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Plasmid loss

Introduction

Getting plasmids to enter bacteria can be a complicated process. Making them stay there is even more so. For this purpose, antibiotic resistance is usually put on the plasmid for selective pressure, which decreases the chance of plasmid loss. However, there might be a few problems when using these procedures: For example, formation of Satellite Colonies (colonies of plasmid-free bacteria that can grow due to the degradation of extracellular antibiotics by the plasmid-containing bacteria) which mainly occur due to over-incubation. This can result in a competition between the plasmid-containing and the plasmid-free bacteria, which can lead to overgrowth of the later.

Although this scenario is not very common while working in the lab, it can still occur more frequently in uncontrolled environments (i.e. when trying to use synthetic biology products outside of the lab). Therefore, before considering the use of Synthetic Biology in commercial products, we must take into account scenarios that don't necessarily happen regularly in the lab. In this document, we raise the problem of plasmid loss, describe some common solutions and problems, model it and suggest solutions.

Plasmid loss

Plasmid loss happens when a bacterial replication results in a plasmid-containing bacterium and a plasmid-free bacterium (Fig. 1b). This type of replication is usually uncommon [1]. However, in some cases (i.e. in satellite colonies, described in the introduction), the plasmid-free bacteria can grow and replicate on their own, which aggravate the problem of plasmid loss (Fig 1c).

Common solutions to plasmid loss and problems

Antibiotic resistance

Antibiotic is often used to prevent from plasmid-free bacteria to live in the medium used in the experiment (because antibiotic resistance is put on the plasmid). This mechanism, when designed and used properly, can eliminate the chance of plasmid-free colonies alongside plasmid-containing colonies.

A typical problem with this solution is degradation of the antibiotics in the medium, creating space for unwanted bacteria to grow. Let us take for example Ampicillin selection on E.coli. The Ampicillin is inactivated by secretion of beta-lactamase expressed from the plasmid. However, if the transformation is incubated long enough, the secretion will cause a formation of a circle, usually bigger than the colony itself, in which there is no presence of Ampicillin. In this circle, due to the lack of selective pressure, unwanted colonies (usually made of plasmid-free E.coli, but of other species of bacteria as well) can easily grow and compete with "our" bacteria (the ones that contain the plasmid). In order to overcome this problem, a stronger and more durable antibiotics is often use. However, the fact that this happens only if the transformation is incubated longer than recommended suggests that this result may be unavoidable in long periods of time.

Plasmid integration

Another common solution is plasmid integration – insertion of the plasmid genetic content directly into the bacterial genome. Since the bacterial genome is not lost in replication, there is no chance of losing the genetic additional genetic content that was inserted. However, this type of insertion is a bit more difficult than plasmid insertion, thus is not always the preferred method.

When modeling plasmid loss, we deal with 2 populations – plasmid-containing bacteria and plasmid-free bacteria.

Assumptions

• 1. At t=0 (initial time), we have only plasmid-containing bacteria.
• 2. At t=0, the colony has reaches the beginning of the exponential growth rate
• 3. Horizontal gene transfer is negligible
• 4. Divergence in replication characteristics among each population is negligible (e.g. each replication of the same kind has the same probability of happening and takes equal time)
• 5. A plasmid-containing bacterium replicates in replication type 2 (which result in plasmid loss) with a probability p and in replication type 1 with probability (1-p).
• 6. Each plasmid-negative bacterium replicates in type 3 replication.

Methods

We write an ODE (Ordinary Differential Equation) for each population type. Please note the following remarks:

• 1. The plasmid-containing population growth rate is only affected by the rate of replication type 2. Explanation: Type 3 is not relevant here, and type 2 doesn't increase or decrease the size of this population.
• 2. The plasmid-containing population growth rate is only affected by replications type 2 and 3 rates. Explanation: Type 1 is not relevant here. Type 3 is the "self" replication type, and type 2 causes the plasmid loss, thus adding a bacterium to this population each time
• 3. There is no consensus regarding the equality between the doubling times of the two populations. Here, we describe the more general case, in which these two can be different or equal.

Notations

• N⁺-number of plasmid-containing bacteria.
• N^--number of plasmid-free bacteria.
• N- total number of bacteria.
• T⁺-doubling time of plasmid-containing bacteria.
• T^--doubling time of plasmid-free bacteria.
• P-probability of replication type 2.
• T_max-maximal number of bacteria.

Equations

Results

After writing down the equations, we obtain the following graphs. In general, all the graphs have the same behavior. In each graph there are 2 lines. The red one represents the ratio between the plasmid-free population and the total population. The blue one represents the ratio between the plasmid-containing population and the total population. The graphs were obtained from a numerical ODE solver which we've written in Excel.

Discussion