Three kinds of enzymes are needed to turn cellulose into glucose. Enzymes Exoglucanase and Endoglucanase are placed in bottom layer of bioflim, and β-glucosidase are in layer 2. layer 3 could be used to further degrade glucose to further degraded molecule such as alcohols. Our application is a combination of stratified bioflim and bio-conversion of celluloses which is a part of the project of Team Edinburgh of igem 2011. Therefore their model (with matlab) could be used to describe our system with some adjustment.

Based on the results of former sections, the expression of proteins in the bottom layer and middle layer are determined by only factor of time, and they would become a constant after the system reached a steady-state. Thus we could set the concentration of these three enzymes to be two constants E_1 and E_2, where Exoglucanase and Endoglucanase are of same concentration C_1 in bottom layer, and concentration of β-glucosidase is C_2 in layer 2. Team Edinburgh has already built a matlab based model in their igem 2011 project for free floating enzyme approach in simulating conversion of celluloses. Their method could be applied in our model. Assumptions of their models are listed below.

1 Underlying assumption: cellulose, cellobiose, and glucose concentrations change continuously with time.
2 Rate equations assume enzyme adsorption follows the Langmuir isotherm model.
3 Glucose and cellobiose, which are the products of cellulose hydrolysis, are assumed to "competitively inhibit enzyme hydrolysis".
4 All reactions are assumed to follow the same temperature dependency Arrhenius relationship (shown below). However, in reality it should be different for every enzyme component, "because of their varying degrees of thermostability, with β-glucocidase being the most thermostable. Hence the assumption is a simplification of reality".
5 Conversion of cellobiose to glucose follows the Michaelis-Menten enzyme kinetic model.
6 Concentration of xylose is assumed to be zero all the time for simplify.
7 Diffusion is assumed to be linear. That is, the concentration of cellobiose in layer 2 is assumed to be a constant times c (assumed to be 0.6) its concentration in layer 1.
8 The temperature of the system is a constant. (30℃)
9 Concentration of celluloses in middle layer is assumed to be zero (no diffusion).

Cellulose to cellobiose reaction with competitive glucose and cellobiose inhibition in bottom layer.

Cellulose to glucose reaction with competitive glucose and cellobiose inhibition in bottom layer.

Cellobiose to glucose reaction with competitive glucose and cellibiose inhibition in bottom layer.

Cellobiose to glucose reaction with competitive glucose and cellibiose inhibition in middle layer.

The Langmuir Isotherm model mathematically describes enzyme adsorption onto solid cellulose substrates.

Cellulose mass balance in bottom layer

Cellobiose mass balance in bottom layer

Glucose mass balance in bottom layer

Glucose production in middle layer

Overall glucose production is equal to g+G.

Overall glucose production is equal to g+G.

In bottom layer, we set E_1 to be 1g/kg, E_2 to be 0.01g/kg, and concentration of cellulose to be 100g/kg. Without considering the reaction in middle layer, the concentrations in bottom layer are shown in following graph.

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