Mechanisms for Plasmid Maintenance
Alexander J. H. Fedorec
CoMPLEX, University College London
Supervisors: Dr. Chris Barnes, Prof. Geraint Thomas
Acknowledgements: Tanel Ozdemir
August 28, 2014
Modern sequencing and manufacturing techniques have made it possible
to design and encode specific functions in to DNA. This allows the manufacture of complex genetic circuitry for use in a therapeutic setting. One
of the current methods of introducing such circuits in to an animal is by
implementing the circuit on a plasmid and transforming bacteria with that
plasmid. The bacteria are then introduced to the animal and the bacterial
machinery will carry out the processes encoded on the plasmid. A primary
concern when using this method is the loss of plasmids from the bacterial
population, due primarily to competition from plasmid-free bacteria. Currently, an antibiotic resistance gene is placed on the plasmid and antibiotics
are used in the environment in order to kill bacteria that lack the plasmid.
Evidently, in a medical setting it is not appropriate to keep a patient on an
antibiotics regime if it can be avoided. To that end, we build a modelling
system in which one can observe how plasmid maintenance is affected by
altering the various mechanisms which may control it. Further, we outline
a toxin-antitoxin system which may prove to be an effective mechanism to
control the growth of plasmid-free bacteria.
1
Synthetic Biology
Synthetic biology aims to develop engineered system that behave in a predictable manner with components adapted from existing biological solutions. Much current research is invested in developing and characterising the
building blocks from which we can make more complex systems for medical
(Ruder et al. 2011, Bacchus et al. 2013, Weber & Fussenegger 2011b), computational (Ausländer et al. 2012, Regot et al. 2010) and industrial purposes
(Keasling 2008). Optogenetic systems have been developed as a new tool
to aid in neuroscience research (Konermann et al. 2013), allowing specific
functionality to be switched on using precise beams of light. In a medical setting, synthetic biological systems have already been produced that
1
aid blood-glucose homeostasis in mice (Ye et al. 2011) and prevent cholera
virulence (Duan & March 2010).
Modularisation and Characterisation of Components As these systems are biological, the problems inherent in making predictions in stochastic systems apply. If the system, for example, relies on a small number of
molecules being detected by a small number of receptors, the random motion
of molecules within a cell will create large variety in outcomes. Although
such problems abound in the field, one of the primary goals of synthetic
biology is to produce robust, well characterised modules that can be treated
in a similar way to the components of electronic circuits. As such, toggle switches have been developed that respond to several different stimuli
(Kramer et al. 2004, Deans et al. 2007). Oscillators have been developed
that have tunable frequencies (Tigges et al. 2009). Logic gates have been
designed and “wired” together to create simple circuits (Ausländer et al.
2012) but a great deal of work is required to characterise the behaviour of
each component in a variety of environments and determine the effects that
other components may have when interacting in a circuit.
‘Architectural’ Considerations The ‘architecture’ most used for synthetic biological systems is the transcriptional gene network, designing relationships of inhibition and promotion of each constituent gene. Other
architectures, translational and post-translational, are described in (Khalil
& Collins 2010).
In addition to the choice of circuit architecture one must consider the
type of cells that will host the circuit. Research is being carried out using
both prokaryotic and eukaryotic cells. Mammalian qorum-sensing mechanisms (Weber & Fussenegger 2011a), tunable oscillators (Tigges et al. 2009)
and optogenetic devices (Ye et al. 2011) have all been developed. However, eukaryotic cells tend to have increased complexity, adding difficulty to
attempts to produce predictable behaviour.
When using bacterial cells there is a further consideration that must be
made; to introduce the circuit in to the cell by altering the host chromosome
or by transformation with a plasmid. The introduction of DNA in to the
chromosome of a cell is a cumbersome process, though the introduction of
CRISPR-Cas systems has, and will continue to, make the process more easily
achievable (Mali et al. 2013).
Our research aims at developing systems for microbiome therapeutics by
colonising the gut with our engineered bacteria. Escherichia coli Nissle 1917
is a very well studied bacteria which is approved for human consumption
and has been shown, without any genetic manipulation, to aid in the development of “enhanced natural immune responses” (Cukrowska et al. 2002)
and for the maintenance of ulcerative colitis remission (Kruis et al. 2004),
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among other things. By using plasmids as a vector, we avoid the need to
alter Nissle chromosome, allowing us to use the system therapeutically. Further, containment of the circuit on a plasmid provides more controllability
in terms of transcription and replication.
There are, however, potential downsides to using plasmids. The primary
issue being that of plasmid maintenance, which we describe later. There is
the possibility of mutation of the plasmid DNA. A single point mutation can
change a medium copy number plasmid to a high copy number plasmid (LinChao et al. 1992). Change in copy number can have drastic consequences on
the behaviour of the system and the survival or proliferation of the host cell.
There is also the concern of horizontal gene transfer, either from parts of the
plasmid getting incorporated in to the host chromosome or from the plasmid
being transferred between bacteria. It is believed that most recombinant
plasmids are non-conjugative (Ganusov & Brilkov 2002) i.e. they are not
capable of transfer to other cells, due to the lack of tra genes. However,
particularly with the use of antibiotic resistance genes in synthetic biology
we must be concerned by the possibility of transferring this ability to other
cells (Schuurmans et al. 2014).
