Introduction and Motivation

In order to suppress transcription of the mecA gene, we need a way to deliver our silencing system to the antibiotic resistant cells. To accomplish this, the CRISPRi and sRNA genes are cloned into an engineered conjugative plasmid in closely related cells. The cells with the engineered plasmid are referred to as “donor” cells, and they are introduced to the target population, also known as the “recipients”. The plasmid will transfer from donors to recipients via conjugation, and we refer to these recipient cells with the conjugative plasmid as “transconjugants”. The transconjugants will, in time, be able to retransmit this plasmid. Thus, the system will propagate throughout the infection, thereby disarming the antibiotic resistant cells.

(fig. 1 - diagram of transmission and retransmission of plasmid)

We model the propagation of our plasmid for a number of reasons. Most importantly, we want to determine the optimal time to apply antibiotics to the infection. To do so, we track the total number of donor, recipient, and transconjugant cells and define the “fall time” as the time it takes the number of recipients to reach 10% its initial value. In addition to determining when to apply methicillin, the model can be used to find how large of a conjugation rate is needed and what initial concentration of donor cells is needed to spread the plasmid at a fast enough rate.

Unfortunately, modeling such a system poses quite a bit of difficulty. The primary obstacle is that we consider S. aureus growth on a solid surface (e.g. a lab plate, or on your skin) and not in a well mixed environment. This forces us to abandon more traditional models of conjugation (such as from Levin, Stewart, and Rice) and develop a spatial model instead. We took two main approaches to developing such a model: an agent-based model (ABM), and a partial differential equation (PDE) model.

The ABM is a stochastic simulation of conjugation through a population of donor and recipient cells. The main advantages of this model are that it accounts for randomness of plasmid transfer, as well as likely maintaining accuracy on small-scale areas. Its main drawback is that it is computationally expensive to model a large number of cells, and thus cannot model populations on large scales.

The PDE Model addresses large scale populations through deterministic methods. Being a differential equation system, a wide variety of mathematical tools to analyze the system are readily available, and it is more computationally efficient than the Agent-Based Model.

Model Formation

We have developed a novel model for bacterial conjugation on flat surfaces. By treating each bacterium as an “agent” that has properties associated with its type (either Donor or Recipient), and letting the group of agents interact over a prescribed period of time, one can make qualitative and quantitative conclusions about their behavior.

The model is based on the idea of having hexagonal cells that each may or may not be occupied by a donor (e.g. modified S. epidermidis) or a recipient (e.g. MRSA). An individual bacterium may divide into an empty neighboring cell and if the cell is a donor, conjugate with an adjacent recipient. We assume that the conjugative plasmid represses methicillin-resistance 100% (i.e. all donor cells will die upon introduction of antibiotic).

The hexagonal grid was used over a traditional square grid because hexagons offer more routes for conjugation and division to occur, as well as the fact that hexagons form a tighter packing structure than squares. Because bacteria are capable of very tight packing on flat surfaces, the hexagonal framework more accurately captures interaction between neighboring bacteria - an important consideration when modeling conjugation.

Conjugation Images