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LIFE PHASES IN A BACTERIAL CULTURE
R. E. Buchanan
From the Bacteriological Laboratories of Iowa State College, Ames, la.
Several important contributions to our knowledge of the numbers
of bacteria present in a culture medium at certain stages in the devel-
opment of the culture have been made recently, but with the exception
of the work of Slator (1917), apparently there has been no effort to
coordinate these results and develop a complete mathematical theory
of such changes in numbers. The present paper is an attempt to ana-
lyze the results of these authors and to present certain phases which
have apparently been neglected heretofore.
When bacteria, particularly cells from an old culture, are inoculated
into a suitable culture medium, as broth, the bacteria will at first
remain unchanged in numbers ; then multiplication begins, the numbers
increase at first slowly, then more rapidly until a certain minimum
average generation time is reached ; this after a time begins to increase,
and there is a negative acceleration in growth, which finally ceases ; the
numbers remain constant for a time, then the bacteria begin to die off.
Lane-Claypon 1 recognized four periods or phases in the life of a bacterial
culture as follows :
1. Initial period of slow growth or even no growth.
2. Period of regular growth.
3. Period during which numbers remain more or less stationary.
4. Period during which the numbers of living bacteria are decreasing.
It would seem, however, that the life phases are somewhat more
complex than indicated by the preceding statement. A study of the
results secured by various authors indicates that seven relatively dis-
tinct periods may be differentiated. These may be recognized easily
by plotting the logarithms of the numbers of bacteria against time.
Chart 1 is such a plot with the seven phases indicated. It will be noted
that points designating the beginning and end of each phase are points
where a curve changes to a straight line and vice versa.
Received for publication Feb. 22, 1918.
1 Jour. Hyg., 1909, 9, p. 239.
R. E. Buchanan
These various growth phases may be designated as follows :
1. Initial Stationary Phase. — During this phase the number of bacteria remains
constant, and the plot is a straight line parallel to the x axis indicated by 1 — a.
2. Lag Phase, or Positive Growth Acceleration Phase.- — During this phase
the average rate of increase in numbers per organism increases with the time,
giving rise to the curve a — b. This increase in rate of growth, per organism
does not continue indefinitely but only to a certain point determined by the aver-
age minimal generation time per organism under the conditions of the test.
3. Logarithmic Growth Phase. — During this phase the rate of increase per
organism remains constant, in other words, the minimal average generation
time is maintained throughout the period. This gives rise to the straight
line b — c.
4. Phase of Negative Growth Acceleration. — During this phase the rate of
growth per organism decreases, that is, the average generation time is increased.
Chart 1. — Diagrammatic plot of logarithms of numbers of bacteria present in a culture.
The bacteria continue to increase in numbers, but less rapidly than during the
logarithmic growth phase. This is the curve c — d.
5. Maximum Stationary Phase. — During this period there is practically no
increase in the numbers of bacteria. The plot gives a straight line d — e
parallel to the x axis. The rate of increase per organism is zero and the
average generation time infinity.
6. Phase of Accelerated Death. — During this phase the numbers of bacteria
are decreasing, slowly at first and with increasing rapidity, until the estab-
lishment of a logarithmic death rate. In the terminology used in the growth
phases, the average "rate of death per organism" is increasing to a certain
maximum. This gives the curve e — f .
7. Logarithmic Death Phase. — During this phase the "rate of death per organ-
ism" remains constant, the plot of the logarithms gives a straight line with a
negative slope. This is represented by f — g.
Life Phases in a Bacterial Culture 111
If death and life in the bacterial cell could be regarded as reversible proc-
esses, we might expect the appearance of an eighth phase, a negative accelera-
tion of the death rate.
It may be noted that the seven phases previously defined are in a
sense arbitrary. The curve, if it could be plotted with data absolutely
accurate, would probably be smooth ; in other words, the portions des-
ignated as straight lines are probably curves, but with curvature so
slight that they may be treated mathematically as straight lines without
the introduction of any error commensurate in value with the inevitable
These various life phases of the bacterial culture will be discussed
in some detail.
