Mechanisms for Plasmid Maintenance

Alexander J. H. FedorecCoMPLEX, University College London

Supervisors: Dr. Chris Barnes, Prof. Geraint ThomasAcknowledgements: Tanel Ozdemir

August 28, 2014

Modern sequencing and manufacturing techniques have made it possibleto design and encode specific functions in to DNA. This allows the manu-facture of complex genetic circuitry for use in a therapeutic setting. Oneof the current methods of introducing such circuits in to an animal is byimplementing the circuit on a plasmid and transforming bacteria with thatplasmid. The bacteria are then introduced to the animal and the bacterialmachinery will carry out the processes encoded on the plasmid. A primaryconcern when using this method is the loss of plasmids from the bacterialpopulation, due primarily to competition from plasmid-free bacteria. Cur-rently, an antibiotic resistance gene is placed on the plasmid and antibioticsare 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 anantibiotics regime if it can be avoided. To that end, we build a modellingsystem in which one can observe how plasmid maintenance is affected byaltering the various mechanisms which may control it. Further, we outlinea toxin-antitoxin system which may prove to be an effective mechanism tocontrol the growth of plasmid-free bacteria.

1 Synthetic Biology

Synthetic biology aims to develop engineered system that behave in a pre-dictable manner with components adapted from existing biological solu-tions. Much current research is invested in developing and characterising thebuilding blocks from which we can make more complex systems for medical(Ruder et al. 2011, Bacchus et al. 2013, Weber & Fussenegger 2011b), com-putational (Auslander et al. 2012, Regot et al. 2010) and industrial purposes(Keasling 2008). Optogenetic systems have been developed as a new toolto aid in neuroscience research (Konermann et al. 2013), allowing specificfunctionality to be switched on using precise beams of light. In a medi-cal setting, synthetic biological systems have already been produced that

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aid blood-glucose homeostasis in mice (Ye et al. 2011) and prevent choleravirulence (Duan & March 2010).

Modularisation and Characterisation of Components As these sys-tems are biological, the problems inherent in making predictions in stochas-tic systems apply. If the system, for example, relies on a small number ofmolecules being detected by a small number of receptors, the random motionof molecules within a cell will create large variety in outcomes. Althoughsuch problems abound in the field, one of the primary goals of syntheticbiology is to produce robust, well characterised modules that can be treatedin a similar way to the components of electronic circuits. As such, tog-gle switches have been developed that respond to several different stimuli(Kramer et al. 2004, Deans et al. 2007). Oscillators have been developedthat have tunable frequencies (Tigges et al. 2009). Logic gates have beendesigned and “wired” together to create simple circuits (Auslander et al.2012) but a great deal of work is required to characterise the behaviour ofeach component in a variety of environments and determine the effects thatother components may have when interacting in a circuit.

‘Architectural’ Considerations The ‘architecture’ most used for syn-thetic biological systems is the transcriptional gene network, designing re-lationships of inhibition and promotion of each constituent gene. Otherarchitectures, translational and post-translational, are described in (Khalil& Collins 2010).

In addition to the choice of circuit architecture one must consider thetype of cells that will host the circuit. Research is being carried out usingboth prokaryotic and eukaryotic cells. Mammalian qorum-sensing mecha-nisms (Weber & Fussenegger 2011a), tunable oscillators (Tigges et al. 2009)and optogenetic devices (Ye et al. 2011) have all been developed. How-ever, eukaryotic cells tend to have increased complexity, adding difficulty toattempts to produce predictable behaviour.

When using bacterial cells there is a further consideration that must bemade; to introduce the circuit in to the cell by altering the host chromosomeor by transformation with a plasmid. The introduction of DNA in to thechromosome of a cell is a cumbersome process, though the introduction ofCRISPR-Cas systems has, and will continue to, make the process more easilyachievable (Mali et al. 2013).

Our research aims at developing systems for microbiome therapeutics bycolonising the gut with our engineered bacteria. Escherichia coli Nissle 1917is a very well studied bacteria which is approved for human consumptionand has been shown, without any genetic manipulation, to aid in the devel-opment 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 toalter Nissle chromosome, allowing us to use the system therapeutically. Fur-ther, containment of the circuit on a plasmid provides more controllabilityin terms of transcription and replication.

There are, however, potential downsides to using plasmids. The primaryissue being that of plasmid maintenance, which we describe later. There isthe possibility of mutation of the plasmid DNA. A single point mutation canchange a medium copy number plasmid to a high copy number plasmid (Lin-Chao et al. 1992). Change in copy number can have drastic consequences onthe 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 theplasmid getting incorporated in to the host chromosome or from the plasmidbeing transferred between bacteria. It is believed that most recombinantplasmids are non-conjugative (Ganusov & Brilkov 2002) i.e. they are notcapable of transfer to other cells, due to the lack of tra genes. However,particularly with the use of antibiotic resistance genes in synthetic biologywe must be concerned by the possibility of transferring this ability to othercells (Schuurmans et al. 2014).

