CHEM E 467 Biochemical Engineering
Contents
Enzyme Kinetics
| Description | Equations |
|---|---|
| Arrhenius equation | $k = A\exp\left(\dfrac{E_A}{RT}\right)$ |
| Power law | $r_P = k\prod C_i^{\nu_i}$ |
| Linearized 1st order rate law | $\ln\left(\dfrac{[A]}{[A]_0}\right) -kt$ |
Michaelis-Menten kinetics
| Description | Equations |
|---|---|
| Overall reaction | $\ce{E + S -> E + P}$ |
| Reaction mechanism | $\ce{S + E <=>[\mathit{k}_1][\mathit{k}_{-1}] ES} \newline \ce{ES ->[\mathit{k}_{2}] P + E}$ |
| Enzyme balance | $\ce{[E_T] = [E] + [ES]} \newline \ce{[E] = [E_T] - [ES]}$ |
| Pseudo-steady-state approximation | $\begin{aligned}r_{\ce{ES}} &= k_1\ce{[S][E]} - k_{-1} \ce{[ES]} - k_2 \ce{[ES]} = 0 \\ r_{\ce{ES}} &= k_1\ce{[S]([E_T] - [ES])} - k_{-1} \ce{[ES]} - k_2 \ce{[ES]} = 0 \end{aligned} \newline \ce{[ES]} = \dfrac{k_1 \ce{[S][E_T]}}{k_1 \ce{[S]} + k_{-1} + k_2} = \dfrac{\ce{[E_T][S]}}{K_M + \ce{[S]}}$ |
| Turnover number (# substrates converted to product per unit time on one enzyme at saturation) | $k_{\mathrm{cat}} = k_2$ |
| Michaelis-Menten constant (attraction of enzyme of its substrate, [Substrate] which rate of rxn is 1/2 max) | $K_M = \dfrac{k_{\mathrm{cat}} + k_{-1}}{k_1}$ |
| Maximum rate | $V_{\max} = k_{\mathrm{cat}} \ce{[E_T]}$ |
| Michaelis-Menten equation Rate of reaction |
$\begin{aligned}r_{\ce{P}} &= k_{2} \ce{[ES]} \\ &= \dfrac{k_1 k_{2} \ce{[S][E_T]}}{k_1 \ce{[S]} + k_{-1} + k_2} \\ &= \dfrac{k_{\mathrm{cat}} \ce{[S][E_T]}}{K_M + \ce{[S]}} \\ &= \dfrac{V_{\max} \ce{[S]}}{K_M + \ce{[S]}} \end{aligned}$ |
| Lineweaver-Burk equation | $\dfrac{1}{r_{\ce{P}}} = \dfrac{K_M}{V_{\max}}\dfrac{1}{\ce{[S]}} + \dfrac{1}{V_{\max}}$ |
| Eadie-Hofstee equation | $r_{\ce{P}} = V_{\max} - K_M\dfrac{r_{\ce{P}}}{\ce{[S]}}$ |
| Hanes-Woolf equation | $\dfrac{\ce{[S]}}{r_{\ce{P}}} = \dfrac{K_M}{V_{\max}} + \dfrac{1}{V_{\max}}\ce{[S]}$ |
Enzymatic inhibition
Competitive inhibition
| Description | Equations |
|---|---|
| Reaction mechanism | $\ce{E + S <=> ES} \newline \ce{ES -> E + P} \newline \ce{E + I <=> EI} \text{ (inactive)}$ |
| Reaction rate | $r_{\ce{P}} = \dfrac{V_{\max}\ce{[S]}}{\ce{[S]} + K_M \left[1 + \dfrac{\ce{[I]}}{K_I}\right]}$ |
| Lineweaver-Burk form $\uparrow K_I, \uparrow \text{slope}$ |
$\dfrac{1}{r_{\ce{P}}} = \dfrac{1}{\ce{[S]}} \left[\dfrac{K_M}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right] \right] + \dfrac{1}{V_{\max}}$ |
Uncompetitive inhibition
| Description | Equations |
|---|---|
| Reaction mechanism | $\ce{E + S <=> ES} \newline \ce{ES -> E + P} \newline \ce{ES + I <=> ESI} \text{ (inactive)}$ |
| Reaction rate | $r_{\ce{P}} = \dfrac{V_{\max}\ce{[S]}}{K_M + \ce{[S]} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]}$ |
