Mathematical Modeling

Having a kinematic model that mimics the behavior of the ceiling tile manufacturing plant's process water is a valuable design and analysis tool to investigate potential solution strategies such as the proposed synthetic biology solution of engineered yeast. As a first step towards this goal of a simulated design and testing environment, basic mathematical models are developed here to characterize both the kinetics of yeast cell growth and the enzymatic reaction of starch degradation by alpha-amylase. Such a kinetic model for microbial growth and product formation must account for the different phases of growth, most importantly an exponential growth phase during cell seeding,a stationary phase relating to steady-state operation, and a death or cell decline phase associated with conditions of nutrient deficiency or toxic products.

Model Formulation

Several models can be found in the literature [1-4] to describe cell growth, substrate utilization, and product formation. The most widely used models are simpler unstructured models that incorporate the Monod equation to represent the logistic growth characteristics typical of microbials. This model is mathematically derived for cell growth with the following differential equations.

Cell Growth

The rate of cell growth is dependent on cell concentration \(X\). Expressed as

\[\dfrac{dX}{dt} = \mu_g X\]

the specific growth rate \(\mu_g\), having units of hr ^-1, is defined as

\[\mu_g = \dfrac{1}{X} \dfrac{dX}{dt}\]

Accounting for the loss of cells from cell death and endogenous metabolism, the cell death rate \(R_d\) and the net cell growth can be written as

\[R_d = K_d X\]

\[\dfrac{dX}{dt} = \mu_g X - K_d X\]

Rearranging terms, the net specific growth \(\mu\) is

\[\mu = \mu_g - K_d = \dfrac{1}{X} \dfrac{dX}{dt}\]

Taking integrals on both sides

\[ \int_0^t \mu \,dt = \int\dfrac{1}{X}\,dX \] \[\ \ln X = \mu t + C \qquad \text{where}\qquad C = \ln {X_0} \] \[\ \ln \frac{X}{X_0} = \mu t \] and \[\ X = X_0 e^{\mu t}\]

where \(X_0\) is the initial cell concentration. It follows from the above equation that a graph of \( \ln \frac{X}{X_0}\)plotted against time will display \(\mu\) in its instantaneous slope. When growth is exponential, \(\frac{dX}{dt} = \mu_g X \) and the graph is linear with a slope of \( \mu_g\). At steady state conditions, \(\frac{dx}{dt} = 0 \), and cell growth approaches cell loss, so that \(\mu_g X = K_d X\). During this stationary phase, the graph plateaus with horizontal slope.

In the cell death or decline phase where \(K_dX >> \mu_g X\), the graph decreases with negative slope. With first-order kinetics for cell decline,

\[\ \dfrac{1}{X} \dfrac{dX}{dt} = -K_d\qquad \text{ }\qquad \ln \frac{X}{X_0} = -K_d t \] and \[\ X = X_0 e^{-k_d t}\]

Substrate Utilization

Substrate is utilized for producing new cells and for maintaining cells. If the maintenance coefficient is \(m\) and the dimensionless yield factor \(Y_{X / S}\) is the ratio of new cells mass to substrate consumed, that is

\[\ Y_{X / S} = \dfrac {\Delta X}{\Delta S}\] then similar to the cell growth rate expression of equation (?), the rate of substrate utilization can be expressed as \[\dfrac {dS}{dt} = -\mu_g \frac {1}{Y_{X / S}} X - mX \]

Product Formation

Similar to cell growth and substrate utilization, products formed can be expressed by a first-order differential equation. The specific rate of product formation, \(\frac{1}{X}\frac {dP}{dt}\) is proportional to the specific rate of cell growth \( \mu_g \) during the exponential growth phase whereas in the non-growth steady-state phase it is a constant, \(\beta\). If the yield ratio \(Y_{P / X}\) is defined as

\[\ Y_{P / X} = \dfrac {\Delta P}{\Delta X}\] then the rate of product formation is expressed as \[\dfrac {dP}{dt} = -\mu_g Y_{P / X} X - \beta X \]

Monod Equation for Limited-Substrate Growth

More often than not, cell growth saturates when a substrate becomes limited. If this growth-limiting substrate is S, the kinetics are well described by the Monod equation:

\ \[ \mu_g = \frac {\mu_{max} S}{K_s + S}\]

where \(K_s\) is called the saturation coefficient. The Monod equation has the same form as another equation that describes the kinetics of enzymatic reactions and called the Michaelis-Menten equation, which is derived in the next section. The Monod equation is suited for microbial growth such as yeast where the growth is not rapid and when cell concentrations are low.

Michaelis-Menten Kinetics for Enzymatic Reactions

Conceptualization of Processes with Feedback

Model Simulation with MATLAB

Future Modeling

A model to include the bacteria prevalent in the ceiling tile process water and wastewater will be developed. To reflect both organisms, bacteria and yeast, the present model will be extended to incorporate the bacteria model and mathematical simulations will be run to gather insight on biomass, starch, and butyric acid kinematics under varying plant conditions of DO, temperature, pH, and starch input starch fed through the recycling process.

References

1. Shuler, M. L., & Kargi, F. (2002). Bioprocess engineering: Basic concepts. Chapter 6. Upper Saddle River, NJ: Prentice Hall.

2. Zangirolami, T.C., Carlsen, M., Nielsen, J., & Jørgensen, S.B.. (2002). Growth and enzyme production during continuous cultures of a high amylase-producing variant of Aspergillus Oryzae. Brazilian Journal of Chemical Engineering, 19(1), 55-68

3. Wang, L., D. Ridgway, T. Gu and M.Y. Murray, 2009. Kinetic modeling of cell growth and product formation in submerged culture of recombinant Aspergillus niger. Chem. Eng. Com., 196:481-490.

4. Akpa J (2012) Modeling of a Bioreactor for the Fermentation of Palm wine by Saccaharomyce cerevisiae (yeast) and lactobacillus (bacteria). Bioresource Technology 3: 231-240.

Team:BroadRun-Baltimore/Model

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