Dlebruck & Luria original experiment

Indiana University, Bloomingion, Indiana, and
va’enderbilt Universily, Nashville, Tennessee
Received May 29, 1943
W HEN a pure bacterial culture is. attacked by a bacterial virus, the cul- ture will clear after a few hours due to destruction of the sensitive ceIls
by the virus. However, after further incubation for a few hours, or sometimes
days, the culture will often become turbid again, due to the growth of a bacterial
variant which is resistant to the action of the virus. This variant can be
isolated and freed from the virus and will in many cases retain its resistance
to the action of the virus even if subcultured through many generations in the
absence of the virus. While the sensitive strain adsorbed the virus readily, the
resistant variant will generally not show any affinity to it.
The resistant bacterial variants appear readily in cultures grown from a
single cell. They were, therefore, certainly not present when the culture was
started. Their resistance is generally rather specific. It does not extend to
viruses that are found to differ by other criteria from the strain in whose presence
the resistant culture developed. The variant may differ from the original
strain in morphological or metabolic characteristics, or in serological type or in
colony type. Most often, however, no such correlated changes are apparent,
and the variant may be distinguished from the original strain only by its resistance
to the inciting strain of virus.
The nature of these variants and the manner in which they originate have
been discussed by many authors, and numerous attempts have been made to
correlate the phenomenon with other instances of bacterial variation.
The net effect of the addition of virus consists of the appearance of a variant
strain, characterized by a new stable character-namely, resistance to the
inciting virus. The situation has often been expressed by saying that bacterial
viruses are powerful “dissociating agents.” While this expression summarizes
adequately the net effect, it must not be taken to imply anything about the
mechanism by which the result is brought about. A moment’s reflection will
show that there are greatly differing mechanisms which might produce the
same end result.
D’HERELLE (1926) and many other investigators believed that the virus
by direct action induced the resistant variants. GRATIA(1 921), BURNET(I 929),
and others, on the other hand, believed that the resistant bacterial variants
are produced by mutation in the culture prior to the addition of virus. The
*Aided by grants from the D A Z IF~OU NDATIOFNOR MEDICALR ESEARCHan d from the


Theory by M. D., experiments by S. E. L.
GENETICSa 8: 491 November 1943
virus merely brings the variants into prominence by eliminating all sensitive
Neither of these views seems to have been rigorously proved in any single
instance. BURNET’S(1 929) work on isolations of colonies, morphologically
distinguishable prior to the addition of virus, which proved resistant to the
virus comes nearest to this goal. His results appear to support the mutation
hypothesis for colony variants. It may seem peculiar that this simple and important
question should not have been settled long ago, but a close analysis of
the problem in hand will show that a decision can only be reached by a more
subtle quantitative study than has hitherto been applied in this field of research.
Let us begin by restating the basic experimental finding.
A bacterial culture is grown from a single cell. At a certain moment, the
culture is plated with virus in excess. Upon incubation, one finds that a very
small fraction of the bacteria survived the attack of the virus, as indicated by
the development of a small number of resistant colonies, consisting of bacteria
which do not even adsorb the virus.
Let us focus our attention on the first generation of the resistant variantthat
is, on those bacteria which survive immediately after the virus has been
added. These survivors we may call the “original variants.” We know that
these bacteria and their offspring are resistant to the virus. We may formulate
three alternative hypotheses regarding them.
a. Hypothesis of mutation to immunity. The original variants were resistant
before the virus was added, and, like their offspring, did not even adsorb it. On
this hypothesis the virus did not interact at all with the original variants,
the origin of which must be ascribed to “mutations” that occur quite independently
of the virus. Naming such hereditary changes “mutations” of course
does not imply a detailed similarity with any of the classes of mutations that
have been analyzed in terms of genes for higher organisms. The similarity
may be merely a formal one.
b. Hypothesis of acquired immunity. The original variants interacted with
the virus, but survived the attack. We may then inquire into the predisposing
cause which effected the survival of these bacteria in contradistinction to the
succumbing ones. The predisposing cause may be hereditary or random. Accordingly
we arrive at two alternative hypotheses-namely,
bl. Hypothesis of acquired immunity of hereditarily predisposed individuals.
The original variants originated by mutations occurring independently of the
presence of virus. When the virus is added, the variants will interact with it,
but they will survive the interaction, just as there may be families which are
hereditarily predisposed to survive an otherwise fatal virus infection. Since we
know that the offspring of the original variants do not adsorb the virus, we
must further assume that the infection caused this additional hereditary
change. .
bz. Hypothesis of acquired immunity-hereditary aj[er infection. The original
variants are predisposed to survival by random physiological variations in
size, age, etc. of the bacteria, or maybe even by random variations in the
point of attack of the virus on the bacterium. After survival of such random
individuals, however, we must assume that their offspring are hereditarily
immune, since they do not even adsorb the virus.
These alternative hypotheses may be grouped by first considering the origin
of the hereditary difference. Do the original variants trace back to mutations
which occur independently of the virus, such that these bacteria belong to a
few clones, or do they represent a random sample of the entire bacterial population?
The first alternative may then be subdivided further, according to
whether the original variants do or do not interact with the virus. Disregarding
for the moment this subdivision, we may formulate two hypotheses:
I. First hypothesis (mutation): There is a finite probability for any bacterium
to mutate during its life time from “sensitive” to “resistant.” Every
offspring of such a mutant will be resistant, unless reverse mutation occurs.
The term “resistant” means here that the bacterium will not be killed if exposed
to virus, and the possibility of its interaction with virus is left open.