I. Initial Stationary Phase
Spore-producing bacteria exhibit this growth phase particularly
well. If a suspension of bacterial spores be placed in a suitable culture
medium microscopic observation will show that growth does not appar-
ently begin immediately. There can be no increase in numbers until
the spores have germinated and begun to multiply. Samples of equal
volume taken during this period show no increase in numbers. While
this stage is most prominent with sporulating organisms it is by no
means always absent in nonspore formers, as is shown by results of
Lane-Claypon, 1 Penfold, 2 and others. In other words, there is evi-
dence that in old cultures of many bacteria the cells are in a relatively
dormant stage, the physiologic equivalent of sporulation though with-
out the spore morphology. When such cells are planted in a suitable
medium there will be an appreciable interval before a single cell will
have resumed growth sufficiently to divide. During this phase the rate
of increase per cell would be zero, and the average generation time
infinity. The equation of the curve which represents this phase
b = B
where b = Number of bacteria after time t,
B = Initial number of bacteria.
Little work has been done on this phase. The conditions which determine
its length are probably those influencing the length of other phases.
It should be noted that some cultures will not show this phase at all. If
there are any actively dividing bacteria in the inoculum at the time of inocula-
tion it will be absent or very transitory. It is therefore probable that the phase
of the culture from which the transfer is made will affect the length of this
2 Jour. Hyg., 1914, 14, p. 215.
112 R. E. Buchanan
II. Lag Phase, or Phase of Positive Growth
This phase apparently has not been differentiated from the pre-
ceding by previous writers. This is illustrated by the definition of
latent period given by Chesney : 8 "By latent period or lag is meant the
interval which elapses between the time of seeding and the time at
which maximum rate of growth begins." The necessity for differen-
tiation of the two phases is not urgent except when the first is long
The lag phase may be defined as that period elapsing between the
beginning of multiplication and the beginning of the maximum rate
of increase per organism.
The phenomenon of bacterial lag was apparently first noted by Miiller,* and
was later studied by Rahn 5 and by Coplans." Penfold 2 gave the first adequate
discussion of the various theories which might be suggested to explain the
phenomenon. Chesney 8 later made a careful study of the lag phase with spe-
cial reference to the growth of the pneumococcus. A mathematical analysis of
the lag phase was given by Penfold and Ledingham' and elaborated by Slator. 8
Theories of Bacterial Lag. — Penfold 2 has enumerated some nine
different theories as to the cause of bacterial lag, all of which he dis-
cards as inadequate. Inasmuch as certain of these have been main-
tained by other writers, and perhaps some discarded hastily, they will
be briefly summarized and reasons for discarding given under the
following seven heads.
1. The organism must excrete some essential substance into the medium
before maximal growth can occur. Experiments show, however, that sub-
cultures taken from cultures showing maximal growth do not show any lag
2. Adaptation to a new medium requires time. This must be discarded,
inasmuch as transfers to the same medium may show lag.
3. Some of the bacteria transferred are not viable, and die off early. Inas-
much as enumeration is by plating and not by direct counting, the organisms
not viable would never be enumerated.
4. Bacteria may agglutinate and plating would then be an enumeration of
clumps and not of individual bacteria. While this may be a factor in some
cases, it cannot explain the lag which still persists when adequate precautions
against confusion from this source have been taken.
» Jour. Exper. Med., 1916, 24, p. 387.
* Ztschr. f. Hyg. u. Infektionskr., 1895, 20, p. 245.
5 Centralbl. f. Bakteriol., 1906, Abt. 2, 16.
* Jour, of Path, and Bacteriol., 1909, 14, pp. 1-27.
* Jour. Hyg., 1914, 14, p. 242.
s Ibid., 1917, 16, p. 100.
