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INFORMATION MOVEMENT IN A BACTERIAL NETWORK
The goal of this project is to identify a probabilistic model for propagation of information (mobilizable plasmids transmission) in a dynamic population (batch culture). The motivation for this goal is identifying movement of the transmissible plasmid as to determine the distribution of E. coli having received the information. Accordingly, the transient response of information must be interpreted as it spreads throughout the system via conjugation. This model serves as a generalized case of bacterial growth and conjugation that may be utilized for a specific system in which conjugation acts a means of forming a bacterial communication network.
One can consider the model design smilar in principle to an epidemiological system. For instance, the plasmid (representative of the disease in the epidemiological model) possesses multiple transmission modes. It is capable of vertical transmittance from parent to offspring and also horizontal transmittance from one cell type (donor) to another cell type (recipient). Upon transmission of plasmid from donor to recipient, the "infected" cell changes phenotype to now becoming capable of transmitting the mobilizable plasmid as well (or becoming infectious).
- The population is well mixed; trajectories of cells are therefore ignored
- The carrying capacity of the system is determined by the size of the vessel
- The cells of one population type are identical in cell size to all others
- Individual cell growth is ignored; rather we assume instantaneous fission after which daughter cells are of equal size as previous existing parent cell
- Assume cells have different replication rates depending upon population type
- Assume the replication rates vary with time dependent on age of culture
- All donor cells are capable of conjugating with any recipient cell
- All transfers result in recipient cell becoming a donor cell (capable of transmission of plasmid)
- There is no latent period between moment of transmission and moment of becoming capable of transmission.
- No cell death (purely birth process); not interested in system in death phase
- Cell replication rates increase with number of plasmids contained in the cell
- Only two cells can conjugate
- Transfer of only one plasmid at a time during conjugation event
- Transfer is unidirectional from donor to recipient
- Rolling replication mechanism for conjugation ensures one copy will remain in the donor after transfer.
Growth phases of the system are modeled based on the concentration of the system and time of initial inoculation of culture. The replication is considered to occur instantaneously at the generation times which are determined by the amount of plasmid in each cell. The model incorporates three states of either open, replicating, or conjugating. The open state represents the time during which either conjugation or replication can occur. During the lag phase, the transition to a replication state has a probability of approximately zero as the cells are adapting to their environment. After 2.5 hours, the cells have entered the log phase to experience a doubling at there specified generation time +/- 1% as ln(xt)=ln(xo) + u*t, where u=max replication rate and x is the number of cells. The rate of replication decreases as the culture approaches the carrying capacity designated. As the carry capacity is reached and stationary phase is initiated, the rate of replication, u, again approaches zero.
The conjugation events were modeled using the Gillespie algorithm. The Gillespie algorithm determined the waiting time for the next conjugation event and the type of conjugation event which occurred. The two types of conjugation events, u, in the system included: a1) recipient cells and donor cells with mobilizable plasmids incorporated into their chromosome; a2) recipient cells and donor cells with separate mobilizable plasmids. At this point, the conjugation event was either determined to be of transfer state or no-transfer state. The system allows the Gillespie algorithm to be simplified to au=cu*xD*xR given:
au represents the conjugation rate
xD represents number of donor cells
xR represents the number of recipient cells
Cu represent stochastic transfer rate constant
Cu=transfer rate * concentration
The value of au is then used to determine the waiting time for the next reaction as t = (1/ao)ln(1/r) where ao is the total conjugation rate of the system (ao=a1+a2) and r is a uniformly distributed variable from (0,1). The algorithm given a larger conjugation rate or concentration of cells will results in a greater chance the reaction will happen in the next step of the simulation.
Set values for system parameters
Set initial total number of cells in culture at inoculation
Set seed percentage of donor cells
Set simulation time to 0
Calculate values for a1 & a2
Generate uniformly distributed random variables
Calculate wait time for next event, dt
Calculate index of the next event
Calculate the transfer state
Adjust the number of cells in the system (due to replication or conjugation event)
Calculate the fraction of cells with information
Set the time of the simulation to t + dt
End when time reaches 48 hours or concentration of recipient cells equals zero
Interpretation of Results
Simulation has provided us with a look at the nature of the response times for plasmid transmission through a bacterial network. Results indicate faster response times, noted by the lower time constants, for larger seed percentages of donor cells. This is intuitive as the transfer rate is concentration dependent. These values for the time constant were shown to decrease almost linearly for an increase in seed percentage. One can also see a faster response time for an increase in the total initial concentration of cells in the system. This represents a faster spreading of plasmid throughout the bacterial network by inoculating an environment with a higher concentration of cells. We would like to see a faster spread of plasmid throughout the system though initial total concentrations are preferred to be lower as this is the purpose of bacterial growth culture (ie. creating many cells from few). If one did not care about how much bacteria was being utilized to transfer plasmid, the largest initial concentration would be used. This is seen from results indicating a gradual increase in the abundance of F plasmid bearing cells after a 48 hour culture period for larger total initial concentrations. It should be noted that the initial concentration would necessarily be less than 10^9 cells/mL as this is the assigned maximum concentration to allow sustained stationary phase growth, and there would be no purpose of such a culture in practice. To recapitulate, a higher initial concentration will provide a higher fraction of cells containing the mobilizable plasmid at stationary phase. Lower initial concentrations may be used, though in order to attain the maximum information propagation by the end of stationary phase a higher seed percentage must be used. The result of the lower initial concentration will be an increase in the final fraction of cells without carrying information (mobilizable plasmids) representing the error in a perfect broadcast in a bacterial communication network. The simulation results have been successful in demonstrating the information movement and have identified the competition for mobilizable plasmids apparent by the sigmoid transient response
Validation of the model is identifiable through flow cytometry experimentation in which the plasmid information will represent a gene for a fluorescent protein. The fluorescent protein will act as a marker for identification of the cell type as donor. Those cells not exhibiting this fluorescent marker are counted as recipient. Counting of these cells in cultures flowed into analysis system at various time points for various initial concentration and seed percentages provide a histogram of the distribution of these cell types in the culture. In this sense, the information content can be assayed providing an accurate method for comparing model generated transient responses of mobilizable plasmid transmission in the bacterial network.
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