2. Second hypothesis (acquired hereditary immunity): There is a small finite
probability for any bacterium to survive an attack by the virus. Survival of
an infection confers immunity not only to the individual but also to its offspring.
The probability of survival in the first instance does not run in clones.
If we find that a bacterium survives an attack, we cannot from this information
infer that close relatives of it, other than descendants, are likely to survive the
The last statement contains the essential difference between the two hypotheses.
On the mutation hypothesis, the mutation to resistance may occur
any time prior to the addition of virus. The culture therefore will contain
“clones of resistant bacteria” of various sizes, whereas on the hypothesis of
acquired immunity the bacteria which survive an attack by the virus will be
a random sample of the culture.
For the discussion of the experimental possibility of distinction between
these two hypotheses, it is important to keep in mind that the offspring of a
tested bacterium which survives is resistant on either hypothesis. Repeated
tests on a bacterium at different times, or on a bacterium and on its offspring,
could therefore give no information of help in deciding the present issue. Thus,
one has to resort to less direct methods. Two main differences may be derived
from the hypotheses:
First, if the individual cells of a very large number of microcolonies, each
containing only a few bacteria, were examined for resistance, a pronounced
correlation between the types found in a single colony would be expected on
the mutation hypothesis, while a random distribution of resistants would be
expected on the hypothesis of acquired hereditary immunity. This experiment,
however, is not practicable, both on account of the difficulty of manipulation
and on account of the small proportion of resistant bacteria.
Second, on the hypothesis of resistance due to mutation, the proportion of
resistant bacteria should increase with time, in a growing culture, as new
mutants constantly add to their ranks.
In contrast to this increase in the proportion of resistants on the mutation
hypothesis, a constant proportion of resistants may be expected on the hypothesis
of acquired hereditary immunity, as long as the physiological conditions
of the culture do not change. To test this point, accurate determinations of
the proportion of resistant bacteria in a growing culture and in successive subcultures
are required. In the attempt to determine accurately the proportion
of resistant bacteria, great variations of the proportions were found, and results
did not seem to be reproducible from day to day.
Eventually, it was realized that these fluctuations were not due to any uncontrolled
conditions of our experiments, but that, on the contrary, large
fluctuations are a necessary consequence of the mutation hypothesis and that
the quantitative study of the fluctuations may serve to test the hypothesis.
The present paper will be concerned with the theoretical analysis of the
probability distribution of the number of resistant bacteria to be expected on
either hypothesis and with experiments from which this distribution may be
While the theory is here applied to a very special case, it will be apparent
that the problem is a general one, encountered in any case of mutation in uniparental
populations. It is the belief of the authors that the quantitative study
of bacterial variation, which until now has made such little progress, has been
hampered by the apparent lack of reproducibility of results, which, as we shall
show, lies in the very nature of the problem and is an essential element for its
analysis. It is our hope that this study may encourage the resumption of quantitative
work on other problems of bacterial variation.
The aim of the theory is the analysis of the probability distributions of the
number of resistant bacteria to be expected on the hypothesis of acquired
immunity and on the hypothesis of mutation.
The basic assumption of the hypothesis of acquired hereditary immunity
is the assumption of a fixed small chance for each bacterium to survive an attack
by the virus. In this case we may therefore expect a binomial distribution
of the number of resistant bacteria, or, in cases where the chance of survival
is small, a Poisson distribution.
The basic assumption of the mutation hypothesis is the assumption of a
fixed small chance per time unit for each bacterium to undergo a mutation to
resistance. The assumption of a fixed chance per time unit is reasonable only
for bacteria in an identical state. Actually the chance may vary in some manner
during the life cycle of each bacterium and may also vary when the physiological
conditions of the culture vary, particularly when growth slows down on
account of crowding of the culture. With regard to the first of these variations,
the assumed chance represents the average chance per time unit, averaged
over the life cycle of a bacterium. With regard to the second variation, it
seems reasonable to assume that the chance is proportional to the growth rate
of the bacteria. We will then obtain the same results as on the simple assumpMUTATIONS
tion of a fixed chance per time unit, we agree to measure time in units of division
cycles of the bacteria, or any proportional unit.
We shall choose as time unit the average division time of the bacteria, divided
by In 2, so that the number Nt of bacteria in a growing culture as function
of time t follows the equations
(1) dNt/dt = N,, and Nt = Noet.
We may then define the chance of mutation for each bacterium during the
time element dt as
(2) adt,
so that a is the chance of mutation per bacterium per time unit, or the “mutation
If a bacterium is capable of different mutations, each of which results in
resistance, the mutation rate here considered will be the sum of the mutation
rates associated with each of the different mutations.
The number dm of mutations which occur in a growing culture during a
time interval dt is then equal to this chance (2) multiplied by the number
of bacteria: or
(3 ) dm = adtN,,
and from this equation the number m of mutations which occur during any
finite time interval may be found by integration to be
(4) m = a(Nt - No)
or, in words, to be equal to the chance of mutation per bacterium per time unit
multiplied by the increase in the number of bacteria.
The bacteria which mutate during any time element dt form a random
sample of the bacteria present at that time. For small mutation rates, their
number will therefore be distributed according to Poisson’s law. Since the
mutations occuring in different time intervals are quite independent from each
other, the distribution of all mutations will also be according to Poisson’s law.
This prediction cannot be verified directly, because what we observe, when
we count the number of resistant bacteria in a culture, is not the number of
mutations which have occurred, but the number of resistant bacteria which
have arisen by multiplication of those which mutated, the amount of multiplication
depending on how far back the mutation occurred.
