import numpy as np
import scipy.integrate
import bokeh.models
import bokeh.plotting
import bokeh.io
bokeh.io.output_notebook()
We first set up the parameters. Most are measured and reported, at least approximately, in the Lee, et al. paper. The parameters $k_8$, $k_{\text{-}8}$ and $k_{12}$ are unknown. However, $K_\mathrm{d} = k_{\text{-}8}/k_8$ is known, so we only have two parameters to set.
# Key for names
names = ['Axin complex', 'Axin-βcat', 'Axin-βcat*', 'βcat*', 'βcat']
# Define known parameters
c_A = 50 # nM (given by fixed GSK-3 concentration)
k9 = 206 # 1/min
k10 = 206 # 1/min
k11 = 0.417 # 1/min
Kd8 = 120 # nM
# Unknown parameters
km8 = 10 # 1/min
k12 = 100 # nM/min
# k8 determined form Kd8 and km8
k8 = km8 / Kd8 # 1/nM-min
# Set up params for ODE
params = np.array([k8, km8, k9, k10, k11, k12])
For our initial conditions, we will assume that we have no $\beta$-catenin and that we have a fixed amount of APC/Axin/GKC-3.
# Initial conditions
c0 = np.array([c_A, 0, 0, 0, 0])
Now we define the time derivative. We use the functional form that is necessary to pass into scipy.integrate.odeint().
def dcdt(c, t, k8, km8, k9, k10, k11, k12):
"""
Time derivative of concentrations.
c = (c3, c8, c9, c10, c11)
"""
# Unpack concentrations and parameters
c3, c8, c9, c10, c11 = c
k8, km8, k9, k10, k11, k12 = params
# Build derivatives
deriv = np.empty(5)
deriv[0] = -k8*c3*c11 + km8*c8 + k10*c9
deriv[1] = k8*c3*c11 - (km8 + k9)*c8
deriv[2] = k9*c8 - k10*c9
deriv[3] = k10*c9 - k11*c10
deriv[4] = -k8*c3*c11 + km8*c8 + k12
return deriv
Now, we just have to specify what time point we want and then run the solver!
# Set up time points and solve
t = np.linspace(0, 15, 400)
c = scipy.integrate.odeint(dcdt, c0, t, args=(k8, km8, k9, k10, k11, k12))
Finally, we can plot the solution.
# Set up canvas on which to paint plot
p = bokeh.plotting.figure(plot_width=650, plot_height=400,
x_axis_label='time (min)', y_axis_label='conc (nM)',
border_fill_alpha=0, background_fill_alpha=0)
# Specify colors
colors = ('#e41a1c','#377eb8','#4daf4a','#984ea3','#ff7f00')
# Add glyphs
for i, name in enumerate(names):
p.line(t, c[:,i], line_width=3, color=colors[i], legend=name)
# Place legend
p.legend.location = 'right_center'
bokeh.io.show(p)