2 Plasmid Maintenance

A problem of great importance for systems which use plasmids as vectorsregards plasmid maintenance. This is the problem of making sure that, oncell division, both daughter cells carry at least one copy of the plasmid. Asit is the plasmid that codes for the system we have designed, we need tomaintain a population of bacteria that contain the plasmid for the systemto carry on working.

Differential growth rates Although the probability of producing plas-mid free daughter cells can be very low, the effects of plasmid loss arecompounded by the difference in growth rates between plasmid-free andplasmid-bearing cells. Although the exact link between growth rate anddivision interval is still disputed, some believing there is a threshold massfor each cell (Abner et al. 2014) while others believe there is target massincrement (Amir 2014), increased metabolic burden does slow growth. Sinceplasmid-bearing cells code for more processes, there is an increased burdenon the host bacterium’s metabolism, leading to slower growth and a de-creased rate of cell division. As such, if it is not possible to stop plasmidloss altogether, it is essential to prevent the plasmid-free population fromexpanding.

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2.1 A Simple Bacterial Population Model

In order to show how plasmid loss probability and differential growth ratesaffect the ratio of plasmid-bearing to plasmid-free cells within a bacterialpopulation, we construct a simple model. We assume there are no limit-ing effects of the environment on the maximum sustainable population, forexample resource limitations. This means that there will be constant ex-ponential growth of the two populations rather than a levelling off. Thiscan be reproduced experimentally by growing the populations in a mediumand diluting regularly to maintain exponential growth (Sezonov et al. 2007).The model takes the form of the differential equations:

dX+

dT− = γX+ − λγX+ = (1− λ)γX+

dX−

dT− = 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 = T−

T+

where T− and T+ are the generation times of plasmid-free and plasmid-carrying cells. This model states that a bacterium, depending on whether itis plasmid-free or plasmid-carrying, has a defined ‘lifetime’ before it under-goes mitosis, given by T− or T+. If a plasmid-free cell divides, both of itsdaughter cells will be plasmid-free as there are no plasmid to pass along. Ifa plasmid-bearing cell divides, one of the daughter cells is guaranteed to beplasmid-bearing as the plasmids within the parent have to go somewhere.The other daughter cell has a probability of becoming plasmid-free given byλ. 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 thedifferential growth rate, γ, affects the proportions of plasmid-bearing cellswithin the population.

2.2 Mechanisms for Plasmid Maintenance

The model above shows how plasmid loss and and bacterial growth canchange the population ratios. As previously discussed, we need a methodfor ensuring that plasmids that we insert into bacteria survive stably in thepopulation.

The primary method of plasmid maintenance in synthetic biology is se-lection using antibiotics. This method works by attaching an antibioticresistance gene to the plasmid. When a bacterium is transformed with theplasmid it gains protection from the associated antibiotic. If the environ-ment in which the bacteria are grown contains the antibiotic, those cellswhich contain the plasmid survive but those which have lost the plasmiddie. 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 bacteriawhich aid in several processes (Yatsunenko et al. 2012). By using antibi-otics, many of these populations will be eradicated. Further, the consistent,unnecessary use of antibiotics has lead to the evolution of antibiotic resis-tance strains of bacteria, some of which are pathogenic (Arias & Murray2009). If we can develop a system that allows us to avoid the use of an-tibiotics, particularly for use in vivo, then it will be a major step in thedevelopment of synthetic biology for therapeutics.

There are a number of possible alternatives to the use of antibiotics forplasmid maintenance, some of which can be used in tandem. They can beclassified as taking one of two approaches to plasmid maintenance: avoidplasmid loss or prevent plasmid-free cells from growing.

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2.2.1 Avoid Plasmid Loss

The affect of these approaches is to reduce the value of λ in equations 1. Asλ tends to 0 the population dynamics are increasingly reliant on differentialgrowth rates. If one can guarantee an initial population solely containingplasmid-bearing cells, if plasmid loss can be prevented altogether, the pop-ulation will remain entirely plasmid-bearing.

Chromosome Editing The most obvious way to avoid the problem ofplasmid maintenance altogether is to avoid the use of plasmids. It is possibleto integrate the designed system directly in to the chromosomal DNA of thebacteria rather than having it extra-chromosomally encoded on a plasmid.Although stability of the designed DNA is gained there are a number ofproblems that this approach introduces as we have previously discussed.