If, however, the premise of the mutation hypothesis can be proved by other
means, the prediction of a Poisson distribution of the number of mutations
’ We assume that the number of resistant bacteria is at all times small in comparison with the
total number of bacteria. If this condition is not fulfilled, the total number of bacteria in this
equation has to be replaced by the number of sensitive bacteria. The subsequent theoretical developments
will then become a little more complicated. For the case studied in the experimental
part of this paper the condition is fulfilled.
may be used. to determine the mutation rate. It is only necessary to determine
the fraction of cultures showing no mutation in a large series of similar cultures.
This fraction PO, according to theory, should be:
( 5 ) po = e-m.
From this equation the average number m of mutations may be calculated,
and hence the mutation rate a from equation (4).
Let us now turn to the discussion of the distribution of the number of
resistant bacteria.
The average number of resistant bacteria is easily obtained by noting that
this number increases on two accounts-namely, first on account of new mutations,
second on account of the growth of resistant bacteria from previous
mutations. During a time element dt the increase on the first account will be,
by equation ( 3 ) : adtNt. Nt, the number of bacteria present at time t, is
given by equation (I). The increase on the second account will depend on the
growth rate of the resistant bacteria. In the simple case, which we shall
treat here, this growth rate is the same as that of the sensitive bacteria, and
the increment on this account is p dt, where p is the average number of resistant
bacteria present at time t. We have then as the total rate of increase
of the average number of resistant bacteria dp/dt = aNt+p and upon integration
(6) p = taNt
if we assume that at time zero the culture contained no resistant bacteria.
It will be seen that the average number of resistant bacteria increases more
rapidly than the total number of bacteria. Indeed the fraction of resistant
bacteria in the culture increases proportionally to time. This, as pointed out
in the introduction, is a distinguishing feature of the mutation hypothesis
but unfortunately, as will be seen in the sequel, is not susceptible to experimental
verification due to statistical fluctuations.
The resistant bacteria in any culture may be grouped, for the purpose of
this analysis, into clones, taking together all those which derive from the
same mutation. We may say that the culture contains clones of various age
and size, calling “age” of a clone the time since its parent mutation occurred
and “size” of a clone the number of bacteria in a clone at the time of observation.
It is clear that size and age of a clone determine each other. If, in particular,
we make the simplifying hypothesis that the resistant bacteria grow
as fast as the normal sensitive strain, the relation between size and age will be
expressed by equation (I), with appropriate meaning given to the symbols.
The relation implies that the size of a clone increases exponentially with its
age. On the other hand, the frequency with which clones of different ages may
be encountered in any culture must decrease exponentially with age, according
to equations (3) and (I).
Combining these two results-namely, that clone size increases exponentially
with clone age and that frequency of clones of different age decreases
exponentially with clone age-we see that the two factors cancel when the
average number of bacteria belonging to clones of one age group is considered.
In other words, at the time of observation we shall have, on the average, as
many resistant bacteria stemming from mutations which occurred during the
first generation after the culture was started as stemming from mutations
which occurred during the last generation before observation, or during any
other single generation.
On the other hand, for small mutation rates it is very improbable that any
mutation will occur during the early generations of a single or of a limited number
of experimental cultures. It follows that the average number of resistant
bacteria derived from a limited number of experimental cultures will, probably,
be considerably smaller than the theoretical value given by equation (6), and,
improbably, the experimental value will be much larger than the theoretical
value. The situation is similar to the operation of a (fair) slot machine, where
the average return from a limited number of plays is probably considerably
less than the input, and improbably, when the jackpot is hit, the return is
much bigger than the input.
This result characterizes the distribution of the'number of resistant bacteria
as a distribution with a long and significant tail of rare cases of high numbers of
resistant bacteria, and therefore as a distribution with an abnormally high variance.
This variance will be calculated below.
For such distributions the averages derived from limited numbers of samples
yield very poor estimates of the true averages. Somewhat better estimates of
the averages may in such cases be obtained by omitting, in the calculation of
the theoretical averages, the contribution to these averages of those events
which probably will not occur in any of our limited number of samples. We
may do this, in the integration leading to equation (6), by putting the lower
limit of integration not at time zero, when the cultures were started, but at a
certain time t o , prior to which mutations were not likely to occur in any of our
experimental cultures. We then obtain as a likely average r of the number of
resistant bacteria in a limited number of samples, instead of equation (6),
r = (t - to)aNt.
It now remains to choose an appropriate value for the time interval t - to.
For this purpose we return to equation (4, in which it was stated that the
average number of mutations which occur in a culture is equal to the mutation
rate multiplied by the increase of the number of bacteria. Let us then choose
t o such that up to that time just one mutation occurred, on the average, in a
group of C similar cultures, or
I = aC(Nt, - NO).
In this equation we may neglect No, the number of bacteria in each inoculum,
in comparison with Nto, the number of bacteria in each culture at the critical
time to. We may also express Nto in terms of Nt, the number of bacteria at the
time of observation, applying equation (I) :
N = N,e-(t-to). t o
We thus obtain
(7) t' - to = ln(NtCa).
Equations (6a) and (7) may be combined to eliminate t-to and to yield
a relation between the observable quantities r and Nt on the one hand and
the mutation rate a on the other hand, to be determined by this equation:
(8) r = aNtln (NtCa).
This simple transcendental equation determining a may be solved by any
standard numerical method. In figure I, the relation between r and aNt is
plotted for several values of C.
FIGUR1E.- The value of aNt as a function of r for various values of C. The upper left hand part
of the figure gives the curves for low values of aNt and of r on a larger scale. See text.
