CoCalc -- 4717.sagews

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## This file investigates the steady-state reached for a cell culture growing in a CSTR (aka a 'chemostat').
## It displays the steady-state analytical solution and graphs the functional dependence

var('mu Y K_m mu_max X_in S_in V_in V_out X_0 S_0')
(X,S,t) = var('X,S,t')

## For steady-state, V_in = Vout == V
var('V')
V_in = V
V_out = V

## Define mu
mu = (mu_max*S)/(K_m+S)

## Solve for the steady-state cell and substrate density as f(process variables)
dXdt = mu*X+X_in*V_in-X*V_out==0
dSdt = (-1/Y)*mu*X+S_in*V_in-S*V_out==0
ss = solve([dXdt,dSdt],X,S)

## Parse out the equations for X and S from the solution matrix
X_ss(mu_max,K_m,Y,X_in,S_in,V) = ss[0][0].rhs()
S_ss(mu_max,K_m,Y,X_in,S_in,V) = ss[0][1].rhs()

## Show the analytical expressions for S and X at steady-state
show(ss[0][0])
show(ss[0][1])

\renewcommand{\Bold}[1]{\mathbf{#1}}X = \frac{{\left({\left(K_{m} + S_{\mbox{in}}\right)} V - S_{\mbox{in}} \mu_{\mbox{max}}\right)} Y + 2 \, V X_{\mbox{in}} - X_{\mbox{in}} \mu_{\mbox{max}} - \sqrt{X_{\mbox{in}}^{2} \mu_{\mbox{max}}^{2} + {\left(S_{\mbox{in}}^{2} \mu_{\mbox{max}}^{2} + {\left(K_{m}^{2} + 2 \, K_{m} S_{\mbox{in}} + S_{\mbox{in}}^{2}\right)} V^{2} - 2 \, {\left(K_{m} S_{\mbox{in}} \mu_{\mbox{max}} + S_{\mbox{in}}^{2} \mu_{\mbox{max}}\right)} V\right)} Y^{2} + 2 \, {\left(S_{\mbox{in}} X_{\mbox{in}} \mu_{\mbox{max}}^{2} + {\left(K_{m} \mu_{\mbox{max}} - S_{\mbox{in}} \mu_{\mbox{max}}\right)} V X_{\mbox{in}}\right)} Y}}{2 \, {\left(V - \mu_{\mbox{max}}\right)}}

<html><script type="math/tex; mode=display">\newcommand{\Bold}[1]{\mathbf{#1}}S = -\frac{{\left(K_{m} X_{\mbox{in}} \mu_{\mbox{max}} - \sqrt{K_{m}^{2} V^{2} Y^{2} + 2 \, K_{m} S_{\mbox{in}} V^{2} Y^{2} - 2 \, K_{m} S_{\mbox{in}} V Y^{2} \mu_{\mbox{max}} + S_{\mbox{in}}^{2} V^{2} Y^{2} - 2 \, S_{\mbox{in}}^{2} V Y^{2} \mu_{\mbox{max}} + S_{\mbox{in}}^{2} Y^{2} \mu_{\mbox{max}}^{2} + 2 \, K_{m} V X_{\mbox{in}} Y \mu_{\mbox{max}} - 2 \, S_{\mbox{in}} V X_{\mbox{in}} Y \mu_{\mbox{max}} + 2 \, S_{\mbox{in}} X_{\mbox{in}} Y \mu_{\mbox{max}}^{2} + X_{\mbox{in}}^{2} \mu_{\mbox{max}}^{2}} K_{m}\right)} V - {\left(K_{m} S_{\mbox{in}} V \mu_{\mbox{max}} - {\left(K_{m}^{2} + K_{m} S_{\mbox{in}}\right)} V^{2}\right)} Y}{X_{\mbox{in}} \mu_{\mbox{max}}^{2} - {\left(X_{\mbox{in}} \mu_{\mbox{max}} + \sqrt{K_{m}^{2} V^{2} Y^{2} + 2 \, K_{m} S_{\mbox{in}} V^{2} Y^{2} - 2 \, K_{m} S_{\mbox{in}} V Y^{2} \mu_{\mbox{max}} + S_{\mbox{in}}^{2} V^{2} Y^{2} - 2 \, S_{\mbox{in}}^{2} V Y^{2} \mu_{\mbox{max}} + S_{\mbox{in}}^{2} Y^{2} \mu_{\mbox{max}}^{2} + 2 \, K_{m} V X_{\mbox{in}} Y \mu_{\mbox{max}} - 2 \, S_{\mbox{in}} V X_{\mbox{in}} Y \mu_{\mbox{max}} + 2 \, S_{\mbox{in}} X_{\mbox{in}} Y \mu_{\mbox{max}}^{2} + X_{\mbox{in}}^{2} \mu_{\mbox{max}}^{2}}\right)} V + {\left({\left(K_{m} + S_{\mbox{in}}\right)} V^{2} + S_{\mbox{in}} \mu_{\mbox{max}}^{2} - {\left(K_{m} \mu_{\mbox{max}} + 2 \, S_{\mbox{in}} \mu_{\mbox{max}}\right)} V\right)} Y + \sqrt{K_{m}^{2} V^{2} Y^{2} + 2 \, K_{m} S_{\mbox{in}} V^{2} Y^{2} - 2 \, K_{m} S_{\mbox{in}} V Y^{2} \mu_{\mbox{max}} + S_{\mbox{in}}^{2} V^{2} Y^{2} - 2 \, S_{\mbox{in}}^{2} V Y^{2} \mu_{\mbox{max}} + S_{\mbox{in}}^{2} Y^{2} \mu_{\mbox{max}}^{2} + 2 \, K_{m} V X_{\mbox{in}} Y \mu_{\mbox{max}} - 2 \, S_{\mbox{in}} V X_{\mbox{in}} Y \mu_{\mbox{max}} + 2 \, S_{\mbox{in}} X_{\mbox{in}} Y \mu_{\mbox{max}}^{2} + X_{\mbox{in}}^{2} \mu_{\mbox{max}}^{2}} \mu_{\mbox{max

/script> }}}

################################################
####   Unnecessarily Fancy Plotting Stuff   ####
################################################
## Create a plot object
a = plot([],figsize=[6,4])

## Set the plot parameters
title = "[cell], [substrate] as f(dilution rate)"
xmin = 0
xmax = 0.08
ymin = 0
ymax = 10

## Add a title to the plot
a += text(title,(xmax/1.8,ymax),color='black',fontsize=15)

## Add the desired lines to the plot
a += plot(X_ss(0.1,2,0.5,0,10),color='orange',legend_label='[cell]')
a += plot(S_ss(0.1,2,0.5,0,10),color='purple',legend_label='[substrate]')

## For more information on plots in general, evaluate 'plot?'
## For a list of legend options, evaluate 'a.set_legend_options?'
## For a list of Sage predefined colors, evaluate 'sorted(colors)'

a.set_axes_range(xmin,xmax,ymin,ymax)
a.axes_labels(['dilution rate','concentration'])
a.axes_label_color('grey')
a.set_legend_options(ncol=2,borderaxespad=5,back_color='whitesmoke',fancybox=true)
show(a)