Team:British Columbia/Model
Modelling
In our system, the mixed culture dynamics is competition between E. coli and C. crescentus for substrate, and commensalism, as E. coli requires the conversion of cellulose to glucose to take place. The culture dynamics have been modeled using a system of four ordinary differential equations: one for each species, one for glucose consumption and formation, and one for cellulose degradation. To model our environment, the individual species growth, enzyme expression, and enzyme activity were experimentally determined. Using these values, ordinary differential equations for a mixed culture were developed assuming that both E. coli and C. crescentus behaved competitively and that cellulose degradation by expressed cellulase enzymes on the C. crescentus surface did not favor C. crescentus due to diffusion gradients.
To model substrate limited growth, Monod kinetics provide a relation between substrate concentration and species growth. The Monod relationship, seen in Equation 1, depends on two constants. The maximum specific growth rate of the culture, μmax, is the growth rate when substrate is in excess. The saturation constant, Ks, is the concentration at which the specific growth rate is half the maximum specific growth rate.
The maximum specific growth rate and saturation constant can be obtained for our system by evaluating the growth rates on a range of substrate concentrations for each strain desired in the model. With a series of biomass concentration values over a range of time, plotting the natural logarithm of each biomass concentration produces a linear plot with a slope equal to the strain growth rate on substrate. The length of the lag phase can be obtained as the initial length of time observed before the linear phase occurs. The growth rates for each initial substrate concentration can be plotted against the initial substrate concentrations to produce a Monod growth kinetics plot and obtain the μmax and Ks constants. Differential equations using these constants can be made for the change in species concentration as a function of time.
Substrate inhibition caused by the presence of glucose can cause the growth equations to be inconsistent, however at low concentrations, the growth is unaffected by substrate and the points can be fit to a model. Using the Monod growth equation over this range, we can obtain the growth of each strain for any substrate concentration within this range.
To model the rate of cellulose degradation into glucose through the expressed enzyme activity, a modified Michaelis-Menten relationship is used. Since cellulase enzymes are expressed on the surface of C. Crescentus, a relationship between glucose formation, cell concentration and enzyme expression must be made. To accomplish this, cellulase activity was tracked for two different cell concentrations, for a number of different substrate concentrations. The inverse of substrate degradation rate can be plotted against the inverse initial substrate concentrations to produce a Lineweaver-Burk plot. The slope of this plot is the Michaelis-Menten constant, KM, divided by the maximum rate of consumption, Vmax. Typically, Vmax is a constant related to enzyme concentration and kinetics, however in the consortium system, enzyme concentration changes as a function of time based on the concentration of C. crescentus in the system. Instead, Vmax is a combination of enzyme activity, the number of enzymes expressed per C. crescentus, and the concentration of C. crescentus, seen in Equation ###. The activity and expression number can be assumed constant, represented by β. To determine the constants in our Michaeles-Menten kinetics, two Lineweaver-Burk plots are created with two different cell concentrations. The difference between the y axis intercept values can be used to determine the constant β, and the slope can be used to determine KM. Using this information, the ordinary differential equation describing cellulose degradation can be developed.
When the production of glucose through cellulose degradation and the consumption due to the growth of E. coli and C. crescentus are combined, the ordinary differential equation describing glucose utilization can be developed.