Estimates of a obtained from equation (8) will be too high if in any of the
experimental cultures a mutation happened to occur prior to time to. From the
definition of t o it will be seen that this can be expected to happen in little more
than half of the cases.
While we have thus obtained a relation permitting an estimate of the mutation
rate from the observation of a limited number of cultures, this relation is
in no way a test of the correctness of the underlying assumptions and, in particular,
is not a test of the mutation hypothesis itself. In order to find such
tests of the correctness of the assumption we must derive further quantitative
relations concerning the distribution of the number of resistant bacteria and
compare them with experimental results.
Since we have seen that the mutation hypothesis, in contrast to the hypothesis
of acquired immunity, predicts a distribution of the number of resistant
bacteria with a long tail of high numbers of resistant bacteria, the determination
of the variance of the distribution should be helpful in differentiating between
the two hypotheses. We may here again determine first the true variance-
that is, the variance of the complete distribution-and second the likely
variance in a limited number of cultures, by omitting those cases which are
not likely to occur in a limited number of cultures.
The variance may be calculated in a simple manner by considering separately
the variances of the partial distributions of resistant bacteria, each
partial distribution comprising the resistant bacteria belonging to clones of
one age group. The distribution of the total number of resistant bacteria is
the resultant of the superposition of these independent partial distributions.
Each partial distribution is due to the mutations which occurred during a
certain time interval d7, extending from (t-.) to (t-T+dr). The average
number of mutations which occurred during this interval is, according to
equation (3),
(9) dm = aN,dT = aNte-rd7.
These mutations will be distributed according to Poisson’s law, so that the
variance of each of these distributions is equal to the mean of the distribution.
We are however not interested in the distribution of the number of mutations
but in the distribution of the number of resistant bacteria which stem from
these mutations at the time of observation-that is, after the time interval r.
Each original mutant has then grown into a clone of size er. The distribution
of the resistant bacteria stemming from mutations occurred in the time interval
d7 has therefore an average value which is er times greater than the average
number of mutations, and a variance which is e2r times greater than the variance
of the number of mutations. Thus we find for the average number of
resistant bacteria:
dp = aNtdr,
and for the variance of this number
vanp = aNterd7.
From this variance of the partial distribution, the variance of the distribution
of all resistant bacteria may be found simply by integrating over the appropriate
time interval-that is, either from time t to time o (7 from o to t), if
the true variance is wanted, or from time t to time t o (7 from o to t- to), if the
likely variance in a limited number of cultures is wanted. In the first case we
obtain :
(10) var, = aNt(et - I).
In the second case we obtain:
(1oa) var, = aNt[e(t-to) - I].
Substituting here the previously found value of (t-to) and neglecting the
second term in the brackets, we obtain:
(11) varr = Ca2Nt2.
Comparing this value of the likely variance with the value of the likely
average, from equation (8), we see that the ratio of the standard deviation
to the average is:
(12) d\/vBT,/r = &/In (NtCa).
It is seen that this ratio depends on the logarithm of the mutation rate and
will consequently be only a little smaller for mutation rates many thousand
times greater than those considered in the experiments reported in this paper.
In the beginning of this theoretical discussion we pointed out that the
hypothesis of acquired immunity leads to the prediction of a distribution of
the number of resistant bacteria according to Poisson’s law, and therefore to
the prediction of a variance equal to the average. On the other hand, if we compare
the average, equation (8), with the variance, equation (11), (not, as above,
with the square root of the variance), we obtain
( 1 4 varr = rNtCa/ln (NtCa).
Equation (Iza) shows that the likely ratio between variance and average is
much greater than unity on the hypothesis of mutation, if (NtCa), the total
number of mutations which occurred in our cultures, is large compared to
It is possible to carry the analysis still further and to evaluate the higher
moments of the distribution function of the number of resistant bacteria, or
even the distribution function itself. The moments are comparatively easy to
obtain, while the calculation of the distribution function involves considerable
6 In some of the experiments reported in the present paper we did not determine the total
number of resistant bacteria in each culture, but the number contained in a small sample from
each culture. In these cases the variance of the distribution of the number of resistant bacteria
will be slightly increased by the sampling error. The proper procedure is here first to find the
average number of resistant bacteria per culture by multiplying the average per sample by the
volume of culture
volume of sample’
second, to evaluate the mutation rate with the help of equation (8); third, to figure the likely
variance for the cultures by equation (11); fourth, to divide this variance by the square of the
ratio (13) to obtain that part of the variance in the samples which is due to the chance distribution
of the mutations. The experimental variance should be greater than this value, on account of
the sampling variance. The sampling variance is in all our cases only a small correction to the
total variance, and it is sufficient to use its upper limit, that of the Poisson distribution, in our
calculations. Consequently, when comparing the experimental with the calculated values, we
first subtract from the experimental value the sampling variance, which we take to be equal to
the average number of resistant bacteria.
mathematical difficulties. An approximation to the beginning of the distribution
function-that is, to its values for small numbers of resistant bacteriamay
be obtained by grouping mutations according to the bacterial generation
during which they occurred. For instance, the probability of obtaining seven
resistant bacteria may be broken down into the sum of the following alternative
events: (a) seven mutations during the last generation; jb) three mutations
during the last generation and two mutations one generation back; (c) three
mutations during the last generation and one mutation two generations back;
(d) one mutation during the last generation and three mutations one generation
back; (e) one mutation during the last generation, one mutation one generation
back and one mutation two generations back.
The probability of each of these events depends only on the mutation rate
and on the final number of bacteria.