Active Partitioning A mechanism that occurs naturally in many lowcopy-number plasmids is active partitioning. As we show later, the randomdistribution of plasmids within a host leads to a binomial distribution ofcopy number between daughter cells. For high copy number plasmids theprobability of a daughter cell ending up plasmid-free is very low. However,with low copy number plasmids, random distribution within the cell leadsto high rates of plasmid loss. Several plasmids have been discovered thatcarry mechanisms that direct the positioning of the plasmids either to thepoles or quarter-cell positions of the host, thus ensuring even distributionof plasmids (Gerdes et al. 2000, Ross & Thomas 1992). As an example ofthe efficacy of such mechanisms, the par mechanism on the P1 plasmid iseffective enough that with a copy number of 3− 4 (Li & Austin 2002), thereis a loss rate of about one in 104 divisions (Li et al. 2004).

High Copy-number Rather than the careful positioning of plasmids be-fore cell division undertaken in active partitioning, one could take the ap-proach of producing such large numbers of plasmids that the probabilityof plasmid loss is vanishingly small. If plasmids are distributed randomlythroughout the cell, the probability of plasmid loss is a function of the num-ber of copies within it (Summers 1991). Problems arise from the potentiallylarge increase in metabolic burden on the host system leading to the exac-erbation of differential growth rates if any plasmid-free cells do exist. Webuild a model of this approach later on and discuss some of the problemsthat 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-freecells rather than reducing plasmid loss. In all of these systems the presence

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of plasmids prevents the death of the host cell. However, there are majordifferences in the approaches.

Environmental Selection This is the approach taken when using an-tibiotic resistance but applies to any system in which an environmental cuecauses the death of cells that lack an in-built resistance to it. As with antibi-otics, unless the cue can be targeted to the particular strain of bacteria usedas 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 one-self with the effects of the selective substance on other populations of cells,the cost of removal of the antibiotic from any final product could proveinhibitive (Friehs 2004).

Complementation of Chromosomal Mutation By removing an essen-tial part of the chromosomal DNA and encoding it on the plasmid, the bac-terium is only viable while holding the plasmid. This doesn’t prevent plas-mid loss but does prevent plasmid-free cells from out-growing their plasmid-bearing counterparts. This has the advantage over placing the entire systemon the chromosomal DNA that there is only a minor change to the bacterialDNA. Mutations which cause the cell to re-establish the knocked-out genecan cause a problem (Friehs 2004), though the author was unable to findstatistics on the rate at which this can occur. Systems created so far haveused the ssb genes (Porter et al. 1990) and proBA genes (Fiedler & Skerra2001).

Post-Segregational Killing In a similar manner to environmental se-lection and chromosomal mutations, a post-segregational killing mechanismdoesn’t prevent plasmid loss but instead makes cells that have lost a plas-mid unviable. The difference with this season is that the plasmid plays anactive role in killing the daughter cells as well as keeping them alive. Thisworks with a toxin-antitoxin (TA) mechanism. There are several types ofTA mechanism but they all work on the premise that a toxin is producedby the plasmid but its action is nullified by an antitoxin, also produced bythe plasmid. The toxin has a greater stability than the antitoxin and sodegrades over a longer period. In a plasmid-bearing cell, the production ofboth toxin and antitoxin are balanced so that the negative effects of thetoxin are not realised. If, on division, a plasmid-free daughter cell is pro-duced, it will initially contain balanced levels of toxin and antitoxin but asthe antitoxin degrades faster, an imbalance will develop and the toxin willhave 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 ofeach type 4 and type 5 have been detailed (Goeders & Melderen 2014). The

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differences relating to whether the toxin and antitoxin are RNAs or proteinsand 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 firstform of plasmid maintenance mechanisms; the effects of which will be toreduce λ. In order to accommodate the effects of the second form of plasmidmaintenance mechanism, removal of plasmid-free cells, we must extend ourmodel. The model described in equation 1 can be expanded by includinga probability of cells dying. There are two states that a bacterium can bein: plasmid-bearing and plasmid-free, so we might consider two differentvalues for the probability of dying depending on whether the cell is in oneor the other state.Indeed, this may be all that is needed to incorporate theenvironmental selection and chromosomal mutation mechanisms. However,

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the post-segregational killing mechanism described above attempts to killcells that make a transition from plasmid-bearing to plasmid-free. As suchwe must also take this in to account with a further parameter. As statedpreviously, there is limited possibility of a cell transitioning from plasmid-free to plasmid-bearing so we have no need to include the correspondingparameter.

