Virginia Tech/bacteria model

The first step to modeling the spread of an epidemic is to model the population itself.

We need to understand how a population of E. coli grows before we can understand how an infection spreads through it. It is very easy to grow E. coli. It only needs some media, a warm environment, and some aeration. The problem is generating the growth curve. It is possible to measure the OD600 over a period of time, but we wanted to generate data with number of bacteria.

Collecting Experimental Data

Generating the Calibration Curve

The team decided to generate OD600 vs Time, OD600 vs CFU/mL, and CFU/mL vs Time curves. It was also decided to grow two different strains of bacteria C600 and LE392 in both LB and TNT media.

This was accomplished using the following procedure:

ATCC C600 and LE392 were regenerated from a previous overnight culture. These cultures were used to inoculate four Corning Tubes labeled: "LE392 in TNT," "LE392 in LB," "C600 in TNT," and "C600 in LB." (These were the four different growth curves generated). Each culture was allowed to grow for at 37oC and 220 rpm. Aliquots were taken every 30 minutes starting at T=0. An OD600 was taken from each of these aliquots. Serial dilutions were made of each aliquot and they were then plated to grow overnight. This was done for a total of four hours.

The plates provided us with colony forming units per mL (CFU/mL), which could then be graphed against OD600. This provided the calibration curve necessary to determine number of cells per well.

Bacterial Growth measured by Plate Reader

Measuring Bacterial Growth

After producing a calibration curve to convert OD600 measurements to total number of cells, we decided to use the new plate reader to measure growth of LE392 in a single well. The new plate reader allows for temperature control, shaking, and can take numerous readings very quickly. Thus, we could follow the growth of LE392 in 96 wells without a considerable amount of work. The only problem was the small volume in each well made it easy for evaporation to occur, but this was compensated for by adding sterilized MilliQ water part of the way through the experiment.

Simulating Bacterial Growth

Population growth curves tend to have a sigmoid growth trajectory. In order to create a sigmoid growth curve, we start with Michaelis Menten's model that describes enzyme kinetics with a limiting substrate. The Michaelis Menten reaction equation is

Michaelis Menten’s equation

Reading the equation from left to right, the enzyme E uses one substrate S and produces the intermediate ES. ES then creates a product P and releases the enzyme E. There are initial concentrations for E and S. The E continuously gets replenished by ES; however, the limiting substrate S gets used up. Once the concentration of S runs out, the product P stops being produced. Michaelis Menten’s equation can be modified to represent the doubling of bacteria with a limiting substrate to create a sigmoid growth curve.

Modified Michaelis Menten’s equation

Reading the equation from left to right, the bacterium consumes a limiting substrate such as food and produces the intermediate X. X produces two bacterial cells. The growth trajectory begins exponentially but flattens out once the food substrate is depleted. When this model is simulated, the sum of the intermediate X and the bacteria give a sigmoid curve. In order to use this growth curve to match experimental data, three rate constants, k1, k-1, and k2 must be picked. It is difficult to do this since the rate constants do not directly control the characteristics of the curve such as slope and peak. All three values are interdependent and the correct ratios must be picked between these constants to produce a usable sigmoid curve. In order to control the characteristics of the curve, it must be converted to differential equations using Monad’s model for bacterial growth. Monad identified the similarities between bacterial growth and Michaelis Menten’s enzyme kinetics. Monad’s equation states

Monad's equation

In terms of B(bacteria), X(intermediate), and S(substrate), the Monad equation can be written as

Monad's equation

The rate k2 controls the exponential growth and the rate km controls the peak of the curve.

km

Once the km rate and k2 rate are chosen, k1 and k-1 can be evaluated using the above relation.

km

The trajectories become roughly linear after about 220 minutes into the growth. This linear growth might be a result of evaporation in the liquid media. Since the bacteria requires a temperature of 37 degrees Celsius to grow, evaporation is the most likely cause.

Derivations

Monad’s equation can be directly derived from the reaction equation

Modified Michaelis Menten’s equation

The derivation for Monad’s equation:

At steady state:

This equation models the growth of bacteria; however, a second differential equation is needed to show the decline of the food substrate as the bacteria consumes it. The substrate S stays constant so there is unlimited food. dS/dt must be derived to create a sigmoid curve.

The derivation for dS/dt (the change of the food substrate):

The dS/dt equation in addition to the dB/dt equation shows the interaction between the substrate and the bacteria with differential equations. By changing the values of k2, which controls the exponential growth, and km, which controls the peak of the curve, the experimental data can be simulated.