The grouping of mutations according to the bacterial generation during
which they occurred, and the assumption that the bacteria increase in simple
geometric progression, simplify the calculation sufficiently to permit numerical
computation. On the other hand, the classes with two, four, eight, etc., mutants
are artificially favored by this procedure, so that a somewhat uneven
distribution results, with too high values for two, four, eight, etc., resistant
bacteria (see fig. 2 ) .
The material used for our experimental study consisted of a bacterial virus
CY and of its host, Escherichia coli B (DELBRUCK and LURIA 1942). Secondary
cultures after apparently complete lysis of B by virus a show up within a few
hours from the time of clearing. They consist of cells which are resistant to
the action of virus CYb, ut sensitive to a series of other viruses active on B.
The resistant cells breed true and can be established easily as pure cultures.
No trace of virus could be found in any pure culture of the resistant bacteria
studied in this paper. The resistant strains are therefore to be considered as
Tests were made to see whether the resistance to virus a was a stable character
of the resistant strains. In the first place, it was found that virus a is
not appreciably adsorbed by any of the resistant strains. In the second place,
when a certain amount of virus CY is mixed with a growing culture of a resistant
strain, no measurable increase of the titer of virus a occurs over a period of
several hours. This is a very sensitive test for the occurrence of sensitive bacteria,
and its negative result for all resistant strains shows that reversion to
sensitivity must be a very rare event.
Morphologically at least two types of colonies of resistant bacteria may be
distinguished. The first type of colony is similar to the type produced by the
sensitive strain both in size and in the character of the surface and of the
edge. The second type of colony is much smaller and translucent. The difference
in colony type is maintained in subcultures. Microscopically the bacteria
from these two types of colonies are indistinguishable. They also do not differ
from each other or from the sensitive strain in their fermentation reactions on
common sugars and in the characteristics of their growth curves in nutrient
broth. In particular, the lag periods, the division times during the logarithmic
phase of growth, and the maximum titers attained are identical for the sensitive
strain and for the two variants. Both variants, therefore, fulfill the requirements
for the applicability of the theory developed above.
In the presentation of our experimental results we have lumped the counts
of the two types of colonies together, because: (I) theoretically, this is equivalent
to summing the corresponding mutation rates; (2) experimentally, we
are not certain whether each of these types does not actually comprise a diversity
of variants; (3) experimentally, no correlation appeared to exist between
the occurrence of these variants, which shows the independence of the causes
of their occurrence.
Cultures of B were grown either in nutrient broth (containing .5 percent
NaC1) or in an asparagin-glucose synthetic medium. In the latter, the division
time during the logarithmic phase of growth was 35 minutes, as compared
with rg minutes in broth. In synthetic medium, the acidity increased during
the time of incubation from pH 7 to pH 5.
In cultures of strain B, between IO-^ and 10-6 of the bacteria are found
usually to give colonies resistant to the action of virus a when samples of such
cultures are plated with large amounts of virus. In order to be reasonably certain
that the resistant bacteria found in the test had not been introduced into
the test culture with the initial inoculum, the test cultures were always started
with very small inocula, containing between 50 and 500 bacteria from a growing
culture. Thus any resistant bacterium found at the moment of testing
(when the culture contains between IO* and 5x10b~ac teria/cc) must be an
offspring of one of the sensitive bacteria of the inoculum.
All platings were made on nutrient agar plates. The plating experiments for
counting the number of resistant bacteria in a liquid culture of the sensitive
strain were done by plating either a portion or the entire culture with a. large
amount of virus a. The virus was plated first, and spread over the entire surface
of the agar. A few minutes later the bacterial suspension to be tested was
spread over the central part of the plate, leaving a margin of at least one centimeter.
Thus all bacteria were surrounded by large numbers of virus particles.
Microscopic examination of plates seeded in this manner showed that lysis
takes place very quickly; only bacteria which at the time of plating were in
the process of division may sometimes complete the division. The resistant
colonies which appear after incubation are therefore due to resistant bacterial
cells present at the time of plating.
The total number of bacteria present in the culture to be tested was determined
by colony counts in the usual manner.
The resistant colonies of the large type appear after 12-16 hours of incubation,
the colonies of the small type appear after 18-24 hours, and never reach
half the size of the former ones. Counts were usually made after 24 and 48
A Test of the Reliability of the Plating Method
In our experiments we wanted to study the fluctuations of the numbers of
resistant bacteria found in cultures of sensitive bacteria. It was therefore
necessary to show first that the method of testing did not involve any unrecognized
variables, which caused the number of resistant colonies to vary from
plate to plate or from sample to sample.
Therefore, parallel platings were made using a series of samples from the
same bacterial culture. If our plating method is reliable, fluctuations should
in this arrangement be due to random sampling only, and the variance from a
series of such samples should be equal to the mean.
Table I gives the results of three such experiments. It will be seen that in
The number of resistant bacteria in different samples from the same culture.
EXP. NO. loa EXP. NO. 11a EXP. NO. 3
all three cases variance and mean agree as well as may be expected. There is
therefore no reason to assume that the method of sampling or plating introduces
any fluctuations into our results besides the sampling error.
Fluctuations of the Number of Resistant Bacteria in Samples from a
Series of Similar Cultures
As pointed out in the introduction and in the theoretical part, the hypothesis
of acquired immunity and the hypothesis of mutation lead to radically
different predictions regarding the distribution of the number of resistant
bacteria in a series of similar cultures. The hypothesis of acquired immunity
predicts a variance equal to the average, as in sampling, while the mutation
hypothesis predicts a much greater variance.
