248
THE KINETICS OF GROWTH
BY J. GRAY, M.A.,
King's College, Cambridge.
(From the Zoological Laboratory, Cambridge.)
{Received 12th September 1928.)
(With Eight Text-figures.)
THE relationship between the increase in size of a growing organism and the period
of time occupied in the process largely depends upon the nature of the environment
in which the growing cells are placed. If a bacterium or a fibroblast is isolated in a
suitable culture medium, the cell will divide and grow at a constant rate as long as
the external conditions remain the same. If, on the other hand, the environment
is changed, the rate of growth also changes, being accelerated in some cases and
retarded in others. As far as is known, the rate of growth of isolated cells does not
change with the age of the culture but for any particular species remains fixed in
any given set of external conditions. In marked contrast to this is the behaviour
of cells occupying their normal position in the body of an animal. In this case, the
rate of growth of individual cells and of the whole population of cells forming the
organism exhibits a marked decline with the advancing age or size of the organism,
and this decline is, partially at any rate, independent of the external conditions of
the animal's environment. It is possible to imagine that if we were able to adjust
the medium surrounding isolated cells in vitro so that the chemical and physical
conditions were precisely the same as in the body, we would observe a course of
growth identical in quantity, if not in form, with that observed in vivo. Conversely, from a knowledge of those factors which influence growth in vitro we
ought to be able to deduce the nature and distribution of the factors which
control the growth of the whole body. Unfortunately we know very little of the
factors controlling growth in vivo and we cannot, as a rule, interfere with the
normal growth rate except by varying the amount and type of foodstuffs available for conversion into new tissue, or in certain cases by altering the temperature of the organism. It is quite clear that the organised growth of a whole
organism is controlled by a series of factors which have not yet been isolated, and
which play no obvious part when a homogeneous tissue is excised from the body
and grown in vitro. For example, the roles played by the "organiser" and by the
hypophysis are not yet fully understood, and as yet we are unable to reproduce in
vitro a type of biological environment for our homogeneous cultures in vitro which
approximates in any real way to the internal environment of an animal's body.
For this reason we tend to fall back on less direct methods of attack and endeavour
The kinetics of growth
249
to define the nature of the factors controlling the rate (if not the form) of normal
growth by a careful observation of the way in which the rate changes during the
life of a normal animal. The results of such enquiry are curious in that the highly
complex changes going on in the body appear to receive an adequate quantitative
expression by algebraic equations of a comparatively simple type. If the course of
growth in vivo can be expressed by a curve of a particular geometrical form and
by no other curve, it is obvious that we have established a fact of fundamental
importance, and can proceed at once to give quantitative expression to the factors
controlling the growth rate at every instance during the life of the growing organism
although we may be unable to isolate these factors as definite physical or chemical
entities. Since, at present, it is impossible to isolate these factors experimentally,
and since it is comparatively easy to construct an ordinary growth curve, it becomes
of immediate importance to know precisely how far the deductions drawn from this
curve are really valid, and how far they are fallacious.
Having constructed a curve of growth which shows as accurately as possible
the weight or size of an organism at different moments of time (care being taken
to maintain a constant external environment) it is possible to express the observed
data in a more compact form by means of an algebraic equation, and as long as this
equation is regarded simply as an empirical and convenient way of expressing
known facts, the procedure is often useful if not particularly instructive. It is,
however, important to go further than this and enquire what type of growth factors
could conceivably result in the existence of the observed equation of growth. In
other words, by differentiating the equation of a growth curve, an expression is
sought for the factors controlling the rate of growth at particular instants of time.
The algebraic and graphical treatments of growth data are liable to cause confusion of thought unless they are handled with considerable care, and it seems,
perhaps, worth while to consider certain elementary facts associated with these
methods before considering their immediate application.
THE VALIDITY OF GROWTH FORMULAE.
To express the results of a prolonged series of observations in the form of a
compact algebraic equation is always convenient and may be of importance. Thus
it is of value to have an equation by which the size of a particular type of horse or
cow can be calculated for any particular age. At the same time, the utility of the
formula must be assessed by the degree of accuracy given by its results; if the values
given by the formula are substantially different to those of direct observation the
formula is of limited value; if the errors of the equation are of the same order of
magnitude as the observational errors involved in collecting the data, the formula
is of correspondingly greater value. It is clear that the value of an empirical formula
depends solely on this relationship—or in other words upon its ability to reproduce
observed figures.
A second desideratum of an empirical formula is that it should hold good over
a maximum range of observations, or at any rate its range of utility should be
accurately defined. Fig. 1 illustrates this point. Let each point on the curve indicate
250
J.
GRAY
an experimental observation. The smooth curve drawn through all these points is
a curve calculated from the formula x = k Jt. If for experimental reasons it had
been necessary to confine observations to the period 24-36 hours, the observed
points within this period could have been joined by a line which is so nearly straight
that its total curvature would fall within the experimental errors of the observations.
Similarly, for the opening stages of the experiment, the points can be found by
It is clear that, of the three
an experimental curve of the type kt = loge a
a — so
equations, only one applies to the whole range of observations and therefore
8
I
10
1
I
12 14
I
16
!
1 1
18 . 20 22 24
I
26
I
28
I
30
I
32
I
I
34 36
Fig. i.
only this particular curve provides an adequate empirical formula. As long as
formulae are entirely empirical, we may simply select that particular one which is
most suitable for the range in which we are immediately interested, but as a rule
the wider the range of applicability the more useful is the formula.
Finally, it may be remembered that in an empirical formula of growth there are
only two symbols which have any real meaning, viz. the symbol denoting time and
the symbol denoting weight or size, all the others—which may be as numerous as
one likes—are all meaningless and can have values assigned to them which give
the best representation of the observed experimental data.
Let us now assume that, for a particular growth curve, the size of the organism
can be expressed within suitable limits of error, as a function of its age by means
of a particular equation. It does not follow that this equation is the only one which
The kinetics of growth
will fit the same facts, for as long as each observation is liable to error there must
be more than one curve which will express these facts within the required limits.
This is a matter of small importance as long as the desired equation is purely
empirical; naturally, if one equation fits the facts better than another we select
the former, but if both are equally good any selection is purely arbitrary. This is
not the place to consider methods of fitting equations to particular curves, but there
can be little doubt that more caution is required than is sometimes displayed in the
graphical analysis of biological growth. In the first place, the method necessarily
involves a known standard of experimental error since in any growth curve there
is no such thing as an observed point on a true curve. All observations are liable
to error and since the error in observing weight is usually greater than that when observing time, the observations of weight should strictly be recorded on a growth curve
as lines and not as points. The length of each recorded line should define the limits
of error of the particular observation. The shorter are the lines (i.e. the more accurate
the observations) the more strictly is it possible to define the growth curve, and
hence the greater is the accuracy with which it is possible to construct an empirical
formula.
If, as is very occasionally the case, the size of an organism (*), when plotted
against its age (t) results in the possibility of drawing a straight line through all the
observational lines, then without further investigation we can construct an empirical formula of the type x + c = kt, since the straightness of a line can be tested
by a suitable ruler. Most growth curves are, however, not of this nature. They are
quite clearly curves and the nature of their curvature cannot be tested by such
simple methods. For this reason, it is customary to re-arrange the facts before
plotting them, in the hope that one particular type of re-arrangement will result
in a straight line graph. If this procedure is adopted, it is essential to bear in mind
that any change in the units concerned may involve an important modification of
the degree of accuracy with which an equation can be tested by graphical methods.