Therefore, we introduce ω+ and ω− the probability of death for plasmid-bearing and plasmid-free cells and ω+−, the probability of death for theplasmid-free daughter cell of a plasmid-bearing cell.

dX+

dT− = (1− ω+)γX+ − (1− ω+)λγX+ − ω+γX+

dX−

dT− = (1− ω−)X− − ω−X− + (1− ω+)(1− ω+−)λγX+ (2)

The probability of cell death, while being grown in a medium designedto support exponential growth, and with the population being regularlydiluted so that the growth remains in the exponential phase, is negligible.As such there is no intrinsic necessity for the death parameters; they areneeded only to describe the effects of the plasmid maintenance mechanisms.This allows us to simplify the above equations to reflect the effects of themechanisms we are interested in. Our research interest is in the use ofpost-segregational killing mechanisms rather than chromosomal mutationor environmental selection. Post-segregational killing only affects plasmid-free cells with plasmid-bearing parents. Therefore, we can remove ω+ andω− from the equations, leaving us with:

dX+

dT− = (1− λ)γX+

dX−

dT− = X− + (1− ω+−)λγX+ (3)

The dynamics of the model described by equation 3 are shown in figure3. The value of ω+− will depend on the effectiveness of the toxin-antitoxinmechanism and it is our aim to develop models for this mechanism in thefuture.

3 Experimental Parametrisation

The parameters for this model will depend on the host bacteria and heplasmids used, as well as the medium in which they are grown. The aimof our study is to use Escherichia coli Nissle 1917 though we initially useE. coli DH5a as it is known to be easy to manipulate for our purposes.

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(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-freebacteria, T−, from the literature. It is, however, simple to ascertain thisfigure by measuring the time for the population to double. As mentionedpreviously, it is important to maintain exponential growth to avoid boundaryartefacts. A study by Sezonov et al. (2007) showed that for E. coli K-12grown in Luria-Bertani medium the exponential growth phase ends at anoptical density, OD600 of 0.3. Experiments were carried out to determinethe growth rate of the DH5a strain, as well as the DH5a strain transformedwith the plasmid shown in figure 13. The results are shown in figure 4.

Calculating T+ is more complicated. If we grow bacteria, transformedwith the plasmid of interest, with antibiotic resistance attached, in the LBmedium containing antibiotic, as shown in figure 4, we will give a growthcurve in which:

dX+

dt=

(1− λ)X+

T+(4)

There won’t be any growth of the plasmid-free population as those bacte-ria will be killed by the antibiotic. One can see that we can’t easily calculateT+ unless we have an idea of the value of λ. We may be able to attain anvalue for λ through the method outlined below, after which it is simple togain the value for T+.

A value for λ is also complicated to find. A number of methods have beendescribed (Ganusov & Brilkov 2002, Lau et al. 2013), with most utilising thesame intuition: if one considers the curve of the fraction of plasmid-bearingcells in a population over time, during the initial stages the increase in theratio of plasmid-free population will predominantly be due to plasmid loss

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Figure 4: The red points show the OD600 values for growth of plasmid-freeE. coli DH5a cells in LB medium. The blue points show the OD600 valuesfor growth of E. coli DH5a cells carrying the plasmid shown in figure 13and grown in LB medium containing kanamycin. The top graph shows theentire experimental period of measurement at the end of which the OD600 ofboth population is close to 1. The doubling time for each population fromthis curve is 43.424 minutes for the plasmids-free cells and 47.921 minutesfor the plasmid-bearing cells. However, if we take in to consideration thatexponential growth may stop at an OD600 of 0.3 we gain different values42.015 and 52.344.

from the plasmid-bearing population rather than differential growth rates.This relies on the experiment being initialised with a population withoutany plasmid-free cells, i.e. grown under antibiotic selection. Evidently, thiscan’t be carried out with a plasmid carrying a mechanism for killing cells

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that are born plasmid-free from a plasmid-bearing parent. As such, it mustbe carried out with the basic plasmid on to which the mechanism of interestwill later be attached. This may lead to a small amount of error as themechanism to be studied will likely have a metabolic burden on the hostbacterium and, therefore, affect its growth rate. However, since we arelimiting ourselves to looking at the initial phase of the curve, precisely withthe view to removing growth rate from consideration, the error should beminimal.

The probability of death, ω+−, can then be found by growing the bacte-ria, transformed with the toxin-antitoxin carrying plasmid in LB medium.As before, the exponential growth phase must be maintained in order toavoid possible confounding effects of competition at a stage of growth whenthere is competition for resources.

4 A More Complex View

The model as described in equation 3 is perhaps overly simple. Treatingeach of the parameters as single values which are given properties of systemwhen initialised in a particular way is naive. In this section we regard theprocesses behind the parameters and show that the form they take is morecomplex than a single value.