Series of five to IOO cultures were set up in parallel with small equal inocula,
and were grown until maximum titer was reached. Three kinds of cultures
were used-namely: (I) 10.0 cc aerated broth cultures; (2) .2 cc broth cultures;
(3) .2 cc synthetic medium cultures.
The results of all tests for the number of resistant bacteria are summarized
in table 2 and table 3.
The number of resistan! bacteria in series of similar cultures.
EXPERIMENT NO. I IO I1 15 16 17 zia zIb
Number of cultures 9 8 10 10 2 0 1 2 I9 5
Volume of cultures, cc 10.0 10.0 10.0 10.0 .2* .2' .2 10.0
Volume of samples, cc .OS .OS .OS .OS .08 .08 .OS .OS
Culture No.
10 29 30 6 I I 0 38
18 41 10 5 0 0 0 28
125 I7 40 I O 3 0 0 35
10 2 0 45 8 0 7 0 107
14 31 183 24 0 0 8 13
27 30 I2 13 5 303 I
3 7 173 165 0 0 0
17 17 23 15 5 0 I
17 57 6 0 3 0
51 10 6 48 15
107 I 0
0 4 0
0 19
0 0
I 0
0 17
0 11
64 0
0 0
Average per sample 26.8 23.8 62 26.2 11.35 30 3.8 48.2
Variance (corrected for
sampling) 1217 84 3498 2178 694 6620 40.8 I171
Bacteriaperculture 3.4X10'0 4 XIO'Q 4 X1ol0 2.gX1olQ 5.6X108 5 XIO~ I . I X I O ~ 3.2X10'~
Mutation rate 1.8X10-8 1.4X10-8 4.1Xio-8 Z.IXIO-8 1.1x1o-6 3.0X10-8 3.3Xro-* 3.0x10-8
L ~ - Standard deviation exp. 1.3 .39 .g5 1.8 2.3 2.7 1 . 7
Average per culture 5360 4760 12400 5240 28.4 . 75 15.1 8440
Average {calc. .35 .33 .33 .37 .94 ,67 1.04 . 2 6

  • Cultures in synthetic medium.

It will be seen that in every experiment the fluctuation of the numbers of
resistant bacteria is tremendously higher than could be accounted for by the
sampling errors, in striking contrast to the results of plating from the same
culture (see table I) and in conflict with the expectations from the hypothesis
of acquired immunity.
We want to see next whether these results fit the expectations fromthe
hypothesis of mutation. We must therefore compare the experimental results
with the relations developed in the theoretical part, keeping in mind that the
theory contains several simplifying assumptions.
First we can compare, according to equation (12), the experimental and the
calculated values of the ratio between the standard deviation and the average
of the numbers of resistant bacteria. These ratios are included in tables 2 and
3. It is seen that the experimental and theoretical values are reasonably close.
However, in all but one case the experimental ratio is greater than the value
calculated from the theory-that is, the variability is even greater than predicted.
Distribution of the numbers of resistant bacteria in series of similar cdfures.
Number of cultures I O 0 87
Volume of cultures, cc .2* .2*
Volume of samples, cc -05 . 2
6- IO
21- 50
51- IO0
201- 500
11- 20
101- 200
Number o j
6- IO
21- 50
51- I00
201- 500
11- 20
101- 200
Number of
Average per sample IO. I2
Average per culture 40.48
Variance (corrected for sampling) 6270
Bacteria per culture
Mutation rate
Standard deviation exp.
Average i calc.
2.8X IO*
2.3X IO-*
2.4 Xro*
2.37 x 106

  • Cultures in synthetic medium.

A part of this discrepancy may be accounted for by the fact that the
time to, mutations occurring prior to which were disregarded by the theory,
was chosen in such a manner that on the average one mutation would occur
prior to time to. This mutation, if it occurs, will of course tend to increase the
variance, and in some of the experiments the high value of the experimental
variance can be traced directly to one exceptional culture in which a mutation
had evidently occurred several generations prior to time to. Unfortunately,
there is no general criterion by which one might eliminate such cultures from
the statistical analysis, because, in a culture with an exceptionally high count
of resistant bacteria, these do not necessarily stem from one exceptionally
early mutation, but may also be due to an exceptionally large number of
mutations after time to.
There may also be other reasons why the observed variances are higher than
the expected ones. First of all, the simplifying assumption that the mutation
rate per bacterial generation is independent of the physiological state of the
bacteria may be too simple. If the mutation rate is higher for actively growing
bacteria than for bacteria near the saturation limit of the cultures, early mutations
and big clone sizes will be favored, and therefore higher variations of the
numbers of resistant bacteria can be expected. Second, the assumption of a
sudden transition from sensitivity to resistance may also be too simple. It is
conceivable that the character “resistance to virus” may not fully develop in
the bacterial cell in which the mutation occurs, but only in its offspring, after
one or more generations. However, if this were the case, cultures with only one
or two resistant bacteria should be relatively rare. The last experiment listed
in table 3, in which the entire cultures were plated, shows a rather high proportion
of cultures with only one resistant bacterium. This seems to show that the
FIGURE z.-Experimental (Experiment No. 23) and calculated distributions of the numbers
of resistpt bacteria in a series of similar cultures. Solid columns: experimental. Cross-hatched
columns: calculated.
character “resistance to virus” in general does come to expression in the bacterial
cell in which the corresponding mutation occurred, as assumed by the
Another way of comparing the experimental results with the theory is to
compare the experimental distribution of resistant bacteria with the approximate
distribution calculated by the method outlined at the end of the theoretical
part. The theoretical distribution has to be calculated from the average
number of mutations per culture given by equation (5). Only experiments
wheqe the whole culture is tested can therefore be used for such a
comparison. This method tests the fitting of the expectations for small numbers
of resistant bacteria, in contrast to the comparison of the standard deviations,
which involves predominantly the cultures with high numbers of resistant
Figure 2 shows the experimental and calculated distributions for Experiment
No. 23; the cultures with more than nine resistant bacteria are lumped
together in one class, since the distribution has not been calculated for values
higher than nine.