The only safe rule is to make sure that the limits of experimental error characteristic of every observation are represented in all regions of the graph by a distance
which can readily be seen by the eye. This standard is by no means easy to maintain
if one or more of the axes of the graph represents the logarithm of a variable.
For example, let us suppose that during the course of growth we make the four observations shown in column 2 of Table I, and that each observation is correct to the nearest
whole number.
Table I.
Time
X
0
3
16
64
356
4096
Xj,
16
64
356
4150
If we plot the logarithm of x against the time, we get a perfectly straight line. Let us
suppose that the logarithmic scale is such that a point, 1 mm. above the point t — o,
J. GRAY
252
log# = 1-2041 is denoted by 1*25, then at this point x = 18. Now consider a similar
point above t = 8, log * = 3-6123, then log * = 3-6523 or x = 4595. In other words, at
the end of the graph 1 mm. of the logarithmic axis represents 250 times the value of the
same distance at the other end of the scale. Now suppose that a series of observations
from experiment give the values shown in column 3, Table I, and that once more the
figures are accurate to the nearest whole number. Plotting them on the same scale as
before, the last observation will be denoted by a point o-i mm. above the line joining the
other three points. To judge the significance of this by eye is quite impossible even if the
graph is constructed with the very greatest care. It is obvious that the errors of graphical
plotting are in this case much too great to bear any real relationship to experimental
errors.
Where a process of growth involves the use of a logarithmic function it is very
unwise to trust to graphical methods unless it can be shown that the ratio of probable
error to the mean is constant during the whole period of growth; it is safer to make
a direct comparison between observed figures and those calculated by means of a
suitable formula. Graphical methods may, of course, often be used as a guide to
finding a suitable formula.
0
t
0
t
Fig. 2. In these figures the observations of a particular experiment are plotted in duplicate,
and the two figures can be superimposed one on the other so that every point on one figure falls
exactly on the corresponding point on the other. It is obvious that only data of a very high
order of accuracy would enable us to assess the relative value of the two equations, although
it will be noted that one curve inflects when * = | a , while the other inflects when * = \a.
If an equation derived from or expressing a growth curve is to be utilised as a
guide to the fundamental processes controlling the rate of growth, a much more
rigorous test must be applied than is necessary for purely empirical representation.
It is now not sufficient to show that a particular equation results in a curve passing
{within a given distance of all the observed points, it is also necessary to show that
|a similar curve cannot be derived from any other type of equation. The extent to
which we can satisfy this condition obviously depends on how accurately we can
establish our data. If every observation were absolutely accurate there would only
be one curve to fit the facts, but the greater are the observational errors the larger
is the number of curves which will fit them. The question therefore arises how
The kinetics of growth
accurate can we make the experiments? Can they be such that the applicable
curves are all of essentially the same type? This cannot be determined by eye.
In Fig. 2 are drawn two sigmoid curves; as far as one can see they both run close
to all the points of one and the same series of observations; one or other curve
might be adequate for empirical purposes. The two curves are, however, essentially
different: one of them belongs to the type kt = log, - ( f ~ x°\ and inflects when
xo{a — x)
x = l-a; the other is represented by the equation \t = log6 a ~ ^ ° and inflects
a — \/xi
when #! = \a (approx.). These two curves will be considered later as they are both
conceivably related to metazoan growth1; at the moment, it should be noted that
only data of very high accuracy could possibly reduce the experimental error to
such small dimensions as to distinguish clearly between the two types.
As already pointed out, as long as an equation of growth is of a purely empirical
nature there can be as many meaningless constants as are convenient, and they
can be given any suitable values. If, however, an equation is based on concrete
conceptions and thereby acquires a real meaning, there should seldom be more
than one indeterminable constant, and all the others should have a real meaning
and be capable of independent measurement. An example illustrating the danger
of assigning arbitrary values to growth "constants" will be found in Appendix I
(p. 272).
With these general principles in mind, we may now survey some of the observed
data of animal growth and consider how far they can be used in framing a quantitative conception of the processes involved.
KINETICS OF UNORGANISED GROWTH.
Perhaps the most fundamental type of organic growth is that found in bacteria
in which each cell grows and reproduces itself at a constant speed as long as the
conditions in which it lives are maintained at a satisfactory and constant level.
Observations of this type of growth were made by Barber (1908), who isolated single
individuals of Bacillus coli from a rapidly growing culture and observed their rate
of reproduction at a constant temperature. Table II shows the time which elapsed
between one generation and another at 37-7° C.
Table II.
Generation time in mins.
18-7
19*3
20'3
2O'3
21-3
2O#O
31*3
19-4
19-8
Average: 20*0
For a considerable period of time therefore every Bacillus reproduced itself
every twenty minutes. Starting from a single bacillus it is obvious that, as long as
1
The first type is that utilised by Robertson, the second type expresses the postnatal growth
cycle of the rat (see p. 269).
17
254
J- GRAY
these conditions hold, the number of bacteria present at any particular multiple of
the generation time can be calculated by a process of compound interest as long
as no bacterium dies, or as long as a fixed percentage of the population dies per
unit of time. Since the rate of growth of each individual is the same, it follows
that the rate of growth of the whole culture will be proportional to the number
of bacteria (x) present at any particular moment
8*-**
Starting from one bacillus, when t = o, x = i
x = eM
(2).
It is important to notice that the succesive generations of Bacillus observed by
Barber were all derived from one original organism, and that as far as is known there
was no material difference in the average size of each cell during the course of the
experiment. It should also be noted that the constancy of the time elapsing between
successive generations was established by direct observation, and is not deduced
by calculations from the total number of cells produced in a given time.
Barber's results are of peculiar importance because they are derived form a
system which is truly comparable to those which exist in simple physico-chemical
reactions. Many of the phenomena of chemical dynamics receive adequate analysis
if the velocity of the reaction is assumed to be proportional to the active mass of
the reacting substances. If a reaction is such that a molecule of a substance A is
converted into other compounds none of which, interfere in any way with the
activity of the original substance, then the velocity of the reaction is proportional
to the number of molecules of A which are present.
If there are a molecules of A present at the beginning of the reaction and of
these x have been destroyed
_
• ,
O X
• ' .'-
f '
UOr
i
*=Tloge
*
a
'
;
•-
<\Of
i
!
.
-
> >J^-
•
,
>-<?/.
,
X.
(3)'
(4).
This equation is so frequently applied to the kinetics of growth that it is advisable to appreciate its full significance. It implies (i) that all the molecules of A
or a fixed percentage of them are taking part in the reaction during the whole
period of their life, (ii) that although the kinetic energy of individual molecules may
vary, yet the population as a whole does not change in tnis respect, but has a constant reaction velocity k. For example, at all times during the reaction the molecules
of A are all disposed on the same variability curve when they are grouped according
to degrees of kinetic activity (Fig. 3). In other words, although they are not all alike,
yet collectively the heterogeneity of the population does not change as the reaction
proceeds; the mean value for the population remains constant, and therefore the
constant k in equation (3) is justified. Now exactly the same conditions hold good
The kinetics of growth
255
in Barber's experiment. Every bacterium is growing, and although the generation
time varies from 18-7-21-2 minutes, there is good reason to believe that the average
of 20-0 holds good throughout the whole experiment. At the same time the change
in the number of bacteria present is only comparable to the change in the number
of the units in a chemical system within certain limits and under certain circumstances.