2
Plasmid Maintenance
A problem of great importance for systems which use plasmids as vectors
regards plasmid maintenance. This is the problem of making sure that, on
cell division, both daughter cells carry at least one copy of the plasmid. As
it is the plasmid that codes for the system we have designed, we need to
maintain a population of bacteria that contain the plasmid for the system
to carry on working.
Differential growth rates Although the probability of producing plasmid free daughter cells can be very low, the effects of plasmid loss are
compounded by the difference in growth rates between plasmid-free and
plasmid-bearing cells. Although the exact link between growth rate and
division interval is still disputed, some believing there is a threshold mass
for each cell (Abner et al. 2014) while others believe there is target mass
increment (Amir 2014), increased metabolic burden does slow growth. Since
plasmid-bearing cells code for more processes, there is an increased burden
on the host bacterium’s metabolism, leading to slower growth and a decreased rate of cell division. As such, if it is not possible to stop plasmid
loss altogether, it is essential to prevent the plasmid-free population from
expanding.
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2.1
A Simple Bacterial Population Model
In order to show how plasmid loss probability and differential growth rates
affect the ratio of plasmid-bearing to plasmid-free cells within a bacterial
population, we construct a simple model. We assume there are no limiting effects of the environment on the maximum sustainable population, for
example resource limitations. This means that there will be constant exponential growth of the two populations rather than a levelling off. This
can be reproduced experimentally by growing the populations in a medium
and diluting regularly to maintain exponential growth (Sezonov et al. 2007).
The model takes the form of the differential equations:
dX +
dT −
dX −
dT −
= γX + − λγX + = (1 − λ)γX +
= X − + λγX +
(1)
Where X + and X − are the number of plasmid-carrying and plasmid−
free bacteria, λ is the probability of plasmid loss on cell division, γ = TT +
where T − and T + are the generation times of plasmid-free and plasmidcarrying cells. This model states that a bacterium, depending on whether it
is plasmid-free or plasmid-carrying, has a defined ‘lifetime’ before it undergoes mitosis, given by T − or T + . If a plasmid-free cell divides, both of its
daughter cells will be plasmid-free as there are no plasmid to pass along. If
a plasmid-bearing cell divides, one of the daughter cells is guaranteed to be
plasmid-bearing as the plasmids within the parent have to go somewhere.
The other daughter cell has a probability of becoming plasmid-free given by
λ. We will discuss in greater detail how λ may be attained later on.
Figure 1 shows how varying the probability of plasmid loss, λ, or the
differential growth rate, γ, affects the proportions of plasmid-bearing cells
within the population.
2.2
Mechanisms for Plasmid Maintenance
The model above shows how plasmid loss and and bacterial growth can
change the population ratios. As previously discussed, we need a method
for ensuring that plasmids that we insert into bacteria survive stably in the
population.
The primary method of plasmid maintenance in synthetic biology is selection using antibiotics. This method works by attaching an antibiotic
resistance gene to the plasmid. When a bacterium is transformed with the
plasmid it gains protection from the associated antibiotic. If the environment in which the bacteria are grown contains the antibiotic, those cells
which contain the plasmid survive but those which have lost the plasmid
die. This method works well in vitro but there are obvious problems when
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(a) λ = {1, 0.1, 0.01, 0.001, 0.0001, 0.00001}, γ = 1
(b) λ = 0.0001, γ = {0.75, 0.8, 0.85, 0.9, 0.95, 1}
Figure 1: Simulations of the ODE model described by equation 1.
used in vivo. A healthy body contains many different strains of bacteria
which aid in several processes (Yatsunenko et al. 2012). By using antibiotics, many of these populations will be eradicated. Further, the consistent,
unnecessary use of antibiotics has lead to the evolution of antibiotic resistance strains of bacteria, some of which are pathogenic (Arias & Murray
2009). If we can develop a system that allows us to avoid the use of antibiotics, particularly for use in vivo, then it will be a major step in the
development of synthetic biology for therapeutics.
There are a number of possible alternatives to the use of antibiotics for
plasmid maintenance, some of which can be used in tandem. They can be
classified as taking one of two approaches to plasmid maintenance: avoid
plasmid loss or prevent plasmid-free cells from growing.
5
2.2.1
Avoid Plasmid Loss
The affect of these approaches is to reduce the value of λ in equations 1. As
λ tends to 0 the population dynamics are increasingly reliant on differential
growth rates. If one can guarantee an initial population solely containing
plasmid-bearing cells, if plasmid loss can be prevented altogether, the population will remain entirely plasmid-bearing.
Chromosome Editing The most obvious way to avoid the problem of
plasmid maintenance altogether is to avoid the use of plasmids. It is possible
to integrate the designed system directly in to the chromosomal DNA of the
bacteria rather than having it extra-chromosomally encoded on a plasmid.
Although stability of the designed DNA is gained there are a number of
problems that this approach introduces as we have previously discussed.
