A simultaneous saccharification and fermentation model for dynamic growth environments

Abstract

Many mathematical models by researchers have been formulated for Saccharomyces cerevisiae which is the common yeast strain used in modern distilleries. A cybernetic model that can account for varying concentrations of glucose, ethanol and organic acids on yeast cell growth dynamics does not exist. A cybernetic model, consisting of 4 reactions and 11 metabolites simulating yeast metabolism, was developed. The effects of variables such as temperature, pH, organic acids, initial inoculum levels and initial glucose concentration were incorporated into the model. Further, substrate and product inhibitions were included. The model simulations over a range of variables agreed with hypothesized trends and to observations from other researchers. Simulations converged to expected results and exhibited continuity in predictions for all ranges of variables simulated. The cybernetic model did not exhibit instability under any conditions simulated.

A. Model summary

A.1 Cybernetic control equations

$$\begin{aligned} r_1&=\mu_1 e_1\left(\frac{G}{K_1+G}\right)\left(1-{\rm {\sc H}}_{G,150}\left(\frac{G-150}{650-150}\right)\right) \\ &\quad \times \left(1-{\rm {\sc H}}_{E,95}\left(\frac{E-95}{150-95}\right)\right) \\ r_2&=\mu_2 e_2\left(\frac{G}{K_2+G}\right)\left(1-{\rm {\sc H}}_{G,150}\left(\frac{G-150}{650-150}\right)\right) \\ &\quad \times \left(1-{\rm {\sc H}}_{E,95}\left(\frac{E-95}{150-95}\right)\right) \\ r_3&=\mu_3 e_3\left(\frac{{\rm EM}}{K_3+{\rm EM}}\right)\left(\frac{{\rm SM}}{K_4+{\rm SM}}\right) \\ r_4&=\mu_4 e_4\left(\frac{G}{K_1+G}\right)\left(\frac{O}{K_5+O}\right) \\ & & \\ {\rm {\sc H}}_{a,b} & = \left\{\begin{array}{rll} 1 & \hbox{if} & a > b \;\;\hbox{\linebreak is the Heaviside function.} \\ 0 & \hbox{if} & a\le b\\ \end{array} \right. \\ \nu^{\prime}_{1} & = \left(\frac{r_1}{\hbox{Max}\left(r_1,r_4\right)}\right)\;\;\;\;\; \leftarrow \hbox{for substitutable pathway $r_1, r_4$} \\ \nu^{\prime\prime}_{1} & = \left(\frac{r_1/{\rm EM}}{\hbox{Max}\left(r_1/{\rm EM},r_2/{\rm SM}\right)}\right)\;\;\;\;\; \leftarrow \hbox{for complementary pathway $r_1, r_2$} \\ \nu_{1} & = \nu^{\prime}_{1}\nu^{\prime\prime}_{1}\;\; \leftarrow \hbox{overall regulation for substitutable and complementary pathways $r_1$} \\ \nu_{2}& = \left(\frac{r_2/{\rm SM}}{\hbox{Max}\left(r_1/{\rm EM},r_2/{\rm SM}\right)}\right)\;\;\;\;\; \leftarrow \hbox{for complementary pathway $r_2, r_1$} \\ \nu_{3} & = 1 \;\;\;\;\; \leftarrow \hbox{for pathway $r_3$} \\ \nu_{4} & = \left(\frac{r_4}{\hbox{Max}(r_1,r_4)}\right)\;\;\;\;\; \leftarrow \hbox{for substitutable pathway $r_4, r_1$} \\ u^{\prime}_{1} & = \left(\frac{r_1}{r_1+r_4}\right)\;\;\;\;\;\;\; \leftarrow \hbox{for substitutable pathway $r_1, r_4$} \\ u^{\prime\prime}_{1}& = \left(\frac{r_1/{\rm EM}}{r_1/{\rm EM}+r_2/{\rm SM}}\right)\;\;\;\;\;\;\; \leftarrow \hbox{for complementary pathway $r_1, r_2$} \\ u_{1} & = u^{\prime}_{1}u^{\prime\prime}_{1}\;\;\;\;\;\;\; \leftarrow \hbox{overall regulation for both pathways $r_1$} \\ u_{2} & = \left(\frac{r_2/{\rm SM}}{r_1/{\rm EM}+r_2/{\rm SM}}\right)\;\;\;\;\;\;\; \leftarrow \hbox{for complementary pathway $r_2, r_1$} \\ u_{3} & = 1 \;\;\;\;\;\;\; \leftarrow \hbox{for pathway $r_3$} \\ u_{4} & = \left(\frac{r_4}{r_1+r_4}\right)\;\;\;\;\;\;\; \leftarrow \hbox{for substitutable pathway $r_4, r_1$} \\ \end{aligned}$$

