Journal of General Microbiology (1987), 133, 1871-1880.
Printed in Great Britain
1871
A Mathematical Method for Analysing Plasmid Stability in
Micro-organisms
By N E I L L S. C O O P E R , ' M I C H A E L E . B R O W N 2
A N D C E L I A A. C A U L C O T T 2 *
Department of Atmospheric Physics, The University, Oxford, UK
2Department of Fermentation, Celltech Ltd, 244 Bath Road, Slough, Berks SLI 4DY, UK
(Received 29 September 1986; revised 22 January 1987)
A mathematical model describing the instability of plasmids in micro-organisms has been
developed. The model is based on the assumption that the overall causes of plasmid instability
are described by the segregational instability of the plasmid, R (i.e. the rate at which plasmid-free
cells are generated from plasmid-bearing cells), and the growth rate difference, d p (i.e. the
difference in growth rate between plasmid-free and plasmid-bearing cells). A method for
determining the values of R and d p (accompanied by 95% confidence limits) for any plasmidbearing micro-organism is described. This method is based on the observation that, depending
on the plasmid, various exponential patterns of plasmid instability are observed. The stability of
Escherichia coli lB373(pMG169), where d p 9 R , and E. coli RV308(pHSG415), where R % dp,
are analysed in order to demonstrate the method.
INTRODUCTION
Interest in the study of plasmid instability in micro-organisms has increased with the
observation that many factors, both genetic and physiological, can alter the ability of a given
host to retain a plasmid (Wouters et al., 1980; Noack et al., 1981;Walmsley et al., 1983; Jones &
Melling, 1984; Primrose et al., 1984; Summers & Sherrat, 1984; Caulcott et al., 1985). In parallel
with these observations has been the development of various models for the mathematical
analysis of the instability of plasmids in bacteria (Bailey et al., 1983; Kim & Ryu, 1984; Noack et
al., 1984; Seo & Bailey, 1985). These models have not, in general, been used by the many workers
in the field. However, such models can provide valuable information about the nature of
plasmid instability, and with some models it may be possible to predict in advance whether a
plasmid will be unstable and if so, to what extent (Ataai & Shuler, 1985).
Any simple mathematical model of the instability of a plasmid in a micro-organism has
principally to consider two factors. These are the rate at which plasmid-free cells are produced
by the plasmid-bearing cells and the difference in the growth rates of the plasmid-bearing and
the plasmid-free cells. (The former factor is comprised of at least the segregation rate of the
plasmid and the mutation rate.)
Such a model was constructed by Moser (1958) to describe the genetic instability of a bacterial
population in continuous culture. The selective advantage conferred on the host cell by a
mutation and the rate at which such mutations occurred were the two factors considered to
affect the stability of the population. This model was used by Walmsley et al. (1983) to interpret
qualitatively their data describing the behaviour of various yeast plasmids in Saccharomyces
cerevisiae AH22. Kim & Ryu (1984) developed a similar model to Moser (1958) which they then
used to analyse data on the effect of expression of trp genes from the plasmid pVH5 on
the stability of that plasmid in Escherichia coli MV12. By fitting their data with a modelproduced curve, they could find approximate values for the growth rate difference and the
segregation coefficient. Noack et al. (1984) developed a simple version of Moser's model to
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M. E . B R O W N A N D C . A . CAULCOTT
determine the segregational instability and growth rate difference for cells grown in continuous
culture. This model, which only considered the relationship between the two parameters when
the growth rate difference was the dominant effect, was used to analyse some of their previously
published data.
In order to utilize fully the potential of mathematical models which describe the instability of
plasmids, we have developed a technique based on the observation (Moser, 1958; Walmsley et
a/., 1983; Caulcott, 1984) that the behaviour of unstable microbial populations follows one of
three patterns, each of which includes an exponential section. Hence, by taking the natural
logarithm of the appropriate parameter a graph with a substantial straight section is produced
from observed data. This new graph can then be analysed using standard linear regression
techniques, and from this analysis the growth rate difference and segregational instability can be
determined, accompanied by 95 % confidence limits. This paper describes a mathematical
method for carrying out such an analysis, and applies the method to data obtained from plasmidbearing cells grown in continuous culture. The method can also be directly applied to batch
cultures.
