Team:Edinburgh/Cellulases (MATLAB model)

Cellulases (MATLAB model)

The most important part of the biorefinery is the reactor where cellulose is converted to glucose. But accurately predicting how much is converted, using synergy between enzymes is difficult without literature to provide the ordinary differential equations (ODE's) and the kinetic parameters. Therefore this model only looks at the free floating enzyme approach (non-synergy). It is deterministic i.e non random and is set by a series of initial conditions.

Assumptions

The mathematical model is based on the ODE's and kinetic parameters outlined in [http://onlinelibrary.wiley.com/doi/10.1021/bp034316x/full Kadam et al, 2004]. The following are its assumptions and basis:

• Rate equations assume enzyme adsorption follows the Langmuir isotherm model
• Glucose and cellobiose which are the products of cellulose hydrolysis, were assumed to, 'competitively inhibit enzyme hyrolysis' [http://onlinelibrary.wiley.com/doi/10.1021/bp034316x/full Kadam et al, 2004]
• Assume all reactions follow the same temperature dependency Arrhenius relationship. However it should be different for every enzyme component, 'because of their varying degrees of thermostability, with beta-glucocidase being the most thermostable. Hence the assumption is a simplification of reality' [http://onlinelibrary.wiley.com/doi/10.1021/bp034316x/full Kadam et al, 2004]
• Conversion of cellobiose to glucose follows the Michaelis-Menten enzyme kinetic model

Equations

Rate Equations

Cellulose to Cellobiose Reaction with Competitive Glucose, Cellobiose and Xylose Inhibition.

Cellulose to Glucose Reaction with Competitive Glucose, Cellobiose and Xylose Inhibition.

Cellobiose to Glucose Reaction with competitive Glucose, Cellibiose and Xylose Inhibition.

Constants

k_nr - reaction rate constant for reaction n EnB is the bound concentration for exo and endo-beta-1,4-glucanase for reaction n
Rs - substrate reactivity parameter
S - substrate reactivity at a given time (g/kg)
G2 - concentration of cellobiose
G - concentration of Glucose
X - xylose concentration
KnIG2 - inhibition constant for cellobiose at reaction n
K1IG - inhibition constant for Glucose at reaction n
K1IX - xylose inhibition constant for reaction n
Note: Assuming no xylose inhibition therefore X=0

Langmuir Isotherm

The Langmuir isotherm model mathematically describes enzyme adsorption onto solid cellulose substrates. Even tough the Langmuir model is based on uniform binding sites and no interaction between the adsorbing molecules, it is not valid for cellulase adsorption onto cellulose. 'Nevertheless the Langmuir formulation remains useful for mathematically describing the phenomenon of enzyme adsorption' [http://onlinelibrary.wiley.com/doi/10.1021/bp034316x/full Kadam et al, 2004]

Constants

exo and endo-beta-1,4-glucanase, i=1
beta-glucosidase, i=2
Eimax - Maximum mass of exo and endo-beta-1,4-glucanase (i=1) or beta-glucosidase (i=2) that can be absorbed onto a unit of mass substrate
Kiad - Dissociation constant for enzyme i
EiF - Free enzyme concentration for enzyme i
S - Substrate reactivity at a given time (g/kg)

Mass Balances

Cellulose mass balance

Cellobiose mass balance

Glucose mass balance

Arrhenius Equation

Arrhenius equation is an empirical relationship which is used to model the temperature dependent reaction rate constant. Note: T1 is set at 45 °C

Constants

Kir - Reaction rate constant of reaction i
Eai - Activation energy of reaction i
R - Universal gas constant

Construction of Model

The model was constructed using the numerical programme [http://www.mathworks.co.uk/products/matlab/index.html MATLAB]. A script file was generated which holds the variable dictionary, constants, temperature dependency equations, ODE45 and plot command.

A separate function file to the script is created as script files can only operate on the variables that are coded into their m-files. Rs, the substrate reactivity parameter changes at every iteration because it is dependent on the S the substrate concentration at a given time, the substrate being cellulose. Therefore S at the first iteration is So, the initial substrate concentration. At the second iteration S is the previous value calculated by the ODE, and so on. After each step the new value of S, is fed into the function file and is used for calculating the reaction rate constant for cellobiose and glucose etc.

Rs, substrate reactivity equation

ODE45 calls on the function file to calculate the rate equations and then substitute them into the respective mass balance. A numerical integration is performed and the results can be seen below. ODE 45 is used as it is more accurate than other solvers [http://www.mathworks.co.uk/help/techdoc/ref/ode23.html (Mathworks)]. It is based on an explicit Runge-Kutta formula and is a one step solver.

Results

Figure 1: Graph of cellulose degradation over time with maximum β-glucosidase.
Initial conditions: Cellulose - 100 g/kg, Glucose - 0.01 g/kg , Cellobiose - 0.01 g/kg.
Enzymes: Exo/endo-glucanase - 0.01 g/kg, β-glucosidase - 1 g/kg
Temperature 35° C

Initial conditions: Cellulose - 100 g/kg, Glucose - 0.01 g/kg , Cellobiose - 0.01 g/kg.
Enzymes: Exo/endo-glucanase - 0.01 g/kg, β-glucosidase - 1 g/kg

The above graph is set with beta-glucosidase at its maximum concentration of 1g/kg and exo/endo glucanase at its lowest to show what effects different enzyme concentrations has cellulose, glucose and cellobiose levels. The result is consistent with what is expected. β

References

• Kadam KL, Rydholm EC, McMillan JD (2004) [http://onlinelibrary.wiley.com/doi/10.1021/bp034316x/full Development and Validation of a Kinetic Model for Enzymatic Saccharification of Lignocellulosic Biomass]. Biotechnology Progress 20(3): 698–705 (doi: 10.1021/bp034316x).