August 26, 2009 – Herbert Sauro
Introduction to Negative Feedback Control
Probably the most famous modern device that employed negative feedback was the governor. Thomas Mead in 1787 took out a patent on a device that could regulate the speed of
windmill sails. His idea was to measure the speed of the mill by the centrifugal motion of
a revolving pendulum and use this to regulate the position of the sail. Very shortly afterwards in early 1788, James Watt is told of this device in a letter from his partner, Matthew
Boulton. Watt recognizes the utility of the governor as a device to regulate the new steam
engines that were rapidly becoming an important source of new power for the industrial
revolution. Figure 1 illustrates an engraving of a governor from a early book entitled ”An
Elementary Treatise on Stream and the Steam-engine by Clark and Sewell published in
1892.
Figure 1: Figure 1. A typical steam-engine governor.
The operation of the governor is simple, its purpose is to maintain the speed of a rotating
engine at a constant predetermined value in spite of changes in load and steam pressure.
The vertical axel of the governor is connected to the rotation of the steam engine. As the
steam engine, for one reason or another, speeds up, the rotation increases, thereby causing
the centrifugal pendulums to swing out. A linkage transmits this motion to the stream
valve in such a manner that the flow of steam is reduced thus slowing down the engine.
If the engine slows down too much, as a result of a sudden load, the flyweights will swing
back and the steam value is opened so that the steam engine can accelerate. The governor
was a highly successful device and it is estimated that by 1868, 75,000 governors where
in operation (A History of Control Engineering, 1800-1930 By Stuart Bennett, 1979).
This description of the governor illustrates the operational characteristics of negative
feedback. The output of the device, in this case the steam engine speed, is ”fed back” to
control the rate of steam entering the steam engine and thus influence the engine speed.
Although the governor is an example of one of the earliest negative feedback systems in
modern times, the concept actually goes back much further in history. There is documen-
tary evidence to show that the ancient Greeks were aware of the concept and used it in
a wide variety of ways to control different mechanisms. Probably the most famous of
these was the use of floats in water clocks to maintain a steady flow of water which could
be used to measure time. The earliest recorded water clock that used negative feedback
was described by Ktesibios who probably lived between 285 and 247 BC in Alexandria.
Further work was done by Philon and particularly Heron (13 AD) who left us with an
extensive book (Pneumatica) detailing many amusing water devices that employed negative feedback.
In more recent times negative feedback has been used extensively in the electronics industry to confer, among other things, electrical stability to electronic devices. In fact without negative feedback considerable swathes of modern technology would not be able to
function. Given the ubiquity of negative feedback in man-made devices it should therefore come as no surprise to discover that living systems employ feedback at many levels,
ranging from gene regulatory network, signaling, network, metabolic networks to neural
networks and hormonal control systems.
It is possible to do a simple analysis which illustrates some of the essential properties
conferred by negative feedback. We can represent a negative feedback system using the
following block diagram:
+
-
Figure 2: Block diagram illustrating negative feedback. yo is the output, yi is called the
reference or set point that the output, yo , must match. d is a disturbance acting on the
controller A. k represents the fraction of output yo returned to yi as feedback.
The block diagram shown above can be expressed in algebraic form:
yo = (A + d)(yi − kyo )
where it is assumed that the disturbance d adds to the controller. By rearrangement we
obtain:
yo =
yi (A + d)
kd + kA
If we assume that the gain in the controller, A, and the feedback, k are strong, that is
Ak 0, then the expression is simplified to:
yo =
yi
k
(1)
This equation highlights a number of effects, the first is that the controller A, and any
disturbances d are eliminated from the equation and that the output yo is a linear function
of the set point yi . The performance of the feedback is therefore dependent on the quality
of the feedback mechanism, k and is independent of either the controller or any disturbances. In relation to actual devices, such as a stream engine, this is a desirable property.
It means that the performance of the steam engine is independent of the load and any
component variation in the construction of the engine, the only requirement is that the
feedback mechanism is reliable.
Classical control theory has an extensive framework for analyzing feedback systems,
however the terminology and sometimes the methodology does not always translate easily to biological systems. In this section we will examine the use of control coefficients
and elasticities to understand the properties of negative feedback.
v1
X0
v2
S1
v3
S2
X1
Figure 3: Simple negative feedback loop. v1 , v2 and v3 are the reaction rates. S2 acts to
inhibit its own production by inhibition of v1 .