4.1 Plasmid loss probability λ

The plasmid loss probability, as described up until now, is the abstract ideathat 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 plasmid-bearing; are all plasmid-bearing cells equal? In reality, there exists a numberof different plasmids, each carrying one of a variety of origin-of-replicationmechanisms which determine how the plasmid replicates within the hostbacterium and therefore how many copies of the plasmid the bacterium willcarry.

One of the mechanisms that plasmids have developed in order to reduceplasmid loss is to increase their replication in a host. It is intuitive thatif a plasmid has a high “copy number”, on cell division there is a greaterlikelihood of both daughter cells containing at least one plasmid. Thereare obvious downsides to replicating a large number of times: the increaseddemand on the host cell for resources to perform the replications and the in-creased metabolic burden of each plasmid trying to undertake the processesthat it codes for.

In order to calculate plasmid loss probability from copy number we beginwith some assumptions. Firstly, we assume that plasmids within a cell aredistributed randomly and that a cell divides in to two daughters of equal

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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 popu-lation of plasmid-carrying bacteria, all with the same number of plasmids,this would be the plasmid loss probability. However, because plasmid repli-cation is a stochastic process, as is cell division, the number of plasmidswill not be equal throughout the population. Our next assumption is thatthere is a Poisson distribution of plasmid copy number within the popula-tion 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 aplasmid-free daughter cell given by equation 5. As such, if we multiply thePoisson 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 equation6, 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 produceplasmid-free daughter cells.

Λµ =∑n

f(µ)λ(n) (6)

where f(µ) = µne−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 isshown in figure 5. It is clear that without considering plasmid copy numbervariance, underestimation of plasmid loss is very large; at a mean copynumber of 12 the method taking in to account variance gives a plasmid lossprobability ten times greater than that without variance.

There are, however, other factors involved in the regulation of plasmidloss. In some circumstances plasmids form dimers or other oligomers (Sum-mers & Sherratt 1984). This effectively reduces the copy number of plasmidswithin a cell, producing a greater probability of plasmid loss. Further, dueto the initiators of replication randomly selecting origin-of-replication sitesfor the replication of plasmids, a dimer has a twofold greater chance of beingreplicated, which leads to selection for larger oligomers (Summers & Sher-ratt 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 cellmay be applicable to high copy number plasmids but is confounded whenlooking at the stability of low copy number plasmids. Their stability ismuch greater than the expectation from random partitioning. Many low

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Figure 5: By defining the plasmid copy number distribution among thebacterial population as following a plasmid distribution one can see thelarge increase in plasmid-loss compared with a system in which there is nodistribution of copy number. This is due to the greatly increased probabilityof plasmid loss in cells in which the copy number is at the lower end of thedistribution.

copy number plasmids have active partitioning systems in which plasmidsare 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 plasmidcopy number within the population of bacteria. Rather than making thisassumption we can determine how the variation in copy number with a newmodel. Here we make the same assumptions as previously, that plasmidsare distributed randomly within a cell and cell division is perfectly in half.With these assumptions we expand the above model, not just to look atthe probability of a daughter cell being produced with no plasmids, but theprobabilities of all possible copy numbers.

When a cell divides, each plasmid within the cell can end up in eitherdaughter cell 1 with probability p, or daughter cell 2 with probability 1− p.The probability mass function for the number of plasmids in daughter cell1, just after division, is the binomial distribution B(N, p) where N is thetotal number of plasmids within the parent cell and p is the probability ofa plasmid being in daughter cell 1 after division. Since we have made theassumption that cell division produces equal sized daughter cells, there isan 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 ofdivision will be equal to the binomial distribution given above multiplied by

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Figure 6: The distribution of plasmid copy number from an initial popu-lation with copy number 50. At each division the plasmids are binomiallydistributed between each daughter cell. As the generations progress, thevariance in copy number increases and the mean skews towards lower copynumber.

twice the initial population size, X0. At this point we make another sim-plifying assumption: the number of plasmids within a cell doubles betweenbirth and division i.e. each plasmid replicates once. Just before the seconddivision, we have sub-populations of cells with plasmid copy number equalto 2n where n = 0, 1, 2...N . The cell count of each population is equal toB(n;N0, p).2X0 i.e the proportion of cells with half the current plasmid copynumber, after the last division, multiplied by the current number of cells.each of these populations will produce a different distribution of copy num-bers after the second division which, when summed together and weightedby their population size, will give the copy number distribution for the entirepopulation.

Figure 6 shows how the distribution of plasmids within the populationchanges after each generation. One can see how the variance quickly in-creases 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 infigure 7.

This model shows that there must be mechanisms in place to reign inthe increase in copy number variance after cell division. As mentioned pre-viously, low copy number plasmids use active partitioning which makes our

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assumption of random plasmid distribution void. Further, our assumptionthat plasmids replicate once between divisions is naive. There are severalmechanisms that have been discovered that regulate the replication of plas-mids, not just making sure that the copy number is not too low but alsothat it doesn’t get too high (Scott 1984).