The customary unit of growth in a bacterial population is one bacterium, and
if all the bacteria present were to divide synchronously, the curve relating time to
number of bacteria present would consist of a series of steps, although the points
denoting the number of bacteria present at successive periods of twenty minutes
would all lie on a smooth logarithmic curve
§
03
I
Kinetic energy
Fig. 3-
The apparent discontinuity of growth of such a population is due to the unit
which has been chosen to denote growth—namely an individual bacterium, and
such a population does not conform to the rule that at any instant all the bacteria
or a fixed percentage of them are taking part in the production of a new individual.
On the contrary, there will be periods when none of the bacteria are dividing, whilst
every twenty minutes there would be moments at which each bacterium has just
produced another separate individual. If, however, we define growth as an increase
in the amount of living material, it is obvious that growth is a continuous process
which must be measured not in terms of individual bacteria but in terms of smaller
units within individual cells, and the velocity at which these units change in number
cannot be measured in terms of individual cells except in purely arbitrary and
discontinuous units—viz. an amount of new tissue equal to and present as one
complete bacterium.
As long as one complete new bacterium is manufactured by each member of a
population in twenty minutes and all are dividing synchronously, the number
present at the end of each complete period of twenty minutes can, as shown above,
be expressed by a logarithmic curve comparable to that of a first order chemical
reaction, but the actual velocity of growth (when this is defined as the rate of increase
17-2
256
J. GRAY
of living matter) may follow a law of quite a different nature—bearing no obvious
relationship to a logarithmic system. In a simple first order chemical reaction the
units involved are of the smallest possible dimensions since only complete molecules
can take part in the reaction, and until comparable units are available for the observation of growth we cannot say how far growth is controlled by a first order
reaction or by some other means. The true velocity of growth can only be followed
within a single cell or within a population which is dividing synchronously, and
in both cases we must adopt a satisfactory
unit for the measurement of "growing"
material. So far this does not appear to
have been done in the case of isolated cells
growing in vitro. In other words, until the
growth curve can be plotted for intercleavage periods we cannot determine the
nature of the growth process itself1.
Although no bacterial population ever
exhibits synchronous division, the above
arguments hold good in other systems. In
any culture of actively growing bacteria,
there can be little doubt that at any inI
stant the population could be expressed by
a curve comparable to Fig. 3 if the population is divided up in terms of numbers
of individuals exhibiting a given phase of
division. In such cases there will always be
a fixed percentage of the total cells which
is about to produce new independent
I
bacteria, and for this reason the number
Time
of bacteria present at any instant when
Fig. 4.
plotted as a function of time will produce a logarithmic curve quite irrespective
of the way in which the actual growth process occurs within the individual cells.
The work of Lane-Clayton (1909), Slator (1913), and more recently of Richards
(1928) illustrates clearly the logarithmic nature of growth within actively growing
cultures of bacteria and yeast. In every case the conditions have been the same,
viz. the population has been homogeneous in the sense that the average generation
time of all the individuals has been constant, and has been heterogeneous in the
sense that the cells at any one moment have been in different phases of the reproductive process, and this heterogeneity has been constant throughout the experiment. For these two reasons, the resultant growth curve is logarithmic although
it throws no real light on the nature of the reaction which increases the amount
of growing material. In other words, growth curves of this type indicate the rate
of increase of a population of cells, just as a first order reaction may indicate the
rate of increase or decrease in a population of chemical molecules, without in either
1
This has been attempted with Echinus eggs (Gray, 1927).
The kinetics of growth
257
case defining the nature of the process which goes on within the units themselves.
The available data are all summed up in Barber's observation that successive
divisions do not influence the time required for the production of a new bacterium
if the environmental conditions are constant.
Since these experiments illustrate to some extent the principles of graphical analysis,
one of them may be considered in some detail. The results as given by Lane-Clayton are
recorded in Table HI.
Table I I I .
Time
Hours
Mins.
2
0
3
4
6
40
5O
10
No. of bacteria
N
at 420 C.
87
2,876
36,675
739,200
log10iV
1 94
3-46
4'56
5-87
<D*
a
!
3
1
no
S
.£2
a
m
a
bO
O
Time in hoars.
Fig. 5. (From Lane-Clayton.)
Graphically the results are shown in Fig. 5 (420 C ) , and from the slope of t
the value of k is approximately 0-94. If judged by the proximity of the points to one auu
the same straight line, observation and theory appear to agree with considerable accuracy.
If, however, the course of growth follows the compound interest law, then, without using
graphical methods, it is possible to calculate the value of k for the period between each
successive observation from the formula
(5)k (t2 - y = log N2 - log Nt
J. GRAY
258
The values so obtained are shown in Table IV, column 2. If we accept an average
value of k of 0-946, we can calculate the number of bacteria present on the assumption
Table IV.
Time
Hours
Mins.
2
0
3
4
6
4O
5O
10
Calculated
value of k
Calculated no. of
bacteria present if
k =0-946
Observed no.
of bacteria
0-9116
0-9476
0-9783
(87)
3,270
41,400
753,400
87
2,876
36,675
739,2oo
that exactly 87 were present at the beginning. These figures are shown in column 3. It is
quite clear that the validity of the compound interest law is subjected: to a rigorous test if
we compare the observed numbers of bacteria with the calculated figures, and are in a
position to decide whether or not the discrepancies revealed do or do not fall within the
limits of accuracy of the experimental counting on the one hand and the errors involved
by calculating logarithmic values on the other. Presumably, in this particular case, the
observed and calculated figures agree within the limits of these errors, but this could not
be determined accurately simply by estimating the position of points on a graph by
means of the human eye or by inspection of the calculated values of k given in Table
IV, column 2. It is therefore important to avoid the presentation of facts in such a form
as makes it difficult to detect divergence between observed and calculated values; they
ought always to be presented in a form which clearly exhibits a real basis of comparison
in terms of a measurable unit. The deceptive properties of logarithmic plotting can be
realised from the fact that an error of o*oi in the logarithm of 5*87 involves an error in the
antilogarithm of approximately 17,000 bacteria.
Precisely the same analysis applies to Slator's (1913) observations on the rate of
reproduction of yeast cells. From Table V it would appear that the course of growth is
Table V.
Time in hours
0
No. of yeast cells
per c.c.
90,100
17*3
4,660,000
0
255,000
1,390,000
2,200,000
7-0
9-0
0
35
1,360
3,550,000
Calculated value of
0-434 k
___
o-ioo
—
0-105
0-104
—
0-098
Average 0-102
very accurately represented by the formula kt = loge Njn where N is the number of yeast
cells at time t and where n is the number of cells present at the beginning of the experiment. In these cases the value of k varies between 0*105 an& 0*098, and is apparently
very constant about an average of 0*102. If, however, we use this average value for
calculating the number of cells present at the end of each experiment, it is clear that
the agreement between observed and calculated values is by no means so complete
(Table VI).
The kinetics of growth
259
Table VI.
No. of yeast cells per c.c.
