Curve fitting to coupled ODEs - curve-fitting

Question

have a question on curve fitting / optimizing. I have three coupled ODEs that descibe a biochemical reaction with a disappearing substrate and two products being formed. I've found examples that have helped me create code to solve the ODEs (below). Now I want to optimize the unknown rate constants (k, k3 and k4) to fit to the experimental data, P, which is a signal from product y[1]. What would be the easiest way of doing this?
Thanks.
import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
# Experimental data
P = [29.976,193.96,362.64,454.78,498.42,517.14,515.76,496.38,472.14,432.81,386.95,
352.93,318.93,279.47,260.19,230.92,202.67,180.3,159.09,137.31,120.47,104.51,99.371,
89.606,75.431,67.137,58.561,55.721]
# Three coupled ODEs
def conc (y, t) :
a1 = k * y[0]
a2 = k2 * y[0]
a3 = k3 * y[1]
a4 = k4 * y[1]
a5 = k5 * y[2]
f1 = -a1 -a2
f2 = a1 -a3 -a4
f3 = a4 -a5
f = np.array([f1, f2, f3])
return f
# Initial conditions for y[0], y[1] and y[2]
y0 = np.array([50000, 0.0, 0.0])
# Times at which the solution is to be computed.
t = np.linspace(0.5, 54.5, 28)
# Experimentally determined parameters.
k2 = 0.071
k5 = 0.029
# Parameters which would have to be fitted
k = 0.002
k3 = 0.1
k4 = 0.018
# Solve the equation
y = odeint(conc, y0, t)
# Plot data and the solution.
plt.plot(t, P, "bo")
#plt.plot(t, y[:,0]) # Substrate
plt.plot(t, y[:,1]) # Product 1
plt.plot(t, y[:,2]) # Product 2
plt.xlabel('t')
plt.ylabel('y')
plt.show()

Answer 1

Edit: I made some changes to the code in order to show how to fit to the experimental data of all ODEs.
Like this:
import numpy as np
from scipy.integrate import odeint
import matplotlib.pyplot as plt
from scipy.odr import Model, Data, ODR
# Experimental data
P = [29.976,193.96,362.64,454.78,498.42,517.14,515.76,496.38,472.14,432.81,386.95,
352.93,318.93,279.47,260.19,230.92,202.67,180.3,159.09,137.31,120.47,104.51,99.371,
89.606,75.431,67.137,58.561,55.721]
# Times at which the solution is to be computed.
t = np.linspace(0.5, 54.5, 28)
def coupledODE(beta, x):
k, k3, k4 = beta
# Three coupled ODEs
def conc (y, t) :
a1 = k * y[0]
a2 = k2 * y[0]
a3 = k3 * y[1]
a4 = k4 * y[1]
a5 = k5 * y[2]
f1 = -a1 -a2
f2 = a1 -a3 -a4
f3 = a4 -a5
f = np.array([f1, f2, f3])
return f
# Initial conditions for y[0], y[1] and y[2]
y0 = np.array([50000, 0.0, 0.0])
# Experimentally determined parameters.
k2 = 0.071
k5 = 0.029
# Parameters which would have to be fitted
#k = 0.002
#k3 = 0.1
#k4 = 0.018
# Solve the equation
y = odeint(conc, y0, x)
return y[:,1]
# in case you are only fitting to experimental findings of ODE #1
# return y.ravel()
# in case you have experimental findings of all three ODEs
data = Data(t, P)
# with P being experimental findings of ODE #1
# data = Data(np.repeat(t, 3), P.ravel())
# with P being a (3,N) array of experimental findings of all ODEs
model = Model(coupledODE)
guess = [0.1,0.1,0.1]
odr = ODR(data, model, guess)
odr.set_job(2)
out = odr.run()
print out.beta
print out.sd_beta
f = plt.figure()
p = f.add_subplot(111)
p.plot(t, P, 'ro')
p.plot(t, coupledODE(out.beta, t))
plt.show()
In case you were using peak-o-mat (http://lorentz.sf.net) which is an interactive curve fitting program based on scipy, you could add your ODE model and save it to userfunc.py (see the customisation section in the docs):
import numpy as np
from scipy.integrate import odeint
from peak_o_mat import peaksupport as ps
def coupODE(x, k, k3, k4):
# Three coupled ODEs
def conc (y, t) :
a1 = k * y[0]
a2 = k2 * y[0]
a3 = k3 * y[1]
a4 = k4 * y[1]
a5 = k5 * y[2]
f1 = -a1 -a2
f2 = a1 -a3 -a4
f3 = a4 -a5
f = np.array([f1, f2, f3])
return f
# Initial conditions for y[0], y[1] and y[2]
y0 = np.array([50000, 0.0, 0.0])
# Times at which the solution is to be computed.
#t = np.linspace(0.5, 54.5, 28)
# Experimentally determined parameters.
k2 = 0.071
k5 = 0.029
# Parameters which would have to be fitted
#k = 0.002
#k3 = 0.1
#k4 = 0.018
# Solve the equation
y = odeint(conc, y0, x)
print y
return y[:,1]
ps.add('ODE',
func='coupODE(x,k,k3,k4)',
info='thre coupled ODEs',
ptype='MISC')
You would need to prepare your data as a text file with two columns for time and experimental data. Import the data into peak-o-mat, enter 'ODE' as fit model, choose appropriate initial parameters for k,k3,k4 and hit 'Fit'.