Bacterial Growth and Antibiotic Resistance Netlogo Model Kyle Hartson and Ashok Prasad Preface Netlogo is an agent-­‐based modeling system, which can be used to program and model countless biological phenomena. This particular model has the ability to simulate several simple bacterial ecologies observed in nature and in experimental data. Figure 1. Screenshot of the model The following is a list of these situations, along with the parameters used to acquire these simulations and comparisons with experimentally observed data. Exponential Growth One of the defining characteristics of bacteria is their ability to reproduce quickly and produce exponential growth curves when unrestrained by resources. This model allows for the recreation of this curve. This is achieved by using a variation of the Gillespie algorithm. First, the overall growth rate of the wild types is calculated by subtracting the death rate from the growth rate, and the absolute value of this number is multiplied by the total number of wild types present. If we refer to this previous value as A0 (as it is named in the code), then a time step is calculated by generating a random exponentially-­‐distributed floating-­‐point number having a mean of (1 / A0). After this amount of time has passed, the next “reaction” will occur (a “reaction” being either a bacteria reproducing by dividing into two identical bacterium, or dying) which in the case of exponential growth will be a “birth reaction” since our A0 term is positive. After a set amount of time steps has passed, the simulation moves forward one tick and carries out all birth reactions simultaneously. With both the Resources and Antibiotics switches turned OFF, and the Mutation Rate set to 0, we can observe exponential growth in the Wild Type population. Figure 2. Conditions for exponential growth This curve was obtained by setting Initial Bacteria Population to 100, Birth Rate Wild Type to 0.5, and Death Rate Wild Type to 0.4. Exponential Growth 16000 Bacteria Cells 14000 y = 99.618e0.0395x R² = 0.99988 12000 10000 8000 Wild Types 6000 Exponential Trendline 4000 2000 0 1 11 21 31 41 51 61 71 81 91 101 111 121 Ticks Figure 3. Exponential growth curve produced by model under above conditions It can be seen that we can get a really nice fit to the exponential growth curve from the simulation results. Note that the simulation is fully stochastic and realistic, i.e. a single bacterium becomes two daughter bacteria. One could envision class exercises in which the consequences of exponential growth are explored using this simulation. By changing how fast bacteria divide, we get different rates of growth. Similarly students could play with death rates and get quantitative understanding of the effective growth rate. Students could export the raw data and use Excel or a similar program to fit it to an exponential trendline. Thus the data generated can be used for teaching statistical techniques too. More advanced students could be introduced to the concept of noisy variations around a trendline, and understand stochastic fluctuations by measuring the variance and standard deviation of the system around its exponential trendline. Logistic Growth (Carrying Capacity) When Resources are introduced into the system, the bacterial growth follows a logistic curve and the population reaches a carrying capacity. With the Resources switch turned to on, we can observe this exact trend. To obtain these results, the model utilizes a population-­‐limiting factor in the calculation of A0. When the Resources switch is active, A0 is multiplied by (1 -­‐ (N + Z) / Carrying Capacity), where N and Z are the total number of wild types and mutants present, respectively. This term dictates that as the total bacterial count approaches the Carrying Capacity, A0 will tend to 0, which leads the program to favor death reactions until the population is again under this capacity. This graph shows Wild Type population with the Carrying Capacity set to 1500, and the Initial Bacteria Population set to 100. The time scale has been adjusted so that 1 tick = 4 minutes. Figure 4. Initial conditions for logistic growth Viable Cells Carrying Capacity 1600 1400 1200 1000 800 600 400 200 0 Bacteria Trendline 0 2 3 5 7 8 10 12 13 15 17 Time (hours) Figure 5. Growth curve for wild type population under logistic conditions If we compare this to a chart from “Modeling of the Bacterial Growth Curve” By M. H. Zwietering et al, we can see that the data is a close fit. Figure 6. “Growth curve of L. plantarum at 40.0°C” This simulation provides a quantitative introduction to the notion of “carrying capacity”. Class exercises could have students changing birth rates, death rates and carrying capacity to understand the interplay between them. More advanced exercises could also study fluctuations as well as use the data generated to generate non-­‐linear fits in external software such as Excel or Matlab. Exporting plot data from Netlogo is easily achievable by navigating to File > Export > Export