Population Explosion: Modeling Phage Growth Jean Douthwright, Frank Percival, Julyet Aksiyote Benbasat, Tia Johnson, and John R.

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1 41 Jean Douthwright, Frank Percival, Julyet Aksiyote Benbasat, Tia Johnson, and John R. Jungck Video I: The Microbial Universe Bacteria never die, they just phage away - Mark Mueller Bacteriophage (phage) are viruses that infect bacteria. Like other viruses, they are composed of genetic material surrounded by a protein coat. They can be isolated from sewage, from feces, from springs, and from soil. Bacteriophage are highly specific in the type of bacteria that they can infect. This specificity makes them useful in the laboratory and of interest to medical researchers as a potential alternative to antibiotics to control pathogenic bacteria. Once a bacterial cell is infected with one or more bacteriophage, the phage reproduce using the machinery of the cells that they infect (the host cells). Several rounds of viral replication may occur before the infection culminates with the bacteria breaking open (lysing), sending thousands of new bacteriophage into the immediate environment. These progeny phage then move on to infect other bacterial hosts. Thus, unlike the continuous growth observed in mold growing on bread or bacteria in culture, viral numbers increase in pulses latent periods followed by bursts when the population size dramatically increases. Figure 1. The T7 phage life cycle. Because bacteriophage can only reproduce within appropriate bacterial hosts, their rate of increase is tied to the growth of the bacteria. In order to model the increase in phage populations, therefore, you must also study the growth of the bacteria they infect. An understanding of the interdependence of these species is critical to developing an accurate model of population growth. In this activity you will model the way particular viral communities (bacteriophage) interact with their host species (bacteria) within their shared ecological community, using a mathematical model of dependent growth. The interdependence of phage on host bacteria is one of the characteristics that make them attractive as a medical therapy for bacterial disease in humans and other multi-celled organisms. Researchers have found that phage populations in infected individuals expand rapidly, infecting and then killing all available hosts, but that phage numbers crash as they exhaust their supply of hosts. Because these phage are specific to certain strains of bacteria, there is little risk of wiping out helpful bacteria in the host, and less chance of resistance spreading to different species of bacteria. When all of the disease-causing bacteria have been killed, their phage die as well. (See TB and Antibiotic Resistance in Chapter 4 for an activity related to the development of antibiotic resistance.) Phage have been used as medical therapy in Eastern Europe for several decades, and research in their use is growing in the United States. Copyright 2003

2 42 Microbes Count! Titer: A measure of the concentration of a substance in solution. So a high titer lysate is a solution with a high concentration of phage, measured after the bacteria have burst. The Phage Growth Spreadsheet Model To create a preparation of bacteriophage in the lab, you could mix a small amount of phage with the right bacterial host. When the cells lyse, the solution now called a lysate will contain many virus particles. In this process you typically are interested in producing as many virus progeny as possible that is, you would like to produce a high titer lysate. This is the starting point for producing more concentrated and purified bacteriophage stocks for work in the laboratory. Discussions of bacteriophage growth typically focus on a one-step growth curve because it provides a framework for thinking about the basic steps of a virus life cycle. When growing phage in the laboratory, however, bacteria are infected with phage, and several rounds of viral replication may occur before the majority of bacteria in the culture are broken open, releasing many phage progeny and producing a high titer lysate (see Figure 1 above.) Table 1: The Input Parameters for the Phage Growth model. In this activity, you will use a model of phage growth to explore some of the factors that affect phage production and the final phage concentration in a high titer lysate. There are six input parameters for the model, all of which you can vary. These parameters are described in Table 1 below. Six Input Parameters Parameter Description Character Example Value Initial concentration of uninfected bacteria Bacterial growth rate Maximum population size of the bacteria, the carrying capacity Initial concentration of phage, initial phage population Number of phage produced per infected bacterial cell, phage growth rate Maximum number of phage which can be attached to a bacterial cell B 0 Bacterial Doubling Time Maximum Bacterial Population Size, Bact K P 0 Burst Size Phage Binding Sites per Bacterium 2.00E+07 cells/ml 30 minutes 2.00E+09 cells 5.00E+03 phage/ml 200 progeny phage/cell 250 phage/cell In this activity you will use a Microsoft Excel spreadsheet to manipulate the model parameters and graph your results. The results of the experiment that you set up when you enter values for the input parameters are calculated using equations already entered into the spreadsheet. These data are displayed in a table and plotted in a graph. To see the model, open the file called PhageGrowth.xls, located in the Modeling Phage Growth section of the Microbes Count! CD. The first row in each column of the data table contains a short description of how the values in the column are calculated. For additional information, move your cursor over the heading of the column. A box describing the calculation will pop up. You should study these explanations carefully so that you will understand

