Team:Nottingham/Model

Modelling

Aim: To model the interaction between bacteria and phage to gain insight into how we can best engineer and deliver phage to be an effective treatment against Clostridium difficile infection.

Figure 1: Lytic and lysogenic phage life cycle. After a bacterium has been infected by a lytic bacteriophage, the viral genome material is transcribed, translated and replicated using the bacterial cellular machinery to produce viral proteins. The proteins are assembled to make viral particles and the genomic material is packaged into the virions. Once the bacterial cell reaches capacity, the host cell lyse, resulting in the release of the viral particles. These viruses can go off to infect other uninfected bacterial cells. Bacteria can also be infected by lysogenic bacteria. However, transcription and translation of the viral genetic material is repressed. It is instead integrated into the host genome where it remains and is replicated with the host genome. Upon induction, however, the lysogenic life cycle is switched to the lytic life cycle.

The diagram above shows the two possible life cycles of phages. Following infection, phages following the lytic life cycle will hijack the host cell machinery to produce multiple copies of the phage proteins. These proteins are then assembled into multiple phage progeny which burst out of the host cell and go on to infect other bacterial cells. In the lysogenic life cycle, phages can integrate their genome into the host cell chromosome upon infection, where they can remain dormant for long periods of time as prophages. When conditions are favourable, usually due to host cell stress, prophage induction can occur. This is where prophages can excise from the host cell chromosome and enter the lytic life cycle leading to the production of progeny phage particles.

Click on the different sections to learn more about our modelling efforts and remember to refer back to the above diagram to help understand how our ODE models reflect the phage life cycles.

Summary

Adapted a lytic model from literature to represent our initial plan for SBRC phage (no induction model).

Adapted model to include prophage induction

Determined all equilibrium points of temperate phage models and relevant stability conditions.

Estimated SBRC phage decay rate after noticing correlation between decay rate and burst size in a previous phage study.

Determined other SBRC phage and C. difficile parameters from our experimental data and a literature search.

Investigated and compared models analytically and numerically to determine which best suited our purpose.

Proposed a model for the correlation between phage concentration and induction rate. Although this model is not influential for SBRC phage, it should be noted by other teams studying different phage.

The main analytical result of our modelling work was the derivation of a condition that determines whether effective temperate phage therapy treatment can occur when prophage induction is allowed:

$\frac{(b-1)k_a}{d_P}>\frac{u}{c(u-q_0-d_X)}.$

Our estimated parameters satisfy this inequality so provided our antisense RNA and dCas9 contructs are effective at suppressing toxin production in lysogens this should lead to a stable population of non-toxigenic C. difficile and SBRC phage which would help to prevent reinfection. We also noticed with our induction model that initial phage dose is less crucial to effective treatment. This is particularly important since we gathered from Dr Cath Rees that high phage titre, are difficult to produce and would lead to a costly therapy. Therefore, the ability of our phage to be effective at a low initial dose should lead to a more affordable treatment. From these results we decided to allow SBRC phage to undergo prophage induction at its natural rate instead of preventing induction as we had initially planned.

Lytic phage model

Phage therapy that has been used to date has focused on using lytic phage to kill pathogenic bacteria. In this section we based our model of lytic phage-bacteria interaction on the delayed differential equations proposed by Campbell before adapting these equations in order to model a temperate phage.

Click on the link below to see results.

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Temperate phage models

Why consider a temperate phage?

One of the biggest issues surrounding Chlostridium difficile infection, is that of reinfection. We originally planned on using a temperate phage, modified so that it could not enter the lytic cycle, with the hope that a stable population of non-toxigenic lysogens (see lab pages for how we developed non-toxigenic C. difficile) would be formed that could outcompete toxigenic C.difficile strains [10]. Hence, we wanted to model the interaction between our engineered temperate phage and bacteria to see if a steady state could be reached and what conditions would be necessary for this to occur. We also wanted to see the time difference in effective treatment between a lytic and temperate phage to ensure that we did not sacrifice speed of treatment in order to prevent reinfection. We later decided to see what would happen if we allowed for phage to undergo induction.

Click on the links below to see our results for both models.

Model without induction Model with induction Induction as a function of phage concentration