It is reasonable to assume that copy number will have an impact ongrowth rates of the host cell and therefore inter-division times. We have notbuilt this in to the model, however, it is intuitive that the skew towards lowcopy number plasmids will occur faster.

5 An Agent-Based Model of Bacterial PopulationDynamics

The first model that we described with equation 3 contained several sim-plifications. We did not concern ourselves with the process of plasmid loss,just that there was a probability that a plasmid-free daughter cell couldbe produced. Greater consideration was then given to this process in thesecond model. We considered how copy number might affect plasmid loss.It was shown that merely concerning ourselves with a mean copy numbercan introduce large errors when there is variance of copy number withina population. Again, several assumptions were made particularly with re-gard to the distribution of copy number within the population. Finally,we showed that without any mechanisms to regulate plasmid replication orcontrol plasmid dispersion, variance in copy number increases and the meantends towards lower copy number as generations progress.

Indeed, there are mechanisms which exist to control copy number withinindividual plasmids and between daughter cells on cell-division. Large metabolicburden is likely to prevent very high copy numbers proliferating. Activepartitioning and post-segregational killing exist to prevent plasmid loss andthe proliferation of plasmid-free cells. In the simple model first outlinedin equation 3 the post-segregational killing mechanism is incorporated into probability of death parameter ω+−, active partitioning would have aneffect on the plasmid loss probability λ and to some extent the differentialgrowth parameter γ is related to metabolic burden.

In order to more explicitly model some of these mechanisms and theireffects we build an agent based model. In agent-based modelling, agentsfollow simple rules or behaviours. The interactions between the agents cancreate complex, population level patterns in the system. It is a techniquewhich has been used in behavioural ecology (Evers et al. 2012), economics(Tesfatsion 2002) and sociology (Squazzoni 2008). The validation of anagent-based model was elegantly demonstrated by Sellers et al. (2007). Fur-ther, they demonstrated that discrepancies in the model can highlight areasin which there is insufficient data or weaknesses in the empirical processes.

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Research by Bryson et al. (2007), justified the use of agent-based modellingas an experimental tool and demonstrated that the models can be easilyadapted 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 beable to incorporate some of the more complex behaviours produced by spe-cific mechanisms, we begin by producing a simple model that can reproducethe dynamics demonstrated by the model of equation 3. The agents in ourinitial model are the bacteria. Each agent can be plasmid-free or plasmid-bearing. If an agent is plasmid-free, it survives for a time, T−, after which itundergoes division. Since it is plasmid-free, both of its daughter cells will beplasmid-free. Each of these daughter cells then behave in the same manner.

The behaviour of plasmid-bearing cells is slightly more complex. In asimilar way to plasmid-free cells, plasmid-bearing cells survive for a time,T+, before undergoing division. At division, one daughter cell will always beplasmid-bearing whereas the other daughter cell has a probability, λ, that itwill 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-basedmodel.

Agent number limits In our ODE simulations we used initial valuesfor the number of plasmid-bearing cells, X+

0 = 104, and plasmid-free cells,X−

0 = 0 (or X−0 = 1 when λ = 0). For our agent-based model simulations we

use the same initial values. However, due to resources needed to calculatethe behaviour of such a large number of agents, we had to put a limit onthe bacterial population size. We implemented this limit in the form of a‘dilution’ of the population when it grew larger than 105 agents. This wasdone by selecting at random a proportion of the population to die. We willdiscuss the impact of this in greater detail later. Although this may seeman arbitrary procedure, it can be viewed as analogous to the dilution ofthe population that one has to undertake in empirical experimentation withbacteria in order to maintain exponential growth. In those circumstancesdilution must occur before the population reaches an OD600 of 0.3 (Sezonovet al. 2007), which equates to ∼ 108 cells/ml. In our circumstances theupper limit was dictated by the necessity to run a number of simulations ina ‘reasonable’ period of time.

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Figure 8: The behaviour of each agent in the agent-based model is depen-dent on whether it is plasmid-bearing or not. The plasmid-bearing parentshave slightly more complex behaviour as one of their daughter cells can beplasmid-free and if so it may die. Modifications were made so that the time-to-divide was sampled from a geometric distribution with mean T+ or T−.We also introduce dilution of the population, though this is a macro-leveleffect rather than a micro-level behaviour.

Inter-division interval Values for T− and T+ didn’t have to be specifiedin the ODE model; we were solely concerned with the relationship betweenthem, γ = T−

T+ . 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+ = T−

γ .