METHODS
Microbial strains andplasmids. The E. coli strains RV308 and 1B373were transformed with plasmids pHSG415
(Hashimoto-Gotoh et al., 1981) and pMG 169 (Celltech) respectively, and maintained on L-agar supplemented
with chloramphenicol (10 mg 1-l).
Media. For starter cultures of E. coli, L-agar supplemented with chloramphenicol (10 mg 1-l) and L-broth
[supplemented with chloramphenicol (20 mg 1-I) for E. coli RV308(pHSG415)] were used. E. coli
RV308(pHSG415)was grown in continuous culture in a defined salts medium (Caulcott et al., 1985)to give glucose
limitation. E. coli lB373(pMG169)was grown in the medium of Evans et al. (1970) also to give glucose limitation.
Antibiotics were not added to the medium used for continuous culture of either strain.
Chemostar experiments. For continuous culture experiments E. coli RV308bHSG415) was grown in a 1 litre
LH500 series fermenter vessel (LH Fermentation) at 30 "C, with the pH maintained at 6.5 by the addition of 3 MNaOH. The fermenter was stirred continuously at 700 r.p.m. and aerated with one volume of air per culture
volume per minute. The dissolved oxygen tension did not fall below 50% saturation. Using the appropriate defined
salts medium described above, the cells were grown under glucose limitation (all other nutrients being in excess)
at a dilution rate of 0.20 h-l.
E. coli 1B373(pMG169) was grown in a 2.5 1 fermenter vessel (MBR Bioreactor AG) at 34 "C with the pH
maintained at 7.0 by the addition of 3 M-NaOH. The fermenter was stirred continuouslyat 700 r.p.m. and aerated
with one volume of air per culture volume per minute. The dissolved oxygen tension did not fall below 30%
saturation. Using the appropriate defined medium the cells were grown under glucose limitation (all other
nutrients being in excess) at a dilution rate of 0.25 h-*.
In both cases a single colony of the desired strain was taken from the appropriate selective agar plate and
inoculated into 50 ml of liquid starter culture medium in a 250 ml conical flask with steel spring baffle. The culture
was incubated on an orbital shaker (30 "C, 220 r.p.m.) until stationary phase was reached. The cells were then
inoculated into a fermenter vessel containing the appropriate defined medium. When the culture reached an
optical density at 600 nm of at least 60% of the maximum supported by the medium, the pump was turned on and
the system run as a chemostat. In the case of E. coli RV308(pHSG415) chloramphenicol(20mg 1-I) was added to
the medium during the initial batch phase in the fermenter. The feed medium was antibiotic-free.
When the total biomass in the fermenter was constant, it was assumed that the initial transient growth phase
had ceased. Thereafter, the number of generations in the steady state was calculated.
Assay for plasmid instability. Samples were withdrawn from the continuouscultures, diluted and plated onto Lagar. One hundred single colonies were picked onto L-agar containing either carbenicillin (100 mg 1-1) [E. coli
lB373@MG169)]or chloramphenical(10 mg 1-I) [E. coli RV308(pHSG415)] and also onto L-agar as a control.
The number of colonies growing on the test plate was expressed as a proportion of the number growing on the
control plate. This value was taken to represent the proportion of the population which carried the plasmid.
In addition, a more sensitive technique, capable of detecting plasmid-free cells as 0.01% of the total cell
population, was used with E. coli 1B373(pMG169). Samples were withdrawn from the continuous culture and
various dilutions plated out onto L-agar. Duplicate dilutions were incubated both at 30 and 42 "C. Cells carrying
the plasmid pMG169 were unable to form colonies at 42 "C and those colonies which did appear were therefore
plasmid-free. The number of colonies growing at 42 "C was expressed as a proportion of the number growing at
30 "C. This value was taken to represent the proportion of the population carrying the plasmid.
Notation for the mathematical model. As with any mathematical description of a biological system, there is a
plethora of symbols, which are defined in Table 1. They are used in the model to describe features of an unstable
plasmid-bearing microbial culture.