Active Partitioning A mechanism that occurs naturally in many low
copy-number plasmids is active partitioning. As we show later, the random
distribution of plasmids within a host leads to a binomial distribution of
copy number between daughter cells. For high copy number plasmids the
probability of a daughter cell ending up plasmid-free is very low. However,
with low copy number plasmids, random distribution within the cell leads
to high rates of plasmid loss. Several plasmids have been discovered that
carry mechanisms that direct the positioning of the plasmids either to the
poles or quarter-cell positions of the host, thus ensuring even distribution
of plasmids (Gerdes et al. 2000, Ross & Thomas 1992). As an example of
the efficacy of such mechanisms, the par mechanism on the P1 plasmid is
effective enough that with a copy number of 3 − 4 (Li & Austin 2002), there
is a loss rate of about one in 104 divisions (Li et al. 2004).
High Copy-number Rather than the careful positioning of plasmids before cell division undertaken in active partitioning, one could take the approach of producing such large numbers of plasmids that the probability
of plasmid loss is vanishingly small. If plasmids are distributed randomly
throughout the cell, the probability of plasmid loss is a function of the number of copies within it (Summers 1991). Problems arise from the potentially
large increase in metabolic burden on the host system leading to the exacerbation of differential growth rates if any plasmid-free cells do exist. We
build a model of this approach later on and discuss some of the problems
that arise from the sole use of high copy number for stability.
2.2.2
Plasmid-free Cell Removal
These mechanisms attempt to kill or prevent the growth of plasmid-free
cells rather than reducing plasmid loss. In all of these systems the presence
6
of plasmids prevents the death of the host cell. However, there are major
differences in the approaches.
Environmental Selection This is the approach taken when using antibiotic resistance but applies to any system in which an environmental cue
causes the death of cells that lack an in-built resistance to it. As with antibiotics, unless the cue can be targeted to the particular strain of bacteria used
as the plasmid host, there will be consequences for other cell populations.
Even in an an industrial bio-reactor where one doesn’t have to concern oneself with the effects of the selective substance on other populations of cells,
the cost of removal of the antibiotic from any final product could prove
inhibitive (Friehs 2004).
Complementation of Chromosomal Mutation By removing an essential part of the chromosomal DNA and encoding it on the plasmid, the bacterium is only viable while holding the plasmid. This doesn’t prevent plasmid loss but does prevent plasmid-free cells from out-growing their plasmidbearing counterparts. This has the advantage over placing the entire system
on the chromosomal DNA that there is only a minor change to the bacterial
DNA. Mutations which cause the cell to re-establish the knocked-out gene
can cause a problem (Friehs 2004), though the author was unable to find
statistics on the rate at which this can occur. Systems created so far have
used the ssb genes (Porter et al. 1990) and proBA genes (Fiedler & Skerra
2001).
Post-Segregational Killing In a similar manner to environmental selection and chromosomal mutations, a post-segregational killing mechanism
doesn’t prevent plasmid loss but instead makes cells that have lost a plasmid unviable. The difference with this season is that the plasmid plays an
active role in killing the daughter cells as well as keeping them alive. This
works with a toxin-antitoxin (TA) mechanism. There are several types of
TA mechanism but they all work on the premise that a toxin is produced
by the plasmid but its action is nullified by an antitoxin, also produced by
the plasmid. The toxin has a greater stability than the antitoxin and so
degrades over a longer period. In a plasmid-bearing cell, the production of
both toxin and antitoxin are balanced so that the negative effects of the
toxin are not realised. If, on division, a plasmid-free daughter cell is produced, it will initially contain balanced levels of toxin and antitoxin but as
the antitoxin degrades faster, an imbalance will develop and the toxin will
have an effect. Not all TA systems are bactericidal, some are bacteristatic
(Wright et al. 2013).
Five types of TA system have been identified, though only one case of
each type 4 and type 5 have been detailed (Goeders & Melderen 2014). The
7
differences relating to whether the toxin and antitoxin are RNAs or proteins
and whether the antitoxin prevents toxin activity or inhibits its synthesis,
as shown in figure 2.
Figure 2: The different types of toxin-antitoxin system. (Wen et al. 2014)
2.3
Extended Bacterial Population Model
The λ term in our initial model is able to incorporate the effects of the first
form of plasmid maintenance mechanisms; the effects of which will be to
reduce λ. In order to accommodate the effects of the second form of plasmid
maintenance mechanism, removal of plasmid-free cells, we must extend our
model. The model described in equation 1 can be expanded by including
a probability of cells dying. There are two states that a bacterium can be
in: plasmid-bearing and plasmid-free, so we might consider two different
values for the probability of dying depending on whether the cell is in one
or the other state.Indeed, this may be all that is needed to incorporate the
environmental selection and chromosomal mutation mechanisms. However,
8
the post-segregational killing mechanism described above attempts to kill
cells that make a transition from plasmid-bearing to plasmid-free. As such
we must also take this in to account with a further parameter. As stated
previously, there is limited possibility of a cell transitioning from plasmidfree to plasmid-bearing so we have no need to include the corresponding
parameter.
Therefore, we introduce ω + and ω − the probability of death for plasmidbearing and plasmid-free cells and ω +− , the probability of death for the
plasmid-free daughter cell of a plasmid-bearing cell.
dX +
dT −
dX −
dT −
= (1 − ω + )γX + − (1 − ω + )λγX + − ω + γX +
= (1 − ω − )X − − ω − X − + (1 − ω + )(1 − ω +− )λγX +
(2)
The probability of cell death, while being grown in a medium designed
to support exponential growth, and with the population being regularly
diluted so that the growth remains in the exponential phase, is negligible.
