## This sheet simulates two reversible first-order chemical reactions in series:
## A1 + A2 <-> A3 -> A4 (S+E ⇌ ES ➙ P+E)
## The initial value problem is solved by Runge-Kutta and the dynamics are graphed
reset()
## Initial Conditions
var('t A1 A2 A3 A4')
ivar = t
vars = [A1,A2,A3,A4]
ics = [0] #starting at t = 0
ics.append(2.0) #A1_0 = S0
ics.append(0.01) #A2_0 = E0
ics.append(0.00) #A3_0 = ES0
ics.append(0.00) #A4_0 = P0
## Kinetic Parameters
k1 = .1
k_1 = .1
k2 = .5
k_2 = 0
## Equations
rxns = []
rxns.append(diff(A1,t) == k_1*A3 - k1*A1*A2)
rxns.append(diff(A2,t) == k_1*A3 - k1*A1*A2 + k2*A3)
rxns.append(diff(A3,t) == k1*A1*A2 - k_1*A3 - k2*A3)
rxns.append(diff(A4,t) == k2*A3 - k_2*A4)
## Calculation Parameters
end_points = 3000
stepsize = 1
steps = end_points/stepsize
des = []
for i in range(0,len(rxns)):
des.append(rxns[i].rhs())
sol = desolve_system_rk4(des,vars,ics,ivar,end_points=end_points,step=stepsize)
## Process the output into a form that can be graphed
sols=[]
for i in range(0,len(sol[0])-1):
sols.append([])
for i in range(steps):
for j in range(0,(len(sol[i])-1)):
sols[j].append([sol[i][0],sol[i][j+1]])
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################################################
#### Unnecessarily Fancy Plotting Stuff ####
################################################
### 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)'
### USER-DEFINED PARAMETERS
title = "Enzymatic Reaction Kinetics" ## Graph Title
legend_labels = ['[S]','[E]','[ES]','[P]'] ## Legend Labels
colors = ['red','orange','yellow','green','blue'] ## Legend Colors
x_label = 'time' ## X-axis label
y_label = 'concentration' ## Y-axis label
### Create a plot object
a = plot([])
a.axes_labels([x_label,y_label])
# a.set_axes_range(xmin=0,xmax=150,ymin=0,ymax=10)
### Add the desired lines to the plot
for i in range(0,len(sols)):
a += list_plot(sols[i],color=colors[mod(i,len(colors))],legend_label=legend_labels[i])
### Set the plot parameters
a += text(title,(a.xmax()/1.8,a.ymax()),color='black',fontsize=15)
#a.axes_label_color('grey')
#a.set_legend_options(ncol=round(len(legend_labels)/2),borderaxespad=5,back_color='whitesmoke',fancybox=true)
#show(a)
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b = plot([])
for i in range(1,3):
b += list_plot(sols[i],color=colors[mod(i,len(colors))],legend_label=legend_labels[i])
title = "Free Enzyme and Enzyme-Substrate complex"
b += text(title,(b.xmax()/1.8,b.ymax()),color='black',fontsize=15)
b.axes_labels([x_label,y_label])
show(b)
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### Re-solve the problem using the Michaelis-Menten kinetic model
### (Briggs-Haldane derivation, applying PSSH to [ES])
## Initial Conditions
var('t S P')
ivar = t
vars = [S,P]
S0 = ics[1]
P0 = ics[4]
E0 = ics[2]
## Kinetic Parameters
Vmax = (k2*E0)
Km = ((k_1 + k2) / k1)
## Equations
rxns = []
rxns.append(diff(S,t) == -Vmax*S/(Km+S))
rxns.append(diff(P,t) == Vmax*S/(Km+S))
## Calculation Parameters
## Same as above!
des = []
for i in range(0,len(rxns)):
des.append(rxns[i].rhs())
sol = desolve_system_rk4(des,vars,[0,S0,P0],ivar,end_points=end_points,step=stepsize)
## Process the output into a form that can be graphed
sols=[]
for i in range(0,len(sol[0])-1):
sols.append([])
for i in range(steps):
for j in range(0,(len(sol[i])-1)):
sols[j].append([sol[i][0],sol[i][j+1]])
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################################################
#### Unnecessarily Fancy Plotting Stuff ####
################################################
### 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)'
### USER-DEFINED PARAMETERS
#title = "Enzymatic Reaction Kinetics" ## Graph Title
legend_labels = ['[S]MM','[P]MM'] ## Legend Labels
colors = ['salmon','lightgreen'] ## Legend Colors
#x_label = 'time' ## X-axis label
#y_label = 'concentration' ## Y-axis label
### Create a plot object
#a = plot([])
#a.axes_labels([x_label,y_label])
#a.set_axes_range(xmin=0,xmax=150,ymin=0,ymax=10)
### Add the desired lines to the plot
for i in range(0,len(sols)):
a += list_plot(sols[i],color=colors[mod(i,len(colors))],legend_label=legend_labels[i],size='4')
### Set the plot parameters
#a += text(title,(a.xmax()/1.8,a.ymax()),color='black',fontsize=15)
a.axes_label_color('grey')
a.set_legend_options(ncol=3,borderaxespad=5,back_color='whitesmoke',fancybox=true)
show(a)
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show('Vmax = '+ Vmax.str())
show('Km = '+ Km.str())
#plot?
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