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Table 1. Definition of symbols
Symbol
Pp+(=l - p J
PP+
M = P - - P+)
M
S
R (=M+S)
P-0, P+o
P*
Definition
Proportion of population comprising plasmid-free cells
Proportion of population comprising plasmid-bearing cells
Growth rate of p- cells (sen-')
Growth rate of p+ cells (sen-')
Growth rate difference (sen-'); also called selective advantage/disadvantage
Mutation rate (sen-l)
Segregation rate (sen-')
Rate of conversion from plasmid-bearing to plasmid-free cells. This is also
described as segregational instability
Dilution rate (h-l)
Dilution rate (gen-l). D = In 2 from its definition
Doubling time (h)
Time (in generations) since steady state of chemostat culture is achieved
Values of p-, p+ at t = 0
Limiting value of p- in case (c), dp < 0; ldpl % R
Notes.
(1) p is the specific growth rate of a cell, and is variable, unlike kax,
the maximum growth rate, which is
constant for a cell line under a given set of conditions.
(2) p-ep- +p+-p+= D.
(3) The values ofp- andp, are between 0 and 1;to convert to percentages multiply by 100. To convert from dp
to a percentage, multiply by (100/ln 2) = 144.93, and to convert from dp (per generation) to dp (h-l) divide by Td.
R E S U L T S A N D DISCUSSION
The mathematical model
It has been shown (Moser, 1958; Walmsley et al., 1983; Caulcott, 1984) that there are two
important physiological parameters which are involved in the generation of plasmid-free cells in
a population which is initially plasmid-bearing. These are the growth rate difference between
the two strains (defined here as d p ) and the rate at which plasmid-free cells are generated from
plasmid-bearing cells (R). It can be mathematically shown that, assuming R and d p are constant,
at time t the plasmid-free proportion (p-) of a given microbial population is
(The mathematical proof of this is given in Appendix A.) The above equation (1) can be
simplified by making appropriate approximations to model the behaviour of plasmid-bearing
systems. First note that, in general, p - o 6 1, 7 so equation (1) becomes:
The expression forp- can be simplified into one of three categories (Moser, 1958; Walmsley et
al., 1983; Caulcott, 1984).
(a) Growth rate difference much greater than segregational instability. (Mathematically this
can be expressed as dp 9 R.)
(b) Growth rate difference of similar size, or smaller than, segregational instability (dp I R).
(c) Negative growth rate difference of magnitude greater than segregational instability
(dp < 0 and Idpl B R).
In case (a), we can neglect the R term in the exponent and approximate the denominator by dp,
leading to
R
p - x (p-,
edP.1- -
+
The signs
2)
+ and % hereDownloaded
denote differences
of at least a factor of ten.
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N. S . COOPER, M . E . B R O W N A N D C. A . CAULCOTT
In case (b), we can neglect the p-,.dp term and then a binomial expansion leads to
so defining
For each case, the idealized behaviour of the cell population is depicted graphically (Figs 1,3
and 5). The right-hand graph in each of these figures is derived from the data in the left-hand
graph, but is transformed to produce a substantially linear plot. In addition, experimental
observations representing two of the situations are presented (Figs 2 and 4) accompanied by the
appropriate analysis of the data.
Growth rate direrence much greater than segregational instability (dp % R)
E. coli lB373(pMG169) was grown in continuous culture as described above, and assayed for
plasmid instability. Using the more sensitive technique, it was found that there was a gradual,
exponential increase in the proportion of the population which was plasmid-free (Fig. 2a). It
was therefore appropriate to plot the natural logarithm of the proportion of plasmid-free cells
against time (Fig. 2 b). In the linear region of this graph t > l / d p and so equation 3 (a)reduces to
and so the slope (m)and intercept (c) of the straight line can be expressed as
and therefore
dp = m ;
R = m(ec - P - ~ )
(4)
p - , is taken as being the proportion of the population which is plasmid-free at the beginning of
the experiment. The linear regression analysis used to calculate m and c was also used to
calculate their 95% confidence limits (see Appendix B). These values can then be used to derive
the 95% confidence limits for R and dp. The results for pMG169 are given in Table 2.
It can be seen that the growth rate difference between the plasmid-free cells and the plasmidbearing cells is significantly greater than the rate at which plasmid-free cells arise (Table 2).