As such there is no intrinsic necessity for the death parameters; they are
needed only to describe the effects of the plasmid maintenance mechanisms.
This allows us to simplify the above equations to reflect the effects of the
mechanisms we are interested in. Our research interest is in the use of
post-segregational killing mechanisms rather than chromosomal mutation
or environmental selection. Post-segregational killing only affects plasmidfree cells with plasmid-bearing parents. Therefore, we can remove ω + and
ω − from the equations, leaving us with:
dX +
dT −
dX −
dT −
= (1 − λ)γX +
= X − + (1 − ω +− )λγX +
(3)
The dynamics of the model described by equation 3 are shown in figure
3. The value of ω +− will depend on the effectiveness of the toxin-antitoxin
mechanism and it is our aim to develop models for this mechanism in the
future.
3
Experimental Parametrisation
The parameters for this model will depend on the host bacteria and he
plasmids used, as well as the medium in which they are grown. The aim
of our study is to use Escherichia coli Nissle 1917 though we initially use
E. coli DH5a as it is known to be easy to manipulate for our purposes.
9
(a) λ = 0.0001, γ = 0.85, ω +− = {0.9, 0.99, 0.999, 0.9999, 0.99999}
Figure 3: Simulations of the ODE model described by equation 3.
We could take some initial values for the growth rate of the plasmid-free
bacteria, T − , from the literature. It is, however, simple to ascertain this
figure by measuring the time for the population to double. As mentioned
previously, it is important to maintain exponential growth to avoid boundary
artefacts. A study by Sezonov et al. (2007) showed that for E. coli K-12
grown in Luria-Bertani medium the exponential growth phase ends at an
optical density, OD600 of 0.3. Experiments were carried out to determine
the growth rate of the DH5a strain, as well as the DH5a strain transformed
with the plasmid shown in figure 13. The results are shown in figure 4.
Calculating T + is more complicated. If we grow bacteria, transformed
with the plasmid of interest, with antibiotic resistance attached, in the LB
medium containing antibiotic, as shown in figure 4, we will give a growth
curve in which:
dX +
(1 − λ)X +
=
(4)
dt
T+
There won’t be any growth of the plasmid-free population as those bacteria will be killed by the antibiotic. One can see that we can’t easily calculate
T + unless we have an idea of the value of λ. We may be able to attain an
value for λ through the method outlined below, after which it is simple to
gain the value for T + .
A value for λ is also complicated to find. A number of methods have been
described (Ganusov & Brilkov 2002, Lau et al. 2013), with most utilising the
same intuition: if one considers the curve of the fraction of plasmid-bearing
cells in a population over time, during the initial stages the increase in the
ratio of plasmid-free population will predominantly be due to plasmid loss
10
Figure 4: The red points show the OD600 values for growth of plasmid-free
E. coli DH5a cells in LB medium. The blue points show the OD600 values
for growth of E. coli DH5a cells carrying the plasmid shown in figure 13
and grown in LB medium containing kanamycin. The top graph shows the
entire experimental period of measurement at the end of which the OD600 of
both population is close to 1. The doubling time for each population from
this curve is 43.424 minutes for the plasmids-free cells and 47.921 minutes
for the plasmid-bearing cells. However, if we take in to consideration that
exponential growth may stop at an OD600 of 0.3 we gain different values
42.015 and 52.344.
from the plasmid-bearing population rather than differential growth rates.
This relies on the experiment being initialised with a population without
any plasmid-free cells, i.e. grown under antibiotic selection. Evidently, this
can’t be carried out with a plasmid carrying a mechanism for killing cells
11
that are born plasmid-free from a plasmid-bearing parent. As such, it must
be carried out with the basic plasmid on to which the mechanism of interest
will later be attached. This may lead to a small amount of error as the
mechanism to be studied will likely have a metabolic burden on the host
bacterium and, therefore, affect its growth rate. However, since we are
limiting ourselves to looking at the initial phase of the curve, precisely with
the view to removing growth rate from consideration, the error should be
minimal.
The probability of death, ω +− , can then be found by growing the bacteria, transformed with the toxin-antitoxin carrying plasmid in LB medium.
As before, the exponential growth phase must be maintained in order to
avoid possible confounding effects of competition at a stage of growth when
there is competition for resources.
4
A More Complex View
The model as described in equation 3 is perhaps overly simple. Treating
each of the parameters as single values which are given properties of system
when initialised in a particular way is naive. In this section we regard the
processes behind the parameters and show that the form they take is more
complex than a single value.
4.1
Plasmid loss probability λ
The plasmid loss probability, as described up until now, is the abstract idea
that a daughter of a plasmid-bearing bacterium may end up plasmid-free.
What we haven’t discussed up until this point is what it means to be plasmidbearing; are all plasmid-bearing cells equal? In reality, there exists a number
of different plasmids, each carrying one of a variety of origin-of-replication
mechanisms which determine how the plasmid replicates within the host
bacterium and therefore how many copies of the plasmid the bacterium will
carry.
One of the mechanisms that plasmids have developed in order to reduce
plasmid loss is to increase their replication in a host. It is intuitive that
if a plasmid has a high “copy number”, on cell division there is a greater
likelihood of both daughter cells containing at least one plasmid. There
are obvious downsides to replicating a large number of times: the increased
demand on the host cell for resources to perform the replications and the increased metabolic burden of each plasmid trying to undertake the processes
that it codes for.
