Mathematical model

Because we have to know about the relation of microbial growth rates in an aqueous environment with the concentration of a limiting nutrient, we used monod equation for design cheater and cooperators mathematical model. In our case, limiting nutrient will be cellobiose. Also, in our experiment we have to find out the quantificated information of the dependence of the growth rate on substance. Finally, we can compare the growth rates of two E.coli strains.

[Fig 1] Monod Equation

This is basic monod Equation. S is glucose released from cellobiose by cooperators. S can be derived from multiplication of q, p, x. q refers the number of totla cell diversitiy, p is the number of total cell diversitiy, and x is the frequency of the cooperator

[Fig 2] Cooperator Equation

[Fig 3] Cheater Equation

Then, we modified monod equation for Cooperator and Cheater. _𝜇_0_ is specific growth rate without glucose and c is the cost of the cooperation. Cost of the cooperation will decrease the growth rate of the cooperator. And 𝛿 is advantage of cooperation. The cheater will not get the benefit of the cooperation. So we should minus from the cheater’s growth rate. And, we think that disadvantage comes from the glucose capture efficiency. Cheater could not catch efficiently like cooperator. So we give disadvantage to cheater by multiply (1 – 𝜀) .

[Fig 4] relative fitness

And we divided _𝜇_𝑐_ by _𝜇_𝐷_ to express the relative fitness of cooperator.

Glucose Capture Efficiency

[Fig 5] glucose capture efficiency

And we esitmated the efficency of glucose capture as divide glucose flux by glucose creation rate. We assumed that the glucose molecules produced by b glucosidase directly diffuse into media because it is located in the outer membrane of E.coli. And this will cause temporally increased of glucose concentration by a local cloud of glucose at the surface. And this will benefit to influx rate of the glucose into the cell.

[Fig 6] Cooperator Equation & Cheater Equation