# Leanne Kuwahara-Week 12

## Purpose

To gain a better understanding of chemostat models and to modify the chemostat MATLAB scripts to simulate a 2-nutrient chemostat model.

## Protocol

### Equations

Biomass rate of change: dx/dt = x[r(y/y+K)(z/z+L)] - qx

Glucose rate of change: dy/dt = -Ex[r(y/y+K)(z/z+L)] - qy + qu

Ammonium rate of change: dz/dt = -Fx[r(y/y+K)(z/z+L)] - qz + qv

• q: dilution rate (1/hr)
• At equilibrium: q = specific growth rate
• V: culture volume (L)
• u/v: feed concentration (g/L)
• r: MAX specific growth rate (1/hr)
• K/L: [substrate] at half MAX specific growth rate (g/L)
• E/F: efficiency of converting nutrients to biomass (unitless)
• x(t): residual biomass (g)
• y(t): residual glucose (g)
• z(t): residual ammonium (g)

AT EQULILIBRIUM production rate = consumption rate

• rate of change = ZERO!
• r(y/y+K)(z/z+L) = q
• For glucose: x = u-y/E
• For ammonium: x = v-z/F

### Tai et al. (2007) Questions

1. The yield quantity Y(glu/x) is a ratio of what quantities? Relate this to the 2-nutrient model developed in class.
• The amount of glucose converted to biomass (Pizzaro et al., 2008; Manesh, 2012)
• g of biomass produced : g of glucose consumed
2. The flux q(glu) is a ratio of what quantites? Relate this to the 2-nutrient model developed in class.
• glucose consumption rate (Sauer et a., 1999; Manesh, 2012)
• specific growth rate : Y(glu/x)
3. Convert the residual glucose and ammonium from mM to g.
• mmol/L * culture volume (1L) / 1000mmol * MM of: glucose (180.156g/mol); ammonium (18.039g/mol)
4. Determine the length of time the chemostat was operated before data was collected.
• Biomass dry weight, metabolites, dissolved oxygen, and gas profiles were constant for at least 3 volume changes before sampling.
• Since culture volume was 1L, 3L was divided by the dilution rate (0.03 h-1)
5. Determine as many model parameters as you can from the given data.

### Two-Nutrient MATLAB Simulation

• In chemostat_dynamics_TwoNutrient (function)
1. Added parameters v, L, and F
2. Added variable z (for ammonium--changed y to be for glucose only)
3. Added differential equation dzdt for ammonium and altered the specfic growth rate parameter to include ammonium (ryz)
• ryz = r*(y/(K+y))*(z/(L+z))
• dzdt = q*(v - z) - fp*ryz*x;
4. Added state variable dSdt (3) to be dzdt
• In chemostat_script_TwoNutrient (script)
1. Added an initial ammonium concentration and added it as an initial state for the differential equation
2. Added variable v, L and F and assigned them to a parameter vector
3. Added ammonium into the legend

PARAMETERS:

1. C-limited parameters:
• q = 0.03hr-1
• u = 25g/L
• v = 5g/L
• r = 0.46hr-1
• K = 0.778g/L
• L = 15.91g/L
• E = 14.3 g(glu)/g(dw)
• F = 2.23 g(NH4)/g(dw)
• x0 = 1g/L
• y0 = 2g/L
• z0 = 1.11g/L
2. N-limited parameters:
• q = 0.03hr-1
• u = 46g/L
• v = 0.65g/L
• r = 0.46hr-1
• K = 0.778g/L
• L = 15.91g/L
• E = 25 g(glu)/g(dw)
• F = 0.526 g(NH4)/g(dw)
• x0 = 1g/L
• y0 = 20g/L
• z0 = 3.5g/L

## Results

### Tai et al. (2007) Paper Questions

1. Y(glu/x) = g of biomass produced / g of glucose consumed
• biomass = x
• glucose consumed (y) = u - residual glucose
2. q(glu) = specific growth rate / Y(glu/x)
• specific growth rate = q
3. Residuals:
• Glucose:
• Glu limited (12C): 0.50g
• Glu limited (30C): 0.054g
• Amm limited (12C): 16.2g
• Amm limited (30C): 15.3g
• Ammonium:
• Glu limited (12C): 1.17g
• Glu limited (30C): 1.11g
• Amm limited (12C): 0.027g
• Amm limited (30C): 0.0036g
4. It took approximately 100hr for the chemostat to reach a steady state
5. Model parameters (from Tai et al. papers 2005 & 2007):
• q = μ = 0.03hr-1
• V = 1L
• u(glu) = 25g/L (C-limited); 46g/L (N-limited)
• v(NH4+) = 5.0g/L (C-limited); 0.65g/L (N-limited)
• r = ???
• r > q
• r was set to be the same as Lievense's data in the Boyd paper. This group used a glucose concentration similar to ours and determined r using a Lineweaver-Burk plot.
• r = 0.46hr-1
• K = (ry/q)-y (g/L)
• L = (rz/q)-z (g/L)
• E = 1/Y(glu/x)
• F = (v-z)/x

### MATLAB Manipulation

Plots give residual concentrations IN chemostat after conversion to biomass

Figure 1. Plot of two-nutrient chemostat simulation of carbon limited cultures at 30C. All data was obtained from the Tai et al. papers (2005, 2007) except for values r, K, and L. The r was determined to be the max specific growth rate given by Lievense in the Boyd paper and was used to determine K and L. This graph depicts the residual concentrations of what is in the chemostat at equilibrium.

Figure 2. Plot of two-nutrient chemostat simulation of nitrogen limited cultures at 30C. All data was obtained from the Tai et al. papers (2005, 2007) except for values r, K, and L. The r was determined to be the max specific growth rate given by Lievense in the Boyd paper and was used to determine K and L. This graph depicts the residual concentrations of what is in the chemostat at equilibrium.

## Conclusion

The purpose of this assignment was to modify the chemostat MATLAB scripts to simulate a 2-nutrient chemostat model using the data provided from the Tai et al. papers (2005, 2007). The model does show the system going to a steady state for both the C and N-limited experiments at 30C, however the steady state values (residual concentrations) were different from those provided in the paper. In our model, for the C-limited culture at 30C, the steady state concentration of glucose was around 2.7g/L, and the steady-state concentrations of ammonium and yeast were around 1.5g/L. This is opposed to the Tai (2007) paper which had the residual glucose to be 0.054g/L, residual ammonium at 1.11g/L and the cell population at 1.74g/L. In the N-limited cultures at 30C, our model shows that glucose had not reached a steady state at 100hr, and the steady-state ammonium and cell population to be around 1g/L. The Tai (2007) paper states the residual glucose to be 15.3g/L, residual ammonium to be 0.0036g/L, and the cell population to be 1.228g/L. In both cultures, the yeast steady-state was the most similar to that provided by Tai et al. (2007), however the glucose and ammonium residual concentrations were not accurate. This may be because the Tai et al (2007) paper did not specify what the maximum specific growth rate (r) was, which was required for our model. In the model run, r was obtained from an outside paper (Boyd, 1985) which provided an r-value under conditions similar to those in the Tai et al (2007) paper.

## Acknowledgements

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