Demo of various features of MM kinetics and turbidity

gistfile1.py

from scipy.integrate import odeint
import numpy as np
import pylab as pylab


def mm_rate(z,k):
    "The Michealis-Menten reaction rate a concentration z and rate parameter k"
    s,e,c,j = z
    k_bind,k_unbind,k_cat = k
    return np.array([-k_bind*s*e+k_unbind*c,
                     -k_bind*s*e+k_unbind*c+k_cat*c,
                     +k_bind*s*e-k_unbind*c-k_cat*c,
                                           +k_cat*c])
#an example of the kinetics
s_0, e_0 = 2.0, 1.0
z_0 = [s_0, e_0, 0.0, 0.0]
k = [1.0,1.1,.2]

T = 30
N_points = 1000
t = np.linspace(0,T,N_points).T

z = odeint(lambda zed,t: mm_rate(zed,k),z_0,t)

pylab.plot(t,z)
pylab.legend(['$s$','$e$','$c','$j$'])
pylab.xlabel('time')
pylab.ylabel('concentration')
pylab.savefig('kinetics_example.png')
pylab.clf()

#example of how turbidity depends on initial concentration

def turbidity_time_series(Z_0,turbid_fun):
    return np.array([turbid_fun(odeint(lambda zed,t: mm_rate(zed,k),z_0,t)) for z_0 in Z_0]).T

turbid_funs = [lambda z: 1- z[:,3]/z[0,0],
               lambda z: 1- z[:,3]**2/z[0,0]**2,
               lambda z: 1- np.sqrt(z[:,3]/z[0,0])]

s_0 = 2.0
E_0 = [0.5, 1.0, 2.0, 4.0, 8.0]
Z_0 = [[s_0, e_0, 0.0, 0.0] for e_0 in E_0]

simple_turbidity = turbidity_time_series(Z_0,turbid_funs[0])

pylab.plot(t,simple_turbidity)
pylab.xlabel('time')
pylab.ylabel('turbidity')
pylab.legend(['$e_0$ = '+str(e_0) for e_0 in E_0])
pylab.savefig('simple_turbidity_example.png')        
pylab.clf()

# demonstration of how the turbidity calibration function matters
turbidities = np.array([turbidity_time_series(Z_0,fun)[:,3] for fun in turbid_funs]).T

pylab.plot(t,turbidities)
pylab.xlabel('time')
pylab.ylabel('turbidity')
pylab.legend(['flat','upward curved','downward curved'])
pylab.savefig('turbidity_calib_example.png')