In order to calculate plasmid loss probability from copy number we begin
with some assumptions. Firstly, we assume that plasmids within a cell are
distributed randomly and that a cell divides in to two daughters of equal
12
volume. With these assumptions it is easy to show, as in Summers (1991),
that the probability of producing a plasmid-free daughter cell on division is:
λ = 2(1−n)
(5)
where n is the number of plasmids within the parent cell. For a population of plasmid-carrying bacteria, all with the same number of plasmids,
this would be the plasmid loss probability. However, because plasmid replication is a stochastic process, as is cell division, the number of plasmids
will not be equal throughout the population. Our next assumption is that
there is a Poisson distribution of plasmid copy number within the population around a mean which is a property of the plasmid’s origin-of-replication.
Each sub-population carrying n plasmids has a probability of producing a
plasmid-free daughter cell given by equation 5. As such, if we multiply the
Poisson distribution function, f (µ), by the probability of plasmid loss, λ(n),
and sum over a range of n that covers the whole distribution, see equation
6, we get the probability of plasmid loss for a given mean copy number, µ.
This is equivalent to the proportion of the bacterial population that produce
plasmid-free daughter cells.
Λµ =
X
f (µ)λ(n)
(6)
n
n −µ
e
where f (µ) = µ n!
A comparison between the situation in which there is no variation in copy
number and one in which there is a Poisson distribution of copy number is
shown in figure 5. It is clear that without considering plasmid copy number
variance, underestimation of plasmid loss is very large; at a mean copy
number of 12 the method taking in to account variance gives a plasmid loss
probability ten times greater than that without variance.
There are, however, other factors involved in the regulation of plasmid
loss. In some circumstances plasmids form dimers or other oligomers (Summers & Sherratt 1984). This effectively reduces the copy number of plasmids
within a cell, producing a greater probability of plasmid loss. Further, due
to the initiators of replication randomly selecting origin-of-replication sites
for the replication of plasmids, a dimer has a twofold greater chance of being
replicated, which leads to selection for larger oligomers (Summers & Sherratt 1984). As such, a mechanism known as the multimer resolution system,
present in some plasmids, acts to resolve the oligomers in to monomers
(Summers 1991).
Further, our assumption of random distribution of plasmids within a cell
may be applicable to high copy number plasmids but is confounded when
looking at the stability of low copy number plasmids. Their stability is
much greater than the expectation from random partitioning. Many low
13
Figure 5: By defining the plasmid copy number distribution among the
bacterial population as following a plasmid distribution one can see the
large increase in plasmid-loss compared with a system in which there is no
distribution of copy number. This is due to the greatly increased probability
of plasmid loss in cells in which the copy number is at the lower end of the
distribution.
copy number plasmids have active partitioning systems in which plasmids
are actively distributed to each of the daughter cells.
4.1.1
Binomial Plasmid Dispersion Model
In the model described above, we assumed a Poisson distribution of plasmid
copy number within the population of bacteria. Rather than making this
assumption we can determine how the variation in copy number with a new
model. Here we make the same assumptions as previously, that plasmids
are distributed randomly within a cell and cell division is perfectly in half.
With these assumptions we expand the above model, not just to look at
the probability of a daughter cell being produced with no plasmids, but the
probabilities of all possible copy numbers.
When a cell divides, each plasmid within the cell can end up in either
daughter cell 1 with probability p, or daughter cell 2 with probability 1 − p.
The probability mass function for the number of plasmids in daughter cell
1, just after division, is the binomial distribution B(N, p) where N is the
total number of plasmids within the parent cell and p is the probability of
a plasmid being in daughter cell 1 after division. Since we have made the
assumption that cell division produces equal sized daughter cells, there is
an equal chance of a plasmid going in to either daughter cell, p = 0.5.
If we start with a population of cells all with the same copy number, N0 ,
the distribution of copy number in the population after the first round of
division will be equal to the binomial distribution given above multiplied by
14
Figure 6: The distribution of plasmid copy number from an initial population with copy number 50. At each division the plasmids are binomially
distributed between each daughter cell. As the generations progress, the
variance in copy number increases and the mean skews towards lower copy
number.
twice the initial population size, X0 . At this point we make another simplifying assumption: the number of plasmids within a cell doubles between
birth and division i.e. each plasmid replicates once. Just before the second
division, we have sub-populations of cells with plasmid copy number equal
to 2n where n = 0, 1, 2...N . The cell count of each population is equal to
B(n; N0 , p).2X0 i.e the proportion of cells with half the current plasmid copy
number, after the last division, multiplied by the current number of cells.
each of these populations will produce a different distribution of copy numbers after the second division which, when summed together and weighted
by their population size, will give the copy number distribution for the entire
population.
Figure 6 shows how the distribution of plasmids within the population
changes after each generation. One can see how the variance quickly increases but also that the mean begins to skew towards a lower copy number.
The effects are more drastic for lower initial copy numbers, as shown in
figure 7.
