# Sarah Carratt: Week 5

## Instructions

1. First, make sure you understand which variables are the state variables (dependent variables that determine the concentrations) and which variables are parameters (e.g., rate constants).
2. Simulate this system with different values for the constants and the initial concentrations of nutrients and cells. The initial nutrient level can be =0, but the constants and the initial cell population size need to be positive.
3. Can you make any observations about how the system behaves?  The matlab models of the enzyme kinetics may be helpful:  this system has two state variables, so you’ll need x(1) and x(2), dxdt(1) and dxdt(2) as in the Michaelis‐Menten substrate/product model.
4. Adapt the system to a logistic growth model. Simulate this system with different values for the constants and the initial concentrations of nutrients and cells. The initial nutrient level can be =0, but the constants and the initial cell population size need to be positive.  Can you make any observations about how the system behaves?
5. Suggest some additional adjustments.  For example, look at the nutrient dependent growth rate in the Malthus model.  Or, think about the waste products the yeast might produce. Are any of them toxic to the yeast?  Where might that lead?

## Student Response

### Part One

STATE VARIABLES:

• n(t) = concentration of nutrient = u - (u - n0) e -Dt
• y = concentration of cells in the mixture = y0ert

PARAMETERS:

• D = 1/time= dilution rate
• u = mass or molar = feed concentration
• V = volume of mixture
• r =growth of cells
• M = an = carrying capacity of cells

OBSERVATIONS:

• Looking at the equation, it appears that this is a standard logarithmic function, but it will only the three of the four phases discussed in class: lag, log, and stationary phases. The decline/death phase isn't achieved. Ultimately, the function should be fairly linear until the initial nutrient is consumed and a carrying capacity can be reached.

### Part Two

OBSERVATIONS:

• With the second problem, the carrying capacity isn't reached. Only the first two phases (lag and log) could really be seen. In fact, the equation should be fairly linear after the initial increase.

• After observing yeast growth, modification could be made to account for the rate of cell death in the culture.