Aim of the Model
Modelling is one of the most important parts of an experiment. It helps us interpolate and extrapolate data with a reasonable level of confidence. To attain these privileges while working with our system and to optimize the software, we came up with the following model.
Model Used
Equations Used
We use the following set of differential equations to model the growth of the bacteria:
Meaning of the Symbols
Variables
\(S\) is the concentration of the substrate
\(X_1\) and \(X_2\) are the concentrations of B. subtilis and E. coli respectively in the system
\(P_1\) and \(P_2\) are the concentrations of the products Product 1 and Product 2 synthesised by B. subtilis and E. coli respectively
\(P_3\) is the concentration of a separate product, Product 3 created by B. subtilis whose production is induced by the presence of Product 1 and Product 2
\(\mu_1\) and \(\mu_2\) are variables which determine the growth rates of B. subtilis and E. coli respectively.
\(\Pi_1\), \(\Pi_2\), and \(\Pi_3\) are variables which determine the production rates of the Products 1, 2 , and 3 respectively
Constants
\(\alpha_1\) and \(\alpha_2\) are the respective rates at which B. subtilis and E. coli consume the substrate resources for sustenance
\(\beta_1\) and \(\beta_2\) are the respective rates at which the bacteria B. subtilis and E. coli consume the substrate resources for reproduction
\(k_{d_1}\) and \(k_{d_2}\) are constants that determine death rates of B. subtilis and E. coli respectively due to competition (internal and external)
\(\gamma_1\) and \(\gamma_2\) are constants that determine the amount of resource returned to the substrate upon the death of B. subtilis and E. coli respectively
\(K_1\), \(K_2\), and \(K_3\) are constants that determine the decay of the Products 1, 2, and 3 respectively
\(\mu_1^*\), \(\mu_2^*\), \(\Pi_1^*\), \(\Pi_2^*\), and \(\Pi_3^*\) are constants (and are equal to values that \(\mu_1\), \(mu_2\), \(\Pi_1\), \(\Pi_2\), and \(\Pi_3\) approach in the limiting case of no light and infinite substrate resources)
\(b_1\) and \(b_2\) are constants that determine the kinetics of the growth of bacteria (they determine \(\mu_1\) and \(\mu_2\))
\(B\), \(R\), and \(G\) are intensities of blue, red, and green lights respectively
\(\sigma_C\), \(\Lambda_C\), \(\Gamma_C\), \(\delta_C\) (\(C = B,R,G\)) are experimentally determined constants in our model
Getting the Equations
Equation {1} determines the change in concentration of (the resources in) the substrate.
Equations {2} and {3} are based on assumptions of competition dynamics. Growth of each bacteria species is hindered by competition due to its own population and population of the other species.
Equations {7} and {8} state that bacterial growth is governed by assumption of Michaelis–Menten kinetics. They also determine how \(B\), \(R\), and \(G\) affect the growth rates of the bacteria according to our model.
Equations {9}, {10}, and {11} reflect our assumptions of how \(B\), \(R\), and \(G\) affect the production of Products 1, 2, and 3.
Interesting Observation
Our model predicts some very interesting facts regarding co-cultures. We ran simulations using the above given state equations, and found that even without optogentic controlling, at some given initial concentrations, we get an approximate co-culture.
Results and Scope
We use our set of equations to model the growth of a co-culture of two bacteria. We make a few assumptions in our model that allows us to simplify an emergent process such as culture growth into a set of differential equations. The substrate in which everything is growing is a finite, depletable resource whose quantity reduces over time; this in turn which affects the bacterial reproduction rate. We verify these predictions using our graphs, where we see that these values tend towards zero after a long time. Bacteria, in the process of growing in a culture, produce some compounds, whose presence may affect rates of production of other compounds. Our model incorporates this possibility at a general level, and the attributes of the corresponding equations can be deduced by experiment. The products themselves interact to induce productions, and have production rates which can be affected by light intensities.
Bacterial growth is dependent on internal competition for resources, competition with other bacteria for resources, and availability of resources. If competition increases, or if the concentration of resources, i.e., the substrate, decreases, the bacterial growth slows down, and vice versa.
Our modified bacteria have growth rates additionally dependent on various light intensities. This gives us control over the growths of the populations in the co-culture, since we can assist or hamper the growth of any bacteria by varying the relevant light intensity. Thus, we control the growth profile of the co-culture, and can use this to optimize industrial processes involving co-cultures.