This model shows that there must be mechanisms in place to reign in
the increase in copy number variance after cell division. As mentioned previously, low copy number plasmids use active partitioning which makes our
15
16
Figure 7: The change in plasmid copy number distribution as generations progress. The plasmids are divided binomially
between each daughter cell and each plasmid replicates once between divisions.
assumption of random plasmid distribution void. Further, our assumption
that plasmids replicate once between divisions is naive. There are several
mechanisms that have been discovered that regulate the replication of plasmids, not just making sure that the copy number is not too low but also
that it doesn’t get too high (Scott 1984).
It is reasonable to assume that copy number will have an impact on
growth rates of the host cell and therefore inter-division times. We have not
built this in to the model, however, it is intuitive that the skew towards low
copy number plasmids will occur faster.
5
An Agent-Based Model of Bacterial Population
Dynamics
The first model that we described with equation 3 contained several simplifications. We did not concern ourselves with the process of plasmid loss,
just that there was a probability that a plasmid-free daughter cell could
be produced. Greater consideration was then given to this process in the
second model. We considered how copy number might affect plasmid loss.
It was shown that merely concerning ourselves with a mean copy number
can introduce large errors when there is variance of copy number within
a population. Again, several assumptions were made particularly with regard to the distribution of copy number within the population. Finally,
we showed that without any mechanisms to regulate plasmid replication or
control plasmid dispersion, variance in copy number increases and the mean
tends towards lower copy number as generations progress.
Indeed, there are mechanisms which exist to control copy number within
individual plasmids and between daughter cells on cell-division. Large metabolic
burden is likely to prevent very high copy numbers proliferating. Active
partitioning and post-segregational killing exist to prevent plasmid loss and
the proliferation of plasmid-free cells. In the simple model first outlined
in equation 3 the post-segregational killing mechanism is incorporated in
to probability of death parameter ω +− , active partitioning would have an
effect on the plasmid loss probability λ and to some extent the differential
growth parameter γ is related to metabolic burden.
In order to more explicitly model some of these mechanisms and their
effects we build an agent based model. In agent-based modelling, agents
follow simple rules or behaviours. The interactions between the agents can
create complex, population level patterns in the system. It is a technique
which has been used in behavioural ecology (Evers et al. 2012), economics
(Tesfatsion 2002) and sociology (Squazzoni 2008). The validation of an
agent-based model was elegantly demonstrated by Sellers et al. (2007). Further, they demonstrated that discrepancies in the model can highlight areas
in which there is insufficient data or weaknesses in the empirical processes.
17
Research by Bryson et al. (2007), justified the use of agent-based modelling
as an experimental tool and demonstrated that the models can be easily
adapted or augmented when part of it is disputed.
5.1
Replication of ODE Model Dynamics
Although our aim with the introduction of agent-based modelling is to be
able to incorporate some of the more complex behaviours produced by specific mechanisms, we begin by producing a simple model that can reproduce
the dynamics demonstrated by the model of equation 3. The agents in our
initial model are the bacteria. Each agent can be plasmid-free or plasmidbearing. If an agent is plasmid-free, it survives for a time, T − , after which it
undergoes division. Since it is plasmid-free, both of its daughter cells will be
plasmid-free. Each of these daughter cells then behave in the same manner.
The behaviour of plasmid-bearing cells is slightly more complex. In a
similar way to plasmid-free cells, plasmid-bearing cells survive for a time,
T + , before undergoing division. At division, one daughter cell will always be
plasmid-bearing whereas the other daughter cell has a probability, λ, that it
will be plasmid-free. If the daughter cell is plasmid-free, it has a probability,
ω, that it will die. These behaviours are outlined in figure 8.
5.2
Modifications to the Agent-Based Model
Due to certain considerations, we had to make alterations to the agent-based
model.
Agent number limits In our ODE simulations we used initial values
for the number of plasmid-bearing cells, X0+ = 104 , and plasmid-free cells,
X0− = 0 (or X0− = 1 when λ = 0). For our agent-based model simulations we
use the same initial values. However, due to resources needed to calculate
the behaviour of such a large number of agents, we had to put a limit on
the bacterial population size. We implemented this limit in the form of a
‘dilution’ of the population when it grew larger than 105 agents. This was
done by selecting at random a proportion of the population to die. We will
discuss the impact of this in greater detail later. Although this may seem
an arbitrary procedure, it can be viewed as analogous to the dilution of
the population that one has to undertake in empirical experimentation with
bacteria in order to maintain exponential growth. In those circumstances
dilution must occur before the population reaches an OD600 of 0.3 (Sezonov
et al. 2007), which equates to ∼ 108 cells/ml. In our circumstances the
upper limit was dictated by the necessity to run a number of simulations in
a ‘reasonable’ period of time.
18
Figure 8: The behaviour of each agent in the agent-based model is dependent on whether it is plasmid-bearing or not. The plasmid-bearing parents
have slightly more complex behaviour as one of their daughter cells can be
plasmid-free and if so it may die. Modifications were made so that the timeto-divide was sampled from a geometric distribution with mean T + or T − .
We also introduce dilution of the population, though this is a macro-level
effect rather than a micro-level behaviour.
Inter-division interval Values for T − and T + didn’t have to be specified
in the ODE model; we were solely concerned with the relationship between
−
them, γ = TT + . In our agent-based model, the simulation runs on discrete,
integer time-steps. Each agent has to know on which time-step it will un−
dergo division. For our initial simulations we set T − = 10 and T + = Tγ .