Life Phases in a Bacterial Culture 113
5. Accumulated products of metabolism may injure the bacterial cell, the
length of the lag phase is the time required to recover from the injury. Penfold
rejects this explanation as inadequate. Chesney, however, insists that lag is
"an expression of injury which the bacterial cell has sustained from its previ-
ous environment." This conception may well be an approximation of the truth,
though probably not entirely accurate.
6. "The inoculum consists of organisms having individually different powers
of growth, and during the lag the selection of the quick growing strain occurs
in response to some selecting agent in the peptone." It is possible that this
might occur in cultures which were not "pure lines," or which contained sev-
eral strains, but there is no proof of its occurrence in pure strains. While there
are undoubtedly some differences in the rates of multiplication of individual
bacteria in the same culture, they are insufficient to account for the great dif-
ferences characteristic of the lag period.
7. Bacteria must overcome an "inertia" before reaching maximal growth
rate. Penfold dismisses this on the basis of certain experiments in which he
chilled rapidly growing bacteria, and thus stopped multiplication, which was
resumed at its former rate when the optimum temperature was restored. While
it is probable that Penfold is correct in discarding inertia in this sense as a
factor, nevertheless a modification of this theory is in the opinion of the writer
the only adequate one suggested.
The explanation favored by Penfold is in fact a variant of the last. He
believes that certain essential constituents of the bacterial protoplasm, probably
synthesized in steps, must be present in the bacterial cell in optimum concen-
tration or at least the intermediate bodies of the steps of the synthesis. When
the bacteria cease growing these intermediate bodies diffuse from the cell and
disappear, and before maximal growth can begin in a new medium these bodies
must again be synthesized. This theory in effect holds that the loss of these
substances gives rise to inertia. During the lag phase the bacteria are gradu-
ally recovering from injury.
It is probable that none of the preceding are wholly satisfactory
explanations of the lag phase. An explanation more in accord with
observed facts may be found in the assumption by the bacterial cells of
a "rest period" comparable to the resting stages so often assumed by
higher forms. It is a well known fact that at certain stages in the life
history of many plants certain cells or tissues are developed which pass
into a resting stage. When these are morphologically well differenti-
ated they are termed spores, sclerotia, etc., in the lower forms, and
seeds, bulbs, tubers, etc., among the higher types. In many other cases
cells or tissues pass into a similar resting stage as a result of cer-
tain environmental influences, without showing marked morphologic
differentiation. These resting cells are usually aroused to renewed
growth and activity only as the result of certain stimuli. The cold of
winter followed by the warmth of spring may be the stimulus which
causes buds to develop. Some seeds will germinate only after the seed
coat has decayed or has been scratched or corroded by acid. Bacterial
spores form at certain stages in the life history of the bacteria, but do
114 R. E. Buchanan
not usually germinate in the parent culture in spite of abundant moist-
ure, food and optimum temperature. Germination takes place under
the stimulus of change to some new medium. It is altogether probable
that most bacteria, whether spore producers or not, enter into such a
resting stage. When not morphologically differentiated as a spore this
resting period is probably more transitory than in a spore, but it is
nevertheless just as real.
What happens, then, when a considerable number of bacteria in the
resting stage are transferred to a medium suitable for development?