However, the value of dp is 0.03 gen-l, which is well within the limits of the experimental error
of growth rate determination in batch culture. It is therefore apparent that a strength of this
mathematical technique is its sensitivity.
Plasmid pMG169 is derived from pMG 168 (Caulcott et al., 1985; Wright et al., 1987), a dual
origin vector which, when grown in E. coli, is capable of significant copy number amplification
on transfer from a growth environment at 34 "C to one at 38 "C. In addition, pMG169 encodes
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Mathematical analysis of plasmid stability
-
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-70
50
100
150
200
No. of generations
Fig. 1. Evolutionof a plasmid-bearing bacterial population when dp is much greater than R.Data were
and p-,, = 0.01. The calculated values of p - with time show a
obtained by using dp = 0-03, R =
sigmoidalpattern (a). Hence, by taking the natural logarithm of thep- values, the 6rst part of the curve
is transformed to a straight line (b).
n
0.45
-6.0
.
I
I
-7.0
180
60
120
180
0
No. of generations
Fig. 2. Stability of E. coli lB373(pMG169). E. coli lB373(pMG169) was grown in continuous culture
with glucose limitation at D = 0-25h-I . Sampleswere removed and assayed for the presence of plasmidfree cells using the very sensitive technique described in Methods. The proportion of the population
which was plasmid-free (a) and its natural logarithm (b) were plotted against generations in continuous
culture. 0 ,p - ; 0 , lnp-.
60
120
Table 2. Statistical analysis of the instability of E. coli IB373(pMG169) grown in continuousculture
Parameter
95 % confidence limits
4
R
rnt
ct
3-06 x
gen-I
8-10 x
gen-I
3-06x 10-2
- 8.3264
2.71 x
-4.09 x
to 3.41 x
to 24.3 x
f0.0035
f0-152
The values of R and dp with their 95% confidence limits can be calculated from the values of the slope (m)and
the intercept (c) obtained from Fig. 2(b), using equation 4 as described in the text.
the gene for chloramphenicol acetyltransferase (CAT) under the control of the trp promoter. At
34°C in the absence of tryptophan in the growth medium, there is detectable, low level
expression of CAT in E. coli 1B373(pMG169). The growth rate difference between E. coli 1B373
and the same strain carrying pMG169 is probably due to this expression of CAT, rather than due
to the actual presence of plasmid DNA, and further work is in progress to verify this. However,
it would be expected that pMG169 would show considerable segregational stability. This is
partly because the origin of replication functional at 34 "C is taken from plasmid pSClO1, which
has been found to be stable in E. coli under similar conditions (accompanying paper : Caulcott et
al., 1987). Furthermore, like pSC101, pMG169 includes a par sequence which should ensure
fidelity of segregation of the plasmid to each potential daughter cell. It is therefore important to
note that the mathematical analysis finds that the segregational instability (R)of pMG169 in
E. coli 1B373 is very low.
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N . S . COOPER, M . E . BROWN A N D C . A. CAULCOTT
+
-6.0
50
100
150
-
-8.01
200
0
No. of generations
I
100
50
1
200
I
150
Fig. 3. Evolution of a plasmid-bearing bacterial population when R is greater than or equal to dp. Data
were obtained by using dp = 0.02, R = 0.02 andp-, = 0.01. The calculated values ofp- with time show
an approximately exponentialdecay towards a value of 1 (a). Hence, by taking the natural logarithm of
the p + (= 1 - p - ) values, all but the initial section of the plot is transformed to a straight line (6).
0
n
10
30
50
I
(b)
-2.6
70
0 10
No. of generations
I
30
1
50
0
70
Fig. 4. Stability of E. coli RV308(pHSG415). E. coli RV308(pHSG415) was grown in continuous
culture with glucose limitation at D = 0.156 h-l. Samples of the culture were removed and assayed for
the presence of plasmid-bearingcells. The proportion of the population which was plasmid-free (a) and
the natural logarithm of the population of plasmid-bearingcells (b)were plotted against the numbers of
generations in continuous culture. 0 ,p - ; 0 , lnp,.