The choice of T − was arbitrary, with consideration given to the runtime
of the simulation against the precision of T + . This meant that the values
of γ that we were able to simulate with were limited to those which would
produce an integer value of T + .
However, this method produced oscillations in the plasmid-bearing/plasmidfree population ratio. This was due to the combined effects of discrete timesteps and the need for dilution. At any one time-step, there is a chance that
one population, either plasmid-free or plasmid-bearing cells, will divide. At
this point, the population that just divided will have increased its population by as much as twofold whereas the other population will not yet have
increased. This changes the population ratio instantly. After some other
19
number of time-steps, the other population will divide, changing the ratio
in the other direction, leading to oscillations.
In order to avoid these oscillations and their potentially confounding
effects, we changed the method by which the agents divided. Previously,
all plasmid-bearing agents divided after T + time-steps and all plasmid-free
agents divided after T − time-steps. It is this rigid allocation of inter-division
interval that leads to blocks of the population dividing simultaneously. Instead, we introduce an allocation of inter-division interval to each agent, at
birth, taken from a geometric distribution with mean T − or T + .
5.3
Inconsistencies Between ODE and Agent-Based Models
Although the agent-based model is inherently stochastic, the average population change of a number of runs should be similar to the results produced
from the ODE model. However, there is not agreement between the two
models.
Effects of Dilution When one population is very small, dilution has the
potential to remove that population from the simulation. If, for example,
the simulation had no plasmid loss from the plasmid-bearing population,
λ = 0, there would be no further opportunity for a plasmid-free population
to arise. Similarly, if the plasmid loss probability is very small, it may
be a long period before a new plasmid-free cell is produced. One method
that we can used to negate this problem is to dilute the two population
independently i.e. remove x% of each individual population rather than x%
of the total population. Figure 9 shows how the two methods compare when
varying γ. The first method is, however, more analogous to what one would
expect to occur in the experimental method.
Small Agent Count Stochastic Effects A further potential source of
difference between the agent-based simulations and the ODE simulations
regards the impact of stochastic events when numbers of agents in a population are small. To illustrate this effect consider the growth, without
plasmid loss or cell death, of a population of plasmid-bearing bacteria. In
the agent-based model, the inter-division interval of each cell is drawn from
a geometric distribution. If we start a simulation with just 1 agent, there is
a probability that the agent’s time to division is large. In figure 10, we can
see how the increasing the initial number of agents in the simulation reduces
the variance in simulation results. We can also see that the means of the
two figures are similarly close to the ODE simulation’s result. This shows
that, although the variance with low numbers of agents is large, if we can
perform enough simulation runs, the average result should describe similar
dynamics.
20
Figure 9: The agent-based model (solid lines) compared with the ODE
model (dashed lines). The left hand column of graphs shows the dynamics
of the agent-based model in which the population is diluted as a whole
rather than as sub-populations, as in the right hand column. The solid lines
in the graphs in the top row are the mean values at each time point of 100
simulations. The bottom row of graphs is a zoomed in view of the dynamics
of each of the 100 simulations with γ = 0.75. The red dashed line shows
the ODE model. One can observe the first dilution occurring just prior to
the fifth time-point on the x-axis. In the left hand graph one can see a
greater variation in impact of the dilution of population ratio. It can also
be seen that, in this first dilution, some of the simulations lose all plasmidfree agents. This is what causes the levelling out of the top left hand graph,
rather than a total population switch, as occurs in the top right hand graph.
5.4
Future Extensions to the Agent-Based Model
Although the simple agent-based model has not yet been completed, future
extensions to it are desired. Due to the implementation of the model, these
extensions are easily carried out either on there own or in concert. Cooper
(1991) describes the points in the division cycle of a cell in which variation
is introduced, as in figure 11. We can consider each point and include
mechanisms in the agent-based model which take these in to account. The
suggestions below deal with these elements of variation and introduce other
mechanisms which may be of interest.
21
Figure 10: The exponential growth of a plasmid-bearing population, without
plasmid loss or cell death. The blue line in each graph is the mean of 1000
simulations of the agent-based model. The red line is the results of the ODE
model with the same parameters. In the graph on the left, their was only
one agent at the start of the simulation whereas in the graph on the right,
the simulations were initiated with 100 agents. The shaded region in each
graph is the 95% interval.
Plasmid Copy Number Currently the model only refers to the cell’s
plasmid-bearing status as either true or false. Including a copy number
for each agent would allow for much greater exploration of various plasmid
maintenance mechanisms. This parameter could effect several other new
mechanisms as detailed below.
Active partitioning Active partitioning could be implemented with a
parameter to determine its efficacy. With a high efficacy plasmid partitioning could be perfect between the two daughter cells but as the parameter
decreases, more error creeps in to the process.
Plasmid replication control The model of binomial division outlined
previously assumed each plasmid replicates once between each division cycle.
We showed that there must be other extant mechanisms in order to prevent
the increase in copy number variance. This is likely to be due to replication
control mechanisms preventing replication of plasmids once the copy number
is too high and making sure enough cycles of replication occur to prevent
the copy number falling too low. The mechanism would likely rely on the
current copy number of the bacterium to control the rate of replication of
the plasmids within that bacterium.