The choice of T− was arbitrary, with consideration given to the runtimeof the simulation against the precision of T+. This meant that the valuesof γ that we were able to simulate with were limited to those which wouldproduce an integer value of T+.

However, this method produced oscillations in the plasmid-bearing/plasmid-free population ratio. This was due to the combined effects of discrete time-steps and the need for dilution. At any one time-step, there is a chance thatone population, either plasmid-free or plasmid-bearing cells, will divide. Atthis point, the population that just divided will have increased its popula-tion by as much as twofold whereas the other population will not yet haveincreased. This changes the population ratio instantly. After some other

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number of time-steps, the other population will divide, changing the ratioin the other direction, leading to oscillations.

In order to avoid these oscillations and their potentially confoundingeffects, we changed the method by which the agents divided. Previously,all plasmid-bearing agents divided after T+ time-steps and all plasmid-freeagents divided after T− time-steps. It is this rigid allocation of inter-divisioninterval that leads to blocks of the population dividing simultaneously. In-stead, we introduce an allocation of inter-division interval to each agent, atbirth, 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 popu-lation change of a number of runs should be similar to the results producedfrom the ODE model. However, there is not agreement between the twomodels.

Effects of Dilution When one population is very small, dilution has thepotential 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 populationto arise. Similarly, if the plasmid loss probability is very small, it maybe a long period before a new plasmid-free cell is produced. One methodthat we can used to negate this problem is to dilute the two populationindependently i.e. remove x% of each individual population rather than x%of the total population. Figure 9 shows how the two methods compare whenvarying γ. The first method is, however, more analogous to what one wouldexpect to occur in the experimental method.

Small Agent Count Stochastic Effects A further potential source ofdifference between the agent-based simulations and the ODE simulationsregards the impact of stochastic events when numbers of agents in a pop-ulation are small. To illustrate this effect consider the growth, withoutplasmid loss or cell death, of a population of plasmid-bearing bacteria. Inthe agent-based model, the inter-division interval of each cell is drawn froma geometric distribution. If we start a simulation with just 1 agent, there isa probability that the agent’s time to division is large. In figure 10, we cansee how the increasing the initial number of agents in the simulation reducesthe variance in simulation results. We can also see that the means of thetwo figures are similarly close to the ODE simulation’s result. This showsthat, although the variance with low numbers of agents is large, if we canperform enough simulation runs, the average result should describe similardynamics.

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Figure 9: The agent-based model (solid lines) compared with the ODEmodel (dashed lines). The left hand column of graphs shows the dynamicsof the agent-based model in which the population is diluted as a wholerather than as sub-populations, as in the right hand column. The solid linesin the graphs in the top row are the mean values at each time point of 100simulations. The bottom row of graphs is a zoomed in view of the dynamicsof each of the 100 simulations with γ = 0.75. The red dashed line showsthe ODE model. One can observe the first dilution occurring just prior tothe fifth time-point on the x-axis. In the left hand graph one can see agreater variation in impact of the dilution of population ratio. It can alsobe seen that, in this first dilution, some of the simulations lose all plasmid-free 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, futureextensions to it are desired. Due to the implementation of the model, theseextensions 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 variationis introduced, as in figure 11. We can consider each point and includemechanisms in the agent-based model which take these in to account. Thesuggestions below deal with these elements of variation and introduce othermechanisms which may be of interest.

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Figure 10: The exponential growth of a plasmid-bearing population, withoutplasmid loss or cell death. The blue line in each graph is the mean of 1000simulations of the agent-based model. The red line is the results of the ODEmodel with the same parameters. In the graph on the left, their was onlyone 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 eachgraph is the 95% interval.

Plasmid Copy Number Currently the model only refers to the cell’splasmid-bearing status as either true or false. Including a copy numberfor each agent would allow for much greater exploration of various plasmidmaintenance mechanisms. This parameter could effect several other newmechanisms as detailed below.

Active partitioning Active partitioning could be implemented with aparameter to determine its efficacy. With a high efficacy plasmid partition-ing could be perfect between the two daughter cells but as the parameterdecreases, more error creeps in to the process.

Plasmid replication control The model of binomial division outlinedpreviously assumed each plasmid replicates once between each division cycle.We showed that there must be other extant mechanisms in order to preventthe increase in copy number variance. This is likely to be due to replicationcontrol mechanisms preventing replication of plasmids once the copy numberis too high and making sure enough cycles of replication occur to preventthe copy number falling too low. The mechanism would likely rely on thecurrent copy number of the bacterium to control the rate of replication ofthe plasmids within that bacterium.

