Contents

CHEM E 467 Biochemical Engineering

2023-04-29

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	$a = \frac{1}{4}(w\gamma_S - c\gamma_X - fj\gamma_P)$