It is seen that the fitting for small values is satisfactory. In particular, the
number of cultures with one resistant bacterium very closely fits the expectation.
The classes with two, four, eight, etc., resistant bacteria are bound to
be favored in the theoretical distribution, as explained in the theoretical part.
The results shown in figure 2 also confirm the assumption that the discrepancy
between experimental and calculated standard deviations must be
due to an excess of cultures with large numbers of resistant bacteria.
Summing up the evidence, we may say that the experiments show clearly
that the resistant bacteria appear in similar cultures not as random samples
but in groups of varying sizes, indicating a correlating cause for such grouping,
and that the assumption of genetic relatedness of the bacteria of such groups
offers the simplest explanation for them.
Mutation Rate
As pointed out in the theoretical part of this paper, mutation rates may be
estimated from the experiments by two essentially different methods. The
first method makes use of the fact that the number of mutations in a series of
similar cultures should be distributed in accordance with Poisson’s law; the
average number of mutations per culture is calculated from the proportion
of cultures containing no resistant bacteria at the moment of the test, according
to equation (5).
There are two technical difficulties involved in the application of this
method. In the first place, rather large numbers of cultures have to be handled
and conditions have to be chosen so that the proportion of resistant bacteria
is neither too small nor too large. In the second place, the entire cultures have
to be tested, which means, in our method of testing, that cultures of rather
small volume have to be used and great care must be taken to plate as nearly
as possible the entire culture.
Experiment No. 23 (see table 3) permits an estimate of the mutation rate
by this method. Out of 87 cultures, no resistant bacteria were found in 29
cultures, a proportion of .33. From equation ( 5 ) we calculate therefore that the
average number of mutations per culture in this experiment was 1.10. Since
the total number of bacteria per culture was 2.4X108, we obtain as the mutation
rate, from equation (4),
a = .47 X IO-^ mutations per bacterium per time unit

.32 X IO-* mutations per bacterium per division cycle.

This calculation makes use exclusively of the proportion of cultures containing
no resistant bacteria. It is therefore inefficient in its use of the information
gathered in the experiment.
The second method makes use of the average number of resistant bacteria
per culture. The relation of this average number with the mutation rate was
discussed in the theoretical part of this paper and was found to be expressed
by equation (8). The mutation rates calculated by this method for each experiment
are collected in table 4.
Values oj mutation rate from different experiments.
Mutations per bacterium
per time unil
I , 8X 10-8
I . 4X IO-^
2. I x 10-8
I. I x 10-8
3 .OX 10-8
3. OX 10-8
2.3X IO-^
2.4X 10-8
3.3 x IO-'
Average 2.45 x 1 6 '

  • Cultures in synthetic medium.

It will be seen that the values of the mutation rate obtained by the second
method are all higher than the value found by the first method. This discrepancy
may be traced back to the same cause as the discrepancy between
the calculated and observed values of the standard deviation of the numbers
of resistant bacteria. This, we found, was due to an excess of early mutations,
giving rise to big clones of resistant bacteria. These big clones do not affect
the mutation rate calculated by the first method, but they do affect the results
of the second method, which is based on the average number of resistant bacteria.
One sees in table 4 that the mutation rate calculated by the second method
does not vary greatly from experiment to experiment. In particular, it will be
noted that there is no significant difference between the values obtained from
cultures in broth and from cultures in synthetic medium, notwithstanding the
considerable difference of metabolic activity and of growth rate of the bacteria
in these two media. This shows that the simple assumption of a fixed small
chance of mutation per physiological time unit is vindicated by the results.
It may also be noted in table 4 that there is no significant difference between
the mutation rates obtained from IO cc cultures and those obtained from .2 cc
cultures, or between the experiments with many and those with few cultures.
The variability of the value of the mutation rate seems to be solely due to the
peculiar probability distribution of the number of resistant bacteria in series
of similar cultures predicted by the mutation theory.
At this point an experiment may be mentioned by which it was desired to
find out whether or not mutations occur in a culture after the bacteria have
ceased growing. A culture was grown to saturation and was then tested repeatedly
for resistant bacteria and for total number of bacteria over several
days. The proportion of resistant bacteria did not change, even when the
sensitive bacteria began to die, showing that the resistant bacteria have the
same death rate in aging cultures as the sensitive bacteria.
We consider the above results as proof that in our case the resistance to
virus is due to a heritable change of the bacterial cell which occurs independently
of the action of the virus. It remains to be seen whether or not this is
the general rule. There is reason to suspect that the mechanism is more complex
in cases where the resistant culture develops only several days after lysis
of the sensitive bacteria.
The proportion of mutant organisms in a culture and the mutation rate are
far smaller in our case than in other studied cases of heritable bacterial variation.