Time in hours
Calculated
Observed
0-434 (0-102) =log10 N/n
90,100
4,660,000
255,000
1,390,000
2,200,000
1,360
3,550,000
0
I7'3
0
7-0
9-0
0
35
5,240,000
1,320,000
2,111,000
5,052,000
The two examples given illustrate the extreme care which must be exercised in the
use of formulae which involve a logarithmic term, and show that the only real test consists
in a comparison of observed and calculated values. If the differences between such values
are within the errors of experimental observation, we may conclude that the results of
the experiments themselves are not definitely against the theoretical conceptions on
which the formula is basfid.
The law of compound interest obviously applies to the growth of a population
of cells as long as all the bacteria are reproducing themselves at the same rate; i.e.
they all produce a new individual by a process of unknown velocity in the same
period of time. If the initial culture is heterogeneous in the sense that some bacteria
grow and divide in a shorter time than others, then the rate of increase of the whole
population cannot be expressed by a simple equation. Suppose we start with two
bacteria in a culture, one of which divides every 20 mins. and the other every 40
mins. Then the number of bacteria present after 4 hours will be 4160, of which 4096
Table VII.
Time
Bacterium A
B
Total
0'
1
1
2
2O'
2
4O'
4
60'
80'
16
4
100'
I2O'
8
32
64
8
—
20
—
72
2
6
140'
128
160'
256
16
180'
512
2OO'
IO24
32
2048
—
272
—
IO56
—
22O'
24O'
4096
64
4160
are derived from the more active of the two original organisms. In a generalised case,
a heterogeneous population of No cells can be subdivided into a series of homogeneous categories, and in each category all the individuals will have the same
generation time. Thus, in one category there will be % individuals with a coefficient
of growth k±, in another n2 individuals with a characteristic k2, and so on. The
total number of bacteria present at the beginning of growth is No and this is equal
to the sum n1 + n2 + nz...nn, and after a given time t, the total number of cells
(Nt) will be given by the expression
Nt = nxeW + ntfW + n ^ ...nneh^
(6).
It is clear that unless some information is available concerning the relationship of
«i, n2, n3... and &l5 k2, k3... it is impossible to foretell the course of growth of a
J. GRAY
260
heterogeneous population. From Table VII it is, however, clear that as growth goes
on in a heterogeneous population, the heterogeneity becomes less because the
percentage of rapidly growing cells very rapidly increases; hence if we subculture
a rapidly growing population, we very quickly obtain a homogeneous culture which
will obey the compound interest law. The experiments of Lane-Clayton and of
Richards are of peculiar interest as in each case the cultures were of a homogeneous
type, thereby justifying the use of a growth constant k in equation (i). In almost
every other biological system this condition is by no means so clearly established,
for we have no evidence that we are dealing with either a population which is
homogeneous or with one whose heterogeneity is always the same.
If the number of bacteria is allowed to increase in any given volume of medium,
there comes a time when the compound interest law breaks down. Sooner or later
the growth rate declines and eventually sinks to zero: after this, there is a rapid
decline in the number of bacteria present. The factors responsible for this breakdown of the logarithmic law of population growth are not completely known.
Graham-Smith (1920) has shown that under certain conditions the amount of food
stuffs available plays an important role. In a particular culture medium GrahamSmith found that Staphylococcus aurens reproduces itself until there are approximately 10 million organisms per o-oi c.c. of medium, and this figure is independent
of the number of bacteria originally inoculated into the medium.
Table VIII.
No. of bacteria
in original
inoculum
Maximum no. of
bacteria obtained
Original no.
of bacteria
Mfnrimnm no.
520
1392
1784
5660
9,248,000
10,606,000
9,280,000
9,872,000
34-400
4-300
4*20
8,720,000
5'9
reached
8,544,ooo
8,496,000
7,584,000
There is therefore an upper limit to the density of bacteria which can be obtained
in any given culture. This upper limit is largely dependent on the concentration of
nutrient substances in the medium (Table IX)—see also Penfold (1912).
Table IX.
Relative concentration
of food
Maximum no. of
bacteria obtained
IOO
25,840,000
20,300,000
15,760,000
9,416,000
4,273,006
75
50
35
10
Unless the food supply is maintained at a uniform level, the maximum population for any particular medium is not stable for any significant period of time but
The kinetics of growth
261
the number of organisms rapidly declines. This is almost certainly due in part to the
fact that the foodstuffs available are being steadily depleted in order to maintain
the normal activities of the organism. For each concentration of food there is a
characteristic maximum density of population, and unless the food is maintained
at this level, the density will fall with falling concentration of food.
The far-reaching effect of food supply upon growth rate is seen when identical
cultures are grown at different temperatures. The higher the temperature the more
intense are the katabolic processes of the organisms, and consequently a higher
concentration of food is required to maintain a maximum population. In other
words, with identical cultures the food supply begins to run short sooner at a high
temperature than at a low, and consequently the maximum density at a high temperature is lower. This is illustrated by Graham-Smith's figures (Table X).
Table X.
Incubation
temperature
(°C.)
Maximum
density of
bacteria
Time in days
required to reach
maximum density
17
27
37
18,272,000
15,164,000
10,448,000
8
5
3
In other words, at the higher temperatures the maximum density is more rapidly
reached, but the maximum itself is lower than at lower temperatures.
Even when the food supply of a bacterial culture is very rich, the growth rate
of the culture will still eventually approximate to zero, and from this we may infer
that other factors in addition to lack of food are also operative in reducing the
growth rate. How far these factors represent toxic substances produced by the
bacteria themselves has not been fully investigated but forms an important subject
of research. (See Richards, 1928.)
Qualitatively, however, the effect of a scarcity of food on the rate of growth
of bacteria is for certain types and concentrations of media well established, and
an attempt has been made to express the phenomena in a quantitative form
(McKendrick and Pai, 1911). These authors assumed that throughout life the food
supply was always a limiting factor. Under such conditions the rate of reproduction
K^j would be proportional to the amount of food (/) and to the number of bacteria
present (x) as long as the death rate is constant during the whole period
It was further assumed that whenever a bacterium is formed a definite amount of
food material was utilised in the process, so that if x bacteria have been formed, k±x
units of food have disappeared: if there were a units of food present at the beginning
of the experiment, then, when x bacteria are present, the amount of food remaining
262
J. GRAY
is a — k (x — xQ), where x0 is the number of bacteria inoculated into the fresh
medium. Under these conditions
8/
= k (a — kxx + &i#0) x
(8).
If a + kxXQ = A
xa
XQ
I si.
X)
or if ^0 is very small compared to x,
i .
ka
xa
—r
x0 (a - ki xx)
= 7-102,,—-.