If we were to examine the culture microscopically we would find that
the bacteria would not all begin development at once, probably for the
same reason that seeds placed under uniform favorable conditions for
growth do not all germinate at the same instant. Cell division will
occur in a few cells first, followed by larger and larger numbers at suc-
ceeding intervals of time until a maximum has been reached and
passed, and at last all the cells have "germinated." As soon as a cell
has actually germinated, there would seem to be no a priori reason why
the cells should not thereafter multiply rapidly, showing practically at
once a minimum generation time. There is no more reason to suppose
that the length of time it takes a bacterial cell to germinate will affect
its subsequent rate of growth than 10 assume that plants derived from
seeds slow in sprouting grow more slowly than those from seeds
soonest sprouted. After any cell had once "germinated" then, it would
proceed to increase in numbers in geometrical progression. Theoret-
ically the lag period would continue until the last viable cell had started
to multiply; practically, however, it ceases before this as the rapid
increase in the bacteria which have germinated soon makes the ungerm-
inated cells such a small fraction of the whole number that their
inclusion is within the limits of error of measurement of the numbers
MATHEMATICAL ANALYSIS OF THE LAG PERIOD
The lag period has been previously defined as that period during which
there is an increase in the average rate of multiplication of the bacteria, an
increase from zero to some constant which is maintained during the succeeding
period. Another statement is that it includes the period during which there is
a decrease in the average generation time. It should be noted that when
used in this sense, the term generation time means the time required for the
bacteria to double in numbers, if they continued growing at the same rate. At
any given instant during this period there will be some cells not multiplying at
all, these at that particular instant would have an infinite generation time, and
the term average generation time would cease to have any meaning. In other
words, during this period there is an acceleration in the rate of growth.
Life Phases in a Bacterial Culture 115
Let us first examine the equation of growth if the rate of increase per organ-
ism should remain constant, that is, the average generation time should not
b = number of bacteria after time t,
B = initial number of bacteria.
It is evident that the rate of increase in number of bacteria at any instant will
vary directly as the number of bacteria, or expressed in terms of the calculus :
db/dt = kb where k is a constant. Now
Therefore k is the rate of growth per organism, or the velocity coefficient of
growth. On integration this becomes,
In b = kt + constant of integration.
The constant of integration is found to be In B by taking t = 0. The equa-
tion then becomes
In b/B = kt
This may be interpreted as the equation of a straight line, hence when lnb is
plotted against t, a straight line with slope k will be secured.
The curve showing the number of bacteria after any time may be derived
from the above equation
In b/B = kt
b/B = e"
and b = Be kt
This equation may be derived without resort to the calculus as follows :
Let n = number of generations in time t
g = generation time.
At the end of time t one organism will have produced 2° bacteria, then
b = B2°
Now n = t/g
... b = B 2V*
Let 2V* = e*
then b = Be kt
and k = 1n2/g
Since the rate of growth varies inversely as the generation time, k may be
regarded as this rate of growth per organism, or the velocity coefficient.
Inasmuch as rate of growth per organism is a function of time, it is a mat-
ter of interest to determine just what the relationship may be existing between
them. Penfold and Slator have suggested relationships empirically determined
from experimental data. Apparently there has been no attempt to derive the
relationship from theoretical considerations.
Assume that the lag phase represents the time required for all the viable
bacteria planted to "germinate." Take as the time of "germination" the instant
that the cell first divides to form two individuals. It is assumed that as soon
as an organism begins dividing its rate of increase is at once constant. Let
this be k'.
Let w = number of bacteria that are dividing after time t, = progeny of all
bacteria that have germinated within time t
z = number of bacteria that have not germinated.
116 R. E. Buchanan
The rate of increase per organism [f(t)] at any instant is given by the follow-
ing equation :
f(t)= = k'. 1/(1 + z/w) (1)
w + z
It is apparent that if the numbers of bacteria "germinating" during each unit
of time are plotted against time, a curve may be secured resting on the x axis
at both ends, one of the forms of a probability curve. The general equation
for such a curve has been shown by Karl Pearson to be
y = c(l + x/aO m i(l — x/a,) m =
in which mi and m 2 are constants, c equals the maximum ordinate, and — aj
and a 2 are intercepts of curve with x axis.
Let y be the number of bacteria germinating at time t, then
y = c (1 + t/a,) ra i(l — t/a 2 ) m 2
and ai + a 2 is the total length of the lag period.
