Variation of the times with the network length



In [18]:

    
%matplotlib notebook
from LinearChain import *
import numpy as np
import matplotlib.pyplot as plt
from matplotlib import rc
from scipy.integrate import ode
from scipy.integrate import quad
from scipy import stats
import pandas as pd

Parameters for the runs



In [8]:

    
N_max = 100
conc_ref = 1
a = 2
b = 1
KS = 2
KP = 2
Ns = np.arange(2,N_max)
sizes = []

## Returns a linearised set of parameters or MMH reactions
ConcSS = 1
F = (a*ConcSS/KS - b*ConcSS/KP) / (1 + ConcSS/KS + ConcSS/KP)
S = (1 + ConcSS/KS + ConcSS/KP)

alpha = a*KS
beta = b*KP
A_c = (alpha-F)/(S*KS)
B_c = (beta+F)/(S*KP)
A_t = alpha/(S*KS)
B_t = beta/(S*KP)

Compute the times for sizes ranging between 2 and 99 metabolites



In [9]:

    
data = pd.DataFrame({"tau_MA":np.zeros(N_max-2),"T_MA":np.zeros(N_max-2),"tau_MM_c":np.zeros(N_max-2),"T_MM_c":np.zeros(N_max-2),"tau_MM_t":np.zeros(N_max-2),"T_MM_t":np.zeros(N_max-2)})
data.index = np.arange(2,N_max)
for n in range(2,N_max):
    sizes.append(n)
    conc_ss = np.ones(n+2)*conc_ref
    
    ## Mass action
    param_vec_MA = np.array([a,b]*(n+1)) ## repeat [a,b] n+1 times
    J_MA = compute_jacobian(n,param_vec_MA)
    data.loc[n].tau_MA = 1/(a+b - 2*np.sqrt(a*b)*np.cos(np.pi/(n+1)))
    data.loc[n].T_MA = compute_lifetime(J_MA,norm_type=1)

    ## Michaelis=Menten-Henri        
    param_vec_MM_c = np.array([A_c,B_c]*(n+1)) 
    param_vec_MM_t = np.array([A_t,B_t]*(n+1))
    J_MM_c = compute_jacobian(n,param_vec_MM_c)
    J_MM_t = compute_jacobian(n,param_vec_MM_t)
    data.loc[n].tau_MM_c = 1/(A_c+B_c - 2*np.sqrt(A_c*B_c)*np.cos(np.pi/(n+1)))
    data.loc[n].T_MM_c = compute_lifetime(J_MM_c,norm_type=1)
    data.loc[n].tau_MM_t = 1/(A_t+B_t - 2*np.sqrt(A_t*B_t)*np.cos(np.pi/(n+1)))
    data.loc[n].T_MM_t = compute_lifetime(J_MM_t,norm_type=1)

sizes = np.array(sizes)

PLot the figures

For the $\tau$s



In [57]:

    
exec(open("fig_settings.py").read())
cross_over = crossover_size([A_c,B_c])
select_points = np.arange(0,len(sizes),2)
N_max = sizes[-1]+1
fig = plt.figure()
ax = fig.add_axes([0.11, 0.11, 0.86, 0.87])
index_relax = 0

ax.scatter(sizes[select_points],data.tau_MM_c[select_points],label=r"$\tau_c^{MMH}$",marker="s",s=ms_size,color=myblue)
fig.show()
ax.scatter(sizes[select_points],data.tau_MM_t[select_points],label=r"$\tau_t^{MMH}$",marker="D",s=ms_size,color=mygreen)
ax.scatter(sizes[select_points],data.tau_MA[select_points],label=r"$\tau_t^{MA}\;\&\;\tau_c^{MA}$",s=ms_size,color=myred)