Be
{
v (io
This equation gives an accurate representation of the observed course of reproduction, for there can be little doubt that it expresses the observed data within
the limits of experimental error; nevertheless, it cannot possibly be accepted as an
adequate representation of the known facts. Equation (8) assumesfivethings: (i) that
S2x
at no moment will the growth be strictly logarithmic since -~-§ falls steadily from the
beginning of incubation, (ii) the maximum number of bacteria will not be reached
until the whole of the foodstuffs have entirely disappeared, (iii) that no food material
is required for the maintenance of a bacterium once it has been formed, (iv) that all
the bacteria react to adverse conditions to the same extent and that the death rate
when food is scarce is not higher than when food is plentiful, (v) that having
approached the maximum density the population can be divided by sub-culturing
into fresh medium, and each bacterium will at once begin to grow as rapidly as the
original culture in its early life. Now it is extremely improbable that any of these
assumptions are true; on the other hand, they are at variance with fairly well
established facts. It therefore follows that equation (io) cannot be of any real value,
although as an empirical equation it may have some practical use. This example
illustrates the fact that although the observed growth of a population of cells can
be represented by a simple equation, it by no means follows that we are justified in
assuming that the factors controlling growth are equally simple. That the food supply
is an important factor determining the rate of growth we already know from direct
experiment, and our knowledge is not only not increased but is liable to be misinterpreted by the use of "simple" quantitative conceptions which are not based
on facts.
If McKendrick and Pai's assumptions are true in so far that the rate of reproduction
is proportional to the number of bacteria (x) and to the concentration of food (y) present
in the culture (and this is constant in volume), then
Sx
The kinetics of growth
263
But the food is decreasing partly owing to the formation of new bacteria and partly
owing to the nutritive needs of these bacteria, so that,
Hence
or
kfr + y)-^
loge (ky + k1) = k(x0+ y0) - k± loge (ky0 + kj
t=
jj—
(11),
...(12),
&
-kz-kj
where
c = k (x0 + j 0 ) — £x loge (ky0 + &x) and z = x + y.
In other words, it is possible to express the number of bacteria in terms of the quantity
of food present, but it is not possible to calculate with any ease the number of bacteria
present at any particular moment. For a practical application of equation (11) see Gray,
1929.
SUMMARY OF CONCLUSIONS CONCERNING UNORGANISED GROWTH.
The data derived from an observation of the growth of unicellular organisms
are of peculiar importance since the systems concerned are very much more simple
than those which exist in the body of an organised metazoon. Not only is it possible
to grow bacteria in media of known and controllable composition, but the units
of growth (i.e. all the organisms in a culture) may, in special cases, be essentially
similar to one another. The facts indicate that as long as the external environment
is maintained at a satisfactory and constant level, each cell of a homogeneous
population multiplies at a rate which is constant within certain definable limits;
there is no indication that the powers of growth are declining with the age of the
culture. Old cultures exhibit a reduced growth rate because the external conditions
automatically change as time proceeds. Given a constant medium, however, the
rate of reproduction and the rate of "accidental" death remain constant, and it is
possible to define the reproduction rate as a constant peculiar to the species.
On the other hand, if the medium varies owing to a diminution of food supply
or to the accumulation of the products of growth, the effect on the growth rate is
clearly of an inhibitory nature. At the same time, the available data do not as yet
enable us to form a quantitative expression of this change. The relationship between
reproduction and concentration of food is not linear (see Penfold, 1912), and we do
not know the precise relationship between lack of food and the death rate of the
population. Presumably some bacteria are more sensitive to adverse conditions than
others, and the death rate will be related to adverse conditions by a variability curve
comparable to that of higher organisms. The net result of our present iack of knowledge makes it impossible to construct a differential equation for reproduction which
takes adequate cognisance of the known qualitative facts, and from this it follows
that, although it is possible to express the final growth curve by means of a simple
formula, this formula cannot throw any real light on the underlying causes of
growth.
If we define growth as an increase in the amount of living material going on
within individual cells, we cannot determine its nature or velocity from a study of
264
J. GRAY
the number of cells present in an ordinary culture, or from a study of the total
activity of the whole culture. The process of growth in a suitable medium may or
may not follow the course of a first order reaction, but until it has been observed
within a single cell its nature will remain unknown.
The essential features exhibited by a growing culture of bacteria, are also seen
in metazoon cells grown in vitro, although the requisite conditions for growth are
more specific, and the known facts are even less complete than in the case of bacteria
(see Carrel (1923), and Fischer (1925) for numerous references). The rate of
growth of tissue cells in vitro depends in part upon the concentration of embryonic
extract present in the medium; as long as this is constant (in addition to other
environmental factors) the cells will grow and multiply at a uniform rate. Until
we know, however, the precise manner in which the embryonic extract is destroyed
or absorbed by growing cells it is premature to express the facts as a differential
equation, or to apply them to growth in situ.
We may conclude, therefore, that as long as a population of cells is of constant
homogeneity, and as long as the external conditions remain at a constant and favourable level, the rate of reproduction, like other biological characteristics, also remains
constant. When, however, the population is not homogeneous, but changes its
heterogeneity as time proceeds, or whenever the conditions of the environment are
changing automatically owing to the process of growth itself, then our data are
as yet an inadequate basis for formulating a quantitative conception of the changes
in reproduction rate which ensue. Differential equations which are derived from
an observed reproduction curve can only be regarded as empirical expressions and
throw no real light on the underlying factors controlling the growth rate.
GROWTH OF METAZOON CELLS IN VIVO.
The growth of a cell inside the body of a living organism is very different
to that of a similar cell isolated in a nutrient medium containing embryonic
extract. As long as a cell is in its natural position in the body, its growth is
determined both qualitatively and quantitatively not only by the size or age of the
organism but also by the particular position occupied by the cell and the particular
relationships of its neighbours. The normal environment of a cell inside a rapidly
growing embryo is therefore extremely complex and quite unlike that of a cell
grown in vitro. With one or two exceptions differentiated cells cannot be cultivated
as such in vitro and since the bulk of an animal's body is composed of differentiated
cells, it follows that evidence derived from in vitro experiments does not as yet throw
much light on the growth which normally occurs in the body as a whole. When
we are able to cultivate differentiated cells (e.g. muscle and glandular epithelia)
as such in vitro, then it will be possible to see how far the presence of one type of
cell is able to modify the rate of growth of others, and thereby form some real conception of the processes of growth in vivo. At present we have to restrict our analysis
to points of comparative detail. A priori, it is extremely improbable that a system so
complex should conform in any fundamental way to the behaviour of a simple
system of strictly definable heterogeneity.
The kinetics of growth
265
As shown above, the so-called logarithmic growth curve of unicellular populations is observed so long as there is a fixed percentage of the total number of units
undergoing change at any instant; this fixed percentage can be expressed as a
fraction of the total number of cells present. In the case of a metazoon the total
number of cells present (except in certain specified tissues) is no indication of the
amount of material present; for this reason it is customary to dispense with the
cell as a unit of metazoon growth.
The simplest possible conception of a growing metazoon suggests a number of
different units {e.g. tissues) each growing at its own characteristic rate. The total
growth of such a system would represent the summation of the growths of the
various parts. Some of these parts may grow at the expense of others and some of
the growing units may give rise to material which is not capable of growth. Unless
therefore there is some factor which controls the total amount of growth made by
all the units at each particular moment, it is impossible to apply to metazoon growth
the simple algebraic treatment applicable to a homogeneous population of bacteria
Food
,a
> Specific tissue 1
>b
>
„
„
2
>X<
growing in a constant environment. It is very doubtful whether evidence in favour
of such a controlling factor or "master reaction" can be derived legitimately from
the apparent ease with which an equation with only two variables can be made to
express the size of a growing organism. The conception of a master reaction as one
possessing a simple chemical nature is of vital importance in any quantitative study
of growth as it enables us to ignore to some extent the heterogeneous nature of its
ultimate products. If such a reaction exists, it will have a definable velocity constant, since the units involved will comprise a molecular population to which chemical
principles can be applied legitimately. From a purely biological point of view, it
seems reasonable to regard the whole process of organised growth as the result of
a coordinating mechanism whereby the velocity of individual processes are mutually
dependent on each other to a marked degree (see Gray, 1929). At the same time
it is by no means easy to express the biological facts in terms of chemical units.