The total number of bacteria which will germinate in time dt is ydt. Since
the number of bacteria developing after time t from one organism is 2 l / s , those
which will develop after time t from those beginning growth during time dt
is y2V g dt, and the total number of bacteria developed after time t from those
starting growth within that time is
Therefore w = c(l + t/ai) m i(l — t/a 2 ) m 2 2V*dt
— a >
The total number of bacteria which "germinate" within time t is
The total number not germinated is
B — [ ydt
. • . z = B — ydt
The relationship probably existing between rate of growth per organism and
t may be shown by substituting the values secured for w and z in the
f (t) = k'
B— T c(l + t/aO m i(l — t/a.) m 2dt
1 + - J -^
c(l + t/ai) m i(l — t/a 2 ) m 2 2V«dt
— a i
All efforts to simplify this expression or put it into usable form have thus far
failed. The only points where the exact relationship is known are when t = 0,
Life Phases in a Bacterial Culture 117
f (t) =0, and when t = ai + a 2 , f(t) = k'. It is evident that the relationship
existing between rate of growth per organism and t during the lag period is
The problem may also be attacked by the empirical derivation of a formula
for a plotted curve by a critical examination of the data of the lag phase. This
has been done by Ledingham and Penfold. These authors first reduced all
figures to a seeding of 1, that is, the numbers of bacteria found at successive
stages of the lag phase were divided by the initial number of bacteria. The
logarithms of these numbers were plotted against the logarithms of times. This
gave a curve which appeared to be logarithmic. The logarithms of the log-
arithms of the numbers of bacteria were then plotted against the logarithms
of the times. These points were found to lie approximately on a straight line.
If n is the slope of this line, and c the intercept with the x axis, the equation
of the line is
log (log b)
n = (2)
(log t)— c
From this they derive the equation
t n = k log b
Since In b = In 10. log b
In b = In lO.tVk = k't"
where = k'
Therefore b = e k ' tn and for a seeding of B bacteria b = Be k ' tB
It is evident that this equation and the equation for regular growth
b = Be kt
are special forms of the equation
b = Be**"
in which j«=f (t). In the equation for constant rate of growth per organ-
ism, /* = 1, and in the Ledingham- Penfold equation m = f" 1 , and in the equation
of initial stationary phase /* = and b = B. The equation developed,
b = Bek't"
has two constants which must be evaluated for each particular experiment.
An equation of this general form was tested out by Ledingham and Penfold
(1914) on data from eight series of experiments, and was found to give remark-
ably consistent results. The value of n in these experiments varied from 1.56
to 2.7 six being below 2.0. The value of k in the equation
f = k log b/B,
varied from 2329 to 1,045,000.
The tables given by Chesney for increase of bacteria during the lag period
afford an opportunity for testing independently the validity of the Ledingham-
Penfold equation, or its generalized form.
Slator after a study of the data of Penfold (1914) concluded that in every
experiment recorded there existed a relationship between the two constants
n and k such that an equation could be derived in which there would appear
118 R. E. Buchanan
but one undetermined constant n. He found by examination that the follow-
ing relationship always held:
= constant = 2.024
k = 105.7"
Substituting the value for k in the Ledingham-Penfold equation
f = k log b/B
f = n 105.7" log b/B = 105.7" log bVB"
Slator uses the general form of equation
kf = log b/B
This becomes (.00945 )°f = log bVB".
While the equation as developed holds for the work of Penfold, Slator gen-
eralizes into the form
Ct" = log b°/B n
in which C might have some value other than .00945. This can be put into the
form of the equations
bn = Bnl0C n t°
or bn = B n ek°f
The advantage of Slator's generalized equation over that of Ledingham and
Penfold, at least for the lag period, is not apparent.
MEASUREMENT OF LAG
A numerical expression indicating the amount of lag may be
secured in either of two principal ways: (a) An expression may be
secured which will involve directly the length of the lag period, this
may be termed "period of lag measurement;" (b) an expression may
be secured which will give a numerical value to the degree of depres-
sion of rate of multiplication at any time during the progress of the
lag period. This may be termed the "time index of lag."
