# Alondra Vega: Week 5

## Instructions

• Make sure you understand which variables are the state variables and which are the parameters.
• Simulate the system with different values for the constants and the initial concentrations of nutrients and cells. The initial nutrient level can equal 0, but the constants and the initial cell population size need to be positive. Can you make any observations about how the system behaves? The matlab model of the enzyme kinetics may be helpful: this sytem has two state variables, so you will need x(1) and x(2), along with dxdt(1) and dxdt(2) as in the Michaelis-Menten substrate/product model.
• Adapt the system to a logistic growth model. we will start with a nutrient dependent carrying capacity, M = a*n. The carrying capacity increases linearly with the nutrient level.
• Simulate this system with different values for the constants and the initial concentrations of nutrients and cells. The initial nutrient level can equal 0, but the constants and the initial cell population size need to be positive. Can you make any observations about how the system behaves?
• Suggest some additional adjustments. For example, look at nutrient dependent growth rate in the Malthus model. Or, think abouty the waste products the yeast might produce. Are any of them toxic to the yeast? Where might that lead?

## Part 1

• The state variables are:
• n(t) = u - (u - n0) e -Dt = concentration of nutrients
• y = concentration of cells in the mixture = y0ert
• The parameters are:
• D = 1/time= dilution rate
• u = mass or molar = feed concentration
• V = volume of mixture
• If we make the initial conditions for both state variables, the curve that we get is similar to a logarithmit curve. It has that shape. It goes up to what we made our u value in this case. If we change them to be 1 and 2, the graph changes to a line where it curves at about 0.1. When we change the initial conditions to be 10 and 15 we get a horizontal line at 0 and then it breaks at around 9.3. Just to clarify my parameters were set to Vmax=2, k=0.1, r=10, D=0.65 and u=2. When working with the program, I realized that changing these parameters did not change the shape of the graph much. What did change were where the graphs were positioned. For the most part, it was a linear relationship, except when both initial conditions of the state variables are zero.

## Part 2

• Both of the curves created followed the same shape. The start at the highest u value and then fall down, looking similar to an inverse exponential function. It seems that it levels off a little before reaching zero. My parameters were left the same to be able to compare both functions.
• Compared to the Malthus model, this graph does not have the sigmoidal effect. This may be due to the fact that too many nutrients in the yeast cells cause the cell to die. After it uses up the nutrients that it needs, the excess nutrients can become toxic to the cell leading to one of the top 5 cell functions:cell death.