Metabolic burden The inter-division interval currently is taken from a
distribution with mean T + or T − . Since the metabolic burden that the
plasmids impart on the host will affect the growth rate and therefore the
division rate of its host, connecting division time to copy number would
be interesting. Different plasmids have different metabolic burdens so the
22
Figure 11: An initial population of cells with equal mass (starting at the
arrow) have normally distributed mass synthesis rates (1). DNA synthesis
begins when the mass reaches a required (noisy) threshold mass (2). The
combination of these two elements of variation creates a distribution of “time
at initiation” (3) which approximates to a reciprocal normal distribution.
There is a normally distributed variation in the DNA replication-segregation
sequence after synthesis has begun (4). The variation due to division in to
daughter cells of uneven sizes must also be taken in to account (5). The
combination of these variations produces a “total interdivision time distribution” (6). (Cooper 1991)
implementation would need to provide a parameter for tuning the degree of
impact that copy number has on the growth rate.
Uneven daughter size So far we have also assumed that the parent cell
divides perfectly in half, creating to equal sized daughter cells. Evidently
this is unlikely to occur as mitosis is a biological process which is inherently
imperfect. The size of a cell at birth may also impact the division time of the
cell. There is some disagreement in the literature as to whether cells divide
after their mass reaches a (noisy) threshold (Abner et al. 2014) or whether
there is a given mass that the cells attempt to put on before dividing (Amir
2014).
23
6
Planned Experimentation
In the laboratory, we have begun to explore certain mechanisms to improve
plasmid stability. Our main focus regards toxin-antitoxin post-segregational
killing mechanisms but we also explore variations in the origin-of-replication
on the plasmids which controls copy number.
6.1
Toxin-antitoxin Mechanisms
We have four different toxin-antitoxin systems to work with; hok/sok, kid/kis,
txe/axe and zeta/epsilon. These pairs cover a range of variation in toxinantitoxin mechanisms.
Name
hok/sok
kid/kis
txe/axe
zeta/epsilon
6.2
Type
I
II
II
II
Bactericidal
√
√
Source
E. coli plasmid R1
E. coli plasmid R1
Enterococcus faecium plasmid pRUM
Streptococcus pyogenes plasmid pSM19035
Origin-of-Replication
The origin-of-replication controls copy number which is of crucial importance
to plasmid stability. We have two origins-of-replication: SC101 and BR322.
SC101 should produce a copy number of ∼ 5 and BR322 a copy number of
∼ 20. There is also a further origin-of-replication, derived from the pUC
plasmid, which has a copy number of ∼ 500 − 700 which may be interesting
to use in these experiments though it is unlikely to be useful in a therapeutic
setting due to the huge metabolic burden that it would place on the host
bacteria.
6.3
Plasmid Design
The plasmid that we use for our experiments, shown in figure 13, has three
important parts to it. There is a GFP reporter which allows us to determine
which cells contain plasmid by viewing which cells are glow under a fluorescent microscope. The GFP reporter is subject to a very strong constitutive
promoter, OXB20. It has kanamycin resistance built in to it. Although we
are trying to get away from the use of antibiotic resistance, for the purposes
of some experiments it is important to be able to start with an initial population of bacteria which are all plasmid-bearing. The simplest way of doing
this is using antibiotics though we could use a flow cytometry machine to
separate cells which express the GFP which is also attached to the plasmid.
24
Figure 12: The plasmid we use in our experiments. This shows the
kanamycin resistance, the GFP reporter flanked by the strong constitutive promoter OXB20 and the strong terminator RrnG, and the origin-ofreplication derived from pSC101 plasmid.
7
Conclusion
The field of synthetic biology, although relatively new, has already shown
its potential. Several studies have shown potential therapeutic benefits of
various engineered circuits as well as a huge number of applications in industrial settings. Before the field can move further in to the medical realm
certain bio-safety considerations must be addressed.
Plasmids provide a useful vehicle for synthetic biological systems and
bacterial hosts can provide the machinery to run the processes. However,
the use of antibiotics for the maintenance of engineered plasmids in host
bacteria is not a viable solution in a medical setting and perhaps not in an
industrial one.
There are several mechanisms which plasmids harness in order to increase there stability. We have built a model to show the folly of relying
on high copy number without incorporating the necessary replication controls; variation in copy number quickly increases and the population tends
to plasmid loss.
We built a simple ordinary differential equation model into which the
25
effects of all plasmid maintenance mechanisms could be introduced. Further,
we have begun to lay out the from work for an agent-based model n which we
can more explicitly model each mechanism. This has the potential to allow
us to build individual models of each mechanism, for example looking at the
dynamics of the transcriptional gene network involved in toxin-antitoxin
behaviour. It also allows us to observe the stochastic effects of low numbers
of bacteria, plasmids or any other agent that we wish to insert in to the
system.
Finally, we have outlined a series of experiments with two goals: to
provide parameters for our models so that we may go on to use them to
make predictions, and to empirically demonstrate the effectiveness of various
toxin-antitoxin systems for plasmid maintenance.
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A
Additional agent-based model results
29
Figure 13: Further dynamics of the agent-based model. The solid line shows
the mean of 100 agent-based model simulations and the dashed line shows
the ODE model results using the same parameters. The top graph has
γ = 1 and ω +− = 0. The bottom graph has γ = 1 and λ = 1. The initial
population size for both models was X + = 104 and X − = 0.
30