Team:Lethbridge HS/Model
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:
1. 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. 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.
2. Burst Size: the number of phages produced per infected bacteria.
3. 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.
4. 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:
1. 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.
2. Influx of Bacteria: An increase in bacteria population caused by the entry of external bacteria into the system.
3. Natural Death Rate of Bacteria: Bacteria death caused by factors other than infection.
4. Natural Death Rate of Phages: Phages that become unable to infect more bacteria.
5. 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.
6. the number of phages produced per infected bacteria.
Equations
Results and Interpretations
Figure 5: Bacteria-phage dynamic in continuous time. The initial bacteria population reproduces and gradually increases. Once the bacteria population reaches the threshold (the peak observed) for mass action, the high abundance of susceptible bacteria makes the probability of phage infection increase greatly, at which time (about 1.25 hours after inoculation as shown in figure 5b) the phage population increase drastically and overwhelms the bacterial population. However, the bacteria population is never zero at any given moment due to the constant influx and the intrinsic rate of natural increase. Any presence of bacteria, though, results in the bacteria being immediately infected by the phage population, which explains the constant gradual increase observed in the phage population following its drastic increase around 1.25 hours and the appearance of the bacteria population to be zero.
Discussion
The gradual increase of phage populations proves that it is possible for our system to be sustainable. However, the extinction of the bacterial population (or, at least, the appearance of extinction, as the bacterial population is not actually zero but appears to be zero because at any moment, the presence of bacteria induces immediate infection and lysis) deviates from reality. The model suggests that the introduction of a very small amount of phage into a population of bacteria will cause the bacteria population to reach an equilibrium centered around zero. We do not believe this to be the case in real ecosystems. Hence, we carried out experiments where we collected real data on the interaction between bacteria and phage populations.
EXPERIMENTAL RESULTS
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:
1. 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.
2. 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.
3. 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.
COPPER BINDING EXPERIMENT
Another important component of our system is the metal-binding peptide, whose purpose is to sequester the floating metal ions before they are precipitated. To gain an understanding of how efficient our metal binding protein, CUT A, can bind to copper, we conducted an experiment where we added the same amount CUP I to solutions with different concentrations of copper. The figure below shows the result of the experiment.
For details on the interpretation of this result, refer to our result page.(click here ) Unexpectedly, the result did not show a consistent decrease in absorbance (indicative of a decrease in copper concentration) over time as the enzymes bind to copper. In fact, the concentration of copper appeared to have increased closer to the end of data collection. To understand this peculiar result, we proceeded to construct a model.
COPPER BINDING MODEL
Purpose:To determine how efficiently CUT A, our copper-binding protein, can bind to copper ions.
Definitions of Parameters and Variables:
1. Concentration of Free Copper in Solution
2. Concentration of Free Enzyme in Solution
3. Concentration of Enzyme-Ion Complex
4. Rate of Forward Reaction (Kforward): The rate at which free copper ions bind to CUT A. This is dependent on the concentration of free copper and CUT A.
5. Rate of Reverse Reaction (Kreverse): The rate at which bound copper ions are released. This is dependent on the concentration of free copper and bound copper.
Equations
Results and Interpretation:
Copper-binding models showing the concentrations of free enzyme (CUT A), formed complex (CUT A – Copper complex), and free copper. The change in concentration of free enzyme and formed complex over time as the enzyme binds to free copper is quite evident, and the result shows that the higher the initial concentration of copper (i.e. the higher the concentration of CuSo4), the faster the rate of change. However, there is no detectable change in the concentration of free copper at all regardless of the initial concentration. Why? We realized that this is because our copper concentration was too high compared to our enzyme concentration. The lowest copper concentration we used in the experiment was 0.2365 molar, whereas the enzyme concentration used was 1 micromolar, or 1E-6 molar. The tiny amount of enzyme is too insignificant to cause a change in the concentration of free copper.
Improve
With further research in elementary kinetics, we realized that our enzyme, CUT A, is capable of binding to six copper ions simultaneously. Previously, we have assumed that there is a 1:1 ratio of enzyme to copper ions. This assumption turns out to be false. With multiple binding, the new equations are as follows (where n represents the number of substrates each enzyme can bind to):
Results and Interpretation:
Copper-binding models where each free enzyme (CUT A) may bind to six substrates (copper ions). Compared with previous results, the rate of change of the concentrations of free enzyme and ion-enzyme complexes are faster, but there is still no detectable change to the concentration of free copper. This is expected as the difference between the copper concentration and enzyme concentration is too great (of a 6th order). Successful removal of copper ions will require a much higher concentration of enzymes.
REFERENCES
1. Ellis, EM; Delbruck, MA (1939). The Growth of Bacteriophage. The Journal of General Physiology. Journal 22 365–384.
2. Fields BN, Knipe DM, Howley PM (2007). Fields virology: Part 1. Philadelphia: Wolters Kluwer Health/Lippincott Williams & Wilkins.
3. Caims BE, Timms AN, Jansen VI, Connerton IA, Payne RO (2009). Quantitative Models of In Vitro Bacteriophage–Host Dynamics and Their Application to Phage Therapy.
4. Claudio Altafini Lecture in ODEs Model in Systems Biology Retrieved from https://www.sissa.it/fa/altafini/teach/SISSA07/lect07D-prn-2.pdf