# Leanne Kuwahara-Week 11

## Purpose

To simulate a chemostat using MATLAB and compare the results to a steady state outcome.

## Background

• Chemostat: a reactor used for growing cell cultures
• Nutrients (media) added at a constant inflow rate
• Nutrients held at fixed concentrations
• Effluent removed at a rate equivalent to the inflow rate
• Effluent concentration DEPENDS on time
• Volume in chemostat held constant
• Assumed that contents of chemostat are well mixed (uniform concentration throught reactor)
• Parameters:
• Q: volumetric inflow rate (vol/time)
• V: volume in chemostat (vol units)
• q = Q/V: dilution rate (1/time)
• u: feed concentration (concentration units)
• y(t): concentration of nutrients
• Rate of change of nutrients = inflow rate - outflow rate - rate consumed in tank
• Modeling Conservation of Nutrient Mass (dy/dt):
• qu: inflow rate (assumed to be CONSTANT over time)
• qy(t): outflow rate
• Er(y/K + y)x(t): consumption rate/rate of metabolism
• Michaelis-Menten model
• x: concentration of yeast cells
• E: unit conversion rate between biomass and nutrient concentration
• r: Maximum rate of system (Max specific growth rate)
• K: Nutrient concentration at which the reaction rate is half of Vmax (r)
• Modeling Cell Population (dx/dt):
• Net growth rate is dependent on nutrient level [y(t)]
• Consumption model depends on size of population
• In EQUILIBRIUM, models are set equal to ZERO
• x CANNOT be 0 (as this would mean the cell population was extinct)
• Nutrient concentration at equilibrium: y = qK/(r-q)
• Cell concentration at equilibrium: x = (u-y)/E
• The steady-state nutrient concentration is INDEPENDENT of the feed rate (u), while the cell population DEPENDS LINEARLY on u

## Protocol

1. Solving for steady-state cell biomass (x) and nutrient mass (y):
• Parameters used:
• q = 0.10 (1/hr)
• u = 5 (g/L)
• E = 1.5
• r = 0.8 (1/hr)
• K = 8 (g)
• Parameters were plugged into y = qK/(r-q) and x = (u-y)/E to solve for cell biomass and nutrient mass (g) at equilibrium
2. The mass calculated in number 1 were divided by 2 (L) to obtain the steady-state concentration (g/L) of cells and nutrients
3. MATLAB files chemostat_script.m and chemostat_dynamics.m were used to simulate the system dynamics
• Same parameters were used in simulation
• Results from simulation were compared to the calculated values
• yyaxis left/right did not work
• Do no think this function is compatible with the 2014 version of MATLAB as it was released in 2016
• plotyy function did not work
• Over-layed a second y and x-axis
• plot2axes function did not work
• gave an error

## Results

1. Calculated cell and nutrient mass:
• Cell biomass = 2.6g
• Nutrient mass = 1.1g
2. Concentration in a 2L chemostat:
• [cells] = 1.5g/L
• [nutrients] = 0.5g/L
3. MATLAB simulation

Figure 1. Plot generated in MATLAB simulation of a chemostat experiment.

• Plot demonstrates the chemostat goes to equilibrium around 50hr
• Plot shows equilibrium masses consistent with the calculated values (cells ~ 3g, nutrients ~ 1g)

Questions(Points of Confusion:

• I do not understand what in the code is assigned to the y-axis and what is assigned to the x-axis
• what are the units?? hours and g/L???

## Conclusion

This aim of this experiment was to simulate a chemostat using MATLAB. The program produced a plot that successfully went to equilibrium, as expected with a chemostat experiment. Initially the cell population dramatically increased as the nutrient supply decreased. Once the nutrient concentration was near zero, the cell population peaked, and then decreased to a stead state around 3g. As the cell population decreased, nutrient concentration began to increase and also reached a steady state around 1g. These results demonstrate that MATLAB can be used to simulate chemostat experiments.

## Acknowledgements

Except for what is noted above, this individual journal entry was completed by me and not copied from another source.

## References

• Dahlquist, K. & Fitpatrick, B. (2019). "BIOL388/S19: Week 11" Biomathematical Modeling, Loyola Marymount University. Accessed from:Week 11 Assignment Page