The equation to the left is the simplest model possible to model cell growth and just displays an exponential increase in cells with respect to time. This is a bad model because E.coli has an upper limit to the number of cells that can survive in a given location before they begin to fight for the same resources and some will die.
The equation on the right is the Gompertz function which is similar to the first model expect the growth of the cells decreases as the number of cells reaches a limit. This is a much better model as it takes into account the upper limit of cell concentration possible in a given volume.
The graph to the left of the image shows the population of cells as a function of time under the first model and the second shows the Gompertz model. As you can see there is a drop off on the second model.
Our idea of forcing different cell types to live together in close vicinity will alter this model however. Taking for example forcing 3 cell types together;
A higher concentration of cell type A could lead to a reduction in cell type B, or an increase in B could increase C.
It is important for cells to have a starting point where they are all stuck together instead of just 3 different cell types being put into a solution and let grow until they come together, as the point of the test is to see how cells grow when they are together with different cell types, and to see if it is possible to control the overall concentrations of the cell types by just changing the arrangement of the original cell cluster.
Because we plan to be able to use any cell types for any of the colonies it is important to come up with a model which can explain the cell growth of all 3 cell types over a period of time.
Where:
GA = Total cell growth of A with respect to time
nA = Cell growth of A with no external forces acting
SB = Concentration of cell type B
CBA = Constant of affect (how B affects A)
SC = Concentration of Cell type C
CCA = Constant of affect (how A affects B)
The constants of affects will be determined experimentally and the concentrations can then be found by rearranging the equations. In order to obtain the constants you would need to stick cells together with known concentrations, for example to find CCA you would stick C to A and monitor how quickly A grows. CCA would then be equal to this value divided by the original growth rate of A.
This equation combines the above two and is the final form of the cell growth of A. The growth of B and C can be found by substituting out A for B or C appropriately.
This graph shows how A grows depending on the affect constants which are displayed at the bottom of the graph. As you can see a negative affect constant decreases cell growth exponentially and a positive affect constant has the opposite effect.
From this graph we can infer that changing the affect constants (by using different cell types) will change the cell growth rate of A. However there always appears to be a limit to the maximum or minimum growth rate as shown by the clear asymptotes of the graph.