(a) Period of Lag Measurement: Three suggestions have been
made as to methods of measuring lag in terms involving the length of
the lag period. These have been defined by Penfold.
1. The actual length of the lag period may be measured.
2. Coplans (1909) measured the restraint of growth in terms of
minimum generation time. It may be expressed by the formula
f — ng
where t = length of lag period
n — number of generations during lag period,
g — minimum generation time.
Life Phases in a Bacterial Culture 119
3. The average generation time for the first part of the period may
be compared with that of any succeeding period.
(b) Time Index of Lag: The degree or amount of lag at any
instant during the lag period may have a numerical value assigned to it
in either of two ways ; the ratio of the generation time at any instant
to the minimal generation time characteristic of the logarithmic period
of increase may be determined, or, the rate of change or increase per
organism at any given instant during the lag period may be compared
with the similar rate of increase per organism during the logarithmic
period. Inasmuch as the rate of growth must vary inversely as the
generation time, it is evident that these two methods of expressing
results will have a constant ratio.
1. Measurement of Lag by Comparison of Generation Times. — The
problem is to secure the ratio of the generation time of the bacteria at
any time during the lag phase to the minimal generation time. It
should be recalled that the term generation time as used here is not a
time average, but that length of time required for the bacteria present
to double in number if the average rate increase per individual
It was earlier developed that the expression
b = B 2V«
represents the equation of growth if the rate of increase per individual remains
constant. Differentiating and solving for g,
g = b In 2 dt/db (1)
The value of dt/db may be determined for the lag phase by differentiation of
either of the equations
b = Bekt" (2)
or b" = Bn e k°t» (3)
db/b = knf-'dt
dt/db = 1/bknf- 1 (4)
Substituting the value of dt/db in (1)
b In 2
g = = In 2/knt"- 1
The ratio between the value of generation time as determined by this formula
during the period of lag and the minimum value of g as determined during
the logarithmic period gives a numerical index to the degree of lag at any instant.
If the equation
bn = Bnek°t"
be chosen as the more general for the lag period (as developed from the work
of Slator), the expression for generation time becomes:
g = In 2/k ,, t ,, - 1
120 R. E. Buchanan
2. Measurement of Lag by Comparisons of Rates of Increase Per
Cell. — The work of Slator suggests the possibility of measuring lag at
any instant during the lag phase by a comparison of the rates of
increase per cell with similar rates for the logarithmic period.
This may be determined from either lag phase equation
b = Bett-
or t>n = Bne k °t n
db/dt = bknt"" 1
The rate of increase per organism at any instant is therefore
= knf- 1
The corresponding rate of increase per organism during the logarithmic
The ratio knt"~'/k' gives the numerical index desired.
If the second equation of the lag phase be employed the ratio becomes
k n t n -7k'
It may be noted that the so-called "constant of growth" during the lag period,
the expression — r — , used by Slator and termed z is directly proportional to
the ^ of the equation
b = Be/*"
III. The Logarithmic Phase
The logarithmic phase of bacterial growth in a culture is that time
during which there is a maximum rate of growth per organism, that is,
the time during which a certain minimum generation time is main-
tained. The various relationships which define this period have for the
most part been developed in the discussion of the lag phase. They are
as follows :
If B = number of bacteria at beginning of logarithmic period,
b = number of bacteria after time t,
n = number of generations in time t,
g = generation time,
k = velocity coefficient of growth,
b = B 2" = B 2V« = Be kt
t In 2
In b— In B
In b— In B
k = 1/t. In. b/B
Life Phases in a Bacterial Culture 121
This phase of bacterial growth has perhaps been more investigated than
any other. The mathematical relationships during this period are comparatively-
simple. It is evident that any effect of change of environmental conditions
on the rate of increase of bacteria will be manifested through a change in the
generation time. For every variable in the environment there is of course an
optimum for each kind of organism, that is, a condition or concentration such
that the generation time is minimal.
