MODELLING

The evolution of our bacteria-phage dynamic model helped us gain a better understanding of the interaction between a bacteria population and a phage population and its impact on the viability of our design. After defining a variety of parameters and making several assumptions, we showed that it is possible for our system of bacteria and phages to be self-sustainable. Comparing our model with our experimental results, we developed a second model where we accounted for additional factors such as a possible mutation in the bacteria’s DNA that results in resistance against phage infection. Furthermore, we modelled the copper-binding efficiency of CUP I (our copper-binding protein) to estimate the optimal ratio of enzyme and copper concentrations that would result in the most efficient binding in the implementation of our system.

DISCRETE TIME MODEL

Purpose:Given an initial Multiplicity of Infection (MOI) and infection onset point (during a bacteria lifecycle), determine how the populations of bacteria and phages change over discrete time intervals.

Assumptions:

• There is no delay in infection
• All bacteria are susceptible to infection, and all infections are successful.
• All bacteria death is caused by infection (i.e. there is no natural death)

Definitions of Parameters and Variables:

Multiplicity of Infection (MOI): The MOI represents the initial ratio between the number of phages and number of bacteria at the time of inoculation. It is a decisive factor in calculating the probability that a bacteria will be infected by at least one phage particle. The equation that relates the MOI to this probability is:

where P is the probability, and m represents the multiplicity of infection. (reference Wikipedia page) Although it is possible that a bacterium is infected with more than one phage particles, a research study conducted by Ellis and Delbruck suggests that bacteria infected with multiple phage particles had similar burst sizes than bacteria that were infected with only one phage. (insert reference)

Burst Size: the number of phages produced per infected bacteria.

Lysis Time: the time it takes for a phage to infect and lyse a bacteria host.

Both the burst size and the lysis time depend on the point during a bacteria lifecycle when inoculation begins. One research study conducted by Zachary Storms and Tobin Brown shows how the burst size—the squares, and the lysis time—the circles, vary with when the infection starts. Storms and Brown suggest that the burst size is the highest and the lysis time is the shortest if the infection starts when the bacteria cell enters the division stage, because at that point the cell has the richest intracellular resources.

Bacteria doubling time: the time it takes a bacteria population to double in size.

Equations:

Results and Interpretation:

The following graphs are constructed with an initial bacteria population of 1,000 and an initial phage population calculated according to the MOI used. However, the actual numbers of bacteria and phages do not influence the trends observed in the graph, as it is the ratio between these numbers (the MOI), not the actual numbers, that matters.

Discussion:

Although this model clearly shows the influence of the initial MOI on the bacteria/phage dynamics, it suggests that even with a very small MOI, the bacteria and phage populations always become extinct. This is not realistic because in many real-world instances, bacteria and phages co-exist. We realized that the problem with this model is that it treats time as a discrete variable. In discrete time, phages reproduce much faster than bacteria so that the bacteria population is wiped out regardless of the initial values. Hence, we proceeded with a different approach, treating time as continuous.

CONTINUOUS TIME MODEL

Purpose:Given initial populations of bacteria and phages (and hence an MOI), understand how the populations interact and change over continuous time.

Assumptions:

• There is no delay in infection. (All infections are instantaneous.)
• All infections are successful (infected bacteria subsequently die of infection)
• New bacteria are introduced from the external environment at a constant rate. This is a reasonable assumption because bacteria concentration in the water that enters our multi-cistern system should be consistent.

Definitions of Parameters and Variables:

Intrinsic Growth Rate of Bacteria (r): aka. Intrinsic rate of natural increase or the Malthusian parameter. This rate describes the maximum theoretical rate of increase of a population per individual. Using this parameter to account for bacteria population growth instead of the traditional doubling time allows us to investigate the interactions between phages and bacteria in continuous time.

Intrinsic Growth Rate of Bacteria (r): aka. Intrinsic rate of natural increase or the Malthusian parameter. This rate describes the maximum theoretical rate of increase of a population per individual. Using this parameter to account for bacteria population growth instead of the traditional doubling time allows us to investigate the interactions between phages and bacteria in continuous time.

Influx of Bacteria: An increase in bacteria population caused by the entry of external bacteria into the system.

Natural Death Rate of Bacteria: Bacteria death caused by factors other than infection.

Natural Death Rate of Phages: Phages that become unable to infect more bacteria.

Contact factor (c): This parameter describes how efficiently a given phage population can infect a given bacterial population. Although the computation of c is very complex and involves many factors as described above (such as the time during a bacteria lifecycle when inoculation begins), the use of empirical evidence of the value of c from past researches simplifies this process without compromising the accuracy of our model.