Even in their simplest form these facts suggest a degree of complexity far beyond
anything yet investigated in inanimate systems. We might conceivably depict the
course of growth by the system illustrated in Fig. 6; the raw materials entering the
growing body are converted first into a substance Xy and from this there are derived
a series of compounds a, b, cy d, each of which can be converted into the other or
266
J. G R A Y
into specific and permanently differentiated tissues. In such a system the total
production of specific tissue might be proportional to X, whereas the amount of
any one tissue would also depend on the amount of other compounds of the series
a-d. The conception of a master reaction of this type restricts our conception of
growth to quantitative principles, and would usually imply that if one organ were
excised (and were not regenerated) the rest would grow with increased speed. In
actual practice the effect of removing parts of the body in some cases has no effect
on the growth rate of the remainder although, in others, compensating hypertrophy
or regeneration may occur. To some extent the heterogeneity of the whole body
can be eliminated by investigating the growth of specific organs (Scammon, 1925); in
such cases, however, we are liable to ignore the fact that the rate of growth of one
organ depends on the behaviour of another, so that we are still dealing with an
extremely complicated system. If the existence of a master reaction is regarded as
intrinsically probable, we are still faced with the difficulty of determining the
amount of the active principle present at a given time. Unless the active mass of X
bears some simple relationship to a measurable unit, we are no nearer to a quantitative estimate of growth.
Putting these difficulties on one side for the moment, we may examine the
possibility of expressing the growth rate of the whole metazoon body in terms of
simple measurable units. It is universally admitted that the factors controlling the
growth rate are not as yet capable of direct measurement. In a chemical system, on
the other hand, the factors controlling the rate of reaction are, under standard
conditions of environment, the active masses of the reacting substances, and these
are at times proportional to the weight of the substances in unit volume of medium:
these can be determined with accuracy. When, therefore, we express the number
of molecules undergoing change in terms of the total number of molecules present,
we are dealing with measurable quantities. In the same way we can express the
number of growing bacteria in terms of the total number of organisms present.
When, however, we consider metazoon growth we cannot do this, for it involves
the power of expressing the amount of new tissue formed in terms of units which
cannot directly be observed. We are, in fact, trying to express the amount of new
tissue formed per unit of time per unit of growing tissue. It is impossible to do
this in most cases of metazoon growth because there is no obvious way of knowing
how far all the organism is growing. In practice we assume that all the organism
or a fixed percentage of it is growing, and therefore express the growth rate in
terms of new tissue formed per unit weight of the whole organism. Minot (1908),
was perhaps the first to use this method, and his formula for the growth rate was
100
Where Wt is the weight of the organism at the beginning of the observational unit
of time, and W2 is the weight of the end of this period. As pointed put by Brody
(1927), this method gives erroneous values unless the period of time is extremely
short, and unless the growth rate is changing very slowly. In practice it is very
The kinetics of growth
267
difficult to fulfil the conditions necessary for accurate results. Minot's formula is
however an attempt to express the growth rate in terms of measurable quantities;
it leads to erroneous results because growth is a continuous and cumulative process.
All attempts to overcome this latter difficulty are based on the assumption that
the amount of growing tissue in an organism is directly proportional to the whole
weight of the organism. In some cases (embryos, larvae, etc.) this assumption does
not appear to be altogether unreasonable, but we usually encounter difficulties.
If the amount of growing tissue is to be assumed to be proportional to the
weight of the whole animal, should this weight include the water content of the
body or not? In the case of young fish both these standards lead to the same result
because the percentage of water in the organism does not vary materially during
a prolonged period of growth (Gray, 1926). In the chick, on the other hand, an
estimate of the growth rate based on wet weight figures may differ materially
from that based on dry weight figures (Murray, 1925); in this case we do not know
whether the percentage of water in the "growing" tissues changes, or whether the
percentage of water in the whole animal changes with the intensity of fat storage,
bone formation and similar processes.
In a few cases, it is tempting to adopt a purely physiological standard for the
estimation of growth. For example, we find that for a significant period of time
the amount of oxygen consumed per gram of growing fish (Salfno fario) remains
constant (Gray, 1927), and from this we might conclude that the amount of living
respiring material is actually directly proportional to the weight of the animal.
Sooner or later, however, this criterion breaks down and the intensity of respiration
falls. In the chick, Murray found still more irregular phenomena.
If we are prepared to accept the rate of respiration as a measure of the amount of
living tissues in very young embryos, we reach an interesting conclusion by studying the
rate of growth during the segmentation stages of Echinus. It will be seen from Fig. 7
that the specific growth rate as measured by (8R/Bt)jR (where R is the rate of respiration
of the whole organism) rises during the early stages of development; in other words
there is a "lag" phase during which the specific growth rate is increasing to a maximum.
It will be remembered that where a bacterium is removed from a medium in which growth
is very slow, and is placed in a favourable medium, there is also a lag phase. It looks as
though the same principle applies to a metazoon egg, since after the conditions for
growth again become favourable (by fertilisation), there elapses a period of time before
the specific growth rate reaches its maximum. Later on, in both cases, the rate falls off
unless the medium is artificially maintained at a satisfactory level. These data from
Echinus eggs obviously provides another example of the way in which more than one
differential equation of growth can be derived from one and the same set of data (see
Gray, 1927) and how each differential equation gives a different conception of the nature
of the growing system.
We now have two main difficulties in obtaining a satisfactory expression for
the total growth rate of a metazoon, (i) the heterogeneity of the system undergoing
change, (ii) the difficulty of expressing the growth rate in terms of rational and
measurable units. Nevertheless, it is claimed (Robertson, Brody, and others)
that the form of the observed growth curve is sufficiently simple to justify the
268
J. GRAY
belief that the whole process of growth is controlled by a master reaction involving
two variables, each of which is a linear function of the size of the whole organism.
"Incidentally, but not primarily, the applicability of the equation of the chemist
to the time relations of growth may be taken as further substantiating evidence
that growth is limited by a chemical reaction" (Brody, 1926, p. 234). It is at this
point that the principles of graphical analysis, discussed in the first section of this
paper, come into prominence. Both Brody and Robertson (1923) claim that a
typical growth curve can be expressed by particular equations, but the equations
they use differ from each other. Robertson claims that the curves are of a symmetrical sigmoid type, although more than one such curve may be included during
the whole life cycle. Brody (1925-7), on the other hand, divides each sigmoid curve
2«4r
2*3
ft!
5 2-2
S
2«0
3
4
Time in hours
Fig. 7.
8
into a number of distinct exponential phases, each of which is presumably more or
less independent of the other.
