Calculation of $t_r$ for Simultaneous Synthesis of Cells and Products

TASK FOR FINAL GRADE

Rate equations for the growth of eukaryotic cells, the reactants consumption and products formation are given below.

$\frac{d{C}_x}{dt}={}_m(1-\frac{C_p}{C_{p_{\text{m}\text{a}\text{x}}}}{)}^m(\frac{C_S}{K_S+C_S})C_x$

$r_S=-\frac{1}{Y_{\frac{x}{S}}}\frac{d{C}_x}{dt}$

$r_P=\frac{1}{Y_{\frac{x}{P}}}\frac{d{C}_x}{dt}$ Data given

$K_s,=1\mathrm{.}6k\frac{g}{m^3},Y_{\frac{x}{S}},=0\mathrm{.}06k\frac{g}{k}g$

$C_{x_0},=0\mathrm{.}1k\frac{g}{m^3},{}_m,=0\mathrm{.}24h^{-1}$

$C_{P_0},=0k\frac{g}{m^3},(Y_{\frac{x}{P}}),=0\mathrm{.}16k\frac{g}{k}g$

$C_{P_{\text{m}\text{a}\text{x}}},=100k\frac{g}{m^3},m,=2$

Calculation of $t_r$ for Simultaneous Synthesis of Cells and Products

TASK FOR FINAL GRADE

$(1)$ Calculate the change of $C_x,C_P,$ and $C_S$ as a function of time when $C_{S_0}=100k\frac{g}{m^3}.$

$(2)$ What is the role of reactant concentration on specific growth rate of cell?

$(3)$ If we increase the reactant concentration, check whether rate of cell growth increases or not.

$(4)$ Show the effect of initial reactant concentration on ${}_{\text{n}\text{e}\text{t}}.$

Hints:

Hints: $1)$ Find and expression for $\frac{d{C}_x}{dt}$ as function of $C_x$ only $(+$ constants$)$

$2)$ Integrate this expression for different $C_x$ and $C_{s0}$

$3)$ Get different plots for $C_x,C_s$ and $C_p$ vs time