The possibility of investigation of such rare mutations is in our case
merely the result of the method of detecting the mutant organisms. In other
cases, the variants are detected by changes in the colony type which is produced
by the mutant organism, either in the pigmentation or in the character
of the surface or the edge of the colony. Often, colonies of intermediate character
occur, and it is difficult to decide whether they are mixed colonies or stem
from bacteria with intermediate character. This is particularly true of cases
where, the mutation rate is high and where reverse mutation occurs. Fairly
high mutation rates, however, are a prerequisite of any study of colony variants,
since the number of colonies that can be examined is limited by practical
The study of mutations causing virus resistance is free of these difficulties.
The segregation of the mutant from the normal organisms occurs in the onecell
stage by elimination of the normal individuals, and the character of the
colony which develops from a mutant organism is of secondary importance.
Owing to the total elimination of the normal individuals, the number of organisms
which may be examined is very much higher than for any other method;
more than 108 bacteria may be tested on a single plate. Since the mutations
to virus resistance are often associated with other significant characters, the
method may well assume importance with regard to the general problems of
bacterial variation.
It must not be supposed that the peculiar statistical difficulties encountered
in our case are restricted to cases of very low mutation rates. The essential
condition for the occurrence of the peculiar distribution studied in the theoretical
part of this paper is the following: the initial number of bacteria in a culture
must be so small that the number oj mutations which occur during thefirst division
cycle o j the bacteria is a small number. This will always be true, however great
the mutation rate, if one studies cultures containing initially a small number of
In a series of very interesting studies of the color variants of Serratia marcescens,
BUNTING(1 g4oa, 194ob, 1942; BUNTINGan d INGRAH1A94M2) succeeded
to some extent in obviating the statistical difficulties by always using
inocula of about IOO,OOO bacteria. In some of her cases this number was sufficiently
high to result in numerous mutations during the first division cycle
of the bacteria. In other cases the number was apparently not high enough,
since the author reports troublesome variations of the fractions of variants in
successive subcultures. In those cases where the size of the inocula was high
enough, the author succeeded in deriving reproducible values for the mutation
rates from the study of single cultures, followed through numerous subcultures.
In these cases it is sufficient to apply the equations of the theory referring to
the average numbers of mutants as a function of time. It is clear, however,
that this method is applicable only in cases of mutation rates of at least IO+
per bacterium per division cycle.
In our case, as in many others, the virus resistant variants do not exhibit
any striking correlated physiological changes. There is therefore little opportunity
for an inquiry into the nature of the physiological changes responsible
for the resistance to virus. Since the offspring of the mutant bacteria, when isolated
after the test, are unable to synthesize the surface elements to which the
virus is specifically adsorbed in the sensitive strain, one might suppose that
this loss is a direct effect of the mutation. However, it is also conceivable that
the loss occurs upon contact with virus, since it is detected only after such
contact (hypothesis bl). In some of the cases studied by BURNET (192g),
where the mutational change to resistance is correlated with a change of phase,
from smooth to rough or vice versa, the change of the surface structure must
be a direct result of the mutation, since the mutant colonies may be picked up
prior to the resistance test and, when tested, exhibit the typical change of
affinity of the surface structure. These findings make it more probable that
the loss of surface affinity to virus is a direct effect of the mutation.
The alteration of specific surface structures due to genetic change is a phenomenon
of the widest occurrence. The genetic factors determining the antigenic
properties of erythrocytes are well known. There is evidence (WEBSTER
1937; HOLME1S9 38; STEVENSONS, CHULTZa, nd CLARK1 939) that resistance
or sensitivity to virus in plants and animals is correlated with, or even dependent
on, genetic changes, possibly affecting the antigenic make-up of the
cellular surface. The proof that resistance to a bacterial virus may be traced
to a specific genetic change may assume importance, therefore, with regard to
thegeneral problems of virus sensitivity and virus resistance.
The distribution of the numbers of virus resistant bacteria in series of similar
cultures of a virus-sensitive strain has been analyzed theoretically on the basis
of two current hypotheses concerning the origin of the resistant bacteria.
The distribution has been studied experimentally and has been found to
conform with the conclusions drawn from the hypothesis that the resistant
bacteria arise by mutations of sensitive cells independently of the action of
The mutation rate has been determined experimentally.
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strain 274. J. Bact. 40: 57-68.
1940b The production of stable populations of color variants of Serratia marcescens #274
in rapidly growing cultures. J. Bact. 40: 69-81.
1942 Factors affecting the distribution of color variants in aging broth cultures of Serratia
maicescens #274. J. Bact. 43: 593-606.
BUNTINGM, . I., and L. J. INGRAH1A94M2 , The distribution of color variants in aging broth
cultures of Serratia marcescens #274. J. Bact. 43: 585-591.
BURNET, F. M., 1929 Smooth-rough variation in bacteria in its relation to bacteriophage.
J. Path. Bact. 32: 15-42.
DELBRUCKM,. ,a nd S. E. LURIA1, 942 Interference between bacterial viruses. I. Arch. Biochem.
I: 111-141.
GRATIAA, ., 1921
D’HERELLEF,. , 1926 The Bacteriophage and Its Behavior. Baltimore: Williams and Wilkins.
HOLMES, F. O., 1938 Inheritance of resistance to tobacco-mosaic disease in tobacco. Phyto-
STEVENSOFN. ,J ., E. S. SCHULTaZn, d C. F. CLARK1,9 39 Inheritance of immunity from virus X
WEBSTERL,. T., 1937 Inheritance of resistance of mice to enteric bacterial and neurotropic virus
Studies on the d’Herelle phenomenon. J. Exp. Med. 34: 115-131.
pathology 28: 553-561.
(latent mosaic) in the potato. Phytopathology 29: 362-365.
infections. J. Exp. Med. 65: 261-286.

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