As already pointed out, it is impossible to say how far a particular sigmoid
curve is the only one capable of expressing the observed facts unless the experimental data are known with extreme accuracy, and it is almost certain that this
degree of accuracy is not forthcoming for many of the cases quoted by Robertson in
support of his equation. Robertson's equation in its differential form is of the type
Sx
and, if this be true, the integrated curve is symmetrical and must inflect when the
animal's growth is half completed. By selecting suitable values of kx and k and by
using an appropriate number of superimposed curves there can be no doubt that
an equation of this type can be shown to express the facts. Unless, however, there
are definite experimental reasons for adopting this procedure, the equation has no
The kinetics of growth
2 6g
real meaning unless its advocates can prove that no other equation will fit the
facts.
That the same group of observed facts can be subjected to entirely different types of
graphical analysis is illustrated by the growth curve of the rat. It can be resolved into a
series of curves of the type selected by Robertson (1923) or it can be regarded as a single
curve inflecting when the animal is not one-half grown but one-third its full size (Davenport, 1927).
The whole curve can, within the limits of reasonable error, either be expressed by an
equation of the type advocated by Crozier (1926) or by that illustrated in equation (15)
orif
From equation (16) it might be inferred that the rate of growth is determined by three factors.
(i) It is directly proportional to the weight of the whole organism (x).
(ii) It is directly proportional to the amount of essential foodstuffs {h^j^/x) conveyed
to each unit of tissue, if the total of such foodstuffs available is determined by the surface
of a digestive or absorptive organ, this surface varying with the two-thirds power of the
weight of the animal.
(iii) A continuous process of wear and tear (whose intensity has an average constant
value (&2) per unit weight of tissue), going on during the whole period of life.
A priori, none of these assumptions is clearly improbable, on the other hand they are
not established facts. The only means of assessing the value of the hypothesis lies in
further experimental analysis and not by an elaborate selection of arbitrary values of
constants which will reproduce the data already known. For this particular example
some experimental analysis is available; if equation (16) be based on reality then the
amputation of a limb or tail should lead to an increase in the weight of the remaining
organism by further growth, since the limit of growth is partly determined by the factor
k^x. Such compensating growth does not occur, and consequently the equation cannot
be valid, however nicely it represents the course of normal growth when expressed in an
integrated form.
Returning to Brody's exponential equations (1925), it is sufficient to point out
that any curve can be expressed as a series of straight lines, or exponential curves
if suitable limits are selected. Unless, therefore, there is good independent evidence that the whole growth cycle is divisible into a finite number of successive and
different processes, the process of subdivision of the growth curve is purely arbitrary.
We therefore reach the conclusion that it is by no means easy to deduce from
known growth curves a proper understanding of the factors controlling the rate of
metazoon growth. We are compelled to examine these factors by direct methods.
Direct investigation of internal growth factors is very far from complete, but
from a study of the growth of tissues in vitro it seems fairly clear that young embryos
contain a supply of a growth promoting substance which is greater than that possessed by older embryos. In the adult animal this substance appears to be absent
except in certain cells. Now if the whole of an embryo possesses the same potential
powers of growth in all its parts, it is not unreasonable to suppose that the growth
18
270
J- G R A Y
rate is proportional to the weight of the embryo (x) and to the concentration or
amount of the growth promoting substance (y)
Sx
If y decreases from a finite value to zero, it follows that the integrated growth
curve will be sigmoidal in form quite independently of the manner in which the
decrease in y occurs. The rate of disappearance of the growth promoting substance
will only determine the point of inflection and symmetry of the curve; the essential
point is that it will always be sigmoidal. The simplest possible conception of the
way in which y disappears is embodied in the hypothesis that each gram of newly
formed tissue uses up in its formation the same amount of growth promoting
substance. In other words, if there are a units of this substance present at any
time, then after x grams of new tissue have been formed there will be left a — kx
units of growth promoting substance. If we also assume that every unit of new tissue
formed has the same potential powers of growth as any of the pre-existing tissue,
then if x is the total number of grams of tissue at any moment, the potential powers
of growth are k±x. If the rate of growth is proportional to the amount of growing
tissue and to the amount of growth providing substance, then
k ^ i k )
(18).
In this way it is possible to reach Robertson's fundamental equation, basing it to
some extent on established facts. There is however no real evidence to show the
precise manner in which the growth promoting substance disappears during
development, and there is consequently no real justification for writing equation
(17) in the form of equation (18).
It is perhaps worth noting that a sigmoidal growth curve would result if the
growth rate were reduced owing to the formation of a growth inhibiting substance
which combines reversibly with the tissues and slows their growth. Such a hypothesis, if based on experimental facts, would harmonise with the observed powers
possessed by tissues to recover from wounds. (See Appendix II.)
From the point of view elaborated in this paper, the comparison of metazoon
growth with the behaviour of comparatively simple chemical reactions meets with
three main difficulties. Firstly, a series of observations which approximate to a
sigmoid curve can only be expressed in the form of a specific differential equation
when the accuracy of the observations reaches a very high level. Until such data
are available it is impossible to determine how far they can only be expressed by
the highly specific curves applicable to chemical systems. Secondly, there is no
direct method of determining the active mass of the growing substance or of the
other factors involved in the reaction: these may be proportional to the weight of
the organism although no definite proof exists. Thirdly, the growing system is,
known to be statistically heterogeneous, and in the absence of reliable evidence to
the contrary, it is intrinsically improbable that the system will behave like a system
whose heterogeneity is constant.
The kinetics of growth
271
The physico-chemical conception of growth enables us to depict certain facts
against a simple and attractive background, but the practice of expressing observations in their simplest apparent form is undoubtedly liable to cause confusion, particularly when we know that the systems providing these observations
are themselves extremely complex. There seems little doubt that our knowledge
of the kinetics of growth can be advanced only by a direct study of factors which
directly control growth rate and by expressing these factors in a quantitative form.
In this way, a real differential equation will be made available, and this, in its
integrated form, will harmonise with the results obtained by observing the size of
the organism at selected moments of time. This position has not yet been reached.
For the present, the whole of the facts indicate that the size and form of a growing
organism are the resultants of a large number of obscure factors all dependent on
each other and not adequately represented in any system of unorganised growth,
and still less so in a simple physico-chemical system. A premature attempt to express
the growth rate by a differential equation involves the grave danger of confusing
rather than of clarifying our knowledge of the facts. At the same time, a real advantage is gained by attempting to put together the known facts concerning growth
factors in such a way as will indicate whether or not they are capable of giving an
adequate picture of observed facts, or whether a search must be made for further
factors. In other words, it is legitimate to work from a differential equation in
which the terms represent real entities; it is not very profitable to base our ideas
on a differential equation which is solely derived from an observed curve of growth.
The known facts of growth in vivo and in vitro seem to indicate quite clearly
that as an organism increases in size or age the environment for growth becomes
less favourable for those tissues still capable of growth. Until the cause of this
phenomenon has been subjected to direct quantitative study, it is unlikely that we
shall find an equation for any particular growth curve which is more than an
empirical representation of observed data.
SUMMARY.
A quantitative expression for the growth rate of living systems can only be
obtained when the systems are homogeneous (or of known heterogeneity), and
when the external conditions are constant. When more than one type of cell is
present, or when the external conditions are varying, the known data (concerning
even unorganised growth), are insufficient for the construction of a real equation
defining the rate of growth in a given set of conditions existing at a particular
instant of time. An equation representing the size of a population of cells or of an
organism in terms of age, yields, on differentiation, a quantitative but empirical
representation of the factors controlling the rate of growth, but since more than one
equation can always represent a typical growth curve within the limits of probable
error, a selection of one particular equation rests solely on the intrinsic probability
of its differentiated form. The degree of probability can only be established by
direct experiment.
