optimum value, the effects of cutting off negative

delay times is seen, and again the achieved value of n falls -- but not as

fast as the reference signal falls.

This is where the analysis stands now. Obviously there are more

explorations of relationships that need to be done. The response of the

system to changes in the reference signal is not like that of a standard

control system; one can't set the reference signal to any desired value,

and have n be kept equal to it. There is an optimum setting of the

reference signal, equal to the maximum experienced rate of change of

concentration. As the reference signal is raised above the optimum value,

the loop gain declines.

One way the model could be changed would be to have a zero-error tumbling

delay that is nonzero, so that both positive and negative errors could be

accommodated. The data on E. coli taken by Koshland suggest that this would

be realistic: normal E. coli tumble at some baseline rate in a uniform

concentration.

Also, a more realistic model of the tumbling mechanism could be attempted.

As I see it, tumbling occurs when some but not all of the flagellae reverse

direction: this desynchronizes the spins of the flagellae. If the reversal

point is different for different flagellae, the common driving signal will

lead to a tumble when it falls close to zero. Furthermore, it is observed

that when the rate of change of concentration goes far negative, ALL the

flagellae reverse, and the bacterium actually backs up, with the flagellae

synchronized in the reverse direction of spin. The synchronization, I would

guess, is probably achieved mechanically, since the flagellae interlock to

form a single spiral when they are all spinning the same way in either

direction. This would allow us to give meaning to negative error signals

without having to introduce artificial constraints, and tumbling would be

automatic when the driving signal goes to zero.

I'm sorry about my method of successive approximations by which I arrive at

a system analysis -- I'm afraid it has led some of you on a wild goose

chase. I had hoped, I suppose, that others would start from scratch and do

their own analyses to see how they compared with mine, instead of taking my

first attempts as gospel and trying to understand them (or shoot them down).

Best,

Bill P.