| Lineweaver-Burk form $\uparrow K_I, \uparrow \text{intercept}$ |
$\dfrac{1}{r_{\ce{P}}} = \dfrac{1}{\ce{[S]}} \dfrac{K_M}{V_{\max}} + \dfrac{1}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]$ |
Noncompetitive (mixed) inhibition
| Description | Equations |
|---|---|
| Reaction mechanism | $\ce{E + S <=> ES} \newline \ce{ES -> E + P} \newline \ce{E + I <=> EI} \text{ (inactive)} \newline \ce{ES + I <=> ESI} \text{ (inactive)} \newline \ce{S + EI <=> ESI} \text{ (inactive)}$ |
| Reaction rate | $r_{\ce{P}} = \dfrac{V_{\max}\ce{[S]}}{(\ce{[S]} + K_M) \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]}$ |
| Lineweaver-Burk form $\uparrow K_I, \uparrow \text{slope}, \uparrow \text{intercept}$ |
$\dfrac{1}{r_{\ce{P}}} = \dfrac{1}{\ce{[S]}} \dfrac{K_M}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right] + \dfrac{1}{V_{\max}} \left[ 1 + \dfrac{\ce{[I]}}{K_I}\right]$ |
Cell Growth Kinetics
| Description | Equations |
|---|---|
| Nomenclature | $S$ = substrate mass $P$ = product mass $X$ = cell mass |
| Mass balance | $\Delta S + \Delta P = \Delta X$ |
| Yield coefficient (mass basis) | $Y_{X/S} = -\dfrac{\Delta X}{\Delta S} = \dfrac{[X] - [X]_0}{[S]_0 - [S]}$ |
| Exponential growth | $[X] = [X]_0 \exp(\mu_{\max}t)$ |
| Doubling time | $t_d = \dfrac{\ln 2}{\mu}$ |
| Exponential death | $[X] = [X]_0 \exp(-k_d t)$ |
| Optical density (absorbance) | $\mathrm{OD}_\lambda = A_\lambda = \log \dfrac{I}{I_0}$ |
Cell Reaction Stoichiometry
Elemental (atomic) balance
| Description | Equations |
|---|---|
| Cell reaction | $\ce{C_wH_xO_yN_z + a O_2 + b NH_3} \newline \ce{-> c CH_\alpha O_\beta N_\delta\ + d CO_2 + e H_2O + f C_jH_kO_lN_m}$ |
| Respiratory quotient | $\mathrm{RQ} = \dfrac{\text{mol } \ce{CO2} \text{ produced}}{\text{mol } \ce{O2} \text{ consumed}} = \dfrac{d}{a}$ |
| Biomass yield (mass basis) | $Y_{X/S} = \dfrac{\Delta X}{\Delta S} = \dfrac{\text{g cell}}{\text{g substrate}}$ |
| Product yield (mass basis) | $Y_{P/S} = \dfrac{\Delta P}{\Delta S} = \dfrac{\text{g product}}{\text{g substrate}}$ |
| Substrate utilization rate (SUR) | $\mathrm{SUR} = \dfrac{\text{mol substrate}}{\text{time}}$ |
| Oxygen utilization rate (OUR) | $\mathrm{OUR} \equiv A = a\ \mathrm{SUR}$ |
| Carbon dioxide evolution rate (CER) | $\mathrm{CER} \equiv D = d\ \mathrm{SUR}$ |
Electron balance
| Description | Equations |
|---|---|
| Cell reaction | $\ce{C_wH_xO_yN_z + a O_2 + b NH_3} \newline \ce{-> c CH_\alpha O_\beta N_\delta\ + d CO_2 + e H_2O + f C_jH_kO_lN_m}$ |
| Degree of reduction | $\gamma_i = \dfrac{(\text{Valance \# of element } i) (\text{\# atom in element } i)}{\text{\#} \ce{C} \text{ atom}}$ |
| Stoichiometric coefficient of oxygen | $a = \frac{1}{4}(w\gamma_S - c\gamma_X - fj\gamma_P)$ |