Metabolic burden The inter-division interval currently is taken from adistribution with mean T+ or T−. Since the metabolic burden that theplasmids impart on the host will affect the growth rate and therefore thedivision rate of its host, connecting division time to copy number wouldbe interesting. Different plasmids have different metabolic burdens so the

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Figure 11: An initial population of cells with equal mass (starting at thearrow) have normally distributed mass synthesis rates (1). DNA synthesisbegins when the mass reaches a required (noisy) threshold mass (2). Thecombination of these two elements of variation creates a distribution of “timeat initiation” (3) which approximates to a reciprocal normal distribution.There is a normally distributed variation in the DNA replication-segregationsequence after synthesis has begun (4). The variation due to division in todaughter cells of uneven sizes must also be taken in to account (5). Thecombination of these variations produces a “total interdivision time distri-bution” (6). (Cooper 1991)

implementation would need to provide a parameter for tuning the degree ofimpact that copy number has on the growth rate.

Uneven daughter size So far we have also assumed that the parent celldivides perfectly in half, creating to equal sized daughter cells. Evidentlythis is unlikely to occur as mitosis is a biological process which is inherentlyimperfect. The size of a cell at birth may also impact the division time of thecell. There is some disagreement in the literature as to whether cells divideafter their mass reaches a (noisy) threshold (Abner et al. 2014) or whetherthere is a given mass that the cells attempt to put on before dividing (Amir2014).

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6 Planned Experimentation

In the laboratory, we have begun to explore certain mechanisms to improveplasmid stability. Our main focus regards toxin-antitoxin post-segregationalkilling mechanisms but we also explore variations in the origin-of-replicationon 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 toxin-antitoxin mechanisms.

Name Type Bactericidal Source

hok/sok I√

E. coli plasmid R1kid/kis II E. coli plasmid R1txe/axe II Enterococcus faecium plasmid pRUMzeta/epsilon II

√Streptococcus pyogenes plasmid pSM19035

6.2 Origin-of-Replication

The origin-of-replication controls copy number which is of crucial importanceto 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 pUCplasmid, which has a copy number of ∼ 500− 700 which may be interestingto use in these experiments though it is unlikely to be useful in a therapeuticsetting due to the huge metabolic burden that it would place on the hostbacteria.

6.3 Plasmid Design

The plasmid that we use for our experiments, shown in figure 13, has threeimportant parts to it. There is a GFP reporter which allows us to determinewhich cells contain plasmid by viewing which cells are glow under a fluores-cent microscope. The GFP reporter is subject to a very strong constitutivepromoter, OXB20. It has kanamycin resistance built in to it. Although weare trying to get away from the use of antibiotic resistance, for the purposesof some experiments it is important to be able to start with an initial popu-lation of bacteria which are all plasmid-bearing. The simplest way of doingthis is using antibiotics though we could use a flow cytometry machine toseparate cells which express the GFP which is also attached to the plasmid.

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Figure 12: The plasmid we use in our experiments. This shows thekanamycin resistance, the GFP reporter flanked by the strong constitu-tive promoter OXB20 and the strong terminator RrnG, and the origin-of-replication derived from pSC101 plasmid.

7 Conclusion

The field of synthetic biology, although relatively new, has already shownits potential. Several studies have shown potential therapeutic benefits ofvarious engineered circuits as well as a huge number of applications in in-dustrial settings. Before the field can move further in to the medical realmcertain bio-safety considerations must be addressed.

Plasmids provide a useful vehicle for synthetic biological systems andbacterial hosts can provide the machinery to run the processes. However,the use of antibiotics for the maintenance of engineered plasmids in hostbacteria is not a viable solution in a medical setting and perhaps not in anindustrial one.

There are several mechanisms which plasmids harness in order to in-crease there stability. We have built a model to show the folly of relyingon high copy number without incorporating the necessary replication con-trols; variation in copy number quickly increases and the population tendsto plasmid loss.

We built a simple ordinary differential equation model into which the

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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 wecan more explicitly model each mechanism. This has the potential to allowus to build individual models of each mechanism, for example looking at thedynamics of the transcriptional gene network involved in toxin-antitoxinbehaviour. It also allows us to observe the stochastic effects of low numbersof bacteria, plasmids or any other agent that we wish to insert in to thesystem.

Finally, we have outlined a series of experiments with two goals: toprovide parameters for our models so that we may go on to use them tomake predictions, and to empirically demonstrate the effectiveness of varioustoxin-antitoxin systems for plasmid maintenance.

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A Additional agent-based model results

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Figure 13: Further dynamics of the agent-based model. The solid line showsthe mean of 100 agent-based model simulations and the dashed line showsthe ODE model results using the same parameters. The top graph hasγ = 1 and ω+− = 0. The bottom graph has γ = 1 and λ = 1. The initialpopulation size for both models was X+ = 104 and X− = 0.

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