Linear Regression and Linear Programming Models

1. E. coli bacteria have an initial concentration in a 2.0 L bioreactor of 27 g cells/L. The growth medium contains two carbon sources, both of which are limiting, and ample supplies of oxygen, ammonium nitrate (NH4NO3) as a nitrogen source, and all other macro and micronutrients. The bacteria prefer to grow on glucose (C6H12O6, supplied at 100 g/L), so it is consumed first, followed by glycerol (C3H8O3, supplied at 75 g/L). Since the E. coli were already growing on glucose, the lag time associated with glucose consumption is only 15 minutes. The lag time associated with glycerol consumption is 40 minutes.

Further information: The chemical formula for E. coli in this bioreactor is CH1.666N0.20O0.57. The specific growth rate on glucose is 0.42 h-1, and on glycerol is 0.27 h-1. The yield of cells on glucose is 0.46 g cells/g glucose, and yield on glycerol is 0.38 g cells/g glycerol. You may ignore the length of time that the cells are in deceleration phase.

A. What is the concentration of cells when they reach the endogenous metabolism phase after consuming both carbon sources?

B. At what time after cell growth begins in the bioreactor will the cell concentration first reach it’s highest concentration? C. Determine the stoichiometric coefficients for cell growth on each limiting carbon source.

2. For problem #1

A. How much total oxygen (O2) is consumed during the growth phase on glucose (g/g)?

B. How much total oxygen (O2) is consumed during growth on glycerol (g/g)?

C. At what minimum rate must oxygen be supplied to the reactor so that it does not limit the growth rate? (assume the oxygen supplied is fully available to the cells and we don’t have to consider mass transfer)

3. The E. coli growth in problem 1 is now transitioned to a 5.0 L continuously-stirred tank reactor, but the feed only contains glucose as the carbon source, and the glucose is still the limiting nutrient. The initial cell concentration is 96 g cells/L The glucose is now supplied in a feed stream at 65 g/L, entering the tank at 1.6 L/h (the dilution rate, D). The specific growth rate in the continuous rector follows the Monod equation, with ∝max = 0.42 h-1 and a saturation constant (Ks) of 12.5 g/L.

A. Use the Monod equation to determine the specific growth rate for the cells. (assume the reactor has an initial glucose concentration of 65 g/L, and that the cells are in the exponential growth phase)

B. Using the initial concentration of cells and the growth rate found in part A, what is the rate of new cell growth, dX/dt, in the bioreactor?

C. At what rate do the cells initially consume glucose (in g/h, considering the reactor volume)? (assume the cells are growing in the exponential growth phase and all glucose is used to produce new cells- no product is formed and maintenance energy is negligible). (Use the stoichiometry and yield factors from problem 1).

D. Determine which of these three events will occur in the reactor: (i) the cell concentration will stay the same (it is at steady state), (ii) the cell concentration will increase (cells are growing faster than they are leaving the reactor), or (iii) the bioreactor reaches washout conditions (washout is where the flowrate through the reactor causes the cells to leave faster than they are growing)

E. What flowrate of 65 g/L glucose feed is needed to run the chemostat bioreactor at steady state with a constant cell concentration of 96 g cells/L?