This equation 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.
This 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.
This shows how the DNA strands come together. Three double stranded strings of DNA are denatured and then when slowly cooled will come together to form the Y shape. However after the denaturing each strand of DNA has an equal chance of bonding to the original piece of DNA as it does to the correct origami side. Therefore the more complex the structure the less likely it is that that structure will fully form.
The image to the left shows the population of cells as a function of time under the first model and the second shows the Gompertz 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.
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.