20

3

Regulation and Control

Fig. 3.1 Schematic diagram of immediate effects (cf. Sect. 3.3) for a regulatory mechanism. The

components are as follows: D, disturbance (from the environment); T, the hardware or the mech-

anism; R, the regulator; and E, the essential variables (output). Arrows represent communication

channels along which information passes; those with solid shafts must exist, and those with dashed

shafts may exist. See the text for further explanation

water bath. T then represents the electric heater and the bath itself with a circulator. E

represents the water temperature (measured with a thermometer)upper TT , R represents the

switch controlling the power supplied to the heater, and D represents the disturbances

from the environment. A typical event is the immersion of a flask containing liquid

at a temperature lower than that of the bath. Sophisticated baths may be able to sense

the temperature and mass of the flask before it has been immersed (channel D right arrow→

R), or at the moment of its placement (channel T right arrow→R), but typically the heater

is switched on if the temperature falls below the target value upper T 0T0 (channel E right arrow→R).

This is called regulation by error (Ashby 1950). Most living cells, as well as higher

organisms, appear to operate according to this principle.

The canonical representation of the thermostat is

down arrow StartLayout 1st Row 1st Column a 2nd Column b 2nd Row 1st Column a 2nd Column a EndLayout comma ↓^{a b}

a a ^,

(3.1)

where state aa represents upper T equals upper T 0T = T0 (within the allowed uncertainty) and state bb repre-

sents upper T less than upper T 0T < T0.

In the case of a bacterium, aa may, for example, represent [HgSuperscript 2 plus²⁺] = 0 (square

brackets denoting concentration) and bb may represent [HgSuperscript 2 plus Baseline right bracket greater than 0²⁺] > 0. D in Fig. 3.1

now corresponds to mercury ions in the environment of the cell, T corresponds to

the proteins able to sense mercury ions and the gene expression machinery able to

synthesize mercury reductase, and R corresponds to the transcription factor binding to

the mercury reductase gene promoter sequence. In stochastic matrix representation,

we have

StartLayout 1st Row 1st Column right arrow 2nd Column a 3rd Column b 2nd Row 1st Column a 2nd Column 1 3rd Column 0 3rd Row 1st Column b 2nd Column 1 3rd Column 0 EndLayout period

→a

b

a 1

0

b 1

0

.

(3.2)

More realistically, however, we might have