3 43 how this simulation models the growth of the bacteriophage population. You will find it much easier to appreciate the complexity of the interactions between the phage and their bacterial hosts if you spend some time working through these calculations. (See the section at the end of this activity for more information on using a spreadsheet model.) This model of phage growth was developed for use in conjunction with laboratory work involving the production of high-titer phage lysates, but even without that context, there are a host of questions that you can explore with using the spreadsheet model. Description of the phage growth model All models are oversimplifications of the phenomena that they are used to represent. Simple models provide two enormous advantages: first, parsimonious models can often account for the major patterns and most of the observed variation in actual experiments; second, if the predictions of the models fail, they provide excellent templates for heuristically investigating what went wrong and why. Our phage model includes the following assumptions: Free phage cannot reproduce; however, they can persist indefinitely. The bacteria grow logistically (see Modeling More Mold in Chapter 1 and Modeling Microbial Growth in Chapter 3 to explore the difference between linear, exponential, and logistic growth). That is, they cannot reproduce at high rates forever because they run out of nutrients and hence reach a carrying capacity in their ecosystem. Thus, their populations eventually reach an equilibrium of births and deaths. In our model, we have an additional cause of death other than starvation lysis due to phage infection. Furthermore, we assume that infected bacteria do not reproduce. We assume that the phage bind randomly to a finite number of discrete receptors on the surface of their host bacteria and that the number of phage bound to a bacterium at one time (the MOI, or multiplicity of infection) is a function of both the number of free phage and the number of currently uninfected bacteria. We don t count phage that are inside of bacterial cells and are not yet released. We assume that all phage that can infect bacteria do so: free phage can exist therefore only when all phage-binding sites of all remaining bacteria are occupied. There is simultaneous lysis of infected cells and adsorption of phage to new host cells. There is a set time of 20 minutes for phage replication. All infected bacteria lyse synchronously at 20-minute intervals, even if they became infected just one minute ago. Our model is a null model. Basically, this means that we assume that in the absence of specific knowledge of how multiple phage would bind to a bacterium, separate phage behave independently of one another. That is, no synergism or antagonism between phage is assumed.

4 44 Microbes Count! In each time interval, the binding of phage to bacteria are independent events. The number of phage bound is an integer. That is, we do not count fractions of phage nor do we count partial binding; phage must be fully bound to be infective. The probability that a phage binds is small and is proportional to the number of phage, phage receptor sites, and bacteria. The probability that more than one phage binds to a single bacterium in a given interval is small compared to the chance of just one binding. (This is similar to a model of radioactive decay and is called an exponential decay function.) An easy way to visualize the null model is to imagine taking a bag of marbles and dropping them over a rectangular array of boxes say one box on each square of a chess board. The probability of any box getting more than one marble in any one time interval is proportional to the number of marbles in your bag. The best way to describe this model is by using a Poisson distribution to represent the infection, growth, and lysis. This function gives the proportion of bacterial cells that are infected with exactly x cells: where: (λt) is the probability that one phage binds to a bacterium; t is the time elapsed; and x is the number of phage bound We estimate lambda (λ) by using the zeroth class: One of the nice aspects of using an exponential model such as the Poisson distribution is that any number raised to the zeroth power is equal to one; therefore, we are able to make a good estimate of the value of the coefficients in our model by simply looking at how many boxes on our chess board have no marbles or by observing how many bacteria are uninfected (which we presume also means that they have no phage attached). For the rest of the model, we are simply tallying who is born and who dies. The three populations that we are concerned with total number of bacteria, number of infected bacteria, and number of free phage are plotted as a function of time (Figure 2.)

5 45 Questions/Exercises 1. Consider a bacterium as a rectangular box with dimensions of 1µm x 1µm x 2µm and the dimensions of a rectangular T7 phage as 60nm x 60nm x 80nm (as given by The Encyclopedia of Virology). What would be the required viral population to occupy the entire volume of the bacterium, i.e., what is the maximum burst size determined by geometry? Figure 2. Examples of the three generated values total bacteria, infected bacteria, and free phage illustrating how all three interact as a function of burst size and doubling time. 2. Examine the table below: T ime (minutes) B acteria (per ml) Phage (per ml) x 10.0 x x x lysis (Based on Molecular Biology and Biochemistry: Problems and Applications by David Freifelder, 1978). a. What is the starting concentration of bacteria, B 0? b. What is the starting concentration of the phage, P 0? c. What is the multiplicity of infection (MOI)? MOI = P 0 /B 0 d. What is the bacterial doubling or generation time? The phage generation time is 20 minutes. e. What is the burst size of the phage (the number of phage produced in each host cell)? f. Why in this model do all of the bacteria lyse when they do? g. How many phage do you expect at lysis?