Plot. The program then allows you to select a location to save the .csv file. Natural Selection (Genetic Mutations) An important biological principle is genetic variation, a difference in genetic information among members of a population brought on by mutations. These differences give rise to organisms with a higher fitness than others in the population, which are then naturally selected for survival. We can model this by setting the Mutation Rate in the model. This slider controls the probability of a mutation occurring when a Wild Type divides, creating a Mutant. Figure 7. Initial conditions for natural selection model If we then leave the birth and death rates the same for the Wild Types, but give the Mutant population a higher fitness by making its birth rate higher, say 0.7, we naturally select it for survival and the Mutant population will thrive, as seen in this graph. Natural Selection Bacterial Cells 1200 1000 800 600 Wild Types 400 Mutants 200 0 1 21 41 61 81 101 121 Ticks Figure 8. Growth curves for wild type and mutant bacterial populations Here, Carrying Capacity was left at 1500, and the Mutation Rate slider was set to 100. We can see that as time goes on, the Mutant population with the higher fitness outcompetes the Wild Types and flourishes. Additional classroom exercises here could involve playing with mutation rates and growth rates to get an idea of how mutations can get fixed in populations by outcompeting native species. Antibiotic Response When we introduce antibiotics into the system by turning the Antibiotics switch ON, we can explore the effects of drugs on bacterial populations. The first situation we can model is the response by a Wild Type population to the administering of antibiotics. In this model, the antibiotics work by multiplying a term with the death rate of the Wild Types. This term is one plus a ratio of the concentration of the drug divided by the concentration added to the half-­‐maximal concentration. This is sometimes called a hyperbolic function, or a Michealis-­‐
Menton function and is a standard way of representing a saturating process. In this way, the death rate is maximal when concentration is maximized, and decreases back to baseline values as concentration decays. This decay is obtained by setting the antibiotic concentration to follow the exponential decay function with the Decay Constant chosen by the user. Figure 9. Antibiotic slider conditions With an Initial Concentration (antibiotics) of 60, a Decay Constant of 0.15, Max Kill Rate of 2, and Half Maximal Concentration of 15, we see the following behavior. Bacterial Cells Antibiotic Response 1.20E+06 1.00E+06 8.00E+05 6.00E+05 4.00E+05 2.00E+05 0.00E+00 Bacteria 0 1 2 3 4 5 6 7 Time (hours) Figure 10. Growth curve of wild types under antibiotic influence The cell count and time has been scaled to match the experimental data presented below, taken from “New In Vitro Model to Study the Effect of Antibiotic Concentration and Rate of Elimination on Antibacterial Activity” by S. Grasso et al. Figure 11. “Antibacterial activity of cefazolin at different starting concentrations, C(o): 5, 10, and 20 pg/ml decreasing with half-­‐lives of 30, 60, and 120 minutes ,against E.coli N 4242 and Klebsiella SSG30.” Thus the simulation of antibiotic response is able to reproduce actual experimental and theoretical data available in the research literature. This simulation can be used to play with a number of what-­‐if scenarios. For example it can be used to get an understanding of the importance of antibiotic dosage. Running the model with the parameters shown in Fig. 12 will make the bacteria all die before the mutant has time to take root in the population. However decreasing the Antibiotic concentration to 25 instead of 100 will lead to the bacterial population escaping from the effect of the antibiotic, and the exponential growth of the antibiotic resistant mutant. Moving the antibiotic concentration to 40 results leads mostly to death of the population, but occasionally to survival, underscoring the importance of stochastic fluctuations in these processes. Advanced students could explore these fluctuations that are important in infections. Figure 12: A screenshot of the parameters that lead to death of bacteria. The chart shows bacterial population plummets to zero before the m utation has time to take root in the population. The growth of antibiotic resistance is due to both the emergence of mutations as well as horizontal gene transfer. In this simulation we have not incorporated horizontal gene transfer, but plan to do so in future editions of this software. Works Cited Grasso B., Meinardi G., de Carneri I., and Tamasala V. (1978.) New in vitro model to study the effect of antibiotic concentration and rate of elimination of antibacterial activity. Antimicrob. Agents Chemother. 13: 570–576. Zweitering, M. H., Jongenburger, I. Rombouts, F. M., and van’t Riet, K. (1990). Modeling of the bacterial growth curve. Appl. Environ. Microbiol. 56: 1875-­‐
1881.