The early development of the plasmid-free population within the culture appears to be
anomalous, and cannot easily be fitted to the model described above. However, the results
obtained during the period 50 to 70 generations are believed to be due to the appearance of a
mutant plasmid-bearing strain (M. E. Brown, unpublished results). This mutation appears to
confer a growth advantage to the cells, leading to a temporary modification in the kinetics of the
appearance of plasmid-free cells. Once the plasmid-bearing population has become dominated
by the mutant form, the predicted kinetics of plasmid loss are restored, and the data can be
mathematically analysed as described above.
Growth rate difference of similar size, or smaller than, segregational instability (Idpl I R)
E. coli RV308(pHSG415) was grown in continuous culture as described above and assayed for
plasmid instability. Graphical analysis of the data (Fig. 4a) showed that there was a rapid,
initially linear, increase in the proportion of the cell population growing in the chemostat which
was plasmid-free, similar to the idealized Fig. 3(a). The data were replotted as the natural
logarithm of the plasmid-bearing cells (Fig. 4b) and the slope (m)and intercept (c) of the straight
section used to derive the values of R and dp. In this exponential phase t > l/(dp R), and
therefore equation 3(b) can be approximated by
+
and thus
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Table 3. Statistical analysis of the instability of' E. coli R V308(pHSG415)grown in continuous
culture
Parameter
R
dp
mt
ct
0.028 gen-l
0.0053 gen-l
-0.033
0.138
95% confidence limits
0.023 to 0.034
0.0028 to 0.0086
+ 0.0044
70.0569
t The values of R and dp with their 95 % confidencelimits can be calculated from the values of the slope (m)and
the intercept (c) obtained from Fig. 4(6), using equation 5 as described in the text.
m = -(dp+R);
c=ln(l+$)
Therefore solving for dp and R leads to
dp=rn(e-'- 1);
R = -m.e-'
(5)
Again, 95% confidence limits for m and c can be determined using the linear regression analysis
described in Appendix B. Using these values the 95% confidence limits for R and dp can then be
derived. The values of R and dp so calculated are given in Table 3.
When performing this analysis it is found that, in some cases, there is a clear deviation from
the straight line during the first few generations of continuous culture (Fig. 3b). This is due to the
effect of the growth rate difference. The more noticeable the deviation, the greater the value of
ldpl/R. If the deviation is downwards (as in Fig. 3b) dp is positive whereas if the deviation is
upwards dp is negative. In the case of E. coli RV308(pHSG415) it can be seen that the growth
rate difference is an order of magnitude less than the segregational instability, and that the
Occurrenceof plasmid-free cells can be primarily attributed to the segregational instability of the
plasmid. The probable explanation of this instability is the nature of pHSG415. It is a low-copynumber plasmid derived from pSClOl (Hashimoto-Gotoh et al., 1981), with a temperaturesensitive origin of replication and, more significant when discussing plasmid instability, almost
entirely absent stability functions (par) (Tucker et al., 1984).Thepar sequence has been shown to
in some way ensure segregation of at least one copy of a plasmid into each potential daughter cell
prior to cell division (Meacock & Cohen, 1980). It is known, from probability theory, that such a
function will not be significant with plasmids which are at a high copy number (Meacock &
Cohen, 1980). However, the absence of apar sequence or equivalent from a plasmid which is
only present in the cell at three to five copies per chromosome will undoubtedly increase the
probability of generating plasmid-free cells. Therefore, the segregational instability of
pHSG415 in E. coli RV308 is probably attributable to the low copy number of the plasmid and
the lack of a complete par sequence (see also accompanying paper: Caulcott et al., 1987).