There is need for careful mathematical study of the effect of temperature
changes, changes in concentration of nutrients, of hydrogen ions, of inhibiting
substances, etc., on the rate of growth. It will be noted that the equation
k = 1/t . In . b/B
is one form of the expression for the value of the velocity coefficient of a
monomolecular reaction. It has been shown that a similar (not identical)
expression holds for the logarithmic death period of bacteria. Will the fol-
k = 1/tC" . In b/B
hold where C is the concentration of some nutrient or inhibiting substance,
and n a constant?
The temperature coefficient per degree or per 10 degree rise in temperature
is in need of study, particularly near the minimum and maximum growth tem-
peratures. This temperature coefficient over certain ranges has been deter-
mined for some bacteria. Lane-Claypon gives the value per 10° and 35° as
2 to 3 with B. coli. Similar results were secured from 20° to 30° by Hehewerth
(1901) and Barber (1908).
IV. Phase of Negative Acceleration of Growth
It is a matter of common laboratory observation that bacteria do
not long maintain their maximum rate of growth, the logarithmic phase
does not usually persist more than a few hours in quick growing types
of bacteria. The average generation time apparently lengthens until
at the close of the period the bacteria are no longer dividing.
The general equation of this portion can be written, as for the pre-
b = Be/*"
During this phase the /a varies as some function of the time, from the
1 of the logarithmic period to 0. Apparently the exact relationship
between fi and t during this phase has not been studied. It is apparent
that as t increases /i must decrease, but a mathematical characterization
has not been successful. The reasons for the decreased rate of growth
per organism are complex. Among them may be enumerated the
1. The average rate of growth per cell will decrease with the
increase in concentration of the injurious products of metabolism.
2. The average rate of growth per organism will decrease with
decrease in the available food supply, or with some single limiting fac-
tor of this food supply.
122 R. E. Buchanan
3. As the period progresses a larger and larger proportion of the
cells go into the "resting stage" and are withdrawn from those dividing
4. It is probable that before this period is completed some cells die.
Slator has suggested that the curve might be described by
bn = BnealW
where n, k and a are constants suitably adjusted. Until further data
are accumulated an attempt to evaluate these constants will prove diffi-
cult. From analogy with preceding and succeeding equations, it is
possible that the growth equation of this phase might be
b = Bek^ n and i* = f"-'
V. The Maximum Stationary Phase
During this period there is theoretically no change in the total num-
ber of bacteria present. If we still employ the useful general expression
b = Be""
fi during this time remains zero, and the number of bacteria is
Persistence of this phase must mean the balancing of increase and
death. The rate of increase of bacteria must be such as to quite
exactly make good the loss from death.
Investigations as to the length of this phase, and the influence of
environment upon it are needed. With some organisms the phase is
very transitory if it can be said truly to occur at all, with other forms
apparently it persists for some time before there is marked any ten-
dency to decrease in numbers.
VI. Phase of Accelerated Death Rate
Sooner or later the number of bacteria which die in a unit time will
exceed the increase. In other words, as soon as bacteria reach the
"resting stage" we may assume that they begin to die off, but they do
not all reach this stage at the same instant. For some time there is
an acceleration in the rate of death. The jak of the equation
b = Be""
varies from zero to the velocity coefficient (constant) of the logarith-
mic death period. It also becomes negative in sign. It increases
numerically in value during this period as time increases. During this
stage the curve apparently is just the reverse of that of the lag period.
Life Phases in a Bacterial Culture 123
It is not improbable that the equation of the curve during this period
will be found to be
b = Be-k'f
When t — 0, b = B. As t increases, b will be found to decrease more
and more rapidly. Data are not at hand to prove the reliability of this
equation. This stage probably does not persist long in most cultures,
the velocity coefficient of death soon reaching a certain maximum.