18-3
272
J-
GRAY
In the growing body of a metazoon the conditions of growth are extremely
complex, and it is difficult to express the growth rate of the whole organism in
terms of rational units. Graphical treatment of the data underlying a typical growth
curve is liable to produce errors of considerable magnitude, and often tends to
confuse the facts. The units which compose a metazoon's body form a very heterogeneous system, in which the rate of growth of one organ is dependent on that of
others. It is, therefore, intrinsically improbable that the behaviour of such a system
should conform to that of a simple chemical system in which the variables are
few in number and capable of accurate analysis. The conception of growth as a
simple physico-chemical process should not be accepted in the absence of very
rigid and direct proof; at present, it rests on the results of a process of graphical
analysis which is often, if not always, of a relatively inaccurate nature.
APPENDIX I.
The significance of growth "constants."
The danger involved by the assignment of arbitrary values to the constants of an
equation of growth may be illustrated by the following example: Ledingham and
Penfold (1914) obtained the following figures of the growth rate of bacteria during such
conditions as exhibited a marked lay-phase (Table XI).
Table X I .
Time in mins.
No. of bacteria
0
45
60
80
100
120
150
180
To account for these figures we might
217
make the following hypothesis. At the beginning of the experiment let us imagine that
all the 217 bacteria are unable to reproduce,
but as time goes on, they gradually recover
from the effects of their past history. All
the bacteria will not recover at the same
<u
rate, some will recover after a short time >,
and some after a longer time; at the end °
of three hours, however, the whole culture is I
known to be growing at a steady rate. Now, o
it is reasonable to suppose that the course I
of recovery of the whole population will
follow a sigmoid probability curve such as
is shown in Fig. 8. As a first approximation
we can replace this curve by the dotted
line AC. Let the number of inactive or
non-growing bacteria at any time be y.
217*5
287
345
470
718
1362
2535
7610
Time in minutes
Fig. 8.
The kinetics of growth
Then
= 217 ( 1 —
^73
) where t = time
.(19).
180/
Let the number of growing bacteria at time t be x,
8x ,
217 ,
,.
then
-KZ = kx -i—~ where k is a constant
o*
.(20),
180
.(21).
The total number of bacteria at time t is x + y
.(22).
Giving A the arbitrary value of 0-0285, the calculated values of x + y are those shown
in Table XII, column 3.
Table XII.
No. of bacteria
Time in mins.
0
5
t60
80
100
120
150
180
Observation
Calculated
217-5
387
345
470
718
1363
3535
7610
317
k =0-0285
Calculated
k =0-0385
317
306
375
336
489
783
1,533
1335
3019
7i3i
31,899
It might well be that the calculated figures are sufficiently near those actually observed
to fall within the limits of experimental error. If however k = 0-0285, then, when the
whole population is growing at its full maximal rate, the number of bacteria should be
doubled in a period of time equal to — ~ - , viz. 24-3 mins. By actual experiment, however,
the generation time at the end of three hours was found to be 18 mins. and if this is true
k must equal 0-0385, and if this be so the calculated figures are shown in Table XII, column
4, and it is quite obvious that the original hypothesis breaks down.
APPENDIX I I .
The significance of the sigmoid growth curve.
The sigmoid nature of a typical metazoon growth curve expresses the fact that up to
a given point the increase in weight of the organism per unit time at first increases, after
which it declines. This can be expressed algebraically by the equation
where ^ represents the rate of increase in weight of the organism; x represents a factor
which increases with the weight (w) of the organism, and y is a factor which decreases as
x increases. If the weight of an organism reaches a maximum value, then y must approach
zero. The integrated growth curve will be sigmoidal whatever be the rate or manner in
274
J- GRAY
which x andy change in value; the precise form of the curve will however depend on these
processes.
The principles outlined in this paper suggest that x may increase and y may decrease
in a variety of ways and yet in every case the integrated growth curve will express the
observed size of an animal within the limits of experimental error. The advocacy of one
specific differential equation of growth over others must therefore depend on independent
external evidence. For example, Richards (1928) has shown that under certain conditions
the sigmoidal form of the growth curve for yeast is directly associated with the generation
of alcohol. Since the effect of the alcohol is reversible, we might imagine that it unites
with actively growing cells in a reversible manner, rendering them incapable of growth.
Under such circumstances it might be possible to advocate a differential equation of the
type given below.
Sx
kx
,
k
where c is a constant involving the threshold value of alcohol inhibition. The integrated
growth curve would then be
a — be
.
x
k
F.
k — ak, — kxb (x — cY]
— i——J——r~r l°ffe
i—n
7 ~i~r^ l°ge
S
J
I•••
k — akx + kxbc
XQ k± \k — akx + kxbc) \_
k — «%
J
The sigmoid nature of this curve can be appreciated by re-differentiating (24) and
by putting a — be = a
1
ka
=
,
^ v
2
(a + 6x) " *
The first term will decrease and when x = T A / 7-— a
tne
^5J>
daily increments in
weight will cease to increase, and at this point the growth curve will show an inflection.
h **"
The limit of growth will be reached when x = It would be interesting but laborious to apply an equation of this type to Richards'
data.
BIBLIOGRAPHY.
BARBER, M. A. (1908). Journ. of Infectious Diseases,
BRODY, S. (1925). Journ. Gen. Physiol. 8, 233.
5, 379.
(1926). Journ. Gen. Physiol. 9, 235.
(1927). Journ. Gen. Physiol. 10, 637.
CARREL, A. (1923). Journ. Exp. Med. 38, 407.
CROZIER, W. J. (1926). Journ. Gen. Physiol. 10, 53.
DAVENPORT, C. B. (1927). Journ. Gen. Physiol. 10, 205.
FISCHER, A. (1925). Tissue Culture. London.
GRAHAM-SMITH, G. S. (1920). Journ. of Hygiene, 19, 133.
GRAY, J. (1926). Brit. Journ. Exp. Biol. 4, 215.
(1927). Brit. Journ. Exp. Biol. 4, 313.
(1929). Brit. Journ. Exp. Biol. 6,
LANE-CLAYTON, J. E. (1909). Journ. of Hygiene, 9, 239.
LEDINGHAM, J. C. G. and PENFOLD, W. J. (1914). Jovrn. of Hygiene, 14, 242.
MCKENDRICK, A. G. and PAI, K. M. (191 I ) . Proc. Roy. Soc. Edin. 3 1 , 649.
MINOT, C. S. (1908). The problem of age, growth, and death. New York.
MURRAY, H. A. (1925). Journ. Gen. Physiol. 9, 39.
PENFOLD, W. J. and NORRIS, D. (1912). Journ. of Hygiene, 12, 527.
RICHARDS, O. W. (1928). Annals of Botany, 42, 271.
(1928 a). Journ. Gen. Physiol. 11, 525.
ROBERTSON, T. B. (1923). The Chemical Basis of Growth and Senescence. Philadelphia
and London.
SCAMMON, R. E. (1925). Proc. Soc. Exp. Biol. and Med. 23, 238, 687, 809.
SLATOR, A (1913). Biochem. Journ. 7, 197.