6 46 Microbes Count! 3. You are given: - a bacterial stock culture that is 5 X 10 8 cells/ml, and - a phage stock that is 1 X 10 9 phage/ml a. How would you set up the experiment as in the table from Problem 2, using a 10 ml total volume? b. Sketch a graph plotting the bacterial and phage concentrations as a function of time. What assumptions have you made? 4. Open the Excel spreadsheet PhageGrowth.xls and enter B 0, P 0, the bacterial doubling time, and the burst size of the phage, into the proper cells of the spreadsheet. Push the enter key to observe the graph. Compare and contrast this new graph with the graph you sketched in question Sketch a graph of infected vs. uninfected bacterial populations, and a graph plotting the bacterial population as a function of phage concentrations. What assumptions have you made? This graph is called a phase portrait or plot of your data. A phase portrait plots two population sizes as the X and Y coordinates of each point on the graph; thus, you need to remember that time is a hidden variable and would be represented along the z-axis. It may help you to understand what is going on if you add an arrow along a path or multiple arrows over different portions of a graph to indicate the direction of time. a. What did you observe? b. How do you relate this to the biological mechanisms involved here? c. How are bacteria and phage populations similar to or different from foxes and rabbits as predators and prey in terms of their dynamics? 6. Analyze the phase portraits in Figure 3 below. (See Question 5 for a discussion of phase protraits.) a. Why are the phage populations jumping discretely? b. Why do the phage populations asymptotically level off? c. In the other phage graph, Phage vs Viable Bacteria, why is just the opposite occurring? d. If you only looked at the two bacterial populations with respect to one another, what might you infer about the mechanism of the mysterious infective agent?

7 47 Using the Phage Growth spreadsheet model To begin your explorations, open the Microsoft Excel file called Phage Growth.xls. (This file is included in the Modeling Phage Growth section of the Microbes Count CD.) You should see a window containing a graph and some additional information. If you do not see a graph, click on the Input and Graph tab at the bottom of the Excel window. Figure 3. Three phase portraits demonstrating the relationship between two populations, independent of time. The Input and Graph window This window is where you will set the initial conditions for your experiments and see the results displayed in graphical format. The input parameters are displayed in the top left of the window. To enter a new value, click on the cell containing the current value for the parameter. The contents of that cell will be displayed in

8 48 Microbes Count! the bar below the Excel tool bar at the top of the screen. Type in a new value and press Enter. The graph will change to reflect the new value. The Data Table window To see the data table for your experiment, click on the Data Table tab at the bottom of the window. The table in this window contains the data generated by the experiment that you set up when you set the input parameters in the previous window. In the course of the simulation the values for each time interval are calculated using the values from the previous interval and then plotted in the graph. The first row in each column contains a short description of how the values in the column are calculated. For additional information, move your cursor over the heading of the column. A box describing the calculation will pop up. Software Used in this Activity Microsoft Excel Platform Compatibility: Macintosh and Windows Additional Resources Available on the Microbes Count! CD Text A PDF copy of this activity, formatted for printing The PhageGrowth.xls model file Related Microbes Count! Activities Chapter 1. Modeling More Mold Chapter 3. Modeling Microbial Growth Chapter 4. TB and Antibiotic Resistance Unseen Life on Earth Telecourse Coordinates with Video I: The Microbial Universe Relevant Textbook Keywords Bacteriophage, Growth rate, High titer lysate, Lysis Related Web Sites (accessed 3/21/03) American Society for Microbiology Bacteriophages in the news: Microbes Count! Website

9 49 Phage applications Unseen Life on Earth: A Telecourse Bibliography Abedon, S. T., T. D. Herschler, and D. Stopar. (2001). Bacteriophage latentperiod evolution as a response to resource availability. Applied and Environmental Microbiology 67: Rabinovitch, A., H. Hadas, M. Einav, Z. Melamed, and A. Zaritsky. (1999). Model for bacteriophage T4 development in Escherichia coli. Journal of Bacteriology 181: Schnaitman, C. A. (2002). Phage biology: coming of age. Science 298:2329. Stansfield, W. D. (1991). Schaum s Outlines Genetics, 3 rd ed. McGraw-Hill: New York. p Stone, R. (2002). Stalin s forgotten cure. Science 298:728. References Freifelder, D. (1978). Molecular Biology and Biochemistry Problems and Applications. W. H. Freeman and Company: San Francisco, CA. Webster, R. G. and A. Granoff, editors (1994). Encyclopedia of Virology, Vol. 3. Academic Press, Harcourt Brace & Company, Publishers: London. Figure and Table References Figure 1. Courtesy of Joshua Tusin Figure 2. Figure 3. Screenshot from Excel Screenshot from Excel