VII. Logarithmic Death Phase
It was first shown by Madsen and Nyman and later by Chick that
when bacteria are subjected to the action of unfavorable environment
such as the presence of disinfectants they die off in accordance with
the law which governs monomolecular reactions. If the logarithms of
the numbers of surviving bacteria after various lengths of time are
plotted against time, they will be found to lie on a straight line. The
slope of this line is negative. This slope is the velocity coefficient of
— k = 1/t . In b/B
or k = 1/t . In B/b
The equation of the curve of the surviving bacteria is
b = Be-"
or B = be"
This behavior of the bacteria has been abundantly verified by experi-
mentation. It has been found to be of great service in the evaluation
The effect of concentration of disinfectants has been developed
principally by the work of Paul, Bierstein and Reuss, and of Chick 9
and the results generalized by Phelps. It is found that a change in the
concentration of a particular disinfectant will change the velocity
coefficient of the death rate in accordance with the following
k = k'C"
where k' is the velocity coefficient of the original, and k the velocity
coefficient with new concentration C, and n is a constant. For a differ-
ent concentration the equation then becomes,
k'= — In —
and the equation of the curve of surviving bacteria becomes
b = Be- k c n t
» Jour. Hyg., 1912, 12, p. 414.
124 R. E. Buchanan
Determination of the values of k and n for a disinfectant and a com-
parison of these values with those determined for some standard, as
phenol, constitute efficient characterization of the disinfectant.
The Rideal-Walker and the Hygienic Laboratory phenol coefficients of dis-
infectants are determined by the use of facts inherent in these formulae. If
the same concentration of two disinfectants are to be compared, we may place
the same number, B of bacteria per unit of solution in each, and determine
the length of time it takes to reduce the number of living bacteria to less than
one per loop. Under these conditions the time required to change b to a cer-
tain number b' is determined. The only undefined quantities left are t and k
in the equations
b' = Be-*"'
b' = Be"*'""
.-. k't' = k"t"
or k'/k" = t'7t'
that is, the velocity coefficients are inversely proportional to the times required
for "disinfection." By determining variations in the values of these ratios
with different concentrations one may approximate the values of n in the
If, in addition, the effect of heat be determined in accelerating the death rate
of the bacteria, a relatively complete diagnosis of the characteristics of the
disinfectant is at hand.
1. There are at least seven life phases during the development of a
culture of bacteria.
2. The general equation which represents the curve of the plot of
numbers of bacteria against time is b = BeM kt .
3. During the first or initial stationary stage /* is equal to zero, b is
equal to B and there is no change in the numbers of bacteria.
4. During the second or lag phase /* is a function of time, increasing
with time from to 1. The relationship between /* and time is com-
plex, but it is approximated by the equation
n = f- 1
and the growth curve equation becomes
b = Beta"
5. During the third or logarithmic growth phase /u, = 1 and the
b = Be"
where k is related to the minimum generation time as follows :
and the equation of the growth curve is
b = Be" = Be (t "■"/*
Life Phases in a Bacterial Culture 125
6. During the fourth period or phase of negative growth accelera-
tion ju. decreases from the 1 of the logarithmic period of growth to 0.
It is a function of time, decreasing with time. The relationship is
complex, and no satisfactory evaluation of /* in terms of constants and
time has been secured. It is possible the equation of growth may
assume the form
b = Bekt-" and n = r"" 1
7. During the fifth period or maximum stationary phase /x remains
equal to and b equals B.
8. During the sixth period or phase of accelerated death rate //.
varies from to — 1. From analogy with the lag phase, the equation
of growth during this phase may be
b = Be-kf, and n = — t"- 1
9. During the seventh period or phase of logarithmic decrease fi
remains constant at -1, the growth curve having the equation
b = Be""
10. The lag phase is interpreted as the time during which bacteria
are gradually emerging from a resting stage. It is not improbable that
the numbers of bacteria which emerge at various successive periods of
time are distributed in accordance with some probability curve.