In one of our experiments, we added in the same amount to phages to different concentrations of bacteria cultures. In another of our experiments, we added in different concentrations of phages to the same concentration of bacteria. We monitored the change in the optical density of the cultures over 24 hours to investigate the influence of the initial concentrations of bacteria and phages on how the populations interact. Growth of bacteria cultures in the presence of phages. Figure 6a shows that different concentrations of phages have some influences on the growth curve of bacteria. For each group of bacteria, the highest concentration of phages results in the least amount of growth. (the lowest concentration at the end of data collection.) However, in no instances did the bacterial population perish completely. Similarly, as figure 6b shows, without the presence of phages, the bacteria population grows before reaching a carrying capacity, at which point the growth curve levels off. However, even in the presence of phages, similar growth curves can be observed. There is no sign that the presence of a phage populations causes the bacteria population to become extinct, which our previous models suggest. In fact, the culture with the same concentration of bacteria as the negative control (0.01OD) but with the presence of phages had a slightly higher concentration of bacteria at the end of data collection. Clearly, bacteria and phages can and do coexist. After obtaining these results, we proceeded to improve our model in hopes of explaining the coexistence of bacteria and phages.

SIRV MODEL

What is SIRV?

An SIRV model (Susceptible, Infected, Recovered, and Vaccinated) for the spread of a virus, in general, divides the viral host population into four classes: those susceptible to be infected, those that have already been infected, those that have recovered from an infection, and those that have grown resistance to the virus and thus not susceptible to be infected. The difference between our SIRV model and the earlier simple continuous time model is that with the SIRV model, we take into consideration the possibility that certain individuals from a strain of bacteria may, over time, become resistant to a type of phage through genetic mutation. This is in keeping with the law of natural selection and evolution, and we hoped that this model may help to explain why bacteria and phages co-exist in many instances.

Purpose:Taking into account the possibility of genetic mutations and virus resistance, we improved our previous continuous-time model in hope of explaining the co-existence of bacteria and phages.

Assumptions:

• All the assumptions from our previous Continuous Time Model.
• No recovery is possible. Unlike other viruses that attack animals and plants, phage infection of bacteria is relatively fast, and a recovery is unlikely. Therefore, we did not consider the possibility that an infected bacterium may revert to a susceptible state.

Definition of Parameters and Variables:

All parameters and variables from the Continuous Time Model.

**The intrinsic growth rate and natural death rate of bacteria will apply to both the susceptible class and the resistant class of the bacterial populations in this model. The prototype below will demonstrate this.

Mutation rate: the mutation rate represents the probability that an individual from a population of bacteria will develop resistance against the infection of a certain type of phage through genetic mutation.

Probability of successful lysis: Although it is assumed that no infection may be reverted, there is a small probability that an infected bacterium may not be able to produce phages (The bacteria may have defective enzymes, for example), in which case the infected bacteria will not be lysed. This parameter, the probability of successful lysis, is involved in computing the number of infected bacteria and number of phages produced at any given moment.

Equations

Results and Interpretation:

SIRV model for the interaction between bacteria and phage populations. As susceptible bacteria replicate and grow, they reach the threshold (the peak observed) for mass action, where the probability of phage infection increases greatly, at which point most susceptible bacteria become infected, causing a drastic increase in the population of infected bacteria. Then, as infected bacteria are lysed, their decrease is accompanied by a surge in the phage population, which reaches 2.0e9 about six hours after inoculation. As with the previous model, although the number of susceptible bacteria decreases greatly, their population is never truly zero due to the constant influx and intrinsic rate of natural increase. At any given moment, however, the entry of susceptible bacteria into the system is followed immediately by the infection of these bacteria by phages. Therefore, the number of phages continues to increase gradually, and the total number of dead bacteria also gradually increases.

Bacteria that have grown resistance to phage infection are not affected with the concentration of phages. Their growth resembles the natural bacteria lifecycle with a log phase (20-25 hrs after inoculation) following a lag phase (0-20 hrs after inoculation). Without a limiting factor (figure 9a), the log phase will continue indefinitely, as the figure on the left shows. With a limiting factor (figure 9b), the population will reach their carrying capacity, and the curve levels off into a stationary phase, as the figure on the right shows. Additional resistant bacteria may also be introduced to the system through mutation of the susceptible population.

Discussion:

This model explains the fact that bacteria and phages may co-exist in the natural environment—over time, some individuals of a bacteria population undergo genetic mutation and become resistant to phage infection. These individuals are subsequently selected for, and the bacteria population evolves. The “disappearance” of susceptible bacteria can be partly explained by evolution, and partly by our assumption that phage infection is instantaneous. A better way to construct the model would have been to use a system of delayed differential equations (DDE), but that is beyond our current ability.

The model suggests that the evolution of the bacterial population may pose a problem to the implementation of our system. If we use the same strain of bacteria and their corresponding phages over long periods of time, the entire population may grow resistance to the phage, meaning that our engineered phages will no longer be able to reproduce and our system no longer functional. To correct this problem, we may need to introduce another strain of bacteria that is vulnerable to phage infection and that competes with the strain that has developed resistance. It is also possible that the phage population will coevolve with bacteria, or that the resistant bacteria may mutate again and lose the resistant gene.

In the future, we will continue to improve our model to incorporate more aspects of our system. For example, we may model the influence of decreasing phage numbers as a result of ELP precipitation. Modelling the interaction between bacteria and phages populations has helped us enormously in understanding the viability, efficiency and sustainability of our design.