A.2 Dynamic mass balance equations

$$ \begin{aligned} \hbox{Saccharification:}\;\;\;\frac{{\rm d}{\rm GP}}{{\rm d}t} & = \left({\rm GP}_0-{\rm GP}\right)D+\left(\frac{V_{{\rm max}}{\rm GP}}{K_{{\rm gp}}+{\rm GP}+K_gG}\right) \\ & \quad \times {\rm GA}.{\rm GA}_{\rm Activity}\left(0.874{\rm pH}^3-9.709{\rm pH}^2-2.734{\,\rm pH}+203.088 \right)0.01 \\ \hbox{Cell mass:}\;\;\;\frac{{\rm d}X}{{\rm d}t} & = \left(r_3-D\right)X+0.1212\;\eta_3\left({\frac{T-15}{30-15}}\right) \\ & \quad -\eta_4\left({\rm {\sc H}}_{T,33}{\rm e}^{\left({\frac{T-33}{41-33}}\right)}\right)-{\rm {\sc H}}_{C_{{\rm aa}},5}{\rm {\sc H}}_{4.71,{\rm pH}}0.05 \\ \hbox{Glucose:}\;\;\;\frac{{\rm d}G}{{\rm d}t} & = \;{\rm {\sc H}}_{{\rm GP},0}\;\frac{{\rm d}{\rm GP}}{{\rm d}t}-\left(\frac{r_1\;\nu_1}{Y_1}+\frac{r_2\;\nu_2}{Y_2}+\frac{r_4\;\nu_4}{Y_3}\right)X+\left(G_0-G\right)D \\ \hbox{Ethanol:}\;\;\;\frac{{\rm d}E}{{\rm d}t} & = \left(\phi_1\;\frac{r_1\;\nu_1}{Y_1}\right)X-E.D \\ \hbox{Oxygen:}\;\;\;\frac{{\rm d}O}{{\rm d}t} & = -\left(\phi_2\;\frac{r_4\;\nu_4}{Y_3}\right)X\;+\;\left(K_{{\rm La}} O^*-D.O\right) \\ \hbox{Energy precursors:}\;\;\;\frac{{\rm d}{\rm EM}}{{\rm d}t} & = \left(r_1\;\nu_1+r_4\;\nu_4-\frac{r_3\;\nu_3}{\alpha_1}\right)\;X-\eta_1\left(\frac{1229.56\;C_{{\rm aa}}}{10^{{\rm pH}-4.71}+1}\right)-EM.D \\ \hbox{Structural precursors:}\;\;\;\frac{{\rm d}{\rm SM}}{{\rm d}t} & = \left(r_2\;\nu_2-\frac{r_3\;\nu_3}{\alpha_2}\right)\;X -\eta_2{\rm {\sc H}}_{{\rm pH},5.0}(0.725{\rm pH})-{\rm SM}.D \\ \hbox{Enzyme 1 for pathway $r_1$:}\;\;\;\frac{{\rm d}e_1}{{\rm d}t} & = u_1\left(\frac{G}{K_1+G}\right)-e_1\;\beta + \alpha^\ast -e_1.D \\ \hbox{Enzyme 2 for pathway $r_2$:}\;\;\;\frac{{\rm d}e_2}{{\rm d}t} & = u_2\left(\frac{G}{K_2+G}\right)-e_2\;\beta + \alpha^\ast -e_2.D \\ \hbox{Enzyme 3 for pathway $r_3$:}\;\;\;\frac{{\rm d}e_3}{{\rm d}t} & = u_3\left(\frac{{\rm EM}}{K_3+{\rm EM}}\right)\left(\frac{{\rm SM}}{K_4+{\rm SM}}\right)-e_3\;\beta + \alpha^\ast -e_3.D \\ \hbox{Enzyme 4 for pathway $r_4$:}\;\;\;\frac{{\rm d}e_4}{{\rm d}t} & = u_4\left(\frac{G}{K_1+G}\right)\left(\frac{O}{K_5+O}\right)-e_4\;\beta + \alpha^\ast -e_4.D \\ \hbox{Acetic acid:}\;\;\;\frac{{\rm d}C_{{\rm aa}}}{{\rm d}t} & = -D.C_{{\rm aa}} \\ & \quad +\left({\rm {\sc H}}_{5.0,{\rm pH}}(0.1056) +{\rm {\sc H}}_{{\rm pH},5.0}\left(0.0533{\rm pH}-0.1782\right)\right)\frac{{\rm d}X}{{\rm d}t} \\ \hbox{Glycerol:}\;\;\;\frac{{\rm d}C_{{\rm gy}}}{{\rm d}t} & = -D.C_{{\rm gy}} \\ & \quad +\left({\rm {\sc H}}_{5.0,{\rm pH}}(4.018)+{\rm {\sc H}}_{{\rm pH},5.0}\left(0.416{\rm pH}-1.40\right)\right)\frac{{\rm d}X}{{\rm d}t} \\ \end{aligned} $$