Team:BroadRun-Baltimore/Model
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Mathematical Modeling
Model Formulation
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
\[\mu_g = \dfrac{1}{X} \dfrac{dX}{dt}\]If the rate of cell loss from cell death and endogenous metabolism is \(K_d\), 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 cell 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 of limited substrate growth are commonly described by the Monod equation [1-4]:
\[ \mu_g = \frac {\mu_{max} S}{K_s + S}\]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
For enzyme-catalyzed reactions such as starch degradation by alpha-amylase, reaction kinetics are modeled with the Michaelis-Menten rate expression. The Michaelis-Menten kinetics applies to a single substrate \(S\) and single enzyme \(E\) reaction. If \(ES\) is the enzyme-substrate complex and \(P\) is the product, the reaction is shown by
\[S + E\mathrel{\mathop{\rightleftharpoons}^{k_{1}}_{k_{-1}}}ES\mathop{\rightarrow}^{k_{2}}P + E\]By writing three coupled differential equations (not shown here) for the rate of product formation, \(\dfrac{d}{dt}C_p\), the rate of enzyme-substrate formation, \(\dfrac{d}{dt}C_{ES}\), and the rate of substrate consumption, \(\dfrac{d}{dt}C_s\), and solving these equation the velocity of the forward reaction defined as \(\nu\) according to \[ \nu = rate \: of \: S \mathop{\rightarrow} P = \dfrac{d}{dt}\left [ C_p \right ] \]
one arrives at the well known Michaelis-Menten rate equation
\[ \nu = \frac{\nu_{max} C_s}{K_m + C_s} \]where \(K_m\) is an equilibrium concentration given by
\[\ K_m=\frac{k_{-1}+k_2}{k_1} \]\(C_s\) is the substrate concentration and \( \nu_{max}\) is the maximum rate given by
\[ \nu_{max} = K_2 C_E(0)\]It should be noted that of the two model parameters, while \(K_m\) is an intrinsic constant, \(\nu_{max}\)depends on the initial enzyme concentration \(C_E(0)\).
Since the rate of product formation \(\nu\), equals the rate of substrate consumption.
\[ \nu = \dfrac{d}{dt}C_p\ = -\dfrac{d}{dt}C_s \]
It can be seen from the equation that when
\[ C_s = K_m \qquad \text{ }\qquad \nu = \frac {1}{2} \nu_{max} \]
Model Simulation with MATLAB
The two parts of the model describing the kinetics for cell growth and the enzymatic reaction is summarized below. The first three differential equations are related to cell growth and are obtained by substituting for \(\mu_g\) in equations ? ? and > with equation ?, the Monod equation. The fourth differential equation is the Micaelis-Menten equation for the enzyme reaction.
\[\dfrac{dX}{dt} = \frac {\mu_{m} S}{K_s + S} X - K_d X\] \[\dfrac{dS}{dt} = - \frac {\mu_{m} S}{(K_s + S)Y_{X / S}} X - M X \] \[\dfrac {dP}{dt} = \frac {\mu_{m} Y_{P / X} S}{K_s + S} X + \beta X \] \[\dfrac {dC_s}{dt} = - \frac{\nu_{max} C_s}{K_m + C_s} \]
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 4 and 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.