The pathway in Figure 3 shows a simple negative feedback, a common motif in many
metabolic pathways. Here we see a downstream species, S2 controlling the first step in
the pathway, v1 . Control is often achieved using allosteric enzymes which have distinct
binding sites for the controlling species that are separate from the main active site.
When comparing the block diagram to the biological pathway in Figure 3 it may not be
apparent how the two representations can be matched. It is however possible to pair each
component in the block diagram to an equivalent component in the biological pathway.
Thus the output, yo in the block diagram corresponds to the concentration of S2 . The
negative feedback component k corresponds to the interaction of S2 with the allosteric
enzyme in the first step. The set point, yi is more problematic but it is most likely embedded in the kinetic characteristics of the allosteric enzyme. Finally the controller A is
represented by the steps v1 and v2 . The load on the system is represented by the last step,
v3 and other disturbances can be assigned to v1 , v2 and the input concentration, Xo .
Just as we did earlier, we can derive the flux and concentration control coefficients in
terms of the elasticities. In the equations below we show the results with and without
feedback to illustrate the difference. Note that the feedback elasticity is given by: ε12 .
With Feedback
Without Feedback
CEJ 1 =
ε21 ε32
ε21 ε32 − ε11 ε32 + ε11 ε22 − ε21 ε12
CEJ 1 =
ε21 ε32
ε21 ε32 − ε11 ε32 + ε11 ε22
CEJ 2 =
−ε11 ε32
ε21 ε32 − ε11 ε32 + ε11 ε22 − ε21 ε12
CEJ 2 =
−ε11 ε32
ε21 ε32 − ε11 ε32 + ε11 ε22
CEJ 3
ε11 ε22 − ε21 ε12
= 2 3
ε1 ε2 − ε11 ε32 + ε11 ε22 − ε21 ε12
CEJ 3
ε11 ε22
= 2 3
ε1 ε2 − ε11 ε32 + ε11 ε22
We can see that the addition of feedback adds a new term to the denominator and to the
numerator for CEJ 3 . If we make the feedback elasticity, ε12 larger we can see that the demand flux control coefficient tends to unity. That is, all control moves out of the feedback
loop. In terms of the steam engine analogy, it is equivalent to being able to change the
demand on the steam engine without loss of power. This can be seen more clearly if we
look at the concentration control coefficients.
CEJ 3 → 1
CES23 → 0
In the list below we give the three control coefficients with respect to S2 . Looking at CES23
we see that as the feedback strength is increased (ε12 ) in magnitude, the control coefficient
tends to zero. This means that the feedback locks the concentration of S2 in to a very
narrow range in response to changes in demand.
CES21 =
ε21
ε21 ε32 − ε11 ε32 + ε11 ε22 − ε21 ε12
CES22 = −
ε11
ε21 ε32 − ε11 ε32 + ε11 ε22 − ε21 ε12
CES23 =
ε11 − ε21
ε21 ε32 − ε11 ε32 + ε11 ε22 − ε21 ε12
What about the set point? If we look at CES21 and again examine the limit as the strength of
the feedback increases, this control coefficient tends to:
CES21 = −
1
ε12
Although the term on the right has a negative sign, we should recall that the feedback
elasticity is itself negative so that overall the control coefficient is positive. By a slight
rearrangement we can also see:
δS2 = −
1
δE1
∂v1 /∂S2
This equation should be compared with equation 1. Assuming that the feedback elasticity
does not vary significantly, the set point is a function of the activity of the first step, E1
although it is weak.
In summary, we can see from this basic analysis that negative feedback confers stability
on the signal species, S2 in the face of either disturbances inside the loop or downstream
of the loop. In addition we see that the performance of the pathway is largely governed
by the value of the feedback elasticity and the set point by the activity of the first step.
Classically, allosteric enzymes have been considered as flux controllers. The analysis here
gives the opposite picture. Allosteric enzymes, when part of a negative feedback loop are
very poor controllers of flux but instead allow distal steps to be good controllers of the
flux. For demand driven systems this is a logical arraignment.