In the production of beer, ingredients consisting of hops and barley are fermented by yeast to produce a very refreshing beverage

In the production of beer, ingredients consisting of hops and barley are fermented by yeast to produce a very refreshing beverage. The main sugar components of the ingredients are glucose (G), maltose (M), and maltotriose (N). The following model equations are adapted from the paper by Gee and Ramirez (J. Inst. Brew., 100:321-329, 1994), describing the growth of yeast on the three sugars:

dXdt=μXX

dGdt=-μ1X

dMdt=-μ2X

dNdt=-μ3X

where

μ1=μGGKG+G

μ2=μMMKM+MKG'KG'+G

μ3=μNNKN+NKG'KG'+GKM'KM'+M

μX=YXGμ1+YXMμ2+YXNμ3

In the above, YXG

, YXM

 and YXN

 are the unknown yield coefficients of biomass (yeast in this case) from the different sugars: glucose, maltose and maltotriose, respectively. The remaining parameters in the model are given in the table below (in the appropriate unit).

Parameters	Value
μG	0.02554
μM	0.02266
μN	0.03913
KG	0.7464
KM	40.97
KN	250
KG'	5.356
KM'	13.17

1. (10 point) Looking at the model equations, what can you say about the dependence of yeast growth on the three sugars: glucose, maltose, and maltotriose? Think about what the different terms in the parentheses describe in terms about yeast growth rate. 

2. (20 points) Given the attached time-series data, perform maximum likelihood (ML) estimation to obtain the unknown yield constants:  YXG

   

   , YXM

   

    and YXN

   

   . These yields typically span between 0.1 and 0.5. The measurement data have a baseline variation of 0.01 and a time-dependent variation with 20% coefficient of variation. That is, the data variance is equal to 0.01+0.2*data point2

   

   .  Give the ML parameter estimates and plot the model prediction vs. experimental data.

3. (20 points) Evaluate the confidence intervals for the parameter estimates

4. (20 points) What can you say about the a priori and practical identifiability of the parameters?

5. (30 point) Perform optimal design of experiments using D-optimal and modified-E optimal criteria to find the optimal initial concentrations of ingredients and yeast (biomass) for the experiment. Note that the sugars in the ingredients have a fixed molar ratio G:M:N of 1:3:0. Also note that glucose concentration should NOT exceed 200 mol/m³ and the initial biomass should NOT exceed 100 mol/m³. Compare and contrast the two criteria.