ax.yaxis.labelpad = 4
ax.xaxis.labelpad = 0 
ax.yaxis.tick_left()
ax.xaxis.tick_bottom()
L = np.int(cross_over/2) +2
ax2 = fig.add_axes([0.65, 0.67, 0.3, 0.23])
ax2.scatter(sizes[:L]**2,data.tau_MM_c[:L],marker="s",s=ms_size/1.5,color=myblue)
ax2.scatter(sizes[:L]**2,data.tau_MM_t[:L],marker="D",s=ms_size/1.5,color=mygreen)
ax2.scatter(sizes[:L]**2,data.tau_MA[:L],s=ms_size/1.5,color=myred)
ax2.yaxis.labelpad = 4
ax2.xaxis.labelpad = 1.5
ax2.yaxis.tick_left()
ax2.xaxis.tick_bottom()
ax2.set_xlim([0,np.max(sizes[:L]**2)*1.])
ax2.set_ylim([0,data.tau_MM_c[L-1]*1.05])
ax2.set_xlabel(r'$N^2$')
ax2.set_ylabel(r'$\tau$',rotation=0)
ax2.xaxis.set_ticks(sizes[np.arange(0,L,2,dtype=int)]**2)
ax2.xaxis.set_ticklabels([ r"$%s^2$"%str(x) for x in sizes[np.arange(0,L,2,dtype=int)]])
ax2.yaxis.set_ticks(np.arange(0,data.tau_MM_c[L-1],4,dtype=int))
ax.legend(loc=4,frameon=0,framealpha=0,ncol=2)
ax.set_xlabel(r'$N$')
ax.set_ylabel(r'$\tau$',rotation=0)
vec = np.ravel([data.tau_MA,data.tau_MM_c,data.tau_MM_t])
ylim_up = np.max(vec)*0.93
ylim_up += 0.12*ylim_up
xlim_up = (N_max-1)*1.02
ax.set_xlim([0,xlim_up])
ax.set_ylim([0,ylim_up])
fig.savefig("./figures/relaxation.pdf")

Plot the lifetimes



In [77]:

    
fig2 = plt.figure()
ax = fig2.add_axes([0.11, 0.11, 0.86, 0.86])
index_relax = 1
ax.scatter(sizes[select_points][:-6],data.T_MM_c[select_points][:-6],label=r"$T_c^{MMH}$",marker="s",s=ms_size,color=myblue)
ax.scatter(sizes[select_points][:-1],data.T_MM_t[select_points][:-1],label=r"$T_t^{MMH}$",marker="D",s=ms_size,color=mygreen)
ax.scatter(sizes[select_points],data.T_MA[select_points],label=r"$T_t^{MA}\;\&\;T_c^{MA}$",s=ms_size,color=myred)
ax.yaxis.labelpad = 4
ax.xaxis.labelpad = 0
ax.yaxis.tick_left()
ax.xaxis.tick_bottom()
ax2 = fig2.add_axes([0.19, 0.65, 0.3, 0.29])
ax2.scatter(sizes[:L]**2,data.T_MM_c[:L],marker="s",s=ms_size/1.5,color=myblue)
ax2.scatter(sizes[:L]**2,data.T_MM_t[:L],marker="D",s=ms_size/1.5,color=mygreen)
ax2.scatter(sizes[:L]**2,data.T_MA[:L],s=ms_size/1.5,color=myred)
ax2.yaxis.labelpad = 4
ax2.xaxis.labelpad = 1.5
ax2.yaxis.tick_left()
ax2.xaxis.tick_bottom()
ax2.xaxis.set_ticks(sizes[np.arange(0,L,2,dtype=int)]**2)
ax2.xaxis.set_ticklabels([ r"$%s^2$"%str(x) for x in sizes[np.arange(0,L,2,dtype=int)]])
ax2.yaxis.set_ticks(np.arange(0,data.T_MM_c[L-1],4,dtype=int))
ax2.set_xlim([0,np.max(sizes[:L]**2)*1.1])
ax2.set_ylim([0,data.T_MM_c[L-1]*1.4])
ax2.set_xlabel(r'$N^2$')
ax2.set_ylabel(r'$T$',rotation=0)

ax.legend(bbox_to_anchor=[1.02,0.3],frameon=0,framealpha=0)
ax.set_xlabel(r'$N$')
ax.set_ylabel(r'$T$',rotation=0)
vec = np.ravel([data.T_MA,data.T_MM_c,data.T_MM_t])
ylim_up = np.max(vec)*0.85
ylim_up += 0.05*ylim_up
xlim_up = (N_max-1)*1.02
ax.set_xlim([0,xlim_up])
ax.set_ylim([0,ylim_up])
fig2.show()
fig2.savefig("./figures/lifetime.pdf")



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