Negative growth rate diflerence of magnitude greater than the segregational instability
(dp < 0; IdpI % R)
There will be some instances in which a plasmid is not stably maintained within the host cell,
yet does confer a growth rate advantage to the host. For example, Walmsley et al. (1983) found
that the plasmid pJDB219 was unstable in S. cerevisiae AH22. However, the plasmid was not
lost from the population, and 92% of the cells remained plasmid-bearing (see Fig. 6 of Walmsley
et al., 1983). It was found that the presence of pJDB219 imparted a growth rate advantage to the
host cells. Thus the data were the result of an unstable plasmid which was capable of conferring
a growth rate advantage to the cells. Similar results might be observed if a micro-organism
containing an unstable plasmid which coded for a specific drug resistance was grown in the
presence of that antibiotic. In such a situation, should the value of R be sufficiently small, the
proportion of plasmid-free cells will tend towards a constant proportion of the population which
is less than 1 (Fig. 5a). This constant value is R/ldpl (here calledp*). If the value ofp-, is less
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N . S . COOPER, M. E. BROWN A N D C . A . CAULCOTT
h
I
4
I
.3
C
c
No. of generations
Fig. 5. Evolution of a plasmid-bearing bacterial population when d p is negative. Data were obtained
by using dp=O-Ol, R=0.001 and p-,,=O.Ol. The calculated values of p - with time show an
approximately exponential decay towards a constant, p*, where p* = R / d p (a). For this example,
p* = 0.1. Hence, by taking the natural logarithm of (p* - p - ) , all but the initial section of the plot is
transformed to a straight line (b).
thanp* then segregational instability will dominate the growth rate advantage and there will be
an initial increase in the proportion of the population which is plasmid-free (Fig. 5a). (This is the
situation observed by Walmsley et al., 1983.). However, it is possible that p-o is greater than
p*, in which case there will be an initial decrease in the plasmid-free population in the
chemostat. In order to analyse the data (irrespective of the relationship between p-o and p*) a
plot of In Ip- (t) - p* I against time was made, giving a straight line (Fig. 5 b). The slope (m) is the
exponent in equation 3(c) and hence
Therefore
R
m = R - IdpI while p* = IdP I
The 95% confidence limits for R and dp could also be calculated using the method described in
Appendix B.
It has thus been shown that in each of the three categories of plasmid stability the use of
appropriate axes will result in a plot with a straight section. In all cases this straight line can be
used to estimate the segregational instability of the plasmid and the growth rate difference due to
the plasmid. This linear regression calculation will also yield 95 % confidence limits on R and dp.
APPENDIX A : THE MATHEMATICAL MODEL
For a constant volume of bacteria growing in continuous culture, with proportionp, plasmidbearing and p- plasmid-free cells, the growth of the plasmid-bearing population is equal to the
loss due to dilution at rate D, the gain from growth at rate p+ and the loss due to segregational
instability R. An equivalent equation applies to the growth rate of plasmid-free cells.
Mat hema t ically these are represented by
dP+
-- -DP+ +P+P+ -RP+
dt
+
+
-dp- - - Dp- p-pRp+
dt
Also, by definition, the sum of the plasmid-bearing cells and the plasmid-free cells is the total
population: p- +p+ = 1. Addition of A1 and A2 yields
O = -D+p+p++p-p-
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(A3)
Mathematical analysis of plasmid stability
1879
By use of the definition d p = p- - p+, equations for the growth rates of the plasmid-free and
plasmid-bearing cells can be derived
p- = D + p + d p
(A4)
p+ = D - p - d p
(A51
Substituting from A4 and A5 into A1 gives, after rearrangement
Assuming that R and d p are constant in time, this differential equation is of the Bernouille form
(Dwight, 1961) and can therefore be solved by setting z = l / p + .The solution can be rearranged
to show that
A P P E N D I X B: L I N E A R R E G R E S S I O N
If there are n data points ( x , y , ) , (x2y 2 ) ,. . . (x, y,,), then using the notation
1x = x , + x2 + . . . + x,
etc.
we can define
" , c 1Y
y = - n1- x ;
c
ssxy=
The least squares linear regression, y = mx
xy-
c
1
n
x cy
+ c, for these n data points is given by
m = SSxy/SSx
c=jJ-mE
With 95% confidence limits
where
and tn-2 is the f-statistic for n - 2 degrees of freedom. These results are standard
(see, e.g., Strike, 1981) but are reproduced here to be easily accessible to those who wish to use
the analysis method outlined in this paper.
When calculating the 95% confidence limits of R and d p it must be borne in mind that, for
instance, the largest value of R may occur for the smallest value of m,hence R and d p must be
calculated using all possible combinations of the 95% confidence limits of m and c, and the
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N . S . COOPER, M. E. BROWN A N D C . A . CAULCOTT
maximum and minimum results used. Also note that the 95% confidence limits on R and dp are
not symmetric about the predicted value due to the complex formula relating R and dp to m and
C.
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